FREGE, HILBERT, AND STRUCTURALISM 

 

 

Mark Burke 

 

 

 

 
Thesis submitted to the 

Faculty of Graduate and Postdoctoral Studies 

In partial fulfillment of the requirements 

For the Ph. D. in Philosophy 

 

 

 

 

 

 

 

 

 

 

 

Department of Philosophy 

Faculty of Arts 

University of Ottawa 

 
 
 
 
 
 
 
 
 
 
 
 

© Mark Burke, Ottawa, Canada, 2015 



ii 

 

Table of Contents 
 

  
 

Abstract .......................................................................................................................................... iii 

Acknowledgements ........................................................................................................................ iv 

Introduction ......................................................................................................................................v 

Chapter 1: The development of a structural view of geometry .......................................................1 

Chapter 2: Frege and Hilbert on axiomatic systems ......................................................................47 

Chapter 3: Problems with Frege‘s views .....................................................................................108 

Chapter 4: Benacerraf, abstract structures, and structuralism ......................................................155 

Bibliography ................................................................................................................................213 

 

             

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



iii 

 

Abstract 
 

The central question of this thesis is: what is mathematics about? The answer arrived at by the 

thesis is an unsettling and unsatisfying one. By examining two of the most promising 

contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of 

giving us a conclusive answer to our question. The conclusion is arrived at by a combination of 

historical and conceptual analysis. It begins with the historical fact that, since the middle of the 

nineteenth century, mathematics has undergone a radical transformation. This transformation 

occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier 

images of geometry understood it as the science of space. In the wake of the emergence of 

multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to 

the description of physical space, the old picture of Euclidean geometry as the sole correct 

description of physical space was no longer tenable. The first chapter of the dissertation provides 

an historical account of some of the forces which led to the destabilization of the traditional 

picture of geometry. The second chapter examines the debate between Gottlob Frege and David 

Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting 

that Hilbert‘s views are ultimately unsatisfying. The third chapter continues to probe the work of 

Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. 

The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an 

impasse: the traditional understanding of mathematics cannot hold, but neither can the two most 

promising modern accounts. The fourth and final chapter of the thesis investigates mathematical 

structuralism—a more recent development in the philosophy of mathematics—in order to see 

whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued 

that the contemporary debate between ‗assertoric‘ structuralists and ‗algebraic‘ structuralists 

recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that 

neither of the two most promising contemporary accounts  can offer us a satisfying philosophical 

answer to the question ‗what is mathematics about?‘. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



iv 

 

Acknowledgements 
 

This thesis represents a radical (and initially daunting) departure from my earlier work in 

philosophy. During those innumerable dark nights of the soul when I began to question the sanity 

of both myself and the entire history of mathematics, I have always been fortunate to find myself 

surrounded by an incredibly supportive and inspiring group of people. 

 

First and foremost I would like to thank my partner in life, Anna Candido, without whom this 

project would unquestionably have been abandoned long ago. I‘ve relied perhaps too much on 

Anna‘s always kind words and on her willingness to entertain the ludicrous late-night questions 

of a philosopher at odds with himself and his subject. I cherish every day spent with her. 

 

I would also like to thank my loving family—my parents, Danny and Carol, and my sister Kristy. 

Against all reason they have supported my interest in philosophy since the very beginning. I owe 

to them any strength of character and persistence required for this project—and a lot more 

besides! 

 

Philosophically, this thesis has benefitted greatly from my conversations with colleagues and 

great friends at the University of King‘s College in Halifax. Though only rarely touching on 

questions of mathematics, the hours (days, months, years…) spent arguing about general 

philosophical matters with Darren Dyck, Matthew Furlong, Christopher Rice, and Saša 

Stanković have shaped my entire approach to philosophy, as well as some of the specific 

contents of this thesis. Often in these conversations, it has struck me that we are all working on 

the same problems from different angles. I only hope, in the future, to benefit them as much as 

they have me. 

 

Outside of philosophy, my close friendships with George Blott, Peru Dyer, and Scott Harber 

have helped me remain (if I have remained) sane while going through the endless ringer of 

academia. The hours we‘ve spent together producing and destroying art have been some of the 

finest of my life. 

 

Finally, I would like to thank my supervisor, the always patient, always critical, always kind Paul 

Rusnock. My first encounter with Rusnock was in a course he taught on Gottlob Frege. Prior to 

this encounter, I had intended to write my doctoral dissertation in the area of ethical and political 

philosophy. Afterwards, I became deeply engrossed in the strange forms of rigor and precision 

found in logic and mathematics. Without his willingness to endure my questions (which 

invariably missed the mark) and to educate me (in metalogic, modal logic, set theory, group 

theory, even category theory) this project would never have come to fruition. Of course none of 

my own imprecisions, and none of the faults of this thesis, can be attributed to him.



v 

 

Introduction 
 

What is mathematics about? This seems a rather straightforward question, but even our naïve 

first approximations seem to miss the mark. A common response to the question is that 

mathematics concerns number. But then we have geometry—a venerable old branch of 

mathematics which needn‘t concern itself with number at all. And, besides, what exactly are 

numbers, in any case? Since its earliest development, mathematics has been a richly varied 

science, including at different times geometry, arithmetic, algebra, mechanics, analysis, logic, 

and many other branches besides. And yet, despite this rich variety, most philosophers and 

mathematicians have tended to treat mathematics as a unified science, sharing, in some sense, a 

common content. Throughout much of its history, mathematics has been seen as a bastion of 

stability amongst the sciences, many of which change significantly in the wake of a new 

discovery or mode of thinking. But mathematics, too, has faced its share of instability. 

Throughout the eighteenth century, European mathematicians struggled to understand the exact 

meaning of the calculus, and devoted a great deal of effort to the taming of the infinitesimal. In 

the nineteenth century, the very nature of geometry was called into question by the emergence of 

non-Euclidean, projective, and other geometries. The nineteenth and twentieth centuries have 

witnessed perhaps the most fundamental crises so far in the shape of the set-theoretic paradoxes 

and Gödel‘s incompleteness results. Each of these events (among others) has altered our 

conception of mathematics as a science in significant ways, such that it is highly unlikely that a 

contemporary mathematician would answer the question ‗what is mathematics about?‘ in the 

same way that Euclid or Archimedes might have. 

 

My goal in this dissertation is not to canvass all possible views of the nature of mathematics, but 

to examine a major division amongst these views through the lens of a few privileged exemplars. 

The division is between what I call the ontologically-fixed conception of mathematics and the 

relational or free-floating or purely structural conception of mathematics. 

 

In the first chapter, I investigate a large scale shift in understanding regarding the nature of 

geometry from the time of Euclid to the publication of David Hilbert‘s Die Grundlagen der 

Geometrie in 1899. My choice in this chapter of geometry, rather than a more obviously abstract 

mathematical field like algebra, is a deliberate one. The reason for this choice is that geometry, 

of all the branches of mathematics, has always seemed the most concrete and, therefore, the least 

likely to give way to an abstract structural approach. Unlike algebra, whose exact subject matter 

is and was a matter for debate, geometry has historically been understood to have an obvious and 

determinate subject matter (i.e., physical space). Thus, when an abstract structural view emerges 

even in geometry it lends credence to the claim that mathematics as a whole is best construed as 

the science or study of structures. (As we will see in the fourth chapter of the dissertation, where 

we examine Benacerraf‘s non-uniqueness problem, the other bastion of determinate content 

within mathematics—basic arithmetic—has come to be understood structurally as well). The 

direct focus of this first, geometric chapter is the relatively narrower problem of understanding 

shifting historical attitudes toward the parallel postulate. Various attempts to situate this 

postulate relative to the system of Euclid‘s Elements are considered until, in the 1830s, we begin 

to see a radical alteration in the conception of geometry as a science. Ultimately, I will argue 

that, in the wake of the development of mathematically consistent and physically applicable 



vi 

 

models of non-Euclidean geometries (among other factors), there was no longer an adequate 

philosophical account of the exact nature of geometrical science.  

 

In the second chapter I turn to a debate that seems to promise such an account. On the one hand, 

we have the philosopher and logician Gottlob Frege, who argues that mathematics has a 

determinate content, that it is a science whose truths are absolute, and that mathematical terms 

have determinate referents. On the other side of the debate we have David Hilbert, one of the 

most influential mathematicians of modern times, and one of the major forces behind the modern 

axiomatic approach to the subject. For Hilbert, mathematics does not have a determinate content 

in the sense that Frege demands. Instead, it concerns uninterpreted formal systems, which can be 

given content through a procedure called ‗interpretation‘, but which are free-standing when left 

alone. For Hilbert, questions of ontology and of absolute truth and reference miss the point of 

(modern) mathematics entirely. At the end of the chapter, we consider some rather significant 

criticisms of Hilbert‘s view made by Frege. The result is that profound doubts have been cast on 

Hilbert‘s answer to the question ‗what is mathematics about‘.  

 

The third chapter proceeds by investigating the chief alternative to Hilbert‘s relational view: 

namely, Frege‘s ontologically fixed view. Frege‘s interesting logicist programme involved the 

reduction of basic arithmetic to pure logic. For him, an important part of this project concerned 

fixing the references of mathematical terms in his unique ontology of functions and objects. The 

chapter gives an account of the motivations for and contents of Frege‘s ontological hierarchy and 

then moves on to discuss some serious problems with his account of the nature of mathematics. 

More specifically, the chapter examines two problems pertaining to the fixing of references: the 

concept horse problem, which seems to leave Frege unable to refer to one of the basic elements 

of his ontology, and the Julius Caesar problem, which threatens to undermine his capacity to fix 

the references of mathematical terms as basic as ‗number‘. The conclusion of the chapter shows 

that, while Frege can escape the concept horse problem reasonably well, his program ultimately 

founders on the Julius Caesar problem. This leaves us at an impasse: neither Hilbert‘s view nor 

Frege‘s view can satisfactorily explain what mathematics is really about. 

 

In the fourth and final chapter I continue my investigation of this question by examining some 

relatively recent attempts to answer (or otherwise dissolve) the Caesar problem faced by Frege. 

The focus of the first half of the chapter is on a 1965 paper by Paul Benacerraf which rejects an 

analogue of the Caesar problem insofar as it is based upon a fundamental misunderstanding of 

the nature of mathematics. The positive position Benacerraf briefly develops after this 

dissolution of the Caesar problem is strikingly reminiscent of Hilbert‘s purely relational view. 

His claim is that mathematics is not about the kind of ontology that Frege was interested in but 

instead about abstract structures. This view has come to be called ‗structuralism‘. In the end, 

Benacerraf‘s particular form of structuralism is shown to be subject to deep difficulties and so 

we must turn to more recent work in the philosophy of mathematics to answer our question. 

Some of the most promising contemporary work follows in the wake of Benacerraf‘s programme 

and can be called ‗structuralist‘ in much the same way. In the contemporary setting, I shall argue, 

two main kinds of structuralism may be distinguished. On the one hand we have assertoric 

structuralism (exemplified in this chapter chiefly by the work of Stewart Shapiro). Assertoric 

structuralism contends, like Frege, that mathematics has a determinate content and that the 

claims of mathematics are meaningful assertions about this content. Unlike Frege, however, 



vii 

 

assertoric structuralists claim that this content is comprised of abstract structures. I go on to 

argue that Shapiro‘s brand of assertoric structuralism ends up enmeshed in the same sort of 

difficulties that troubled Frege‘s view. 

 

On the other hand, we have algebraic structuralism (exemplified here by Steve Awodey and John 

Bell). Algebraic structuralism, like Hilbert‘s earlier position, rejects the idea that mathematics 

has a single determinate content in favour of a view which relativizes this content to a particular 

system, framework, or structure. Algebraic structuralism in its modern form is often developed 

using the tools of the branch of abstract algebra called category theory. On the algebraists‘ view, 

there are multiple different local categories within which mathematics can be done, each with its 

own ontology and logic. Thus, for them, Frege‘s dream of absolutely fixing the references of 

mathematical terms is just that—a dream, and a misleading one. Like Hilbert, however, the 

algebraic structuralists are forced to embrace relativized notions of truth, reference, and meaning. 

In the conclusion to the chapter, I argue that neither assertoric nor algebraic structuralism offer 

us a significant philosophical advance over the impasse of the Frege-Hilbert debate, and we are 

left with no way to adjudicate between competing, and deeply opposed, conceptions of the nature 

of mathematics. 

 

In short, the thesis illustrates an historical and conceptual oscillation between two distinct 

pictures of mathematics. One in which mathematics is understood as a science having a unique 

content and clear referents, the other in which both content and reference vary depending upon 

one‘s local situation, system, or framework. We are left, then, with no clear answer to the 

question ‗what is mathematics about,‘ but a clearer diagnosis of our dilemma. 

 



1 

 

Chapter 1: The development of a structural view of geometry 
 

§0. Introduction 

 

The point of this chapter is to explain why a structural approach to geometry emerged. This can 

appear to be a purely historical question, but I believe it is crucial for the philosophical project of 

this thesis as well. The project of the thesis, recall, is to investigate what mathematics is about. 

Since the 19
th

 century, there have been two broad approaches to this question. The first centers 

on discovering exactly which objects lie at the basis of all mathematical reasoning. This 

approach is much older and is found already in the work of Euclid. The second approach 

suggests that mathematics concerns structures and attempts to explain what structures are and 

how they relate to one another. This approach is new—beginning in the 19
th

 century—but is 

dominant in most areas of contemporary mathematics and the philosophy of mathematics. In this 

chapter, I will explain why this second view emerged at all. 

 

It is impossible to evaluate and understand the various forms of mathematical structuralism if we 

have no idea what problems they were originally intended to address. The success of the 

structural approach to mathematics since the beginning of the twentieth century has made many 

of these initial problems difficult even to formulate: the status of Euclid‘s parallel postulate, for 

instance, does not strike us as a pressing problem which cuts to the very heart of geometry. It is 

now mostly an historical curiosity. The reason why we are not deeply troubled by the parallel 

postulate is because a new mode of thinking about geometry has emerged which eliminates both 

the problem of the parallel postulate and the basis for our deep interest in it. This new mode of 

thinking is not particularly concerned with the specifics of Euclidean geometry because it 

understands this geometry as just one among infinitely many. It is the relations between 

geometries and other sorts of ‗structures‘ which occupy the new mode of thinking. This approach 

has been wildly successful. It has been so successful, in fact, that to speak meaningfully about 

modern mathematics seems to require us to accept the conceptual vocabulary of structuralism. 

We breathe the air of structuralism almost without notice. 

 

But, as Frege notes already in his correspondence with Hilbert and elsewhere, the ‗modern‘ 

approach to mathematics is by no means clear itself. I think it is very difficult to see that this 

approach might be fundamentally incoherent or unclear if we already ask and answer every 

question by employing its conceptual vocabulary. In this first chapter, then, I want to draw 

attention to the peculiarity of structuralism by examining the non-structuralist questions which 

prompted its emergence in the first place. It is difficult to avoid speaking even about the history 

of mathematics in structural terms. I think a careful and detailed attention to the problems which 

led to structuralism and the philosophical views which generated these problems is the best 

approach. 

 

The chapter begins with an examination of the conceptual structure of Euclid‘s influential text, 

the Elements. After examining Euclid‘s work directly, we then turn to a series of criticisms and 

concerns pertaining to the parallels axiom. This historical account of the troubles engendered by 

this axiom is intended to illustrate the slow shift away from the object-centered account of 

geometry we find in Euclid towards the structure-focussed views more characteristic of 



2 

 

contemporary mathematics. This historical portrait culminates in an examination of the unstable 

philosophical position of geometry just prior to the publication, in 1899, of Hilbert‘s Grundlagen 

der Geometrie. The chapter should show that, at the end of the 19
th

 century, an entirely new way 

of doing mathematics had emerged, but its philosophical basis had not been clearly articulated. 

Chapter two will consider Hilbert‘s work, which, to many, brought a new philosophical and 

conceptual clarity for modern mathematics. 

 

This historical matter is relevant not merely as background information, but as a necessary 

element for understanding the concerns discussed by Frege and Hilbert (to be examined in 

subsequent chapters). Without an appreciation of both the traditional picture of geometry and the 

instability created by its demise, we cannot grasp many of the important subtleties encountered 

in the Frege-Hilbert correspondence. In particular, without grasping changes in the common 

understanding of geometry throughout the nineteenth century, the motivations for and 

advancements embodied within Hilbert‘s work disappear entirely. Moreover, without 

understanding the crucial differences between Frege‘s position vis-à-vis axiomatization and the 

traditional Euclidean picture of geometry, we cannot fully appreciate how different even Frege‘s 

supposedly traditional picture of geometry is from that of his predecessors.  

 

The ideas of both Frege and Hilbert have been very influential over the last century. The 

prominence of their respective views on logic and mathematics has made it easy for many 

commentators to read their works somewhat anachronistically. For instance, even astute readers 

of Hilbert tend to attribute a fully model-theoretic understanding of axiomatics to his earlier 

work.
1
 But, as we shall see in the second chapter, during the earlier part of his career Hilbert had 

not yet developed clear enough logical and semantic views for this to be the case. By illustrating 

the links between Hilbert‘s work and that of his predecessors (which is partially accomplished in 

this chapter), such anachronism becomes easier to avoid. In Frege‘s case, too, commentators 

have tended to misconstrue his works based on anachronistic readings. When examining the 

Frege-Hilbert correspondence, for instance, many commentators have found Frege‘s criticisms to 

be pedantic or old-fashioned in a way that simply misses the point of ‗modern‘ axiomatics.
2
 

Careful attention to prior developments in geometry and axiomatics, however, helps to illustrate, 

first, that Frege‘s views are not quite as conservative as they appear and, second, that his 

criticisms of the supposedly model-theoretic work of Hilbert are more sophisticated than has 

usually been assumed. 

 

A more general benefit of this thoroughly historical way of proceeding is that it allows us to see 

that seemingly straightforward mathematical terms mean and have meant different things for 

different people, and that these differences in meaning have not always been noticed. In my 

initial attempts to address the problem of structure in mathematics I myself rather naively 

presupposed that there was some widespread agreement about what ‗structure‘ might mean. As 

further historical enquiry has shown the meaning of common mathematical terms like ‗axiom‘ 

and ‗structure‘ is far from settled even today (though there is much less debate about the nature 

of axioms today, as the question seems to many to have been settled).  

 

                                                 
1
 See, for example, [Weyl 1970, 264-268]. 

2
 E.g., [Torretti 1984] and [Freudenthal 1962]. 



3 

 

§1. Euclid and the Parallel Postulate 

 

It is chiefly from Euclid‘s Elements that the traditional understanding of geometry as the science 

of space and spatial objects derives. And it is chiefly from the geometries which differ from 

Euclid‘s own that we derive the cluster of views which hold that geometry is concerned 

primarily with spatial structures. At the outset of this chapter, I want to establish as clearly as 

possible the content of the traditional Euclidean picture of geometry. I am not interested in the 

purely mathematical aspects of Euclid‘s geometrical thinking, but in showing the philosophical 

principles and assumptions which underlie this thinking. As our historical analysis will show, 

many of these principles remained tacit, vague, and unclear for hundreds and even thousands of 

years. Indeed, one useful way of understanding the structure-focussed view of geometry which 

dominates the world of mathematics today is to see it as an attempt to avoid the incoherence and 

limitations of the Euclidean picture of geometry. In this chapter, I intend to flesh out what I take 

to be the central features of this incomplete picture so that we can better understand why 

alternative views emerged at all.  

 

1.1 The origins and importance of Euclid‘s Elements 

 

The thirteen books of Euclid‘s Elements are one of the marvels of ancient Greek mathematics, 

and, indeed, of rigorous thinking more generally. For nearly two thousand years, they remained a 

benchmark for certitude and provided an ideal to be lived up to by all other scientific treatises. 

Euclid‘s axiomatic method has also been emulated in a variety of fields, from ethics to 

economics, in addition to a pervasive and massive influence within almost every sub-discipline 

of mathematics, from geometry to analysis. It is indeed difficult to overestimate the importance 

this geometrical textbook has had for the development of scientific thought over the last two and 

a half millennia.   

 

But, despite its singular position within the history of thought, Euclid‘s most famous work was 

not the first of its kind, only the most successful. We know from Proclus (410-485), and 

obliquely from Eudemus (c. 370 BC- c. 300 BC) and Simplicius (c. 490-560), that Hippocrates 

of Chios (fl. latter half of the 5
th

 century BC) also wrote a treatise on geometry, entitled the 

Elements, about a century before Euclid‘s.  We also know from Proclus that the two Platonists 

Leon and Theudius of Magnesia (c. 4
th

 century BC) composed elements, though nothing from 

either of these works has survived and little else is known of their authors. Despite the 

disappearance of these competitors, it is clear both that Euclid‘s Elements was part of a tradition 

of Greek mathematical textbook writing and that it was considered by many of his 

contemporaries to be the pinnacle of that tradition. We can surmise its position at the head of the 

queue both from the number of favourable references in ancient sources and from the fact that its 

competitors have not survived. My interest here is not exclusively in the superiority and 

idiosyncrasy of Euclid‘s work. I focus on Euclid chiefly because his work has indisputably been 

the most influential and because his work has been thought by many to represent the central 

features of the classical Greek attitude toward mathematics. This Greek attitude, or, rather, the 

Greek methodological and philosophical approach to mathematics, has shaped the history and 

content of mathematics to such an extent that many of its assumptions remained invisible for 

centuries. By looking closely at Euclid and the problems stemming from his work, we will gain a 



4 

 

clearer historical and philosophical understanding of the subsequent attack on the traditional 

picture of mathematics as a science concerned with certain types of objects. 

 

So, let us turn now to the thing itself and discuss the aims, format, and content of Euclid‘s best-

known work.  

 

1.2 The aims of the Elements. 

 

As noted above, the Elements was a member of a literary tradition which likely began in the fifth 

century B.C. with Hippocrates of Chios. Euclid‘s Elements, presumably like the lost members of 

the tradition, is a compendium of mathematical results—in this case those pertaining to plane 

geometry, proportions, number theory, incommensurable quantities, and the geometry of solids. 

As Kline notes, ―[t]he Elements [...] is as much a mathematical history of the age just brought to 

a close as it is the logical development of a subject‖.
3
 Accordingly, it is likely that very few of 

the theorems in the Elements are the original production of Euclid himself; rather, Euclid is 

responsible chiefly for the arrangement of the work and, probably, for a few of the proofs as 

well. While there is still debate as to the exact status of the Elements within the mathematical 

community of Euclid‘s time,
4
 we can, with Proclus (to whom we owe a valuable commentary on 

Euclid), nevertheless discern two principal aims in the work. The first of these is the 

investigation and development of the theory of the ―cosmic bodies‖
5
 (i.e., the five regular or 

Platonic solids) in the final three books—this aspect of the work likely represented the height of 

Greek geometrical knowledge of the time. The Elements presents a clear path to the development 

of this branch of geometry by way of a rigorous and lucid presentation of the methods and results 

required for its elaboration.  

 

The second aim of the Elements is more general and pedagogical in character. Of this aim, 

Proclus writes: 

 

Of the purpose of the work with reference to the student we shall say that it is to lay before 

him an elementary exposition [...] and a method of perfecting his understanding for the 

whole of geometry. If we start from the elements, we shall be able to understand the other 

parts of this science; without the elements we cannot grasp its complexity, and the learning 

of the rest will be beyond us. The theorems that are simplest and most fundamental and 

nearest to first principles are assembled here in a suitable order, and the demonstrations of 

other propositions take them as the most clearly known and proceed from them.
6
 

 

                                                 
3
 [Kline 1972, 56]. 

4
 The debate concerns Euclid‘s intended audience; i.e., whether he was chiefly interested in communicating to 

already-sophisticated mathematical peers, or if the book was intended as a sort of catechism for prospective students 

of mathematics. Kline writes of the matter that ―The purpose of Euclid‘s Elements is in question. It is considered by 

some as a treatise for learned mathematicians and by others as a text for students. Proclus gives some weight to the 

latter belief‖ [Kline 1972, 57]. The block quote from Proclus on this page is partial evidence for his belief in the 

pedagogical intent of the Elements, though the inclusion of relatively complex work on the platonic solids seems to 

indicate that Euclid may have been writing for already-knowledgeable mathematicians. 
5
 [Proclus 1970, 70]. 

6
 [Proclus 1970, 71]. 



5 

 

Thus, Proclus thinks, by studying the basic elements of geometry, the aspiring geometer is led by 

degrees toward ever more complex theorems and a correspondingly deepened understanding of 

the nature of the subject. Ultimately all of these resources are marshalled in the development of 

the theory of the platonic solids, combining both aims. Since the structure of the Elements is, in 

large part, the source of its unique influence, I turn now to a closer examination of that structure. 

 

1.3 The format of the Elements 

 

Book I of the Elements sets the stage for the austerity and strictness of the rest of the work. It 

begins abruptly with a list of twenty-three definitions. There is no preamble or explanation, and 

these definitions are followed by two more lists: the first contains five postulates, while the 

second contains five ‗common notions‘ or axioms. After these lists Euclid proceeds to prove a 

number of propositions by employing the common notions, postulates, and acceptable modes of 

reasoning (which are never enumerated within the work itself). The following twelve books are 

all similar in character; occasionally, new definitions are introduced, though no new postulates or 

common notions appear. So, in sum, we find within the work five postulates, five common 

notions, a relatively small number of definitions, and a large number of theorems with 

accompanying proofs and figures. All of this is remarkably similar to a modern mathematics 

textbook—deceptively so. 

 

1.3.1 Definitions. 

 

Euclid‘s definitions can be divided into two distinct types. Those of the first type, exemplified by 

his definition of the circle,
7
 are directly employed within proofs. These definitions define key 

geometrical concepts in terms of simpler ones. Thus, the circle is defined in terms of points, 

straight lines, planes, etc., and is appealed to in the proof of the first theorem of Book I. The 

second sort of definition has a less obvious role to play. These definitions (exemplified by 

Euclid‘s definition of ‗point‘ and ‗line‘), are never directly employed to prove a proposition, nor 

are they referred to once they have been introduced. What, then, is their function? Clearly, one 

role is that of a propaedeutic—they introduce the reader to the most basic sorts of objects (points, 

lines, planes, etc.) about which Euclid will reason. While modern mathematical definitions tend 

to be of the abbreviatory sort, Euclid‘s second group of definitions might better be aligned with 

what Frege, for instance, called ‗elucidations‘ [Erläuterungen].
8
 These are pre-systematic 

remarks which help guide the reader toward a proper understanding of the basic elements to be 

treated within the system. Once these elements become clearly fixed, the function of the 

elucidations is achieved and they can simply be ignored. Thus, for instance, Euclid‘s definitions 

of a point as ―that which has no part‖
9
 or a line as ―breadthless length‖

10
 help the reader to 

                                                 
7
 ―A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among 

those lying within the figure are equal to one another‖ [Euclid, I, Definition 15; Heath 1956, 183]. Euclid then 

defines the centre of a circle as the aforementioned point [Euclid, I, Definition 16; Heath 1956, 183]. 
8
 Cf., e.g., [Frege 1914, 207] and [Frege 1971, 6-8]. I discuss Frege‘s use of the term Erläuterung in more detail in 

the second and third chapters of the thesis. Earlier in the 19
th

 century Bernard Bolzano (1781-1848) employed the 

term Verständigung [understanding] to much the same effect as Frege‘s Erläuterung; cf. [Bolzano 1975, §9] where 

he lays out the rules for the use of these sorts of explications. 
9
 [Heath 1956, 155]. 

10
 [Heath 1956, 158]. 



6 

 

understand precisely which objects he has in mind, though the definitions themselves play no 

direct role when he later proves propositions which contain the terms at issue. Ian Mueller, in a 

paper which highlights the divergences between modern axiomatic geometry and Euclid‘s 

approach, writes that 

 

The definitions should be looked at as attempts to make clear the meanings of the terms to 

be used before argumentation begins, that is, to make clear the nature of the objects to be 

studied. That the most fundamental definitions (e.g. of point, line, straight line) succeed 

only with persons who already have some idea what the objects in question are does not 

really matter if these definitions are taken to represent preliminary agreements among 

people of presumably normal intelligence.
11

 

 

As Mueller suggests, Euclid‘s second type of definition should help to focus the reader‘s mind 

on the appropriate sorts of objects so that official argumentation can begin. The reader may 

already possess ideas of these objects, in which case the definitions serve the simple function of 

directing the mind to what it already knows, or of representing this knowledge. In the case where 

the reader does not possess appropriate ideas, the definitions cannot fix the reference of the terms 

in question; they serve, again, as helpful hints which can (at best) direct sufficiently capable 

readers to the sorts of things that Euclid has in mind. Beyond this preliminary role in establishing 

the particular sorts of objects about which Euclid will reason, definitions of the second type play 

no direct role within the system itself. Importantly, also, these definitions do not insist upon the 

existence of the objects which they define; the attribution of existence is reserved for the 

postulates.
12

  

 

1.3.2 Postulates and common notions (axioms). 

 

Unlike some of Euclid‘s most basic definitions, the postulates and common notions are directly 

employed in order to prove propositions, and their special status relative to the rest of the system 

is of particular importance for understanding the nature of Euclid‘s work. Like the use made of 

definitions, the division of unproved assumptions into the two categories of common notions and 

postulates is not unique to Euclid. There was already by Euclid‘s time a robust methodological 

tradition, headed by Aristotle, which made such a division.  

 

In general, common notions (Aristotle called them ‗common items‘ or ‗common opinions‘) are 

primitive, unprovable principles which do not contain reference to the specific notions of a given 

science. Assent to these common notions is not demanded because these notions act as 

constraints on reasoning in general, and, thus, their rejection would be tantamount to the 

rejection of deductive reasoning as such. Axioms, on the contrary, are claims for which assent is 

demanded and which are specific to the purposes of the particular science at issue. The idea is 

that we must accept certain facts as basic in order to get the deductive machinery going within a 

                                                 
11

 [Mueller 1969, 294]. 
12

 This is one of the central elements of the Aristotelian conception of definition. Aristotle writes, for instance, that 

―what the primitives and what the items proceeding from them mean is assumed; but that they are must be assumed 

for the principles and proved for the rest‖ [Aristotle 1994, I. 10, 76a 31-35, emphasis mine]. 



7 

 

particular area. Aristotle writes of the difference between definitions, postulates, and common 

notions as follows: 

 

An immediate deductive principle I call a posit if it cannot be proved but need not be 

grasped by anyone who is to learn anything. If it must be grasped by anyone who is going 

to learn anything whatever, I call it an axiom (there are items of this kind); for it is of this 

sort of item in particular that we normally use this name. A posit which assumes either of 

the parts of a contradictory pair—what I mean is that something is or that something is 

not—I call a supposition. A posit which does not I call a definition. Definitions are posits 

(arithmeticians posit that a unit is what is quantitatively indivisible), but they are not 

suppositions (for what a unit is and that a unit is are not the same).
13

  

 

Thus, common notions are unprovable principles which guide deductive reasoning in general, 

which needn‘t be explicitly accepted for that reason. Postulates are unprovable principles, assent 

to which is demanded for the purposes of a particular science. Definitions, finally, are postulates 

which make no existence claims. They inform us what the basic elements of the system are, but 

leave it open whether or not these elements exist.  

 

These divisions are found in Euclid‘s work. There the common notions, for example, make no 

mention of the special geometrical objects defined, but concern only more general concepts like 

‗part,‘ ‗whole,‘ and ‗equality‘. Euclid‘s
14

 five common notions are as follows: 

 

1. Things which are equal to the same thing are also equal to one another. 
2. If equals be added to equals, the wholes are equal. 
3. If equals be subtracted from equals, the remainders are equal. 
4. Things which coincide with one another are equal to one another. 
5. The whole is greater than the part.

15
 

 

As we can see, no mention is made here of the objects defined in the first twenty-three 

definitions (e.g., points, lines, planes, etc.). These are general principles that Aristotle and Euclid 

would likely have assumed to be implicitly accepted as constraints upon deductive reasoning by 

anyone of normal intelligence. The postulates, by contrast, are specifically geometrical in 

character. They are as follows: 

 

Let the following be postulated:  

 

1. To draw a straight line from any point to any point 
2. To produce a finite straight line continuously in a straight line 
3. To describe a circle with any centre and distance 
4. That all right angles are equal to one another 

                                                 
13

 [Aristotle 1994, I. 2 72a 15-24]. 
14

 There is some debate as to the historical origin of the common notions in Euclid, with the suggestion that some or 

all of them were added by later commentators. For relevant discussion, see [Heath 1956, 221-222]. 
15

 [Heath 1956, 155]. 



8 

 

5. That if a straight line falling on two straight lines make the interior angles on the 
same side less than two right angles, the two straight lines, if produced 

indefinitely, meet on that side on which are the angles less than the two right 

angles.
16

 

 

In contrast to the common notions, the postulates make free use of the primitive terms of 

geometry. Their function is to insist that we accept some basic claims from which all other 

geometrical theorems will be derived. These claims are not proved in the Elements. Ideally, too, 

they should not be provable using the resources which Euclid makes available there; the 

postulates are basic claims (accepted as true), which provide the basis for Euclid‘s proof system. 

Thus, there should be a sharp separation between the theorems of the system (which are provable 

using only basic principles and previously proven theorems) and the axioms themselves.  In 

addition to the importance of their unprovability, their truth is crucial as well. For, in contrast to 

some conceptions of mathematical truth in which truth is only defined relative to the consistency 

of a given axiomatic scheme (and, hence, undefined for axioms in isolation),
17

 the Greeks 

demanded that the axiomatic basis of a science be independently true as well. As we shall see, 

the common view for the next two millennia was, precisely, that (most of) Euclid‘s postulates 

were not only understood to be true, but necessary and indubitable as well. Given the demand for 

rigor which characterized early Greek geometry, it is no surprise that even early readers of the 

Elements had their interest piqued by the fifth postulate, the so called parallel postulate, which is 

considerably more complex than the previous four. It is by examining the parallel postulate and 

the philosophical problems which stem from it that we start to see problems with the traditional 

Euclidean picture of geometry. 

 

1.4 The parallel postulate 

 

Despite widespread agreement that the fifth postulate, like the others, was a truth, there was early 

on a serious debate as to its status as a postulate within the system. Despite its status as a 

postulate, many felt that the parallel postulate should be provable from the other four postulates, 

a belief which gave rise to a remarkably long and complex history, beginning as early as Ptolemy 

and continuing well into the 19
th

 century. Even in 1767 D‘Alembert called the status of the 

parallel postulate the ―the scandal of geometry‖.
18

 And as late as 1813 Gauss—one of the men 

chiefly responsible for overturning the view of Euclidean geometry as the science of space—

remarked that ―In the theory of parallels we are still no further along than Euclid,‖ and that the 

theory of parallels was a rather ―shameful part of mathematics‖.
19

 

                                                 
16

 [Heath 1956, 154-155]. 
17

 This is the case for, e.g., Hilbert‘s influential conception of mathematical truth. Hilbert writes in response to 

Frege‘s somewhat more traditional view of mathematical truth, that ―If the arbitrarily postulated axioms [of 

geometry] together with all their consequences do not contradict one another, then they are true and the things 

defined by these axioms exist. For me, this is the criterion of truth and existence‖ [Hilbert, in Frege 1971, 12]. I 

discuss the differences between these two conceptions of mathematical truth in more detail in the second chapter. 
18

 ―La définition et les proprietés de la ligne droite, ainsi que des lignes parallèles, sont donc l’écueil, et pour ainsi 

dire le scandale des élémens de géométrie‖ [d‘Alembert 1759-1767; reprint 1805, volume 2, 331]. 
19

 ―In der Theorie der Parallellinien sind wir jetzt noch nicht weiter als Euklid war. Diess ist die partie honteuse der 

Mathematik, die früh oder spät eine ganz andere Gestalt bekommen muss‖ [Gauss, dated 27 April 1813; in Gauss 

1863-1929, Band VIII, 166]. 



9 

 

 

Though there was little argument over the status of the first four postulates, the provability of the 

fifth was, from Euclid‘s time onward, a matter of some contention. As Heath notes, ―[f]rom the 

very beginning, as we know from Proclus, the Postulate was attacked as such, and attempts were 

made to prove it as a theorem or to get rid of it by adopting some other definition of parallels‖.
20

 

It is worth quoting the relevant passage from Proclus at some length to illustrate the troubling 

character of the postulate as it was understood relatively early in its history. Proclus writes that 

the fifth postulate 

 

[...] ought to be struck from the postulates altogether. For it is a theorem—one that invites 

many questions, which Ptolemy proposed to resolve in one of his books—and requires for 

its demonstration a number of definitions as well as theorems. And the converse of it is 

proved by Euclid himself as a theorem. But perhaps some persons might mistakenly think 

that this proposition deserves to be ranked among the postulates on the ground that the 

angles‘ being less than two right angles makes us at once believe in the convergence and 

intersection of the straight lines. To them Geminus has given the proper answer when he 

said that we have learned from the very founders of this science not to pay attention to 

plausible imaginings in determining what propositions are to be accepted in geometry. 

Aristotle likewise says that to accept probable reasoning from a geometer is like 

demanding proofs from a rhetorician. And Simmias is made by Plato to say, ―I am aware 

that those who make proofs out of probabilities are impostors.‖ So here, although the 

statement that the straight lines converge when the right angles are diminished is true and 

necessary, yet the conclusion that because they converge more as they are extended farther 

they will meet at some time is plausible, but not necessary, in the absence of an argument 

proving that this is true of straight lines. That there are lines that approach each other 

indefinitely but never meet seems implausible and paradoxical, yet it is nevertheless true 

and has been ascertained for other species of lines. May not this, then, be possible for 

straight lines as for those other lines? Until we have firmly demonstrated that they meet, 

what is said about other lines strips our imagination of its plausibility. And although the 

arguments against the intersection of these lines may contain much that surprises us, 

should we not all the more refuse to admit into our tradition this unreasoned appeal to 

probability? [...] These considerations make it clear that we should seek a proof of the 

theorem that lies before us and that it lacks the special character of a postulate. But how it 

is to be proved, and with what arguments the objections to this proposition may be met, we 

can only say when the author of the Elements is at the point of mentioning it and using it as 

obvious. At that time it will be necessary to show that its obvious character does not appear 

independently of demonstration but is turned by proof into a matter of knowledge.
21

  

 

For Proclus, then, there are a few concerns weighing against the unprovability of the fifth 

postulate. Primarily, he claims that the postulate is, rather, a theorem—a theorem which Ptolemy 

                                                 
20

 [Heath 1956, 202]. 
21

 [Proclus 1970, 191-193]; in the final sentences here, Proclus alludes to what would be a common tactic used in 

attempts at proving the fifth postulate, namely using only the first four postulates and the first twenty-eight theorems 

of Euclid which occur before the first use of the parallel postulate is required. 



10 

 

had already attempted to prove. That it is a theorem seems to be made evident by the fact that 

Euclid proves a proposition which is equivalent to the postulate‘s converse, namely proposition  

17 of Book I of the Elements. This proposition states that ―In any triangle two angles taken 

together in any manner are less than two right angles‖.
22

 Euclid proves this proposition by 

employing the second postulate and the previously proved proposition 13 of Book I. 

Accordingly, Proclus finds it suspicious that a principle whose converse is provable using only 

the resources of the first four postulates should not itself be provable from those resources. 

Nevertheless, Euclid‘s proof of proposition 17 of Book I rests upon the unstated further 

assumption that two straight lines meet, if they meet at all, in at most one point—an assumption 

quite as strong as the parallel postulate itself.
23

 Proclus takes it to be the case that the existence of 

a proof of the postulate‘s converse speaks in favour of the provability of the postulate itself. This 

seems a reasonable enough assumption; nevertheless, when we take into consideration the fact 

that Euclid‘s proof requires the assumption of a principle which is equal in complexity and 

obscurity to the parallel postulate, it is no longer obvious that the existence of this proof can act 

as evidence either for or against the provability of the parallel postulate itself.  

 

In addition to the supposed provability of its converse, Proclus also finds fault with the parallel 

postulate insofar as it seems to have been given the status of a postulate for rather weak 

epistemological reasons. He rejects the view that our immediate assent to a geometrical claim 

can justify us in assigning that claim the status of a postulate. For a proposition to be considered 

a postulate, our immediate assent, based upon its undeniable plausibility, is not enough. Beyond 

the mere fact of the layman‘s acceptance of the proposition as obvious, it must also be deemed 

necessary in the sense that its opposite cannot be the case. But, as Proclus rightly notes, there are 

species of lines other than straight lines (e.g., hyperbolas and their asymptotes) which do not 

obey the dictates of the parallel postulate; accordingly, our intuitive acceptance of the postulate 

seems to be ill-founded or, at the very least, in need of supplementation. Thus, Proclus thinks, we 

should expect a proof that straight lines, unlike hyperbolas and their asymptotes, do in fact obey 

the parallel postulate. 

 

For these reasons, then, we find attempts as early as Ptolemy to prove the fifth postulate by 

means of the other four. In what follows, I would like to examine a few of these attempts to 

prove, reformulate, or resituate the parallel postulate relative to the rest of Euclid‘s system. My 

chief aim here is to illustrate that the unstable position occupied by the parallel postulate slowly 

led to the rise of a new understanding of the science of geometry in which ‗structure‘ became 

increasingly prominent. Though I mention other figures, I will focus chiefly on Ptolemy, Proclus, 

Khayyam, and Saccheri, as elements of their work were central to the later emergence of non-

Euclidean geometry and its destructive effect on the traditional conception of geometry as the 

science of space or spatial objects.  

 

 

 

 

                                                 
22

 [Heath 1956, 281]. 
23

 This assumption is provably false within elliptic geometry, for instance, unless the further condition that antipodal 

points are taken to be equivalent is added. 



11 

 

1.4.1 Ptolemy and Proclus 

 

As noted above, we know of Ptolemy‘s attempted proof of the fifth postulate by way of Proclus. 

Ptolemy‘s assault on the parallel postulate was likely the first which attempted to prove the 

troublesome claim by assuming an equivalent principle. (This would turn out to be a popular 

tactic over the centuries). In Ptolemy‘s case this assumption was that ―through any point only 

one parallel can be drawn to a given straight line‖.
24

  

 

Figure 1: Proclus and parallels 

 
Proclus‘ own proof, too, rests on a disguised equivalent of the very postulate to be proved. In 

Proclus‘ case, this is an axiom taken from Aristotle.
25

 This axiom is stated as follows: ―If, from a 

single point two straight lines making an angle are produced indefinitely, the interval between 

them when produced indefinitely will exceed any finite magnitude‖.
26

 From this principle he 

deduces the proposition that a straight line which meets one of a pair of parallels will also meet 

the other. He proves this proposition as follows. Take two parallels, AB and CD, and another 

straight line EF which intersects AB at a point G. The distance between a point moving along the 

line segment GF and the line CD will decrease indefinitely when the distance of the moving 

                                                 
24

 [Heath 1956, 206]. 
25

 The axiom is taken from Aristotle‘s well-known arguments for the finitude of the universe in De Caelo (cf. 

[Aristotle 1971, I. 5, 271b 28ff.]). Guthrie‘s translation has: ―The following arguments make it plain that every body 

which revolves in a circle must be finite. If the revolving body be infinite, the straight lines radiating from the centre 

[of the circular path of the body] will be infinite. But if they are infinite, the intervening space must be infinite. 

―Intervening space‖ I am defining as space beyond lines. This must be infinite. In the case of finite lines it is always 

finite, and moreover it is always possible to take more than any given quantity of it, so that this space is infinite in 

the sense in which we say that number is infinite, because there exists no greatest number. If then it is impossible to 

traverse an infinite space, and in an infinite body the space between the radii is infinite, the body cannot move in a 

circle. But we ourselves see the heaven revolving in a circle, and also we established by argument that circular 

motion is the motion of a real body‖ [Aristotle 1971, I.5., 271
b 
28-272

a 
7]. 

26
 [Proclus 1970, 371.10]. 



12 

 

point from G is increased indefinitely. But, since the distance between the two parallel lines is 

finite, the straight line EF must meet CD (see Figure 1). 

 

But this reasoning, too, relies upon a rather strong assumption; namely, the assumption that the 

distance between any two parallel lines always remains finite. And, again, this assumption is 

provably equivalent to the parallel postulate itself.  This argument would later be criticized by 

Christopher Clavius (1538-1612)—whom Saccheri read and referred to
27

—on the fairly 

reasonable grounds that the Aristotelian axiom itself requires proof. As Heath notes, Clavius 

 

points out that, just as you cannot assume that two lines which continually approach one 

another will meet (witness the hyperbola and its asymptote), so you cannot assume that two 

lines which continually diverge will ultimately be so far apart that a perpendicular from a 

point on one let fall on the other will be greater than any assigned distance; and he refers to 

the conchoid of Nicomedes which continually approaches its asymptote, and therefore 

continually gets farther away from the tangent at the vertex; yet the perpendicular from any 

point on the curve to that tangent will always be less than the distance between the tangent 

and the asymptote. Saccheri supports the objection.
28

 

 

Subsequent attempts at proof often rested on the assumption of similarly disguised equivalents of 

the parallel postulate. Occasionally mathematicians would explicitly offer reformulations of the 

postulate in seemingly simpler terms. The English mathematician, John Wallis (1616-1703), for 

instance, knowingly assumed that transformations which alter only the size of a figure do not 

affect its angles or the proportions of its sides.
29

 Somewhat surprisingly, this assumption, too, is 

equivalent to Euclid‘s parallel postulate—an equivalence made apparent in more recent times by 

the impossibility of constructing similar triangles of different size within some of the more 

familiar non-Euclidean geometries. 

 

These attempts to prove the postulate invariably end up reintroducing an equally troublesome (or 

equivalent) claim in its place. There are innumerable examples of this problem, a few of which 

are mentioned above. There was a general feeling that the parallel postulate was somehow 

unsatisfying, but there was no real consensus as to what feature of the postulate was at issue. As 

yet, there was no significant attempt to alter the Euclidean picture of geometry. Postulate-

tinkerers were not revolutionaries but refiners: they sought to perfect Euclid rather than overturn 

him. But, over time, more and more difficulties with the Euclidean picture began to pile up. 

 

                                                 
27

 Saccheri discusses Clavius at [Saccheri 1733, 83ff.]; incidentally, Saccheri corrects an error in Clavius‘ attempt to 

employ the bisection of a circle by its diameter as a means of proving the parallel postulate—unbeknownst to 

Saccheri, he corrects Clavius in precisely the same place and manner (see [Saccheri 1733, 91]) that Leibniz had a 

half century earlier. On this, see [De Risi 2007, 250-252, note 109]. 
28

 [Heath 1956, 208]. 
29

 To be more precise, Wallis assumed that for any triangle ABC there exists a triangle DEF having the same angles 

and proportions (i.e., a triangle which is similar to ABC), but differing in size, or, that there exist two similar but 

unequal triangles. (See [Wallis 1693]). Wallis was influenced in his work on parallels by the 13
th

 century Arab 

geometer, Nasir al-Din al-Tusi (1201-1274), so much so that he commissioned a translation of his work, which 

exploited basic facts about triangles to ‗prove‘ the fifth postulate. [Wallis 1693] contains Wallis‘ own proof as well 

as a rendition of al-Tusi‘s results; cf. also [Bonola 1955, 10-12] for a brief resumé of al-Tusi‘s attempted proof.  



13 

 

1.4.2 Omar Khayyam  

 

A rather important addition to the picture begins to emerge with the work of the Persian 

polymath Omar Khayyam (1050-1123). The Commentary on the Difficulties of Certain 

Postulates of Euclid’s Work, Khayyam‘s monograph on the parallel postulate, is interestingly 

different than earlier work insofar as its aim was not to prove the postulate directly, but rather to 

derive it explicitly from an equivalent claim. In other words, where Ptolemy and Proclus, for 

instance, had unknowingly employed equivalent axioms to prove Euclid‘s, Khayyam knowingly 

attempted to replace the rather complex version we find in Euclid with his own, hopefully 

simpler and intuitively more acceptable principle. Instead of proving that the postulate was 

straightforwardly a theorem in Euclid‘s system, Khayyam sought to add a simpler version of the 

postulate to that system, thereby eliminating questions about the complexity and strangeness of 

the parallel postulate. He would then, ideally, show that Euclid‘s complex postulate could be 

derived from his simpler, more acceptable reformulation. His reformulation was as follows: 

 

Two convergent straight lines intersect and it is impossible for two convergent straight 

lines to diverge in the direction in which they converge.
30

 

 

He also noted something which Saccheri was later to exploit to great success: namely that, if we 

choose to leave Euclid‘s fifth postulate out of the Elements entirely, we are presented with three 

possible situations. Recall that Euclid‘s formulation of the parallel postulate runs as follows: 

 

That if a straight line falling on two straight lines make the interior angles on the same side 

less than two right angles, the two straight lines, if produced indefinitely, meet on that side 

on which are the angles less than the two right angles.
31

 

 

So, in the properly Euclidean case, if two straight lines, both perpendicular to a third, cross 

another, fourth, line, then the sum of the internal angles where the two perpendiculars meet the 

fourth line will be 180º. Khayyam noted two other possibilities made available if we do not 

accept Euclid‘s postulate (both of which Saccheri explored in greater detail). He noted, namely, 

the case in which the sum of the internal angles is greater than 180º (the obtuse case) and the 

case where the sum of the internal angles is less than 180º (the acute case). Indeed, it is precisely 

these replacements for Euclid‘s parallel postulate which would later provide the basis for 

hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case).  The three cases 

can all be represented visually (see Figure 2) by so-called Saccheri quadrilaterals (sometimes 

called Khayyam-Saccheri quadrilaterals), i.e., quadrilaterals which possess at least two right 

angles at their base and whose opposing sides are equal.  

 

Khayyam would end up dismissing both non-Euclidean cases because he believed they led to 

contradictions, but his reasoning employed certain unstated assumptions provably equivalent to 

the parallel postulate itself. Despite his own misgivings, Khayyam‘s work on multiple geometric 

systems heralded an entirely new approach to the problem of the parallel postulate. Over time the 

misgivings disappeared and multiple geometries came to replace the view of Euclid‘s geometry 

                                                 
30

 [Khayyam, quoted in Rosenfeld and Youschkevitch 1996, 130]. 
31

 [Heath 1956, 155]. 



14 

 

as the only one possible. But it would take a tremendous amount of time before this could 

happen. Why? Why were Khayyam‘s successors, too, reticent to embrace these other cases as 

consistent alternatives to Euclid‘s geometry? Why didn‘t something like the structural approach 

to geometry emerge in 11
th

 century Persia? 

 

 

Figure 2: Khayyam-Saccheri quadrilaterals 

 
 

1.4.3 Girolamo Saccheri 

 

The best place to look for an answer to this question is the Italian Jesuit Girolamo Saccheri, who 

seems to have possessed detailed knowledge of elements of (both) hyperbolic and elliptic 

geometry. And yet, unlike later geometers like Gauss, Bolyai, and Lobachevsky, he fell short of 

discovering these non-Euclidean geometries. An explanation of this near-success is illustrative of 

the small steps by which the movement away from the traditional Euclidean picture of geometry 

occurred. The details here are important precisely because they help to illustrate the deep hold of 

the Euclidean picture even in the 18
th

 century. 

 

In his 1733 work on Euclid (Euclides ab omni naevo vindicatus [Euclid Vindicated of Every 

Flaw]) Saccheri set out to prove the fifth postulate. His approach was similar to that of Khayyam 

in that it attempted, by reductio, to illustrate the impossibility of the two alternatives to the 

parallel postulate—namely the acute and obtuse cases isolated by his predecessor. His arguments 

against the obtuse case are rather more convincing than those against the acute. In the first 

instance he shows that the ‗elliptical‘ version of the parallel postulate leads to a contradiction, 

namely the conclusion that all straight lines are finite (a claim which, on one reading,
32

 directly 

                                                 
32

 As Stäckel and Engel note in their edition of Halsted‘s translation of Saccheri, there is insufficient precision in 

Saccheri‘s own text to decide between a few possible readings. They write, The Proposition on Exterior Angles (Eu. 

I. 16) that is being used here presupposes that the straight line is of infinite length. An alternative assumption, 

suggested by Riemann […] is that the straight line is unbounded. This property is compatible with the hypothesis 

that the straight line is infinite (open) as well as with the hypothesis that it is finite (closed). If the first of these two 



15 

 

contradicts Euclid‘s second postulate).
33

 In the acute case, however, such a contradiction is not 

so easy to achieve. Saccheri‘s attempt to destroy the acute hypothesis (and, thus, hyperbolic 

geometry as a real alternative to Euclid‘s) is instructive. It shows the deep, and tacit, 

commitment to the view of geometry as the science of space and the ways in which this 

commitment acted as a conceptual barrier to the development of a relational or structural 

approach to geometry. Let us turn, briefly, to his conclusions, ignoring for the moment some of 

the particulars of his reasoning. Though very few geometers wrote philosophical treatises 

outlining their view of the nature of geometry, in arguments like Saccheri‘s we can tease out 

some of the underlying assumptions. After a long and involved chain of theorems derived from 

the assumption that the acute hypothesis is true,
34

 Saccheri eventually arrives at the following 

proposition: 

 

Proposition XXXIII 

 

The hypothesis of acute angle is absolutely false; because repugnant to the nature of the 

straight line. 

 

PROOF. From the foregoing theorem may be established, that at length the hypothesis of 

acute angle inimical to the Euclidean geometry has as outcome that we must recognize two 

straights AX, BX, existing in the same plane, which produced in infinitum toward the parts 

of the points X must run together at length into one and the same straight line, truly 

receiving, at one and the same infinitely distant point a common perpendicular in the same 

plane with them. But since I am here to go into the very first principles, I shall diligently 

take care, that I omit nothing objected almost too scrupulously, which indeed I recognize to 

be opportune to the most exact demonstration.
35

 

 

Saccheri here simply abdicates any further investigation of the acute-angled hypothesis. Why? 

Because, as he may simply have assumed from the start, this hypothesis is repugnant to the 

Euclidean conception of the straight line. To the modern eye, this is a rather surprising surrender 

after several pages of technical and sophisticated argumentation. Indeed, Bonola characterizes 

this move as an attempt ―to come to a decision,‖ in which we witness Saccheri ―trusting to 

intuition and to faith in the validity of the Fifth Postulate rather than to logic‖.
36

 George Halsted, 

one of Saccheri‘s English translators and a great populariser of early works of non-Euclidean 

                                                                                                                                                             
hypotheses is taken, then, as Saccheri shows, the hypothesis of the obtuse angle leads to a contradiction. If the 

straight line is, on the other hand, taken to be (merely) unbounded, which allows the second possibility – that the 

straight line is finite (closed) – then the Proposition on Exterior Angles (Eu. I. 16) cannot be invoked, and Saccheri‘s 

proof becomes insufficient. Indeed, if this second possibility is assumed to be a closed line of finite length, then a 

contradiction can be shown not to occur‖ [Saccheri 1986, 246 (editors‘ notes)]. 
33

 Accordingly, modern elliptic geometry rejects both the second and the fifth of Euclid‘s postulates. (The third 

postulate also fails to hold, unless it is slightly modified such that one can construct a circle of radius equal to any 

given line segment rather than the less restricted claim we find in Euclid that one can construct a circle of any 

radius). 
34

 For a reasonably brief synopsis of the steps involved in this part of Saccheri‘s reasoning, [Bonola 1955, 38-44] is 

helpful. 
35

 [Saccheri 1733, 173]. 
36

 [Bonola 1955, 43]. 



16 

 

geometry in general, makes the following remarks on this somewhat hasty rejection of the acute 

angle hypothesis: 

 

Remember De Morgan‘s saying: ―As to writing another work on geometry, the middle 

ages would as soon have thought of composing another New Testament...... [Euclid‘s] 

order of demonstration was thought to be necessary and founded in the nature of our 

minds;‖ and remember that Saccheri‘s book contains not merely ―another work on 

geometry‖ but another geometry, a thought so tremendous, so unorthodox, that its 

discovery in his book by these great Church Dignitaries would have doomed Saccheri to 

death.
37

 

 

Despite the rather hyperbolic tone, there is some substance to Halsted‘s remarks here. The notion 

that Euclid‘s venerable geometry might be wrong—or, worse, that it might simply be one 

geometry among many—was almost unthinkable in the early eighteenth century.
38

 Bonola 

characterizes Saccheri‘s rejection here as a matter of faith in the time-tested Euclidean view of 

geometry. I think it is perhaps more accurate to say that Saccheri‘s enquiry was undertaken with 

the fairly reasonable assumption that geometry was the science of space and that there was no 

room in such a science for two valid but mutually exclusive claims. Nevertheless, Halsted seems 

to harbour the suspicion that Saccheri, against the spirit and limitations of his time, may very 

well have been a closet non-Euclidean. Pursuing this theme, he continues: 

 

There exists in the library of Modena, Italy, a Manuscript Biography of Saccheri. Dr. 

Emory McClintock, the able President of the New York Mathematical Society, writes of 

Saccheri: ―He confessed to a distracting heretical tendency on his part in favour of the 

‗hypothesis anguli acuti,‘ a tendency against which, however, he kept up a perpetual 

struggle. After yielding so far as to work out an accurate theory anticipating 

Lobatschewsky‘s doctrine of the parallel-angle, he appears to have conquered the internal 

enemy abruptly, since, to the surprise of his commentator Beltrami, he proceeded to 

announce dogmatically that the specious ‗hypothesis anguli acuti‘ is positively false.‖ Of 

course, no such confession occurs or could have occurred in the book itself; for with it the 

book could never have been printed. If such a statement occurs in the Manuscript 

Biography, that must have been the work of a personal friend written after Saccheri‘s 

death. [...] The sudden dogmatic assertion above mentioned does occur, first on page 70 of 

the work, a quarto. But this, after seventy quarto pages of rigorous logic and elegant 

demonstration to establish a non-Euclidean geometry, may be looked upon as something 

like the stucco for the king‘s inspection with which the immortal architect in Egypt 

covered the stone bearing his own name.
39

  

 

We may wish, with Halsted and McClintock, to read into Saccheri a more tolerant conception of 

the nature of geometry than is present in his work. However, such tolerance was not merely a 

                                                 
37

 [Halsted 1894, 150]. 
38

 Even early forms of (what would later be called) projective geometry, or Lambert‘s startlingly abstract ‗spherical 

geometry‘, were not considered true alternatives to traditional Euclidean geometry, but special cases of it. In the 

case of projective geometry, its potentially ‗extra-Euclidean‘ quality only became exploited in the works of Arthur 

Cayley in the mid-19
th

 century. 
39

 [Halsted 1894, 150-151]. 



17 

 

matter of submission to Euclid as an unquestioned authority. Saccheri, like his contemporaries, 

worked from within a tradition in which the indubitable, intuitive foundation of Euclidean 

geometry was taken as a matter of course. Indeed, the very title and intent of the work—to 

vindicate Euclid of all possible blemishes—is illustrative of Saccheri‘s acceptance of the 

traditional view of Euclidean geometry as the correct science of physical space. Without 

replacing this belief in the certainty of Euclidean geometry as a description of physical space, it 

is highly unlikely that anyone, even a geometer as astute as Saccheri, would have been capable 

of accepting hyperbolic geometry as anything more than a confusingly resistant collection of 

falsehoods, or perhaps an amusing formal game.  All of this, combined with Saccheri‘s text 

itself, speaks against any hidden affinity for hyperbolic geometry as a real alternative to 

Euclidean geometry. And so it is that, after seventy quarto pages, we find an exasperated 

Saccheri throwing up his hands at the futility of pursuing the acute angle hypothesis any further. 

 

Nevertheless, Halsted is correct when he finds the end of this lengthy and elegant chain of 

reasoning somewhat abrupt: Saccheri simply refuses to accept the possibility that a geometry 

based on the acute angle hypothesis (what we would call ‗hyperbolic geometry‘) can be taken 

seriously, and he rejects the acute angle hypothesis out of hand. All of this seems rather puzzling 

from a modern point of view in which the notion of multiple geometries with numerous different 

applications is a commonplace. But we must consider the traditional view of geometry as the 

science of physical space, a science whose absolute certitude was thought to have been 

unassailable because grounded in our direct experience and intuition of that space. To help 

explain Saccheri‘s rather abrupt rejection of a potential alternative to Euclidean geometry, I shall 

turn in the following section to a more focussed discussion of the philosophical and conceptual 

background which framed the tacit basis of Euclidean geometry as late as the mid-nineteenth 

century. It is only when the assumptions underlying this picture became visible that they were 

called into question. And it is only when the philosophical assumptions of Euclidean geometry 

were called into question that the structural approach to geometry began to emerge as a 

philosophical and mathematical alternative. 

 

§2. Geometric intuition and the emergence of non-Euclidean geometry 

 

Now, as we‘ve just seen, Saccheri‘s attempts to prove the parallel postulate brought him 

extraordinarily close to the discovery of hyperbolic geometry; a discovery which would, a 

century later, make Lobachevsky and Bolyai deservedly famous. Given his proximity to 

revolutionary geometric results, it is worth pausing to discuss the source of Saccheri‘s ‗failure‘, 

though we may call it such only in hindsight. What prevented Saccheri from exploiting his 

potential discoveries in the same way that these men would do so well—why did he halt at the 

cusp of non-Euclidean geometry? What was the origin of his ‗faith‘ in the primacy of Euclid‘s 

geometry?  

 

There are several answers to these questions here. As we are interested in the emergence of 

structural thinking in geometry, one is of primary importance. This is that that Saccheri, unlike 

later geometers, had not yet begun to question the nature and role of his own spatial intuition as a 

guarantor of the validity of geometry, nor of geometry as a unified science of physical space. In 

other words, Saccheri, like his predecessors Euclid, Proclus, Khayyam, and others, never 

seriously doubted the assumption that Euclidean geometry contained a true (if idealized) 



18 

 

description of the laws governing the physical space of our world, nor did he question the 

assumption that the certainty of its axioms was grounded in our immediate intuitive grasp of the 

properties of that space. Taking Euclidean geometry as the science of space, it made little sense 

to inquire after ‗alternative‘ geometries – for to be an alternative was straightforwardly to be 

false. Saccheri, of course, was not alone in these assumptions. Indeed, even in Gauss, one of the 

earliest mathematicians to take seriously the possibility of an alternative to Euclidean geometry, 

we find reverence for the intuitive basis of geometry. He writes in an 1816 review of a work on 

parallels, for instance, that: 

 

A great part of the text [of Schwab] turns on the contention against Kant that the certainty 

of geometry is not based on intuition but on definitions and on the principium identitatis 

and the principium contradictionis. Kant certainly did not wish to deny that use is 

constantly made in geometry of these logical aids to the presentation and linking of truths: 

but anybody who is acquainted with the essence of geometry knows that they are able to 

accomplish nothing by themselves, and that they put forth only sterile blossoms unless the 

fertilizing living intuition of the object itself prevails everywhere. Schwab‘s contradiction 

seems moreover to rest in part only on a misunderstanding: at any rate, it seems to us that 

in the sixteenth paragraph of his work (which uses precisely the faculty of intuition from 

beginning to end, and which claims at the end to prove that ―[the postulates of Euclid can 

be resolved into more general ones, based not on sensation and intuition, but on 

understanding]‖) Schwab must have imagined something else than did the Königsberg 

philosopher in this terminology for two different branches of the faculty of knowledge.
40

 

 

The view that the certainty of basic geometrical claims was based on intuition (in 

contradistinction to the apparently non-intuitive character of, e.g., arithmetical claims) was 

popular throughout the seventeenth, eighteenth, and nineteenth centuries. Its widespread 

popularity, however, does not mean that it had a clear or even a consistent meaning. Given that 

(most) mathematicians are not directly interested in the epistemological aspects of mathematics, 

it is perhaps not surprising that we find very few direct explanations of the notion of intuition in 

mathematical thinking. Nevertheless, it is mentioned almost everywhere in the 18
th

 and 19
th

 

centuries as an essential part of mathematics. 

 

The view that intuition (whatever it might be) was a guarantor of the truth and necessity of 

Euclidean geometry was not spelled out in detail by the mathematicians who held the view. In 

Saccheri, we find only the remark that the acute-angled hypothesis is inimical to the nature of the 

Euclidean straight line. The acute-angled hypothesis was untenable, therefore, only because it 

opposed Euclidean geometry and Euclidean geometry was the sole science of space. If Euclidean 

geometry is the only game in town, we cannot recognize the acute-angled hypothesis as true 

because it contradicts what we already know to be true. But what exactly was the basis of the 

truth of Euclid‘s geometry? Saccheri, like most other mathematicians, did not take up this 

                                                 
40

 [Gauss 1816, in Ewald 1999, 299-300]; Gauss‘s conception here of the sterility of any geometrical reasoning not 

fertilized by intuition clearly echoes Kant‘s remark that ―mathematics must first present all its concepts in intuition, 

and pure mathematics in pure intuition; that is, it must construct them. If it proceeded in any other way, it would be 

impossible to take a single step; for mathematics proceeds, not analytically by dissection of concepts, but 

synthetically, and if pure intuition be wanting there is nothing in which the matter for synthetical judgments a priori 

can be given‖ [Kant 1950, 30]. 



19 

 

question, but an answer was given in much greater philosophical detail in the work of Immanuel 

Kant (1724-1804). Subsequently, Kant‘s philosophy became, with varying degrees of 

explicitness, the de facto conceptual framework for many mathematicians, philosophers, and 

laymen interested in the nature of geometry and geometrical reasoning. Even despite Kant‘s own 

lack of mathematical expertise, his account of the intuitive basis of geometry seemed to capture 

the prevailing view of geometry as a spatial science, and was, accordingly, widely adopted and 

influential in variously altered forms well into the twentieth century. For instance, we find Paul 

Bernays, as late as 1930—well after the development of a seemingly anti-Kantian, non-intuitive 

approach to geometry—writing that:  

 

We can separate Kant‘s basic idea that mathematical knowledge, and in general the 

successful application of logical inference, rests on intuitive evidence, from the particular 

formulation that Kant gave to this idea in his theory of space and time. In this way we 

simultaneously gain the possibility of doing justice to the very elementary character of 

mathematical evidence and to the level of abstraction of the mathematical attitude.
41

 

 

Thus Kant‘s basic claim—that the intuition of space lies at the basis of geometric knowledge and 

reasoning—was taken up in forms which Kant himself could not possibly have conceived. As 

we‘ll see below in more detail, the Kantian view of geometry was surprisingly flexible in the 

face of the radically new results and approaches to geometry developed throughout the 

nineteenth century. Before we turn to these results, however, it will be useful to explore Kant‘s 

views on geometry, especially given their influence over the last two centuries or so.  

 

2.1 Kant‘s philosophy of geometry 

 

Despite a brief early interest in the possibility of higher-dimensional geometries,
42

 Kant‘s most 

important contribution to the philosophy of geometry was in the systematic philosophical form 

he lent to the traditional belief in the certainty of Euclidean geometry. He helped to cement the 

traditional view in two ways. First, he offered a somewhat plausible (if complex) description of 

the seemingly contradictory combination of intuition and apriority involved in geometric 

certainty; and, second, he tacitly worked within the universe of Euclidean geometry and its 

‗space‘—thus connecting his novel view of the nature of physical space with the familiar 

particulars of three-dimensional, Euclidean space. These views are found at various points 

throughout his work, though perhaps most prominently in the Critique of Pure Reason, where he 

writes that 

 

Geometry is a science which determines the properties of space synthetically and yet a 

priori. What, then, must be our representation of space, in order that such knowledge of it 

may be possible? It must in its origin be intuition; for from a mere concept no propositions 

can be obtained which go beyond the concept—as happens in geometry. [...] Further, this 

intuition must be a priori, that is, it must be found in us prior to any perception of an object 

                                                 
41

 [Bernays 1930-1931, in Mancosu 1998, 243]. These two approaches to geometry (the intuitive and the logical) are 

even more explicitly outlined and separated in [Hilbert and Cohn-Vossen 1999, iii-v], with the focus of the work 

being the development of the intuitive approach. 
42

 Cf. [Kant 1747, in Ewald 1999, 133-134]. 



20 

 

and must therefore be pure, not empirical, intuition. For geometrical propositions are one 

and all apodeictic, that is, are bound up with the consciousness of their necessity; for 

instance, that space has only three dimensions. Such propositions cannot be empirical or, in 

other words, judgments of experience, nor can they be derived from any such judgments 

[...]. How, then, can there exist in the mind an outer intuition which precedes the objects 

themselves, and in which the concept of these objects can be determined a priori? 

Manifestly, not otherwise than in so far as the intuition has its seat in the subject only, as 

the formal character of the subject, in virtue of which in being affected by objects, it 

obtains immediate representation, that is, intuition, of them; and only in so far, therefore, 

as it is merely the form of outer sense in general. Our explanation is thus the only 

explanation that makes intelligible the possibility of geometry, as a body of a priori 

synthetic knowledge.
43

 

 

Here Kant makes at least two important claims regarding the nature of geometry. First, he 

characterizes it as a science of space. And, second, he suggests that this science is both synthetic 

and a priori—i.e., that its truths can be known prior to all experience and that knowledge of 

these truths requires some form of intuitive input. This remark would have appeared 

contradictory to a philosophical public for whom intuition was a type of experience, which 

meant that it was, by definition, already a posteriori. To avoid this apparent contradiction, Kant 

introduces the concept of a pure intuition, which would become one of the cornerstones of his 

mature critical philosophy. There are two such pure intuitions (space and time) which are, for 

Kant, the epistemological sources of geometry and arithmetic, respectively. He divides up 

intuitions and concepts as follows: 

 

Both intuition and concepts are either pure or empirical. They are empirical if they contain 

sensation (sensation presupposes the actual presence of the object); they are pure if no 

sensation is mixed in with the presentation. Sensation may be called the matter of sensible 

cognition. Hence pure intuition contains only the form under which something is intuited, 

and a pure concept contains solely the form of the thought of an object as such. Only pure 

intuitions or concepts are possible a priori; empirical ones are possible only a posteriori.
44

 

 

As one of Kant‘s main interests is to explain the certainty of geometrical reasoning—by which is 

meant, tacitly at least, the rigorously developed axiomatic presentation we find in Euclid—he 

begins here by contrasting the source of geometrical knowledge with the error-prone sources of 

knowledge within the ‗empirical‘ sciences. These other sciences, along with other unscientific 

areas of human endeavour, are rooted in experience and the empirical—they deal with the 

specific matter of cognitions. On Kant‘s view, the so-called matter of these cognitions, what he 

called ‗sensations‘, can only be delivered through our contact with the physical world. For Kant, 

this places the truths related to these areas of cognition solely within the realm of the a 

posteriori. The sciences of pure intuition (e.g., geometry), by contrast, deal exclusively with the 

form of cognition, and this form is given prior to all possible experience. Indeed, Kant takes 

great pains to argue that there must be such a form, prior to experience, for experience to be 

possible in the first place. So, geometrical intuition differs from empirical intuition in that it does 

                                                 
43

 [Kant 1787, B40-41]. 
44

 [Kant 1787, A50-51/B74-75]. 



21 

 

not come into contact with the specific matter or content of sensations, but concerns only their 

form. 

 

Another distinction Kant is interested in making is that between pure intuitions and pure 

concepts. This distinction allowed him to distinguish analytic a priori knowledge from the 

peculiar synthetic a priori knowledge he wished to attribute to geometry and arithmetic. For 

Kant, if geometry were to be construed as having its basis in a pure concept of space, it would be 

a purely analytic discipline, in which no new truths other than those already contained in its basic 

concept(s) could be found. A purely conceptual geometry would be very similar, for Kant, to the 

discipline of logic. If it was in fact the case that geometry was merely a formal discipline, this 

would not be a problem. However, geometry is construed by Kant as a science of space, which 

has experiential import. By introducing the new category of pure intuition, Kant attempts to 

break the close connection he and others had previously thought to exist between analyticity and 

apriority, and thus to allow geometry formal necessity as well as the capacity to help explain the 

world of spatial experience. 

 

With these more general distinctions in mind, let us now turn to Kant‘s remarks on the specific 

qualities of our pure intuition of space. Early on in the Transcendental Aesthetic, Kant writes that 

 

Space is not an empirical concept that has been abstracted from outer experiences. For the 

presentation of space must already lie at the basis in order for certain sensations to be 

referred to something outside me (i.e., referred to something in a location of space other 

than the location in which I am). And it must similarly already lie at the basis in order for 

me to be able to present [the objects of] these sensations as outside and alongside one 

another, and hence to present them not only as different but as being in different locations. 

Accordingly the presentation of space cannot be one that we take from the relations of 

outer appearance by means of experience; rather, only through the presentation of space is 

that outer experience possible in the first place.
45

 

 

Here, Kant suggests that in order for us to present the objects of experience as distinct from 

ourselves (or from each other) we must already possess some notion of space within which these 

differences can be located and made apparent. Thus, this pure intuition is prior to and engenders 

the possibility of our outer experience. Following this line of thought, Kant further characterizes 

space as 

 

a necessary a priori presentation that underlies all outer intuitions. We can never have a 

presentation of there being no space, even though we are quite able to think of there being 

no objects encountered in it. Hence space must be regarded as the condition for the 

possibility of appearances, and not as a determination dependent on them. Space is an a 

priori presentation that necessarily underlies outer appearances.
46

 

 

Or, stated more clearly with respect to the notion of intuition, he writes that ―Space is nothing 

but the mere form of all appearances of outer senses; i.e., it is the subjective condition of 

                                                 
45

 [Kant 1787, A23/B38]. 
46

 [Kant 1787, A24/B38-39]. 



22 

 

sensibility under which alone outer intuition is possible for us‖; space is thus conceived as ―a 

pure intuition in which all objects must be determined‖.
47

 In sum, then, for Kant space is a pure 

intuition which determines the form of all outer appearance. Because it determines the form of 

appearances without itself being determined by a further appearance or experience, this intuition 

is both a priori and necessary for any possible experience. 

 

Since, for Kant and those following the Euclidean tradition, geometry was the science of space, 

all of these claims about the nature of space were intended to support the view that the existing 

body of geometrical truths was certain, necessary, and knowable a priori. Kant writes that 

 

On this a priori necessity [i.e., that of the pure intuition of space] rests the apodeictic 

certainty of all geometric principles and the possibility of geometry‘s constructions. For if 

this presentation of space were a concept acquired a posteriori, drawn from general outer 

experience, then the first principles for determining [things] in mathematics would be 

nothing but perceptions. Hence they would have all the contingency that perception has; 

and it would then precisely not be necessary for there to be only one straight line between 

two points, but this would be something that experience always teaches us. By the same 

token, what we take from experience has only comparative universality, viz., through 

induction. Hence all we could say is: as far as we have been able to tell until now, no space 

has been found that has more than three dimensions.
48

  

 

So, the crucial source of our geometrical knowledge is the pure intuition of space. As Kant‘s 

explanations here are intended to show, the concept of a pure intuition is meant to safeguard the 

necessity of mathematical (i.e., arithmetical and geometrical) reasoning while at the same time 

making sense of the obviously synthetic character of specifically geometric reasoning. By 

countenancing an epistemological source other than pure reason and sensory input, Kant allows 

for what he had formerly taken to be a contradictory combination of synthetic knowledge with 

                                                 
47

 [Kant 1787, A26/B42]. 
48

 [Kant 1787, A25]. At this point in the second edition of the Critique of Pure Reason, Kant slightly amends this 

paragraph as follows: ―By the same token, no geometric principles—e.g., the principle that in a triangle two sides 

together are greater than the third—are ever derived from universal concepts of line and triangle; rather, they are all 

derived from intuition, and are derived from it moreover a priori with apodeictic certainty‖ [Kant 1787, B39]. It is 

interesting to contrast the first edition quotation with Kant‘s 1747 remarks, in which he appears to countenance 

precisely the empirical sort of geometry that he rejects here. One of Kant‘s translators, Werner S. Pluhar, makes the 

following remark: ―But, as Kant has indicated, from this original intuition of space [i.e., the pure intuition of space] 

concepts can be formed, including such concepts as those of empirical space, relative space, Euclidean space, 

mathematical space‖ [Kant 1787, B40, translator‘s footnote 48]. While there is some truth to Pluhar‘s remark here—

we can form less general concepts of space from the pure intuition of space, concepts of space limited by, e.g., 

epistemological, mathematical, or physical criteria—it seems partially misleading with respect to Kant‘s own views 

to suggest that we might develop what appear to be more general conceptions of space, like those we find in the 

theory of relativity or in other non-Euclidean domains of mathematics. The suggestion is misleading because it 

seems that, for Kant‘s critical philosophy at least, our pure intuition of space was necessarily three-dimensional and 

(very likely) ‗Euclidean‘ in other details as well. Unfortunately, a detailed discussion of the specific qualities of the 

pure intuition of space is not forthcoming in Kant, and it is not clear if the object of this intuition is identical with 

Euclidean space—though Kant‘s offhand remarks regarding certain theorems of Euclidean geometry incline one to 

think as much. This view is perhaps reinforced by Kant‘s claim that ―We have no one who has exceeded Aristotle or 

enlarged his pure logic (which is in itself fundamentally impossible) just as no mathematician has exceeded Euclid‖ 

[Kant 1992, 438]. 



23 

 

the a priori.
49

 In addition to anchoring the putative necessity of geometrical truths, Kant also 

helped to reaffirm the privileged position of Euclid‘s geometry as the true description of the 

characteristics of (our pure intuition of) space. Kant, like most of his contemporaries, was of the 

opinion that the true science of space had already been given, by and large, in Euclid‘s Elements. 

Though he was likely aware of the nagging problem of the parallel postulate,
50

 this did not 

hinder his appreciation of Euclidean geometry a jot.
51

 He insists, as we see above, that it is not 

mere chance or empirical fact that there can be only one straight line through any two points, or 

that space is three-dimensional. Instead, he provides us with a philosophical basis for 

understanding how it is that these geometric truths, already present in Euclid, are necessary and a 

priori without being deducible in a purely formal manner. 

 

Thus, Kant provided an appealing attempt to explain the certainty of geometry by rooting it in 

pure intuition. This move accomplished two things. First, it explained geometrical reasoning‘s 

separation from the error-ridden arena of everyday experience and, second, it unveiled a realm of 

certainty distinct from the (apparently empty) realm of analytic truths. Thus, geometry was at 

once creative and assured; a neat epistemological trick. In addition to this explanation of 

geometry‘s certitude, Kant also reinforced the view that this certitude was the exclusive property 

of the Euclidean view of space. For Kant, our basic intuition of space could only be understood 

in one way, and that way was paved by Euclid. (As we will see, however, the very general way 

in which Kant characterized his views of geometry—i.e., with very little direct reference to the 

specific claims of Euclidean geometry—provided a certain measure of resilience for those views 

in the face of seemingly contradictory evidence provided by non-Euclidean geometries). Kant‘s 

views on geometry gave many geometers (e.g., Gauss) an explicit philosophical basis for their 

own views. This gave the philosophy of geometry a common language to work with. This 

resulted, on the one hand, in a firmer entrenchment of the classical Euclidean position (as we can 

see in the remark from Gauss quoted at the beginning of §2 of this chapter). On the other hand, it 

also gave those unsatisfied with Euclidean geometry some specific claims to attack. If geometry 

                                                 
49

 While it is commonly held (indeed it was held by Kant himself) that he was the discoverer of the synthetic a priori 

character of mathematical judgments (see [Kant 1783, §2]), [Rusnock 2013] prevents a convincing argument that he 

was preceded in this discovery by (at least) John Locke, as evidenced by the following passage from Locke‘s Essay 

Concerning Human Understanding: ―We can know then the Truth of two sorts of Propositions with perfect 

certainty; the one is, of those trifling Propositions, which have a certainty in them, but ‗tis but a verbal Certainty, 

but not instructive. And, secondly, we can know the Truth, and so may be certain in Propositions, which affirm 

something of another, which is a necessary consequence of its precise complex Idea, but not contained in it. As that 

the external Angle of all Triangles is bigger than either of the two opposite internal Angles; which relation of the 

outward Angle, to either of the opposite internal Angles, making no part of the comple Idea, signified by the Name 

Triangle, this is a real Truth, and conveys with it instructive, real Knowledge‖ [Locke 1690, Book IV, Chapter viii, 

§8]. 
50

 Kant corresponded with Lambert, and indeed called him ―the greatest genius in Germany‖ [Kant to Lambert 

December 31, 1764; Kant 1999, 81]. It seems likely that Kant would have been familiar with Lambert‘s 1766 work 

on parallels, at least in passing. Given the prominence of the problem of parallels for Euclidean geometry at the 

time, it seems unlikely that Kant would have been entirely ignorant of the existence of the problem, given his 

passing familiarity with Euclid and his interest in the nature of geometry and its epistemology. These remain, 

however, speculative remarks, as, to my knowledge, Kant provides us with no material on the basis of which we can 

make more definitive claims about his views. 
51

 Though Kant only rarely mentions Euclid or his Elements by name, he recapitulates Euclidean formulae and 

proofs at several points. See, for instance, his rendition of Proposition 32 of Book I of the Elements at [Kant 1787, 

A716-717/B744-745]. 



24 

 

is the science of space, and we know space through pure intuition, perhaps this intuition is not 

Euclidean, or perhaps this intuition does not include the parallel postulate as an essential part. 

Kant‘s philosophy of geometry both solidified the traditional conception of geometry and pinned 

it in place. 

 

2.2 The emergence of non-Euclidean geometry: Gauss, Bolyai, and Lobachevsky
52

 

 

While geometers from Khayyam to Saccheri were unable or unwilling to countenance the so-

called ‗acute hypothesis‘ as a viable alternative to Euclid‘s geometry, a century later we find no 

less than three brave souls willing to do so. The men in question were Carl Friedrich Gauss, 

János Bolyai, and Nikolai Lobachevsky, each of whom arrived at relatively similar conceptions 

regarding hyperbolic geometry independently of the others. Gauss was the first to countenance 

non-Euclidean geometry as a worthwhile scientific endeavour, but, due to his notorious reticence 

to publish, he was also the last of the three to have his findings published.
53

 Lobachevsky was 

the earliest of the three to have published, though his work had very little influence at the time 

given his isolated position at the University of Kazan, far from the centers of nineteenth century 

mathematical activity. Bolyai‘s work appeared in a rather timely manner, but, like that of 

Lobachevsky, it did not have the immediate impact one would have expected, a circumstance 

likely due to the relative obscurity of the author, the fact that his results were published in an 

appendix to a work of his father‘s, and the still-formidable status of Euclidean geometry with its 

alleged bedrock of intuitive certainty recently reinforced by Kant. In what follows, I will trace 

some elements of the early development of non-Euclidean geometry, beginning with one of its 

earliest forms in the notebooks and correspondence of Gauss. 

 

2.2.1 Carl Friedrich Gauss 

 

It is widely supposed that Gauss was the first to seriously consider non-Euclidean geometry.
54

 

Due, however, to his unwillingness to publish and his fear of the ―clamour of the Boeotians‖
55

  

over the suggestion that we ought to take seriously any geometry other than Euclid‘s, almost 

nothing of his work on the subject was known publicly until after his death in 1855. Collected, 

this material is rather thin, comprising scattered remarks in letters, a few notebook sketches, and 

two papers.
56

 From the letters, however, we find suggestions from Gauss that he had expended a 

                                                 
52

 In this sketch of a rather complex conceptual history I must remain somewhat selective. After Saccheri and prior 

to the publication of Lobachevsky‘s results, work of undoubted importance was accomplished in the theory of 

parallels by several men. Of particular note are works by Lambert, by the German jurist Ferdinand Karl Schweikart 

(1780-1859), and two publications by Schweikart‘s nephew, Franz Adolf Taurinus (1794-1874). 
53

 Gauss is known to have followed the maxim ―Pauca sed matura‖ (Few, but ripe), which was exemplified in his 

attitude toward publishing; despite the lateness with which his thoughts on non-Euclidean geometry appeared, they 

nevertheless became rather more influential than those of either Bolyai or Lobachevsky, due in large part to his 

reputation. 
54

 Cf. [Kline 1972, 877-878], [Ewald 1999, 297], [Gray 1979, 76], and [Dunnington 1955, 174-190]. 
55

 Cf. Gauss‘ letter to Friedrich Bessel (1784-1846) of the 27
th

 of January, 1829: ―But I shall not for a long time 

work up for public consumption my very extensive investigations into these matters [i.e., non-Euclidean geometry], 

and this will possibly not occur in my lifetime, since I fear the clamour of the Boeotians if I were to express my 

opinion completely‖ [Gauss 1863-1929, Band VIII, 200; translated in Ewald 1999, 301]. 
56

 His correspondence, beginning at the close of the eighteenth century, with Farkas Bolyai, Gerling, Taurinus, and 

Bessel, is of particular note in connection with his researches into non-Euclidean geometry. The papers in question 



25 

 

considerable amount of mental energy upon the subject of non-Euclidean geometry, even if he 

had not deigned to commit much to paper in any definitive form.  

 

Gauss was a long-time correspondent with the Hungarian mathematician Farkas Bolyai, the 

father of János Bolyai. After years of toil, the latter had finally produced a detailed work on non-

Euclidean geometry. This work, The Science Absolute of Space: Independent of the Truth or 

Falsity of Euclid’s Axiom XI (which can never be decided a priori), was published in 1831 as an 

appendix to a work of his father. Subsequently, his father forwarded a copy to his friend Gauss in 

1832.
57

 Gauss‘ reply, from the 6th of March 1832, was as follows: 

 

If I commenced by saying that I am unable to praise this work (by Janos), you would 

certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to 

praise myself. Indeed the whole contents of the work, the path taken by your son, the 

results to which he is led, coincide almost entirely with my meditations, which have 

occupied my mind partly for the last thirty or thirty-five years. So I remain quite stupefied. 

So far as my own work is concerned, of which up till now I have put little on paper, my 

attention was not to let it be published during my lifetime. Indeed the majority of people 

have not clear ideas upon the questions of which we are speaking, and I have found very 

few people who could regard with any special interest what I communicated to them on 

this subject. To be able to take such an interest it is first of all necessary to have devoted 

careful thought to the real nature of what is wanted and upon this matter almost all are 

most uncertain. On the other hand it was my idea to write down all this later so that at least 

it should not perish with me. It is therefore a pleasant surprise for me that I am spared this 

trouble, and I am very glad that it is just the son of my old friend, who takes the precedence 

of me in such a remarkable manner.
58

 

 

As is not terribly surprising, János was rather irked by Gauss‘ reply; Gray writes that he ―never 

really forgave the Prince of Geometers. He was incredulous that Gauss should have made his 

discoveries before him, and became so convinced of plagiarism that he never published again‖.
59

 

We are perhaps in a better position than Bolyai to estimate Gauss‘ intellectual honesty on the 

matter.  

 

Whether or not the specific findings later contained in Bolyai‘s work were known to Gauss, it 

does seem clear that he had begun to take seriously the problems with the traditional conception 

of Euclidean geometry as the only possible science of space. From scattered remarks in his 

correspondence it is evident that Gauss was dissatisfied with the current understanding of 

geometry, particularly with the belief that the theorems of Euclidean geometry were 

straightforward truths about the physical world.
60

 Moreover, he was cognizant that whatever new 

                                                                                                                                                             
are [Gauss 1816] and [Gauss 1822], both of which were published locally in the Göttingensche gelehrten Anzeige. 

Both the notebook material and the relevant correspondence can be found in [Gauss 1863-1929, Band VIII, 157-

268]. 
57

 Farkas Bolyai had actually sent Gauss a copy of the work in 1831, but due in part to the instability caused by the 

cholera pandemic then making its way across Europe, the book never arrived. 
58

 [Gauss 1863-1929, Band VIII, 221; translated in Gray 1989, 97]. 
59

 [Gray 1989, 97-98]. 
60

 See, e.g., [Gauss 1863-1929, Band VIII, 159, 200-201; translated in Ewald 1999, 299, 301]. 



26 

 

approach was developed, it would likely alter radically our conception of geometry as an a priori 

science grounded in pure intuition, not to mention our understanding of the connection between 

geometry and the physical world. His distance from accepted tradition is further evidenced in an 

1817 letter to the German astronomer Heinrich Wilhelm Matthäus Olbers (1758-1840), in which 

Gauss writes:  

 

I come more and more to the view that the necessity of our geometry cannot be proved... 

Perhaps we shall come to another insight in another life into the nature of space, which is 

unattainable for us now. But until then one must not rank Geometry with Arithmetic, 

which is truly a priori, but with Mechanics.
61

   

 

Here we see the dual effects of Kant‘s philosophy of geometry. Gauss speaks, like Kant, in the 

language of the a priori and the a posteriori but he also comes to doubt that traditional geometry 

is merely an elucidation of the a priori, pure intuition that Kant had suggested. Given the 

problems in establishing the parallel postulate, Gauss, aware of Kant‘s philosophy but 

considerably more knowledgeable about geometry than Kant, is ready to (privately) consider that 

the application of Euclidean geometry to the physical world might be an a posteriori affair after 

all. There are inklings of the empirical grounding of geometrical axioms which we will later find 

in the work of Gauss‘ student Riemann. But, from Gauss himself, we do not find much more in 

his non-Euclidean works than isolated theorems and hints toward a new vision of the subject; for 

more substantial results, we must turn to the work of the two men whose names have since 

become most closely associated with the birth of non-Euclidean geometry, namely, Bolyai and 

Lobachevsky. 

 

2.2.2 Bolyai and Lobachevsky 

 

Both Bolyai and Lobachevsky were isolated from the main mathematical centers of their time, 

with the result that they and their works were little known until the surge of interest in non-

Euclidean geometries prompted by the publishing of Gauss‘ work after his death. Indeed, the two 

men remained isolated even from each other for many years; Gray writes that 

 

The work of Bolyai and Lobachevsky is astonishingly similar, and yet each remained in 

ignorance of the very existence of the other until some years after their work was 

published. The Hungarian was aware through his father of the Western European work on 

non-Euclidean geometry, although he appears to have largely gone his own way. The 

Russian kept apart from the controversy, and his earliest publication on the subject, in the 

Kazan Messenger for 1829, passed completely unnoticed.
62

 

 

Let us turn now to a discussion of some elements of the conceptual content of their work and the 

ways in which that work diverges from the traditional understanding of geometry as grounded in 

a basic, Euclidean intuition of physical space. Here we will begin to see the philosophical 

vacuum that begins to open up in the wake of the mathematical dismantling of the primacy of 

                                                 
61

 [Gauss 1817, translated in Gray 1989, 76; also in Gauss 1863-1929, Band VIII, 177]. 
62

 [Gray 1979, 96]. 



27 

 

Euclidean geometry. It is in this vacuum that the structure-focussed approach to mathematics 

begins to come into focus. 

 

The very title of Bolyai‘s work, The Science Absolute of Space: Independent of the Truth or 

Falsity of Euclid’s Axiom XI, indicates his belief that he had penetrated more deeply into the 

nature of space than had Euclid or his followers. His own early frustrations— as late as 1821 he 

was convinced ―that the parallel postulate must hold‖
63

— certainly encouraged his resolve. In a 

remarkable letter of 1823 to his father, he writes: 

 

I have now resolved to publish a work on parallels ... I have not yet completed the work, 

but the road I have followed has made it almost certain that the goal will be attained, if that 

is at all possible: the goal is not yet reached, but I have made such wonderful discoveries 

that I have been almost overwhelmed by them, and it would be the cause of continual 

regret if they were lost. When you see them, you too will recognize them. In the meantime 

I can say only this: I have created a new universe from nothing. All that I have sent you till 

now is but a house of cards compared to a tower. I am as fully persuaded that will bring me 

honour, as if I had already completed the discovery.
64

 

 

His researches were not published until 1831, and even then ―passed straight into obscurity,‖
65

 

despite this effusive self-praise. Only much later was his approach to geometry noticed and 

properly evaluated by others. 

 

One of the key conceptual innovations that Bolyai introduced was his clear use of the notion of 

‗absolute‘ geometry (often also called neutral geometry, or ‗pangeometry‘ by Lobachevsky). He 

used this term to refer to the collection of geometrical theorems which remain true independently 

of the parallel postulate, i.e., those which can be proved from Euclid‘s other axioms alone. One 

of the remarkable achievements of his work was the tacit ascription of independent importance to 

the absolute or neutral perspective from which we can make decisions about the status of the 

parallel postulate. This perspective had been implicitly employed as early as Khayyam‘s 

discussion of the three possible cases which arise when we suspend judgment regarding the 

parallel postulate, but neither Khayyam nor Saccheri had dwelt overmuch on the nature of this 

neutral geometrical perspective itself. What is novel in Bolyai is not solely his discussion of 

theorems from within this perspective, but also his analysis of which of these theorems remained 

true when the parallel postulate was denied (or, correspondingly, accepted). Thus, he makes a 

distinction between theorems which are ‗absolutely‘ true—i.e., those which are true whether we 

accept, reject, or suspend judgment upon the parallel postulate—and those which are true only 

given the further assumption/denial of the parallel postulate. This is a remarkable change in 

attitude towards the ‗acute angled hypothesis‘ from that of Saccheri, who, recall, thought of this 

hypothesis as inimical to the nature of the Euclidean straight line and, therefore, as absolutely 

false. From within this neutral perspective, Bolyai proves a number of theorems. He also goes on 

to work within a geometry in which the parallel postulate is denied, namely a hyperbolic 

geometry developed from the assumption that the acute angled case of Saccheri is true.  

                                                 
63

 [Gray 1979, 97]. 
64

 [J. Bolyai to F. Bolyai, 3 November 1832; quoted in Gray 1979, 97]. 
65

 [Gray 1979, 98]. 



28 

 

 

As Gray notes, Bolyai and Lobachevsky produced works which were ‗astonishingly similar‘. 

Several of the theorems proved by Bolyai are proved analogously by Lobachevsky. 

Lobachevsky‘s ‗imaginary geometry‘ was, like Bolyai‘s, based upon the assumption that 

Saccheri‘s hypothesis of the acute angle holds. One of the notable differences between their 

works is that Lobachevsky did not limit his geometry to representations in the plane. He offers, 

for example, the figure of the horosphere (a sphere of infinite radius with constant negative 

curvature) as a surface upon which a non-Euclidean geometry holds when its basic terms are 

given appropriate interpretations. Such spherical ‗models‘ of hyperbolic and other geometries 

would later play a role in making non-Euclidean geometries acceptable and scientifically useful. 

 

Lobachevsky, perhaps even more than Bolyai, was also quite well aware of the implications 

which his pan-geometrical perspective had for the traditional, Kantian underpinnings of 

Euclidean geometry. He writes: 

 

The fruitlessness of the attempts made, since Euclid‘s time, for the space of 2000 years, 

aroused in me the suspicion that the truth, which it was desired to prove, was not contained 

in the data themselves; that to establish it the aid of experiment would be needed, for 

example of astronomical observations, as in the case of other laws of nature. When I had 

finally convinced myself of the justice of my conjecture and believed that I had completely 

solved this difficult question, I wrote, in 1826, a memoir on this subject.
66

 

 

Here we see that the continual failure of Lobachevsky‘s early (1815-1817) attempts to prove the 

parallel postulate brought him to the conclusion which Gauss had suggested already in 1799. 

Namely, that the basis of the truth of Euclidean geometry was not to be found in our a priori 

intuition of space but, rather, upon empirical considerations. Two thousand years of history had 

shown that establishing the truth of at least some aspects of geometry did require what Kant 

would call the ‗matter‘ of sensation (i.e., astronomical observation). Thus, Euclidean geometry 

could not be considered an a priori science. Accordingly, Lobachevsky‘s work on Euclidean 

geometry ―compelled [it] to take its place again among the experimental sciences‖.
67

 With his 

work on pangeometry, however, Lobachevsky yet secured the a priori status of something 

recognizably geometric: ―... Pangeometry becomes a branch of analysis, including and extending 

the analytical methods of ordinary geometry‖.
68

 Lobachevsky‘s pangeometric approach detached 

an important portion of Euclidean geometry from the intuitive underpinnings which had 

previously linked geometry to physical space. This conceptual move partially enabled the shift 

away from the conception of geometry as the science of physical space towards a more abstract 

understanding of geometry as the science of a number of possible spaces. From the pangeometric 

perspective, as from those of Gauss‘s astral geometry and Bolyai‘s absolute geometry, Euclidean 

geometry is treated as a special case whose applicability to the physical world cannot be proven 

a priori, but must instead be shown by empirical means. The peculiarity of the parallel postulate 

in relation to Euclid‘s other postulates finally makes sense. Unlike the other postulates (whose 

                                                 
66

 [Lobachevsky, quoted in Bonola 1955, 92]; the memoir referred to is his Exposition succincte des principes de 

Géométrie. 
67

 [Bonola 1955, 93]. 
68

 [Lobachevsky, quoted in Bonola 1955, 94]. 



29 

 

apriority and certainty were not yet to be challenged), the parallel postulate‘s truth was 

connected to some peculiarity about our physical world, a peculiarity which could not be derived 

from our intuition of space, or from the nature of reason, but instead had to be measured. This 

initial step, shared in common by Gauss, Bolyai, and Lobachevsky, allowed the dismantling of 

the Kantian dogma of the intuitive basis of geometrical necessity to begin. It also planted the 

seeds for the emergence of geometrical spaces—plural—conceived abstractly without direct 

reference to any intuitive or physical manifestations.  

 

2.3 Challenges to Kant 

 

With their work, Bolyai and Lobachevsky performed a key conceptual manoeuvre which would 

eventually help enable a significant alteration (one might say revolution) in the way that 

geometry was perceived. Their countenancing of alternative geometries was a necessary first 

step in the dismantling of the traditional picture, but it was not sufficient.  

 

It was a necessary step because without presenting a coherent and reasonably consistent 

alternative to Euclid‘s geometry, there could be no question of abandoning the view of geometry 

as the science of space. If geometry is about space, and there is one true description of space, 

then any putative alternatives to this description must necessarily be false. But, if it can be shown 

that a non-Euclidean form of geometry is a viable description of space, then the question must be 

asked: how can we know which alternative is the correct one? Prior to the work of Gauss, Bolyai, 

and Lobachevsky, this question seemed literally nonsensical. But this step was not enough to 

usher in a completely new understanding of the nature of geometry overnight.  

 

Many mathematicians remained certain of geometrical truths, and this certainty seemed to 

require explanation, given the apparent similarity between geometry as a concrete science (the 

science of space) and the error-prone empirical sciences. Kant‘s views seemed to offer an 

appealing path towards explaining this apparently contradictory combination of certainty with 

the concrete. Though it is a commonplace to insist that the development of consistent non-

Euclidean geometries overturned the certainty with which the Kantian view of geometry was 

held, this is a somewhat misleading claim. The claim is misleading insofar as it fails to take into 

account the remarkable resilience which those views possessed in the face of seemingly 

recalcitrant evidence. And, even more, such a claim can seem to anachronistically 

mischaracterize the nature of the relevant counter-evidence as understood by proponents of the 

Kantian view. 

  

As we‘ve seen, prior to the emergence of non-Euclidean geometries, Kant had attempted to 

ground the apriority of geometry (i.e., Euclidean geometry) in its intuitive necessity. The basis of 

geometry was our pure intuition of space, and, I have argued, this intuition itself was at least 

tacitly assumed to be Euclidean in nature. For Kant and the legion of geometers who took his 

view to be a more or less accurate description of their pre-theoretic conception of geometry, our 

pure intuition of space presented (with certainty and necessity) a space which was three-

dimensional, isotropic (i.e., whose geometrical properties and relations are invariant with respect 

to direction, location, or size), which obeyed the parallel postulate, etc.  In other words, the form 

of our pure intuition of space was taken to be Euclidean. Thus, while the logical possibility of 

non-Euclidean geometries would likely have been admitted by Kant, such geometries would 



30 

 

nevertheless have been impossible for us to intuit (as the pure intuition of space already 

contained specific geometric content) and could therefore not be true in the sense that Euclidean 

geometry was understood to be true. And, given Kant‘s popular dictum that thoughts without 

(intuitive) content are empty,
69

 it seems likely that he would have seen non-Euclidean geometries 

as interesting but scientifically vacuous constructions—merely ‗logical‘ constructions which 

could not be applied to the realm of experience insofar as that realm is always structured and 

made possible by our pure, Euclidean intuition of space. Thus, non-Euclidean geometries could 

not be considered sciences at all, at least not in the sense of a theory applicable to some empirical 

subject matter.
70

 Far from sciences, these descriptions of alternative ‗spaces‘ would be better 

understood as arbitrary logical games, with no application to reality. Indeed, something like this 

seems to have been Frege‘s view, inspired by Kant, as late as the turn of the century:  

 

The question at the present time is whether Euclidean or non-Euclidean geometry should 

be struck off the role of the sciences and made to line up as a museum piece alongside 

alchemy and astrology. If one is content to have only phantoms hovering around one, there 

is no need to take the matter so seriously; but in science we are subject to the necessity of 

seeking after truth. There it is a case of in or out! Well, is it Euclidean or non-Euclidean 

geometry that should get the sack? That is the question. Do we dare treat Euclid‘s 

elements, which have exercised unquestioned sway for 2000 years, as we have treated 

astrology? It is only if we do not dare to do this that we can put Euclid‘s axioms forward as 

propositions that are neither false nor doubtful. In that case non-Euclidean geometry will 

have to be counted amongst the pseudo-sciences, to the study of which we still attach some 

slight importance, but only as historical curiosities.
71

 

 

Frege here puts forward the view that one must inevitably choose between Euclidean and non-

Euclidean geometries—with the heavy bias being in favour of Euclid. This view was quite 

common during the early years of non-Euclidean geometries and was based upon two key 

presuppositions. The first is the (Aristotelian) view of distinct sciences differentiated according 

to classes of truths related to particular domains of entities. Geometry, on this view, was the 

science which studied the class of truths related to spatial objects, whose characteristics are 

idealized in points, lines, planes, etc. And, second, it is based upon the view that Euclidean and 

                                                 
69

 The oft-quoted remark is to be found early on in the Critique of Pure Reason, in the introductory preamble to the 

transcendental logic: ―Thoughts without content are empty; intuitions without concepts are blind‖ [Kant 1787, 

A51/B75]. 
70

 Kant writes in the Prolegomena, for instance, that ―Pure mathematics, and especially pure geometry, can have 

objective reality only on condition that they refer merely to objects of sense‖ [Kant 1950, 34]. Geometry‘s way of 

fulfilling this requirement is by referring only to the necessary spatial form of the objects of sense.  Thus, a 

geometry constructed upon an alternative concept of space (i.e., one which differs from our pure intuition of space) 

can never connect with objective reality insofar as it fails to refer in any way to the objects of sense, which are 

determined exclusively by way of our pure intuition of space. So, if our pure intuition of space is Euclidean, then 

any alternative conception of space which hopes to achieve objective reality mustn‘t contradict any of the postulates 

of Euclidean geometry. On such a view, alternate geometries which endeavour to attain the status of sciences can 

only do so on the condition that they are special cases of Euclidean geometry. Given these views, we might consider 

Kant a kind of proto-structuralist about geometry, but I think this would be to ignore the fact that for Kant such 

‗structures‘ would be empty and meaningless (something which all but the most extreme mathematical structuralists 

would find deeply troubling). 
71

 [Frege c.1900, 169]. 



31 

 

non-Euclidean theories were both geometries – thus, that they were competing descriptions of 

the same domain of objects. So, accordingly, if a science is a (true) description of some 

collection of objects, and if there are two sciences which purport to describe the same aspects of 

the same collection of objects, it seems to be the case that either the two sciences say the same 

things in different ways (in which case one will do), or it will be the case that one of the sciences 

is simply false—as Frege implies here. Alberto Coffa describes the early prevalence of this 

supposed competition between Euclidean and non-Euclidean geometries as follows: 

 

When non-Euclidean geometries first appeared on the scene, they were naturally 

understood as rivals of Euclid, and their primitives were therefore taken to designate the 

same things as Euclid‘s, whatever they might be. Presumably, since both Euclid‘s and 

Lobachevskii‘s were geometries, they had to be interpreted as attempts to identify the class 

of truths concerning the very same domain of entities. Their points, straight lines, and so 

on had to be the same in both theories.
72

 

 

Given the long tradition and apparent intuitive plausibility supporting Euclidean geometry, it is 

not surprising, then, that early work on the legitimation of non-Euclidean geometry focussed on 

connecting it in some way to ‗real‘ geometry, i.e., Euclidean geometry. One of the best known, 

and most influential, examples of such an attempt was Eugenio Beltrami‘s (1835-1900) model of 

hyperbolic geometry constructed within a fragment of Euclidean space. Somewhat surprisingly, 

such attempts to connect non-Euclidean geometry to the legitimacy of Euclidean geometry were 

instrumental in the (eventual) dismantling of the Kantian picture. Beltrami‘s model is developed 

in two separate papers of 1868. These papers were subsequently generalized by Felix Klein 

(1849-1925) in 1871. Thus, the resulting model of hyperbolic geometry is usually called the 

Beltrami-Klein model.
73

 Given that the emergence of this model played an important role in 

challenging the pervasive Kantian view of geometry, it will be helpful to discuss some of the key 

conceptual moves employed in its construction. 

 

2.3.1 The Beltrami-Klein model of hyperbolic geometry 

 

The Beltrami-Klein model begins within a two-dimensional Euclidean space, upon which we 

construct an open disc (i.e., all the points contained within the radius of a circle γ, excepting the 

points which lie on the circumference of the circle). Now, we interpret the ‗lines‘ of 

Lobachevsky‘s hyperbolic geometry as the open chords of this open disc (i.e., the chords of γ 

excluding the endpoints which lie upon the circumference of γ). We then consider two ‗lines‘ 

(i.e., open chords) a and b to be parallel if they have no point within γ in common. If the 

Euclidean lines which extend the open chords intersect outside γ, the hyperbolic ‗lines‘ are 

nevertheless called parallel as long as they have no point within γ in common (see Figure 3). 

Here the open disc represents the hyperbolic plane, the open chords of γ represent hyperbolic 

lines, and the relation holding between two lines which do not have a point within the hyperbolic 

plane in common replaces the Euclidean parallel relation. Other basic Euclidean concepts are 

translated with varying degrees of familiarity: a ‗point‘ is a Euclidean point within the open disc; 

such a point ‗lies on‘ a hyperbolic line in just the same way it would lie on a Euclidean line; a 

                                                 
72

 [Coffa 1986, 7]. 
73

 The references are to [Beltrami 1868a], [Beltrami 1868b], and [Klein 1871], respectively. 



32 

 

hyperbolic point lies ‗between‘ two others in the familiar way, etc. The key difference here is 

that the points and lines of the hyperbolic plane all lie within the open disc. (Hyperbolic 

congruence and perpendicularity are not as easily defined, as they require some tinkering with 

rather tricky metrical concepts. I will delay discussion of these difficulties until the following 

subsection.) 

 

Figure 3: Parallel lines within the Beltrami-Klein model 

 
 

Essentially, then, this method of modelling shows that we can realize Lobachevsky‘s hyperbolic 

geometry inside the space of traditional Euclidean geometry. With this manoeuvre, Beltrami 

illustrated not only that non-Euclidean geometry was consistent,
74

 but, also, that it was strictly 

reducible to Euclidean geometry—an apparent vindication of the Kantian view that all geometry, 

properly speaking, was Euclidean in the final analysis. So, while Beltrami had provided a model 

of non-Euclidean geometry, he also reinforced the Kantian view insofar as his model was 

constructed within Euclidean geometry. Somewhat oddly, then, the first real vindication of non-

Euclidean geometry as more than a tangle of counter-intuitive propositions at the same time 

bolstered the confidence of those who believed geometry was ultimately grounded in our 

Euclidean intuition of space. Coffa describes the situation as follows: 

 

Despite appearances, the doctrines put forth could have brought nothing but comfort to 

Kantian souls. For Beltrami‘s ultimate goal was not so much to interpret as to reduce 

hyperbolic geometry to Euclidean geometry and to argue that there was no more geometric 

sense in the former than it could derive from the latter.
75

 

 

And, further, that 

 

                                                 
74

 Consistent, of course, only relative to the consistency of Euclidean geometry – which was not in question at the 

time given the widely held belief in the supposed epistemological guarantee of spatial intuition (and the assumption 

that the physical world itself is consistent). 
75

 [Coffa 1991, 48]. 



33 

 

Far from posing any threats to Kant‘s philosophy, Beltrami‘s work was consistent with and 

possibly even grounded upon it. Kant had never doubted the logical consistency of non-

Euclidean geometries. He would surely have said of hyperbolic geometry that it is 

impossible but not logically impossible (since its ―negation,‖ Euclidean geometry, is not 

logically necessary but only intuitionally necessary). So the fact that there is an 

interpretation of hyperbolic geometry is hardly surprising, nor is it surprising that the 

interpretation has to be given in terms of Euclidean intuitable notions. Nor is it surprising 

that wherever that reduction to Euclidean intuition fails, we must abandon the project of 

giving an interpretation to Lobachevski‘s theory. One could hardly find a more appealing 

package of good news for Kantians in a geometric monograph.
76

 

 

To the Kantian geometer of the time, then, any scientific importance we might wish to lend to 

new forms of geometry was determined entirely by the possibility of connecting those 

geometries to Euclid‘s intuitively guaranteed geometry. Once such a connection was made 

evident—as in Beltrami‘s case—Kantian scruples were satisfied and normal geometric business 

could continue, for a while at least.
77

  

 

But, despite the increased assurance lent to the Kantian understanding of geometry by Beltrami‘s 

model, subsequent work by Felix Klein on hyperbolic models, influenced heavily by the British 

mathematician Arthur Cayley (1821-1895), was to prove instrumental in dislodging the Kantian 

view.  

 

2.3.2 Cayley‘s projective concept of distance and Klein‘s generalizations 

 

Let‘s begin our discussion of Klein and Cayley by first returning to the discussion of the 

interpretation of hyperbolic congruence and perpendicularity within the Beltrami-Klein model. 

Now, one might begin by attempting to treat congruence for angles in the normal manner, i.e., by 

introducing some metric for angles and then calling two angles congruent just in case they 

possess the same number of degrees according to this metric. The problem with this approach in 

our case is that we cannot employ a Euclidean metric. If we employ a usual Euclidean measure 

of length, for instance, then all our hyperbolic lines will have to be considered finite. This is so 

because once we import a Euclidean metric, the circumference of the circle within which the 

hyperbolic plane is embedded becomes a measurable part of the hyperbolic universe, whereas in 

the model it is importantly outside that universe.
78

 It seems that the use of any standard 

Euclidean metric will result in the collapsing of the horizon of the non-Euclidean world into that 

world, thus eliminating the very distinction required for the model to function in the first place. If 

we cannot employ any of the usual Euclidean approaches toward measurement, how then do we 

discuss congruence and perpendicularity within the hyperbolic plane?  

                                                 
76

 [Coffa 1991, 49]. 
77

 This reading of the import of Beltrami‘s work should not suggest that Beltrami saw himself as proving the 

relative consistency of hyperbolic geometry. This is chiefly because the notion of a relative consistency proof was 

not at all clearly formulated, by Beltrami or anyone else, in 1868.  If it is misleading, then, to suggest that Beltrami 

intended to prove the consistency of hyperbolic geometry relative to Euclidean geometry, it is nevertheless true to 

state that his work did not immediately call into question the primacy or truth of Euclidean geometry. See [Gray 

2007, 219-220] for more detailed remarks to this effect. 
78

 See [Kvasz 1998, 155f.] for a useful discussion of this problem. 



34 

 

 

The standard approach, developed by Beltrami and further specified by Klein, runs roughly as 

follows. We say two hyperbolic lines (i.e. open chords of the circle γ) are ‗perpendicular‘ in the 

following two cases: 

 

1. If one of the two Euclidean chords a1 and b1 is a diameter of the circle γ, then the 
hyperbolic lines a and b (i.e., a1 and b1 excluding their endpoints) are said to be 

hyperbolically perpendicular iff they are perpendicular in the usual Euclidean sense. 

 

2. If neither a1 nor b1 is a diameter of the circle γ, then a and b are perpendicular iff the 
Euclidean line which extends a passes through the pole of b, where by ‗pole of b‘ is 

meant the (Euclidean) point of intersection of the tangents of the radius of γ with the 

endpoints of b1 (Figure 4). 

 

Figure 4: Hyperbolic perpendicularity in the Beltrami-Klein model 

 

 
 

Now, what is interesting here is that the definition of hyperbolic perpendicularity requires that 

we extend hyperbolic lines to points at infinity, i.e., ‗Euclidean‘ points which do not lie within 

the hyperbolic plane itself. The use of points at infinity is an important technique of projective 

geometry, which Cayley had examined and generalized in the 1859 paper from which Klein 

drew. In contrast to Beltrami‘s efforts to legitimate non-Euclidean geometry by embedding it 

within intuitively acceptable Euclidean geometry, one of Cayley‘s main goals was to eliminate 

the elevated position occupied by Euclidean geometry in favour of a more general standpoint. He 

famously found such a standpoint within projective geometry, to which we now turn. 

 

The paper which was to influence Klein‘s approach to geometry was Cayley‘s ―Sixth Memoir 

Upon Quantics‖. In that paper, Cayley argues that the projective properties of geometrical 



35 

 

figures are their most fundamental, and, further, that the whole of geometry can be reconstructed 

in exclusively projective terms. Projective geometry was a burgeoning field of study within 

Cayley‘s time, but it was still generally treated as a sub-field of properly Euclidean geometry. 

Historically, it had been developed by Gérard Desargues (1591-1661) and others in order to help 

explain very concrete—hence Euclidean—problems of perspective in Renaissance painting. In 

the early nineteenth century there was something of a projective renaissance begun by the French 

mathematician Jean-Victor Poncelet (1788-1867), which gathered steam throughout Germany, 

France, and Italy.
79

 

 

While Beltrami himself had used projective techniques in the construction of his model of non-

Euclidean geometry, his focus was never explicitly upon the medium of projection, or on the 

nature of the projective plane itself. As we saw above, the Beltrami-Klein model employs certain 

points outside the circle γ in order to define the concept of perpendicularity. These points at 

infinity are not part of the hyperbolic universe, but are nevertheless still situated within the 

Euclidean plane into which the hyperbolic plane is projected. The relationship between these 

points and the lines of the hyperbolic plane is not describable from within the hyperbolic 

universe itself, but requires a more general perspective. For Beltrami, this perspective was 

satisfyingly that of traditional, Euclidean geometry. Cayley, on the contrary, developed a point of 

view from which Euclidean geometry itself appeared to be merely a special case, a specific type 

of metric geometry. Conceptually, the role played by Cayley‘s projective geometry is quite 

similar to that of ‗neutral geometry‘ in the works of Bolyai and, more powerfully, Lobachevsky. 

 

One of the key results of Cayley‘s 1859 paper is his celebrated treatment of the distance 

function. This treatment of the concept of distance involves characterizing it solely in projective 

terms (he writes that ―a chief object of [this 1859 paper] is the establishment, upon purely 

descriptive principles, of the notion of distance‖
80

); effectively, Cayley elides all the peculiarities 

of the Euclidean plane by mapping it onto the projective plane and back onto the Euclidean plane 

once more. He introduces the distance function relative to a particular figure he called the 

Absolute. The Absolute functioned as a yardstick against which distance could be measured 

irrespective of the specificities of a given space. More precisely, the process of projecting from a 

given metric plane into the projective plane and back again to the metric plane allowed Cayley to 

define the Absolute (and hence distance) by appealing only to projective properties—its non-

projective properties were essentially erased by the successive projective transformations. Thus, 

Cayley provided a method for characterizing distance as a purely projective concept. By doing 

so, he successfully illustrated that we can treat Euclidean geometry as a particular form of the 

more general projective geometry, and thus we have his well-known remark that ―Metrical 

geometry is thus a part of descriptive geometry, and descriptive geometry is all geometry‖.
81

  

                                                 
79

 Poncelet‘s two-volume work, Traité des proprieties projectives des figures ([Poncelet 1822]) was the instigator of 

this trend; throughout the middle third of the nineteenth century, algebraic techniques were fruitfully applied to 

projective geometry by, e.g., Christian von Staudt (1798-1867) and his school in Germany, and by the Italian 

geometer Luigi Cremona (1830-1903). The algebraic approach to projective geometry was instrumental in the 

establishment of the school of Alfred Clebsch (1833-1872) at Göttingen, which would prove to be so influential in 

the development of geometry towards its more recognizably modern forms in the work of Riemann, Klein, Hilbert, 

and others. (On the German side of these developments, see [Ziegler 1985]). 
80

 [Cayley 1859, 61]. 
81

 [Cayley 1859, 90]; Cayley uses the term ‗descriptive geometry‘ for what we call projective geometry. 



36 

 

 

Cayley restricted his investigation to the properties of a particular type of Absolute (he employed 

a specific degenerate conic section), which limited the applicability of his work considerably. 

Even still, his work allowed him to establish the important result that Euclidean metric geometry 

can be included within projective geometry. Though this was already quite a general standpoint, 

it would become further generalized by Felix Klein, beginning in the 1870s with his famous 

Erlanger Programm. One of the central ideas of Klein‘s papers of the early 1870s was to vary 

the form of the Absolute itself in order to investigate other possible metric structures, with other 

sorts of distance functions, all still characterized solely in terms of their projective properties. In 

effect, Klein combined Cayley‘s projective insight into the metric structure of Euclidean 

geometry with an appreciation for the variety of metric geometries made evident in the works of 

Lobachevsky and Bolyai.
82

 Ultimately, this led Klein to classify different metric geometries 

according to the existence of isomorphisms between those geometries and particular projective 

transformation groups. Later, with the help of group-theoretic work by his good friend Sophus 

Lie (and perhaps also an increasing appreciation of the geometric insights of Riemann), Klein 

extended this program to include transformation groups more generally.  

 

From the point of view of someone like Gauss, it would likely have seemed impossible for 

Klein‘s view of a group-theoretically classified system of interrelated geometries to have arisen 

over the brief span of a few decades. And, indeed, it was only because of a remarkable series of 

profound conceptual mutations combined with seemingly minor shifts of technical emphasis that 

such alteration was at all possible. From Beltrami‘s limited projection of the hyperbolic plane 

into the Euclidean plane we arrive (with Klein) at a radically more general understanding of the 

logical relations amongst a wide variety of geometries. For Beltrami, hyperbolic geometry was 

contained within true, Euclidean geometry; for Cayley, Euclidean geometry was contained 

within true, projective geometry; for Klein, these isolated logical relations between two 

geometries were replaced altogether by a higher standpoint, from which many different 

geometries stand in many different logical relations according to a group-theoretic typology.
83

 

With the development of the full-blown Erlanger Programm we have a clear example of the 

                                                 
82

 Klein was well aware that the influence of Cayley on his work was decisive. He wrote, well after the development 

of his Programm, that ―This whole manner of viewing the subject [i.e., the viewpoint of the Erlanger Programm] 

was given an important turn by the great English geometer A. Cayley in 1859. Whereas, up to this time, it had 

seemed that affine and projective geometry were poorer sections of metric geometry, Cayley made it possible, on the 

contrary, to look upon affine geometry as well as metric geometry as special cases of projective geometry‖ [Klein 

1939, 134]. For a clear explanation of the technical details of Klein‘s generalization of Cayley‘s approach to metric 

geometry [Gray 2007, 225-226] is helpful. 
83

 It should be noted here that, despite the central role Klein‘s Erlanger typology had in re-orienting understanding 

of the nature of geometry, Klein himself still resisted the fully abstract conception of geometries developed purely 

from postulate sets which is characteristic of the most modern approaches. In contrast to Riemann (about whom, 

more below), Klein‘s typology was meant to organize—rather than produce—geometrical knowledge. Because 

Klein was most interested in treating projective geometry as the ultimate foundation of all geometry, he was not 

inclined to include in his typology geometries which could not be easily characterized in projective terms (e.g., 

elliptic and various differential geometries). After meeting the British mathematician William Clifford (1845-

1879)—who translated Riemann‘s Habilitationsvortrag and was very impressed by the possibility of geometries of 

discontinuous spaces—Klein became more tolerant of a wider variety of geometries, though he was still convinced 

of the fundamental place occupied by the perspective of projective geometry. (For some insightful remarks on 

Klein‘s position and its development, see [Corry 2004a, 33f]). 



37 

 

increasing focus on structure as a central, organizing principle within geometry. It is important to 

note, though, that the initial developments which led to this shift (e.g., Cayley‘s and Beltrami‘s 

respective works) were not in the least self-conscious attempts to introduce structural thinking 

into geometry. Beltrami was concerned to illustrate the legitimacy of non-Euclidean geometry by 

connecting it to the intuitively acceptable objects of Euclidean geometry, while Cayley sought a 

deeper understanding of the science of space by embedding the objects of Euclidean geometry in 

the more fundamental properties of projective space. And, while Klein helped shift the focus 

away from the competition for the rights to our intuition of space and toward the more interesting 

(and, eventually, far more influential) problem of determining the relationships between a variety 

of possible geometries, he too was still concerned that the source of our knowledge of these 

geometries should be found in intuition. So, while each of the moments addressed in the 

historical story I give above is crucial to the development of a structural, non-Aristotelian 

conception of geometric science, none of them, not even the highly abstract system of Klein, can 

be said to fully or self-consciously embody such a conception. 

 

2.3.3 Physics and geometry 

 

In addition to the challenges posed to Kant‘s philosophy of geometry by mathematicians, there 

was significant trouble raised on the physical front as well. In order to see this, let‘s take a (very 

brief) detour through the pre-Gaussian understanding of the relation between geometry and 

physics. 

 

For many physicists and mathematicians prior to the acceptance of non-Euclidean geometry, the 

mathematical study of geometry was understood to be separate from physics, though whether 

this separation was a matter of logical priority or a methodology varied from thinker to thinker. 

Thus for instance we find Isaac Newton (1643-1727) claiming that, while geometry is, in a sense, 

a part of (universal) mechanics, it was nevertheless importantly distinguished by its peculiarly 

perfect method. As he notes in the preface to his Principia,  

 

To practical mechanics all the manual arts belong, from which mechanics took its name. 

But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so 

distinguished from geometry, that what is called perfectly accurate is called geometrical, 

what is less so is called mechanical.
84

 

 

So, while Newton‘s geometry was a part of mechanics insofar as it pertained to the study of the 

mechanical principles which animated the physical world, it was nevertheless that specific part 

of mechanics which abstracted from the particularity and inaccuracy of our actual measurements 

of the physical world to consider the exact principles, proportions, and demonstrations which 

made them possible in the first place. As Guicciardini puts it,  

 

Rather than excluding mechanics from the realm of geometrical exactness, Newton 

proposed to subsume geometry under mechanics. Geometry [for Newton] is founded upon 

                                                 
84

 [Newton 1845, lxvii]. 



38 

 

mechanics, since the description (or the construction) of geometrical objects appertains to 

mechanics.
85

 

 

The important difference between Newton‘s view and the more straightforward (and later) view 

that physical geometry is subject to the dictates of physics, is that Newton held that the object of 

geometry was the perfectly precise realm of absolute space—a space whose perfection is foreign 

to the imprecision and relativity of the space we actually experience. As he puts it, 

 

Absolute space, in its own nature, without regard to anything external, remains always 

similar and immovable. Relative space is some movable dimension or measure of the 

absolute spaces; which our senses determine by its position to bodies; and which is 

vulgarly taken for immovable space.
86

 

 

The imperfect measurement and theorization of this absolute space was the domain of practical 

mechanics, while the objects of geometry were generated by an appreciation of the perfect or 

idealized motions which lay beneath our imperfect measurements and observations. In this sense, 

then, geometry at least partially shared its domain with practical mechanics, and both of them 

were subsumed under the general heading of rational or universal mechanics. Despite the fact 

that the perfect objects of geometry are, in this loose sense, generated by our imperfect 

observations of the physical world, they remain methodologically separate from them. 

 

Now, while Newton‘s interest is not otherwise in the elaboration of a complete philosophy of 

geometry, his general views dovetail (in some respects) with the later Kantian philosophy of 

geometry. Newton suggests that geometry is a component of rational (or universal) mechanics, 

and that its objects are generated by our observations of motion. This is not to say that they are 

generated by any particular motions or observations, but by the perfect motions which are 

possible within absolute physical space. An obvious and important distinction between Kant‘s 

account of space and Newton‘s understanding of physical space is Newton‘s eagerness to 

attribute the perfection and endurance of physical space to God‘s good graces.
87

 Kant, by 

contrast, argues that the nature of physical space is determined by our pure intuition of space. Of 

course, on Kant‘s view, for all we know, this pure intuition itself is generated by some noumenal 

God. For our purposes here, however, this difference in the origins and sustenance of space is not 

terribly important. What is important is the notion that, for both men (and many others), 

geometry is, in some respects, prior to any specific physical investigation. We may personally 

arrive at the investigation of (absolute) space through our empirical considerations. But, as both 

Newton and Kant hold, the objects of geometry are not themselves solely empirical in character. 

 

This view of geometry as importantly separate (in method or content) from physics endured in 

various forms for quite some time. It was not really until after the advent of non-Euclidean 

geometries that the question of the empirical investigation of geometry became a live possibility. 

It was only really when the idea of geometry as connected to empirical investigation became 

                                                 
85

 [Guicciardini 2009, 297]. 
86

 [Newton 1854, 77]. 
87

 Though this isn‘t directly stated in this manner in the Principia, this view is made implicitly clear throughout, and 

particularly in the General Scholium. 



39 

 

possible—as in the (probably apocryphal and very early) tale of Gauss‘ mountaineering 

expeditions aimed at establishing the curvature of space
88

—that the connection between Kant‘s 

pure intuition and space came to be challenged by in earnest by physicists. 

 

Prior to this conceptual move (or ‗discovery,‘ if we prefer), on Kant‘s account, our conception of 

space was understood to be entirely synthetic a priori and, therefore, not in need of any external 

input from physics. Newton‘s view is not entirely dissimilar: we take the a posteriori 

deliverances of measurement and empirical physics, and from them we gain access to the 

geometrically perfect realm of absolute space. 

 

Afterwards, the ‗pure‘ intuitive grasp of space no longer seems a reasonable option. The 

properties of space are no longer given to us but instead become a matter for investigation. What 

this meant for physicists, in particular, was that space might end up being nothing like Euclid, 

Kant, and Newton imagined. And, given the development of strange new geometries by Riemann 

and others, it might turn out to be stranger than anyone could possibly have imagined. 

Ultimately, the shift towards a conception of the geometry of physical space as a matter for 

physical investigation prompted both physicians and mathematicians to develop interesting new 

geometries and interesting ways of testing their applicability to physical space. In the end, then, 

Riemann‘s weird new geometries—which, to Kant and those of like mind might have seemed 

mere flights of fancy or amusing but scientifically inert games—ended up providing 

mathematical descriptions of physical space in the works of men like Einstein and Minkowski. 

So it is that we find Einstein remarking in 1921 that 

 

It is clear that the system of concepts of axiomatic geometry alone cannot make any 

assertions as to the behaviour of real objects of this kind, which we will call practically-

rigid bodies. To be able to make such assertions, geometry must be stripped of its merely 

logical-formal character by the coordination of real objects of experience with the empty 

conceptual schemata of axiomatic geometry. To accomplish this, we need only add the 

proposition: solid bodies are related, with respect to their possible disposition, as are bodies 

in Euclidean geometry of three dimensions. Then the propositions of Euclid contain 

affirmations as to the behaviour of practically-rigid bodies. Geometry thus completed is 

evidently a natural science; we may in fact regard it as the most ancient branch of physics. 

Its affirmations rest essentially on indication from experience, but not on logical inferences 

only. We will call this completed geometry ‗practical geometry,‘ and shall distinguish it in 

what follows from ‗purely axiomatic geometry.‘ The question whether the practical 

geometry of the universe is Euclidean or not has a clear meaning, and its answer can only 

be furnished by experience. […] I attach special importance to the view of geometry, 

which I have just set forth, because without it I should have been unable to formulate the 

theory of relativity.
89

  

 

                                                 
88

 As the tale goes, Gauss measured the angles of the triangle formed between three mountain peaks in order to 

determine the curvature of physical space. While it is certain that Gauss did engage in just such a survey, there is 

little evidence to support the idea that his aim was such a geometrical test. More serious tests of the geometry of 

physical space would have to wait until considerably later. 
89

 [Einstein 2002, 387-388]. 



40 

 

As Einstein makes clear here, a prerequisite of his own fundamental advances in physical theory 

was the decoupling of geometry from the moorings of Kantian pure intuition. After severing the 

connection between the a priori certainty of Euclidean geometry and the physical investigation of 

space, physicists like Einstein were free to apply (or attempt to apply) a variety of axiomatic 

geometries to a particular physical problem. And, from the mathematical point of view, 

geometers like Minkowski had a new physical incentive to develop entirely new forms of 

geometry. The result, particularly throughout the first quarter of the twentieth-century was an 

incredibly fruitful series of collaborations between physicists and mathematicians.
90

 

 

2.4 Some philosophical lessons 

 

The overall aim of this chapter has been to illustrate the content of the traditional Euclidean 

picture of geometry, and the conceptual steps through which this view of geometry as a science 

of idealized physical objects was eventually overturned in favour of a picture of geometry as a 

science interested in a variety of abstract structures whose relations to physical space were not of 

particular mathematical interest. Though we have yet to explore this second picture, with Felix 

Klein‘s work, we have arrived at an important threshold between these two conceptions of 

geometry. 

 

I began by describing several aspects of Euclid‘s Elements. This description is intended to 

illustrate that his work, and the tradition stemming from his work, exemplified a conception of 

science which was importantly distinct from the structure-focussed conception of mathematics 

which emerges just after our historical survey with the work of Hilbert and others. Rather than 

the study of formal structures, traditional Euclidean geometry was understood by many of its 

practitioners as a concrete science distinguished by the sort of objects it studied—namely spatial 

objects. In Euclid, we find clear evidence of this object-oriented view: the Elements begins with 

a series of definitions, which are intended to familiarize the reader with the particular objects 

about which geometers reason (e.g., points, lines, circles, etc.). It then proceeds to the 

presentation of a number of basic truths or axioms about these objects, truths which must be 

accepted before geometrical argumentation begins. 

 

My own exposition focussed upon one of these axioms, namely, the parallel postulate. This 

postulate was chosen as a historical focal point largely because the attempts to prove it gave rise 

to a number of developments that proved central to the emergence of a structural view of 

geometry. In hindsight, it is certainly true that history could have taken quite a different route, 

and geometers might have focused more of their energies on proving, e.g., the Archimedean 

axiom, or any other of Euclid‘s axioms for that matter. Had Euclid phrased his axiom of parallels 

                                                 
90

 This sort of collaboration is perhaps most evident in the interactions between Minkowski and Einstein, about 

which Einstein writes the following: ―The generalization of Relativity theory has been made much easier through 

the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize 

clearly the formal equivalence of the space-like and time-like co-ordinates, and who made use of it in the building 

up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the 

‗Absolute Differential Calculus,‘ which were based on the researches of Gauss, Riemann and Christoffel on the non-

Euclidean manifold, and which have been shaped into a system by Ricci and Levi-civita, and already applied to the 

problems of theoretical physics‖ [Einstein 1916, 89] 

 



41 

 

more simply (imitating Wallis‘ formulation, perhaps) the history of geometry would surely have 

been quite different. Despite the fact that the centrality of the parallel postulate for the history of 

geometry is a mere historical accident, however, it is worthwhile, from a philosophical 

standpoint, to study attempts at proving the postulate insofar as such a study reveals a number of 

deep and important conceptual shifts in the understanding of geometry as a science. Through this 

history, we can witness the slow and largely accidental shift away from the traditional 

understanding of geometry as the science characterized by its study of the genus of spatial 

objects towards the modern understanding of geometry as a science which studies a variety of 

abstract structures and relations between those structures. 

 

From the array of attempts to prove the postulate, I have focused my attention on a mere handful. 

We began with those of Ptolemy and Proclus for two reasons. First, because they are some of the 

earliest attempts of which we know. And, second, because they are exemplary of a common sort 

of ‗proof‘ of the parallel postulate, the sort which appeals to a disguised equivalent of that very 

postulate. The unwitting character of these appeals is in direct contrast to attempts at 

simplification which show that the postulate is equivalent to some simpler and intuitively more 

appealing statement. Wallis‘ proof that the parallel postulate is equivalent to the claim that space 

is everywhere homogenous, or that there exist similar triangles which differ in size, is an 

example of the latter sort of approach. 

 

From Ptolemy and Proclus I turn to the work of Omar Khayyam. For my purposes, the chief 

conceptual importance of Khayyam‘s work lies in his brief exploration of the logical possibility 

of geometries in which the parallel postulate does not hold. Though Khayyam is quick to dismiss 

these alternative non-Euclidean geometries, his suggestion was later exploited in much more 

detail by Girolamo Saccheri. With Saccheri‘s work, we have an example of the detailed 

exploration of logically possible geometries which do not obey Euclid‘s parallel postulate. One 

of these possibilities, based upon an ‗acute-angled‘ hypothesis, is explored in particular detail. 

Despite the effort which Saccheri devotes to the development of this non-Euclidean geometry (as 

we would understand it today), he abruptly rejects the coherence of this alternative in favour of 

traditional Euclidean geometry. To a modern eye, this rejection is rather surprising—it seems as 

if he simply decides to stop arguing. This is because we are intimately familiar with the 

structural approach to geometry which involves dealing with multiple geometries and multiple 

spaces as a matter of course. So why does Saccheri not take the opportunity to describe just this 

sort of alternative geometry? 

 

My claim was that an implicit cluster of assumptions about the nature of geometry as a science 

guided Saccheri away from the implications of his own detailed work and toward a more 

traditional understanding of the status of logically possible alternative geometries. I discuss the 

philosophy of geometry of Immanuel Kant as a particularly influential codification of many of 

these implicit assumptions about geometry. Of central importance among these assumptions are 

the claims that geometry is the science of space, and that geometrical truths are certain and 

necessary because they are delivered to us by a special faculty of intuition somehow exempt 

from empirical doubt. Given the widespread acceptance of these sorts of claims, then, Saccheri‘s 

refusal to countenance the coherence of a geometry alternative to Euclid‘s is not terribly 

surprising. In order to do so, Saccheri would have had to reject or seriously alter the view that 

geometry as he understood it was the science of space: for if there is only one science of space, 



42 

 

and there are conflicting claims about the nature of that space, one of those claims must 

necessarily be rejected. Given the traditional weight lent to the theorems of Euclidean geometry, 

it is not at all surprising that Saccheri would have chosen Euclid‘s geometry over his own upstart 

alternative (a sentiment which was echoed much later by Frege). Even if he could have rejected 

the view that there is only one science of space, Saccheri would still have had to face the 

epistemological problem implicit in the view that the truths of geometry were certain, and that 

this certainty was grounded somehow in our very apprehension of the nature of space. No small 

challenge, then. 

 

It would take another hundred years or so before we find mathematicians capable of overturning 

or questioning these implicit assumptions. Gauss, Bolyai, and Lobachevsky all began to call into 

question the view that Euclidean geometry was the true description of physical space, and in so 

doing they slowly opened the door for the view that the applicability of the theorems of geometry 

to physical space was itself an empirical question which could not be decided on purely a priori 

grounds. This move required a huge conceptual shift away from the understanding of geometry 

as the science of space. From the work of these three men, a view developed within which it was 

possible to explicitly investigate questions pertaining to particular sorts of spaces (e.g., Euclidean 

or hyperbolic space). The development of a ‗neutral‘ geometric perspective from which claims 

about the structure of a particular kind of space could be decided was a crucial element in the 

larger story of the emergence of a structure-focused understanding of geometry. 

 

With Beltrami‘s work on modelling non-Euclidean geometry within the Euclidean plane, we 

have a much more explicit instance of the tendency to discuss the properties of particular kinds 

of geometrical structures. Beltrami‘s work is also important from a philosophical standpoint for 

the fact that it was one of the earliest instances in which relations between what we might 

anachronistically call ‗geometrical structures‘ were discussed with technical sophistication. 

Beltrami‘s illustration that we can model hyperbolic geometry by embedding hyperbolic space 

within a fragment of Euclidean space is highly suggestive of a more general perspective from 

which both Euclidean and hyperbolic ‗space‘ appear to be special cases. Despite Beltrami‘s own 

understanding of his model as an illustration of the conceptual priority of Euclidean geometry, 

his work had the effect of drawing geometric attention to relations between different spaces, or 

different logically possible spatial structures. 

 

In the hands of Felix Klein, the gestures towards this more general perspective—within which 

different spaces and relations between those spaces are examined—became far more 

sophisticated. Relying upon earlier work of Cayley in projective geometry, Klein extended 

Beltrami‘s work by characterizing a whole host of different geometrical structures (different 

‗spaces‘) in terms of a limited group of projective properties. 

 

All of this helps to illustrate the slow unravelling of an object-oriented approach in favour of a 

more structure-focussed view within geometry. Early attempts to prove the parallel postulate 

within this traditional object-oriented understanding of geometry were largely unsuccessful. 

When we arrive at the work of Girolamo Saccheri, we have the first real hints that progress in the 

theory of parallels might require overturning one or many of the implicit assumptions which 

undergirded the traditional view of geometry. With Gauss, Lobachevsky, and Bolyai, these hints 

are given a more tangible form and the assumption of the applicability of Euclidean geometry to 



43 

 

physical space is made into an empirical matter. From this questioning of the a priori certainty of 

the Euclidean nature of physical space, the traditional understanding of geometrical research as 

the exploration of the properties of idealized physical space was slowly supplanted by a view of 

geometry as the logical development of various abstract structures. With the work of Cayley, 

Beltrami, and, eventually, Klein, this structure-focussed view became increasingly more explicit.  

 

2.5 Bernhard Riemann‘s n-dimensional manifolds 

 

While there was, as we‘ve seen, a lengthy history leading to the eventual development and 

legitimation of the specific non-Euclidean geometries created by Bolyai and Lobachevsky, and 

to the subsequent establishment of Klein‘s Erlanger Programm as a partial result of attempts to 

model those geometries, it is difficult to avoid treating Bernhard Riemann‘s 1854 

Habilitationsvortrag, ―On the Hypotheses Which Lie at the Basis of Geometry,‖ as an example 

of an almost entirely isolated development.
91

 In that short lecture, prepared at the behest of his 

doctoral supervisor Gauss, Riemann produced a radical re-imagining of the nature of geometrical 

science and its relationship to physical space. We‘ve seen how difficult it was to achieve even 

minor headway in the advancement of non-Euclidean geometries even up to Gauss‘s death in 

1855. Given these difficulties, the dramatic progress represented by Riemann‘s 1854 paper is all 

the more surprising. The first few paragraphs, which set the tone for the rest of the work, are 

worth quoting at length here: 

 

It is known that geometry assumes, as things given, both the notion of space and the first 

principles of constructions in space. She gives definitions of them which are merely 

nominal, while the true determinations appear in the form of axioms. The relation of these 

assumptions remains consequently in darkness; we neither perceive whether and how far 

their connection is necessary, nor, a priori, whether it is possible. [...] From Euclid to 

Legendre (to name the most famous of modern reforming geometers) this darkness was 

cleared up neither by mathematicians nor by such philosophers as doubtless concerned 

themselves with it. The reason of this is doubtless that the general notion of multiply 

extended magnitudes [ausgedehnten Grösse] (in which space-magnitudes are included) 

remained entirely unworked. I have in the first place, therefore, set myself the task of 

                                                 
91

 Riemann himself refers to some remarks made by Gauss in a paper on biquadratic residues (i.e., [Gauss 1831]) 

and the works of the philosopher Johann Friedrich Herbart (1776-1841) as his main influences in this respect, 

though he admits their influence was rather limited. Gauss‘s paper presents a rather general approach to the theory 

of numbers in which relations across various number domains (natural, real, complex, etc.) are treated via the 

concept of a manifold. The generality implied by the concept of a number manifold is likely the particular element 

in Gauss‘s paper which partially inspired Riemann‘s own generalizing manoeuvres in geometry, though there is 

little there to suggest the specific application to ‗space‘ that we find in Riemann. The philosophical views of 

Herbart—who worked at Göttingen from 1833 until his death in 1841—significantly influenced Riemann‘s 

philosophical view of the world. Particularly relevant for us is Herbart‘s view that the so-called ‗pure‘ intuitions of 

Kant (i.e., space and time) are not ‗pure‘ in Kant‘s sense but are, in fact, based upon the mind‘s own attempt to 

understand causal experiences. Thus, they are not pre-experiential, a priori sources of knowledge, but the result of 

our attempts to make sense of experience in a coherent fashion. Riemann, as we will see, significantly developed 

Herbart‘s philosophical point within a more strictly mathematical setting. (For some of Herbart‘s views which 

influenced Riemann, see [Herbart, 1824-1825]—a work that deals with the problem of Kant‘s pure intuition among 

other things, and a work which Riemann is known to have read. For a careful analysis of the specific influences of 

Herbart on Riemann‘s work, see [Scholz 1982]). 



44 

 

constructing the notion of a multiply extended magnitude out of general notions of 

magnitude. It will follow from this that a multiply extended magnitude is capable of 

different measure-relations, and consequently that space is only a particular case of a triply 

extended magnitude. But hence flows as a necessary consequence that the propositions of 

geometry cannot be derived from general notions of magnitude, but that the properties 

which distinguish space from other conceivable triply extended magnitudes are only to be 

deduced from experience. Thus arises the problem, to discover the simplest matters of fact 

from which the measure-relations of space may be determined; a problem which from the 

nature of the case is not completely determinate, since there may be several systems of 

matters of fact which suffice to determine the measure-relations of space—the most 

important system for our present purpose being that which Euclid has laid down as a 

foundation. These matters of fact are—like all matters of fact—not necessary, but only of 

empirical certainty; they are hypotheses. We may therefore investigate their probability, 

which within the limits of observation is of course very great, and inquire about the justice 

of their extension beyond the limits of observation, on the side of both of the infinitely 

great and of the infinitely small.
92

 

 

In this relatively brief excerpt we have a startlingly clear formulation of what would become the 

basis of the ‗modern‘ conception of geometry. Recalling Saccheri‘s exasperation and Farkas 

Bolyai‘s desolation, it is mildly scandalous to witness the casual air with which Riemann 

overturns the centuries-old understanding of geometrical science. Instead of the study of the 

idealized elements of an intuitively given, three-dimensional, Euclidean space, Riemann suggests 

that we treat geometry as a subfield of the study of multiply extended magnitudes (or, more 

generally, n-dimensional manifolds [Mannigfaltigkeiten]), some of which have properties similar 

to Euclidean space and some of which are radically different. On this view, the link between 

intuition and Euclidean space (if, indeed, there is one) becomes rather beside the point. We do 

not intuit the truth of the axioms of geometry, but develop them as hypotheses which may or may 

not turn out to be true of empirical reality. The applicability of these axioms to the physical 

world (i.e., their empirical truth) is a matter for empirical determination and not of purely formal, 

pre-experiential necessity delivered by Kantian intuition. It is not surprising that Riemann‘s work 

was partly inspired by the philosopher Johann Friedrich Herbart‘s attempts to reform what he 

perceived to be Kant‘s errors (or hyperbole). Precisely this sort of philosophical view of the 

relationship between physical space and geometric theory was lacking in many earlier attempts 

to explain problems related to Euclid‘s parallel postulate.  

 

On Riemann‘s view, the traditional space of geometry (i.e., three-dimensional Euclidean space) 

becomes one instance of an n-dimensional manifold amongst many others. Rather than 

examining this or that specific manifold, Riemann begins by presenting the concept of an n-

dimensional manifold in its full generality. In order to fix this concept in its most general form, 

he proceeds by distinguishing its more specific instantiations. He thus isolates a number of 

geometric properties by which we might fix particular classes of geometries. Some of the crucial 

properties are: the number of ‗dimensions‘ of a manifold; whether it is discrete or continuous 

and, if continuous, whether or not it is isotropic; its metric relations; its curvature. We can see the 

generality of Riemann‘s point of view by considering how he comes to understand what had, for 

                                                 
92

 [Riemann 1868, translated in Ewald 1999, 652-653]. 



45 

 

two millennia, been the traditional content of the whole of geometry. Three-dimensional 

Euclidean space becomes a specific instance of a three-dimensional, continuous, isotropic 

manifold of constant curvature, where that curvature is zero, and which possesses a particular 

distance function. Now, each of these adjectival descriptors of Euclidean space can be varied, 

producing a distinct class of geometries, each describing particular sorts of spaces. In Riemann‘s 

very general view, Bolyai-Lobachevsky hyperbolic space and traditional Euclidean space are not 

terribly different; indeed they are both examples of continuous three-dimensional manifolds of 

constant curvature—they differ mainly in the fact that hyperbolic space has a constant negative 

curvature, while Euclidean space has a constant curvature of 0. Thus, while we might discuss the 

emergence of non-Euclidean geometry as a troubling development for the intuitively grounded 

Kantian view, it should nevertheless be apparent that the work of Bolyai and Lobachevsky 

(though an important first step) was the first step toward a radically different view of geometry 

which would be made more mathematically explicit in the work of Riemann, and more 

philosophically explicit in the work of Hilbert a half century later. 

 

With Riemann‘s work, even more than with the developments which led to Klein‘s Erlanger 

Programm, a significant blow seemed to have been dealt to the Kantian belief that pure intuition 

grounds geometrical reasoning, though this was not widely realized until considerably later. 

Riemann begins his lecture by suggesting that the application of our geometrical axioms to the 

world is not a matter of necessity but one which is established only by empirical investigation. 

Instead of intuitively grounded, necessary truths about the structure of (the only possible) space, 

we arrive at more-or-less applicable, empirically-testable hypotheses whose truth is not at all 

certain. In addition, some of these hypotheses (e.g., those lying at the basis of higher-

dimensional geometries, or geometries radically different from the more familiar Euclidean case) 

will be difficult if not impossible to ‗intuit‘ at all. In order to salvage the Kantian picture in the 

face of Riemann‘s work, it would seem that we require a considerably expanded understanding 

of the notion of intuition, as well as an alteration in our understanding of the epistemological role 

which such intuition can play within geometric reasoning. And, beyond even Klein, Riemann‘s 

work suggests the systematic development of geometries from various sets of hypotheses and 

slight alterations in coefficients, leading to a typology of spaces and geometries which is entirely 

indifferent to the deliverances of intuition, whatever they may be. Riemannian geometers might 

be interested in the connection between a given geometry and physical space, but they certainly 

needn‘t be, and usually will not be. 

 

2.6 A conceptual vacuum 

 

The increasingly rich assemblage of new geometrical concepts, as well as the rapid development 

of highly abstract concepts in other areas of mathematics like group theory, created a situation in 

which the very nature of geometrical science became unclear. Previously, geometry had been 

seen as a (relatively) concrete science dealing with the laws governing physical space. Its rather 

odd status as an a priori science that, somehow, also applied to the changing world of 

appearances had been temporarily propped up by Kant‘s view of space as a pre-experiential, pure 

intuition. 

  

Having undermined Kant‘s position and the privileged position of Euclidean geometry, 

geometers became more fully capable of questioning what role, if any, intuition played within 



46 

 

the new, generalized science of geometry. As we will see, this loosening of the epistemological 

connection between pure spatial intuition and the logical development of geometry enabled a 

rather significant alteration in the way geometry itself was conceived. We‘ve already witnessed 

the shift from a view of Euclidean geometry as the science of space to the view of Euclidean 

geometry as the description of a very particular sort of three-dimensional manifold. Indeed, the 

distance between the traditional conception of geometry and the more ‗structural‘ approach is 

most evident in the alteration of the concept of space itself. For Euclid and those following in his 

tradition, space was physical space: geometry may have studied its properties by way of 

idealized objects like points and lines, but geometry nevertheless studied the properties of our 

physical space. With the advent of Klein‘s Erlanger Programm, we have a variety of geometries, 

which operate on a variety of different spaces. Initially, these spaces are determined by the 

invariants of particular groups of spatial transformations. In Riemann, we have even more such 

spaces, which are now determined neither by intuition nor by transformation groups, but simply 

by sets of hypotheses containing varying numerical coefficients. Geometers are free to construct 

such spaces at their pleasure, and investigate their properties without ever imagining how or even 

if they can have any physical application. Thus, geometry moved away from being the science 

which determines the laws governing physical space to the science which studies the properties 

of occasionally baffling abstract spaces, conceived as particular sorts of multiply extended 

magnitudes, and the various relations between them. 

 

With these developments in non-Euclidean, projective, and Riemannian geometry, pure 

geometrical science became disconnected from the physical world, and it was no longer 

immediately clear how to understand its results or their purposes. What exactly was geometry? 

What were the abstract structures which it seemed to be studying? For a dyed in the wool 

Kantian, they might have been seen as logically interesting fictions. But for most late 19
th

 

century mathematicians, they seemed to be something quite different.  

 

With Hilbert‘s work on the foundations of geometry, a new conception of mathematics as the 

study of formalized axiomatic systems finally began to emerge. Both Riemann and Helmholtz 

(who helped to popularize and conceptualize Riemann‘s work) had, earlier, offered sophisticated 

explanations of the nature of the new geometry, but it wasn‘t really until the publication of 

Hilbert‘s Grundlagen and its axiomatization of Euclidean geometry that the new approach was 

taken up with gusto by the mathematical community at large. Hilbert‘s work shored up the 

burgeoning axiomatic understanding of the nature of geometry by applying it to the core of the 

old tradition. By showing that Euclidean geometry could be fruitfully axiomatized in the new 

style, Hilbert at once lent credence to the new approach and vindicated the importance of the 

traditional heart of the science. In the following chapter I will show how Hilbert‘s work helped 

to challenge the prevailing Aristotelian conception of science. 



47 

 

Chapter Two: Frege and Hilbert on axiomatic systems 
 

§0. Introduction 

 

In the previous chapter, we examined elements of the historical shift away from a view of 

geometry as the science of idealized spatial objects. The increasing prominence of the formal 

structure of and interrelations among geometric theories (as opposed to their content, 

properly speaking) had a destabilizing effect on the traditional understanding of geometry as 

the science of space. In place of the traditional view of geometry as a science concerned with 

the relations amongst ideal objects like points, lines, and planes, we find, toward the end of 

the nineteenth century, an increasing confusion about the nature of geometry as a science. If 

geometry was not best understood as the idealized study of physical space, what then was its 

proper object? 

 

In this chapter, we will examine two distinct philosophical attempts to understand the nature 

of geometry in light of these developments. We will first turn to David Hilbert‘s influential 

views, as developed in his Grundlagen der Geometrie. Hilbert‘s work in geometry focused 

on the development of the deductive consequences of groups of axioms. His philosophical 

views of geometry give pride of place to the formal characteristics of these axiomatic 

systems.  

 

After examining Hilbert‘s re-imagining of geometry, as well as his philosophical views of 

the nature of axiomatics, we will turn to the views of Frege. Frege, in some ways, was a 

geometric traditionalist. Unlike Hilbert, he held on to the view of axioms as meaningful 

truths about the world, and of geometry as a science with a determinate subject matter. In 

place of uninterpreted formal systems, Frege viewed Euclidean geometry as the study of a 

particular collection of objects and concepts. Though sympathetic to the idea of higher-level 

geometric theories (within which, e.g., Euclidean geometry might fall), Frege was 

nevertheless harshly critical of what he took to be the logical imperspicuity required for 

Hilbert-style axiomatics. In this chapter, we will first examine Hilbert and Frege‘s respective 

views of geometry and axiomatics, then turn to an analysis of Frege‘s criticisms of Hilbert.  

 

The chapter has two chief aims. First, it will call into question the relatively common view 

that Hilbert‘s prescient account of axiomatics simply wins the day when pit against Frege‘s 

old-fashioned, rather curmudgeonly criticisms. This will be accomplished by showing that a) 

Hilbert‘s views are not entirely in line with the more contemporary model-theoretic approach 

to axiomatics and b) that Frege‘s criticisms of Hilbert are not based on a simple 

recapitulation of the traditional Euclidean/Kantian view of geometry. Second, the chapter 

will show that, despite the widespread acceptance of an altered form of Hilbert‘s approach to 

axiomatics, Frege poses deep questions about the clarity and meaningfulness of this 

approach which Hilbert and his defenders leave unanswered. The result of this chapter, when 

combined with the first, is that the traditional Euclidean approach is shown to be incapable 

of addressing the reality of modern mathematics, while the conceptual clarification of this 

new reality offered by Hilbert and others faces significant philosophical difficulties.  

 



48 

 

As will become clearer in subsequent chapters, Hilbert‘s views are, with some exceptions, 

representative of a larger trend in mathematical thought within which abstract mathematical 

structures are characterized in purely relational terms (i.e., as ontologically un-fixed). By 

illustrating the force of Frege‘s criticisms of Hilbert‘s views, then, I hope also to call into 

question the viability of this more general method of characterizing mathematical structures 

in this relational, or free-floating way. We will begin the chapter by differentiating Hilbert‘s 

views of geometry and (mathematical) science from the traditional account stemming from 

Euclid and Aristotle.  

 

0.1 The Euclidean tradition and Aristotelian science 

 

As we‘ve seen above, in the works of Euclid one finds two sorts of definition. Definitions of 

the first type were essentially abbreviations: ‗circle‘ for instance, is defined in terms of the 

more basic notions of point, line, etc. We could just as well cut out the term ‗circle‘ and 

replace it everywhere with its definition in terms of points and lines and no essential 

information would be lost. There were, however, definitions of a different sort, which served 

as pre-systematic guides to orient the reader towards the most basic objects of geometry (i.e., 

the points, lines, and planes which form its subject matter). Through these definitions, 

readers were either oriented towards objects with which they were already familiar or 

introduced to those objects, with varying degrees of success. Within this tradition, axioms 

were construed as truths about a determinate subject matter; in this case, the (idealized) 

elements of physical space. In many cases, too, the truth of these axioms was thought to have 

been grounded in our basic intuition of space—a source of knowledge which came to be 

considered importantly distinct from both experience and pure logic. These specific claims 

about the nature of geometric science were themselves embedded within a broader 

Aristotelian conception of the nature of science in general.  

 

For Aristotle and those following this tradition, human knowledge was divisible into distinct 

sciences, differentiated according to the genus of objects which they study.
1
 Thus, for 

instance, we can conceive of biology as a science distinct from geometry insofar as the 

former studies the genus of living objects, while the latter studies the genus of spatial 

objects. We can establish certain relations (e.g., relations of inclusion, co-extension, etc.) 

                                                 
1
 We find echoes of the traditional Aristotelian concept of sciences (differentiated according to the species of 

objects which they study) within the works of many of the most important thinkers in the history of Western 

thought. We find, for instance, Thomas Aquinas (1225-1274) writing in the Summa Theologica that ―a species 

is a kind of science‖ [Aquinas 1957, 2.57.2, objection 1; 1005b] and that ―according to the different kinds of 

knowable matter, there are different habits of scientific knowledge‖ [Aquinas 1957, 2.57.2, response; 1006a-b]; 

John Duns Scotus (1265/66-1308), too, refers specifically to the concept of science outlined in the Posterior 

Analytics (at 1.2 79b9-12) early on in the Ordinatio (see [Duns Scotus 1950, 141-142]); Leibniz was 

undoubtedly heavily influenced by both Aristotle and the Schoolmen, and his systematic approach to natural 

philosophy owes much to the Aristotelian tradition despite his more general understanding of species and 

essences; Bolzano, a partial follower of Leibniz and Aristotle, calls ―an aggregate of truths of a certain kind a 

science‖ [Bolzano 1837, §1.1] and notes that he ―shall call the class of truths which assert something about the 

constitution of space the science of space (geometry), because these propositions form a separate species of 

truths‖ [Bolzano 1837. §1.1]. Bolzano departs from this Aristotelian tradition, however, in that he allows for 

species of truths which are not determined by classes of objects. As we shall see in what follows, Frege, too, 

held a rather Aristotelian conception of the nature of science.  



49 

 

amongst the sciences themselves by looking to the relations amongst the classes of objects 

which they study: thus geometry is a more general science than biology because all living 

objects are, presumably, also spatial objects, while most spatial objects are not alive. 

Accordingly, geometrical truths have wider application than those of biology. On this 

Aristotelian model, some sciences are said to fall within others, in which case the truths of 

the more general science apply to the objects of the less general, or included, science. In the 

Posterior Analytics Aristotle makes some remarks to this effect which are specifically 

relevant to the mathematical sciences. He writes: 

 

[Y]ou cannot prove anything by crossing from another kind—e.g. something 

geometrical by arithmetic. There are three things involved in demonstrations: one, 

what is being demonstrated, or the conclusion (this is what holds of some kind in 

itself); two, the axioms (axioms are the items from which the demonstrations proceed); 

third, the underlying kind whose attributes—i.e. the items incidental to it in itself—the 

demonstrations make plain. Now the items from which the demonstrations proceed 

may be the same; but where the kinds are different, as with arithmetic and geometry, 

you cannot attach arithmetical demonstrations to what is incidental to magnitudes—

unless [the] magnitudes are numbers. [...] Arithmetical demonstrations always contain 

the kind with which the demonstrations are concerned, and so too do all other 

demonstrations. Hence the kind must be the same, either simpliciter or in some respect, 

if a demonstration is to cross. [...] For this reason you cannot prove by geometry that 

there is a single science of contraries, nor even that [the product of two cubes makes] a 

cube. (Nor can you prove by any other science what pertains to a different science, 

except when they are so related to one another that the one falls under the other—as 

e.g. optics is related to geometry and harmonics to arithmetic.) Nor indeed anything 

that holds of lines not as lines and as depending on the principles proper to them—e.g. 

whether straight lines are the most beautiful of lines, or whether they are contrarily 

related to curved lines; for these things hold of lines not in virtue of their proper kind 

but rather in virtue of something common.
2
 

 

From Aristotle‘s remarks here we can gather some understanding of the traditional model of 

science which was, by and large, the tacit background for geometers working with the 

traditional, Euclidean view of geometry. Sciences are delineated by the genus of objects 

about which they provide demonstrations. The logical relations amongst distinct sciences are 

determined by the relations between the classes of objects studied by those sciences; if the 

classes of objects are entirely disjoint (i.e., if there is no object which is treated in both 

sciences), then there will be no noteworthy relations amongst the sciences which study those 

objects, other than mutual exclusion. Various other relations are possible when there is 

overlap or inclusion amongst the classes of objects studied. Further, the demonstrations of 

distinct sciences cannot be applied to one another, except under the aforementioned 

condition, because any given science, qua science, may only contain reasoning pertaining to 

its proper objects—to ‗cross to a different kind‘ is, in essence, to fail to reason scientifically. 

 

                                                 
2
 [Aristotle 1994, Book I, Chapter 7, 75a37-75b20; translation slightly altered]. 



50 

 

The emergence of consistent non-Euclidean geometries and the development of sophisticated 

methods of classifying those geometries (e.g., Klein‘s Erlanger Programm and Riemann‘s 

differential approach) considerably loosened the link between spatial intuition and 

geometrical truth. Along with the slackened connection between intuition and geometrical 

demonstration came a general uncertainty about the objects of geometry itself. Traditionally, 

geometry had been construed as the science of the (idealized) elements of physical space. So, 

when the intuition which was the putative source of our knowledge of this space and its 

elements fell into disrepute, and the concept of space mutated into something considerably 

more general, it should be no surprise that there arose some confusion over the very nature 

of the science of geometry itself. If geometry could not be characterized as the science of 

spatial objects, then what sort of a science was it? What genus of objects did it study? What 

was geometry about? As we will see, the new approach to geometry offered by Hilbert and 

others—an approach which contained an alternative understanding of the nature of 

definitions and axioms—required a rather different understanding of the nature of (at least 

‗formal‘) science than was available within the Aristotelian tradition. 

 

§1. Hilbert’s Grundlagen 

 

Formally at least, Hilbert‘s Grundlagen der Geometrie bears some similarity to Euclid‘s 

Elements. Within its pages we find a number of definitions and a collection of unproved 

propositions employed to prove theorems, many of which are familiar (at least in name) 

from the history of geometry. Like many mathematicians before him, Hilbert does away with 

Euclid‘s distinction between common notions and postulates and instead speaks of all non-

definitional, unproved propositions simply as axioms. In addition to individual axioms 

Hilbert also speaks of ‗axiom groups‘, which are collections of axioms grouped together 

according to their specific subject matter. In the Grundlagen, there are five such groups, 

which deal with incidence, order, congruence, parallels, and continuity, respectively. But, 

despite these formal resemblances between Hilbert‘s work and that of Euclid, a rather wide 

conceptual gulf separates their respective uses of terms like ‗point,‘ ‗line,‘ ‗plane,‘ 

‗between,‘ ‗definition‘ and ‗postulate‘ (or ‗axiom‘).  

 

1.1 Definitions 

 

In order to see exactly how wide this divergence is, it is instructive to compare Euclid‘s 

definition of ‗point‘ with that of Hilbert. Recall that Euclid defines a point as ―that which has 

no part‖.
3
 Hilbert, like Euclid, presents a number of definitions, some more in line with 

traditional Euclidean definitions, and some which seem quite different. This is partially due 

to the novelty of his axiomatic approach and partly due to confusions regarding the nature of 

definitions. Take his first attempt at a definition of the term ‗point‘ for instance: 

 

DEFINITION. Consider three distinct sets of objects. Let the objects of the first set be 

called points and be denoted by A, B, C, ...; let the objects of the second set be called 

lines and be denoted by a, b, c, ...; let the objects of the third set be called planes and 

                                                 
3
 [Heath 1956, 155]. 



51 

 

be denoted by α, β, γ, .... The points are also called the elements of line geometry; the 

points and the lines are called the elements of plane geometry; and the points, the lines 

and the planes are called the elements of space geometry or the elements of space.
4
 

 

Now, postponing for the moment the question whether or not this paragraph forms Hilbert‘s 

complete definition of ‗point‘, ‗line‘, or ‗plane‘, it is already apparent that his approach is 

quite different from that of Euclid. Rather than an attempt to fix our attention definitively 

upon a certain type of object (e.g., partless wholes), Hilbert begins his definition by situating 

points, lines, and planes within three geometric systems. There is no real attempt at 

reference-fixing here; instead what we have is a very loose indication of the relations in 

which the basic items of the system stand, or, perhaps more accurately, the bare indication 

that these items stand in some relations to certain systems and to each other. The pre-

systematic, introductory role of the definition seems to have disappeared entirely. This is the 

case for Hilbert‘s definitions of other basic concepts and relations as well. He defines 

‗betweenness‘ for example, as follows: ―The points of a line stand in a certain relation to 

each other and for its description the word ―between‖ will be specifically used‖.
5
 Now, taken 

on its own, this definition is hardly satisfactory. And, indeed, Hilbert does not seem to have 

wanted it, or any of his definitions, to be taken in isolation. Instead, he suggests that these 

concepts and their relations are to be further explicated by means of his axiom groups. Thus 

after (provisionally) defining ‗point‘, ‗line‘, and ‗plane‘ by vague reference to the geometric 

systems which they form together, he suggests that his broad initial account will be narrowed 

through the axiomatic characterization of the relations in which these primitive concepts 

stand to each other: 

 

The points, lines and planes are considered to have certain mutual relations and these 

relations are denoted by words like ―lie,‖ ―between,‖ ―congruent.‖ The precise and 

mathematically complete description of these relations follows from the axioms of 

geometry.
6
 

 

Here, again, we can see the difference between the traditional, Euclidean view of geometry 

(as a collection of intuitively grounded truths about idealized elements of physical space) 

and Hilbert‘s view. Rather than unprovable truths about the basic objects of geometry, the 

axioms in Hilbert‘s sense play a quasi-definitional role in which they help to determine the 

relations amongst the basic elements of certain geometrical systems; the nature of those 

elements beyond the relations in which they stand to each other seems to be a matter of no 

intrinsic geometric interest. As we‘ve seen, within the traditional Euclidean dispensation, 

one type of definition played a reference-fixing role prior to the introduction of the axioms, 

and these axioms were understood to be unprovable truths about the objects toward which 

the definitions guided the reader. Within Hilbert‘s work, by contrast, the initial definitions of 

‗line‘, ‗point‘, and ‗plane‘ do not fix (or even attempt to fix) the references of these terms. 

Rather, they serve to indicate, in an extremely general way, that the objects stand in certain 

relations to each other. Neither do these initial definitions themselves determine specifically 

                                                 
4
 [Hilbert 1971, 3]; [Hilbert 1903, 2]. 

5
 [Hilbert 1971, 5]; [Hilbert 1903, 4] . 

6
 [Hilbert 1971, 3]; [Hilbert 1903, 2]. 



52 

 

which relations these are. It is instead left up to the subsequent axiom groups to ‗fix‘ the 

reference of these terms and the relations in which their referents stand more definitively—

though, as we will see, it is precisely the free-floating character of Hilbert‘s points, lines, 

planes, etc. which represents his biggest departure from the traditional view, so that, even 

later in the work, he is not terribly interested in pinning them down.  

 

It is for these reasons that Hilbert‘s approach is remarkably different than the traditional 

Euclidean approach. While many traditional geometers considered the intention of their 

work to be the elaboration of a science of space and spatial objects, it is very difficult to 

construe Hilbert‘s work in this way. Instead of an emphasis on the objects of geometry, 

which are intuitively graspable according to the traditional view, Hilbert emphasizes the 

relations in which these objects might stand (though, as we will see, speaking of a 

determinate group of objects is, perhaps, not an ideal description of Hilbert‘s intention).  

This shift in emphasis had quite radical implications for the conception of geometry as a 

science.  

 

1.2 Axioms and intuition 

 

Despite this divergence from tradition regarding the definition of the primitive, intuitively 

given objects of geometry, Hilbert held on to some aspects of the Euclidean tradition. For 

instance, he still believed that a proper set of geometrical axioms (like his own) ought to be 

developed on the basis of our intuition of space. In this he resembled his colleague Klein, as 

both men favoured a highly abstract approach to geometry but remained insistent that, at 

bottom, geometry was the science of space (however expansive their notion of space), and 

that the ultimate properties of space were graspable by way of intuition. Indeed, more than 

having their certainty grounded in our intuition, Hilbert insisted on several occasions that 

any worthwhile mathematical investigations should have their origin in empirical concerns. 

At the outset of the Grundlagen he writes, for instance, that 

 

Geometry, like arithmetic, requires only a few and simple principles for its logical 

development. These principles are called the axioms of geometry. The establishment 

of the axioms of geometry and the investigation of their relationships is a problem 

which has been treated in many excellent works of the mathematical literature since 

the time of Euclid. This problem is equivalent to the logical analysis of our intuition of 

space.
7
 

 

Thus, for Hilbert the basic material of geometry, given in its axioms, has its origin in our 

intuition of space. The remainder (i.e., the collection of theorems which are not axioms) 

stems from the logical analysis of this intuition, particularly, we may assume, from the 

analysis of the different aspects of this intuition given in Hilbert‘s axiom groups. He writes 

that ―[e]ach of these groups expresses certain related facts basic to our intuition‖.
8
 The 

axioms, then, are not arbitrarily chosen, with indifference to our intuition. Instead, they were 

intended by Hilbert to capture important features of spatial intuition with an eye toward the 

                                                 
7
 [Hilbert 1971, 2]; [Hilbert 1903, 1]. 

8
 [Hilbert 1971, 3]; [Hilbert 1903, 2]. 



53 

 

development of a geometry which ought to be capable of delivering the appropriate set of 

deductive consequences implied by that intuition.
9
 It is only at the stage of logical analysis—

after intuition has been crystallized within the axiom groups—that the purely deductive, non-

intuitive work so characteristic of ‗modern‘ mathematics begins. Hilbert quite clearly 

outlines his intention to capture the important implications of our geometrical intuition of 

space in an 1899 letter to Frege: 

 

It was of necessity that I had to set up my axiomatic system: I wanted to make it 

possible to understand those geometrical propositions that I regard as the most 

important results of geometrical inquiries: that the parallel axiom is not a consequence 

of the other axioms, and similarly Archimedes‘ axiom, etc. I wanted to answer the 

question whether it is possible to prove the proposition that in two identical rectangles 

with an identical base line the sides must also be identical, or whether as in Euclid this 

proposition is a new postulate. I wanted to make it possible to understand and answer 

such questions as why the sum of the angles in a triangle is equal to two right angles 

and how this fact is connected with the parallel axiom. That my system of axioms 

allows one to answer such questions in a very definite manner, and that the answers to 

many of these questions are very surprising and even quite unexpected, is shown, I 

believe, by my Festschrift as well as by the writings of my students who have followed 

it up.
10

 

 

We can see here that, instead of taking the content of Euclidean geometry as somehow 

originating in a set of arbitrary postulates, Hilbert views his axiom groups themselves as the 

components of a successful attempt to capture some pre-systematic intuitive content. In this 

case, the content is delivered by a history stemming from two millennia of geometric 

research, itself grounded (or at least thought to have been grounded) in our intuitive 

apprehension of space.
11

 

 

The most novel elements of Hilbert‘s approach to the axiomatization of geometry had 

important precursors at least as far back as Lambert. One element of the Grundlagen which 

separates it from most prior work in geometry is Hilbert‘s insistence that the deductive 

                                                 
9
 Hilbert remarks, for instance, that ―Axiom I, 7 expresses the fact that space has no more than three 

dimensions, whereas Axiom I, 8 expresses the fact that space has no less than three dimensions‖ [Hilbert 1971, 

4]. Note that Hilbert writes simply ‗space‘ and not ‗Euclidean space‘. 
10

 [Hilbert to Frege, 29 December 1899; excerpt by Frege in Frege 1980, 38-39]; the final sentence refers, 

particularly, to the dissertation of Hilbert‘s student Max Dehn (1878-1952) on Legendre‘s theorem, 

subsequently published as [Dehn 1900]. 
11

 An illuminating account of Hilbert‘s understanding of the ‗life cycle‘ of mathematical theories is given in 

[Corry 2004a]; Corry writes there that Hilbert distinguishes ―three clearly separated stages that, in his view, 

mathematical theories usually undergo in the course of their development: the naive, the formal, and the 

critical‖ and that ―For Hilbert, the need to introduce this method [i.e., the axiomatic method] arises precisely as 

a means to analyze already established theories, to criticize their basic assumptions, and to elucidate their 

logical deductive structure‖ [Corry 2004a, 20]. Given this view of the evolution of mathematical theories, it 

seems likely that Hilbert saw his own axiomatic endeavours as attempts to likewise clarify already-going 

concerns, rather than as efforts to investigate arbitrary domains of inquiry freshly sprung from the head of a 

mathematical Zeus. (Corry seems to suggest as much at [Corry 2004a, 93-94]). This approach to mathematical 

axiomatics is, accordingly, continuous with Hilbert‘s later attempts to axiomatize physics. 



54 

 

structure developed on the basis of intuitively given axiom groups does not require any 

further input from intuition once these groups are established. But as early as 1766 we find a 

remarkably similar view expressed by Lambert in a work on the theory of parallels.
12

 There 

he writes the following rather prescient paragraph:  

 

The difficulties concerning Euclid's 11th axiom [i.e., the parallel postulate] have 

essentially to do only with the following question: Can this axiom be derived correctly 

from Euclid's postulates and the remaining axioms? Or, if these premises are not 

sufficient, can we produce other postulates or axioms, no less evident than Euclid's, 

from which his 11th axiom can be derived? In dealing with the first part of this 

question we may wholly ignore what I have called the representation of the subject-

matter. Since Euclid's postulates and remaining axioms are stated in words, we can and 

should demand that no appeal be made anywhere in the proof to the matter itself, but 

that the proof be carried out—if it is at all possible—in a thoroughly symbolic fashion. 

In this respect, Euclid's postulates are, so to speak, like so many given algebraic 

equations, from which we must obtain x, y, z, etc. without ever looking back to the 

matter in discussion. Since the postulates are not quite such formulae, we can allow the 

drawing of a figure as a guiding thread to direct the proof. On the other hand, it would 

be preposterous to forbid consideration and representation of the subject-matter in the 

second part of the question, and to require that the new postulates and axioms be found 

without reflecting on their subject-matter, off the cuff, so to speak.
13

  

 

As we can see here, Lambert, like Hilbert, paid due homage to the understanding of 

geometry as founded upon some specific content (e.g., the deliverances of spatial intuition). 

He also insisted that much of our subsequent reasoning about geometrical relations can and 

should be divorced from any intuitive or specifically geometrical content. We should think 

entirely abstractly, without reference to intuition, about, e.g., relations of independence 

between Euclid‘s axioms.  

 

Lambert suggests we can treat these axioms as mere symbolic formulae to be manipulated 

using acceptable techniques, just as we manipulate algebraic equations by performing 

allowable operations upon them, very often with indifference to the possible references of 

their terms. This view is, in germinal form, quite similar to Hilbert‘s later treatment of proofs 

as mathematical objects in their own right. For Lambert, intuition plays an important role in 

the development of basic geometrical axioms. Once these axioms are established on the 

basis of intuition, however, geometry can and should be treated as an entirely abstract 

science, decoupled from reference to spatial intuition except at its basis. 

 

A little over a century later, we find a more direct influence on Hilbert in the views of Moritz 

Pasch (1843-1930), particularly as expressed in his 1882 Vorlesungen über neuere 

                                                 
12

 Though completed by 1766, the work was only published in 1786. 
13

 [Lambert 1786, quoted in Torretti 1984, 49]. 



55 

 

Geometrie.
14

 Pasch, like Lambert before him, held that geometry‘s ―foundational concepts 

and propositions should be derived immediately from experience,‖ but made the slightly 

different claim that ―all other concepts and results must be reducible to these alone. ... 

[G]eometry should thus proceed from the beginning as a descriptive science of the world of 

appearances and even in its most complex conceptions it should retain this character.‖
15

 

Thus, while Lambert was content to treat axioms as the intuitive basis for a kind of symbolic 

geometric calculus which itself needn‘t be geometrically meaningful, Pasch sought a firmer 

experiential basis for geometry by insisting on the meaningfulness of both axioms and each 

step in any chain of geometric reasoning. This does not, however, mean that Pasch eschewed 

the abstract algebraic techniques hinted at by Lambert. He writes, for instance that 

 

If geometry is to be truly deductive, the process of inference must be independent in all 

its parts from the meaning of the geometrical concepts, just as it must be independent 

of the diagrams; only the relations specified in the propositions and definitions may 

legitimately be taken into account. During the deduction it is useful and legitimate, but 

in no way necessary, to think of the meanings of the terms; in fact, if it is necessary to 

do so, the inadequacy of the proof is made manifest. If, however, a theorem is 

rigorously derived from a set of propositions [...] the deduction has value which goes 

beyond its original purpose. For if, on replacing the geometric terms in the basic set of 

propositions by certain other terms true propositions are obtained, then corresponding 

replacements may be made in the theorem; in this way we obtain new theorems ... 

without having to repeat the proof.
16

 

 

Here Pasch develops a conception of the deductive structure of geometric reasoning which is 

quite similar to Hilbert‘s later view. He suggests that, while we may employ intuitive 

thinking and diagrams when working our way through a geometric proof, if such devices are 

necessary the proof is insufficiently rigorous. Pasch‘s full view, then, is that the axioms of 

geometry must be grounded in our intuition of space, and that each subsequent 

argumentative step towards a given proof must itself be meaningfully construed on the basis 

of these axioms. But, despite this stepwise connection to intuition, the fully rigorous 

expression of geometric proof—if it is to be deductively valid—should not rely in any 

necessary way on intuitive reasoning or diagrammatic representation. So, while the 

meaningfulness of geometric formulae is guaranteed by their connection to intuitively given 

axioms, the deductive manipulation of those formulae ought to be performed in an algebraic 

manner with complete indifference to meaning. Thus, for both Lambert and Pasch, the 

deductive element of geometric reasoning is in important respects separable from the 

intuitive element required to establish a proper axiom set. 

 

Hilbert‘s view—developed after the advent of differential and various non-Euclidean 

geometries—is almost identical with Pasch‘s and Lambert‘s in these respects. For Hilbert, 

                                                 
14

 For accounts of Pasch‘s work and its influence on subsequent work in geometry, see [Torretti 1984, 210-218] 

and [Corry 2004b, 155ff.]. For Pasch‘s influence specifically on Hilbert, [Corry 2004b, 157-159] and [Toepell 

1986] are useful. 
15

 [Pasch 1887, 129; translation in Rowe 2000, 72]. 
16

 [Pasch 1882, 98].  



56 

 

too, once the intuitive basis is given, the theorems of geometry can be deductively unfurled 

without ever returning to their intuitive ground. His proofs are all conducted (he would 

argue) using purely logical, non-intuitive means. None of these proofs should rely essentially 

upon a feature of a diagram or upon our accepting intuitions other than those formulated in 

the axiom groups. This view, exemplified in the Grundlagen, is more explicitly and 

generally stated in his well-known lecture on the most important mathematical problems for 

the twentieth century. There he states the following:  

 

To new concepts correspond, necessarily, new signs. These we choose in such a way 

that they remind us of the phenomena which were the occasion for the formation of the 

new concepts. So the geometrical figures are signs or mnemonic symbols of space 

intuition and are used as such by all mathematicians. Who does not always use along 

with the double inequality a > b > c the picture of three points following one another 

on a straight line as the geometrical picture of the idea ‗between‘? Who does not make 

use of drawings of segments and rectangles enclosed in one another, when it is 

required to prove with perfect rigour a difficult theorem on the continuity of functions 

or the existence of points of condensation? [...] The arithmetical symbols are written 

diagrams and the geometrical figures are graphic formulae; and no mathematician 

could spare these graphic formulae, any more than in calculation the insertion and 

removal of parentheses or the use of other analytical signs. The use of geometrical 

signs as a means of strict proof presupposes the exact knowledge and complete 

mastery of the axioms which underlie those figures and in order that these geometrical 

figures may be incorporated in the general treasure of the mathematical signs, there is 

necessary a rigorous axiomatic investigation of their conceptual content. Just as in 

adding two numbers, one must place the digits under each other in the right order, so 

that only the rules of calculation, i.e., the axioms of arithmetic, determine the correct 

use of the digits, so the use of geometrical signs is determined by the axioms of 

geometrical concepts and their combinations.
17

  

 

Beyond the clear similarity to Pasch‘s views expressed here, it is illuminating that Hilbert 

does not restrict himself to geometric examples. This is because he does not make any strict 

distinction between the type of work involved in geometric reasoning and that involved in, 

e.g., arithmetic or algebraic reasoning. There is for Hilbert a basic mathematical unity which 

is best expressed by the wide applicability of the axiomatic method to branches of 

mathematics which are, from the point of view of the ‗objects‘ studied, quite distinct. We 

can establish Euclidean geometry axiomatically and prove its theorems purely deductively. 

 

Here it is made clear by Hilbert that, while we may employ aids such as diagrams and 

pictures, even in absolutely strict proofs, our use of these sorts of aids must be licensed by 

the axiomatic presentation of the mathematical system in question. Thus, for instance, we 

may employ a picture of three points on a straight line to represent the notion of 

‗betweenness‘ in a strict proof only insofar as the intended conceptual content of the picture 

can be explicated in terms of the axioms of the system. The meaning of a specific theorem, 

proposition, or diagram formulated from within an axiomatized system is guaranteed by the 

                                                 
17

 [Hilbert 1900a, excerpt translated in Ewald 1999, 1100-1101]. 



57 

 

intuitively grounded axioms themselves. But, just as with Pasch, the actual deductive linkage 

of propositions in geometric and other proofs should be treated with indifference to the 

epistemologically useful meanings of the propositions at issue. For Hilbert, it is only once 

we have established a given theory on an axiomatic basis that we can begin to speak of truly 

rigorous proof; in the case of non-axiomatic theories, our use of intuitive reasoning or 

diagrammatic representation cannot be guaranteed by an axiomatic basis, and can only very 

rarely be dispensed with. Without the exact knowledge engendered by the process of 

axiomatization, we cannot engage in the purely symbolic form of deductive reasoning 

because the meaningfulness of the steps in this formal reasoning cannot be guaranteed, and 

we may very well stray into absurdity or imprecision.   

 

Though it seems clear from what we have just seen that Hilbert himself never held the purely 

postulational view of mathematics as the investigation of deductive structures derived from 

arbitrarily selected postulate sets, the way in which he developed his axiomatic approach to 

geometry was (largely) consistent with such an approach. Some of his remarks, particularly 

in his correspondence with Frege, certainly lead one in this direction, so that it is no surprise 

that such a strong opponent of this sort of project as Frege would have reacted accordingly. 

  

But it was not only his critic Frege who read Hilbert in this manner; many of Hilbert‘s 

admirers within the mathematical community (e.g., Felix Hausdorff, and the school of 

American mathematicians surrounding E. H. Moore in Chicago and Edward Huntington at 

Harvard) developed their own approaches to axiomatics inspired by Hilbert‘s work, 

approaches which were avowedly ‗arbitrary‘ in a way that Hilbert himself continued to resist 

throughout his career.
18

 Hausdorff, inspired in large part by Hilbert, came very close to a 

purely arbitrary understanding of the nature of axiomatics. In the context of Hilbert‘s 

approach to geometry, Hausdorff wrote that  

 

In all philosophical debates since Kant, mathematics, or at least geometry, has always 

been treated as heteronomous, as dependent on some external instance of what we 

would call, for want of a better term, intuition, be it empirical, subjective or 

scientifically amended, innate or acquired. The most fundamental task of modern 

mathematics has been to set itself free from this dependence, to fight its way through 

from heteronomy to autonomy.
19

  

 

The importance of completely detaching mathematics from intuition is further emphasized 

by Hausdorff in course notes from the same period: ―Mathematics totally disregards the 

actual significance conveyed to its concepts, the actual validity that one can accord to its 

theorems. Its indefinable concepts are arbitrarily chosen objects of thought and its axioms 

are arbitrarily, albeit consistently, chosen relations among these objects. Mathematics is a 

science of pure thought, exactly like logic‖.
20

 So, here we have Hausdorff, inspired by the 

Grundlagen, extending Hilbert‘s own view of deduction as indifferent to meaning to include 

the axiomatic basis of a theory as well. Hausdorff attempts to break entirely from the 

                                                 
18

 See, e.g., [Huntington 1902] and [Moore 1902b]. 
19

 [Hausdorff c. 1904, quoted in Corry 2004a, 117]. 
20

 [Hausdorff 1903-1904, quoted in Corry 2004a, 117]. 



58 

 

Kantian understanding of ‗heteronomous‘ mathematics tethered to pure intuitions of space 

and time by suggesting we examine arbitrarily developed postulate sets with no connection 

to intuition whatsoever. On this view, geometry, for instance, would cease to be coupled to 

spatial intuition and, instead, become the systematic exploration of a wide variety of 

structures of a certain type—perhaps those suggested by Riemann‘s coefficient-determined 

classificatory scheme. (Though it is not possible here, a more detailed examination of 

Hausdorff‘s influence on topology and measure theory would illustrate how powerful and 

widespread this view would later become in important branches of geometry). 

 

While Hausdorff‘s views seem at least partially opposed to the intuitively-grounded view 

developed by Lambert, Pasch, and Hilbert, some of Hilbert‘s own remarks (particularly 

those made in correspondence to Frege) directly support this purely postulational approach 

to mathematics. Indeed, in one of the better known paragraphs of their correspondence, 

Hilbert writes to Frege that ―if the arbitrarily given axioms do not contradict one another 

with all their consequences, then they are true and the things defined by the axioms exist. 

This is for me the criterion of truth and existence‖.
21

 Yet, elsewhere, Hilbert seems 

committed to the view that his axioms are anything but arbitrary—for to investigate 

arbitrarily postulated axioms without being guided by a pre-theoretic, pre-axiomatic intuition 

was, for Hilbert, a pointless exercise, and mathematics was certainly not meaningless for 

him, even in the later stages of his career when he began developing his well-known 

formalist programme. He writes later, for instance, that ―We are not speaking here of 

arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by 

arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that 

can only be so and by no means otherwise.‖
22

 For Hilbert, mathematics is different from a 

merely formal game insofar as the rules of mathematics cannot be altered in the way that we 

might alter the rules of a game. For, in constructing a game, we can alter the rules however 

we like, even altering the nature of the ‗logical‘ relations amongst the rules and components 

of the game. But, for Hilbert, once we have established some elements of a conceptual 

system (the axioms), the unfolding of the rest (the theorems) is a matter of necessity, not 

happenstance. Though by the time of the publication of the Grundlagen Hilbert had not yet 

developed a sophisticated picture of the nature of logic and its relation to mathematical 

reasoning, it is clear that he did not understand logic as a merely formal enterprise, but 

instead as an important and meaningful constraint to which mathematical thinking must 

answer. 

 

1.3 Axioms, truth, and existence 

 

Within the traditional understanding of Euclidean geometry, Euclid‘s axioms (barring, 

perhaps, the troublesome parallel postulate) were thought to be basic, unprovable truths 

about physical space. As we‘ve just seen, Hilbert‘s understanding of the role of his axioms 

was partly traditional and partly novel. As within the traditional view, the content of 

Hilbert‘s axiom groups was thought to be given by spatial intuition. Hilbert‘s axioms, too, 

are unproven expressions of intuition on the basis of which a whole deductive structure is 

                                                 
21

 [Hilbert to Frege, 29 December 1899; excerpt by Frege in Frege 1980, 39-40]. 
22

 [Hilbert 1919-20 [1992], 14; quoted in Corry 2006, 138]. 



59 

 

developed. But, in contrast to the tradition, the truth of his axioms is not guaranteed by the 

nature of physical space (though the content of the axioms is suggested by our intuition of 

physical space). Instead, their truth is held to result from their consistency with each other. 

While on the traditional dispensation the truth of axioms was not relativized to a particular 

axiomatic system, on Hilbert‘s view, we cannot speak of the truth of mathematical axioms 

except by reference to their mutual consistency.  

 

In addition to the truth of the axioms, consistency also establishes for Hilbert the existence of 

the objects to which the axioms refer. Thus, for instance, a geometric point is said to exist if 

the axioms which describe points (and the relations between points and other elements of the 

axiomatic system) are consistent. This, perhaps even more than Hilbert‘s relativized 

conception of mathematical truth, is a radical claim. It shows that Hilbert, at this time, was 

willing to countenance the existence of an almost bewildering array of mathematical objects, 

with the only condition being the possibility of providing a consistent axiomatization 

describing the objects. Thus, for Hilbert, the realm of the mathematically existent is 

coextensive with the realm of the logically possible.
23

 The exact relationship between 

mathematical existence and the consistency of axiom sets was not fully worked out in 

Hilbert‘s work of this time—a fact which became clearer in Hilbert‘s own later work on 

direct consistency proofs, and in Frege‘s criticisms of the Grundlagen der Geometrie. 

 

Hilbert‘s views on mathematical existence and truth have a wide range of consequences. One 

of the most important is that the proof of the consistency of a set of axioms becomes a 

crucial question in the development of an acceptable axiomatic system. Within the 

traditional dispensation, the consistency of Euclid‘s axioms (again, excluding the parallel 

postulate) was, for the most part, not a matter for intensive geometric research. Since the 

axioms were understood to be truths based upon spatial intuition, the idea that the 

consistency of the axioms might require any special investigation simply did not occur. It 

didn‘t make much sense to ask whether a collection of obvious truths could all be true 

together—their being independently true of the world was a tacit guarantee that they were 

true when taken together, too. On Hilbert‘s view, however, the grounding role of intuition 

was more heuristic in character: our intuition suggests certain axioms, but their truth is not at 

all grounded in that intuition. Instead, once we have established some collection of axioms 

using our intuitive heuristic, we must investigate and, ideally, prove their consistency to 

justify their truth. Intuition on its own is insufficiently rigorous and requires precise and 

careful axiomatization, followed by the deductive elaboration of consequences. This 

alteration in the concept of mathematical truth goes hand in hand with Hilbert‘s shift away 

from an object-oriented approach to geometry. In contrast to geometrical truth understood as 

a pre-structural, absolute property determined by the nature of space and certain idealized 

                                                 
23

 Bolzano, in a seemingly offhand remark, offers a remarkably similar view as early as 1837. In §352 of the 

Wissenschaftslehre, while discussing how we might tell if an idea is objectual or not (i.e., whether any objects 

fall under it), he writes that ―In order to determine [if an idea is objectual], we must also decide whether the 

attributes that the idea would ascribe to its object presuppose actual existence or not. If not, the idea is to be 

declared objectual. For if its object need not be something existing, it is obvious that such an object could only 

be lacking if the assumption that there were such an object contradicted a purely conceptual truth‖ [Bolzano 

1837, §352]. In effect, Bolzano claims here that particular abstract objects ‗exist‘ (in a restricted sense) just in 

case their existence is not contrary to a conceptual truth. 



60 

 

spatial objects, Hilbert turns to a notion of truth relative to a given axiomatization, 

determined by relations of consistency. This interest in questions of consistency was part of 

a larger shift in interest towards certain structural properties of axiomatic systems. In 

addition to consistency, Hilbert began to focus on other properties of axiomatic systems, 

particularly their simplicity, completeness, and the mutual independence of their axioms. 

 

1.4 Desirable properties of axiomatic systems 

 

One of the more innovative elements of Hilbert‘s axiomatic approach was his specific focus 

on relations amongst the axioms themselves. Within the Euclidean tradition, as we saw, 

there were some concerns about the independence of the axioms relative to each other. In 

particular, the potential dependence of the parallel postulate on the other axioms was a 

matter of intensive investigation. With Hilbert, this concern for the independence of his 

axiom groups became a matter for rigorous logical investigation. Alongside independence 

and consistency, Hilbert was also concerned that a suitable axiomatization be complete, 

simple, and (if possible) categorical. He writes at the outset of the Grundlagen, for instance, 

that ―This present investigation is a new attempt to establish for geometry a complete, and as 

simple as possible, system of axioms and to deduce from them the most important theorems 

in such a way that the significance of the various groups of axioms, as well as the scope of 

the conclusions that can be drawn from the individual axioms, become as clear as 

possible‖.
24

 Though these properties have since become the object of intense study 

throughout mathematics, formal semantics, and logic, at Hilbert‘s time the precise 

formulation of criteria for consistency, independence, completeness, categoricity, and 

simplicity had not yet been given. I will now turn to an examination of Hilbert‘s 

understanding of these notions around the time of the publication of the Grundlagen der 

Geometrie.
25

 

 

1.4.1 Consistency 

 

As we‘ve seen, for Hilbert the consistency of a set of axioms (or axiom groups) was of 

central importance for the establishment of the truth of a set of axioms, as well as the 

existence of the objects referred to by the axioms. Accordingly, proving the consistency of a 

set of axioms becomes a matter for direct mathematical investigation. The consistency of a 

collection of axioms, for Hilbert, amounts to the impossibility of deriving a contradiction 

from the consequences of those axioms taken together.
26

 Although his conception of 

                                                 
24

 [Hilbert 1971, 2]; ―Die vorliegende Untersuchung ist ein neuer Versuch, für die Geometrie ein vollständiges 

und möglichst einfaches System von Axiomen aufzustellen und aus denselben die wichtigsten geometrischen 

Sätze in der Weise abzuleiten, daß dabei die Bedeutung der verschiedenen Axiomgruppen und die Tragweite 

der aus den einzelnen Axiomen zu ziehenden Folgerungen möglichst klar zu Tage tritt‖ [Hilbert 1903, 1]. 
25

 Though Hilbert was instrumental in the subsequent development of proof theory and mathematical logic 

throughout the 1920s and 1930s, I limit my discussion here to his early work insofar as my interest is in 

highlighting the differences between Hilbert‘s early position and that of Frege. In his later work, Hilbert in fact 

took a number of Frege‘s criticisms to heart, so that the gap between Hilbert‘s highly structure-focussed 

position and Frege‘s point of view narrowed considerably. 
26

 ―The necessary task then arises of showing the consistency and completeness of these axioms, i.e. it must be 

proved that the application of the given axioms can never lead to contradictions, and, further that the system of 



61 

 

consistency is clearly proof-theoretic, his method of proving this sort of consistency is quite 

similar to the modern model-theoretic approach: an interpretation in which all the axioms 

hold (i.e., a model of those axioms) is constructed and this construction is taken to directly 

show the consistency of the axioms. Let us take a brief look at the interpretation Hilbert 

employs to prove the consistency of his particular set of axioms for Euclidean geometry. 

 

Hilbert begins by interpreting his axioms within ―the field Ω of all algebraic numbers that 

arise from the number 1 and the application of a finite number of times of the four 

arithmetical operations of addition, subtraction, multiplication and division, and the fifth 

operation √       , where ω denotes a number that results from these five operations‖.
27

 

Points are interpreted as pairs of numbers (x, y) within this field; lines are interpreted as ―the 

ratios (u : v : w) of any three numbers from Ω [...] provided u, v are not both zero.‖
28

 Now, 

having thus interpreted the basic notions of his formalization of Euclidean geometry within 

the algebraic number field Ω, Hilbert proceeds to suggest that his axioms up to and including 

Axiom V, 1 are satisfied. The remaining axiom, the so-called completeness axiom V, 2
29

 is 

satisfied, he claims, only if we employ the specific field of the real numbers in our 

interpretation in place of the subfield Ω.  Having thus interpreted the basic notions of 

Euclidean geometry within the domain of the real numbers (and, initially, within the field 

Ω), Hilbert goes on to suggest that each of his axioms, thus interpreted, still holds. The first 

model of his axiomatization interpreted within the field Ω illustrates that his axioms from I-

IV and including V, 1 are consistent with each other precisely because ―Every contradiction 

in the consequences of [these axioms] would therefore have to be detectable in the arithmetic 

of the field Ω.‖
30

 It is assumed, at this point, that no contradictions arise within the arithmetic 

of this field; likewise, it is assumed that no contradictions can arise within the axioms thus 

modelled. When Hilbert extends his model to the field of all real numbers, the claim is 

similar: his axioms for Euclidean geometry are shown to be consistent upon the assumption 

that no contradiction arises within the arithmetic of the real numbers. Today we see this 

immediately as an example of a relative consistency proof: the consistency of Euclidean 

geometry is proved only relative to the consistency of the real numbers. By the turn of the 

century, however, the forbidding difficulties of providing direct consistency proofs for 

                                                                                                                                                       
axioms is adequate to prove all geometrical propositions. We shall call this procedure of investigation the 

axiomatic method‖ [Hilbert 1900b, in Ewald 1999, 1093]. 
27

 [Hilbert 1971, 29]. 
28

 [Hilbert 1971, 29]. 
29

 This axiom was not included in Hilbert‘s Festschrift version of the Grundlagen, nor in the first edition, but 

appeared only in the first French edition and second German edition. Subsequent to those editions it was further 

narrowed and renamed the axiom of line completeness, and altered to read as follows: ―An extension of a set of 

points on a line with its order and congruence relations that would preserve the relations existing among the 

original elements as well as the fundamental properties of line order and congruence that follows from Axioms 

I-III, and from V, 1 is impossible‖ [Hilbert 1971, 26]; in the 1903 edition it is phrased as follows: ―Die 

Elemente (Punkte, Geraden, Ebenen) der Geometrie bilden ein System von Dingen, welches bei 

Aufrechterhaltung sämtlicher genannten Axiome keiner Erweiterung mehr fähig ist, d.h.: zu dem System der 

Punkte, Geraden, Ebenen ist es nicht möglich, ein anderes System von Dingen hinzuzufügen, so daß in dem 

durch Zusammensetzung entstehenden System sämtliche aufgeführten Axiome I-IV, V 1 erfüllt sind‖ [Hilbert 

1903, 16]. We will have occasion to return to this axiom in the sections on completeness and categoricity 

below. 
30

 [Hilbert 1971, 30]. 



62 

 

axiomatized versions of arithmetic were not yet apparent to Hilbert, as is evident in his 

statement that he was ―convinced that it must be possible to find a direct proof for the 

consistency of the arithmetical axioms, by means of a careful study and suitable 

modification of the known methods of reasoning in the theory of the real numbers‖.
31

 The 

relativity of Hilbert‘s consistency proofs would, as we shall see, form one of the focal points 

of Frege‘s criticism, and would also become one of the chief motivators for Hilbert‘s 

formalist program of the 1920s. 

 

1.4.2 Independence 

 

After establishing the consistency of his axioms to his satisfaction, Hilbert moves on in the 

Grundlagen to prove the independence of some of those axioms from the others. He begins, 

unsurprisingly, with an illustration of the independence of his parallel axiom group. He 

opens this section of the work as follows: ―Having seen the [consistency] of the axioms it is 

of interest to investigate whether they are all independent of each other. In fact it can be 

shown that no essential part of any one of these groups of axioms can be deduced from the 

others by logical inference‖.
32

 At this time in his career, Hilbert (like most mathematicians 

of the time) had not expended much energy attempting to flesh out any systematic 

conception of basic logical notions like inference or consequence. He had, however, taken at 

least some interest in the quasi-logical work in elementary set-theory undertaken by Cantor 

and Dedekind. One thing that is quite clear about his conception of logic at the time, though, 

is that he, like Frege, took it to flow from a source of knowledge distinct from intuition. 

When we find the early Hilbert discussing things in a ‗purely logical‘ manner, it is in order 

to draw an epistemological contrast between knowledge derived from (usually spatial) 

intuition and knowledge given without reference either to intuition or experience. It is only 

given this epistemological distinction that Hilbert can speak of the logical analysis of our 

intuition of space, and maintain the distinction between the intuitions which inform our 

choice of axioms and the subsequent logical work which gives us the deductive structure 

erected on the basis of those axioms. Thus, the independence of axioms from one another 

ought to be provable without reference to spatial intuition, or other forms of experiential 

knowledge, as suggested by both Lambert and Pasch. Let‘s take a brief look at the way in 

which Hilbert goes about proving the independence of his parallel axiom group from the 

remaining groups. 

 

His procedure again seems to involve what we would now understand to be model-theoretic 

techniques. Having already provided a model in which all of his axioms excepting the 

completeness axiom hold (by interpreting his axioms and basic notions in terms of the field 

Ω, or in the field of the real numbers), Hilbert now constructs a model in which all of his 

axioms except the parallel axiom are satisfied. For our purposes here, the details of this 

                                                 
31

 [Hilbert 1900b, in Ewald 1999, 1104]. In another paper of the same year on the concept of number, Hilbert 

writes, again optimistically, that ―To prove the consistency of the above axioms [for arithmetic] one needs only 

a suitable modification of familiar methods of inference. In this proof I also see the proof of the existence of the 

totality of real numbers, or—in the terminology of G. Cantor—the proof that the system of real numbers is a 

consistent (finished) set‖ [Hilbert 1900a, in Ewald 1999, 1095]. A suitable modification and the resulting proof 

were both to remain elusive and, perhaps, impossible after Gödel‘s incompleteness results. 
32

 [Hilbert 1971, 32]. 



63 

 

model are unimportant.
33

 What is important is the relativity of Hilbert‘s proofs. In order to 

prove that the uninterpreted axioms of geometry are independent of one another, Hilbert 

employs the following procedure. First, he interprets the primitive terms of his axiomatic 

system in such a way that a model of his axioms is produced, i.e., an interpretation in which 

all the axioms are satisfied. (This was his way of proving their consistency). He then 

proceeds to construct a different interpretation on which only some of the axioms are 

satisfied, while the axiom to be proven independent is not (or, within which all the axioms 

and the negation of the axiom to be proven independent are satisfied).
34

 This procedure is 

meant to illustrate that there are certain structures within which the axioms as a whole are 

satisfied and other structures within which the axiom to be proven independent is not 

satisfied. Hence, the axiom to be proven independent is not a purely logical consequence of 

the remaining axioms insofar as both it and its negation are consistent with those axioms. 

That an axiom and its negation are both consistent with a set of axioms illustrates, for 

Hilbert, that it is not a deductive, logical consequence of those axioms and, thus, that it is 

independent of them. As we will see, Frege would criticize this approach to independence 

proofs as incoherent, given what he takes to be a proper understanding of the nature of 

logical inference and model-theoretic techniques. Hilbert, for his part, offers no discussion 

about what logic is or how it operates with indifference to intuitive content. 

 

1.4.3 Completeness  

 

In addition to independence and consistency, Hilbert also suggests that any decent axiom 

system should possess the property of completeness. Given the close resemblance between 

Hilbert‘s concepts of consistency and independence with their modern model-theoretic 

equivalents, we may be tempted to align his notion of completeness, too, with its model-

theoretic cousin. But, as Leo Corry correctly notes,  

 

The ―completeness‖ that Hilbert demanded for his systems of axioms should not be 

confused with the later, model-theoretical notion that bears the same name, a notion 

that is totally foreign to Hilbert‘s axiomatic approach at this early stage. Rather it is an 

idea that runs parallel to [physicist Heinrich] Hertz‘s demand for ―correctness‖. Thus, 

Hilbert demanded from any adequate axiomatization that it should allow for a 

derivation of all the known theorems of the discipline in question. The axioms 

formulated in [Grundlagen], Hilbert claimed, would indeed yield all the known results 

of Euclidean geometry or of the so-called absolute geometry, namely that [geometry 

which is] valid independently of the parallel postulate, if the corresponding group of 

axioms is ignored. Thus, reconstructing the very ideas that had given rise to his own 

conception, Hilbert discussed in great detail the role of each of the groups of axioms in 

                                                 
33

 Hilbert‘s interprets the axioms of his geometry, aside from the parallel axioms, in much the same manner as 

Beltrami had prior to him: he limits the points, lines, and planes to the interior of a sphere and suggests we 

―define the congruences of this geometry by aid of such linear transformations of the ordinary geometry as 

transform the fixed sphere into itself‖ [Hilbert 1903, 19]. 
34

 ―By suitable conventions, we can make this ―non-euclidean geometry‖ obey all the axioms of our system 

except the axiom of Euclid (group III). Since the possibility of the ordinary geometry has already been 

established [by Hilbert‘s consistency proof], that of the non-euclidean geometry is now an immediate 

consequence of the above considerations‖ [Hilbert 1903, 19]. 



64 

 

the proofs of two crucial results: the theorem of Desargues and the theorem of Pappus 

[...]. Unlike independence, however, the completeness of the system of axioms is not a 

property that Hilbert knew how to verify formally, except to the extent that, starting 

from the given axioms, he could prove all the theorems he was interested in.
35

 

 

In contrast, then, to the modern understanding of model-theoretic completeness, Hilbert‘s 

view of the concept was a rather more informal, historical one. In his view, guided no doubt 

by his understanding of the non-arbitrary, intuitive basis of any interesting axiom system (in 

our case, guided by his view that the axioms of geometry are based upon our intuition of 

space), an axiom system is complete just in case it captures the theorems one would have 

liked it to capture and no anti-theorems. Even if Hilbert‘s axioms for Euclidean geometry 

were consistent and mutually independent, they would still be unsatisfying if they did not, 

for instance, include Pascal‘s theorem among their consequences. Completeness in Hilbert‘s 

sense is guided by a pre-systematic understanding of the nature and extent of the 

mathematical field one aims to describe via a particular axiomatization. If some interesting 

or important element of this field is not deducible from a particular axiomatization, that is 

good reason to prefer another axiomatization. A complete axiom system is one that captures 

all the theorems that it is (at least tacitly) intended to capture. Which theorems are intended 

is a matter determined largely by the historical trajectory of the field in question up to that 

point. 

 

1.4.4 Categoricity 

 

The explicit model-theoretic notion of categoricity
36

 was not yet developed at the time of the 

publication of the Grundlagen. Nevertheless, Hilbert‘s addition, in the French translation of 

the Grundlagen (and all subsequent editions) of a further axiom, the axiom of completeness, 

seems to reveal that he intended his geometrical axiom system to be something like 

‗categorical‘ in our modern sense. But, again, there are important differences between 

Hilbert‘s view and the standard model-theoretic view of today. 

 

Hilbert introduces the early version of this axiom as follows: 

 

REMARK. To the preceding five groups of axioms, we may add the following one, 

which, although not of a purely geometrical nature, merits particular attention from a 

theoretical point of view. It may be expressed in the following form: 

 

AXIOM OF COMPLETENESS [Vollständigkeit]. To a system of points, straight lines, 

and planes, it is impossible to add other elements in such a manner that the system 

thus generalized shall form a new geometry obeying all of the five groups of axioms. In 

other words, the elements of geometry form a system which is not susceptible of 

extension, if we regard the five groups of axioms as valid.
37

 

                                                 
35

 [Corry 2004a, 95-96]. 
36

 In model-theoretic terms, an uninterpreted theory is said to be categorical iff there exists an isomorphism 

between any two models of the theory. 
37

 [Hilbert 1950, 16]. 



65 

 

 

Unlike the other axioms, each of which states some property of or relation among the 

elements of the system (e.g., lines, points, or planes), this axiom states something about the 

axiomatized system itself, namely that no further elements can be added to the system while 

still maintaining consistency. This axiom, in contrast to the others, has the appearance of 

what we would now call a metatheoretic claim. What is its function here? Hilbert‘s system 

was intended, as we‘ve seen, to present a simple and complete axiomatization of the known 

results of Euclidean geometry, all of which seems perfectly accomplished prior to the 

addition of this further axiom of completeness, so what purpose does it serve? 

 

The name itself sheds light on Hilbert‘s own conception of completeness, which, as we‘ve 

seen, was not obviously identical with any modern concept of deductive or semantic 

completeness. Hilbert‘s intent was to capture all the theorems of geometry, but, further, to do 

so in a way which accurately corresponded to the intuitive understanding of those theorems 

throughout the history of geometry. We can understand Hilbert‘s goal as one of 

systematization: he attempts to clear away irrelevant features of mathematical history in 

order to uncover the underlying logical and mathematical structure—the really interesting 

parts. To do so in a purely abstract manner, as postulation theorists like Huntington, Veblen, 

Hausdorff, and others would later do,
38

 was not Hilbert‘s goal at all; his interest was to 

formalize the properly mathematical part of geometrical history, and this history included 

analytic geometry and its concomitant connection between the real numbers and the line. 

 

What Hilbert‘s axiom attempts to formalize is the (often tacitly) accepted isomorphism 

between the real numbers and the points on a line. By rejecting models of his axioms which 

are not isomorphic to the real numbers (e.g., the ‗incomplete‘ model within the field Ω of 

algebraic numbers outlined above), Hilbert illustrates that his intention is not simply to 

capture the abstract formal structure of Euclid‘s geometry—which is perfectly well 

accomplished within the subfield Ω, and without the completeness axiom—but, also, to 

capture the precise way in which that structure was actually understood and employed by the 

mathematical community since at least the time of Descartes. Without the axiom, we might 

add historically and geometrically irrelevant bits to an ‗incomplete‘ model like the subfield 

Ω. By adding the completeness axiom, Hilbert explicitly forbids such extension, deliberately 

pushing his reader toward 
3
 for historical and intuitive reasons. Here again we can see 

Hilbert‘s divergence from the contemporary notion of categoricity: his intent here is not to 

ensure that all models of the axioms are isomorphic with one another, but, instead, to insist 

as carefully as possible that we employ the historically established interpretation of analytic 

geometry. Thus, though Hilbert‘s axiom of completeness might push his axiomatization 

toward categoricity, this was clearly not his own intention.
39

 

 

                                                 
38

 Cf., e.g., [Huntington 1904], where a ―purely mathematical or abstract point of view‖ is employed in order to 

―consider the construction of a purely deductive theory, without regard to its possible applications‖ [Huntington 

1904, 288]. He then speaks of his postulates (for an ‗algebra of logic‘) as ―simply conditions arbitrarily 

imposed on the fundamental concepts‖ of the theory [Huntington 1904, 290]. 
39

 [Awodey and Reck 2002, 8-13] gives a more thorough account of Hilbert‘s views in relation to the modern 

notion of categoricity. 



66 

 

The axiom of completeness is different in kind from the other axioms and axiom groups. 

While the remaining axioms formalize properties and relations related to the elements of the 

geometrical system, the axiom of completeness dictates that the system itself should obey a 

certain criterion, namely, that any interpretation of the system must have the property of 

being non-extendible. As we will see in detail when we turn to the correspondence between 

Frege and Hilbert, Hilbert‘s use of this isolated metatheoretic statement within his axiomatic 

system is illustrative of a general lack of precision regarding logical and model-theoretic 

concepts. It is precisely this sort of precision that Frege, with his penetrating analyses of 

basic logical concepts, would bring to the table. 

 

1.4.5 Simplicity 

 

Unlike consistency, independence, and completeness, the property of simplicity has only 

rarely been formalized, even today. Though no clear criterion for simplicity has won the day 

even in the contemporary mathematical world, the American postulation theorist E. H. 

Moore came close to providing one. In a 1902 paper on the definition of abstract groups, 

Moore wrote that  

 

From the standpoint of abstract logic the canons of relative simplicity of equivalent 

definitions by sets of postulates are not well established. Perhaps the only established 

canon is this, that a definition is simplified by the omission of a group of postulates 

logically deducible from the remaining postulates. One is tempted to add this, that 

every postulate of a desirably simple definition shall be a simple statement, that is, a 

single and not a multiple statement. The difficulty here would arise in the precise 

formulation of the terms of this second canon, especially in view of the fact that the 

same statement may be made in various forms. At least, a definition is simplified by 

the substitution, for a postulate consisting of an aggregate of independent statements, 

of those statements as distinct postulates. Further, one may add as a third canon this, 

that of two definitions one with the smaller number of postulates is the simpler. As to 

this canon the case now in question [i.e., the definition of the abstract group concept] 

seems to show that the definition with the larger number of independent postulates 

may reveal more immediately the fundamental properties of the object of definition. It 

is, of course, evident that the task of proving the independence of the postulates 

presumably increases with the number of postulates.
40

  

 

The sort of precision sought by Moore here was not much pursued by other mathematicians, 

and simplicity has since remained a largely informal matter within mathematics.  

 

Despite the general lack of precision regarding the notion of simplicity, most 

mathematicians, Hilbert included, take simplicity to be an extremely desirable property of a 

theory or axiomatization. Simplicity, in this respect, is largely an aesthetic notion, though 

one still capable of directing research and methodology. Recall, for instance, that the relative 

complexity of the parallel postulate when compared to Euclid‘s remaining axioms was the 

                                                 
40

 [Moore 1902b, 488-489]. E.V. Huntington makes similar, though less detailed remarks at [Huntington 1904, 

290]. 



67 

 

chief reason that a proof of that postulate was so ardently sought; there is something 

aesthetically displeasing to the mathematical temperament about an unnecessarily 

complicated theory or technique. Leo Corry is again helpful in understanding Hilbert‘s 

particular view of this notion (which agrees broadly with Huntington‘s account). He writes 

that 

 

The [...] requirement [of] simplicity [...] complements that of independence. It means, 

roughly, that an axiom should contain ―no more than a single idea.‖ [...] Nevertheless, 

it was neither formally defined nor otherwise realized in any clearly identifiable way 

within [Grundlagen der Geometrie]. The ideal of formulating ―simple‖ axioms as part 

of this system was present implicitly as an aesthetic desideratum that was not 

transformed into a mathematically controllable feature.
41

 

 

One way of reading this view of simplicity into Hilbert‘s Grundlagen is to take each of his 

axioms as the expression of a single facet of spatial intuition, while the axiom groups are 

collections of axioms which express facets of spatial intuition related to some one more 

general concept (e.g., incidence or order). Thus, the axioms of Hilbert‘s third axiom group 

all express distinct aspects of space related to the concept of congruence. For instance, 

Axiom III, 1 states that ―If A, B are two points on a line a, and A′ is a point on the same or 

on another line a′ then it is always possible to find a point B′ on a given side of the line a′ 

through A′ such that the segment AB is congruent or equal to the segment A′B′.‖
42

 And 

Hilbert himself writes that ―This axiom requires the possibility of constructing segments. Its 

uniqueness will be proved later.‖
43

 So, while we have here a single aspect of spatial 

congruence (i.e., that it is possible to construct congruent line segments in space), it is a 

further and separate matter to prove the uniqueness of segment construction. Similarly, 

Hilbert presents the additivity of segments and the transitivity of congruence in separate 

axioms. Taken together, these axioms represent the presentation of the single concept of 

spatial congruence, while taken singly they each represent a single aspect (a ‗characteristic 

mark,‘ as Frege might say) of that concept. Assuming that our axiom system is complete in 

Hilbert‘s sense (i.e., it captures all the theorems we want it to and no anti-theorems), this 

criterion of simplicity ought to ensure that the axiomatization possesses only as many 

axioms as are required. For, if each axiom is required to clearly represent a single idea, it 

will (ideally) be clear when two axioms express the same idea, thus enabling the easy 

removal of redundant axioms. The result of such a complete and simple system should be a 

minimal collection of axioms from which the maximum number of theorems may be 

derived.
44

 

 

In Hilbert‘s work we can see very clearly the shift away from a geometry which was oriented 

towards the properties of particular sorts of geometrical objects (points, lines, planes, etc.) 

and towards an examination of the structure of axiomatic systems. Hilbert‘s view is an 

interesting hybrid of these two approaches: he maintains the utility of intuition as a heuristic 

                                                 
41

 [Corry 2004a, 95]. 
42

 [Hilbert 1971, 10]. 
43

 [Hilbert 1971, 10]. 
44

 Bolzano offers a similar view at [Bolzano 1837, §221.2, note]. 



68 

 

for determining which axiom sets are worthy of mathematical analysis, while eschewing any 

appeal to intuition in his attempt to provide satisfactory axiomatic characterizations for 

various geometric structures. The main focus of his work, however, was to prove theorems 

about certain geometric structures rather than proving theorems within them.  

 

Hilbert‘s main interest is not in triangles, for example, but with the relationship between 

theorems pertaining to triangles and the postulates of a particular axiomatization. Similarly, 

he is not focussed upon the content of Archimedes‘ axiom as much as its relation to the other 

axioms of Euclidean geometry. As we will see in more detail below, this shift from an 

object-oriented approach toward a structure-oriented one was not without its difficulties. 

 

From Hilbert‘s views I would now like to turn to those of Frege. Both men had complex 

relationships with prior mathematical and philosophical traditions: Hilbert took on some of 

the intuitive elements of geometry with his understanding of the nature of axioms, but 

insisted upon the purely deductive character of geometric proofs and theorems. For his part, 

Frege held more conservative views about the nature of elementary geometry, but was a 

radical innovator when it came to logical precision and mathematical rigor.  

 

§2. Frege 

 

2.1 Frege‘s work in context 

 

Frege had, as early as the late 1870s, become very interested in presenting complex 

mathematical concepts and proofs in a more perspicuous and rigorous manner. As we‘ve 

seen, Hilbert‘s work in geometry attempted to provide a coherent axiomatic framework 

capable of representing a whole host of new and old geometrical results; Hilbert‘s work 

rather successfully eliminated (or, at least limited) geometry‘s reliance upon the intuition of 

space. This work was an evacuation of geometry‘s logically impertinent trappings, resulting 

in a pared-down version of the age-old truths of Euclid. Frege, for his part, had focussed 

more heavily on clarifying the nature and logical structure of certain basic mathematical 

concepts and modes of reasoning (e.g., the concept of number and mathematical induction). 

 

While Frege was quite familiar with the revolutions taking place throughout geometry—he 

wrote a dissertation at Göttingen about geometry in the early 1870s,
45

 as well as a few short 

reviews and a paper dealing with geometrical issues
46

—his views were for the most part 

those of a staunch geometric conservative. He was, for instance, convinced very late in his 

career of the intuitive basis of the truth of the axioms of Euclid‘s geometry, and he 

                                                 
45

 [Frege 1873], supervised by Ernst Schering (1824-1897) who, incidentally, was the editor of Gauss‘ 

collected works. Frege was also quite familiar with the work of the innovative algebraic geometer Alfred 

Clebsch (1833-1872), having attended his lectures on analytic geometry. 
46

 The reviews are: [Frege 1877] (reviewing a work on linear geometry and its applications) and [Frege 1880], 

which reviews a textbook in analytic geometry; the paper is [Frege 1878], which discusses the possibility of 

conceiving of triangles as complex quantities. Prior to his exchanges with Hilbert and Korselt, we find Frege 

returning only briefly to geometrical matters in [Frege 1884a], the text of an 1883 lecture on plane geometry; 

though he did teach geometry, which likely kept him abreast of developments in projective geometry (in which 

he showed interest throughout his career).  



69 

 

characterized these axioms as necessary truths about the physical world.
 
The following 

fragment on Euclidean geometry, which remained unpublished during his lifetime, is 

illustrative of his general approach to geometric questions: ―In Euclidean geometry certain 

truths have traditionally been accorded the status of axioms. No thought that is held to be 

false can be accepted as an axiom, for an axiom is a truth. Furthermore, it is part of the 

concept of an axiom that it can be recognized as true independently of other truths.‖
47

 

Elsewhere, he offered some rather unconvincing arguments about the indispensability of 

Euclidean geometry for getting around in the world:  

 

Space, according to Kant, belongs to appearance. It would be possible for it to be 

differently represented by another rationality entirely different from us. True, we 

cannot even know whether it appears the same to one person as to another; for we 

cannot lay one intuition of space next to another in order to compare them. There is 

nevertheless something objective contained therein; everyone recognizes the same 

geometrical axioms even if only by the fact that he must do so in order to find his way 

in the world. What is objective in it is what is law-governed, conceivable, and capable 

of being judged, what is expressible.
48

 

 

That Frege‘s claims for the experiential or pragmatic necessity of Euclidean geometry here 

miss their mark is shown by the fact that even his contemporaries (e.g., Poincaré and 

Helmholtz) were quite capable of illustrating the compatibility of everyday experience with 

geometries other than Euclid‘s.
49

 In addition to this pragmatic consideration in favour of the 

traditional view of Euclidean geometry, Frege also conceived of physical space as Euclidean 

in nature and he believed we were able to grasp basic truths pertaining to that space (i.e., 

Euclid‘s axioms) a priori by way of pure intuition. Thus, in most respects, he was an 

orthodox Kantian when it came to the philosophy of geometry. Frege‘s fairly traditional 

views might partially explain his initial reaction to Hilbert‘s work on the foundations of 

geometry, but neither that reaction nor its continuation throughout his correspondence with 

Hilbert (and the subsequent series of articles written in response to Korselt) can be fully 

explained simply in light of those views. Indeed, the most interesting and far-reaching of 

Frege‘s criticisms of Hilbert‘s work have very little to do with his own geometrical views 

and quite a lot to do with his more impressive re-imagining of the nature of logic and the 

relations between mathematics and logic.
50

 

 

Before we turn to Frege‘s criticisms of Hilbert I will discuss some elements of one of the 

main mathematical traditions which preceded and informed Frege‘s own work in logic and 

the foundations of mathematics. As we will see, it is the trajectory partially determined by 

this tradition, rather than Frege‘s own views of the nature of geometry, which informed 

many of the most important of his responses to Hilbert. 

 

 

                                                 
47

 [Frege c. 1900, 169]. 
48

 [Frege 1884b, §26].  
49

 Cf., e.g., [Helmholtz 1977]. 
50

 See [Rusnock 1993] for a more detailed argument along these lines. 



70 

 

 

 

2.1.1 The rigorization of mathematics 

 

Alongside the structure-focussed, axiomatic approach to geometry which had developed 

from the late 18
th

 century and throughout the 19
th

 century, there was another equally radical 

and influential movement within analysis and number theory which focussed on tightening 

up mathematical argumentation and providing stricter definitions of basic concepts. Many 

elements of Frege‘s work on the foundations of mathematics and logic belong squarely 

within this tradition and it is worthwhile to outline, at least briefly, some of the aims and 

methods which characterize it. This discussion should help to illustrate that Frege‘s 

criticisms of Hilbert on the foundations of geometry were not merely the result of his own 

somewhat outmoded, Kantian view of geometry but, rather, that they developed mainly as a 

result of his preoccupation with the central themes of a nineteenth century tradition of 

mathematical rigor. 

 

The modern project of injecting rigor into mathematics began in earnest at the close of the 

18
th

 century, and was developed, for the most part, in response to the conceptual morass 

introduced into mathematics by the (then) obscure notions of the infinitesimal calculus. 

From the time of Newton and Leibniz, there had been an uneasy feeling surrounding the use 

of unclear notions like fluxions, derivatives, second and third differentials, vanishing 

quantities, etc. One of the better known (and certainly the most amusing) of the polemics 

deployed against these concepts can be found in Bishop George Berkeley‘s The Analyst. 

There Berkeley famously derides the use of fluxions in the calculus: ―And what are these 

Fluxions? The Velocities of evanescent Increments? And what are these same evanescent 

Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet 

nothing. May we not call them the Ghosts of departed Quantities?‖
51

 In conjunction with his 

mocking tone, Berkeley also offered a detailed critique of several of the central notions of 

the calculus. While he dismissed extant attempts to provide a conceptual basis for the 

successes of the calculus, he did not dismiss those successes themselves. Instead he 

explained them as the felicitous result of two competing errors. While considering the 

obscure solutions to several problems related to the determination of tangents for parabolas, 

Berkeley writes that ―the two errors being equal and contrary destroy each other; the first 

error of defect being corrected by a second error of excess,‖
52

 and, further, that  

 

If you [i.e., the mathematician employing the methods of the calculus] had committed 

only one error, you would not have come at a true Solution of the Problem. But by 

virtue of a twofold mistake you arrive, though not at Science, yet at Truth. For Science 

it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing 

how or by what means.
53

 

 

                                                 
51

 [Berkeley 1734, paragraph XXXV]. 
52

 [Berkeley 1734, paragraph XXI]. 
53

 [Berkeley 1734, paragraph XXII]. 



71 

 

Despite the fact that the results of the calculus were quite often correct, Berkeley resisted 

treating the methods of fluxions and vanishing quantities as scientific ones for the simple 

reason that no one could explain how these methods managed to arrive at truth. Without 

some form of conceptual clarification, the success of the calculus appeared to result only 

from an obscure and inexplicable fumbling toward the truth, a kind of mathematical water-

dowsing. Accordingly, its results could not be called scientific, as it was not at all clear how 

or even if those results could be grounded in scientific reason. 

 

While many mathematicians were more than willing to employ the notions of the calculus as 

long as they delivered results, the scathing criticisms offered by Berkeley and others 

eventually began to weigh on the more philosophically inclined members of the 

mathematical community. Throughout the latter half of the 18
th

 century, there was a great 

push to remedy the situation. Two general approaches were taken: one which attempted to 

deliver the same results as the calculus without employing its more troublesome concepts, 

and one which sought to clarify those concepts and place them on a sounder philosophical 

and scientific footing. Joseph Louis Lagrange (1736-1813), one of the foremost 

representatives of the former method, published an influential text on the theory of functions, 

whose subtitle boasted that it was ―redeemed from all considerations of the infinitely small, 

from vanishing quantities, from limits and from fluxions, and reduced to the algebraic 

analysis of finite quantities.‖
54

 Lagrange, as many others, characterized the new demand for 

rigor as a return to the spirit of precision exemplified by the ancients,
55

 and was so 

concerned with the incoherent conceptual foundations of the calculus that, while serving as 

the director of the mathematical section of the Berlin Academy, he set the following prize 

problem for the year 1784: 

 

It is widely known that advanced geometry regularly employs the infinitely great and 

the infinitely small ... The Academy, therefore, desires an explanation of how it is that 

so many correct theorems have been deduced from a contradictory supposition, 

together with... a truly mathematical principle that may be substituted for that of the 

infinite.
56

 

 

The feeling evidenced here and throughout the mathematical works and correspondence of 

the second half of the 18
th

 century was that there had to be some explanation for the great 

successes, both mathematical and physical, of the calculus. The concepts of the calculus, 

however murkily understood, indisputably led to valid results, and no one could 

convincingly explain how or why. The attempts to avoid the issue by founding the calculus 

without the use of its most troublesome concepts ultimately had less appeal than those 

attempts based on conceptual clarification (it is indeed a rare case where mathematicians 

give up even shoddy tools if they produce beautiful results). It is largely to this work of 

clarification, begun in the late eighteenth century, that we owe the advances in mathematical 

                                                 
54

 [Lagrange, quoted in Grattan-Guiness 2000, 100]. 
55

 ―L‘objet de cet Ouvrage est de donner la théorie des fonctions, considérées comme primitives et dérivées, de 

résoudre par cette théorie les principaux problèmes d‘Analyse, de Géométrie et de Mécanique, qu‘on fait 

dépendre du Calcul différential, et de donner par là à la solution de ces problemes toute la rigueur des 

démonstrations des Anciens‖ [Lagrange 1867, tome 9, 20] 
56

 [Lagrange 1784, quoted in Grattan-Guiness 2000, 101]. 



72 

 

rigor made in the following century by Cauchy, Gauss, Abel, Dirichlet, Bolzano, and others. 

These mathematicians achieved a much stricter understanding of the demands to be met by 

any piece of rigorous mathematical reasoning, one that even Berkeley would likely have 

found unexceptionable.  

 

Bolzano, whose work remained largely unknown during the period, nevertheless provides us 

with an exemplary philosophical account of proper mathematical reasoning in his essay on 

the mathematical method.
57

 Bolzano, like Frege later, was interested in eliminating the 

psychological elements of mathematics in favour of a focus on the objective logical relations 

which held amongst its propositions (Sätze an sich), irrespective of their historical 

epistemological status. Thus, in the works of Bolzano and the other rigorizers, we find an 

insistence upon proving even obvious theorems
58

—a tradition which had slackened 

considerably since the time of the Greeks, perhaps because of the rapid advances in 

mathematical applications after the emergence of the integral and differential techniques of 

Newton, Leibniz, and the Bernoullis. Bolzano was critical of the lax treatment of definition 

within contemporary mathematical reasoning, and constructed a much stricter theory of 

definition than the relatively informal approach that had become popular by the early 19
th

 

century. Inferential gaps, too, were to be filled by the newly rigorized approach: hidden 

assumptions were to be uncovered and made explicit, superfluous appeals to intuition were 

to be dispensed with, and any propositions susceptible of proof (obvious or not) were to be 

proved. If a proposition was assumed but not proved, it was to be expected that its 

unprovability was to be demonstrated or, at least, that this would remain an open problem for 

research. 

 

The appreciation for strict forms of definition and gapless chains of inference exhibited by 

Bolzano and others in the early part of the 19
th

 century was extended considerably in the 

latter half of that century by Karl Weierstrass (1815-1897) and the talented circle of 

mathematicians which formed around him in Berlin. The members of the school were 

instrumental in the so-called arithmetization of analysis, a process which involved 

reconstructing many of the theorems and concepts of analysis in terms of a group of 

acceptable techniques which avoided as much as possible any appeal to (geometric) intuition 

and the murkier concepts of the early calculus. Weierstrass and his school developed and 

popularized rigorous definitions of a number of basic mathematical concepts, chief among 

them were the (ε, δ) definition of a limit, of uniform convergence, and of the uniform 

continuity of a function.
59

 

                                                 
57

 [Bolzano 2004b]. 
58

 Bolzano‘s ‗purely analytic‘ proof of the intermediate value theorem (i.e., [Bolzano 1817]) is an excellent 

example of the application of rigorous forms of reasoning and definition to a seemingly obvious theorem. 
59

 Though Cauchy was the first to offer this form of the definition of a limit (see [Cauchy 1823, 44]), 

Weierstrass and his circle employed it frequently and in a variety of contexts, with the effect that it was 

disseminated more widely; for the Weierstrassian definition of uniform convergence (in a region), see 

[Weierstrass 1880, article 1]; for that of the uniform continuity of a function, see [Heine 1870, 361]. 

(Interestingly, an earlier, almost equally rigorous definition of uniform continuity was implicitly given by 

Bolzano in the 1830s in his Functionenlehre ([Bolzano 1930, §13; English translation available in [Bolzano 

2004a]—some relevant emendations can be found in a manuscript, also available in translation in [Bolzano 

2004a, 573-589]). For a discussion of Bolzano‘s priority in the matter, see [Rusnock 2000, 166-172] and 

[Rusnock and Kerr-Lawson 2004]. 



73 

 

 

During the period of the Weierstrass school‘s influence at Berlin, a rival tradition arose at 

Göttingen centered around Bernhard Riemann. The Göttingers were noted less for their 

strictness and more for their mathematical creativity. Like the early workers in the calculus, 

those following in Riemann‘s wake were less interested in shoring up the rigor of their new 

topological and functional investigations than in extending as far as possible new 

mathematical vistas. Göttingen, in contrast to Weierstrass‘s Berlin, was noted for its more 

liberal use of geometrical and other forms of intuition as a fruitful generator of mathematical 

material. Much of German mathematics at the time was divided between the two schools, to 

the point that very little cross-pollination by way of publications, correspondence, or 

colloquia occurred. Only towards the end of the nineteenth century, when Klein and Hilbert 

were in ascendency at Göttingen did the two traditions begin to run together—though even 

then there were firm distinctions of style and method.
60

 

 

By all appearances, Frege was well-disposed to become a card-carrying member of the 

Riemann school. He completed his doctoral work on geometry in Göttingen, he was quite 

familiar with the work of Clebsch (a highly influential Göttinger), and later worked on the 

geometric representation of complex quantities, a topic which combined two key elements of 

the Göttingen tradition. Despite these connections, however, Frege‘s more mature work 

focussed almost exclusively on the foundations of arithmetic and elementary number theory, 

topics considerably more Weierstrassian than Riemannian.
61

  

 

Even while Frege was clearly inspired by the Weierstrassian project of rigor, he brought a 

peculiarly logic-centred approach to this project which was absent in the Berlin school. 

While the Weierstrassians were concerned to provide analysis with rigorous conceptual 

foundations, Frege was interested in going further by providing strict definitions which 

would reduce complex mathematical notions to purely logical ones. Thus we find, for 

instance, his attempt to define number in terms of the extensions of certain concepts, which 

he understood to be properly logical objects. While the details of Frege‘s efforts to clarify 

the nature of particular mathematical results are not directly relevant to our concerns here, 

his broad conception of the proper nature of mathematical reasoning and definition are. 

Many of his criticisms of Hilbert‘s views in the Foundations of Geometry, for instance, have 

to do with what Frege perceived as failures to observe the proper distinctions between 

definitions, axioms, and explanations, as well as Hilbert‘s improper understanding of the 

nature of definition more generally. Accordingly, let‘s take a look at what Frege has to say 

about the nature of logic, and, following that, what he has to say about Hilbert‘s allegedly 

inadequate understanding thereof. 

 

                                                 
60

 For an account of the Göttingen school with particular emphasis on Klein and Hilbert, see [Rowe 1989]; for 

the Berliners, see [Biermann 1973]. 
61

 Though Frege‘s earliest work (e.g., his dissertation of 1874) dealt with geometrical issues, and he taught 

several courses relating to various geometrical topics, it is apparent that the bulk of his own mental energy was 

expended on his project of rigorizing arithmetic by providing it with solid foundations in logic. This project 

was clearly related to (though Frege was highly critical of) similar efforts by the Weierstrassians. For a view 

which emphasizes Frege‘s connection to the Riemannian tradition at Göttingen, where Frege did his graduate 

work, (rather than the Weierstrass-Berlin tradition), see [Tappenden 2006]. 



74 

 

 

2.2 Frege on the nature of logic 

 

In a fragment from what appears to have been intended as a logic textbook, Frege writes that 

 

Logic is concerned only with those grounds of judgement which are truths. To make a 

judgement because we are cognisant of other truths as providing a justification for it is 

known as inferring. There are laws governing this kind of justification, and to set up 

these laws of valid inference is the goal of logic.
62

 

 

Thus logic is a science which considers the laws of inference in their relation to truths. In the 

modern period, many logicians took this to involve distinguishing various types of 

judgments and the possible inferential relations between them. For Frege, by contrast, the 

aspects of inference in which logic is interested are not at all psychological, or even 

empirical, and hence logic proper is not concerned with the peculiarities of human judgment. 

(At which temperature humans best infer or with which mental images were not, for Frege, 

matters for logic to decide). In the same fragment, he writes the following: 

 

The subject-matter of logic is therefore such as cannot be perceived by the senses and 

in this respect it compares with that of psychology and contrasts with that of the 

natural sciences. Instincts, ideas, etc. are also neither visible nor tangible. All the same 

there is a sharp divide between these disciplines [i.e., logic and psychology], and it is 

marked by the word ‗true‘. Psychology is only concerned with truth in the way every 

other science is, in that its goal is to extend the domain of truths; but in the field it 

investigates it does not study the property ‗true‘ as, in its field, physics focuses on the 

properties ‗heavy‘, ‗warm‘, etc. This is what logic does. It would not perhaps be beside 

the mark to say that the laws of logic are nothing other than an unfolding of the content 

of the word ‗true‘. Anyone who has failed to grasp the meaning of this word—what 

marks it off from others—cannot attain to any clear idea of what the task of logic is.
63

 

 

The references to psychology here are not incidental, as a variety of psychologistic views of 

both logic and mathematics were popular during Frege‘s time, and a considerable portion of 

his work was dedicated to undermining these views and building up his own alternative 

conception.
64

 Psychologism in logic, as the name suggests, is the view that logical laws are 

subject to, deducible from, or in some way reducible to psychological laws, and that logic 

itself studies the particular ways in which human beings infer and make judgments. Instead 

of the laws regulating ideas and the association of ideas found at the heart of psychologistic 

accounts of logic, however, Frege turned to the study of the laws of truth itself. Instead of 

                                                 
62

 [Frege ?1879-1891, 3]. 
63

 [Frege ?1879-1891, 3]. 
64

 Psychologism in logic has been popular in a variety of forms throughout the history of the discipline; in 

Frege‘s time, the Logic of John Stuart Mill ([Mill 1843]) was one of the major sources of the view. In addition 

to Mill, a number of German logicians held psychologistic views, notable among them were Christian von 

Sigwart (1830-1904) and Benno Erdmann (1851-1921). One of the better-known disputes regarding 

psychologism in mathematics can be found in Edmund Husserl‘s psychologistic views on arithmetic in 

[Husserl 1891] and Frege‘s highly critical review of Husserl [Frege 1894]. 



75 

 

studying how humans actually infer (which is, more often than not, erroneously) Frege 

endeavoured to study the nature of correct inference and the relations between inference and 

truth. Since logic, properly construed, is ―the science of the most general laws of truth,‖
65

 

and since truth is not, for Frege, a subjective, mental property, logic in its proper form 

cannot be concerned with the subjective, mental realm.  

 

Frege‘s own view of logic involved a shift away from the subjective ideas of psychologism 

towards what he (rather unfortunately) calls ‗thoughts‘. We will have occasion in the 

following chapter to return in more detail to Frege‘s ‗thoughts‘; here it is sufficient to 

characterize thoughts as those things which are true or false.
66

 For Frege, truth was not at all 

a subjective property, nor, somewhat confusingly, were thoughts mental entities. Since logic 

deals with truth and the relation between thoughts and truth (neither of which are 

psychological or subjective in nature), it cannot be a subjective science. Nevertheless, Frege 

admits that our interaction with logic has to occur within the spatiotemporal world we 

inhabit and, thus, that any thoughts which we endeavour to study scientifically must be 

cloaked in something tangible, e.g., some language or mental process.  

 

Because of its public character, language is invariably the means by which scientific results 

are communicated and formulated. Unfortunately for science, however, the relationship 

between thoughts and their linguistic garb is not always a clear or helpful one. Ordinary 

language can obscure the logical structure of thoughts; indeed, it often systematically 

misleads the unwary away from the true nature of proper inference. Completely different 

phrases may express thoughts which are, from a logical point of view, identical. Similarly, 

identical phrases, uttered or written in different contexts, may express different thoughts. In 

many cases, too, ordinary language is insufficiently precise to pin down a single meaning for 

a particular phrase, such that it may seem to express several different thoughts at once. Frege 

sought to remedy this situation through the creation of a logically perspicuous language. The 

result was his Begriffsschrift, or concept-script.
67

 Frege‘s formal language was based upon a 

cluster of rigorously circumscribed logical concepts and rules of inference.
68

 From this basic 

stock, Frege held that the logical structure of thoughts and their relations to each other could 

be clearly presented while most logically irrelevant properties (i.e., those which do not affect 

the truth or falsity of thoughts or the validity of inferences) could be ignored. In addition to 

the purely symbolic work, the construction of this language also required a significant 

amount of conceptual work: Frege had to isolate the logical concepts from those alien to 

logic, he also had to avoid the ambiguity and obscurity which prevented logical perspicuity 

                                                 
65

 [Frege 1897, 128]. 
66

 ―I mean by ‗a thought‘ something for which the question of truth can arise at all‖ [Frege 1918, 4]. Though 

Frege‘s idiosyncratic use of the word ‗thought‘ to represent the objective content of a judgment has not caught 

on, the basic idea has persisted within contemporary philosophy under the heading ‗proposition‘. Frege 

reserves the German term ‗Satz‘ for what are now often referred to in English as sentences, i.e., verbal or 

written expressions of objective propositions (or Fregean thoughts). 
67

 [Frege 1879]. 
68

 The basic symbols of Frege‘s first system, which express his basic concepts, are: the judgment or assertion 

stroke (§2), the conditional stroke (§5), the negation stroke (§7), the identity symbol (§8), letters which serve as 

variables (§§10-12), and the symbol for generality (§11). In addition to the specific notions expressed by these 

symbols, Frege also discusses the metatheoretic concepts of ‗function‘ and ‗argument‘ (§9). 



76 

 

within ordinary language. To this end, he developed some rather strict ideas concerning the 

relations between language and logic, and carefully distinguished between thoughts and their 

various possible expressions.  

 

2.3 The role of a scientific language 

 

In an interesting paper defending and elaborating upon the role of his Begriffsschrift, Frege 

opens with the following paragraph: 

 

Time and again, in the more abstract regions of science, the lack of a means of 

avoiding misunderstandings on the part of others, and also errors in one‘s own thought, 

makes itself felt. Both [shortcomings] have their origin in the imperfection of 

language, for we do have to use sensible symbols to think.
69

  

 

Thus we find ourselves in something of a quandary. Logic, as noted, studies relations, 

objects, and properties which are not at all sensible. But, nevertheless, given that we are 

experientially bound to our spatiotemporal world, we must employ spatiotemporal means in 

our study of logic. That this is possible, Frege does not doubt, but neither does he explain the 

possibility of interaction between an objective, intangible, non-physical realm and our 

physical world. Despite the lack of any detailed metaphysical or ontological speculation on 

the nature of this relation, Frege does discuss in some detail specific ways in which slippage 

between the two realms can occur and lead us into error. Ordinary language, for instance, is 

very often ambiguous: sentences or individual words can possess multiple meanings, some 

of which can be determined from context, some of which cannot. In addition to being 

ambiguous, words and sentences may also be vague, such that it is not at all clear which 

thoughts (if any) they make tangible. In ordinary, everyday speech ambiguity and vagueness 

are mostly harmless. However, when it comes to scientific uses, where precision and the 

pursuit of truth are paramount, the situation becomes intolerable.
70

  

 

Since ordinary language drives us away from the clarity and precision required to investigate 

the realm of thoughts and to communicate our findings to others, science must either put 

language aside or construct linguistic tools which minimize as much as possible the slippage 

between thoughts and their expressions. And, since Frege admits that (at least for humanity) 

language is a necessary accompaniment of our apprehension of thoughts, science is in need 

of a logically precise language. Indeed, Frege notes that, even despite the obvious difficulties 

of employing the sensible to represent the non-sensible, the use of symbols allows us 

 

[...] to penetrate step by step into the inner world of our ideas and move about there at 

will, using the realm of the sensible itself to free ourselves from its constraint. Symbols 

have the same importance for thought that discovering how to use the wind to sail 

against the wind had for navigation. Thus let no one despise symbols!
71

 

                                                 
69

 [Frege 1882, 83]. 
70

 For an interesting analysis of Frege‘s views on vagueness in light of the modern differentiation between 

semantic, epistemological, and ontological vagueness, see [Ruffino 2003]. 
71

 [Frege 1882, 84]. 



77 

 

 

The importance accorded to symbols here is impaired by the imprecision in ordinary 

language. In order to better avoid this sort of imprecision in his Begriffsschrift, Frege 

developed a sophisticated account of the relations between language and the realm of 

thought. Most relevant for our purposes here are his views on the nature of definitions, 

axioms, and what he called ‗elucidations,‘ to which I now turn.  

 

2.4 Definitions 

 

For Frege, within a regimented language suitable for science, the occurrence of a sign within 

a sentence should indicate that the sense of that sign plays a role within the sense of the 

sentence (i.e., within the thought that the sentence expresses). Thus, a sentence ―is put 

together in such a way that parts of the thought correspond to parts of the sentence‖.
72

 Thus, 

in an ideal scientific language, the structure of thought is perfectly mirrored by the structure 

of the language itself. Of course, with the increasingly complex concepts and thoughts of 

modern science, such a goal is difficult (not to mention irritating) to maintain. To remedy the 

potential annoyance of perfect rigor, Frege allows for definitions as a kind of shorthand, 

though he does so in a way which he believes can maintain the intended correspondence 

between thought and language. He does this by treating definitions as arbitrary stipulations 

which give a simple sign a sense identical to the sense of a complex sign or group of signs.
73

 

Moreover, the sense of the complex sign must already be known. This apparently simple 

understanding of definition has a number of important consequences, particularly when 

framed in relation to Frege‘s other views concerning the nature of logic. 

 

Frege‘s definitions are, for instance, tautologous; given that a definition stipulates that the 

sense of a previously-senseless, simple sign is to be equivalent to that of an already-known 

complex sign, and given that the sense of this complex sign can be ‗read off‘ from the senses 

of its parts in a regimented language, there can be no doubt as to the truth of a definition. 

Nor can such a definition be informative in any positive sense. For all a definition says, on 

this view, is that a sign possesses its own sense, i.e. the sense that we have decided to give it. 

Once we accept a definition, it becomes a tautology. 

 

Another important consequence of Frege‘s understanding of the role of definition is that 

definitions are, at least logically speaking, superfluous. As we‘ve seen, Frege‘s definitions 

stipulate for some sign that it possesses a sense identical to the sense of another sign or 

group of signs. Definition thus occurs on a linguistic level, and definitions amount to 

linguistic conventions by which we agree to equate the senses of two distinct signs (or sign-

complexes). Now, for Frege, any two expressions which express identical senses are 

logically equivalent; indeed, he calls definitions ‗identities‘ or ‗logical identities‘ on several 

occasions.
74

 Thus, in any instance where we might employ one side of a definitional identity 

we could, without any change in truth conditions, employ the other. From a logical point of 

view, then, definition is a mere superfluity. Of course, psychologically or pragmatically, 

                                                 
72

 [Frege 1914, in PW 207]. 
73

 Cf. [Frege 1893, §27], [Frege 1914, in PW 210]. 
74

 At, e.g., [Frege 1893, §33], [Frege 1892, in PW 102], and [Frege 1914, PW 208]. 



78 

 

definitions are often indispensible aids to the continued progress of a given science. This is 

particularly true in mathematics and physics, where linguistically simple terms like ‗limit‘ or 

‗tensor‘ allow scientists access to the mental shortcuts required to produce results in the 

hyper-specialized fields of modern science. Without the ease of operation enabled by 

definitions and their linguistic condensation of complex concepts, the smooth functioning 

and forward progress of science would be greatly stunted. This is nowhere more evident than 

in contemporary mathematics and physics, where whole axiomatic systems are often 

condensed into a few words. Without the conceptual shorthand provided by well-defined 

terms for basic mathematical structures like groups or rings, it would be very difficult for 

mathematicians to work with more complex structures built up from these simpler 

constituents—e.g., cohomology groups or topological rings. Even at the level of group 

theory itself, it would be rather difficult, from both the typesetter‘s and the mathematician‘s 

point of view, to prove theorems pertaining to groups if one were forced to restate the group 

axioms at each occurrence of the term ‗group‘. While Frege might take definitions to be 

logically inert, he does not deny their necessity from a pragmatic point of view. 

 

An important corollary of the logical inertness of Fregean definition is that no such 

definition can be used to prove a proposition which would otherwise remain unprovable. 

Consider again that, for Frege, a definition is mere shorthand which should always be 

eliminable, however tedious such elimination might be in practice. On this view, in any 

instance where we employ a definition within a proof we could just as well employ the 

complex sign or group of signs the definition is intended to abbreviate. This view also tells 

against the informativeness of proper definitions, and against the use of ‗creative‘ definitions 

within mathematics. On this score, Frege writes the following: 

 

In the development of science it can indeed happen that one has used a word, a sign, an 

expression, over a long period under the impression that its sense is simple until one 

succeeds in analysing it into simpler logical constituents. By means of such an 

analysis, we may hope to reduce the number of axioms; for it may not be possible to 

prove a truth containing a complex constituent so long as that constituent remains 

unanalysed; but it may be possible, given an analysis, to prove it from truths in which 

the elements of the analysis occur. This is why it seems that a proof may be possible 

by means of a definition, if it provides an analysis, which would not be possible 

without this analysis, and this seems to contradict what we said earlier. Thus what 

seemed to be an axiom before the analysis can appear as a theorem after the analysis. 

But how does one judge whether a logical analysis is correct? We cannot prove it to be 

so. The most one can be certain of is that as far as the form of words goes we have the 

same sentence after the analysis as before. But that the thought itself also remains the 

same is problematic. When we think that we have given a logical analysis of a word or 

sign that has been in use over a long period, what we have is a complex expression the 

sense of whose parts is known to us. The sense of the complex expression must be 

yielded by that of its parts. But does it coincide with the sense of the word with the 

long established use? I believe that we shall only be able to assert that it does when 

this is self-evident. And then what we have is an axiom. But that the simple sign that 



79 

 

has been in use over a long period coincides in sense with that of the complex 

expression that we have formed, is just what the definition was meant to stipulate.
75

 

 

Frege‘s worry here is the apparent contradiction between his view that a definition cannot 

occur essentially in a proof and the claim that ―it may not be possible to prove a truth 

containing a complex constituent so long as that constituent remains unanalysed‖. Frege‘s 

explanation here, as I read it, is that, while certain definitions seem to perform an essential 

function within proofs, this is illusory, an after-effect of the mistaken belief that the complex 

sign expresses some simple sense. For, on Frege‘s view, either we grasp the sense of the 

complex sign—in which case the logical analysis of that sign‘s meaning only tells us what 

we already know and is thus superfluous—or we do not. If we do not grasp the sense of the 

sign, then we have strayed from Frege‘s vision of a perfectly scientific language in which 

every sign and sentence possesses a clear meaning. We might read Berkeley‘s attack on the 

‗unscientific‘ methods of the early calculus as an example of the sort of thing Frege wishes 

to highlight here: we may appear to be able to prove theorems with the modern definition of 

a limit that we might not have been able to prove while employing Newtonian fluxions, and 

yet we might view fluxions as somehow definable in terms of more precise modern notions. 

The problem, as Berkeley points out, is that the concept of a fluxion (among others) was 

insufficiently precise, such that we cannot really view the definition of integrals and 

derivatives in terms of more rigorous modern concepts to be the kind of ‗logical analysis‘ 

which Frege discusses in this passage. Thus, we do not gain access to new theorems purely 

by means of logical analysis, as the concepts purported to be analysed in this case were, in a 

sense, not full-fledged concepts at all, and, therefore, incapable of being analysed in a purely 

logical manner. None of this, of course, tells against the historical or epistemological 

usefulness of these sorts of definitions, nor against the epistemological connection between 

vague ‗concepts‘ like those of the early calculus and their more precise modern cousins. 

 

But, even granting that we understand the sense of the complex sign without already 

grasping its identity with the sense of a simpler sign given by a logical analysis, Frege notes 

potential problems. If we fail to grasp immediately that two senses are identical, we either 

have a vague apprehension of the sense (which is a psychological or physiological problem, 

and not a logical one), the sense itself is vague (in which case it is no ‗sense‘ at all in Frege‘s 

view), or we have mistakenly shifted the meanings of our terms, such that we take now one 

and now another sense to be the object of our logical analysis. If we are working within a 

scientific language constructed along Fregean lines, in none of these three cases does the 

definition, in its role as a definition, provide us with information that we could not otherwise 

have gained. For Frege, then, the appearance that a definition is essential or otherwise 

informative is an historical accident related to our misapprehension of a particular sense, 

which has nothing to do with the definition in its proper role. 

 

On the face of it, this view seems rather puzzling: for, surely, we can employ a sign properly 

without grasping all the finer characteristics of its sense. On this Frege writes 

 

                                                 
75

 [Frege 1914, PW 209-210]. 



80 

 

The fact is that if we really do have a clear grasp of the sense of the simple sign, then it 

cannot be doubtful whether it agrees with the sense of the complex expression. If this 

is open to question although we can clearly recognize the sense of the complex 

expression from the way it is put together, then the reason must lie in the fact that we 

do not have a clear grasp of the sense of the simple sign, but that its outlines are 

confused as if we saw it through a mist. The effect of the logical analysis of which we 

spoke will then be precisely this—to articulate the sense clearly. Work of this kind is 

very useful; it does not, however, form part of the construction of the system, but must 

take place beforehand.
76

 

 

Frege admits, then, that occasionally the use of a simple sign—a sign which really 

represents, indeed is defined as identical in sense with, some complex expression—becomes 

so entrenched that we can lose sight of the fact that it is identical in sense to some complex 

sign or group of signs. In effect, we fail to grasp the sense of the simple sign fully, and grasp 

only enough of this sense to employ it correctly in certain contexts. Still working under this 

veil, we then perform a logical analysis (i.e., we construct a definitional equality) which 

illustrates that the simple sign has the same sense as a complex sign whose sense is already 

known. But, for Frege, the correctness of the thought which establishes this identity is not 

and cannot be a positive discovery relative to the science at issue. For we can be neither 

correct nor incorrect in our logical analysis until both terms already possess some sense, and 

it is precisely the provision of the simple term with a sense that is performed by a definition. 

If we grasp the sense of the simple term clearly, then we will know immediately whether or 

not it is identical with the sense of the complex sign, which is already known. So, without a 

clear sense already in mind, the so-called logical analysis of the sense of a simple sign by 

means of a definition into a complex sense is entirely impossible—something like this kind 

of analysis is only made possible by our vague, unclear grasp of a sense which is, in itself, 

perfectly precise.  

 

The discovery of logically efficacious definitions already presupposes a prior assumption 

about the equivalence of the senses of the two expressions. Such logical analyses, then, 

simply establish clearly the sense of the signs to be used within the system—they clarify 

what must already be known if we are to work and think scientifically, i.e., precisely.
77

 Such 

work is undoubtedly important, and many of the most fruitful mathematical definitions 

perform this kind of clarificatory work. Though not at all in accordance with Frege‘s 

particular vision of the nature of definition, the early history of group theory developed 

along these lines. The group concept was first employed in the very specific context of 

attempting to illustrate the general insolubility of quintic equations. It was only through great 

mathematical excavation and rigorization that the abstract concept of the group fully 

emerged, such that the factor groups employed by Galois and others in their search for an 

answer regarding quintics were seen to be specific instances of a more general mathematical 

phenomenon. The eventual definition of the group concept, by way of the now well-known 

group axioms, occurred only after the concept had been sufficiently clearly grasped to be 

                                                 
76

 [Frege 1914, PW 211]. 
77

 A more extensive discussion of the ways in which Frege deals with this apparent ‗paradox of analysis‘ can be 

found in [Horty 2007, 36ff.] and [Resnik 1980, 43-47; 181-185]. 



81 

 

employed in a number of seemingly unrelated contexts.
78

 So it is that many mathematical 

concepts are grasped only partially, or ‗seen through a mist‘, and, with this tenuous grasp, 

come to be clarified through what appears to be purely logical analysis. Frege‘s point is that 

this is epistemological, historical, or psychological analysis, and not the sort of systematic 

logical activity it may on occasion appear to be. For Frege, then, this is not really logical 

work insofar as it has no real correlate within the objective realm of thoughts, nor, 

correspondingly, within a perfectly scientific language. 

 

Another important quality of Fregean definitions is their ‗completeness‘. As a result of his 

views on the nature of sense (and the nature of regimented languages suitable for science), 

Frege requires that definitions of concepts
79

  must be ‗complete‘ in the following sense: 

 

A definition of a concept (of a possible predicate) must [...] unambiguously determine, 

as regards any object, whether or not it falls under the concept (whether or not the 

predicate is truly assertible of it). Thus there must not be any object as regards which 

the definition leaves in doubt whether it falls under the concept; though for us men, 

with our defective knowledge, the question may not always be decidable. We may 

express this metaphorically as follows: the concept must have a sharp boundary. If we 

represent concepts in extension by areas on a plane, this is admittedly a picture that 

may be used only with caution, but here it can do us good service. To a concept 

without a sharp boundary there would correspond an area that had not a sharp 

boundary-line all round, but in places just vaguely faded away into the background. 

This would not really be an area at all; and likewise a concept that is not sharply 

defined is wrongly termed a concept. Such quasi-conceptual constructions cannot be 

recognized as concepts by logic; it is impossible to lay down precise laws for them. 

The law of excluded middle is really just another form of the requirement that the 

concept should have a sharp boundary. Any object Δ that you choose to take either 

falls under the concept Φ or does not fall under it; tertium non datur. E.g. would the 

sentence ‗any square root of 9 is odd‘ have a comprehensible sense at all if square root 

of 9 were not a concept with a sharp boundary? Has the question ‗Are we still 

Christians?‘ really got a sense, if it is indeterminate whom the predicate ‗Christian‘ can 

truly be asserted of, and who must be refused it?
80

 

 

As we‘ve seen, a large part of Frege‘s reconstruction of scientific discourse via his work in 

logic and logical notation involves removing ambiguity and vagueness, and insisting upon 

strict divisions among distinct concepts. Remember, too, that, for Frege, definitions establish 

a logical equality between the senses of signs, i.e., they establish that the sense of a simple 

sign is to be identical with the sense of a complex sign. If the sense of the complex sign were 

indeterminate or ambiguous, we would be unable to determine whether or not we were 

associating the proper sense with the simple sign. Thus, the simple sign could not properly 

                                                 
78

 See [Wussing 1984] for an excellent account of the various historical sources leading to the emergence of the 

abstract group concept. 
79

 Frege extends these considerations on the definition of concepts to the definition of relations in §62 of [Frege 

1893]. 
80

 [Frege 1893, §56]; [Frege 1970, 159]. 



82 

 

function as an abbreviation for the complex sign and the definition would be entirely useless 

for the purposes of science. 

 

The requirement that definitions of concepts be complete in this sense also leads Frege to 

reject the relatively common mathematical practice of ‗piecemeal‘ definition. Piecemeal 

definitions, within mathematics, are those which (partially) define a concept for some 

limited domain and then employ this limited, partially defined concept to extend the 

definition to other domains. Thus for instance, we might define addition for the natural 

numbers and employ this definition to extend the ‗same‘ concept of addition to the rational, 

irrational, real, and complex numbers. In a continuation of the train of thought quoted above, 

Frege attacks this form of definition: 

 

Now from [the requirement that definitions be complete] it follows that the 

mathematicians‘ favourite procedure, piecemeal definition, is inadmissible. [...] Such 

piecemeal definition is a procedure comparable to drawing the boundary of a part of a 

surface in bits, perhaps without making them join up. But the chief mistake is that they 

are already using the symbol or word for theorems before it has been completely 

defined—often, indeed, with a view to further development of the definition itself. So 

long as it is not completely defined, or known in some other way, what a word or 

symbol stands for, it may not be used in an exact science—least of all with a view to 

further development of its own definition.
81

 

 

Thus, the practice of defining a term only for the natural numbers and, subsequently, 

extending it bit by bit to other number domains is forbidden by Frege insofar as, strictly 

speaking, the initial ‗partial definition‘ of a concept is senseless. A definition is a stipulation 

that the sense of one sign is to be equated with the sense of another; senses are unambiguous 

entities which populate a realm distinct from the logical imperspicuity of everyday speech 

and thought.
82

 Hence a definition which purports to provide a sign with an ambiguous or 

incompletely determined ‗sense‘ is no definition at all, because there can be no such 

senses.
83

 If all senses are determinate, as Frege holds, and all definitions are mere 

assignations of already-known senses to previously unused signs, then the ascription of an 

indeterminate or incomplete sense is simply nonsense and not a form of definition at all. 

 

In addition to logical superfluity and completeness, Frege‘s stipulative conception of 

definition also commits him to the view that there can only be a single definition for any 

given sign. For consider the two possible consequences of a contrary view. In the first case 

there might arise two distinct definitions of the same sign, i.e., a single sign would be 

assigned two distinct senses. The use of such a sign would be ambiguous. But, according to 

Frege‘s conception of a properly scientific language, which is his only concern when 

discussing definition, such ambiguity must at all costs be avoided. Indeed, one of the 

                                                 
81

 [Frege 1893, §57]; [Frege 1970, 159-160]. 
82

 On the possibility of ontologically vague or ambiguous entities in Frege‘s thought, see [Ruffino 2003, 262-

264]. 
83

 Frege notes other problems with piecemeal definition which I shall not discuss here (e.g., their effect on the 

validity of alleged theorems employing the term thus defined); cf. [Frege 1893, §61].  



83 

 

primary reasons for constructing a regimented language at all is to eliminate the ambiguity 

present in both ordinary language and contemporary scientific discourse. Accordingly, we 

cannot allow multiple senses for a single sign. In the second case, two distinct definitions 

might assign the same sense to a single sign. In this case, we have not ambiguity but 

redundancy. Even more than redundancy, we are forcing upon ourselves a duty which 

otherwise would not present itself, namely the duty of proving that the two supposedly 

consistent definitions are in fact consistent—or in the case of equivalence, that they are 

equivalent.
84

 For these reasons, then, Frege‘s views allow for at most one definition of any 

given term. 

 

2.5 Elucidations 

 

It is by stipulating, arbitrarily, that the sense of the new simple sign is identical to the sense 

of a complex sign or group of signs that Fregean definitions provide signs with senses. Such 

definitions give new signs the same sense as complex signs whose senses are already known. 

Now, it can happen that the sense of such a complex sign was itself given by means of a 

definition, in which case we can ask after the meanings of the simpler signs in terms of 

which it, too, is defined. But, on Frege‘s view, such a chain of definitions must eventually 

come to an end. A science must start with some primitive signs whose senses cannot be 

given in terms of definitions because, in the initial stages of system-construction, there just 

are no simpler signs in terms of which the primitives can be defined. 

 

Given that Frege is interested in creating a scientific language in which every sentence and 

every sign is meaningful, it follows that even the primitives of the system must already 

possess senses. The role of elucidations [Erläuterungen] is to guide the reader toward a 

grasp of the senses which the system-builder intends to express by the primitive signs of the 

system. It is to be assumed that the system-builders themselves are always able to grasp the 

senses of their system-primitives, and that these senses do not fluctuate. As Frege‘s 

translator Kluge notes, however, ―even [elucidations] presuppose the intelligibility of at least 

some of the terms involved in giving them; hence not even here can we escape the necessity 

of simply presupposing [the senses of] some terms as known.‖
85

 For Frege, elucidations 

cannot play a role within the system itself. Instead, like Euclid‘s second type of definition 

(e.g., his definition of ‗point‘), they are relegated to the propaedeutic, which clears the way 

for systematic thinking by preparing the materials out of which the system is to be built. 

Once these elucidations have achieved their function (i.e., once they have managed to guide 

the reader toward the senses intended to be expressed by the primitive signs of the system) 

they can be safely discarded and the construction of the system from its primitive building 

blocks can begin.  

 

 

 

 

                                                 
84

 ―...the same thing may never be defined twice, because it would then remain in doubt whether these 

definitions were consistent with each other‖ [Frege 1893, §33.2]. 
85

 [Kluge 1971, xxvii-xxviii]. 



84 

 

2.6 Axioms 

 

Frege held a traditional view of axioms as unprovable truths which lie at the foundation of a 

particular deductive edifice. Axioms, unlike definitions, offer indispensible support when 

attempting to prove theorems, for different sets of axioms will generally speaking lead to 

different deductive consequences. On Frege‘s view, the axioms of geometry present us with 

a small group of unprovable truths about space which, when combined with suitable modes 

of inference and a grasp of some primitive geometric concepts, allow us to elaborate the 

entire edifice of Euclidean geometry. In a more general attack on the notion of a purely 

formal mathematical system, Frege writes the following of specifically geometrical science: 

 

To be sure, that on which we base our definitions may itself have been defined 

previously; however, when we retrace our steps further, we shall always come upon 

something which, being a simple, is indefinable, and must be admitted to be incapable 

of further analysis. And the properties belonging to these ultimate building blocks of a 

discipline contain, as it were in nuce, its whole contents. In geometry, these properties 

are expressed in the axioms insofar as they are independent of one another. Now it is 

clear that the boundaries of a discipline are determined by the nature of its ultimate 

building blocks.
86

  

 

Here we see Frege‘s view of the purpose of axioms in relation to his understanding of 

deductive theories in general. We begin with a stock of basic concepts and objects (which we 

have, ideally, grasped via elucidations), and the properties/characteristics of these are given 

to us by way of a cluster of axioms, which are truths about those basic concepts and objects. 

In contrast with Hilbert, for whom the truth of geometric axioms was intimately connected to 

their consistency relative to a particular axiomatic system, Frege saw the truth of the axioms 

of geometry as truth simpliciter. These axioms, if true, were true in just the same way that 

any other everyday thought was true.  

 

Here we find one of Frege‘s most fundamental disagreements with the geometric trends of 

the 19
th

 century. For Hilbert and others like him, the truth of Euclidean geometry was not at 

all incompatible with the truth of various other non-Euclidean geometries. This is because 

the truth of an axiomatic system simply results from its internal consistency. Insofar as 

Euclidean, non-Euclidean, Archimedean, non-Archimedean, and many other forms of 

geometry are all internally consistent, then, they are all equally ‗true‘. Frege, in stark 

contrast, takes Euclidean geometry to be the science of space, so that its axioms are truths 

about space, truths which directly contradict the axioms of other candidate geometries like 

the hyperbolic geometry investigated by Bolyai and Lobachevsky. Insofar as Frege 

construed Euclidean geometry to be a theory about space, then, its truth is in direct 

contradiction with the claims of the various non-Euclidean geometries. 

 

 

 

 

                                                 
86

 [Frege 1971, 143]. 



85 

 

§3. The Frege-Hilbert correspondence 

 

Now that we have established some of the context and content of the views of both Hilbert 

and Frege, we can turn more profitably to their correspondence. The correspondence and 

subsequent articles by Frege are an interesting focal point for examining differing 

conceptions of the nature of mathematical structures. Hilbert, for his part, offers a more 

strictly relational point of view regarding axiomatics than he does elsewhere in his own 

work. The viewpoint developed there is similar in many respects to the modern postulational 

point of view, and (some of) Frege‘s criticisms are illustrative of larger problems within the 

more general postulational point of view as well. Frege‘s viewpoint, too, despite its 

idiosyncrasies shares general features with many modern accounts of the notion of structure 

which seek to embed the purely relational concept of structure within a fixed hierarchy of 

some sort (e.g., logic, set theory, category theory, some form of axiomatized structure 

theory, etc.). It is against Hilbert‘s position that Frege‘s own views come into sharpest relief, 

and it is in direct contrast with Hilbert‘s quasi-relational views that we can best understand 

the problems related to the concept of mathematical structure which will interest us in 

subsequent chapters. Let us now turn to the debate. 

 

3.1 The course of the dispute 

 

After having met Hilbert at a scientific convention in Lübeck in 1895, Frege instigated a 

brief exchange regarding the uses and abuses of formalization in mathematics, a topic in 

which Hilbert had apparently expressed some interest when they met. This initial exchange 

quickly petered out, and it wasn‘t until December of 1899 that Frege wrote to Hilbert again, 

having recently received and worked through a copy of the latter‘s Festschrift on the 

foundations of geometry. Having found much confusion in the work, Frege wrote to Hilbert 

for clarification (particularly regarding his use of the terms ‗definition,‘ ‗explanation,‘ and 

‗axiom‘), he also took the opportunity to acquaint Hilbert with his own views on these 

matters. Hilbert‘s response was prompt—dated just two days after Frege‘s letter—and 

contained several attempts to respond to Frege‘s request for clarification. He restated in more 

detail his own understanding of the nature and purpose of axiomatic systems and, towards 

the close of the letter, attempted to illustrate the advantages of his own (loosely model-

theoretic) understanding over the traditional picture he believed to be held by Frege. 

 

Frege replied on the sixth of January, thanking Hilbert for his letter, and beginning by 

rephrasing his newly clarified understanding of Hilbert‘s approach to axiomatization in 

general, and geometry in particular. After agreeing, in broad terms, with Hilbert‘s dismissal 

of earlier forms of ‗genetic‘ or piecemeal approaches to axiomatization, Frege turns once 

more to the perceived lack of clarity in the specifics of Hilbert‘s approach. His criticisms 

here again pertain to Hilbert‘s use of definitions, explanations, and axioms. The tone of the 

letter is admonitory, and Frege describes at length his view of the proper approach to the 

subject at hand. 

 

After this point, there were a few more exchanges between mid-January 1900 and November 

1903, but Hilbert seems to have lost interest in the correspondence, much to Frege‘s chagrin. 

Frege, who was not at all satisfied with the results of their exchanges, suggested that their 



86 

 

correspondence be published—a suggestion which Hilbert apparently rejected, as his views 

had changed and his focus had shifted in the meantime. Undeterred, Frege published a series 

of articles relating to their dispute; Alwin Korselt, a proponent of Hilbert-style axiomatics, 

took up the gauntlet and Frege‘s final two papers in the series are directed at both Korselt 

and Hilbert. I will now turn to Frege‘s criticisms of the views proposed, first by Hilbert, and 

then by Korselt. 

 

3.2 Frege‘s criticisms of Hilbert 

 

3.2.1 Definitions and ‗explanations‘ 

 

One of Frege‘s central complaints against Hilbert pertains to the way in which he employs 

the terms Definition and Erklärung.
87

 As we‘ve seen, Frege‘s work in logic and the 

foundations of arithmetic had led him to erect a strict boundary between pre-systematic 

elucidations and definitions properly speaking. A Fregean elucidation is similar in kind to 

Euclid‘s definition of the term ‗point‘: it is a pre-systematic signpost toward the proper 

meaning of the basic elements of the system. Once we have grasped these basic elements and 

begun to work within the system (of logic, arithmetic, geometry, etc.) we can dispense with 

elucidations altogether. Definitions, for Frege, are a rather different matter. A definition only 

makes sense within a particular system, insofar as it is a proposition which declares that a 

simple term (e.g., ‗continuous‘) is to have the same meaning as a more complicated term, 

proposition, or conjunction of propositions. In this sense, definitions too are dispensable for 

Frege, as we could just as well employ the complex expression in each case—nevertheless 

the role of a definition only makes sense within a particular system, whereas the role of an 

elucidation is fulfilled in the anteroom through which we must pass before entering the 

system. Given the precision with which Frege employed the term ‗definition‘, it is not 

entirely surprising that he found Hilbert‘s treatment of geometrical definitions somewhat 

unclear.  

 

His initial criticism is that Hilbert does not adequately distinguish between his own use of 

the terms ‗definition‘ and ‗explanation‘—some explanations seem, to Frege, to be identical 

in character to definitions properly speaking, while at other times the two seem to diverge 

significantly. In connection with this confusion, he refers to Hilbert‘s axioms of order (found 

in Hilbert‘s axiom group II). Frege writes to Hilbert that ―I [..] cannot take your [definition 

of the concept ‗between‘ via the axioms of order] for a definition, but you also do not call it 

a definition but an explanation [Erklärung]‖.
88

 Indeed, Hilbert does employ both Definition 

and Erklärung throughout, and Frege notes that it is difficult to establish the precise 

                                                 
87

 It is useful to note that the German term ‗Erklärung‘ is very often used as a synonym of ‗Definition‘. The 

point Frege makes against Hilbert is not that he uses the term ‗Erklärung‘ incorrectly, but that his use of the 

term seems to fluctuate, such that Frege is unsure whether Hilbert‘s Erklärungen are (what Frege himself 

would call) elucidations or if they are definitions properly speaking. At several points throughout the 

Grundlagen, it appears as though Hilbert‘s Erklärungen are different in kind from his more usual definitions, 

so that Frege‘s confusion regarding Hilbert‘s use of the term is not surprising, nor as pedantic as it might seem 

at first glance. (In what follows I will occasionally employ the English term ‗explanation‘ in place of the 

German ‗Erklärung‘). 
88

 [Frege 1980, 35]. 



87 

 

difference between the meanings of the two terms, if indeed one is to be found. To this end, 

Frege highlights the many different ways in which the terms are employed in Hilbert‘s text.  

He points, for example, to Hilbert‘s explanations in §4 (‗Consequences of the Axioms of 

Connection and Order‘) as instances where the term ‗Erklärung‘ seems to be employed as a 

synonym for ‗Definition‘. 

 

Here is the first of these explanations: 

 

Erklärung. Es seien A, A′, O, B vier Punkte einer Geraden a, so daβ O 

zwischen A und B, aber nicht zwischen A und A′ liegt; dann sagen wir: die 

Punkte A, A′ liegen in der Geraden a auf ein und derselben Seite vom Punkte 

O, und die Punkte A, B liegen in der Geraden a auf verschiedenen Seiten vom 

Punkte O. Die sämtlichen auf ein und derselben Seite von O gelegenen 

Punkte der Geraden a heiβen auch ein von O ausgehender Halbstrahl; somit 

teilt jeder Punkt einer Geraden diese in zwei Halbstrahlen.
89

 

 

Now, to see Frege‘s point here, contrast this explanation with the following, from the 

previous section: 

 

Erklärung. Die Punkte einer Geraden stehen in gewissen Beziehungen 

zueinander, zu deren Beschreibung uns insbesondere das Wort ‗zwischen‘ 

dient. 

Explanation. The points of a line stand in a certain relation to one another, 

which can be described by the word ‗between‘.
90

 

 

In the first instance, we witness Hilbert providing a precise statement of the conditions under 

which we are licensed to say that a specific relation holds; in the second instance, however, 

we are given a very broad, almost vacuous, statement which indicates only that we can use 

the word ‗between‘ to describe some relation in which points of a line may stand to one 

another, though what this relation is, precisely, is not given by the explanation. Frege writes 

of the latter sort of explanations that they ―are apparently of a very different kind [from the 

explanations of §4], for here the meanings of the words ‗point‘, ‗line‘ ‗between‘ are not 

given, but are assumed to be known in advance. At least it seems so. But it is left unclear 

what you call a point‖.
91

  

 

So, if we are to understand some of Hilbert‘s explanations on the model of definitions, then 

they are by Frege‘s standards quite bad definitions and, really, no definitions at all, insofar as 

they fail to clearly fix the references of the terms which are supposed to be defined. Recall 

that, for Frege, the role of a definition, properly speaking, is to stipulate that the meaning of 

a complex term or proposition is the same as the meaning of a newly introduced, simple 

term. Thus, for example, if Frege‘s work on the concept of number had panned out, in place 

of a host of complex statements about equivalence classes and courses of values, we would 

                                                 
89

 [Hilbert 1903, 6].  
90

 [Hilbert 1903, 4]. 
91

 [Frege 1980, 35]. 



88 

 

have been able to plug in the much simpler term ‗number‘ while still preserving the precise 

meaning of statements which employed the more complex combination of terms. Recall, too, 

that for Frege ‗meaning‘ is understood in terms of his basic distinction between functions (or 

concepts) and objects, and that only sentences which express ‗fully saturated‘ thoughts (in 

his sense) can be considered meaningful or possess a truth value. If the references of 

Hilbert‘s terms are not fixed by way of elucidations or definitions, then the sentences which 

contain those terms cannot express complete thoughts, and hence cannot be true or false by 

Frege‘s lights. 

 

In stark contrast to Frege‘s aims, one of the key features of Hilbert‘s approach to axiomatic 

systems is to be found in the fact that the terms of those systems are not given a definitive, 

fixed interpretation. At most, an axiom system should be ‗categorical‘ in the sense that any 

model of the system will be isomorphic to any other—thus, even a categorical axiom system 

should only ever determine a class of isomorphic models. For Hilbert, there appeared to be 

no point in moving beyond this class of models to a single, uniquely determined model, with 

a particular fixed set of objects (i.e., the Euclidean points, lines, etc.). Precisely the point of 

formalizing geometry was to escape the vagaries required by the elucidatory definitions of, 

e.g., lines as breadthless length—precision and generality were gained by escaping the 

demand for unique, fixed reference and examining the more important structural relations 

determined by a particular axiomatization. For Frege, however, the indeterminacy of 

reference here makes Hilbert‘s definitions useless. Frege asks ‗What do you mean by point?‘ 

and by this question he means ‗To what does your term ‗point‘ refer?‘. Hilbert, on the other 

hand, can only answer Frege‘s question by directing him toward the relations determined by 

the axioms of his system. The further requirement that he precisely fix the reference of his 

basic terms once and for all directly contradicts the very purpose of his axiomatization of 

geometry, as he saw it. Fixing the reference of the terms in more than a merely relational 

way would restrict the system to one particular interpretation, and, accordingly, greatly limit 

the applicability of that system to areas of inquiry not already intended by the developers of 

that interpretation.  He writes in response to Frege that: 

 

[...] it is surely obvious that every theory is only a scaffolding or schema of concepts 

together with their necessary relations to one another, and that the basic elements can 

be thought of in any way one likes. If in speaking of my points I think of some system 

of things, e.g., the system: love, law, chimney-sweep... and then assume all my axioms 

as relations between these things, then my propositions, e.g. Pythagoras‘ theorem, are 

also valid for these things. In other words: any theory can always be applied to 

infinitely many systems of basic elements. One only needs to apply a reversible one-

one transformation and lay it down that the axioms shall be correspondingly the same 

for the transformed things. This circumstance is in fact frequently made use of, e.g. in 

the principle of duality, etc., and I have made use of it in my independence proofs.
92

 

 

We can see here that the idea of fixing the reference of his terms any more determinately 

than up to isomorphism seemed mathematically uninteresting to Hilbert. As we saw above, 

Hilbert‘s interest in geometry was not at all in the properties of a specific collection of 

                                                 
92

 [Hilbert to Frege, in Frege 1980, 40-41]. 



89 

 

objects, but in the structural properties (consistency, simplicity, categoricity, etc.) of 

axiomatic systems themselves. He took a mathematical theory to be a schema which can be 

exemplified in a variety of different concrete systems. For instance, because there exists an 

isomorphism between the basic entities and relations of Euclidean geometry and those of the 

set of ordered triples of real numbers,
93

 the consistency of Euclidean geometry follows from 

the (tacitly assumed) consistency of the theory of ℝ
 3

. But this model-theoretic, structure-

focussed view of Hilbert‘s is predicated upon the belief that the axiomatic structure of 

Euclidean geometry can be discussed with complete indifference to any of its particular 

interpretations, be they number-theoretic or more traditionally geometric in nature. To 

demand to know which things were picked out by his axioms prior to interpretation seemed 

to him almost nonsensical and, at the very least, beside the point. From Hilbert‘s point of 

view, then, Frege‘s remarks seemed, first, to have missed the point of axiomatizing geometry 

entirely and, second, to have demanded a massive step backwards in mathematical thinking.  

 

The conflict between two demands—Frege‘s for rigor based upon a precise understanding of 

certain logical concepts, and Hilbert‘s for mathematical fruitfulness based on generality and 

applicability—was sharpened considerably by Hilbert‘s reply to Frege‘s initial criticisms. In 

his very direct response, Hilbert writes: 

 

I do not want to presuppose anything as known. I think that with my explanation in §1 

I give the definitions of the concepts point, straight line, and plane if one again adds to 

these all the axioms of axiom-groups I-V as characteristics. If someone is looking for 

other definitions of ―point,‖ etc., perhaps by means of circumscriptions like 

extensionless, then of course I would most decidedly have to oppose such an 

enterprise. One is then looking for something that can never be found because there is 

nothing there, and everything gets lost, becomes confused and vague, and degenerates 

into a game of hide-and-seek. If you prefer to call my axioms characteristics of the 

concept that are posited in the ―explanations‖ and consequently exist, I should have no 

objections to this except, perhaps, that it contradicts the custom of mathematicians and 

physicists. Of course I must also be able to do as I please in the matter of positing 

characteristics; for as soon as I have posited an axiom, it will exist and be ―true‖.
94

 

 

Here Hilbert attempts to defuse Frege‘s objections regarding his apparently unclear concept 

of definition in two ways. First, he suggests that if Frege is looking for something like 

Euclid‘s definitions of ‗point‘ and ‗line‘, then he is barking up the wrong tree—such 

definitions play no real mathematical role and have, to Hilbert‘s mind, been rightly 

dispensed with in most modern mathematical works. Their obsolescence is due to their lack 

of clarity, the seeming impossibility of scientifically providing a pre-mathematical definition 

of point, and, most importantly, to the lack of intrinsic mathematical interest in uniquely 

fixing geometric concepts beyond the limits imposed by a categorical formalization. 

Hilbert‘s interest here is in the structure determined by the axioms of Euclidean geometry, 

and not in the nature of the entities which might serve as the reference for the basic terms of 

the axiomatization on one particular interpretation, traditional though it might be. Second, 

                                                 
93

 Either ℝ
2 
or ℝ

3
, for the cases of plane and solid geometry respectively. 

94
 [Hilbert to Frege, in Frege 1971, 11-12]. 



90 

 

Hilbert argues that if Frege still has problems with his understanding of definition, he can 

look to the axiom groups for answers: the full meaning of the concept of point, for example, 

is not given solely through the explanations and definitions of that term which are given by 

Hilbert. Instead, this meaning is given when those definitions and explanations are taken in 

conjunction with the axiom groups. If Frege is still unsatisfied by this characterization of the 

concept of a Euclidean point, then, to Hilbert‘s mind, he is simply asking for something 

which cannot (and, really, should not) be given.  

 

3.2.2 Defining axioms 

 

As we‘ve just seen, in his attempt to respond to Frege‘s criticisms of his definitions, Hilbert 

suggested that, in addition to those propositions marked off by the headings ‗definition‘ and 

‗explanation,‘ Frege should consider his axiom groups themselves as part of the definition of 

basic concepts like point, line, etc. Even more strongly, he writes that ―[...] it is impossible to 

give a definition of a point in 3 lines, since it is only the whole axiom-structure that gives the 

complete definition, and therefore each new axiom alters the concept. ‗Point‘ is always 

something different in Euclidean, non-Euclidean, Archimedean and non-Archimedean 

geometry respectively‖.
95

 Two things are worth noting in this remark. First is Hilbert‘s 

view—dramatically different from Frege‘s—that axioms can play a role in the definition of 

the basic terms of a particular system. Second, is Hilbert‘s admission that different 

collections of axioms result in different basic concepts, such that the term ‗point‘ is 

employed within different axiomatizations to designate quite different objects (i.e., 

Euclidean, non-Euclidean, Archimedean, and non-Archimedean ‗points‘). Both of these 

aspects of Hilbert‘s view have become a mainstay of the model-theoretic conception of 

axiomatics. In any introductory text in group theory or abstract algebra, for instance, the 

term ‗group‘ is defined by the group axioms, and different types of group (Abelian, non-

Abelian, Galois, cyclic, etc.) are given by altering the axiom set in various ways.
96

 This 

procedure is now common throughout other branches of mathematics, various forms of 

geometry included. Frege, however, found Hilbert‘s particular version of this view logically 

imprecise at best, and an example of meaningless formalism at worst.  

 

Frege himself viewed axioms as unprovable truths, and definitions as a kind of logical 

shorthand which helped to fix the meaning of new signs in relation to already established 

signs. Axioms and definitions thus play very different logical roles, and have quite different 

relations to the inferential systems of which they are parts. Fregean definitions are true by 

virtue of stipulation: we simply decree that a new sign is to have the sense of some other 

sign or combination of signs. An axiom, by contrast, was thought by Frege to be true in 

virtue of the way that the world is—in the case of geometry, by virtue of the nature of space 

itself. In his first essay dealing with the foundations of geometry, Frege writes of Hilbert‘s 

understanding of axioms that:  

 

Once the explanation [of the meaning of the term ‗rectangle‘] including the two 

axioms has been posited, the latter may be asserted as true; however, their truth [i.e., 

                                                 
95

 [Hilbert to Frege, in Frege 1971, 12-13]. 
96

 [Scott 1987, 9ff.] and [Deskins 1995, 192ff.] are representative. 



91 

 

the truth of the axioms] will not be founded on an intuition, but on the definition. And 

it is precisely because of this that no real knowledge is contained in [Hilbert‘s 

axioms]—something which undoubtedly is the case with axioms in the traditional 

sense of the word.
97

 

 

The truth of Hilbert‘s axioms, as well as their meaningfulness, depends essentially upon the 

complex definitions of which they are a part. Because the supposed truth of Hilbert‘s axioms 

is not determined by an external check (like our intuition of space, some objective 

conceptual structure, or the very nature of physical space itself), Frege argues that the entire 

edifice constructed upon these axioms is empty and ill-founded.  

 

Furthermore, since for Frege definitions are, at least in principle, always eliminable from any 

chain of logical inferences, they therefore ought not to serve as premises in a proof. Nor can 

they make possible proofs which are otherwise impossible. Axioms, by contrast, must, at 

some point, serve as the premises of some proofs within a system, as they are the foundation 

upon which the entire deductive structure of a particular system is based. Given that the 

axioms of a system must play some inferential role, it is important (for Frege at least) that 

they express complete thoughts.
98

 But, if the sense of a term occurring within a sentence 

expressing an axiom is not fixed, then the axiom itself is not a complete thought, and thus 

can be neither true nor false. Thus, if each axiom or axiom group adds some new element in 

the definition of the basic terms of the system, it is only with an extraordinarily complex 

conjunction of all the axioms, explanations, and definitions that we get a complete thought. 

The result, Frege suggests, is a ―logical edifice [which is] puzzling and opaque in the highest 

degree‖.
99

 In ―On the Foundations of Geometry‖ Frege remarks that  

 

Insight into the logical nature of a mathematical theory is frequently made more 

difficult by the fact that what really ought to be represented as a unitary propositional 

complex is torn apart into apparently independent grammatical propositions. This often 

happens for stylistic reasons, in order to avoid a propositional monstrosity; but this 

cannot be permitted to obstruct one‘s insight into the nature of the case. 
100

 

 

It seems clear that Frege believed Hilbert‘s stylistic choices to have greatly obstructed 

logical insight in this case. So, taken individually, Frege suggests that Hilbert‘s axioms do 

not possess a meaning, nor are they true or false, hence they can play no inferential role in 

the subsequent development of the science of geometry.  

 

 

                                                 
97

 [Frege 1971, 27]. 
98

 Moreover, the axioms must express true thoughts as Frege maintained that proper inferences can only be 

drawn from truths. In a posthumously published fragment, he writes: ―Now if A is the concept of a right-angled 

equilateral pentagon, ‗this A‘ is an inadmissible—a meaningless—proper name, and nothing whatever can be 

inferred either from the sentence ‗This A has the property B‘ or from ‗This A does not have the property B; for, 

strictly, we do not infer from sentences at all, but from thoughts. And only true thoughts are admissible as 

premises of inferences‖ [Frege 1906a, in PW, 177]. 
99

 [Frege to Hilbert, in Frege 1971, 17]. 
100

 [Frege 1971, 72]. 



92 

 

 

 

3.2.3 Equivocations in Hilbert‘s independence and consistency proofs 

 

Hilbert happily accepts that different collections of axioms help to define different basic 

concepts. Thus, despite the similarity between the Euclidean and non-Euclidean concepts of 

‗point‘, because these different geometrical structures are given by different collections of 

axioms, their basic notions are also different. Hence, theorems proven for non-Euclidean 

points are not the same as theorems proven for Euclidean points. Hilbert insists that the 

removal or addition of a single (independent) axiom results in a completely new system of 

basic concepts, a system which will have its own set of deductive consequences. 

 

In his replies to this aspect of Hilbert‘s view, Frege notes two difficulties, both of which 

involve dangerous equivocations. The first sort of equivocation is most apparent in Hilbert‘s 

independence proofs. Recall that Hilbert‘s method of proving the independence of a 

particular axiom from a collection of other axioms involved modelling two distinct sets of 

axioms: one in which all the axioms including the independent axiom hold true, and one in 

which all the axioms save the independent one hold true. This is intended to prove 

something about a particular axiom as well as a particular set of axioms. But, notes Frege, 

using Hilbert‘s own reasoning, such proofs simply cannot achieve this goal. For, in the two 

separate models we are not working with the same basic concepts at all. By removing an 

axiom (as is required for the independence proof to go through) we are altering the meaning 

of the basic terms, if indeed they possess a meaning in Frege‘s sense at all. For example, if 

our intent is to prove the independence of the parallel postulate from the other axioms of 

Euclidean geometry, when we remove the parallel postulate from the larger set, the meaning 

of the basic terms occurring in the axioms (e.g., point, line, etc.) changes. But Hilbert‘s proof 

seems to require that the meanings of these terms remain stable throughout the proof, 

otherwise we are simply constructing two completely unrelated models. It seems, then, that 

Hilbert‘s own construal of his methods directly thwarts his aim of providing independence 

proofs. From Frege‘s point of view, addressing this problem requires either that Hilbert alter 

his methods such that different systems are not treated as one, or he must alter his goal of 

providing an independence proof of the Euclidean version of the parallel postulate from the 

remainder of the Euclidean axioms. 

 

The second, more subtle, form of equivocation is best illustrated by Hilbert‘s consistency 

proofs. Here Hilbert constructs a single model of his axioms, i.e., an interpretation in which 

each axiom holds. The existence of a model is meant to show that the axioms can be true 

together, and are thus consistent. Since Hilbert takes his axioms to be a mere scaffolding 

whose basic terms do not refer to specific objects, from his point of view there appears to be 

no problem with this procedure. To see where the equivocation lies here, we must turn to 

Frege‘s idea of concepts of different levels. A first-level concept is one under which objects 

fall. For instance, all cows fall under the first-level concept ‗x is a cow‘ and all Euclidean 

points fall under the first-level concept ‗x is a Euclidean point‘. Of first-level concepts, Frege 

writes that 

 



93 

 

[...] between objects and (first-level) concepts there obtains a relation of subsumption: 

an object falls under a concept. For example, Jena is a university town. Concepts are 

generally composed of component-concepts—the characteristics. Black silken cloth 

has the characteristics black, silken, and cloth. An object falling under this concept has 

these characteristics as its properties. What is a characteristic with respect to a concept 

is a property of an object falling under that concept.
101

 

 

Analogous to (though still distinct from) the relation of subsumption which obtains between 

first-level concepts and objects is the relation between second- and first-level concepts. 

Frege writes that in this case  

 

[...] one could speak of subsumption; but this relation, although indeed similar, 

nevertheless is not the same as that of the subsumption of an object under a first-level 

concept. I shall say that a first-level concept falls (not under, but) within a second-level 

concept. The distinction between concepts of the first and second levels is just as sharp 

as that between objects and concepts of the first level; for objects can never substitute 

for concepts. Therefore an object can never fall under a second-level concept; such 

would be not false but nonsensical. If one tried something like this linguistically, one 

would get neither a true nor a false thought, but no thought at all. [...] A different 

feature of first-level concepts is expressed by the proposition that if an object falls 

under such a concept, another object distinct from the preceding one also falls under it. 

Here we have a second concept of the second level. From both, as second-level 

characteristics, we can form a third second-level concept within which fall all those 

first-level concepts under which fall at least two distinct objects. The concepts prime 

number, planet, and human being would be such as fall within this second-level 

concept.
102

  

 

Frege notes here that the meaningfulness of our language depends upon our subsuming 

entities of appropriate type under concepts of appropriate type. Thus, we can meaningfully 

state that a certain entity falls under a certain concept only if the first entity is one level 

lower in Frege‘s ontological hierarchy. Any confusion in this regard results in nonsense, 

which is neither true nor false. The second equivocation which Frege examines in Hilbert‘s 

work results from just this sort of confusion between concepts of different levels. 

 

Hilbert‘s proof that Euclidean geometry is consistent, for instance, involves constructing an 

interpretation of the axioms of Euclidean geometry. To Frege‘s mind, this process of 

constructing a model already requires an equivocation between first- and second-level 

concepts. For instance, Euclid‘s first axiom—taken in the traditional sense—is a claim 

which pertains to objects (i.e., Euclidean points and straight lines). Similarly, Hilbert‘s 

model (constructed, either in the field Ω or within the real numbers) requires a discussion of 

objects, insofar as Hilbert treats ‗points‘ in this interpretation as pairs of numbers. By 

illustrating that the real numbers can serve, given an appropriate interpretation, as a model of 

                                                 
101

 [Frege to Liebmann, in Frege 1971, 4]. 
102

 [Frege to Liebmann, in Frege 1971, 5].  



94 

 

Euclidean geometry, Hilbert intends to show that Euclidean geometry is consistent, under 

the reasonable assumption that the real numbers themselves are consistent. 

 

What Frege takes Hilbert to have shown, however, is that both the Euclidean concept of 

‗point‘ and the concept of ‗point‘ understood in terms of the real numbers or the field Ω fall 

under a distinct, second-level concept (which Hilbert also calls ‗point‘). More generally, 

Frege believes that Hilbert‘s consistency proofs pertain, not to Euclidean geometry and its 

first-level concepts, but to a system of second-level concepts which is entirely distinct from 

Euclidean geometry, but within which Euclidean geometry and Hilbert‘s interpretation of 

Euclidean geometry both fall. 

 

Frege‘s reasoning runs as follows. Hilbert presents us with two systems of objects and first-

level concepts: one in which we deal with points, lines, and planes, and the other in which 

we deal with numbers and sets of numbers. From the mere presentation of two systems of 

objects, we cannot reach the conclusion Hilbert wishes to make. Frege rightly notes that we 

must, at the very least, ascend to a higher level from which both the system of Euclidean 

geometry and the system of the real numbers appear as particular cases of a more general 

system. What Frege takes Hilbert to have shown with his consistency proof, then, is not the 

consistency of Euclidean geometry, but the consistency of a second-level system under 

which Euclidean geometry falls. And, most importantly for Frege, from the consistency of 

this more general theory, we cannot derive the consistency of Euclidean geometry itself. In 

one of his replies to Hilbert, Frege writes of this problem as it relates to the more limited 

case of individual propositions: 

 

If a general proposition contains a contradiction, then every particular proposition 

included under it will do likewise. Therefore, from the consistency of the latter, we can 

infer that of the general [proposition], but not vice versa. Suppose we proved that in a 

right triangle with two equal sides, the square on the hypotenuse is twice as large as 

that on one of the other two sides—which is easier than proving the general 

Pythagorean theorem. From this we can now infer that the proposition contains no 

contradiction; that the square on the hypotenuse of a right triangle having two [equal] 

sides is equal to the sum of the squares on the other two sides. From this we can 

deduce further that the general Pythagorean theorem contains no contradiction. But can 

we conclude still further that the Pythagorean theorem is therefore true? I cannot admit 

such an inference from consistency to truth.
103

 

 

Frege‘s claims here hold more generally for systems of propositions as well. Let‘s see how 

this applies in the present case. There are three distinct systems at issue here. First is the 

system of Euclidean geometry itself, second is the ‗model‘ of Euclidean geometry 

interpreted within the real numbers, and third is the more general system of propositions 

within which are included both traditional Euclidean geometry and Hilbert‘s model. For ease 

of reference, I will call this more general system L. Now, from the consistency of Hilbert‘s 

model, it does indeed follow that the system L does not contain a contradiction. It does not, 

however, follow that all of the particular systems which fall within L are consistent.  

                                                 
103

 [Hilbert to Frege, in Frege 1971, 21].  



95 

 

 

More specifically, it does not follow that Euclidean geometry itself is consistent. For Frege, 

Hilbert‘s mistake is to tacitly assume that his claims about L also apply directly to the 

Euclidean system. To Frege‘s mind, the claims made about ‗points‘ in L have nothing 

whatsoever to do with Euclidean points, other than the unfortunate use of the same word to 

describe them. The dangerous equivocation, then, is found in Hilbert‘s fluctuation between 

the general system L and the specific system of Euclidean geometry which falls within L. By 

Frege‘s lights, it is only via this equivocation that Hilbert is able to ‗prove‘ the consistency 

of Euclidean geometry. What he in fact proves is that L is consistent, given that the system 

of the real numbers is consistent.  

 

A further criticism stems from Frege‘s non-relativized conception of truth, in which truth 

does not directly follow from consistency as it seems to for Hilbert. Hilbert takes his 

consistency proof for Euclidean geometry to be a sufficient proof of the truth of Euclidean 

geometry, whereas Frege, who views consistency as merely possible truth, does not at all 

authorize the inference from the merely possible to the actual. Here the crucial difference 

between Hilbert‘s point of view and Frege‘s is that Hilbert does not understand Euclidean 

geometry in Frege‘s more traditional sense. Instead, he views the axioms of Euclidean 

geometry as an uninterpreted conceptual schema which determines a formal structure. This 

structure (or scaffolding) can then be applied to a variety of different systems by way of the 

process of interpretation.  Both Hilbert and his defender Korselt insist upon this. Korselt, for 

his part, emphasizes the potential emptiness of the uninterpreted concepts of a given formal 

theory. To this effect he writes:  

 

A purely formal system is noteworthy as long as the propositions that might occur in it 

do not lead to contradiction; indeed, it remains interesting even as a contradictory 

formal system, as long as this contradiction becomes apparent only at the end of a long 

sequence of inferences about the objects of the formal system. Therefore certain 

concepts of a ―purely formal system‖ may even show themselves to be ―empty‖. But 

on the other hand, a formal system may be ―applied‖ to a given domain only after we 

have assured ourselves of the validity of the principles for that domain.
104

 

 

Here Korselt expresses a key element of the postulational point of view: namely, that 

axiomatized mathematical theories are objects of investigation in their own right, and not 

simply tools employed to study their potential interpretations. To make contact with a 

concrete system, an application is required, and this application lies outside the 

mathematical theory properly speaking. This emphasis on the abstractness of formal theories 

is intended to illustrate that Frege‘s insistence upon fixing precise references for basic terms 

is not at all mathematically important, and that it is in a sense also misguided. For Korselt 

and Hilbert, a more precise characterization of the objects of a theory is only possible once 

we descend from the high generality of the axiomatic theory itself to one of its possible 

interpretations. Within a particular interpretation we can certainly ask after the precise 

reference of a given term, but from the more general formal perspective, this sort of question 

makes no sense. 

                                                 
104

 [Korselt 1903, in Frege 1971, 40-41]. 



96 

 

 

From Frege‘s point of view, however, this formal stance requires a great degree of logical 

perspicuity which is not at all evident in either Hilbert‘s or Korselt‘s responses. From his 

own point of view, it appears impossible to claim (as Hilbert does), that we can prove the 

independence of, e.g., the Euclidean parallel postulate by means of these model-theoretic or 

‗formal techniques‘. It is difficult for Frege to see, that is, how a proof pertaining to one 

system of objects and first-level concepts can show anything whatsoever about a completely 

distinct system of objects and first-level concepts without a clear understanding or 

presentation of the higher-level system which acts as their intermediary.  

 

Moreover, if Hilbert or Korselt wish to highlight some of the features of a more general 

conceptual system (e.g., the system L), under which Euclidean and other systems might fall, 

their method of doing so is both misleading and unclear. In response to Korselt (who defends 

the apparent indeterminacy and ambiguity of Hilbert‘s presentation as a feature resulting 

from a formal point of view), Frege writes that, when referring to the increased generality of 

Hilbert‘s conception of a point, Korselt seems to have 

 

[...] in mind a second-level concept within which, aside from the Euclidean point-

concept still other concepts fall. Of course this second-level concept must also be a 

completely determinate one; but it behaves toward the first-level concepts falling 

within it in a way similar to that in which a first-level concept behaves towards the 

objects falling under it. When we consider the multiplicity of these concepts of the first 

level (point-concepts), we get the notion that we are faced with an indeterminacy or 

ambiguity. This need not be the case here any more than it is in the case of the first-

level concept prime number, where not only 2 but also 3 falls under the latter.
105

 

 

In other words, if Korselt and Hilbert wish to discuss concepts from this higher point of 

view, that does not exempt them from making sure the reference of their basic terms is 

precisely fixed. A second-level concept is just as determinate and precise as a first-level 

concept. For Frege, rigor is lost, not gained, by the logical fiction of ambiguous or 

indeterminate concepts. What Frege objects to here is not the very idea of a higher level 

geometry—to which he is occasionally quite sympathetic
106

—but to the ‗free-floating‘ 

character of that geometry as presented by Hilbert and defended by Korselt. Frege‘s logical 

view of the world involves (as we will discover in more detail in the following chapter) an 

ontological hierarchy within which each entity belongs to a distinct ontological level.
107

 

Hilbert‘s work attempts to avoid this hierarchy by characterizing a structure which is not 

determinately linked to any particular ontological level. To be sure, Hilbert does not intend 

his formal characterization to describe a concrete system of objects and first-level concepts. 

But neither does he intend his work to outline a system of second-level concepts. Instead, 

                                                 
105

 [Frege 1971, 68-69]. 
106

 He writes to Hilbert, for instance, that ―At any rate, your idea of considering Euclidean geometry as a 

special case of a more inclusive theory is valuable even without [successful independence proofs]‖ [Frege to 

Hilbert, in Frege 1971, 15-16]. 
107

 Before our more detailed discussion in the following chapter it might be helpful to note that Frege‘s 

hierarchy begins with all objects at the lowest level, and then ascends to first-level concepts and functions, 

second-level functions, etc.  



97 

 

Hilbert‘s work describes a structure which is itself free from the fixed references of any one 

interpretation, but which may thus be applied to systems of a variety of levels. The 

usefulness of such a free-floating picture of mathematical structures is apparent in the many 

applications of, e.g., abstract group theory, which range from the study of the symmetry 

groups of various particles in quantum physics to the use of groups as a tool to understand 

other abstract algebraic structures (e.g., polynomial rings). As Frege would surely note, 

however, this apparent usefulness does not preclude the emptiness, vagueness, or even the 

inconsistency of the concepts at issue. 

 

3.2.4 Blanchette on Frege‘s conception of analysis 

 

Many commentators
108

 on the Frege-Hilbert correspondence have sided against Frege in his 

estimation of Hilbert‘s project as an almost complete failure.
109

 A common thread has been 

that Frege was short-sighted, too conservative, and blind to the obvious advances evidenced 

in Hilbert‘s work. Patricia Blanchette, in an illuminating paper on Frege, Hilbert, and 

consistency, outlines the reasons why this evaluation of their correspondence has seemed 

attractive: 

 

Frege was clearly of the view that principles of logic can be expressed as syntactic 

transformation rules; his own formal system used precisely the kinds of topic-neutral, 

syntactically specified rules [that Hilbert employed in his Grundlagen]. Given a set of 

sentences in Frege‘s Begriffsschrift notation, a Hilbert-style consistency proof will 

demonstrate that no contradiction is derivable from that set via Frege‘s own rules of 

logic. Hilbert-style independence proofs, similarly, would successfully demonstrate the 

nonderivability in Frege‘s system of one of Frege‘s sentences from a set thereof. It 

looks, therefore, as if Frege‘s assessment of Hilbert‘s demonstrations as ―failures‖ is 

simply wrong.
110

 

 

On this view, Frege‘s own work had set out to achieve the same goals as Hilbert‘s and by the 

same means, so that his rejection of Hilbert‘s views seems simply to have missed their point. 

But, as Blanchette notes,
111

 one crucial difference between Hilbert‘s approach to consistency 

and Frege‘s is that Frege takes it to be the case that thoughts (and not sentences) are the 

things which are proven or disproven, and it is the capacity of sentences to represent the 

logical structure of thoughts which is the subject of analysis. Thus, the derivability or non-

derivability of a sentence from a collection of other sentences is by no means the whole story 

                                                 
108

 See, for instance [Freudenthal 1962] and [Torretti 1984].  Freudenthal writes that ―Frege, rebuking Hilbert 

like a schoolboy, also joins the Boeotians. (I have never understood why he is so highly esteemed today‖ 

[Freudenthal 1962, 618]; Torretti writes that ―To demand like Frege that the meaning of [Hilbert‘s basic, 

uninterpreted concepts] be intuitively elucidated shows a lack of understanding of the nature of logical 

consequence that is indeed astonishing in the founder of modern logic [Torretti 1984, 251] and that ―Frege‘s 

failure to understand abstract axiomatics comes out very clearly in his criticism of Hilbert‘s independence 

proofs‖ [Torretti 1984, 251]. More recently, Stewart Shapiro suggests of Hilbert‘s view that ―Frege did not get 

it, or did not want to‖ [Shapiro 2005, 66]. 
109

 ―Clever and inventive as [Hilbert‘s Grundlagen] is in many points, I think that it is on the whole a failure 

and in any case that it can be used only after thorough criticism‖ [Frege to Liebmann, in Frege 1980, 91]. 
110

 [Blanchette 1996, 322]. 
111

 [Blanchette 1996, 322-323]. 



98 

 

for Frege. If we ignore this difference, then several of Frege‘s criticisms seem considerably 

weaker, pedantic, or even absurd. Keeping this difference in mind, however, we get quite a 

different picture. 

 

Let‘s take a closer look at exactly what is going on here, beginning with an examination of 

Frege‘s understanding of logical analysis. As Blanchette points out in her recent book, 

Frege’s Conception of Logic, the idea of analysis is central to Frege‘s overall logicist 

project.
112

 So what exactly is analysis for Frege, why is it important for his logicist project, 

and why is it relevant for our present purpose of understanding his opposition to Hilbert? 

 

While Frege‘s views of analysis shift in relation to his broader philosophical and logical 

development,
113

 it seems clear that for him analysis is never merely a linguistic affair, but 

essentially involves the realm of thoughts (or conceptual contents). Analysis often involves 

the reduction (in some way) of a complex concept to simpler concepts. Often, such analysis 

is made quite easy by language. Thus, for instance, we might wish to analyze the concept ‗x 

is an orange house‘. In this case, our language makes it very clear that the concept is likely 

not a simple one, and therefore an appropriate subject for continued analysis. Part of the 

reason the concept strikes us as complex is because of our use of the two terms ‗orange‘ and 

‗house‘ in the expression above. A straightforward analysis of this complex concept would 

be to treat it as a concept under which fall all and only those entities which fall under both 

the concept ‗x is orange‘ and the concept ‗x is a house‘. Here the underlying logical 

complexity is obviously represented by the linguistic complexity of our expression. 

 

But language is not always so cooperative, and we can (and often do) represent complex 

concepts with linguistically simple terms. Mathematics, in particular, is filled with this sort 

of thing: the definition of the linguistically simple term ‗group,‘ for instance, involves at 

least four logically complex axioms. This isn‘t a problem if the complexity is recognized by 

those employing the term (as is certainly the case for group theorists using the term ‗group‘). 

But we can also be misled into thinking that linguistic simplicity mirrors logical simplicity. 

Thus, for example, we might think that the basic notions of arithmetic or Euclidean 

geometry are logically simple when they might be ‗simple‘ only relative to a given linguistic 

rendering of those theories. 

 

As Blanchette points out, a significant portion of Frege‘s work involves illustrating that 

some of the apparently simple mathematical concepts of arithmetic are in fact complex. She 

gives a useful example of this phenomenon in relation to Frege‘s analysis of the successor 

function. 

 

She suggests that, for Frege, the relatively simple statement that ―One is the successor of 

zero‖ ought to be analyzed into the obviously far more complex claim that 

 

                                                 
112

 This is the key theme of [Blanchette 2012, chapter 1].  
113

 See [Blanchette 2012, 7-23] for an examination of the role of analysis in Frege‘s works from the 

Begriffsschrift to the Grundgesetze. 



99 

 

(A) For some concept F and some object a falling under F, the extension of the concept 
equinumerous with the concept identical with the extension of the concept 

(equinumerous with the concept (not self-identical)) is the extension of the concept 

equinumerous with the concept F, and the extension of the concept equinumerous 

with the concept not self-identical is the extension of the concept equinumerous 

with the concept falling under F, but not identical with a.
114

 

 

The point of Frege‘s analysis is to show that these two markedly different statements 

ultimately express the same thought. The real logical structure of the thought that one is the 

successor of zero is actually better represented by the more complex statement.  

 

As Blanchette points out, this kind of analysis is crucially important for Frege‘s project of 

showing that arithmetic can be developed on the basis of logic alone. For much of Frege‘s 

project involves illuminating the hidden logical structure beneath the linguistic surface of 

arithmetic. But, as anyone familiar with Frege‘s technical work can attest, such analyses are 

rarely obvious, easy, or even intuitive. Now, this again prompts a certain question: why 

should we take the result of Frege‘s analysis (i.e., the statement (A)) to be a better 

representation of the actual logical situation we are interested in when we speak of the 

successor function? Why should we accept his analysis at all? Given their importance for his 

project, one would imagine that Frege would provide a thoroughgoing account of the nature 

of his analyses and give us reasons for accepting the accuracy of their results. But, as 

Blanchette notes, 

 

Notoriously […] he offers nothing of the sort. Though he repeatedly affirms the central 

importance of analysis, Frege never defines this notion, and—most importantly—

never gives an account of the necessary and sufficient conditions for successful 

analysis. We are left to look at the details of the analyses he gives, and at his various 

scattered remarks about the nature of analysis, to determine what he took himself to be 

providing under the heading of ―analysis‖ or ―elucidation‖ and to assess the 

plausibility of his general approach.
115

  

 

In the present context, our goal is not to provide a general account of Frege‘s notion of 

analysis. What is relevant for us here is how Frege‘s conception of analysis bears on his 

criticisms of Hilbert. There are two areas in his critique which seem to directly involve his 

broad conception of analysis. The first is in relation to Hilbert‘s consistency proofs, the 

second in relation to his independence proofs. (In the following section, we will also see that 

Frege‘s capacity to provide logically accurate analyses plays a significant role in his 

suggested emendations of Hilbert‘s system). 

 

3.2.5 Blanchette on derivability and provability 

 

Frege‘s claim is that if Hilbert wishes to employ a purely uninterpreted system (i.e., one with 

no semantics) then he may discuss relations of derivability between the sentences of his 

                                                 
114

 [Blanchette 2012, 24; emphasis removed]. 
115

 [Blanchette 2012,24]. 



100 

 

system, but not between thoughts. For uninterpreted sentences do not possess any 

determinate reference and are thus incapable of expressing thoughts for Frege.  

 

Now, none of this would matter if Hilbert‘s syntactic brand of consistency always exactly 

overlapped with Frege-consistency, i.e., if the relations between the sentences of his formal 

language perfectly mirrored those between thoughts. But, as Blanchette notes, this is not at 

all the case, and, for Frege, simply cannot be the case. Indeed, she writes that ―The 

inconsistency of a set of thoughts despite the syntactic consistency [i.e., Hilbert-consistency] 

of a set of sentences expressing those thoughts is not simply a farfetched possibility for 

Frege. It is of the very essence of his views on analysis and thoughts that such cases are to be 

expected.‖
116

 Even supposing that Hilbert‘s sentences somehow express thoughts—which, 

for Frege, was already quite a dubious supposition—the problem of equivocating between 

consistency at the level of sentences and consistency at the level of Fregean thoughts would 

have remained. The problem stems from the fact that sentences which are syntactically 

different can yet express the same thought (as we‘ve just seen in relation to Frege‘s analysis 

of the successor function). Blanchette explains that 

 

Because each thought is generally expressible by a number of different sentences, 

there is much more to the relation of provability [which holds between thoughts] than 

is evidenced by the relation of derivability [which holds between sentences]. Where p 

is a thought and s a sentence expressing it, Π a set of thoughts and Σ a set of sentences 

expressing Π: the derivability of s from Σ guarantees that p is a consequence of Π, but 

the fact that s is not derivable from Σ is no guarantee that p is not a consequence of Π. 

For s‘s nonderivability from Σ is entirely compatible with the existence of some sꞌ and 

some Σꞌ expressing p and Π, respectively, such that sꞌ is derivable from Σꞌ. Relations 

between sentences of a well-designed formal system can always, for Frege, provide a 

positive test of provability and consequence, but never a negative test.
117

 

 

(Blanchette‘s terminological distinction here between the relation of provability between 

thoughts and the relation of derivability between sentences is a useful one, which we shall 

employ in what follows). The key point here is that multiple different sentences can express 

the same thought. This is an essential component of Frege‘s philosophy of language and 

(allowing for differences of opinion regarding the nature of ‗thoughts‘) a presupposition 

shared by many philosophers and non-philosophers alike. Simply put: there is, in general, 

more than one way to say the same thing. 

 

But, importantly, the relation of derivability can only hold between particular groups of 

sentences. In the case at hand, derivability claims will pertain to the sentences of Hilbert‘s 

axiomatization in the Grundlagen. At best, for Frege, what Hilbert is capable of showing is 

that a given sentence S is derivable from a sentence or collection of sentences within his 

formal theory, using the allowable syntactic transformations. But, given the fact that the 

same thought can be expressed in different ways, we can very easily imagine a different 

sentence, Sº, which expresses the same thought as Hilbert‘s S but which is not derivable 

                                                 
116

 [Blanchette 1996, 325]. 
117

 [Blanchette 1996, 325]. 



101 

 

from the same sentence or collection of sentences. For example, in the absence of definitions 

of the terms ‗sine‘, ‗continuous‘ etc. the sentence ‗the sine function is continuous at zero‘ is 

not derivable from the sentence ‗the sine function is differentiable at zero‘; with appropriate 

definitions (i.e., ones that adequately express the usual conceptual content), however, the 

derivation will go through. What this shows is that derivability and provability are different 

relations, and, subsequently, if Hilbert wishes to equivocate between the two, he will have 

claimed more than he has actually proved. 

 

Frege‘s criticism, then, is not simply that Hilbert‘s sentences fail to refer for a whole host of 

reasons, but that, even if they did manage to express thoughts, still Hilbert would be 

incapable of proving what he believes he has proved. For Hilbert has not paid sufficient 

attention to the differences between derivability of sentences and the provability of thoughts. 

Ultimately, as Blanchette notes 

 

Frege was right to hold that the method of interpretations holds no promise for 

demonstrating what he took to be consistency and independence results. The point here 

is not merely terminological: Frege takes it that the consistency of a set of thoughts, 

and the independence of thoughts from one another, have important philosophical 

ramifications, none of which is demonstrable via Hilbert‘s method.
118

 

 

3.2.6 Independence and logical simplicity 

 

We‘ve seen why Frege‘s views of analysis and provability led him to reject Hilbert‘s 

interpretation of his consistency proofs (or, better, his derivations of consistency within his 

particular formal system). Let‘s now turn our attention to Blanchette‘s suggestion (in the 

above quotation) that there are important implications for Frege‘s understanding of Hilbert‘s 

independence proofs as well. 

 

As we saw above in Frege‘s example of the successor function, relatively simple expressions 

can go along with considerable logical complexity. Thus, we may very often be incorrect 

when we take linguistically simple expressions to straightforwardly represent logical or 

conceptual simplicity. The goal of many of Frege‘s analyses is to bring out exactly this 

hidden logical complexity, so that its role in inferences can be directly examined. And, in 

fact, one of the chief goals of Frege‘s logicist project is to unearth the hidden logical 

complexity of mathematics in order to prove that basic arithmetic can be founded on pure 

logic alone. Ultimately, this view has two implications. The first is that we can be wrong 

about logical simplicity: we may, for example, take a linguistically simple term like 

‗number‘ to express true logical simplicity, when in fact (at least according to Frege) it 

contains hidden logical complexity. And, second, the revelation of this logical complexity 

can allow us to prove things which we otherwise might not have been able to prove. 

 

This view has direct implications for Hilbert‘s independence proofs. These proofs, recall, 

have the aim of showing that ―no essential part of any of these groups of axioms can be 

                                                 
118

 [Blanchette 1996, 336]. 



102 

 

deduced from the others by logical inference.‖
119

 Recall, too, that each of the axioms 

contains ‗primitive‘ or (putatively) logically simple terms, e.g., ‗point‘. We can see the 

difficulty posed for these proofs by Frege‘s views on simplicity: if the putative logical 

simples of Hilbert‘s system are not in fact simple, but (upon further analysis) have additional 

logical structure, then it may in fact be the case that we can deduce some of the supposedly 

independent axiom groups from one another. 

 

Thus, for instance, if, as Frege later suggests, the putatively simple term ‗point‘ in Hilbert‘s 

axiomatization actually refers—if it is to refer to anything at all— then it refers to a second-

level concept ranging over all first-level point concepts. Though he does not go through with 

the attempt, the implication is that such hidden complexity may very well allow us to prove 

the dependence of some of Hilbert‘s putatively independent axioms. Here, in the end, it 

seems that Frege‘s criticism ultimately amounts to the claim that he offers a better analysis 

of the actual logical situation than Hilbert does. As Blanchette has shown us, however, the 

success of this and other of Frege‘s criticisms relies on the actual superiority of his 

analyses—and Frege himself provides no systematic means of establishing this superiority. 

 

In any case, as Blanchette‘s work helps us to see very clearly, Frege‘s criticism is not simply 

that Hilbert‘s sentences fail to refer for a whole host of reasons, but that, even if they did 

manage to express thoughts, still Hilbert would be incapable of proving what he believes he 

has proved (at least by Frege‘s lights). For Hilbert has not paid sufficient attention to the 

relations between derivability from sentences vs. provability from thoughts, nor of the 

linguistic relativity of his own notion of logical simplicity. Against Torretti‘s view of Frege 

as astonishingly blind to the nature of logical consequence, then, Frege‘s criticisms of 

Hilbert seem to indicate an extremely subtle appreciation of the logical relations between 

sentences and what they are meant to represent (i.e., thoughts in Frege‘s case)—one which 

even modern commentators have failed to notice. And it is just this kind of appreciation 

which Frege lambastes Hilbert for lacking when he claims more than Frege believes he is 

capable of proving. 

 

3.3 Frege‘s proposed emendations of Hilbert‘s project 

 

By 1900 Frege had already expended considerable effort developing his logical and semantic 

concepts: he had re-organized and formalized logic by way of his Begriffsschrift and had 

developed a detailed theory of meaning in a series of papers, most notably Über Sinn und 

Bedeutung. The result of his logical and linguistic work was a theory of meaning based upon 

the notion of a complete thought. For Frege, sensible discourse was only possible when our 

words manage to express what he called complete thoughts, which are those objective 

entities which can be either true or false. If geometry and mathematics are to be understood 

as sciences, then, Frege argued, they must aim at truth. Since truths are a particular kind of 

complete thought, the mathematical sciences (like all the others) must be composed of 

complete thoughts. Complete thoughts result only from the saturation or completion of a 

                                                 
119

 [Hilbert 1971, 32]. 



103 

 

concept
120

 by an entity of an appropriate level (i.e., an object if the concept is a first-level 

concept, or a first-level concept if the concept is of the second-level, etc.). For Frege, then, 

the very conditions of meaningfulness hinge upon carefully distinguishing between entities 

of different ontological levels. This is particularly evident in modern mathematical 

discourse, which deals with relations between entities of different levels more frequently 

than most other areas of human enquiry. 

 

Both Hilbert and Korselt insisted upon the free-floating, schematic aspect of the 

uninterpreted axiomatic system of geometry presented in the Grundlagen. For Frege, with 

his careful semantic and logical distinctions, this system and its lack of fixed references 

could only result in meaningless discourse. Since meaning attaches only to complete 

thoughts, and complete thoughts require the combination of entities of appropriate 

ontological types, any system of sentences which ignores distinctions between entities of 

different ontological types must fail to deliver meaningful claims. Since Hilbert‘s suggested 

reading of his system of geometry as a mere scaffolding ignores precisely these kinds of 

distinctions, it must be meaningless in its current form, according to Frege.  

 

Again, however, this is not to say that Frege rejected the notion that a higher-level system of 

geometry was possible at all. On the contrary, the idea that Hilbert‘s geometry might 

describe a higher level structure was perfectly acceptable for Frege, and he even expended a 

considerable amount of effort attempting to provide a clear logical foundation for just such a 

structure. His key objection to Hilbert, for our purposes, was that Hilbert‘s free-floating view 

required the description of a structure which was completely indifferent to any ontological 

level. Such a system would be meaningless for Frege. More importantly, without fixing the 

references of even ‗structural‘ terms, Frege took it to be the case that Hilbert‘s geometric 

work (e.g., his independence and consistency proofs) ultimately relied on a kind of logical 

sleight of hand whereby claims pertaining to a determinate system of second-level concepts 

were substituted for those pertaining to an equally determinate first-level system, all under 

the guise of an uninterpreted structure which was supposedly free from any particular 

ontological level whatsoever.  

 

Though Frege was dubious of the possibility (and usefulness) of successfully altering 

Hilbert‘s picture of geometry to answer his criticisms, he nevertheless attempted to provide a 

reformulation of this picture in terms that he himself could accept. This reformulation 

entailed embedding Hilbert‘s supposedly free-floating mathematical structures within his 

ontological hierarchy. This embedding first required, however, a method of giving a 

determinate sense to Hilbert‘s axioms, definitions, and theorems. In order to overcome this 

rather serious stumbling block, Frege suggested a way in which Hilbert‘s ‗pseudo-

propositions‘ could be reformulated to express complete thoughts, and thereby attain the 

truth values required of a proper science. Let us examine how he believed this could be done. 

 

                                                 
120

 More generally, complete thoughts result from the saturation of a function of a certain kind by one or more 

entities of the appropriate type(s); Frege viewed (first-level) concepts as functions which take objects as 

arguments and deliver truth values as outputs. I will discuss Frege‘s ontological hierarchy in more detail in the 

following chapter. 



104 

 

For Frege, pseudo-propositions result from the tendency to break up conditional propositions 

into antecedent and consequent parts, neither of which expresses a complete thought. Take 

the following conditional proposition, for instance: 

 

If a is a whole number, then (a x (a – 1)) is an even number.
121

 

 

This propositional complex expresses a complete thought, and utilizes the dummy letter ‗a‘ 

as a means of indicating generality. Now, we can—and often do—split up such conditional 

propositions for stylistic, pedagogical, or other reasons. For example, the above statement 

can be split up into two pseudo-propositions as follows: 

 

a is a whole number. 

(a x (a – 1)) is an even number. 

 

In the complete conditional propositional complex above, we express the thought that if a 

whole number is of the form (a x (a – 1), then it falls under the concept of being even. Here 

the sign ‗a‘ does not refer to a specific number, but serves only to lend generality to the 

proposition; it allows us to claim that any whole number of the appropriate form possesses 

the property of being even. (Frege says that such signs indicate but do not refer). It is 

important that, without the stipulation that a is a whole number, the phrase ‗(a x (a – 1)) is 

an even number‘ does not express the same thought, or, indeed, any thought at all. Without 

carrying over the antecedent condition that a is a whole number into the consequent claim 

that (a x (a – 1) is an even number, we fail to express a complete thought. For in both 

isolated pseudo-propositions, the apparent function of the dummy letter ‗a‘ is not to lend 

generality to the thought expressed. In the consequent pseudo-proposition, for instance, its 

function (or at least its purported function) is to refer to an object and predicate of that object 

that it possesses the property of being an even number. But the expression ‗(a x (a – 1))‘ 

does not refer to any specific object and so the truth or falsity of the pseudo-proposition 

remains indeterminate for Frege. The result of splitting up a conditional proposition into 

separate expressions of its antecedent and consequent components is not two propositions, 

each possessing a truth value, but two pseudo-propositions, neither of which is a complete 

thought and neither of which can be true or false. Frege writes that 

 

[...] we must not let ourselves be deceived by the fact that for stylistic reasons, an 

antecedent pseudo-proposition occasionally occurs in such a form that upon cursory 

examination it appears to be an explanation of one or more letters. For in fact neither 

these putative explanations nor the proposition in which the conclusion is asserted are 

real propositions. Rather, being antecedent and consequent pseudo-propositions, they 

belong inseparably together, so that only the whole constituted of them is a real 

proposition. It would greatly facilitate insight into logical structure if what is a single 

real proposition according to its subject matter were also a unitary propositional 

complex according to its grammar and did not break down into independent 

propositions. To be sure, in our word-languages such propositional complex[es] would 

                                                 
121

 This example is found in [Frege 1971, 72]. 



105 

 

sometimes attain a monstrous length, whereas, because of its perspicuous nature, the 

Begriffsschrift is better suited to the representation of the logical fabric.
122

 

 

Frege came to the conclusion that something like the confusion between real and pseudo-

propositions occurs in Hilbert‘s (and Korselt‘s) attempt to outline a formal axiomatic system 

for geometry. The logical structure of Hilbert‘s axiomatic system is greatly obscured by his 

use of the familiar terms ‗point,‘ ‗line,‘ ‗lies on,‘ ‗between,‘ etc. By employing these familiar 

terms, Frege argues that Hilbert unwittingly engages in a dangerous equivocation. At one 

moment these basic terms are to be understood as referring to specific entities (the concepts 

and objects of traditional Euclidean geometry, or their correlates within Hilbert‘s model of 

Euclidean geometry within the real numbers). At another moment, however, these same 

terms are intended to function as purely uninterpreted dummy signs, meant only to lend 

generality to the propositions in which they occur. Through this confusion the fact that 

Hilbert‘s claims are either meaningless or possess quite different meanings than he intended 

is simply glazed over, and the real logical situation quite obscured. 

 

Frege, though clearly critical of Hilbert‘s endeavour, attempts to clarify the logical situation 

by employing the notion of a pseudo-proposition. Instead of Hilbert‘s ‗defining axioms‘—

which, by Frege‘s lights, do not define at all, as they do not fix the references of their 

respective definienda—we have a system of antecedent pseudo-propositions, none of which 

expresses a complete thought. Similarly, in place of the theorems Hilbert proves on the basis 

of his axioms and definitions, we have a system of consequent pseudo-propositions, none of 

which, again, expresses a complete thought. A theorem or axiom, taken in isolation, does not 

express any thought insofar as it contains signs which do not have a fixed reference. 

Moreover, taken in isolation, these empty signs do not lend generality to the pseudo-

propositions within which they occur any more than did the sign ‗(a x (a – 1)‘ in our 

previous pseudo-proposition.  

 

Frege‘s method for salvaging Hilbert‘s work from this type of meaninglessness involved 

combining his axioms and theorems into appropriate propositional wholes which expressed 

complete thoughts, and whose terms properly indicated or referred. Unfortunately, in the 

case of Hilbert‘s work, these wholes are exceedingly complex: for in order to give sense to a 

particular theorem (consequent pseudo-proposition) we must combine it with an antecedent 

pseudo-proposition which contains all the relevant ‗axioms‘ for the system we are interested 

in. In other words, each actually meaningful theorem will contain, as an inseparable part, an 

entire collection of ‗definitions‘ and axioms connected together via conjunction. Of course, 

expressing this complex logical edifice in ordinary language would be extremely unwieldy—

hence Frege‘s recommendation of his own, more perspicuous, logical formalism in the 

quotation above. For any theorem of Hilbert‘s, reformulated according to Frege‘s 

specifications, we will have a conjunction of conditions (the axioms) followed by the 

statement of the appropriate theorem. Ideally, once we‘ve done this, the appearance of a 

purely relational, or referentially indeterminate system should disappear.  

 

                                                 
122

 [Frege 1971, 73-74]. 



106 

 

Let‘s examine how this would work in the case of the first ‗theorem‘ (really three theorems) 

of Hilbert‘s Grundlagen. These are stated as follows: 

 

THEOREM 1. Two straight lines of a plane have either one point or no point in 

common; two planes have no point in common or a straight line in common; a plane 

and a straight line not lying in it have no point or one in common.
123

 

 

Now, in the traditional Euclidean context, these theorems are perfectly meaningful and 

require no further interpretation. They describe first-level relations amongst objects, and the 

claims they express are inferred directly from the (ostensibly true) axioms. Importantly, too, 

the basic terms of traditional Euclidean geometry are supposed to refer to particular objects 

and concepts: we grasp the reference of a particular term by means of definitions or (what 

Frege would call) elucidations. Hilbert‘s view is quite different, as is Korselt‘s; for both, the 

theorem does not pertain (or at least does not solely pertain) to particular objects or concepts. 

Instead of referring, the basic terms of Hilbert‘s system are intended to lend generality to the 

propositions in which they occur, such that these propositions can (by means of a suitably 

outlined interpretation) hold true of a variety of domains. Until such an interpretation is 

outlined, however, these theorems are of a purely formal character, having no determinate 

reference and no need of one.  

 

Frege was quite opposed to the notion that purely formal, or completely uninterpreted signs 

could be of any use in science—as science concerns itself with truth, and truth adheres only 

to complete thoughts. Moreover, he argued that the appearance that these terms are actually 

completely uninterpreted in Hilbert‘s work was simply an after-effect of poor terminological 

choices. More specifically, Frege argued that the use of the traditional Euclidean 

terminology (e.g., ‗point,‘ ‗lies on,‘ etc.) in the place of proper variables for the purely 

formal system had the effect of obscuring the true logical status of Hilbert‘s claims. By 

replacing this misleading Euclidean terminology in the first of the above theorems with 

appropriately differentiated variables (‗x,‘ ‗y,‘ and ‗z‘ are variables for objects; ‗P,‘ and ‗R,‘ 

variables for relations between objects), we get something like the following: 

 

(H1) Two x standing in relation P to y either stand in relation R to one z or to no z. 

 

Here it becomes quite apparent that, without further clarification of the references of the 

terms at issue, we not only do not have a theorem, we have no thought at all. When the 

purported theorem is more perspicuously laid out to illustrate its generality, it becomes quite 

clear (as Hilbert would accept) that, unlike Euclidean geometry, the basic terms of Hilbert‘s 

system do not seem to have fixed references. But neither is it clear exactly how (or if) they 

serve to lend generality to the theorem. Frege writes that ―here, unless we are to understand 

the words in the Euclidean sense, just about everything is unknown‖.
124

 Frege‘s contention is 

that, either we make the Euclidean claim and Hilbert‘s work does not have the import it is 

supposed to, or we accept that the basic terms are intended to lend generality, in which case 

the result is at best an incomplete or partial thought, and at worst no thought at all.   

                                                 
123

 [Hilbert 1950, 3]. 
124

 [Frege 1971, 90]. 



107 

 

 

Clearly Hilbert‘s intention is not to make the traditional claim that two straight Euclidean 

lines which lie in the same Euclidean plane either have one Euclidean point or none in 

common. Rather, his intention seems to be to claim that for any specific system which 

satisfies his axioms on a particular interpretation, the theorem (whose terms will have 

references tailored to the interpretation) will come out true. Frege‘s ultimate point is that 

statements like this do not require—and indeed are not possible on the basis of—

indeterminate reference. Instead, they are statements which (when suitably reworked along 

his lines) make very clear and fixed reference to second-level concepts and the first-level 

concepts which fall within them. His re-imagining of Hilbert‘s work involves construing the 

supposedly purely formal system as one containing clusters of pseudo-propositions, 

themselves meaningless, which can be made to express genuine thoughts by proper 

combination. Once this has been achieved, the resulting propositions do not contain 

indeterminately referring terms but, instead, terms that refer to second-level concepts and the 

first-level concepts which fall within them. Thus, for instance, supposedly ‗uninterpreted‘ 

statements about points, lines, etc. are taken to express determinate claims about second-

level ‗point-‘ and ‗line-‘ concepts, i.e., concepts within which fall first-level concepts like 

those of traditional Euclidean geometry or Hilbert‘s interpretation of the real numbers.  In 

place of Hilbert‘s own statement of his first theorem, then, we obtain a lengthy conjunction 

of axioms and definitions as antecedent pseudo-propositions combined with the re-

configured statement (H1). By characterizing Hilbert‘s axioms as conditions pertaining to a 

system of second-level concepts, Frege gives them a fixed reference. By including these 

conditions as the antecedent portion of every theorem, he also makes it possible for the 

variables in (H1), for instance, to properly lend generality to the complete second-level 

proposition. Finally, in place of interpretations, inferences are made from these general 

propositions to particular propositions, e.g., from second-level claims about all point-

concepts to claims about particular first-level point concepts.  

 

In Hilbert‘s work (and others like it), many geometers believed they had found the answer to 

their questions about the nature of geometry. Geometry was a science which studied the 

deductive elaboration of particular types of axiomatic structures, irrespective of their 

relationships to physical space. One underappreciated effect of Frege‘s criticisms of Hilbert 

(and Korselt) was to draw in sharper relief the shortcomings of the purely relational or free-

floating understanding of geometrical structures. In Frege‘s opinion, the answers offered by 

Hilbert were merely chimerical, and relied upon various logical confusions in order to 

maintain the appearance of working properly. In place of this relational view, Frege offered a 

picture wherein various mathematical structures are embedded within a single ontological 

hierarchy. One key benefit of such a view was an increase in the precision of claims 

regarding the various relations between different structures. Thus, for instance, by careful 

attention to distinctions between Hilbert‘s axiomatic theory and its different possible 

‗interpretations‘, Frege is able to draw out the logical connections (or lack thereof) between 

the second- and first-level propositions of these respective theories. Of course Hilbert would 

not agree with Frege‘s view that his independence proofs were useless, but (as his later work 

would show) the logical perspicuity Frege brought to the table was an early indicator to 

Hilbert that understanding the logical relations between structures required a careful and 

attentive eye. In the following chapter, I will begin to examine problems with Frege‘s own 



108 

 

attempt to characterize his ontological hierarchy and employ it as a means of explaining the 

nature of (certain elements of) mathematics. 



109 

 

Chapter 3: Problems with Frege’s views 
 

§0. Introduction 

 

In the previous chapter we examined a number of problems pertaining to Hilbert‘s understanding 

of the nature of mathematics and mathematical structures. Hilbert‘s account of axiomatics was 

predicated on the view that we needn‘t fix the references of the basic terms of an axiomatic 

system in order to prove theorems in that system. As we saw, however, Frege raised a number of 

difficult philosophical questions for Hilbert. These chiefly pertained to Hilbert‘s purported 

inability to make it clear precisely what he is talking about in his discussions of mathematical 

structures like those required for his independence proofs. As we saw in that chapter, Frege‘s 

criticisms are themselves predicated on the view that mathematics (and, indeed, any meaningful 

speech whatsoever) is about something in particular, i.e., that it have a determinate content. 

While Hilbert‘s views took it for granted that we could speak about a purely uninterpreted 

system (or ‗scaffolding of concepts,‘ in Hilbert‘s words), the basis of Frege‘s attack was his 

understanding of the importance of fixity of reference and determinately circumscribed concepts. 

These criticisms of Hilbert seem to require that Frege himself have a clear and defensible 

understanding of how it is that we can fix the references of (at least) mathematical terms.  

 

In this chapter we will examine problems relating to Frege‘s own attempts to fix the reference of 

mathematical and logical terms within his ontological hierarchy. The first such problem is the so-

called concept ‗horse‘ problem. While this problem can be solved using resources available to 

Frege, the solution depends entirely upon his ability to clearly pin down the references of basic 

terms within his ontological hierarchy. The second problem, the Julius Caesar problem, is rather 

more difficult, and (I will argue) Frege never provides an adequate solution to it. This problem 

cuts to the heart of Frege‘s fixed-reference theory of mathematical structures, and, as we shall 

see, his inability to solve this obscure difficulty makes many of his criticisms of Hilbert fall flat. 

In the end, despite Frege‘s best efforts, I will argue that his account fails to explain how we can 

pin mathematical structures down within his ontological hierarchy of objects and functions.  

 

Moreover, in the final sections of the chapter, I will argue that the inability of Frege to fix the 

reference of certain mathematical concepts is illustrative of a more widespread difficulty still 

facing the philosophy of mathematics today. While the specific forms of the problems faced by 

Frege are not necessarily faced by the contemporary philosophy of mathematics, the general 

problem of fixing the references of mathematical terms (or explaining why we don‘t need fixed 

reference) is shared by all. The upshot of this chapter, when combined with my analysis of 

Hilbert‘s views, is that neither the free-floating account of structures nor the general view of 

structures as embedded within a single ontological hierarchy is satisfying, and that neither 

picture is capable of satisfactorily explaining what mathematics is about. 

 

Let‘s begin with an examination of Frege‘s ontological hierarchy. 

 

 

 

 



110 

 

§1. Frege’s Ontology 

 

For Frege, the entities
1
 proper to logic have a special ontological status, and are not reducible to 

(or otherwise eliminable in terms of) linguistic, mental, or physical entities. In particular, Frege 

takes the semantic content of meaningful speech and thought to possess independent ontological 

status, such that the things which are true or false (what he calls ‗thoughts‘) are not dependent 

upon the human mind or even the empirical world. This is a central aspect of his logical views, 

and one of the key theses of early analytic philosophy.
2
 Though Frege often insisted upon the 

strict separation of the science of logic from psychological concerns, his fundamental ontology 

is, by and large, indifferent to the distinctions between the mental, physical, and logical realms. 

Numbers, chairs, and ideas, for example, all occupy the same basic ontological niche: they are all 

objects. While Frege‘s ontology is not in this sense ultimately divided into the physical, mental, 

and the logical, it is not therefore unstructured. In fact, it is hierarchically organized into an 

infinite number of strictly separate levels. In what follows I will examine the organization of this 

hierarchy.
3
 

 

1.1 Objects 

 

At the most basic level of Frege‘s hierarchy, there are objects. In addition to the everyday entities 

we are used to calling objects (e.g., tables, dogs, windows, etc.) Frege includes a number of less 

familiar entities under this heading. For example the extensions of concepts, the True, the False, 

numbers, thoughts, ideas, etc. are all taken to be objects. Despite the extreme generality of the 

category, there are some characteristics that all objects have in common which distinguish them 

from other types of entity. On the linguistic level, for instance, objects are referred to by way of 

proper names or declarative sentences; this distinguishes them from functions and concepts, 

which are only (properly) referred to via a grammatical predicate or function-name. Another 

important (if somewhat metaphorical) way in which Frege distinguishes objects from other 

entities is through their completeness, self-subsistence, or ‗saturation‘. In contrast to other 

entities (i.e., functions), objects do not require and, indeed, do not accept any supplementation or 

saturation from other entities. They are already complete entities. In a slightly less metaphorical 

sense, objects have the property that they can only combine with other entities (objects or 

functions) if there is some function serving as combinator or tie. Frege writes, for instance, that 

―An object—e.g., the number 2—cannot logically adhere to another object—e.g., Julius 

                                                 
1
 A terminological note: here, and throughout the thesis, I employ the term ‗entity‘ to indicate only the barest 

ontological status. All entities possess being, in some sense, but not all entities are ‗things‘ or ‗objects‘. I reserve 

these latter terms to refer to a specific type of entity. Thus, I take functions or concepts (as Frege construes them) to 

be entities but not objects. 
2
 In his earlier Wissenschaftslehre [Bolzano 1837], Bolzano presents a very similar view. There Bolzano employs 

the term ‗proposition in itself‘ [Satz an sich] for the objective content of our spoken, written, or thought 

propositions.  Bolzano also held that, though there are such propositions in themselves, they are not actual, and, 

further, that the being of these propositions did not depend at all upon their being thought, or even upon their being 

thinkable. For a comparison of Frege‘s ‗thoughts‘ with Bolzano‘s ‗propositions in themselves‘, see [Künne 1997]. 

G. E. Moore and Bertrand Russell, too, dealt with entities similar in some respects to Fregean thoughts (see, e.g., 

[Moore 1953] and [Russell 1903]). 
3
 [Wells 1951, 542] gives a succinct table which partially articulates the structure of this hierarchy, though it fails to 

include functions of more than two variables. [Mendelsohn 2005, chapter 5] also develops a type-system for Frege‘s 

ontology. 



111 

 

Caesar—without a binding agent. But that may not be an object, but must be something 

unsaturated. A logical combination can only become a whole by having an unsaturated part 

saturated or supplemented by one or more parts‖.
4
 Though slightly misleading, we might picture 

objects as the bricks required to build a logical edifice (a thought), and functions as the mortar 

binding them together—without both components, we end up with something less than the 

building we sought.  

 

1.2 Thoughts, subjective and objective 

 

For Frege, a very important distinction amongst objects is that between thoughts [Gedanken] and 

what we might call subjective thoughts. The latter, which most of us would call simply 

‗thoughts‘, are possessed by individual thinkers. We have subjective thoughts, and they, like us, 

come into and pass out of being. These thoughts are more like ordinary objects (mice, rugs, etc.) 

than the abstract entities Frege calls thoughts. Depending upon one‘s theory of mind, such 

subjective thoughts may or may not be physical in character. For Frege, in any case, it does seem 

to be the case that the subjective element of thinking is limited (in humans at least) to the mind 

of an individual thinker, such that no one other than myself can have direct access to my 

subjective thoughts, for example.  

 

To understand why Frege posits the existence of non-subjective thoughts, we need only recall 

that it often seems to be the case that two individuals share the same thought—perhaps we both 

think that 2+2=4 or that it is a beautiful day. Though our internal picture or subjective grasp of 

these thoughts may differ significantly—I may, for instance, imagine four oranges arranged in a 

certain pattern, while you may imagine a sequence of squares when thinking that 2+2=4—there 

is in these claims some content that we seem to share. The possibility of sharing thoughts (or at 

least part of their content) is central to human communication in general, and crucial to the 

progress of science more specifically. Frege‘s belief was that the objectivity of scientific and 

mathematical truth (which he did not question) required that there be something over and above 

our individual, subjective thoughts which guaranteed the possibility of communication. In his 

many arguments against psychologism in logic,
5
 for instance, Frege contends that, if we restrict 

thoughts to the minds of individuals, then there can be no rational adjudication between 

seemingly contradictory claims. If, for example, you hold that 2+2=5 and I hold that 2+2=4, the 

appearance of contradiction between our views evaporates if our understanding of the meaning 

of terms like ‗2‘ and ‗=‘ is limited to our own minds. There will be your 2 and my 2, each 

possessing its own distinct properties, and there will be no possibility of our deciding which of 

our statements is true, simpliciter. Yours will be ‗true for you‘ while mine will be ‗true for me‘. 

As a succinct statement of Frege‘s objections against this type of psychologism, take the 

following two sentences: ―A criterion for whether a mode of connection constitutes a thought is 

that it makes sense to ask whether it is true or untrue. Associations of ideas are neither true nor 

untrue‖.
6
  

                                                 
4
 [Frege 1903, 372]. 

5
 For a clear exposition of Frege‘s position against psychologism, as well as a fair presentation of the doctrine itself, 

see [Resnik 1980, 25-53]. The first nine chapters of [Husserl 2001] also contain an extremely detailed account and 

refutation of the works of the psychological logicians Frege inveighed against in his own work. 
6
 [Frege c. 1906b, 174]. It is worth noting that Frege does not think that judgments are simply associations of ideas.  



112 

 

 

Because Frege believed in the objectivity of scientific truths, he argued in favour of a realm of 

objective, non-empirical, atemporal entities which are represented or grasped by individual 

thinkers with the aid of various tangible devices like spoken and written language or mental 

processes. In one of the most unfortunate terminological decisions in philosophical history, Frege 

chose to call these entities ‗thoughts‘ [Gedanken].
7
 These objective thoughts differ markedly 

from their subjective cousins. For one, no one ‗thinks‘ a Fregean thought—instead, as he says, 

we apprehend or grasp them. These thoughts are not the possessions of the individuals who 

apprehend them, nor do they come into or pass out of being. For Frege, objective thoughts do not 

even require the possibility of being grasped by a thinker in order to possess being. They are, in 

the strictest sense, independent entities, with properties quite distinct from those of subjective 

thoughts. 

 

Now, such remote entities would be entirely useless for the purposes of science if humans, as the 

limited beings we are, could not somehow encounter them. Frege‘s claim is that we grasp these 

entities and, through various media (e.g., mental processes and languages) we are able to 

represent and communicate their structure to ourselves and others. The nature of the process of 

apprehension is never satisfactorily clarified by Frege and at one point he refers to it as ―perhaps 

the most mysterious of all‖
8
. Nevertheless, after grasping a thought we are, somehow, capable of 

representing or expressing aspects of its structure by way of language or subjective thinking. The 

idea seems to be that certain tangible objects (e.g., written inscriptions or mental ideas) represent 

or express these intangible, causally inert
9
 thoughts, which are then apprehended. Natural 

languages like German and English help us to grasp thoughts by mimicking their logical 

structure as well as the inferential relations between them. But often they do so only imperfectly, 

such that language is as much a barrier to the representation of thoughts as it is a facilitator. The 

intention of Frege‘s Begriffsschrift was to construct a formal language which would help 

eliminate the logically extraneous elements of natural language so that only the inferentially 

important elements of the realm of thoughts were represented. Unfortunately, the relation of 

                                                                                                                                                             
Rather, by the 1890s, at least, Frege understands judgment to be a mode of connection which does involve truth. He 

thinks of judgments as the admission or acknowledgement [Anerkennung] of the truth of a thought, and also as an 

advance ―from a thought to a truth-value‖ [Frege 1892b, 35; in Beaney 1997, 158]. But, importantly here, the 

judgment itself is not true, it is only a mental act whereby the truth-value (a non-mental entity) of a thought (also a 

non-mental entity) is admitted.  
7
 I will occasionally refer to Fregean thoughts as ‗objective thoughts‘ when the distinction between thoughts and 

their subjective, mental counterparts is at issue or in cases where the term ‗thought‘ seems especially misleading. 
8
 [Frege 1914; in Beaney 1997, 246]. 

9
 Though Frege insists upon the non-empirical nature of his ‗thoughts‘, in a later paper he seems inclined to attribute 

them at least some form of causal efficacy. Frege himself was troubled (as are many contemporary philosophers of 

mathematics) by the deep epistemological problems posed by purely abstract entities, as can be seen in the following 

passage: ―A thought, admittedly, is not the sort of thing to which it is usual to apply the term ‗actual‘ [‗wirklich‘]. 

The world of actuality is a world in which this acts [wirkt] on that and changes it again and undergoes reactions 

itself and is changed by them. All this is a process in time. […] And yet what value could there be for us in the 

eternally unchangeable, which could neither be acted upon nor act on us? Something entirely and in every respect 

inactive would be quite unactual, and so far as we are concerned it would not be there [nicht vorhanden]. […] 

Thoughts are not wholly unactual but their actuality is quite different from the actuality of things. And their action is 

brought about by a performance of the thinker; without this they would be inactive, at least as far as we can see. And 

yet the thinker does not create them but must take them as they are‖ [Frege 1918, 76-77; in Beaney 1997, 343-345]. 

(See [Dummett 1982] for a thorough analysis of Frege‘s use of the term ‗wirklich’). 



113 

 

representation or expression between the causal realm of written and spoken inscriptions and the 

causally inert logical realm, though central to Frege‘s project of developing a logically 

perspicuous language, is very little explored or explained by him. What we do get from Frege‘s 

remarks on the topic is that language can help us to clarify the logical structure of the realm of 

thoughts, but that it is rarely, if ever, a perfect mirror of that structure. 

 

Despite the flaws in language, it can be a useful logical tool. All thoughts are structured, and 

Frege seems to think that their structure is best represented by declarative sentences (perhaps 

because these sentences seem to express truths and falsehoods most directly). Frege‘s interest is 

in logic, and hence in the logical structure represented by declarative sentences. This is not to say 

that thoughts do not possess non-logical structure; simply that, in his work, Frege is only 

interested in thoughts construed as logical entities. For him logic is that science which is 

concerned primarily with the most general laws of truth.
10

 Hence, he is interested not in the 

emotional or aesthetic qualities of thoughts, but primarily with their relationship to truth. In his 

logical work, Frege characterizes thoughts specifically as those entities which are either true or 

false; whether or not a particular thought is true or false depends largely upon its internal 

structure, i.e. upon the particular way in which its elements are combined. Frege‘s analysis of the 

structure of thoughts is intimately connected to his ontological hierarchy: each thought is 

decomposable into at least two parts: one which is unsaturated, and one which saturates the 

unsaturated part. This latter part may itself be either saturated (i.e., an object) or unsaturated (i.e., 

a function, concept, or relation of appropriate level), depending upon the nature of the argument 

places of the unsaturated component.
11

 The unsaturated component of a thought must be a 

function or a concept. The bulk of Frege‘s ontological work is spent clarifying the ways in which 

various entities can or cannot combine to form the saturated wholes he calls thoughts. As we will 

see below, it is Frege‘s attempt to develop mathematics on the basis of this ontological work 

which leads him into trouble.
12

  

 

1.3 Truth and truth values 

 

As noted, for Frege thoughts are those things which can be true or false. But how does he 

understand the notions of truth and falsity? Like many of his modern descendants, he treats truth 

and falsity as values taken by certain types of functions (e.g., concepts) when entities of 

appropriate type are input as arguments, and these values are themselves objects. There are, as 

one might expect, two such values, namely the True and the False. The concept ‗x is a bottle,‘ for 

example, maps all and only those objects which are bottles onto the True, while all objects which 

                                                 
10

 ―[L]ogic is the science of the most general laws of truth‖ [Frege 1897, 139; in Beaney 1997, 228]. 
11

 Though only objects are themselves saturated, functions can complete or saturate other functions if they are of the 

appropriate level. The result of such saturation is, again, an object—often a thought. Thus, both objects and 

functions can saturate or complete other functions. 
12

 Perhaps the most direct attack on the coherence of Frege‘s basic distinction between complete and incomplete 

entities can be found in [Resnik 1965]. There Resnik‘s aim is to illustrate that it is Frege‘s metaphysical assumptions 

which lead to the difficulties of his logicist program. In many ways, the present chapter aims to arrive at the same 

claim by different means. Unlike Resnik‘s paper, however, my aim is to extend these metaphysical troubles to 

include post-Fregean work in the philosophy of mathematics (including Resnik‘s own account (s) of structures in 

[Resnik 1981] and [Resnik 1999]). 



114 

 

are not bottles are mapped onto the False.
13

 Though we are not primarily interested in Frege‘s 

linguistic views here, it is interesting to note that he takes the role of declarative sentences to be 

logically identical to (what we would normally call) proper names, and the objects which they 

name or refer to are the True (if the sentence is true) and the False (if the sentence is false).
14

 

 

Fixing precisely which objects the True and the False are is something of a thorny issue for 

Frege and his commentators, but he does seem willing to treat the True and the False as courses-

of-values of functions. In §10 of Grundgesetze, for example, he presents an elaborate argument 

supporting the claim that ―it is always possible to stipulate that an arbitrary course-of-values is to 

be the True and another the False‖.
15

  

 

Aside from these arbitrary stipulations (which are themselves quite problematic),
16

 Frege says 

little about the properties of either truth value, or even his reasons for construing truth and falsity 

in terms of objects (i.e., courses-of-values). This is not entirely surprising, as he notes in several 

places that he takes the concept of truth itself to be something indefinable. He writes, for 

instance: 

 

[...] it would be futile to employ a definition in order to make it clearer what is to be 

understood by ‗true‘. If, for example, we wished to say ‗an idea is true if it agrees with 

reality‘ nothing would have been achieved, since in order to apply this definition we should 

have to decide whether some idea or other did agree with reality. Thus we should have to 

presuppose the very thing that is being defined. The same would hold of any definition of 

the form ‗A is true if and only if it has such-and-such properties or stands in such-and-such 

relation to such-and-such a thing‘. In each case in hand it would always come back to the 

question whether it is true that A has such-and-such properties, or stands in such-and-such 

a relation to such-and-such a thing. Truth is obviously something so primitive and simple 

that it is not possible to reduce it to anything simpler.
17

 

 

Embedded in this paragraph are the beginnings of two distinct arguments in favour of the claim 

that truth is indefinable. These are developed further in one of Frege‘s last works, a 1918 essay 

entitled ―The Thought‖. The first argument is rather limited in scope, addressing as it does only a 

very narrow form of the correspondence theory of truth. Both Wolfgang Künne and Michael 

Dummett have suggested for different reasons that the apparent narrowness of this first argument 

                                                 
13

 For Frege, entities which are not objects (i.e., functions) take neither the True nor the False as a value when input 

as argument into a (first-level) concept like ‗x is a bottle‘; instead, they fail to produce a logical whole at all, 

resulting in nonsense. See [Diamond 1991, chapter 2] and [Goldfarb 1997] for competing accounts of exactly what 

is meant by nonsense here. 
14

 As can be imagined, the view of sentences as names of truth values is a fairly controversial one which has given 

rise to considerable debate. [Heck 1997] gives a fairly reasonable explanation of this peculiar view, suggesting that 

this move expedites Frege‘s treatment of relations in terms of double courses-of-values. 
15

 [Frege 1893, §10]. 
16

 Below, in our discussion of the Julius Caesar objection, we will treat in more detail problems pertaining to the 

association of a function with a unique course-of-values, as well as problems directly related to Frege‘s arguments in 

Grundgesetze §10. 
17

 [Frege 1897, 128-129]. 



115 

 

should not be taken to undermine Frege‘s broader claims about the indefinability of truth.
18

 Let‘s 

take a look at this first argument and then see how it connects up with his larger claims about the 

indefinability of truth. 

 

This first argument is pitted against what Künne calls the ―traditional object-based 

correspondence theory‖ of truth.
19

 On such a view, a statement is true if it corresponds to the 

object which it is about. Thus if I claim that my hat is yellow, the claim is true if it (somehow or 

other) corresponds with the object in question (viz. my hat).  But what is meant here by 

correspondence, and how can such a correspondence give us truth? Frege writes 

 

Now a correspondence is a relation. But this goes against the use of the word ‗true‘, which 

is not a relative term and contains no indication of anything else to which something is to 

correspond. If I do not know that a picture is meant to represent Cologne Cathedral then I 

do not know what to compare the picture with in order to decide on its truth. A 

correspondence, moreover, can only be perfect if the corresponding things coincide and so 

just are not different things. […] It would only be possible to compare an idea with a thing 

if the thing were an idea too. And then, if the first did correspond perfectly with the 

second, they would coincide. But this is not at all what people intend when they define 

truth as the correspondence of an idea with something real. For in this case it is essential 

precisely that the reality shall be distinct from the idea. But then there can be no complete 

correspondence, no complete truth.
20

 

 

Here Frege suggests that the very idea of a correspondence contains a kind of imperfection 

which is foreign to the nature of truth. One of Frege‘s guiding assumptions throughout his work 

is that truth is not a matter of degrees, nor is it relative to a given context, situation, or theory. 

Truth is an unchanging feature of thoughts. Thus, if we suggest that the only means of getting at 

perfect truth is via a correspondence between our claims and reality, then we are doomed to 

failure from the very start. For a picture is, by its very nature, something other than what it 

pictures. But this argument is only really effective against a fairly naïve form of the 

correspondence theory. Frege uses this naïve view as the starting point for a broader criticism of 

correspondence as a means of defining truth. 

 

Immediately following the above-quoted passage he writes: 

 

But could we not maintain that there is truth when there is correspondence in a certain 

respect? But which respect? For in that case what ought we to do so as to decide whether 

something is true? We should have to inquire whether it is true that an idea and a reality, 

say, correspond in the specified respect. And then we should be confronted by a question 

                                                 
18

 Cf. [Dummett 1973, 443] and [Künne 2008, §2]. Dummett holds that, while Frege‘s arguments do not convince us 

of his ultimate claim, they do provide us with sharp new limitations on any attempted definition of truth. Künne 

suggests that Frege‘s arguments about truth are subtler than many commentators have noted, a fact owing, perhaps 

to their enthymematic character. 
19

 [Künne 2008, 12]. 
20

 [Frege 1918, 59-60; Beaney 1997, 326-327]. 



116 

 

of the same kind, and the game could begin again. So the attempted explanation of truth as 

correspondence breaks down.
21

 

 

This is the crux of his first argument against the definability of truth in terms of correspondence. 

The claim here is that any attempt at defining truth in terms of correspondence leaves us no 

better off than when we started. For in order to apply any definition of this form (say, ‗x is true 

iff x corresponds in respect y with reality‘) in the first place, we would have to know whether it 

was true that our idea corresponded with reality in the relevant respect. But to establish that our 

idea corresponds to reality in the relevant respect, we can supply another claim or give another 

idea (the idea of my idea corresponding to reality in the relevant respect), and so on down the 

line. The problem here is not that we encounter an infinite regress, but that our attempts to apply 

the definition never actually succeed; we are always left with the same demand to apply the 

definition again. We find ourselves unable to get anywhere with the definition. 

 

But this narrower argument only attacks certain features of correspondence theories of truth, 

which are far from the only game in town. Following this focussed attack, Frege generalizes the 

argument to include any attempt at defining truth as a property or relation of any kind. He writes: 

 

And any other attempt to define truth also breaks down. For in a definition certain 

characteristics
22

 would have to be specified. And in the application to any particular case 

the question would always arise whether it were true that the characteristics were present. 

So we should be going round in a circle. So it seems likely that the content of the word 

‗true‘ is sui generis and indefinable.
23

 

 

In this more general argument, nothing hinges upon the particularities of correspondence 

relations. Instead, the general nature of definitions is brought into play. A definition involves 

isolating a particular notion by providing characteristics which only that notion possesses. On 

Frege‘s view, when providing such characteristics for ‗true‘, we would have to ask again whether 

it was true that these general characteristics were present in the particular case at hand. But then 

we would have to apply our definition again, resulting in the same question once more. And 

round and round we go. As Künne points out, this isn‘t exactly a vicious infinite regress, but  

 

a kind of vicious circle objection. This does not mean that [Frege] condemns all definitions 

of ‗true‘ as circular definitions A definition is circular if the concept expressed by the 

definiendum is […] expressed by a proper part of the alleged definiens. […] The circle 

Frege is complaining about is not of this kind. Rather, it is a circle one gets into as soon as 

one tries to apply the alleged definiens of ‗true‘.
24

 

 

The success of this argument, then, seems to hinge at least in some part on the capacities of those 

who would attempt to apply it. More specifically, as Künne notes, it hinges on the general 

                                                 
21

 [Frege 1918, 60; Beaney 1997, 327]. 
22

 The German term here is ‗Merkmale‘ which can also be translated ‗marks‘. Below we will have occasion to 

discuss Frege‘s technical understanding of these characteristics or marks in more detail. [Künne 2008, 13] gives 

good reasons for thinking that Frege is not using ‗Merkmale‘ here in his technical sense. 
23

 [Frege 1918, 60; Beaney 1997, 327]. 
24

 [Künne 2008, 13]. 



117 

 

principle ―One cannot decide whether things are thus and so without deciding whether it is true 

that things are thus and so‖, which, he claims, ―Frege seems to regard as self-evident‖.
25

 

Ultimately, Frege does not claim that these arguments are apodeictically certain or even terribly 

convincing, he only holds that given these considerations ―it seems likely‖ that we cannot define 

truth at all. 

 

Despite his acceptance of the impossibility of a proper definition of the concept, he goes on to 

suggest that we can ―bring out the peculiarity of our predicate [i.e., ‗is true‘] by comparing it 

with others‖ and that ―What, in the first place, distinguishes it from all other predicates is that 

predicating it is always included in predicating anything whatever‖.
26

 It would seem, then, that 

Frege‘s many remarks about truth and the predicate ‗is true‘ are to be taken as pre-systematic 

elucidations of the concept. None of these elucidations are particularly helpful in deciding upon 

the relationship between Frege‘s primitive concept of truth and the two objects which serve as 

truth values. Presumably, the extension of the concept of truth is identical with the object Frege 

calls the True, but this assumption is never made explicit and there remains confusion as to the 

relations between the True, and the predicate ‗x is true‘.
27

  

 

1.4 Functions 

 

In addition to the saturated entities Frege calls objects, he also discusses unsaturated entities. 

These he calls functions.
28

 Frege‘s view of functions is closely related to the informal 

mathematical notion of a numerical function. A function in this sense is a rule for, or way of, 

determining one quantity (the value of the function) by means of another quantity (the argument, 

or input).  

 

Within most areas of mathematics functions are only defined for a particular domain, e.g., on the 

natural, real, or complex numbers. Each entity within the domain of a function determines a 

specific value for that function. The range of the function is composed of the collection of 

entities which act as its values. The range and domain may be disjoint, co-extensive, or partially 

overlap with each other. The successor function on the natural numbers, for instance, takes 

natural numbers as inputs and delivers natural numbers as outputs. The square root function 

defined on the natural numbers as arguments, however, takes natural numbers as inputs but 

includes real numbers amongst its outputs. Though, historically, many mathematical domains 

have been defined by the types of objects of which they are composed (e.g., Euclidean space as 

the collection of Euclidean points, the natural number domain as the collection of all natural 

                                                 
25

 [Künne 2008, 14]. 
26

 [Frege 1897, 129]. [Künne 2008, §3] discusses the arguments in favour of a version of this claim in more detail. 
27

 For more detailed examination of Frege‘s views on the indefinability of truth and their development, see the 

remainder of [Künne 2008] as well as [Sluga 2002]; for an account of the relationship between Frege‘s anti-

empiricist epistemology and his conception of truth, see [Ricketts 1996] and [Levine 1996]. 
28

 It should be noted here that the ontological or metaphysical characterization of functions which I outline in this 

section was not always present in Frege‘s work. As [Heck and May 2013] convincingly shows, in the Begriffsschrift 

of 1879, Frege‘s distinction between functions and objects was (at least tacitly) a linguistic one. Later, in the 

Grundgesetze of 1893, Frege is very explicit about the ontological character of this distinction. My remarks here 

concern only this later period in Frege‘s thinking, which he took to be an improvement upon his earlier position (see, 

again, [Heck and May 2013] for detailed argumentation to this effect). 



118 

 

numbers), some mathematical domains have been partially characterized in terms of closure 

under certain types of functions. For a domain to remain closed under a given function the range 

must be a subset of the domain. Thus, for instance, an important part of the definition of a group 

is that a group is closed under the binary group operation. Similarly, the natural numbers are 

closed under addition, but if we include the operation of subtraction, we must extend the domain 

to include all the integers to maintain closure. 

 

Historically, many new mathematical domains—perhaps most notably within number theory—

have been investigated in an effort to understand what type of domain is required to maintain 

closure under a particular operation or type of function. This is one way in which the concept of 

a function has been extended. Beginning with the birth of analysis during the time of Leibniz and 

Newton, the concept of a function has been almost continuously expanded, such that it has come 

to include functions defined for wider and wider number domains, as well as functions which 

employ a much wider array of allowable operations. Frege saw his work as contributing to both 

these ways of extending the notion of a function. First, he extended functions to include a much 

greater variety of possible inputs. Instead of simply the natural, real, or complex numbers, Frege 

viewed functions as accepting any objects whatsoever as arguments and values.
29

 In effect, Frege 

demanded that functions be defined exclusively within the single domain of his ontological 

hierarchy. Second, he extended the allowable operations which were taken to define 

mathematical functions. Just as the Irish mathematician William Hamilton‘s development of 

quaternions required him to countenance non-commutative forms of multiplication, Frege, 

similarly, sought to expand the domain of allowable mathematical functions by including, e.g., 

the symbol for equality in the construction of function-expressions.
30

 

 

Alongside his two-fold extension of the mathematical notion of a function, Frege also attempted, 

in both his formal and informal works, to deepen our understanding of the nature of functions 

and their relations to objects. On the linguistic level, there is the obvious difference between 

symbols which refer to functions and those which refer to objects that function-symbols tend to 

indicate, in one way or another, the need for completion by object-symbols, often of a particular 

type. Frege believed that this orthographic feature of mathematical discourse represented a 

deeper, ontological difference between functions and objects. One key aspect of this difference is 

that, for him, functions possess argument places. Frege writes that he ―is concerned to show that 

the argument does not belong with a function, but goes together with the function to make up a 

complete whole; for a function by itself must be incomplete, in need of supplementation, or 

unsaturated. And in this respect functions differ fundamentally from numbers‖.
31

 Though Frege 

is specifically concerned here with differentiating functions from numbers, his remarks are 

guided by the more general view that functions differ fundamentally from all objects. The 

argument places of functions dictate which form or forms of supplementation a particular 

                                                 
29

After Frege introduces concepts as functions whose values are always truth-values (at [Frege 1891, 139]), he 

writes that ―we must go further and admit objects without restriction as values of functions‖ [Frege 1891, 140]. 
30

 Though Frege‘s remarks on the nature of functions always presume an extremely liberal understanding of the 

allowable types of functions, in practice he rarely refers to the stranger types of function which were gaining in 

mathematical popularity during his career. [Burgess 1997] presents an interesting examination of Frege‘s actual use 

and treatment of specifically mathematical functions throughout his work against the backdrop of broader 

developments in the concept of function of the time. 
31

 [Frege 1891, 133]. 



119 

 

function will accept. There are different levels and degrees of function determined by the nature 

and number of argument places. Each of these levels is entirely distinct from all others, such that 

a function of one level is (according to Frege) as ontologically separate from every other type of 

function as it is from objects. 

 

For Frege, the most basic functions are first-level functions of one argument place. Such 

functions possess a single argument place which can be supplemented or saturated only by an 

object. The result of a first-level function with an object saturating its argument place is another 

object. Thus, for instance, we might have the square function x
2
 (where ‗x‘ indicates an argument 

place). This function takes objects as arguments and delivers objects in turn: if we input the 

number 2 as argument, the value returned by the function is the number 4, another object. There 

are, similarly, functions of the first level with multiple argument places, each of which can take 

any object as an argument.
32

 Second-level functions take first-level functions as arguments, and 

can possess one or more argument places. Third-level functions take second-level functions as 

arguments, and so on. 

 

Perhaps one of Frege‘s most lasting contributions to modern thought was his understanding of 

concepts as particular types of functions, namely those of one argument place whose values are 

always truth-values. Similarly, he viewed relations of all types as functions of two or more 

arguments whose values are always truth-values. Thus, for instance, the concept of a natural 

number is a function which takes objects as arguments and gives truth values as outputs. All 

objects which are indeed natural numbers give the True as value when input into the function, 

while any objects which are not natural numbers give the False as value. Concepts, like other 

functions, can be of different levels. Thus, the concept ‗x is a horse‘ is of the first-level because it 

takes objects as arguments, while the concept ‗F is a concept easily attained‘ is a second-level 

concept insofar as it takes first-level concepts as arguments (we will discuss this in more detail 

below in the section on the concept horse problem). 

 

In order to completely specify what type of function we are dealing with, Frege believes that we 

must carefully indicate the nature and number of its argument places. We may, for instance, have 

two second-level functions which, despite this similarity, are nevertheless fundamentally 

different insofar as they possess a different number of argument-places. For Frege, a second-

level function of two argument places is, despite certain similarities, as ontologically distinct 

from a second-level function of three argument places as it is from an object.
33

 Because this 

hierarchy of functions is (at least in principle) infinitely extendible, and each level of function is 

ontologically separate from every other, Frege‘s ontology ends up being quite rich despite the 

relative simplicity of the general notions involved. We have both an infinite number of levels of 

function and, at each level, an infinite number of different types of functions determined by the 

number of argument places they possess. To further add to this ontological panoply, there are 

also various functions of mixed level which possess two or more argument places of different 

levels. Thus, for instance, we might have a function which possesses one argument place which 

                                                 
32

 E.g., x + y, x = x, ((x + 3) – x)/3, etc. where ‗x‘ and ‗y‘ represent argument places for objects. 
33

 I have found Warren Goldfarb‘s description (at [Goldfarb 1997, 58] ) of these distinctions in terms of different 

types of Velcro particularly useful. 



120 

 

takes objects as arguments and another argument place which takes first-level functions as 

arguments. This does not mean, however, that there are single argument places which accept 

entities of different level, only that a function may possess argument places of multiple different 

levels.
34

 

 

As we‘ve seen, Frege describes objects as saturated and self-standing or self-subsistent. We‘ve 

also seen that for him a key feature of concepts (and functions more generally) is their 

unsaturatedness. In this they differ fundamentally from objects. But are concepts also self-

subsistent for Frege? That is, do concepts possess independent existence in the same way that 

Fregean objects do? 

 

Given the centrality of functions for Frege‘s ontology, the somewhat surprising answer is that 

they do not. In 1882, in a letter to the Swiss philosopher Anton Marty (1847-1914) outlining his 

logical views, Frege writes 

 

A concept is unsaturated in that it requires something to fall under it; hence it cannot exist 

on its own. […] Now I do not believe that concept formation can precede judgement, 

because this would presuppose the independent existence of concepts, but I think of a 

concept as having arisen by decomposition from a judgeable content.
35

 

 

And later, in one of his 1903 articles on the foundations of geometry, he states the view again: 

 

It is clear that we cannot put a concept forward as self-subsisting like an object. Rather a 

concept can only occur in a combination. It can be said that the concept can be 

distinguished in the combination, but cannot be separated out of the combination.
36

 

 

This is in some ways a puzzling view. For Frege certainly does not want to deny that concepts 

and functions have a kind of ontological status. They are not in any sense reducible to objects, 

though it does seem, as Thomas Ricketts convincingly argues, that objects are, in a certain sense, 

logically prior to concepts.
37

 But, as the metaphor of unsaturatedness is meant to elucidate, it is 

of the essence of concepts that they are isolable only in the context of the analysis of the logical 

wholes Frege calls ‗thoughts‘. To treat them as self-standing entities, as already-completed 

wholes, would be to misconstrue them as yet another type of object. And, as we will see in 

greater detail below, one of Frege‘s three guiding principles in his logical and mathematical 

work was ―never to lose sight of the distinction between concept and object‖.
38

 

 

1.5 Relations between functions and objects 

 

                                                 
34

 E.g., the functions ∫  ( )  
 

 
 and [

 

  
 ( )]

   
, both of which take objects and functions as arguments. 

35
 [Frege to Marty, August 29, 1882; in Beaney 1997, 81]. 

36
 [Frege 1903, 372, note 5]. 

37
 Cf. [Ricketts 2010, 163-170]. Ricketts argues that this sort of priority is a self-conscious (if complex) aspect of 

Frege‘s view; [Bergmann 1958] suggests that, self-conscious or not, it leads Frege away from the sort of Platonic 

realism about functions with which he is often associated and towards a ‗hidden nominalism‘ about functions. 
38

 [Frege 1884b, xxii]. 



121 

 

In addition to outlining the properties of functions and objects, Frege also discusses various 

relationships between these ontological categories. One of the most important logical 

relationships between objects and concepts, as noted in Frege‘s letter to Marty, is that of an 

object ‗falling under‘ a concept. Frege says that an object falls under a concept just in case it 

produces the value the True when input into the concept as argument. As a human being, I fall 

under a number of concepts. For instance, I fall under the concept ―x is a corporeal being‖. 

Frege‘s understanding of properties is directly connected to this relationship between concepts 

and the entities which fall under them. The properties (Eigenschaften) of an entity are, for Frege, 

just the concepts under which that entity falls.
39

 

 

In his attempt to understand the nature of number, Frege found it useful to note a distinction 

between the properties of an entity and the marks or characteristics (Merkmale) of the concept 

under which that object falls. Let‘s take a look at what he says, and follow it up with a few 

examples. In §§52-53 of the Foundations of Arithmetic, Frege writes: 

 

[…] our ordinary language does assign number not to concepts but to objects: we speak of 

―the number of the bales‖ just as we do of ―the weight of the bales‖. Thus on the face of it 

we are talking about objects, whereas really we are intending to assert something of a 

concept. This usage is confusing. The construction in ―four thoroughbred horses‖ fosters 

the illusion that ―four‖ modifies the concept ―thoroughbred horse‖ in just the same way as 

―thoroughbred‖ modifies the concept ―horse.‖ Whereas in fact only ―thoroughbred‖ is a 

characteristic [Merkmal] used in this way; the word ―four‖ is used to assert something of a 

concept. […] By properties which are asserted of a concept I naturally do not mean the 

characteristics [Merkmale] which make up the concept. These latter are properties of the 

things which fall under the concept, not of the concept. Thus, ―rectangular‖ is not a 

property of the concept ―rectangular triangle‖; but the proposition that there exists no 

rectangular equilateral rectilinear triangle does state a property of the concept ―rectangular 

equilateral rectilinear triangle‖; it assigns to it the number nought.
40

 

 

We can see here that the distinction between a mark and a concept is quite a useful one for 

Frege‘s dual project of disambiguating natural language and garnering a proper understanding of 

the nature of number. Natural languages like German and English tend to blur the distinction, 

particularly when it comes to number, and Frege believes that this can (and does) lead us to 

incorrect accounts of the actual nature of number. By mistaking the properties of a concept for 

the properties of the object or objects which fall under that concept, we are led to a 

misunderstanding about the role concepts play in numerical applications like counting. 

 

But the distinction is not only useful in the specific context of Frege‘s logicist program in the 

Grundlagen. Eight years later, in his attempt to overcome Benno Kerry‘s criticisms of his 

distinction between concepts and objects, he returns to the vocabulary of marks and properties. 

In ‗On Concept and Object‘ he writes: 

 

                                                 
39

 [Frege 1892a, 201; in Beaney 1997, 189].  
40

 [Frege 1884b, §§52-53]. 



122 

 

The words [‗Eigenschaft‘ and ‗Merkmal‘] serve to signify relations, in sentences like ‗Φ is 

a property of Γ‘ and ‗Φ is a mark of Ω‘. In my way of speaking, a thing can be at once a 

property and a mark, but not of the same thing. I call the concepts under which an object 

falls its properties; thus 

 

‗to be Φ is a property of Γ‘ 

 

is just another way of saying 

 

‗Γ falls under the concept of a Φ‘. 

 

If the object Γ has the properties Φ, Χ and Ψ, I may combine them into Ω; so that it is the 

same thing if I say that Γ has the property Ω, or, that Γ has the properties Φ, Χ and Ψ. I 

then call Φ, Χ and Ψ marks of the concept Ω, and, at the same time, properties of Γ. It is 

clear that the relations of Φ to Γ and to Ω are quite different, and that consequently 

different terms are required. Γ falls under the concept Φ; but Ω, which is itself a concept, 

cannot fall under the first-level concept Φ; only to a second-level concept could it stand in 

a similar relation. Ω is, on the other hand, subordinate to Φ.
41

 

 

Here we see the distinction laid out more explicitly. Frege employs the term ‗mark‘ when 

encountering what we might call a conjunctive or a complex concept. A complex concept, in this 

sense, is one composed of a number of component concepts. Thus we might think of the complex 

concept ‗x is a large orange cat‘ as a conjunction of three component concepts, ‗x is large,‘ ‗x is 

orange,‘ and ‗x is a cat‘. Such complex concepts needn‘t be given in such an obviously 

conjunctive form. We might, for instance, have previously defined the phrase ‗x is a Garfield‘ to 

mean ‗x is a large orange cat,‘ in which case we can still think of ‗x is large,‘ ‗x is orange,‘ and ‗x 

is a cat‘ as component concepts of the complex concept ‗x is a Garfield‘. 

 

Let‘s take another example and work through the relationship between the properties of an entity 

and the marks of a concept under which that entity falls. 

 

Blue guitars—objects in Frege‘s dispensation—fall under two distinct component concepts: ‗x is 

blue‘ and ‗x is a guitar‘. For Frege, this is another (and perhaps a more accurate) way of saying 

that these objects possess the properties of being blue and of being a guitar. 

 

Just as guitars have properties, so too do the concepts under which they fall. Importantly, and 

perhaps obviously, the properties of concepts are usually quite different from the properties of 

the objects which fall under them. For instance, it is not a property of the concept ‗x is a blue 

guitar‘ that it is blue. One property of our complex concept would be that it is not an empty one; 

there have in fact been objects in the world which have been both blue and a guitar. 

 

Frege uses the term ‗mark‘ or ‗characteristic‘ to refer to a quite different relationship between 

concepts and the objects which fall under them. For Frege, some of the properties of the objects 

which fall under a complex concept are to be understood as the marks of that concept. Take a 

                                                 
41

 [Frege 1892a, 201-202; in Beaney 1997, 189-190]. 



123 

 

particular blue guitar as an example. Such an object falls under a plethora of concepts, not all of 

which will count as marks of the complex concept (‗x is a blue guitar‘) we are interested in here. 

For instance, it might fall under the concept ‗x is destructible‘. Now, it is in fact the case that all 

the objects which fall under our complex concept ‗x is a blue guitar‘ also happen to be 

destructible. But for properties to count as marks of a given concept, they have to be component 

concepts of that concept, and ‗x is destructible‘ is not one of the concepts conjoined to form our 

complex concept. The notion of a mark has much more to do with the parts or components of the 

concept than it has to do with the objects which fall under the concept. 

 

Let‘s see how this plays out.  

 

We have an object (say, Rollie Fingers), a complex concept (‗x is a moustachioed baseball 

player), and two component concepts (‗x is a baseball player‘ and ‗x is moustachioed‘).  

 

Our object, Rollie Fingers, has the following three properties: 

 

1. He is a Cy Young award winner. 
2. He is a baseball player. 
3. He is moustachioed. 

 

None of these are properties of the concept ‗x is a moustachioed baseball player‘, which is to say 

that the concept ‗x is a moustachioed baseball player‘ does not fall under (indeed, cannot fall 

under) any of the concepts underlying these three properties. The latter two of these properties 

are, however, marks of the complex concept ‗x is a moustachioed baseball player‘. They are 

marks of this complex concept because they are the components or parts out of which it was 

constructed. The first property and many other properties of Rollie Fingers are not components 

of the complex concept and are not, therefore, marks of that concept. 

 

All of this might seem obvious enough (even pedantic), but when it comes to complex concepts 

of higher levels, or highly abstract ones like those we find in mathematics, it becomes more 

difficult to distinguish between properties and marks without such a clearly defined vocabulary. 

The distinction between a property and a mark is not exactly an ontological one, as a single 

concept can serve as both a mark and a property,
42

 though it cannot be a mark and a property of 

the same entity. It is nevertheless a useful terminological distinction. 

 

Another relation that emerges in Frege‘s discussion of properties and marks is that of 

subordination (Unterordnung). For Frege, a concept F is subordinate to a concept G, when the 

extension of F is contained within the extension of G. Note that this relation need not be 

asymmetrical: co-extensive concepts can be said to be mutually subordinate. A complex concept 

is subordinate to its component concepts. Thus, for instance, the concept ‗x is a positive even 

whole number greater than 7‘ is subordinate to the component concepts ‗x is a positive number‘, 

‗x is a whole number‘ ‗x is an even number‘ and ‗x is greater than 7‘. The relation of 

subordination, however, needn‘t be restricted to complex concepts (i.e., concepts which are 

given as conjunctions of simpler concepts).  

                                                 
42

 Cf. [Frege 1892a, 201; in Beaney 1997, 189]. 



124 

 

 

In contrast to the relations between concepts and marks, the relation of subordination is 

concerned more directly with the entities which fall under the concept than with the components 

of the concept itself. Let‘s take two first-level concepts as an example, say ‗x is a banjo‘ and ‗x is 

a physical object‘. The former is subordinate to the latter not because being a physical object is 

part of a conjunction of concepts which make up the concept ‗x is a banjo‘ but because of certain 

facts about the objects which fall under these two concepts. All the objects which fall under the 

concept ‗x is a banjo‘ also fall under the concept ‗x is a physical object‘, but the converse is not 

true. It is this relation between the objects which fall under the concepts which determines 

subordination. 

 

Frege stresses at a number of points that this relationship of subordination which holds only 

between concepts of the same level is not to be confused with similar relations between entities 

of different levels.
43

 We‘ve already seen the relation of falling under which holds between 

objects and first-level concepts. But Frege also distinguishes a similar relationship between first- 

and second-level concepts which he calls ‗falling within‘. The distinctions between 

‗subordination,‘ ‗falling under,‘ and ‗falling within‘ are of particular importance for 

understanding his remarks regarding Hilbert‘s work on Euclidean geometry. He writes that 

 

Second-level concepts, which [first-level] concepts fall under, are essentially different 

from first-level concepts, which objects fall under. The relation of an object to a first-level 

concept that it falls under is different from the (admittedly similar) relation of a first-level 

to a second-level concept. To do justice at once to the distinction and to the similarity, we 

might perhaps say: an object falls under a first level concept; a concept falls within a 

second-level concept. The distinction of concept and object thus still holds, with all its 

sharpness.
44

 

 

So, for instance, the object Jumbo falls under the first-level concept ‗x is an elephant‘. The first-

level concept ‗x is an elephant‘ falls within the second-level concept ‗F is not an empty concept‘. 

And the first-level concept ‗x is an elephant‘ is subordinate to the first-level concept ‗x is a 

mammal‘. In case of Hilbert‘s geometric researches, Frege would say that Euclidean lines are 

objects which fall under the first-level concept ‗x is a Euclidean line‘, while Hilbert‘s supposedly 

uninterpreted notion of a line is better understood as a second-level concept within which fall 

various first-level concepts, including the concept of a Euclidean line. This is different from the 

relation of subordination which holds between, e.g., the first-level concepts ‗x is a Euclidean 

line‘ and ‗x is a Euclidean parabola‘. Thus, from Frege‘s point of view, much of the confusion in 

Hilbert‘s work stems from the fact that Hilbert confuses the properties of Euclidean points with 

the marks of his second-level concept of point. Though Frege does not often remark upon 

concepts of levels higher than the second, it is presumably the case that the difference between 

falling under and falling within can be analogously extended to each ontological level. Thus the 

relationship between second- and third-level concepts (falling into?) will be similar, though 

ontologically distinct, from that which holds between second- and first-level concepts. The 

                                                 
43

 [Frege 1882; in Beaney 1997, 81]. 
44

 E.g., at [Frege 1884b, §53], [Frege 1892a, 201; in Beaney 1997, 189]. 



125 

 

difficulty of making such distinctions in ordinary language can of course be surmounted through 

various formal devices (e.g., subscripts). 

 

1.6 Courses-of-values and extensions 

 

Frege believed that each function was associated with a (unique) object. These objects are called 

courses-of-values (Wertverläufe) or, if the function happens to be a concept, extensions. Let‘s 

look at the special case of extensions first. Though Frege assumes that it is known what the 

extension of a concept is,
45

 there are, I think, at least two ways of understanding this notion. The 

first is to take as the extension of a concept simply the collection of all entities which produce the 

value ‗the True‘ when input into the concept as arguments, i.e., the collection of all those objects 

which fall under the concept at issue. The second would be to take the extension as a set of 

ordered pairs, where the first term of each pair is any object whatever, and the second either the 

True or the False, depending on whether the object falls under the given concept. For the concept 

‗x is a natural number‘, accordingly, the extension would thus be either: 

 

{0, 1, 2, 3, …}  

 

or (using the usual notation for ordered pairs): 

 

{...<Scipio Africanus, the False>, <2, the True>, <3, the True>, ...} 

 

The first way of understanding extensions of concepts is a fairly common one. But it will not do 

as an account of the nature of courses-of-values in general. This first option cannot help us 

explain courses-of-values for functions which are not concepts, as in such functions there is no 

pairing between the arguments of a function and the True and thus no obvious way to generate a 

list of entities. In the general case, if we are to attempt to pick out a unique course-of-values, we 

need to know which arguments produce which values, a task which can be accomplished if we 

model courses-of-values on the second approach using ordered. 

 

Since Frege claimed that no entity can stand in the relation of identity to any entity of a different 

ontological level, he did not identify functions with their courses of values (as is often done, for 

example, in most versions of contemporary set theory). Given the loose characterization of 

functions as rules for determining quantities by means of other quantities, this refusal seems 

plausible; for a rule is not identical to the pairings it determines. For Frege, too, this makes sense, 

as his ontological hierarchy is built largely upon the fundamental distinction between functions 

and objects. This understanding of the notion of function has the consequence that distinct 

functions might have the same course of values,
46

 or distinct concepts the same extension. Thus, 

for instance, the square function will possess the same course of values as the function x
2
 + 0. 

Similarly, when programming a computer we might have three distinct programs which achieve 

the same goal (i.e., deliver the same output given the same inputs) though at wildly different 

computational speeds and levels of complexity. There are also well-known examples of co-

extensive concepts which seem ‗intensionally‘ quite different (a standard example is the pair of 

                                                 
45

 [Frege 1884b, §69, note 1]. 
46

 Frege admits as much in the specific case of concepts at [Frege 1884b, §68 note 1]. 



126 

 

concepts ‗x is a creature with kidneys‘ and ‗x is creature with a heart‘). Below we will discuss in 

greater detail the way in which Frege believes the course of values of a given function (or the 

extension of a particular concept) is determined—we will also see that this seemingly simple 

concern posed a major problem for his entire philosophical program. 

 

Now that we have a clearer understanding of Frege‘s ontological hierarchy, let‘s begin our 

examination of problems pertaining to that hierarchy. 

 

§2. The concept horse problem 

 

The first problem with Frege‘s philosophy we will examine is the so called concept horse 

problem. This problem relates directly to Frege‘s basic distinction between objects and functions. 

As we‘ve seen above, Frege took it to be the case that the world was exhaustively divided into 

objects and functions,
47

 and that no object could be identical to a function. The concept horse 

problem, first raised by Benno Kerry (1858-1889), seems to attack the possibility of maintaining 

this absolute distinction. In a series of eight articles published between 1885 and 1891, Kerry 

examined the role of intuition within reasoning. At several points in these articles, Kerry 

mentions Frege‘s works, sometimes favourably, sometimes critically. Kerry agreed with Frege, 

for instance, that the relationship between concepts and the objects which fall under them is 

likely irreducible, but argued that ―this [is] in no way bound up with the view that the properties 

of being a concept and of being an object are mutually exclusive‖.
48

 Further, he argued that, on 

Frege‘s own account, we can construct meaningful sentences expressing thoughts which sensibly 

treat concepts as objects, and, thus, that the distinction between the two types of entity is not 

absolute. His arguments against Frege‘s claim that the world is exhaustively divided into these 

two disjoint categories center upon the following putative counterexample: 

 

The concept horse is a concept easily attained. 

 

This relatively simple example seemed, to Kerry, to illustrate a serious problem with Frege‘s 

distinction. For, by Frege‘s own lights, the use of the definite article generally indicates that we 

are speaking of an object.
49

 Thus, if the phrase ―the concept horse‖ refers to anything, it would 

seem to refer to an object and not a concept. Hence, Kerry‘s sentence seems to express a thought 

in which an entity that is obviously a concept (namely, the concept horse) is treated as an object. 

So, following Kerry‘s reasoning, Frege must either drop his insistence that concepts and objects 

are mutually exclusive, or admit that his system contains a contradictory element.  

 

                                                 
47

 ―[A]n object is anything that is not a function‖ [Frege 1891; in Beaney 1997, 140].  
48

 [Kerry, quoted in Frege 1892a, 193; in Beaney 1997, 182]. 
49

 Frege notes of Kerry‘s claim that ―the concept ‗horse‘‖ refers to an object that it is quite correct, and that it ―is in 

full accord with the criterion I gave—that the singular definite article always indicates an object, whereas the 

indefinite article accompanies a concept-word‖ [Frege 1892a, 195; in Beaney 1997, 184]. This criterion is also 

discussed in Die Grundlagen der Arithmetik, where Frege writes, for instance, that ―A general concept word just 

designates a concept. Only with the definite article or a demonstrative pronoun does it function as a proper name of 

a thing, but it then ceases to function as a concept word‖ [Frege 1884b, §51; in Beaney 1997, 102]. 



127 

 

A third rather easy way out of the problem for Frege would be to alter his views so that they 

allow for instances where the use of the definite article does not accurately reflect the logical 

structure of the thought expressed.
50

 Frege might then treat Kerry‘s putative counter-example as 

an exception to a rule which is applicable only in most cases. Frege does admit as much, and 

provides a few such examples of his own. But he subsequently dismisses these rare exceptions as 

unimportant and irrelevant to Kerry‘s main point.
51

 As we will see when we begin to analyze 

Frege‘s response to the problem, this refusal to alter his views of the logical function of the 

definite article is one of the most puzzling aspects of this episode. 

 

Before we get to Frege‘s response, however, let‘s take a closer look at the exact sources of the 

problem in Frege‘s philosophy. The first and most fundamental is his refusal to allow any 

identification of concepts/functions with objects, and vice versa. This means that no function can 

be an object, and no object can be a function. And, since Frege‘s ontological distinctions are 

intended to cut deeper than mere psychological, linguistic, or temporal considerations, there is no 

sense to claims that the very same entity can be considered an object from one point of view and 

a function from another. Putative counterexamples to the contrary are, on Frege‘s view, simply 

the result of confusion or equivocation. This insistence that ontological distinctions are absolute 

is essential both to his own project of reducing arithmetic to logic and to his criticisms of 

Hilbert‘s views. 

 

The second, related, source of the problem lies in the fact that Frege takes his ontological 

distinction to be exhaustive. Every entity is either an object or a function with a determinate 

number of argument places of determinate types. There is no third type of hybrid entity able to 

partially fulfill the roles of both saturated and unsaturated entities. Hence the option of placing 

troublesome entities somewhere in the middle of the function/object distinction is simply not 

available to Frege.  

 

Third, the problem arises because of certain features in Frege‘s theory of language, notably his 

view that proper names of any kind (including those indicated via the use of the definite article) 

can only refer to objects if they refer at all.  

 

2.1 Mendelsohn‘s analysis of the problem 

 

We can understand, I think, Frege not wanting to alter his views regarding the fundamental 

distinction at the heart of his entire theory, so it makes sense that he does not simply invent an ad 

                                                 
50

 An example of such an instance might be the statement ‗The concept unicorn is empty‘. Here the surface 

grammar suggests that we are attributing a property (emptiness) to an object (the concept unicorn), but it seems 

relatively uncontroversial to symbolize the actual logical structure of this sentence, in modern notation, as follows: 

x~Ux, i.e., as a straightforward claim that nothing falls under the concept unicorn. 
51

 Frege admits that there are certain difficulties pertaining to the plural usage of the definite article, as well as rare 

cases of the singular. He writes: ―In the singular, so far as I can see, the matter is doubtful only when a singular 

takes the place of a plural, as in the sentences ‗The Turk besieged Vienna,‘ ‗The horse is a four-legged animal‘. 

These cases are so easily recognizable as special ones that the value of our rule is hardly impaired by their 

occurrence. It is clear that in the first sentence ‗the Turk‘ is a proper name of a people. The second sentence is 

probably best regarded as expressing a universal judgement, say ‗All horses are four-legged animals‘ or ‗All 

properly constituted horses are four-legged animals‘‖ [Frege 1892a, 196; in Beaney 1997, 184-185]. 



128 

 

hoc third ontological category or relax the strictness of the distinction. But it does seem rather 

puzzling that he is so steadfast in his merely linguistic doctrine regarding the definite article. 

What reasons can he possibly have for sticking to his guns here? Frege himself is of very little 

help in answering this question. In a rather infamous passage from ―On Concept and Object,‖ for 

instance, he writes that 

 

By a kind of necessity of language, my expressions, taken literally, sometimes miss my 

thought; I mention an object, when what I intend is a concept. I fully realize that in such 

cases I was relying upon a reader who would be ready to meet me half-way—who does not 

begrudge a pinch of salt 
52

 

 

As Richard Mendelsohn (among others)
53

 notes, this appeal to readers to simply accept the prima 

facie preposterous claim that the concept horse is not a concept is ―much too dogmatic: a 

reasonable doubt about the theory remains as long as Frege is unable to dispel the puzzle.‖
54

 One 

way of doing so would be to explain exactly what is meant by ‗a kind of necessity of language‘ 

here. 

 

He does briefly try to explain this necessity in a footnote, but the explanation itself is confusing 

and quite misleading. In the footnote, Frege writes: 

 

A similar thing happens when we say as regards the sentence ‗this rose is red‘: the 

grammatical predicate ‗is red‘ belongs to the subject ‗this rose‘. Here the words ―The 

grammatical predicate ‗is red‘‖ are not a grammatical predicate but a subject. By the very 

act of explicitly calling it a predicate, we deprive it of this property.
55

 

 

Now presumably what Frege is trying to show here is that we encounter exactly the same 

difficulty with phrases like ‗The concept η‘ as we do with phrases like ‗The predicate x‘. As his 

wording indicates, he takes this difficulty to stem from the subject/predicate structure of (some) 

natural languages. But, as Richard Mendelsohn points out, Frege is just wrong here. And, as a 

result, the analogy that he wishes to construct as a means of explanation here is actually rather 

misleading and unhelpful. Mendelsohn writes that ―contrary to what Frege says, we don‘t, 

simply by calling a given expression a grammatical predicate, deprive it of the property of being 

a grammatical predicate‖.
56

 

 

To see why, consider the following two sentences from Mendelsohn: 

 

(A)  The grammatical predicate ‗is red‘ is not a grammatical predicate.
57

   

(B) ‗The grammatical predicate ―is red‖‘ is not a grammatical predicate.
58

   

                                                 
52

 [Frege 1892a, 204; in Beaney 1997, 192]. 
53

 See, for instance, Crispin Wright‘s aptly titled chapter ―Why Frege Does not Deserve His Grain of Salt‖ in [Hale 

and Wright 2001].  
54

 [Mendelsohn 2005, 73]. 
55

 [Frege 1892a, 196 note H; in Beaney 1997, 185 note H]. 
56

 [Mendelsohn 2005, 73]. 
57

 [Mendelsohn 2005, 73]. Here and below I‘ve replaced Mendelsohn‘s sequential numeration of these sentences. 
58

 [Mendelsohn 2005, 74]. 



129 

 

 

Mendelsohn rightly suggests that (A) is straightforwardly false: ‗is red‘ is indeed a grammatical 

predicate, and (A) falsely states that it is not. There seems to be no problem at all in holding that 

‗is red‘ is indeed a grammatical predicate. Now, (B) is, according to Mendelsohn, equally 

straightforwardly true: the words ―The grammatical predicate ‗is red‘‖ do not form a 

grammatical predicate at all. As Frege points out in his footnote, grammatically, these words 

form a subject not a predicate. And, again, we have no trouble at all seeing this. The problem in 

the footnote is that Frege seems to hold that (B) is helpfully and illuminatingly analogous to the 

troublesome claim that ‗The concept horse is not a concept‘. But, given the straightforward 

falsity of (A), this analogy gets us nowhere. 

 

Mendelsohn writes: 

 

But, in the first place, it is not clear why we should regard (B) rather than (A) as the proper 

analogue to [‗The concept horse is not a concept‘]. In the second place, there is no 

awkwardness about (A), and whatever awkwardness accrues to (B) is readily explained. In 

the third place, since (A) is false, we are able to speak about grammatical predicates, 

whereas on Frege‘s view, we do not seem to be able to say anything intelligible about 

concepts.
59

 

 

So, Frege does not explain why the subject/predicate structure of natural languages necessarily 

leads us astray when speaking about either concepts or predicates. After this attempt at 

explanation, then, we are no closer to an understanding of the exact nature of the problem. 

Unfortunately, Frege himself does not give us much else to go on, and we have to do a bit of 

digging to see exactly which aspects of Frege‘s philosophy are at issue in the problem. 

 

Mendelsohn, for his part, does not try to root the concept horse problem in the infelicities of 

language. He holds, rather, that ―Frege‘s claim that the problems with [‗the concept horse is not 

a concept‘] are the result of a mere awkwardness of language just does not seem to be able to 

capture the depths of his difficulties.‖
60

 So, where are we to look for the real root of the problem? 

Clearly the problem first arises when Frege commits himself to the truth of the thought expressed 

by the sentence: 

 

(Horse) The concept horse is not a concept. 

 

So, if we are to understand the source of the problem, we must first understand what sort of 

reasons Frege could possibly have had for holding that (Horse) is true. Frege himself does not 

give us any compelling argument for his support of this claim. However, Mendelsohn provides a 

reasonable reconstruction of an argument that one might make on the basis of Frege‘s statements 

in ―On Concept and Object‖. 

 

Let‘s take a look at Mendelsohn‘s reconstruction. He begins with the claim that (Horse) is a 

consequence of the principle: 

                                                 
59

 [Mendelsohn 2005, 74]. 
60

 [Mendelsohn 2005, 74]. 



130 

 

 

(C) η is a predicate if, and only if, η stands for a concept.
61

 

 

For Frege, this is a fundamental principle of his philosophy of language. A predicate is just an 

expression which stands for a concept, and something stands for a concept only if it is a 

predicate. This seems a fairly straightforward claim, and uncontroversial given Frege‘s other 

views. So where do we get into trouble and confusion? 

 

Mendelsohn takes it to be the case that Frege fairly obviously holds that 

 

(D) ‗The concept horse‘ is not a predicate 
 

If he did not hold this, then the problem of ‗the concept horse‘ would not have arisen in the first 

place. So, by (C), this also commits him to the claim that 

 

(E) ‗The concept horse‘ does not stand for a concept 

 

Since only predicates are able to stand for concepts, and ‗the concept horse‘ is not a predicate, it 

cannot stand for a concept. Clear enough. Then we also have 

 

(F) That which ‗the concept horse‘ stands for is not a concept 

 

But we also have the obvious identity:  

 

(G) That which ‗the concept horse‘ stands for = the concept horse 

 

And then finally we get (Horse) by substituting ‗the concept horse‘ for ‗that which ‗the concept 

horse‘ stands for‖ in (F). But, following this reasoning, we can easily show that there are in fact 

no predicates at all. Nothing here depends upon the peculiarity of ‗the concept horse,‘ so we can 

simply reiterate the argument for any ostensible predicate we like, with the same result. Nothing, 

then, can be said to stand for a concept. This result is one with which Frege – who speaks of 

predicates and concepts quite happily throughout his work – ought to have been deeply 

unsatisfied. 

 

So, what exactly is wrong with this picture? According to Mendelsohn, something is wrong with 

the very first step in the above argument, with (C). 

 

First, Mendelsohn suggests that throughout ―On Concept and Object‖ everything Frege says 

shows that he treats ‗η is a concept‘ as the name of a first-level function. This means that Frege 

assumes that ‗η is a concept‘ (or, after Kerry‘s example, ‗η is a concept easily attained‘) refers to 

a function which accepts only objects as values. This assumption is evident when, for instance, 

he simultaneously suggests that the sentence ‗The concept horse is a concept‘ is well-formed 

(i.e., has a truth value) and that ‗the concept horse‘ is a proper name. If ‗the concept horse‘ is a 

proper name, then it can only sensibly be combined with first-level function names (or perhaps 

                                                 
61

 [Mendelsohn 2005, 74]. 



131 

 

the names of functions of mixed level) to form a proposition which expresses a complete thought 

bearing a truth value. Since ‗η is a concept‘ is obviously not a name of a mixed level function, it 

must, therefore, be a first-level function name. 

 

The result of all this is that the function denoted by ‗η denotes a concept‘ can only accept objects 

as arguments and, therefore, must always be false, as no object is a concept. So, we can never 

truthfully say of anything that it denotes a concept. 

 

Let‘s look again at (C). In order for the biconditional to be true, either both of the following 

conditions must be met or both must fail to be met: 

 

1. η must be a predicate 
2. η must stand for a concept 

 

Frege would certainly like to hold that there are predicates, as he‘s spent quite a bit of time 

discussing them elsewhere in his work (hence his plea for a pinch of salt). But, by the argument 

we‘ve just seen, the second condition can never be met. So, despite the fact that there are 

predicates (or so Frege‘s body of work seems to assume), (C) tells us that there can be no 

predicates because nothing can really stand for a concept. Since all of this is intolerable, we must 

throw away one or more of the assumptions. Given Mendelsohn‘s reconstruction, the most 

obvious solution is to reject the claim that ‗η is a concept‘ is the name of a first-level function. 

 

We have, then, to look at the reasons why Frege might have held this view in the first place. 

There is, again, a paucity of evidence from Frege himself. Mendelsohn notes that ―The only 

reason he offers, as far as we can tell, is that ‗η is a concept‘ can be sensefully completed by an 

Eigenname [a proper name].‖
62

 But, as Mendelsohn notes, this isn‘t terribly convincing. For, 

elsewhere, Frege is more than happy to construe sentences of very similar form as representing 

completely different logical structures.
63

 More important than specific examples, however, is the 

fact that Frege‘s lifelong project of crafting a logically perspicuous language is predicated on the 

belief that natural language is rife with ambiguity, polysemy, and vagueness of all kinds. In 

many other places in Frege‘s work, we find him rejecting the typical analysis of the logical 

structure of a particular sentence in surprising ways.
64

 So, as Mendelsohn writes,  

 

it is rather surprising that [in ―On Concept and Object‖ Frege] should so staunchly 

maintain that the singular definite article invariably signals an Eigenname, for, as we see, 

he had adopted a more flexible attitude elsewhere, and moreover, his general rule of thumb 

was that the superficial grammar of a sentence was not always an accurate reflection of its 

logical structure. In order to establish that ‗η is a concept‘ is a first-level Funktionsname, 

                                                 
62

 [Mendelsohn 2005, 78]. 
63

 [Mendelsohn 2005, 77-80] presents a clear comparison between Frege‘s account of the logical structure of (H) 

and his account (in the Grundlagen and elsewhere) of the logical structure of sentences like ‗The number 2 exists‘.  
64

 See, for instance, [Frege 1884b, §91], or the preface to the Begriffsschrift, where Frege states the view about as 

clearly as possible: ―If it is a task of philosophy to break the power of words over the human mind, by uncovering 

illusions that through the use of language often almost unavoidably arise concerning the relations of concepts, by 

freeing thought from the taint of ordinary  linguistic means of expression, then my Begriffsschrift, further developed 

for these purposes, can become a useful tool for philosophers‖ [Frege 1879, v-vi; translated in Beaney 1997, 50-51]. 



132 

 

then, it is not enough merely to point to the fact that it can be sensefully completed by an 

Eigenname – especially when the interpretation creates such enormous logical difficulties. 

One must also show that the expression filling the blank space is operating as an 

Eigenname in that context.
65

 

 

But, of course, Frege does not show this, and so there is no real reason to grant him his 

paradoxical grain of salt.  So, taking into account Frege‘s wider philosophical views, his 

explanation of linguistic necessity as the source of the concept horse problem simply makes no 

sense. So now we have to ask: can we solve or avoid the problem in a way which is consonant 

with Frege‘s views, or is Kerry‘s objection disastrous for Frege‘s ontological distinction? 

 

2.2 Frege‘s proxy-object solution 

 

Frege himself offers an ontologically grounded account of the semantic situation presented by 

Kerry. This account involves the introduction of two new types of entity into Frege‘s ontology. 

He writes 

 

In logical discussions one quite often needs to say something about a concept, and to 

express this in the form usual for such predications – viz. to make what is said about the 

concept into the content of the grammatical predicate. Consequently, one would expect that 

the reference [Bedeutung] of the grammatical subject would be the concept; but the 

concept as such cannot play this part, in view of its predicative nature; it must first be 

converted into an object, or, more precisely, an object must go proxy for it. We designate 

this object by prefixing the words ‗the concept‘, e.g.: 

  

‗The concept man is not empty‘. 

 

Here the first three words are to be regarded as a proper name, which can no more be used 

predicatively than ‗Berlin‘ or ‗Vesuvius‘.
66

 

 

The solution here involves describing the linguistic situation in ontological terms. If we take the 

surface grammar here as an indicator of the actual logical structure, then, Frege suggests, we are 

forced to treat ‗the concept man‘ as a phrase which designates an object. But we desperately 

want this phrase to stand in some relation to a concept. Since Frege‘s linguistic views seemed to 

him to forbid from allowing the phrase of the form ‗the concept x‘ to refer to a concept, he 

suggests a different relation entirely. The suggestion is that phrases of the form ‗the concept x‘ 

refer to a new type of entity, not found elsewhere in Frege‘s works, a special sort of proxy 

object. To fully explicate the semantic situation here also requires the introduction of a new 

relation, the relation of ‗representing‘ [vertreten], which differs from the relation of reference. 

Thus Frege has considerably expanded his ontology in order to explain the nature of this 

linguistic confusion. But this also commits him (or so the reader would hope) to a more explicit 

account of the nature of the entities (proxy objects, the relation of representing) which he has just 

                                                 
65

 [Mendelsohn 2005, 78].  
66

 [Frege 1892a, 196-197; in Beaney 1997, 184-185]. Here Geach translates the German ‗vertreten‘ using the phrase 

‗go proxy for‘. This term would more usually be translated ‗represents‘.  



133 

 

introduced. Given the difficulties which Frege has already encountered in relation to his 

ontological hierarchy, it seems that a solution which could avoid introducing new and poorly 

explained entities would be preferable. 

 

2.3 A salt-free solution to the problem 

 

How might we arrive at such a solution? Well, the first step will be to reject the surface grammar 

as a useful indicator of the logical structure of the thought expressed here. As Frege elsewhere 

has shown no hesitation to rehabilitate natural language, it seems entirely reasonable and 

consistent with his practice to suggest that Kerry is simply pointing out an ambiguity or 

vagueness in natural language which needs to be corrected. The question thus arises: how are we 

to construe a sentence like ‗η is a concept‘? 

 

Ultimately, Frege‘s best response seems to be not to grant Kerry his claim in the first place. 

Without further information about what exactly is meant, sentences of the forms ‗η is a concept‘ 

or ‗η is a concept easily attained‘ are ambiguous. We need to know more specifically which level 

of concept is intended. For, as we‘ve seen, Frege is keen to insist that concepts of different level 

and number of argument places are as ontologically distinct from one another as they are from 

objects. The English word ‗concept‘ and the German word ‗Begriff‘ are both insufficiently 

precise to pin down exactly what is meant and, so, without further information or context, we 

have no way of deciding which level of concept is meant. This is a merely linguistic problem, 

akin in many ways to specifying the surname of each member of a group of men named 

‗Gregory‘ for clarity‘s sake.  This sort of confusion might be obviated with subscripts or more 

explicit wording, e.g.,  

 

1
st
 level concept: ‗η is a first-level concept‘ or ‗η is a concept1‘ 

2
nd

 level concept: ‗N is a second-level concept‘ or ‗N is a concept2‘ 

Etc. 

 

By use of similar devices we can avoid vagueness when confronted with seemingly identical 

uses of words like ‗exist‘ in relation to entities of different ontological level (e.g., ‗The Moon 

exists0,‘ ‗The concept horse exists1,‘ etc.). On this reading, the concept horse problem arises 

largely because our natural languages fail to make the appropriate logical distinctions between an 

object‘s falling under a first-level concept and a first-level concept‘s falling within a second-

level concept. On the linguistic level, both are captured by way of the same subject-predicate 

form, and by use of the catch-all term ‗concept‘. This coarse-grained separation of entities 

simply cannot do justice to the logical subtleties of the realm of thoughts.  Again, though, simply 

adding appropriate terminology to the language obviates this form of the problem. Even without 

adding to the natural language, Frege‘s Begriffsschrift is fully capable of stating things like 

‗First-level concepts exist‘. 

 

If we employed this more logically perspicuous approach, it would become immediately 

apparent that Kerry has provided no evidence whatsoever that one and the same entity (viz. the 

concept horse) is simultaneously treated as both an object and a concept. As Mendelsohn notes, 

Kerry simply cannot provide this kind of evidence in a way which ought to convince (a more 

self-consistent version of) Frege: 



134 

 

 

For somewhere along the line Kerry would have to claim that there is an object x and a 

function f such that x = f; and, on Frege‘s view, placing the identity sign thus between an 

object variable and a function variable is incoherent. Actually, it would seem best in this 

situation simply to deny that [‗The concept horse is an object‘] is meaningful at all. It is 

entirely consistent with our interpretation that we adopt a suggestion of Geach‘s that ‗the 

concept horse‘ is never serving as an Eigenname in the sense that ‗the president‘ does. If a 

sentence containing such an expression is genuinely about a concept, then this would be 

captured in the symbolization by our use of Funktionsnamen and second-order quantifiers, 

and the singular term would thus be eliminated. Alternatively, if the term is ineliminable, 

the sentence containing it would be deemed to be nonsense.
67

 

 

In short, Frege himself developed the logical tools to make syntactic distinctions which seem to 

have no previous corollary in natural languages like German and English. In responding to 

Kerry, rather than simply granting him the meaningfulness of sentences which treat objects as 

concepts, he should have referred him to precisely these syntactic distinctions. His unwillingness 

to alter his views of the definite article is both confusing (given his general attitude toward 

language) and detrimental to his own cause (as it renders his chief ontological distinction 

ineffable). As Mendelsohn notes, the superior response would simply have been to reject the 

meaningfulness of claims which identify objects with concepts in the first place and, instead, to 

present a more perspicuous account of the logical structure of the claim that ‗The concept horse 

is a concept easily attained‘. 

 

Having shown, then, that Frege‘s own philosophy contains the resources to address the concept 

horse problem, let‘s move on to a much more difficult problem with his attempts to fix the 

references of mathematical terms. 

 

§3. The Julius Caesar objection 

 

This second problem pertaining to Frege‘s ontological approach to reference-fixing is the so-

called Julius Caesar objection. This objection directly concerns Frege‘s attempt to construe the 

natural numbers in terms of his ontological framework. More generally, it is relevant to his 

understanding of mathematical structures (like Hilbert‘s ostensibly uninterpreted axiomatization 

of geometry) as non-relational, fixed entities embedded within that framework. The first instance 

of this objection is formulated by Frege in §56 of his Grundlagen der Arithmetik. In the previous 

section, Frege gives the following tentative definitions of the numbers 0 and 1, as well as the 

successor function: 

 

It is tempting to define 0 by saying that the number 0 belongs to a concept if no object falls 

under it. But this seems to amount to replacing 0 by ―no‖, which means the same. The 

following formulation is therefore preferable: the number 0 belongs to a concept, if the 

proposition that a does not fall under that concept is true universally, whatever a may be. 

Similarly we could say: the number 1 belongs to a concept F, if the proposition that a does 

not fall under F is not true universally, whatever a may be, and if from the propositions ―a 

                                                 
67

 [Mendelsohn 2005, 80]. 



135 

 

falls under F‖ and ―b falls under F‖ it follows universally that a and b are the same. It 

remains still to give a general definition of the step from any given number to the next. Let 

us try the following formulation: the number (n + 1) belongs to a concept F, if there is an 

object a falling under F and such that the number n belongs to the concept ―falling under 

F, but not a‖.
68

 

 

The chief aim of Frege‘s Grundlagen is to illustrate that arithmetic is a part of pure logic, and 

that its basic objects, concepts, and modes of reasoning are all purely logical in character. A 

crucial component of these logicist theses is Frege‘s attempt to define the natural numbers in 

logical terms. After having argued against a variety of rival conceptions of the natural numbers, 

Frege has begun in §55 to construct his own position. One of the key results of his analysis of the 

grammatical characteristics of statements involving number terms was the view that numbers can 

only be understood to be particular objects. Hence, given his understanding of the ontological 

difference between functions and objects, numbers cannot be construed as functions or concepts. 

Nevertheless, Frege admits that they do have a peculiar relation to concepts. This becomes clear 

in the tentative definitions above: there numbers are construed as ‗belonging‘ to concepts. The 

number 0 ‗belongs‘ to all those concepts under which no objects fall, i.e., concepts whose 

extensions are empty. The number 1 and the successor function are defined in similar fashion by 

reference to concepts with extensions of a particular type. 

 

Now, before we examine the Julius Caesar objection, which Frege raises immediately after 

stating these tentative definitions, let us recall a few fundamental facts about his understanding 

of definitions. Recall that Fregean definitions are, ostensibly, mere logical shorthand: they are 

stipulations that the sense of a previously senseless sign is to be identical with the sense of an 

already given sign. Essentially, definitions are identity statements whose truth is based upon 

stipulation. Such a view requires that the previously given sign in a definition (the definiens) 

must possess a determinate meaning in order for the definition to function properly. This is a 

somewhat obvious point: if a new sign is to possess the same sense as a previously given sign, 

then that previously given sign must have some sense to give. Now, with this in mind, let us turn 

to the Caesar problem.  

 

In §56 of Grundlagen, Frege writes that 

 

The most likely [of the three definitions] to cause misgivings is the last; for strictly 

speaking we do not know the sense of the expression ―the number n belongs to the concept 

G‖ any more than we do that of the expression ―the number (n + 1) belongs to the concept 

F‖. We can, of course, by using the last two definitions together, say what is meant by 

 

―the number 1 + 1 belongs to the concept F‖ 

 

and then, using this, give the sense of the expression 

 

 ―the number 1 + 1 + 1 belongs to the concept F‖ 

 

                                                 
68

 [Frege 1884b, §55]. 



136 

 

And so on; but we can never—to take a crude example—decide by means of our 

definitions whether any concept has the number JULIUS CAESAR belonging to it, or 

whether that same familiar conqueror of Gaul is a number or is not. Moreover we cannot 

by the aid of our suggested definitions prove that, if the number a belongs to the concept F 

and the number b belongs to the same concept, then necessarily a = b. Thus we should be 

unable to justify the expression ―the number which belongs to the concept F‖, and 

therefore should find it impossible in general to prove a numerical identity, since we 

should be quite unable to achieve a determinate number.
69

 

 

Here Frege rejects these tentative definitions for the reason that they do not determine, for each 

object in the universe, whether or not it falls within the scope of the definition. More specifically, 

the last definition (that of the phrase ‗the number (n + 1) belongs to a concept F‘) is 

unsatisfactory because it contains a phrase whose meaning is as unknown as the phrase it is 

intended to define. At best, Frege argues, we can employ the last two definitions together to 

construct a system of subsequent definitions, none of which will possess a determinate sense 

until we clarify what it means for a number n to belong to a concept F in general.   

 

This problem arises for Frege because he believes that proper definitions should determine 

generally whether or not an object falls within the scope of the definition. This is entailed by his 

view that properly meaningful statements pertain always to a single universe of discourse, 

namely the universe, which is characterized by Frege as an ontological hierarchy of objects and 

functions of various levels. Since definitions are intended to fix the references of new terms, and 

reference-fixing involves connecting symbols and specific elements of the ontological hierarchy, 

any definition which fails to determine exactly which entities fall within its scope is no definition 

at all. Hence, any use of the terms supposedly defined in this manner will, ultimately, be 

meaningless or rely upon prior assumptions for their meaning. As Frege‘s goal is to define the 

concepts of arithmetic using only logical concepts, he cannot rely upon any prior assumptions; 

the whole premise of his work is that an analysis of the logical structure of arithmetic vitiates the 

need for any special non-logical assumptions. So the Julius Caesar objection cannot be answered 

simply with a shrug of the shoulders. 

 

While it might seem ludicrous to worry that we cannot tell by means of a definition that Julius 

Caesar is not the number 1, or that he is not a number at all, the objection is a serious one for 

Frege. Frege of course does not doubt that Caesar is not a number; his objection is not the 

epistemological one that, without a proper definition, we do not know that Caesar is not a 

number. Instead, his objection is semantic (or ontological) in nature: without a proper definition, 

the words we ostensibly use to refer to numbers do not clearly refer to any particular object, and 

hence statements employing those words will fail to be properly meaningful.
70

 The objection, 

understood in these terms, is directly relevant to Frege‘s logicist project, a crucial component of 

which hinges upon his ability to fix the reference of number terms in an unexceptionable way. 

                                                 
69

 [Frege 1884b, §56]. 
70

 This semantic objection has an epistemological correlate, however. Frege‘s interest in the Grundlagen and 

elsewhere is to show that arithmetic is reducible to logic by showing, contra Kant, that we can gain arithmetical 

knowledge without resorting to any form of intuition. He argues in favour of this view by showing that the meanings 

of arithmetical statements are actually purely logical in character, and, thus, that we ought to be able to grasp those 

meanings by purely logical means. 



137 

 

More to the point, for our purposes, many of Frege‘s objections to the free-floating, relational 

view of geometric structures proposed by Hilbert are based upon his contention that Hilbert fails 

to properly fix the references of his terms. So, it is imperative that Frege avoid problems related 

to reference-fixing if he is interested in the success of his own work, as well as that of his critical 

attack on Hilbert-style mathematics.  

 

Essentially, then, the problem amounts to giving a determinate sense to propositions which 

contain number words. As number-words can pop up in a huge variety of contexts, this is a 

rather large problem. Frege‘s approach at the outset is to make the problem more manageable by 

beginning with a particular class of propositions, ones which he knows must have a sense in 

order for his project to get off the ground. Given that he has already determined that numbers are 

objects, he believes that propositions which ―express our recognition of a number as the same 

again,‖
71

 i.e., statements of identity, must be given a determinate sense. Whether or not it is 

practicable in every case,
72

 if numbers are objects then it must always be possible (at least in 

principle) to determine if a given number is identical to or different from any other object. In 

other words, if number terms have fixed reference we should—if we properly grasp that 

reference in a particular case—be able to determine whether or not the reference of a sign 

possessing a different sense nevertheless has the same reference as the number term in question. 

This won‘t necessarily solve the problem of determining the sense of all propositions containing 

number terms, but it is at least a necessary first step in the process. Frege states the whole 

predicament as follows: 

 

In our present case, we have to define the sense of the proposition ‗the number which 

belongs to the concept F is the same as that which belongs to the concept G‘; that is to say, 

we must reproduce the content of this proposition in other terms, avoiding the use of the 

expression ‗the number which belongs to the concept F‘. In doing this, we shall be giving a 

general criterion for the identity of numbers. When we have thus acquired a means of 

arriving at a determinate number and of recognizing it again as the same, we can assign it a 

number word as its proper name.
73

 

 

Thus, to solve his problem, Frege must figure out some method of explaining the general notion 

of a number belonging to a particular concept without explaining that notion in terms of the 

truth-value of identity statements which already employ the (as yet referentially unfixed) phrase 

‗the number which belongs to the concept‘. Until this last condition is met, however, the Julius 

Caesar problem remains, and Frege‘s attempt to characterize arithmetic in terms of his 

ontological hierarchy is in danger. What we need, then, is a non-circular way of recognizing an 

                                                 
71

 [Frege 1884b, §62]. 
72

 Even if Frege‘s view that we apprehend logical truths in a purely rational manner ends up being true, it is 

presumably the case that, whatever our rational faculty of apprehension happens to be, it has certain empirical limits 

related to, e.g., the structure of the human brain. While in principle we ought to be able to determine the truth value 

of identity statements pertaining to extremely large numbers which are defined by repeated application of the 

successor function, we might in fact be physically incapable of such determinations. This, though, would be no 

evidence against Frege‘s logicist programme, any more than would be the fact that we tend to learn arithmetic 

through its simplistic physical applications. 
73

 [Frege 1884b, §62]. 



138 

 

object (in our case, a number) as the same again, even if the way in which the object is given 

(i.e., the sense of the term used to refer to the object) is quite different in separate cases. 

 

After assigning himself this task, in §62, Frege attempts to fulfill it in §§64-68 of Die 

Grundlagen der Arithmetik. He begins by constructing a geometric example which is exactly 

analogous to the numerical examples we have examined above. In his examination of this 

example—where the direction of a line is analogous to the number belonging to a concept—

Frege elaborates on the nature of his problem. He writes that: 

 

In the proposition ‗the direction of a is identical with the direction of b‘ the direction of a 

plays the part of an object, and our definition affords us a means of recognizing this object 

as the same again, in case it should happen to crop up in some other guise, say as the 

direction of b. But this means does not provide for all cases. It will not, for instance, decide 

for us whether England is the same as the direction of the Earth‘s axis—if I may be 

forgiven an example which looks nonsensical. Naturally no one is going to confuse 

England with the direction of the Earth‘s axis; but that is no thanks to our definition of 

direction. That says nothing as to whether the proposition ―the direction of a is identical 

with q‖ should be affirmed or denied, except for the one case where q is given in the form 

of ―the direction of b‖. What we lack is the concept of direction; for if we had that, then we 

could lay it down that, if q is not a direction, our proposition is to be denied, while if it is a 

direction, our original definition will decide whether it is to be denied or affirmed. So the 

temptation is to give as our definition: q is a direction, if there is a line b whose direction is 

q. But then we have obviously come round in a circle. For in order to make use of this 

definition, we should have to know already in every case whether the proposition ―q is 

identical with the direction of b‖ was to be affirmed or denied.
74

 

 

This reformulation of the problem cuts to the heart of the matter. Here Frege insists that the way 

in which an object is given to us (e.g., the sense of the particular term we employ to refer to the 

object) is not a property of the object itself. The problem with defining the direction of a line by 

reference to the direction of another line is that, without an explicit understanding of the notion 

of direction, our definition requires that the object in question be given to us in a particular form 

in order for us to recognize it once again as the same. So, if our definition of ‗the direction of a‘ 

is given by the proposition ‗The direction of a is identical with the direction of b‘, then, unless 

the direction of a is given as the direction of b, we simply have no means of re-identifying the 

object in question. Our definition, then, is only useful within contexts where the relevant object 

is given as a direction and proves no help in grasping the general concept of the direction of a 

line. The situation when it comes to Frege‘s attempted contextual definitions of 0, 1, and the 

successor function is exactly parallel: using Frege‘s definitions we can, e.g., grasp what it means 

to say that the number 1 + 1 belongs to a concept if we already understand what it means 

generally for a number to belong to a concept. But, using these sorts of definitions, we get no 

closer to this goal insofar as they require the objects in question to be given in a particular way 

(i.e., already as numbers which belong to particular concepts). When confronted with objects not 

given in the proper way, these definitions—and, Frege suggests, contextual definitions in 

general—simply fail to fulfill their role. If we ask whether England is the same as the direction 

                                                 
74

 [Frege 1884b, §66]. 



139 

 

of a particular line, or if Julius Caesar is identical to the natural number 1, these contextual 

definitions are of no help.  

 

And, again, the problem here is not epistemological for Frege: he readily admits that, 

presumably, we already know that Caesar isn‘t a number, and that England isn‘t the direction of 

a line. The problem is a semantic one. The semantic function of a definition is to lay it down that 

a given sign is to possess the same sense as another already-understood sign whose reference is 

clear.  The problem with contextual definitions like those in Frege‘s examples is that they fail to 

completely fix the reference of the new sign because there are cases where it is unclear if the 

definition applies or does not apply. This is because an epistemological relationship between 

ourselves and an object (i.e., the way in which a sign is given to us) goes proxy for direct 

reference to the object itself. Because the sense of a term is not a feature of the object to which 

that term refers, there are many cases in which the definition will be inapplicable because we use 

a term with a different sense to refer to the same object. In cases like this—not all of which will 

be as absurd as the Caesar or England examples—these purported definitions cannot help us 

recognize an object as the same again and, more importantly, from a semantic point of view they 

simply do not determine the truth conditions for (most) identity statements involving the term in 

question. Frege‘s goal, then, is to provide definitions whose strict application does not require 

that objects be given in a particular way. 

 

In §68 Frege begins to outline his response to this problem. After admitting that he cannot get 

―any concept of direction with sharp limits to its application, nor therefore […] any satisfactory 

concept of Number‖
75

 using the methods of contextual definition, he suggests the following: 

 

If line a is parallel to line b, then the extension of the concept ―line parallel to line a‖ is 

identical with the extension of the concept ―line parallel to line b‖; and conversely, if the 

extensions of the two concepts just named are identical, then a is parallel to b. Let us try, 

therefore, the following type of definition: 

 

 the direction of line a is the extension of the concept ―parallel to line a‖; 

 the shape of triangle t is the extension of the concept ―similar to triangle t‖. 

 

To apply this to our own case of Number, we must substitute for lines or triangles 

concepts, and for parallelism or similarity the possibility of correlating one to one the 

objects which fall under the one concept with those which fall under the other. For brevity, 

I shall, when this condition is satisfied speak of the concept F being equinumerous 

[gleichzahlig] to the concept G; but I must ask that this word be treated as an arbitrarily 

selected symbol, whose meaning is to be gathered, not from its etymology, but from what 

is here laid down. My definition is therefore as follows: the Number which belongs to the 

concept F is the extension of the concept ―[equinumerous] to the concept F‖.
76

  

                                                 
75

 [Frege 1884b, §68]. 
76

 [Frege 1884b, §68]. Austin‘s translation has ‗equal‘ for ‗gleichzahlig‘. Despite Frege‘s insistence that we ought 

not pay attention to the etymology of his neologism, it does seem slightly misleading to render it ‗equal‘, ignoring 

entirely the occurrence of ‗-zahlig‘ in Frege‘s term, as well as the difference between ‗gleichheit‘ (general equality) 

and Frege‘s ‗gleichzahlig’. Accordingly, where it seems appropriate I have employed the term ‗equinumerous‘. 



140 

 

 

Frege‘s strategy here is to get rid of the context-dependence of his previous attempts at definition 

and replace it by direct reference to specific objects: namely the extensions of particular 

concepts. In the case of directions, the relevant extensions are determined by the concept of 

parallelism (which is, presumably, a clearly fixed concept). In the case of numbers, the relevant 

extensions are determined by the concept of one-to-one correlation (which, again, ought to be a 

concept with a clearly defined boundary). By determining explicitly when a number belongs to a 

particular concept, Frege may now use his previous method of characterizing 0, 1, and the 

successor function to explicitly define each of the natural numbers.  

 

Ideally, we ought to be able to answer the Caesar objection directly now. Armed with this new 

definition, are we able to definitively rule out Julius Caesar as a member of the set of natural 

numbers? Certainly, Frege seems to think so. But the case is not so clear. For what Frege has 

done here is to require either that we have a clear understanding of the concept of an extension, 

or else that objects must be given as extensions for us to recognize them as numbers. Clearly, the 

latter option is not ideal, as it simply recapitulates the Caesar problem. But neither is it clear that 

the former is a viable solution to Frege‘s woes. He himself seems not to have been entirely 

unaware that his proposed solution was not ideal, for in a footnote to §68, he writes: ―I believe 

that for ‗extension of the concept‘ we could write simply ‗concept‘‖.
77

  Frege‘s earlier definition 

of the concept of number, namely: 

 

The number which belongs to the concept F is the extension of the concept 

―[equinumerous] to the concept F,‖ 

 

would then become:  

 

The number which belongs to the concept F is the concept ―equinumerous to the concept 

F‖. 

 

In light of Frege‘s other views, the idea that we can simply switch concepts for objects seems a 

rather shocking claim. Frege seems aware of the fact and notes that his suggestion 

 

[...] would be open to the two objections: 

 

1. that this contradicts my earlier statement that the individual numbers are objects, as is 
indicated by the use of the definite article in expressions like ―the number two‖ and by 

the impossibility of speaking of ones, twos, etc. in the plural, as also by the fact that the 

number constitutes only an element in the predicate of a statement of number; 

2. that concepts can have identical extensions without themselves coinciding. 
 

I am, as it happens, convinced that both these objections can be met; but to do this 

would take us too far afield for present purposes. I assume that it is known what the 

extension of a concept is.
78

 

                                                 
77

 [Frege 1884b, §68 note 1]. 
78

 [Frege 1884b, §68 note 1]. 



141 

 

 

First, Frege notes the obvious objection that this would contradict the entire drift of his work up 

to this point: his criticisms of rival views and his analysis of the grammar of statements 

concerning number terms have both led him to the view that numbers are particular objects and 

not concepts at all. The entire Caesar problem itself is phrased by Frege as that of figuring out 

specifically which objects number terms refer to. If he could simply have shrugged off the 

crucial difference between objects and concepts from the start, then why bother doing the work 

in the first place, and why formulate the question in terms of identity statements pertaining to 

objects? It seems that taking this route would suggest that we have at least two competing ways 

to fix the reference of number terms: one on which these terms refer to objects (i.e., extensions) 

and another on which they refer to concepts. But, the whole point of Frege‘s ferretting out a 

response to the Caesar problem as to definitively pin down references in the first place! This 

seems like a serious objection one way or another, either to Frege‘s claim here or to the 

relevance of his previous work supporting the claim that numbers are objects. In any case, it is 

not clear how Frege‘s suggestion here could work given his deep commitment to the division 

between concepts and objects. 

 

The second objection, that different concepts may possess the same extension, seems to suggest 

that, by switching extensions for concepts, we will lose the specificity Frege had gained in his 

response to the Caesar problem. In Frege‘s original definition, particular numbers are identified 

with particular extensions. Despite the fact that different concepts can pick out the same 

extension, the extension itself is a specific object, so the fact that multiple concepts have the 

same extension does not call into question the determinacy given to number terms by reference 

to specific extensions. Now, when we begin to treat numbers as concepts, no further 

indeterminacy seeps in unless we wish to claim that the two accounts are equivalent. Concepts 

are, for Frege, perfectly determinate (if incomplete) entities, so that, once we identify numbers 

with particular concepts, there should be no difficulty determining the truth-value of statements 

concerning numbers. If, however, we wish the two proposed accounts (i.e., the first account 

involving extensions of concepts, and the second involving just concepts) to be equivalent, then a 

further indeterminacy does in fact arise. And this is what Frege notes in his second objection. 

Because different concepts may have the same extension, there will be many different concepts 

which could just as well serve for each of the natural numbers, with no particular reason to 

decide which ones really are those numbers, if any. This is the crucial point. If more than one 

concept can be identified with a particular number, then it is illegitimate to speak of the number 

n on our second account, and hence Frege will have lost the specificity he had been seeking. 

Presumably it is Frege‘s intention to claim that his extension-based account and his unspecified 

concept-based account are both equally serviceable, so that he will indeed have to figure out a 

solution to this second objection as well as the first. 

 

Regardless of Frege‘s ability to answer these two particular objections (which we will not 

examine here), the fact remains that he recognizes a problem. This is evident, first, in his 

willingness to entertain the obviously problematic idea that we might switch extensions for 

concepts and, second, in his acknowledgement of the assumption that it is known what 

extensions of concepts are. Frege is generally wary (and rightly so) of the use of extensions 

throughout his work; in a later section he writes that his ―way of getting over the difficulty 

cannot be expected to meet with universal approval, and many will prefer other methods of 



142 

 

removing the doubt in question. I attach no decisive importance even to bringing in the 

extensions of concepts at all‖.
79

 The assumption that it is known what extensions are (without 

them being given as such) would return to haunt Frege when a problem very similar to the initial 

Julius Caesar objection arose in his later attempt, in Grundgesetze der Arithmetik, to produce a 

formally precise version of the logicist programme discussed in the Grundlagen. At this point, 

however, it seems clear that Frege‘s first attempt at a solution to the Caesar objection is 

something of a stopgap measure, one which depends upon our recognition of extensions of 

concepts without them being given as such. It does not take much effort to recognize the Caesar 

objection lurking in a different guise here; for the problem arose, initially, when Frege could not 

pick out numbers unless they were given in a particular way. Here, he has simply shifted the 

burden onto extensions and required of us the capacity to recognize extensions no matter how 

they may be given.   

 

3.1 An analogue of the Caesar problem in Grundgesetze der Arithmetik 

 

Let us now turn to an examination of the Caesar problem (or, at least, its close analogue) as it 

arises within §10 of Grundgesetze der Arithmetik. Up to this point in the Grundgesetze, Frege 

has attempted to familiarize the reader with his ontological scheme, as well as some of the 

peculiarities of his notation. As far as the formal system, he has so far only introduced the two 

truth-values as objects, and the functions ξ = δ, —ξ, and ┬ ξ. In §9, Frege introduces the smooth 

breathing notation for courses-of-values: the course-of-values of the function Φ(ε) is represented 

as ἐΦ(ε).
80

 He ends that section by noting that ―The introduction of a notation for courses-of-

values seems to me to be one of the most important supplementations that I have made of my 

Begriffsschrift since my first publication on this subject‖.
81

 So, while Frege was obviously quite 

unsatisfied with his use of extensions of concepts in his earlier work, here he seems to stand 

firmly behind the usefulness of the more general notion of a course-of-values. In the section 

immediately following this introduction of the notation for courses-of-values, Frege raises the 

analogue of the Caesar problem and attempts to solve it.  

 

This stretch of the Grundgesetze, entitled ‗The course-of-values of a function more exactly 

specified,‘ contains Frege‘s attempt to answer the nagging questions about extensions left 

unanswered by his proposed solution to the Caesar problem in the Grundlagen. He begins by 

stating a version of that familiar problem, this time applied to expressions for courses-of-values 

rather than directions or numbers. In this case, the particular expression is that for the course-of-

values of an arbitrary function Φ(ε). Frege writes: 

 

Although we have laid it down that the combination of signs  

 

‗έΦ(ε) =  άΨ(α)‘ 

 

has the same denotation as  

                                                 
79

 [Frege 1884b, §107]. 
80

 Note that, due to a seemingly insurmountable typesetting problem, my representation of the smooth breathing 

notation (here and throughout) does not exactly match Frege‘s own. 
81

 [Frege 1893, §9]. 



143 

 

 

‗α(Φ(α) = Ψ(α)),‘
82

 

 

this by no means fixes completely the denotation of a name like ‗έΦ(ε)‘. We have only a 

means of recognizing a course-of-values if it is designated by a name like ‗έΦ(ε)‘, by 

which it is already recognizable as a course-of-values. But we can neither decide, so far, 

whether an object is a course-of-values that is not given us as such, and to what function it 

may correspond, nor decide in general whether a given course-of-values has a given 

property unless we know that this property is connected with a property of the 

corresponding function.
83

 

 

Here, again, we encounter a problem very similar to the Caesar objection: we have the tools to 

recognize the course-of-values of a particular function, but only if that course-of-values is given 

in a particular way. In this case, we are licensed to recognize two courses-of-values, έΦ(ε) and 

άΨ(α), as one and the same only if they are given as courses-of-values (i.e., if they are referred to 

using Frege‘s smooth-breathing notation). In cases where these same entities are given 

differently, Frege‘s contextual definition affords us no means of recognizing their identity. The 

problem however is deeper than the epistemological one of our recognizing the entities as the 

same again: for, if we are given only the contextual definition to characterize courses-of-values, 

statements like ‗έΦ(ε) = Julius Caesar‘ will not possess a truth-value. Note, too, that not all cases 

of this sort will be as absurd as the Caesar example: for, if numbers are identified with extensions 

(as Frege suggests), then statements of the form ‗έΦ(ε) = 2‘ will be of crucial importance for 

establishing the foundations of arithmetic. Using Frege‘s contextual definition, such statements 

are neither true nor false—indeed, they are, by Frege‘s lights, completely senseless until we 

determine more carefully the meaning of terms of the form ‗έΦ(ε)‘.  

 

Frege‘s goal, then, is to provide a general criterion for determining whether an object is a course-

of-values or not. As before, Frege has reduced this problem to one of fixing the reference of 

particular types of identity statements. Ostensibly, if we can determine the truth or falsity of an 

identity statement containing terms which refer to courses-of-values without the condition that 

those terms be of a particular form, then we will have provided a general criterion for 

recognizing courses-of-values. The contextual definition above allows one to determine the truth 

of identity statements concerning entities which are given to us already as courses-of-values by 

stipulating that the reference of these statements (i.e., their truth value) is to be the same as a 

statement whose sense is already grasped (e.g., the statement that the two functions in question 

always take the same value for the same argument). But, crucially, this contextual definition does 

not and cannot determine the truth-value of identity-statements concerning entities which are not 

already given as courses-of-values. We cannot, for instance, determine whether or not Julius 

Caesar is identical to any course-of-values simply by using the definition above. Frege‘s overall 

point in raising these objections is to insist that, without a solution to this sort of problem, the 

terms at issue here are actually senseless, and sentences containing them will therefore fail to 

refer to a truth-value. Hence we are not licensed to (and, by Frege‘s lights—are actually 

                                                 
82

 For ease of typesetting, here and throughout, I use the common modern notation for the universal quantifier (i.e., 

‗‘) in place of Frege‘s idiosyncratic symbolization.  
83

 [Frege 1893, §10]. 



144 

 

incapable of) making inferences which employ the purported ‗thoughts‘ about courses-of-values 

as premises. As we‘ve seen, when using courses-of-values to solve the Caesar problem in the 

Grundlagen, Frege had simply assumed that it was known what the extension of a concept was 

(and, perhaps too, what the course-of-values of a function was). His remarks there, and 

elsewhere, seem to indicate that he was not entirely comfortable with this situation. Now, in what 

was to be the culmination of his life‘s work, Frege appears ready to face the problem head on. 

Whereas previously he attempted to shrug off the Caesar objection via reference to the 

extensions of concepts, here he raises the problem for courses-of-values, and hence for 

extensions as well. His earlier remedy will thus not work here, and so a new tactic is required. 

 

Frege‘s method for avoiding the problem of contextual definition when it comes to courses-of-

values is to pin them down more precisely by clarifying their relationship to functions. More 

specifically, his suggestion is that, for each function, when it is introduced, we ought to 

determine what value it takes for courses-of-values as arguments. Doing so will clarify the truth 

value of any statement containing a course-of-values term, eliminating any indeterminacy left 

unsettled by his previous solution to the Caesar problem. Let‘s examine how this is supposed to 

work. 

 

Since, up to this point in the Grundgesetze, Frege has only introduced three functions, the 

required determinations are relatively brief. The three functions Frege has discussed up to this 

point are: ξ = δ, —ξ, ┬ ξ; i.e., identity, the horizontal, and negation. Essentially, Frege‘s concern 

here is to determine for each of these functions what value it will take when a course-of-values is 

input as argument.  

 

The negation function can be discounted from consideration immediately, for, on Frege‘s 

dispensation, the function ―can be considered always to take a truth-value as argument‖.
84

 This is 

because Frege defines negation as follows: ―The value of the function ┬ ξ shall be the False for 

every argument for which the value of the function —ξ is the True; and shall be the True for all 

other arguments‖.
85

 Further, the function  —ξ is defined as follows:  ―—Δ is the True if Δ is the 

True; on the other hand it is the False if Δ is not the True‖.
86

 So, since the argument for the 

function ┬ ξ is defined in terms of the values taken by the function —ξ (which are, by stipulation, 

always truth values), any argument taken by the function ┬ ξ, even if it is an object other than a 

truth value, can nevertheless be treated as a truth-value.  The function —ξ, too, can be eliminated 

from consideration here insofar as it can be reduced to the function ξ = δ. Frege writes that ―by 

our stipulations the function —ξ has for every argument the same value as the function ξ = (ξ = 

ξ); for the value of the function ξ = ξ is the True for every argument. From this it follows that the 

value of the function ξ = (ξ = ξ) is the True only for the True as argument, and that it is the False 

for all other arguments, exactly as with the function —ξ.‖
87

 So, then, we are left only with the 

function ξ = δ to consider. Frege‘s problem, again, is to determine what value this function gives 

for courses-of-values taken as arguments.  

 

                                                 
84

 [Frege 1893, §10]. 
85

 [Frege 1893, §6]. 
86

 [Frege 1893, §5]. 
87

 [Frege 1893, §10]. 



145 

 

3.2 The restriction problem 

 

At this point in his argument, Frege makes some remarks which are rather difficult to interpret, 

given certain key commitments of his philosophy. He writes: 

 

Since in this way everything reduces to consideration of the function ξ = δ, we ask what 

value this has if a course-of-values occurs as argument. Since up to now we have 

introduced only the truth-values and courses-of-values as objects, it can only be a question 

of whether one of the truth-values can perhaps be a course-of-values. If not, then it is 

thereby also decided that the value of the function ξ = δ is always the False if a truth-value 

is taken as one of its arguments and a course-of-values as the other. If on the other hand the 

True is at the same time the course-of-values of some function Φ(ξ), then it is thereby also 

decided what the value of the function ξ = δ is in all cases in which the True is taken as one 

of the arguments, and likewise if the False is at the same time the course-of-values of a 

certain function.
88

 

 

Here Frege has reduced his problem to the determination of the value taken by the function ξ = δ 

for courses-of-values. His first concern is to determine whether the only two objects hitherto 

introduced in his formal system (i.e., the two truth-values) are themselves courses-of-values. His 

reasoning regarding the values taken by the function ξ = δ if either of these objects is or is not a 

course-of-values is unexceptionable: if neither truth-value is a course-of-values, then the function 

gives the False as value when a course-of-values is taken as one argument and a truth-value as 

the other. But if either truth-value is in fact a course-of-values of some function, then we also 

have a determinate value for the function when one of the truth-values is taken as an argument; 

this value will be the True when both argument places are filled by names for one of the truth-

values (be they referred to by terms of the form ‗έΦ(ε)‘ or otherwise), and the False in all other 

case. This of course requires that we have picked out precisely which courses-of-values the two 

truth-values are. All this is fine, as far as it goes.  

 

The problematic portion of the above quotation is Frege‘s restriction of his consideration to just 

these two objects. His suggestion seems to be that, because he has only introduced the two truth-

values as objects thus far, he need only decide whether either of these is itself a course-of-values 

to determine the value taken by the function ξ = δ for courses-of-values, and thereby, also, to fix 

the reference of the term ‗έΦ(ε)‘ completely. This seems to imply that the functions so far 

introduced only take as arguments the objects which have been introduced by Frege at this point 

in the Grundgesetze; i.e., that Frege‘s functions have a highly restricted domain. Taken at face-

value, this seems to directly contradict Frege‘s views that functions have an ontological status 

which is not dependent upon their being given in a particular way (e.g., in Frege‘s 

Begriffsschrift). Here he seems to be restricting the domain of functions in a way which depends 

upon the status of his formal system. More than this, the restriction also seems to conflict with 

his demand that functions be defined for every object in the universe—a demand which seems to 

be one of the sources of the Caesar problem in the first place. While Frege‘s views allow for the 

                                                 
88

 [Frege 1893, §10]. 



146 

 

‗restriction‘ of functions in a certain sense, such restrictions do not involve the exclusion as 

arguments of objects without names in a particular formal system.
89

 

 

Moreover, it seems to imply that the problem carried over to the Grundgesetze (because of 

Frege‘s inability to provide a general solution to the Caesar objection in the Grundlagen) is in 

fact a pseudo-problem. For, if a particular formalization determines which objects a function 

takes as arguments, then we ought not be concerned with the fact that our definitions cannot 

determine for an arbitrarily given object—not just those which have been introduced into the 

system or given names—whether or not they fall under the definition. If Caesar isn‘t part of the 

system, or if there is no name for Caesar in Frege‘s formalization of number theory, for instance, 

then there simply is no problem in determining the value of the function ξ = 2 for Caesar as an 

argument. In the terms of the Grundgesetze, we may just as well have avoided the problem of 

fixing the reference of terms of the form ‗έΦ(ε)‘ by never introducing such terms in the first 

place. 

 

So what exactly is going on here? Some commentators
90

 have suggested that the best way to 

understand this restriction is to characterize it in what we would now call meta-linguistic terms. 

That is, they suggest that the restriction is a meta-linguistic statement concerning which names 

have thus far been introduced into the system. 

 

But it seems clear from what Frege explicitly states that it is not merely a matter of names: 

 

Since up to now we have introduced only the truth-values and courses-of-values as objects, 

it can only be a question of whether one of the truth-values can perhaps be a course-of-

values.
91

 

 

Taken on its own, this fairly clearly shows that Frege is not concerned with which names have 

been introduced into the system up to this point, but with which objects have been introduced. 

Moreover, if we keep in mind that this restriction is undertaken within the context of the Caesar 

problem, then we have good reason to suspect that the meta-linguistic reading misses the mark. 

 

If Frege was merely interested in the linguistic features of his formal language, the Caesar 

problem could never have arisen in the first place, as it is fundamentally not a linguistic problem 

concerned with which sorts of statements are formulable within the confines of Begriffsschrift. 

The problem is rather concerned with our relation to the references of such statements. Or, as 

Richard Heck puts it, 

                                                 
89

 One way in which a Fregean function might be ‗restricted‘ to, e.g., the natural numbers is by defining the function 

in the intended manner for the natural numbers as arguments (presuming we have a rigorously circumscribed 

concept of natural number to work with), while the function takes on the value the False (or some dummy value) for 

objects which are not natural numbers. The key here is that the function remains defined for all objects in the 

universe, and that the so-called restriction requires a referentially fixed concept of natural number (or whatever) to 

get off the ground. 
90

 Cf., e.g., [Moore and Rein 1986].  
91

 [Frege 1893, §10]. The introduction of objects (rather than names) is equally clear in the German: ―Da wir bisher 

nur die Wahrheitswerthe und Werthverläufe als Gegenstände eingeführt haben, so kann es sich nur darum handeln, 

ob einer der Wahrheitswerthe etwa ein Werthverlauf sei‖ [Frege 1893, §10, emphasis mine]. 



147 

 

 

What the Caesar problem suggests is that we have no understanding of what numbers are 

that supports the idea that they are independent of how they are given to us, that is 

independent of our capacity for thought about them.
92

  

 

As Heck‘s remarks here help to show, the Julius Caesar problem is not merely a bug arising from 

Frege‘s particular choice of formal system. It is instead an essential problem which cuts to the 

core of his attempt to understand the nature of mathematics. If he is avoid the kind of free-

floating, purely formal understanding of mathematics which he sought to combat in his 

engagements with Hilbert, Frege must be able to show that mathematical terms refer to 

something in particular. The Caesar problem arises precisely because Frege is unwilling to allow 

contextual definitions (which do rely on the peculiarities of a particular context, language, or 

formalism) as a means of fixing the references of mathematical terms. Thus, it would be 

incredibly odd for Frege, right at the point when he begins to solve the problem, to restrict his 

solution to the merely linguistic peculiarities of a particular stage of his formal system. 

 

But, even if we reject the meta-linguistic reading, we are still presented with a problem. For, 

while Frege does not restrict his solution to the linguistic realm, he does nevertheless restrict 

himself to considering only the objects which are introduced to his formal system at a given 

stage. Given Frege‘s methodological scruples, such a restriction seems fundamentally at odds 

with his goal of solving the Caesar problem once and for all. How, then, are we to understand 

Frege‘s move here? 

 

3.3 Heck‘s Argument Against Frege‘s ‗Universalism‘ 

 

Let‘s try to reconstruct the situation we seem to be presented with here. If Caesar is in the 

domain of the functions Frege considers here, then of course these functions have to be defined 

for Caesar as an argument. On the reading of Frege I have presented thus far in this dissertation, 

he holds that a function must be defined for every object in the universe. So, straightforwardly, 

Caesar must be in the domain of the functions Frege is about to consider. But the restriction of 

his attention to the few objects he has already introduced seems to indicate that, for Frege, 

Caesar is not in fact in the domain of these functions. This situation leaves us with at least two 

options. 

 

1. Something is wrong in the above reconstruction of the situation.  
2. Frege has (knowingly or otherwise) committed himself to an inconsistent position. 

 

Before we leap to either conclusion, let‘s try to re-evaluate our reconstruction of the situation. 

Richard Heck, in a paper examining §10 of the Grundgesetze offers an alternative account of the 

restriction which involves re-evaluating the ‗universalist‘ component of our reconstruction. That 

is to say, Heck wants to call into question the view that Frege holds that a function always has to 

be defined for every object in the universe. If Heck‘s argument is correct, then Frege‘s restriction 

is an unproblematic one. How, then, does Heck think that we can deny Frege‘s universalism? 

Rather straightforwardly, he states: 

                                                 
92

 [Heck 1999, 266, note 16]. 



148 

 

 

When [Frege] says that we have ‗introduced only the truth-values and value-ranges as 

objects‘, he means to tell us that the quantifiers in his formal language are, at this point, to 

be taken as ranging only over such objects.
93

 

 

For a modern logician, mathematician, or model theorist this sort of restriction of the domain of 

quantification is a matter of course. For Frege, however, this seems difficult, as every ‗model‘ 

must ultimately be embedded within a single ontological hierarchy. So in what sense can the 

domain of quantification be restricted for Frege? 

 

Well, let‘s look at what Heck has to say about the motivations for Frege‘s ‗universalism‘. He 

writes: 

 

Frege understood some of the logicians of his time to be proposing to deal with restricted 

quantification by making use of what the Booleans called ‗universes of discourse‘.
94

 Frege 

objected vehemently to this practice. He insisted, firstly, that sentences involving restricted 

quantification—e.g., ‗Every real number is F‘—can be interpreted as generalized 

conditionals, in now familiar fashion: That is, he argues that there is no need to invoke a 

restricted domain, containing only the reals, for this purpose. He argues, secondly, that 

such sentences should be interpreted as generalized conditionals. Treating restricted 

quantification in terms of universes of discourse makes it difficult to represent the validity 

of certain sorts of inferences: For example, ‗Every real number has a non-negative square‘ 

implies ‗Any number which has a negative square is non-real‘. At the very least, then, 

some way needs to be provided for the ‗universe of discourse‘ to be represented in the 

logical forms of such sentences.
95

  

 

As the italics indicate here, Heck‘s suggestion is that Frege‘s opposition to restricted 

quantification is not fundamental but, instead, relative to particular examples where this sort of 

quantification seems to obscure logical relations rather than clarify them. Heck‘s argument for 

this reading, it seems to me, involves imputing a significant amount of modern, model-

theoretically sophisticated logic into Frege‘s work. While Heck himself admits that this reading 

requires ―only a dollop of anachronism‖
96

, the dollop seems rather a large one. Let‘s look a bit 

closer. 

 

                                                 
93

 [Heck 1999, 272-273]. 
94

 Boole himself introduces the notion of a universe of discourse in The Laws of Thought. There he writes ―In every 

discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, 

there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered 

discourse is that in which the words we use are understood in the widest possible application, and for them the limits 

of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less 

spacious field. […] Now, whatever may be the extent of the field within which all the objects of our discourse are 

found, that field may properly be termed the universe of discourse‖ [Boole 1854, Chapter III, §4]. As will be made 

clearer below, Frege‘s point is that the restriction of quantification is only undertaken within the universe (i.e., 

within his ontological hierarchy), and, since logic is concerned with the most general laws of truth, such restrictions, 

too, have to be cemented clearly within that universe. 
95

 [Heck 1999, 273]. 
96

 [Heck 1999, 274]. 



149 

 

The view is that Frege does not object to all uses of restricted quantification, but only to those 

which involve the confusion of domains of quantification with universes of discourse. Let‘s look 

at what Heck says about universes of discourse. 

 

The logical form of ‗Every number is F‘ is therefore much more nearly: ‗x(Nx;Fx). The 

first argument of the quantifier specifies the ‗universe of discourse‘, in the sense relevant to 

the analysis of restricted quantification.
97

  

 

His suggestion here is that the logical role of ‗Nx‘ is to specify a universe of discourse. On this 

view, presumably, the naked quantifier ‗x‘ indicates the bare universe, while Nx‘ is meant to 

whittle this immense totality down to those entities which fall under the concept N, whatever that 

might be. Thus the universe of discourse, on this view, is always a subset of the universe 

properly speaking. In mathematics and logic, of course, the universe of discourse understood in 

this way is almost never indicated explicitly. Universes of discourse, understood in this way, are 

superfluous on Heck‘s reading of Frege. For, in our example, if the concept N is well-defined, 

then any statement about the predicate-restricted universe of discourse can simply be rephrased 

in terms of a generalized conditional (in a manner similar to that of Frege‘s proposed 

emendations of Hilbert‘s views). 

 

So, Heck admits that Frege has good reasons to eschew the use of ‗universes of discourse‘ 

understood in the sense of predicate-restrictions placed on the universal quantifier. We are left, 

then, to consider the availability of variable ‗domains of quantification‘ to Frege. Heck writes 

that 

 

Obviously, though, even if we make it explicit, in the syntax, that the quantifier can be […] 

‗restricted‘ by some predicate, that does not make domains of quantification, in the sense 

relevant to model-theory, superfluous: One can—indeed, must—still allow for varying 

domains of quantification when defining validity, implication, and the like, for the usual 

sorts of reasons.
98

 

 

This, it seems to me, is where Heck‘s argument requires more than his suggested dollop of 

anachronism, for he has not attempted to show that variation in the domain of quantification (in 

this sense) is required for Frege’s understanding of validity, implication, and the like. This is the 

crucial point in the argument, and it rests on the reader‘s accepting that Frege‘s notions of 

‗validity, implication, and the like‘ are sufficiently similar to their modern model-theoretic 

cousins that they will require varying the domain of quantification ―for the usual sorts of 

reasons‖. But, as I hope to have shown at several points in this thesis, there are good reasons not 

to assume that Frege‘s logical notions are primitive versions of contemporary notions. On the 

face of it, then, it seems suspect to simply import the modern model-theoretic conceptions of 

these notions into Frege‘s view, which elsewhere (e.g., in his correspondence with Hilbert) 

seems highly critical of the philosophical underpinnings of an enterprise quite like model theory 

in important respects. 

 

                                                 
97

 [Heck 1999, 274]. 
98

 [Heck 1999, 274]. 



150 

 

So, it seems, neither the meta-linguistic reading of the restriction nor the anti-universalist reading 

suggested by Heck offer a clear explanation of our problem. Without good reasons for thinking 

otherwise, then, we are left with the suspicion that Frege‘s restriction here is simply in conflict 

with his established views. 

 

3.4 Frege‘s attempted solution to the Caesar problem in the Grundgesetze 

 

Regardless of the status of this restriction relative to the rest of Frege‘s philosophy, he does 

employ it in his attempted solution to the Caesar problem. Let‘s return to our examination of the 

rest of this solution, with the caveat that it may hinge in part on this inconsistency. 

 

After further reducing his problem via the restriction, Frege sets about determining whether or 

not either of the truth-values is a course-of-values. He notes that he cannot decide whether or not 

either of the truth-values is a course-of-values by appeal to his stipulation that ―‗έΦ(ε) = άΨ(α)‘ 

is to have the same denotation as ‗α(Φ(α) = Ψ(α))‘.
99

  

 

Frege then engages in a technical argument intended to illustrate that ―it is always possible to 

stipulate that an arbitrary course-of-values is to be the True and another the False‖
100

 and, hence, 

that the answer to his question is that the truth-values themselves are (or, rather, can be taken to 

be) courses-of-values. Let‘s examine his argument supporting this claim. He writes that: 

 

it is possible to stipulate generally that ‗ῆΦ(ε) = ᾶΨ(α)‘ shall denote the same thing as 

‗α(Φ(α) = Ψ(α))‘ without the identity of έΦ(ε) and ῆΦ(ε) being derivable from this. We 

should then have a class of objects with names of the form ‗ῆΦ(ε)‘, and for whose 

differentiation and recognition the same distinguishing mark held good as for courses-of-

values.
101

 

 

This first step is simply a generalization of our earlier problem: instead of considering the 

singular case, Frege moves to consider an entire class of objects with names of the form ‗ῆΦ(ε)‘. 

After having determined such a class, he defines a function X(ξ) as follows: 

 

X(ῆΛ(ε))   =  the True 

X(the True)  = ῆΛ(ε) 

X(ῆM(ε))    = the False 

X(the False)  = ῆM(ε) 

 

For all other arguments, the value of the function is simply the argument itself. 

 

Now, the functions Λ(ξ) and M(ξ) can either take the same value for every argument or they can 

differ in value for some arguments. In the case where they differ in value for at least one object, 

then the  

 

                                                 
99

 [Frege 1893, §10]. 
100

 [Frege 1893, §10]. 
101

 [Frege 1893, §10]. 



151 

 

[...] function X(ξ) never has the same value for different arguments, hence ‗X(ῆΦ(ε)) = 

X(ᾶΨ(α))‘ also always has the same denotation as ‗α(Φ(α) = Ψ(α))‘. The objects whose 

names were of the form ‗X(ῆΦ(ε))‘ would then be recognized by the same means as the 

courses-of-values, and X(ῆΛ(ε)) would be the True and X(ῆM(ε)) the False.
102

 

 

Essentially, what Frege has done here is to construct a function which licenses him to give the 

truth-values names of the same form as those for courses-of-values, and thus to treat the truth-

values as courses-of-values. But this license does not uniquely pick out which course-of-values 

the truth-values are, for we could very well have defined a function Xˈ(ξ) such that the roles of 

ῆΛ(ε) and ῆM(ε) are reversed, thereby identifying the truth-values with different courses-of-

values. In other words, Frege‘s argument shows only that we can identify the truth-values with 

particular courses-of-values, it doesn‘t allow us to take the further step of picking out precisely 

which courses-of-values these might be, other than arbitrarily. Accordingly, he writes that: 

 

Thus without contradicting our setting ‗έΦ(ε) =  έΨ(ε)‘ equal to ‗α(Φ(α) = Ψ(α))‘ it is 

always possible to stipulate that an arbitrary course-of-values is to be the True and another 

the False. Accordingly let us lay it down that έ(—ε) is to be the True and έ(ε = (┬α(α = 

α)) is to be the False. έ(—ε) is the course-of-values of the function —ξ, whose value is the 

True only if the argument is the True, and whose value for all other arguments is the False. 

All functions for which this holds, have the same course-of-values, and this is by our 

stipulation the True. Accordingly —έΦ(ε) is the True only if the function Φ(ξ) is a concept 

under which falls only the True; in all other cases —έΦ(ε) is the False. Further, έ(ε = ┬α 

(α = α)) is the course-of-values of the function ξ = ┬α (α = α), whose value is the True 

only if the argument is the False, and whose value for all other arguments is the False. All 

functions for which this holds have the same course-of-values, and this is by our stipulation 

the False. Thus every concept under which falls the False and only the False, has as its 

extension the False.
103

 

 

Here Frege stipulates precisely which courses-of-values he wishes to identify with the True and 

the False. His earlier argument was designed to illustrate that it is always possible to identify the 

True and the False with an arbitrary course-of-values, so that is precisely what he does. 

Ostensibly, then, the Grundgesetze version of the Caesar problem is solved, and Frege writes that 

―With this we have determined the courses-of-values so far as is here possible‖.
104

  

 

Let‘s review Frege‘s reasoning thus far. In order to precisely fix the reference of terms of the 

form ‗έΦ(ε)‘, Frege argues that we must determine the value taken by all functions for such 

terms taken as arguments. Now, up to this point Frege has introduced only three functions, and 

consideration of these can, in fact be reduced to the consideration of a single function, namely 

the identity function ξ = δ. Frege‘s problem, then, is to figure out what value this function takes 

for courses-of-values as arguments. Just as he has only introduced a few functions thus far in the 

Grundgesetze, so too he has only introduced a few objects: the True and the False. Frege‘s 

problem, if we allow him his restriction, thus ultimately reduces to figuring out two things: first, 

                                                 
102

 [Frege 1893, §10]. 
103

 [Frege 1893, §10]. 
104

 [Frege 1893, §10]. 



152 

 

whether or not the truth-values are themselves courses-of-values and, second, what values the 

function ξ = δ takes for the truth-values as arguments if they are courses-of-values. To solve his 

problem, then, Frege constructs a function designed to licence him to identify the truth-values 

with arbitrary courses-of-values. With this license, he then stipulates precisely which courses-of-

values they are, thus allowing us to decide the values the function ξ = δ takes for courses-of-

values. At the end of this chain of reasoning, Frege believes he has fixed the reference of terms 

of the form ‗έΦ(ε)‘ as precisely as is possible, and thus that he has solved the Caesar problem as 

it applies to courses-of-values for his system as elaborated up to this point. By stipulating the 

values taken by functions for courses-of-values as arguments whenever new functions are 

introduced, the solution should hold for further elaborations of the system as well. 

 

3.5 An insoluble problem 

 

But, with historical hindsight, we now know that there is trouble yet lurking in Frege‘s project in 

the form of Russell‘s paradox. It is something of a commonplace to suggest that the ultimate 

failure of Frege‘s project rests on the inconsistency which results from this paradox. And, to be 

sure, the discovery of a contradiction in his formal system was a significant blow to Frege‘s 

life‘s work. But, I will argue (in substantial agreement with Richard Heck)
105

 that the 

contradiction is merely an effect of the real failure, which is Frege‘s inability to satisfactorily 

solve the Caesar problem and, hence, his inability to fix the references of mathematical terms. 

 

The proximal source of Russell‘s paradox in Frege‘s work is his use of Basic Law V. This axiom 

can be stated as follows: 

 

έF(ε) = έG(ε) ≡ ∀x(Fx ≡ Gx) 
 

Or, in plain English, the course-of-values of a function F is identical to the course of values of a 

function G if, and only if, the functions F and G take the same values for every object. Frege 

employs this axiom in a number of places throughout the Grundgesetze, but there are only two 

which are ineliminable, and these are the source of the problem. 

 

The two uses of Basic Law V which cannot be eliminated, i.e., which are not merely uses of 

convenience, are connected to Frege‘s attempted derivation of Hume‘s principle. Thus it is only 

in his attempt to prove this principle that Frege‘s system is threatened by Russell‘s paradox.  

 

The term ‗Hume‘s principle‘ was coined by George Boolos,
106

 but the principle itself does have 

its source in Hume. In his 1739 Treatise, Hume gives the following definition of equality 

between numbers: 

 

When two numbers are so combin‘d, as that the one has always an unite answering to 

every unite of the other, we pronounce them equal.
107

 

                                                 
105

 Cf. [Heck 2005]; [Boolos 1993, 230] also clearly establishes the importance of the Caesar problem for 

understanding the collapse of Frege‘s work. 
106

[Boolos 1987a, 137]. 
107

 [Hume 1896, Book I, Part III, §1]. 



153 

 

 

Frege in fact cites this passage from Hume in §63 of the Grundlagen. Hume‘s formulation of the 

principle is less relevant to us than its analogue in the philosophy of Frege. Put in Frege‘s 

language of concepts and objects, this principle holds that the number of objects falling under a 

concept F is the same as the number of objects falling under a concept G if, and only if, the Fs 

can be put in a one-to-one correspondence with the Gs. As noted by Charles Parsons and others, 

the bulk of Frege‘s system of arithmetic is derived from Hume‘s principle and the axioms other 

than Basic Law V.
108

 Moreover, if we drop Basic Law V, there are a number of different 

consistent fragments to be found in Frege‘s system.
109

 So, if the source of Frege‘s problems is 

Basic Law V, and this axiom is only used essentially in a very few cases, why not simply drop it 

and replace it with Hume‘s seemingly very similar principle?
110

  

 

The answer is that, by using only Hume‘s principle in his definition of number, Frege simply 

cannot solve the Caesar problem. Let‘s see why not. 

 

In Frege‘s work, Hume‘s principle first arises in connection with his attempts to fix the meaning 

of identity-statements between numerical terms. As we‘ve seen, fixing the meaning of such 

statements is a crucial step in Frege‘s larger attempt to fix the references of mathematical terms. 

And, at first glance at least, Hume‘s principle seems like an ideal candidate for a solution to this 

problem. ―But,‖ Frege writes, ―it raises at once certain logical doubts and difficulties.‖
111

 The 

chief difficulty, as he sees it, is that Hume‘s principle seeks a special definition of numerical 

identity, or ‗equinumerosity‘.  

 

In some well-known passages from the Grundlagen, Frege attempts to define the notion of the 

direction of a line using the sort of limited, contextual definition suggested by Hume‘s principle. 

He writes of his proposed definition that 

 

If […] we were to adopt this way out, we should have to be presupposing that an object can 

only be given one way [i.e., as a direction for directions, or as a number for numbers]; for 

otherwise it would not follow, from the fact that q was not introduced by means of our 

definition, that it could not have been introduced by means of it. All identities would then 

amount to this, that whatever is given to us in the same way is to be reckoned as the same. 

This, however, is a principle so obvious and so sterile as not to be worth stating. We could 

not, in fact, draw from it any conclusion which was not the same as one of our 

premisses.
112

 

 

Such definitions, then, require objects to be given in a particular way, thus making them relative 

to a given system, language, or formalization. But this is exactly what Frege is seeking to avoid 

(as is evident from his rejection of the formalism suggested by Hilbert and Korselt). In order to 

                                                 
108

 Cf. [C. Parsons 1995, 196ff]; [Boolos and Heck 1998] gives a bit more detail. 
109

 Cf., e.g., [Boolos 1987a], [Boolos1987b], [T. Parsons 1987], [Heck 1996] and  [Wehmeier 1999].  
110

 [Heck 1995] argues convincingly that Frege himself knew that almost all of his uses of Basic Law V were 

matters of mere convenience. 
111

 [Frege 1884b, §63]. 
112

 [Frege 1884b, §67]. 



154 

 

truly solve the Caesar problem, then, Hume‘s principle is insufficient and an explicit definition 

must be given. 

 

And so it is that Frege is put in an impossible position. He must either employ an explicit 

definition at some point in his derivation of Hume‘s principle or he must abandon his search for 

a context-independent way of fixing the references of mathematical terms. 

 

He chose the first route which famously leads to Russell‘s paradox and the inconsistency of his 

system. By rejecting Basic Law V in favour of Hume‘s principle, however, we are no closer to 

solving the Caesar problem. Thus, without Basic Law V, Frege saw no way of actually fixing the 

references of mathematical terms, other than relative to a given system. We are, therefore, left 

again at an impasse in our attempt to understand what mathematics is about. The free-floating, 

structure-focussed road taken by Hilbert seems to lead to the threat of meaninglessness (without 

a background ontology) and the relativisation of truth; the fixed-reference path taken by Frege 

leads either to contradiction and incoherence or back once more to the meaninglessness of 

Hilbert‘s position. 

 

§4. Deadlock 

 

Now, how does the ultimate failure of Frege‘s attempt to solve the Julius Caesar problem affect 

the outcome of his debate with Hilbert over the nature of axiomatics and mathematics more 

generally? 

 

As we‘ve seen above, many of Frege‘s criticisms of Hilbert focus on the fact that Hilbert is 

unable (and unwilling) to pin down the references of the terms of an axiomatic system, prior to 

the process of interpretation. Hilbert‘s view, in fact, was based on the (now widely-held) belief 

that the schematic or uninterpreted character of formal mathematical systems was one of the 

chief benefits of axiomatization. Given the fact that Frege‘s own best efforts were unable to fix 

the references of the key terms of even his modest fragment of mathematics, his criticisms of 

Hilbert might seem to lose much of their persuasiveness. If Frege himself is unable to do what he 

demands of Hilbert, and Hilbert‘s method appeared incredibly productive to the mathematical 

community at large, what is there left to recommend Frege‘s view? Why shouldn‘t we simply 

adopt Hilbert‘s view? 

 

But I believe we are left with a more profound difficulty in the wake of Frege‘s inability to solve 

the Caesar problem. One lesson to be learned from a detailed examination of the Frege-Hilbert 

correspondence is that the difficulties faced by Frege here are much more obviously present in 

Hilbert‘s work. The fact that neither Frege nor Hilbert seem capable of fixing the references of 

the terms of their axiomatic systems does not mean that the uninterpreted, purely formal point of 

view simply wins the day—quite the contrary! We are left, instead, with a much more carefully 

specified problem lying at the heart of both approaches to mathematics.  

 

What Frege‘s Caesar problem shows us, then, is a problem which remains open in the 

philosophy of mathematics. Namely, the problem of specifying exactly what mathematics is 

about. Frege‘s approach to this problem was to suggest that, ultimately, mathematics (or, at least 

a crucial and potentially foundational component of mathematics) was reducible to logic. Since 



155 

 

his view of logic was an ontological one, mathematics, too, had a clear ontological basis: it was 

about some of the more general features of the world. But, as we have just seen, this view is not 

tenable; the Caesar problem shows us that Frege, too, is incapable of pointing out exactly which 

entities mathematics deals with, and it remains unclear what exactly mathematics is about. 

 

Hilbert, by contrast, held the view that (at least in its more mature forms) mathematics is 

ultimately concerned with certain formal structures and their inter-relations. Through our 

examination of the Frege-Hilbert correspondence, however, we have seen that this view is 

plagued with difficulties quite similar to those faced by Frege in his own work. When trying to 

become clear about exactly what a formal structure is, Frege noted several dangerous 

equivocations that he attempted to solve by situating such structures within his 

logical/ontological hierarchy. But, as his inability to solve the Caesar problem shows, this route 

is simply incapable of shoring up Hilbert‘s account of mathematics. Hilbert‘s work throughout 

the 1920s was similarly focussed on the attempt to provide a clear referential basis for higher 

forms of mathematics, though his formalist/proof-theoretic programme, too, faced 

insurmountable difficulties. Just as Frege‘s best efforts led to what he himself saw as a dead-end, 

so too Hilbert‘s own approach came to seem mathematically and philosophically unsatisfying. 

 

In the following chapter, we will examine some of the ways in which more recent work in 

mathematics and the philosophy of mathematics has tended to repeat (rather than solve) this 

dichotomy.



156 

 

Chapter 4: Benacerraf, abstract structures, and structuralism 
 

§0. Introduction 

 

In previous chapters, we have examined two distinct views of mathematics offered by Hilbert 

and Frege in their correspondence and other works. Frege‘s view entails that mathematics deals 

with a specific content, and that the meaningfulness of mathematical statements relies in part on 

reference to an ontological hierarchy of functions and objects. Hilbert‘s view, by contrast, resists 

Frege‘s attempt to fix the references of mathematical terms and treats mathematics as the study 

of the properties of and relations between formal axiomatic structures which cannot (or should 

not) be referentially fixed. As we saw in these chapters, both Hilbert‘s relational view and 

Frege‘s ontologically-grounded view face profound difficulties related to meaning and reference. 

 

The primary difficulty facing Hilbert‘s view was the threat of meaninglessness and mathematics 

as a contentless science. As Frege pointed out in their correspondence, without references for 

their terms, statements within an axiomatic theory would appear to be meaningless and incapable 

of possessing an absolute truth value. While both Hilbert and his defender Korselt argued in 

favour of a view of axiomatic systems as uninterpreted formal structures, Frege countered by 

demanding that even statements within such formal structures require a determinate meaning if 

they are to become relevant for scientific inquiry. On Frege‘s view, the applicability and 

scientific pedigree of mathematics required that it be more than a complex game of symbols. 

Frege‘s suggestion was to embed the referentially unfixed statements of Hilbert‘s uninterpreted 

systems within his ontological hierarchy. But, as we saw, Hilbert resisted this move, viewing it 

as contrary to the very point of axiomatizing a theory in the first place (namely, providing a 

flexible structure applicable in multiple distinct situations).
1
 Thus Hilbert‘s view is faced with a 

dilemma. Axiomatic theories are either purely formal systems and, hence, meaningless prior to 

being given an interpretation, or they possess some specific content and require at least some of 

their terms to have fixed references, thus eliminating their flexibility. Frege saw no way out of 

the first horn of the dilemma, and instead dedicated his attention to the second. 

 

It is worth noting here that, despite his initial hesitance to accept Frege‘s demand for fixed 

reference, Hilbert‘s later work attempted to achieve something quite similar. Throughout the 

1920s and early 1930s Hilbert and his collaborator Paul Bernays sought to embed higher 

mathematics (particularly any mathematics which required the use of infinites) in a ‗contentual‘ 

(inhaltlich) system of stroke-symbols which, it was presumed, were epistemologically and 

semantically accessible.
2
 The guiding idea of Hilbert‘s Program (as it has come to be called) was 

that, even if the statements of higher mathematics are meaningless or unverifiable, given a 

properly constructed syntactic theory of proof we can ensure that the allowable moves in the 

formal system always lead us from contentual truths, through a formal game, and back again to 

contentual truths. By grounding mathematics within his system of stroke symbols, Hilbert sought 

to ensure that any detour through the infinite would always loop back to some verifiable, 

                                                 
1
 Cf. [Frege 1980, 40-41]. 

2
 The most important papers by Hilbert and Paul Bernays which develop proof theory and the formalist programme 

are collected (in English translation) in [Mancosu 1998], with insightful commentary by Mancosu. See [Hilbert 

1919-1920, chapter 9] as well. 



157 

 

meaningful claim in the realm of the finite. Just as Frege sought to give meaning to Hilbert‘s 

purely formal conception of axiomatics by embedding his supposedly uninterpreted statements 

within an ontological hierarchy, so Hilbert‘s later work in proof theory attempted to ground all 

higher mathematics on a fixed foundation of straightforward reference.
3
  

 

Unfortunately, this interesting programme suffered a fate rather similar to that of Frege‘s own 

work on the foundations of mathematics. Whereas the inability to solve the Caesar problem had 

destroyed Frege‘s hope of embedding mathematics within his ontological hierarchy, Hilbert‘s 

program was undermined in fundamental ways when Kurt Gödel—who, somewhat ironically, 

had been a great contributor to proof theory—announced his incompleteness results in the 

autumn of 1930. What Gödel had shown (with some help from John von Neumann)
4
 was that a 

consistency proof of the system of finitary reasoning (i.e., the system of ‗contentual‘ 

mathematics) was impossible without already appealing to transfinite, ‗non-contentual‘ modes of 

reasoning.
5
 This effectively reduced Hilbert‘s Program to another dead end in the search for 

mathematical foundations.
6
 

 

But, just as Hilbert‘s later program faces serious difficulties, so too Frege‘s view (which 

grounded his suggested emendations of Hilbert‘s axiomatization of geometry) is subject to deep 

difficulties. Frege encountered seemingly intractable problems in his effort to fix the references 

of the terms employed in his logicist reduction of basic arithmetic. As we saw in the previous 

chapter, his attempts to solve various forms of the Julius Caesar problem eventually required the 

use of his Basic Law V, leading directly to the inconsistency of his logical system. As a result, 

Frege ultimately abandoned his life‘s work, and published very little in his remaining years. In 

post-paradox work, there were many attempts to avoid the paradoxes implied by the ‗naïve‘ 

understanding of set theory and logic. This is one of the chief aims of Russell and Whitehead‘s 

Principia Mathematica
7
 and an important part of Hilbert‘s program, too. But as we‘ve just seen, 

these projects, too, ran against the insoluble difficulty of Gödel‘s incompleteness results, which 

of course relate directly to Principia. 

 

                                                 
3
 Note that my gloss of Hilbert‘s Program here ignores the important epistemological aspect of the project. In 

addition to seeking consistency and completeness proofs, Hilbert also claimed that his system of stroke symbols was 

―surveyable‖ [cf. Hilbert 1922], so that the age-old question of epistemological access to the abstract entities of 

mathematics disappeared. [Kitcher 1976] presents a fair account of the epistemological side of the project, as well as 

an account of some of its failings.  
4
 See [Gödel 1931] for the initial publication of these results. Von Neumann‘s help first appears in letters written to 

Gödel between November of 1930 and January of 1931. These are  available in [Von Neumann 2005, 123-126]. 
5
 Gerhard Gentzen‘s celebrated 1936 proof of the consistency of first-order Peano arithmetic, for instance, employs 

transfinite induction.  
6
 Hilbert‘s collaborator Bernays almost immediately recognized the destructive significance of the results. For a 

thorough examination of Gödel‘s effects on Hilbert and Bernays‘ programme, see [Tait 2006, §4 and §8]; for a 

broader look at Bernays and Gödel‘s correspondence regarding finitary mathematics and the formalist programme, 

see [Feferman 2008]. Despite the widespread acceptance of the failure of Hilbert and Bernays‘ particular program, 

there have been a number of interesting attempts to develop the program in a more satisfactory form (e.g., [Feferman 

1992] and [Simpson 1988]). Of course the success of the broader idea of a formalized proof theory within logic, 

mathematics, and computer science can hardly be overstated. 
7
 ―A very large part of the labour involved in writing the present work has been expended on the contradictions and 

paradoxes which have infected logic and the theory of aggregates‖ [Russell and Whitehead 1963, vii]. 



158 

 

We are left, then, at something of an impasse. It turns out that both the relational view of Hilbert 

and the ontologically grounded view of Frege face tremendous difficulties, particularly in 

relation to meaning and reference-fixing. We‘ve traced the emergence of a structure-focussed 

view of mathematics and examined two of the most influential ways of explaining what 

mathematical structures might be, but both views face great problems. Without significant 

alteration, neither Frege‘s fixed ontological view nor Hilbert‘s uninterpreted view are capable of 

satisfactorily explaining what much of modern mathematics is about.  

 

Throughout the 1920s and 1930s there was a pervasive feeling that the foundations of 

mathematics had been seriously undermined. This deepening sense of crisis was brought about 

by a variety of factors: the emergence of non-Euclidean geometries, the prominence of the ill-

understood technique of axiomatization, the set-theoretic paradoxes, and, finally, Gödel‘s 

incompleteness results of the 1930s.
8
 But, even despite the widespread belief in a foundational 

crisis, mathematics continued to develop at a rapid pace. Though problems with set theory 

continued to be raised by both philosophers and mathematicians, it was, for the most part, 

employed as an acceptable, if philosophically troublesome, foundation for mathematics.
9
 Set 

theory had been axiomatized by Ernst Zermelo between 1905 and 1908,
10

 and subsequently 

altered by Abraham Fraenkel and Thoralf Skolem in 1922.
11

 The resulting system, Zermelo-

Fraenkel set theory (ZF) acted as the de facto basis for mathematics for three quarters of a 

century, and still figures prominently in foundational work and philosophical debates today.
12

 

Between the 1930s and the 1960s general foundational questions about mathematics were, for 

the most part, formulated in set-theoretic terms. Most mathematicians and philosophers 

interested in foundational questions during this period tended to concern themselves with the 

specific problems of set theory and its basic concepts. All of this set-theoretic work was 

undertaken despite the many unresolved conceptual problems with set theory (many of which 

were quite similar to those faced by Frege‘s logical system). Solutions to these problems moved 

away from the so-called naïve set theory of the late 19
th

 century and towards more explicit 

axiomatic treatments like ZF, or the less widely used Von Neumann-Bernays-Gödel 

axiomatization (NBG) developed between the mid-1920s and the 1950s, though even in these 

systems many conceptual problems remained. The general view among those concerned was that 

‗higher‘ mathematics was ultimately reducible to the basic concepts of set theory, and that 

                                                 
8
 While it is a historical commonplace to note that there was a foundational crisis in mathematics in the 1920s (see, 

e.g., [Weyl 1921] for one of the earlier and best-known expressions of this view), relatively fewer accounts focus on 

the continuity between the factors mentioned above. An interesting exception is [Ferreirós 2008]. As earlier chapters 

in this thesis attest, I hold that the conceptual crisis in mathematics was in full swing already by the time of the 

Frege-Hilbert correspondence, and well before the public debate surrounding the set-theoretic paradoxes. 
9
 This sort of attitude (―We‘ll figure out why it works later‖) is remarkably similar to the rapid expansion in the 

application of the calculus prior to its rigorization in the 1800s. Most mathematicians in both periods remained 

unconcerned with what we might call the ‗philosophical‘ underpinnings of their mathematical tools and were instead 

content to extend their successes as far as possible. In contrast to the calculus, however, it seems to me that we have 

yet to achieve a satisfactory philosophical foundation for modern axiomatics. 
10

 The chief reference here is [Zermelo 1908]; almost all of the key papers surrounding the early axiomatization of 

set theory are available in [Van Heijenoort 1967]. 
11

 For an illuminating history of the development of set theory, see [Ferreirós 1999]. 
12

 For example, Stewart Shapiro‘s theory of structures is self-consciously modelled on ZF (cf., [Shapiro 1997, 93-

97]), though he departs from the usual philosophical understanding of ZF as a foundation. 



159 

 

problems with this reduction were simply a matter of finding the correct axiomatization of set 

theory or the proper understanding of the relevant concepts.
13

   

 

Throughout the first half of the twentieth century (and perhaps a few decades beyond), the 

philosophy of mathematics was dominated by an intense interest in set-theoretic reductionism as 

a means of putting mathematics on a solid foundation. When people (or, at least, philosophers) 

asked ―what is mathematics really about?‖ the answer was, invariably, ―Sets!‖.
14

 

 

0.1 Set-theoretic reductionism and set-theoretic foundationalism 

 

But what does this mean? What exactly is set-theoretic reductionism and how does it relate to the 

desire to properly found mathematics? We‘ll need to answer a few sets of general questions 

about the nature of reduction and foundations in mathematics to understand the role of set theory 

in the post-Fregean world. 

 

First question: what does it mean to reduce one part of mathematics to another? To answer this 

question, let‘s look at the most often discussed reduction: the reduction of the natural numbers 

and their arithmetic to set theory. How is this reduction accomplished? The basic idea is that for 

every arithmetical object, property, or relation of the natural number system, there is a 

corresponding object, property, or relation in the universe of whichever set theory we are 

working with. Thus, the entire sequence of numbers itself has to be replaced by or reduced to 

some collection of sets which bears properties and relations that (somehow) mirror those of the 

natural numbers. In essence, a set-theoretic reduction of the natural numbers and their basic 

arithmetic involves modelling the natural number structure within some particular universe of 

sets. This of course poses a number of questions we‘ve been examining throughout this thesis 

(i.e., what is the natural number structure, how do we figure out the ‗somehow‘ in the previous 

sentence, what are models, etc.). We‘ll return to these concerns below. 

 

                                                 
13

 Of course there were exceptions to this general tendency. [Putnam 1967] attempts to eschew the search for 

foundations in mathematics entirely. An earlier and more focussed rejection of specifically set-theoretic foundations 

can be found in [Skolem 1922], where Skolem first introduces his eponymous ‗paradox‘.  Not really a paradox at all, 

Skolem‘s theorem illustrates the startling relativity of set-theoretic notions. For him, this relativity introduced a 

profound and confusing complexity into set theory which was directly at odds with most set theorists‘ dreams of 

reducing all of mathematics to the notions of set theory. Of foundations, Skolem writes that ―Our only concern, then, 

should be that the initial foundations be something immediately clear, natural, and not open to question. This 

condition is satisfied by the notion of integer and by inductive inferences, but it is decidedly not satisfied by set-

theoretic axioms of the type of Zermelo‘s or anything else of that kind; if we were to accept the reduction of the 

former notions to the latter, the set-theoretic notion would have to be simpler than mathematical induction, and 

reasoning with them less open to question, but this runs entirely counter to the actual state of affairs‖ [Skolem, in 

Van Heijenoort 1967, 299, my emphasis]. Unsurprisingly, Zermelo was not thrilled with Skolem‘s work, initially 

believing it to contain a mistake [cf., Van Dalen and Ebbinghaus 200, 145-146]. 
14

 Wilson makes a similar claim which includes at least the 20
th

 century: ―set theory has become the canonical 

backdrop to which questions of structural existence are referred. So when we go beyond napkins and set up a 

structure […] formally, we need both: (a) to mark the ‗intrinsic‘ elements proper to the internal workings of the 

structure, without extraneous fat; (b) to provide sufficient extrinsic links to set theory so that the existence of our 

structure can be established‖ [Wilson 1993, 209]. Shapiro seems to extend the dominance of set theory even to the 

21
st
 century: ―Within the community of professional mathematicians, if not philosophers, a set-theoretic proof of 

satisfiability resolves any legitimate questions of existence‖ [Shapiro 2005, 72].  



160 

 

Second question: what is the point of reducing one part of mathematics to another? There are at 

least three related answers to this question which are relevant here. We might call them the 

epistemological, the pragmatic/heuristic, and the ontological answers, respectively. I‘ll isolate 

them here for clarity, but it should be noted that they are really rather difficult to separate (as 

Frege‘s work very clearly shows). 

 

The epistemological answer to the question suggests that reducing one area of mathematics to 

another can help to clarify, simplify, or solve epistemological problems. Frege‘s work is 

probably the best example of reductionism undertaken for the sake of epistemology; we might 

also think of parts of Hilbert‘s Program in this way.  One of the central aims of Frege‘s work 

prior to the emergence of Russell‘s paradox was precisely to prove an epistemological point via 

the reduction of arithmetic to logic. By showing that arithmetic can be understood in purely 

logical terms, Frege hoped to show that arithmetic is not a synthetic a priori science, as Kant 

famously argued, but that it is instead a purely analytic discipline. To use the Kantian 

vocabulary, his reduction of arithmetic to logic would have shown that arithmetic had no 

intrinsic connection to the pure intuition of time. So, at least part of his goal was to overturn an 

error he saw in Kant‘s influential understanding of the epistemology of mathematics, with all the 

implications this might have for the philosophy of mathematics. 

 

We might also reduce one part of mathematics to another for pragmatic or heuristic reasons. 

There are numerous instances in the history of mathematics where a concept or technique from 

one branch or sub-discipline has proven essential to the advancement of seemingly unrelated 

work in another branch or sub-discipline of mathematics.
15

 Such cross-pollination is crucial for 

mathematics. But, at least since the early 1800s, mathematics has become a field of increasingly 

isolated specialization, such that someone working in one branch of mathematics might be 

unable even to understand (let alone use) results in another – despite the fact that these results 

may be of dramatic importance for their particular problem. This poses a pragmatic problem: 

how can mathematicians from disparate fields speak to one another? One solution to this 

problem is to recast these disparate fields in a common language which could provide the 

common ground required for fertile cross-disciplinary collaboration. As the category-theorist and 

philosopher F. William Lawvere puts it, ―Unification and simplification are necessary not only 

for the dissemination of results, but also for the coherent advance of research in the diverse 

branches of mathematics.‖
16

 Thus, if mathematicians can cash out their higher-level 

mathematical discussions in terms of a single theory or way of speaking, then the recognition of 

similarities between seemingly disparate parts of mathematics will be made considerably easier. 

By reducing all of mathematics to a single theory (e.g., set theory or category theory), the 

pragmatic reductionist gives all mathematicians a basic vocabulary to communicate with one 

another. A related benefit of this form of reductionism is that, when a surprising connection 

between fields is in fact discovered, it ought to be explicable in the uniform and universal 

language developed by the reduction (e.g., one might explain the powerful connections between 

                                                 
15

 The application of techniques for solving equations to geometric problems via Descartes‘ coordinate system is a 

particularly well-known example of this phenomenon; another fruitful example is the deep connection between 

permutation groups and the roots of polynomial equations which forms the basis of Galois theory. 
16

 [Lawvere 2000, 715]. 



161 

 

algebra and geometry of analytic geometry by showing how both equations and spaces can be 

understood in set-theoretic terms). So pragmatic/heuristic reductionism can help to explain 

connections, and, the better explained such connections are, the better exploited, and the more 

productive the cross-pollination becomes. 

 

Reductionism can also be ontological in character. This sort of reduction aims to show that our 

discussions of higher-level mathematical entities are really and ultimately about something 

simpler (things like sets or categories or numbers). So when we do mathematics what we are 

really doing is engaging with and speaking about this most fundamental ontological layer. 

Ontological reductions in this sense aim to show us that everything other than set theory or 

category theory or (in Frege‘s case) logic is merely a façon de parler, a historically entrenched, 

partially misleading linguistic fiction. Only the final level of the reduction touches reality. One 

clear benefit of this sort of view is ontological parsimony: in place of the bewildering array of 

abstract objects and structures developed by mathematics (Abelian categories, fuzzy sets, 

Hausdorff spaces, Fermat primes, Grothendieck universes, Blotto games, etc., etc.) we have one 

or two sorts of things (e.g., sets, objects and arrows, objects and functions, numbers, etc.). This 

parsimony itself has the added benefit of simplifying semantic and epistemological concerns: 

instead of working things out on a case by case basis, we can solve these sorts of problems in a 

uniform way. Perhaps the chief philosophical benefit, though, is that this kind of ontological 

reduction promises to tell us what mathematics is really about.  

 

Third set of questions: what do we mean when we speak of the foundations of mathematics? 

What are such foundations meant to provide? How do the various forms of reductionism act as 

foundations? In some ways we have given answers to these questions already. 

 

Epistemological reductions are meant to provide (unsurprisingly) epistemological foundations 

for mathematics. Let‘s again take Frege as an example. Part of Frege‘s worry was that 

mathematicians actually have no idea what they are talking about when they engage in proofs. 

Indeed, much of the first half of Frege‘s Grundlagen der Arithmetik is taken up with showing 

that extant views of the concept of number are nonsensical. As a result, proofs based upon these 

views don‘t really prove anything because they are meaningless concatenations of symbols. If we 

can show that important parts of mathematics are reducible to logic, and that logic is 

epistemologically accessible, then we can show that it is possible to make mathematics more 

epistemologically secure. If we know what numbers are, and follow rigorous and gapless proof 

procedures, then we can ensure that mathematicians aren‘t engaging in flights of fancy. We can 

really prove things now. This is how Frege‘s reduction was meant provide an epistemological 

foundation for mathematics. 

 

Pragmatic reductions can provide pragmatic foundation for mathematics. By showing that we 

can, if we so choose, cash out our higher level talk in terms of set- or category-theoretic talk, 

when we collaborate with others we have a firm basis for thinking that this collaboration is 

adding to the stock of mathematical knowledge in general, rather than just the stock of our 

particular nook or cranny of mathematics. This pragmatic reduction guarantees that mathematics 

is not, in fact, splintering apart into a thousand dissonant voices like an abstract Tower of Babel. 

Many of the proponents of category theory as a foundation for mathematics have emphasized its 



162 

 

pragmatic use as a kind of lingua franca for the mathematics.
17

 A pragmatic foundation of this 

sort is meant to give a foundation for the working unity of the science. This won‘t necessarily 

guarantee that the tower is stable, nor that it touches reality, but it can help guarantee that we are 

engaged, in some sense, in the same global project. 

 

Finally, ontological reductions can give us ontological foundations. When we show that a new 

mathematical concept or object can be reduced to the single ongoing ontology of our particular 

reduction, we know that our utterances about these novelties actually refer to something real and 

that they are therefore meaningful and capable of truth and falsity. Ontological reduction aims to 

show that the abstract claims of mathematics have a foundation in reality, in much the same way 

that the claims of physics and zoology have. By showing it is possible to reduce higher-level 

mathematical entities to an acceptable ontology, we provide a means for connecting mathematics 

to the world that we inhabit. Such reductions help to answer the question ―What is mathematics 

about?‖ by answering ―Reality‖.
18

 

 

In this simplified view of the nature of mathematical foundations, we already have three distinct 

approaches. Part of the problem in the philosophy of mathematics over at least the last century 

and a half  has been that these approaches often intermingle to form a confusing tangle, so that 

ontological questions are posed in pragmatic terms, epistemological questions have ontological 

implications, and so on. In this dissertation thus far, I have focussed on the question: what is 

mathematics really about? I have meant this question to be taken ontologically, i.e., I have meant 

to ask which features of reality (if any) does mathematics concern itself with?  If we think of the 

debate between Hilbert and Frege as chiefly an ontological one—and I suggest that we should—

then we have already seen a few answers to this question. Frege‘s answer is that the ontology of 

mathematics (or at least that part of mathematics derivable from basic arithmetic) is just the 

ontological hierarchy of objects and functions. Hilbert‘s suggestion is that mathematics, taken on 

its own, has no ontology—indeed, that this lack of ontology is its chief feature. We‘ve seen, too, 

that both of these answers are deeply problematic. So, again, we‘ve returned to our impasse. 

 

§1. Benacerraf on the natural numbers 

 

In the next few sections I would like to see whether the more recent history of the philosophy of 

mathematics can offer us plausible new ways of answering the ontological question. I‘ve chosen 

as a starting point an influential paper by the American philosopher of mathematics Paul 

Benacerraf. The 1965 paper, ―What Numbers Could Not Be‖ has been cited over eight hundred 

                                                 
17

 Mac Lane‘s well-known textbook ―Categories for the Working Mathematician‖ is perhaps the best known 

application of this view. There Mac Lane ―aims to present those ideas and methods that can now be effectively used 

by mathematicians working in a variety of other fields of mathematical research. […] [C]ategories provide a 

convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, 

and functor category‖ [Mac Lane 1998, vii]. In an appendix to the text, Mac Lane also attempts to show that 

categories ―can be used as a possible foundation for all of mathematics, thus replacing the use in such a foundation 

of the usual Zermelo-Fraenkel axioms for set theory‖ [Mac Lane 1998, 290]. The view of category theory as a 

conceptual language or organizational tool for large parts of mathematics is also developed in a more philosophical 

vocabulary in [Landry 1999] and [Mac Lane 1992]. 
18

 Of course such reductions often entail a fairly expansive notion of ‗reality‘ – thus, for instance, an ontological 

account of the traditional set-theoretic reduction will include as part of ‗reality‘ the entire set-theoretic universe V. 



163 

 

times (something of a feat in the rather cloistered world of the philosophy of mathematics) and it 

seems to have heralded a new way of understanding the problem of foundations in 

mathematics.
19

 

 

At the height of the popularity of set-theoretic reductionism, Benacerraf raised a significant 

conceptual problem with the usual set-theoretic reduction of the natural numbers. He noted what 

was already common knowledge: that, even within Zermelo-Fraenkel set theory, there can exist 

multiple different, equally valid ways in which to define the natural numbers as sets. Benacerraf 

proceeded to argue that the existence of multiple different reductions poses a problem for the 

naïve view of set-theoretic foundationalism, namely the view that there is some unique correlate 

in the set-theoretic hierarchy corresponding to each higher level mathematical domain, in this 

case the natural numbers. If natural numbers are sets, Benacerraf asked, well, just which sets are 

they?  

 

In many ways his is a very charitable approach to the problem. At the outset Benacerraf 

concedes almost everything to reductionism. Sets are real? Fine. Numbers are sets? Fine. All that 

he wishes to know is which particular sets they are. If there is no principled reason for picking 

out particular sets as the real correlates for particular numbers, then there seems to be no reason 

to accept set theory as the final ontological foundation for mathematics in the first place. This, in 

broad strokes, is what has come to be called the non-uniqueness problem. Its effect is to call into 

question the post-Fregean, post-paradox attempts by set theorists and philosophers to explain just 

what it is that mathematics is about. In the following section, I will investigate Benacerraf‘s 

account of this problem in more detail, and also look at his admittedly tentative ‗structuralist‘ 

suggestions toward its solution. The point of looking so closely at Benacerraf‘s paper is to see 

whether his work offers us any clear alternative to the impasse we encountered in the Frege-

Hilbert debate. 

 

1.1 Benacerraf‘s non-uniqueness problem 

 

The focus of Benacerraf‘s paper is a single, relatively straightforward question: if numbers are 

reducible to sets, then which sets are they?  

 

In one sense, this question is as easily answered as it is asked, since in any reduction of the 

natural numbers to the universe of set theory, it is a simple matter to make explicit which 

numbers are identified with which sets. This is indeed the chief point of such reductions. The real 

problem for Benacerraf begins when multiple adequate though extensionally distinct set-

theoretic reductions are available. Benacerraf examines two fairly common approaches, one 

attributed to Ernst Zermelo, the other to John von Neumann. 

 

On the Zermelo reduction, we have the following identifications: 

 

Ø  = 0 

{Ø}   = 1 

                                                 
19

 One of the chief contemporary proponents of structuralism, Stewart Shapiro, suggests that Benacerraf‘s paper 

―provides the standard motivation for structuralism‖ [Shapiro 2005, 61]. 



164 

 

{{Ø}}  = 2 

{{{Ø}}} = 3 

 

Whereas on the von Neumann account, we have: 

 

Ø    = 0 

{Ø}    = 1 

{Ø, {Ø}}   = 2 

{Ø, {Ø}, {Ø, {Ø}}}  = 3 

 

We needn‘t get into great detail about the particulars in each reduction, but it is worth noting two 

general facts here. First, each reduction is entirely adequate in the sense that all the (set-theoretic 

correlates of the) relations and properties of the natural numbers can be deduced in either 

reduction. And, second, many of the ‗same‘ elements in each reduction possess quite different 

properties. For instance, on the von Neumann account, the number 3 has three members, one of 

which is the set {Ø, {Ø}}. On the Zermelo account, the number 3 has a single member, {{Ø}}, 

which is not a member of the von Neumann reduction‘s 3. In short, then, we possess supposedly 

equivalent arithmetic reductions which are both extensionally and intensionally different in many 

respects. Beyond these two reductions, which Benacerraf employs mainly for illustrative 

purposes, there are of course an infinite number of other equally valid set-theoretic reductions of 

the natural numbers. Moreover, we needn‘t restrict ourselves to set theory; we might employ 

category theory, lattice theory,
20

 or even a physical structure of spacetime points to explain just 

what the natural numbers are.
21

 

 

The question, then, is this: if there are multiple equally valid but radically different reductions of 

the natural numbers, which of these is the correct one? Which sets are the natural numbers 

really? Benacerraf writes  

 

We are left in a quandary. We have two (infinitely many, really) accounts of the meaning 

of certain words (―number,‖ ―one,‖ ―seventeen,‖ and so forth) each of which satisfies what 

appear to be necessary and sufficient conditions for a correct account. Although there are 

differences between the two accounts, it appears that both are correct in virtue of satisfying 

common conditions. If so, then the differences are incidental and do not affect correctness. 

Furthermore, in Fregean terminology, each account fixes the sense of the words whose 

analysis it provides. Each account must also, therefore, fix the reference of these 

                                                 
20

 There are several ways to define the natural numbers in terms of many general algebraic structures, not just set 

theory. Since category theory, for example, can model set theory (cf. [Osius 1974]), one could simply employ the 

category-theoretic correlate of, say, the von Neumann reduction in a definition of the natural numbers. Alternatively, 

one can define the natural numbers directly in terms of a category containing a terminal object, a global element, and 

arrows with the requisite properties. Similarly, one can define the natural numbers (including zero) in lattice -

theoretic terms as a semi-lattice  under the operations ‗min‘ and ‗max‘, with 0 as the ‗bottom‘. 
21

 Cf. [Field 1980]. Field likely would not treat his use of this spacetime structure as an attempt to explain what the 

natural numbers are (given that the title of the book is Science Without Numbers), though this structure is intended to 

fulfill the duties the (real) numbers generally perform within the physical sciences. Someone with less interest in 

nominalizing science could certainly employ Field‘s structure in ways very similar to the set-theoretic reductions we 

refer to here. 



165 

 

expressions. Yet, as we have seen, one way in which these accounts differ is in the 

referents assigned to each of the terms under analysis.
22

 

 

At this stage Benacerraf points out that, on the assumption that numbers just are sets (or that they 

are ultimately reducible to sets), we are left with only two possible options. Either both (or all) of 

these accounts are correct or it is not the case that both (or all) are correct. At the outset he 

rejects the idea that multiple referentially-fixed accounts of the natural numbers can be correct in 

the required sense. This is because the point of the ontological reduction is to pick out the exact 

objects which the numbers are, not a group of objects which share certain similarities—unless of 

course that group is a specific set which uniquely fulfills the relevant criteria, but Benacerraf‘s 

point is that this is not in fact possible in the first place. If the numbers are sets, then they must 

be particular sets. Benacerraf‘s position here is straightforwardly ontological in character. Since 

no two distinct entities can also be identical—at least by a plausible and fairly common 

metaphysical assumption
23

—it simply cannot be the case that the number 3 is both the set 

{{{Ø}}} and the set {Ø, {Ø}, {Ø, {Ø}}},  

 

[f]or if the number 3 is in fact some particular set b, it cannot be that two correct accounts 

of the meaning of ―3‖—and therefore also its reference—assign two different sets to 3. For 

if it is true that for some set b, 3 = b, then it cannot be true that for some set c, different 

from b, 3 = c.
24

 

 

So, bypassing the first option of rejecting the indiscernibility of identicals, Benacerraf examines 

the second. If numbers just are sets, and it is not the case that there are multiple, equally correct 

accounts of which sets exactly they are, then there must be a single correct account. If this is the 

case, Benacerraf suggests that there ought to be some reason for preferring this account over all 

the others, which all suffice equally well for the purposes of arithmetic. He writes 

 

We are now faced with a crucial problem: if there exists such a ―correct‖ account, do there 

also exist arguments which will show it to be the correct one? Or does there exist a 

particular set of sets b, which is really the numbers, but such that there exists no argument 

one can give to establish that it, and not, say, [the von Neumann reduction], is really the 

numbers? It seems altogether too obvious that this latter possibility borders on the absurd. 

If the numbers constitute one particular set of sets, and not another, then there must be 

arguments to indicate which.
25

 

 

                                                 
22

 [Benacerraf 1965, 55-56]. 
23

 Though, in the philosophy of mathematics, not a universally held assumption. Cf., e.g., [MacBride 2006], 

[Ketland 2006], [Leitgeb and Ladyman 2008], and [Shapiro 2008].  
24

 [Benacerraf 1965, 56]. 
25

 [Benacerraf 1965, 57-58]. It seems to me that a more reasonable claim (or at least one which is more agnostic 

about the relation between human capacity and the world of mathematics)  would suggest that if we are to believe 

that the numbers constitute one particular set of sets, and not another, then there must be arguments or at least 

evidence to indicate which. In a later appraisal of the argument of the 1965 paper, Benacerraf himself notes ―no red -

blooded realist […] should accept the bald statement that if there isn‘t some a priori proof that some particular 

candidate reduction is the correct one, there can‘t be a ‗correct‘ one‖ [Benacerraf 1996b, 25]. 



166 

 

As this remark shows, Benacerraf‘s dilemma is remarkably similar to Frege‘s Caesar problem. 

Frege, in his attempt to embed the natural numbers within his logical/ontological hierarchy, was 

compelled to provide completely determinate identity statements for each number, such that it 

was clear for every entity in his hierarchy whether or not it was identical with the number in 

question. His problem was to ensure that he picked out exactly which entities he wished to talk 

about, not simply a cluster of entities having some selection of shared properties sufficient to 

satisfy a collection of axioms. He wanted to show, for instance, which exact entity the number 2 

was, not give us a list of different possible entities which sufficed to play the role of the number 

2 up to isomorphism. Given his logical scruples, this desire for strictly fixed reference required 

that his definitions of the numbers exhaustively determined not only which entities they were, 

but also which entities they were not. 

 

Benacerraf‘s aim here is quite similar. He wants to be able to show that, among the many 

arithmetically equivalent reductions of the natural numbers, one and only one is the correct 

reduction—the one which actually tells us which sets the natural numbers are (or, if we prefer, 

what we really refer to when we use number words). Benacerraf, again following Frege, is not 

interested in privileging a particular reduction for purely aesthetic or idiosyncratic reasons, or out 

of convenience; he demands principled argumentation illustrating the superiority of one 

reduction over all others. Since the set-theoretic differences between, e.g., the von Neumann and 

Zermelo reductions are irrelevant from a purely arithmetical point of view they cannot be used to 

militate in favour of either reduction as the superior account. How then can we distinguish 

between the two in anything other than an ad hoc or idiosyncratic fashion? As we saw in the last 

chapter, Frege‘s attempts to solve his version of this problem led him into difficulties which he 

eventually viewed to be insurmountable. Though Benacerraf is not bound to the particularities of 

Frege‘s ontological view of logic, nor to Frege‘s logicist program, his problem here is 

nevertheless almost identical in structure to Frege‘s. So what is Benacerraf‘s strategy for 

avoiding the dead end of Frege‘s approach without falling into the meaninglessness of a purely 

formal approach like Hilbert‘s? How can we determine precisely which sets the natural numbers 

are?  

 

1.2 Restricted identity 

 

Benacerraf outlines a few ways we might deal with the non-uniqueness problem. He begins, 

again in a familiar Fregean mode, to investigate exactly which question it is that we are asking 

here. An important part of what we really want to know is the truth value of certain identity 

statements of the form ‗n = …‘ where n is some number. There are, Benacerraf suggests, at least 

three different types of identity statement we might wish to investigate here. First are the more 

commonplace identities between obviously arithmetical entities, i.e., statements like ‗2 + 2 = 4,‘ 

where each component of the statement is presented in recognizably arithmetical form. Second, 

we have more circuitous but ultimately similar claims, like ‗2 = the number of shoes currently on 

my feet,‘ where we have obvious reference to a numerical entity without the use of a numeral or 

strictly arithmetic term. Neither of these statement-types is particular troubling; in a given 

reduction, the (relative) truth value of these statements will be uncontroversial. The third sort of 

identity statement is rather more problematic. These statements, e.g., ‗2 = Julius Caesar‘, or ‗5 + 

8 = {Ø, {Ø, {Ø}}}‘ are familiar from Frege‘s Caesar problem. Indeed, Benacerraf explicitly 

refers to Frege‘s problem in his discussion of these identities: 



167 

 

 

To speak from Frege's standpoint, there is a world of objects—that is, the designata or 

referents of names, descriptions, and so forth—in which the identity relation had free reign. 

It made sense for Frege to ask of any two names (or descriptions) whether they named the 

same object or different ones. Hence the complaint at one point in his argument that, thus 

far, one could not tell from his definitions whether Julius Caesar was a number.
26

 

 

Benacerraf‘s first move in responding to the non-uniqueness problem is to distance himself 

precisely from Frege‘s requirement that every identity statement is meaningful (and, relatedly, 

that each such statement has a determinate truth value). Benacerraf suggests that identity 

statements of the third type are ‗unsemantical‘ for the reason that there is no context in which 

there are determinate truth-conditions for such statements.
27

 He writes that 

 

If an expression of the form ―x = y‖ is to have a sense, it can be only in contexts where it is 

clear that both x and y are of some kind or category C, and that it is the conditions which 

individuate things as the same C which are operative and determine its truth value.
28

 

  

And, further, that 

 

What determines that something is a particular lamppost could not individuate it as a 

particular number. I am arguing that questions of the identity of a particular ―entity‖ do 

not make sense. ―Entity‖ is too broad. For such questions to make sense, there must be a 

well-entrenched predicate C, in terms of which one then asks about the identity of a 

particular C, and the conditions associated with identifying C‘s as the same C will be the 

deciding ones.
29

 

 

Here again we see Benacerraf attempting to break free of Frege‘s constraint that every identity 

statement must be meaningful. He does so by suggesting that, in order for there to be an identity 

relation between two entities, there must be some context in which it is made clear that there is a 

semantically acceptable category under which both entities fall. In the case where there is no 

such category available, we cannot make sense of the identity relation, and identity statements 

can‘t be given a truth value. Presumably, a statement like ‗2 = Julius Caesar‘ will be neither true 

nor false on this view, but, instead, ‗unsemantical‘. This demand for a uniform semantic category 

within which to subsume the terms of a meaningful identity statement is broadly similar to 

Frege‘s insistence that functions have specific types of argument places which cannot be 

satisfied by the wrong sort of entity. Thus, for instance, we saw Frege engaged in a great deal of 

fancy footwork to avoid the implication (of the concept horse problem) that his view allowed 

meaningful identity statements between objects and concepts. Frege, for his part, attempted to 

explicate as best he could the ontological reasons for these formal conditions. Benacerraf, 

likewise, is presented with the difficulty of specifying (in a uniform way) exactly which 

                                                 
26

 [Benacerraf 1965, 64]. 
27

 Note that by ‗unsemantical‘ Benacerraf does not mean strictly nonsensical, for, as he notes, ―we grasp enough of 

[the] sense [of unsemantical claims] to explain why they are senseless‖ [Benacerraf 1965, 64]. He doesn‘t further 

define this notion, unfortunately. We will return to this difficulty below. 
28

 [Benacerraf 1965, 64-65]. 
29

 [Benacerraf 1965, 65]. 



168 

 

categories are semantically acceptable and which are ‗unsemantical‘. Unlike Frege, Benacerraf 

does not give us any detailed story about how this might be accomplished, though his use of the 

phrase ‗well-entrenched predicate‘ is suggestive that his account would differ significantly from 

Frege‘s. 

 

To avoid this problem entirely, one might simply define a disjunctive category or predicate (e.g., 

‗x is an F just in case x is either a Roman general or a number‘) which includes whichever 

entities we wish to make identity claims about. Benacerraf, however, wishes to reject this 

possibility as well, despite his admitted inability to give a precise account of what he means by 

‗uniform conditions‘ sufficient to prevent such ad hoc counterexamples.
30

  

 

1.2.1 Problems with Benacerraf‘s restriction of identity 

 

The restriction of statements of identity to properly semantical contexts promises to solve the 

non-uniqueness problem, at least in part, by allowing us to ignore thornier questions about 

mathematical objects like Frege‘s Caesar problem. What the Caesar problem and others like it 

show is that we cannot straightforwardly pick out mathematical entities like numbers precisely 

enough to pin them down in an absolute ontological sense. Benacerraf‘s suggestion that we 

restrict the relation of identity to semantical contexts allows us to say, ‗There really isn‘t an 

answer to the question ―Is Julius Caesar a number?‖ because we are dealing here with an ill-

defined, unsemantical context within which the (or an) identity relation can gain no purchase‘. 

The problem, then, simply dissolves and we are free to discuss the truth or falsity of identity 

claims within various well-defined contexts like the Zermelo and Von Neumann reductions. 

 

But there are clear difficulties in this line of thinking. Let‘s take a look at a few of these before 

we proceed to Benacerraf‘s next attempt to address the non-uniqueness problem. 

 

As we‘ve seen, Benacerraf‘s restriction of identity leans heavily on ‗uniform semantic 

conditions‘ and ‗unsemanticality‘. The notion of uniform semantic conditions for the 

meaningfulness of identity statements is meant to protect his account from the introduction of ad 

hoc solutions to the problem. Such solutions would simply dictate by fiat which sorts of identity 

claim are allowed and which are not. But, on this view, anything which was deemed to be 

unsemantical (e.g., ‗Julius Caesar = 2‘) could simply be made meaningful by the introduction of 

an appropriately defined context.
31

 To avoid all this, Benacerraf demands semantic conditions 

which would be applied across the board. But how are we to know which contexts are 

meaningful? 

 

Certainly we would not want to reject all disjunctive categories. One might think of a perfectly 

acceptable category like ‗resident of Canada or resident of the United States‘. What, then, would 

be the basis for rejecting some disjunctive categories over others? Something like ‗naturalness‘ 

might be such a basis: we find it natural to think of ‗residency in the United States‘ and 

                                                 
30

 [Benacerraf 1965, 65 note 12]. 
31

 A purely postulational view of mathematics, like that of Hausdorff, would presumably have no trouble with this 

sort of view. But such a view, of course, would have little trouble with any questions pertaining to the uniqueness of 

mathematical entities. Its difficulties lie elsewhere. 



169 

 

‗residency in Canada‘ as falling under the more general notion of ‗residency‘—after all, we 

already have a word for this category and could likely tell a convincing story about why and how 

this concept emerged in the first place. But perhaps a category under which Roman generals and 

natural numbers both fall is an unnatural one. This, of course, simply pushes the question back 

further by requiring some account of naturalness which, it seems to me, would be equally 

difficult to produce.  

 

Benacerraf‘s use of the phrase ‗well-entrenched predicate‘ might very well indicate that his 

definition of ‗uniform conditions‘ (and, relatedly, of ‗naturalness‘) would be largely historical in 

character: we find it natural to think of categories like ‗mammal‘, ‗residency‘, ‗cat‘, or 

‗temperature‘ but unnatural to think of ‗general or number‘, ‗hairpiece or firing range‘, etc.
32

 

This may be the case simply because we have found it prudent over the course of our history to 

think and speak in terms of various separate entities as mammals, residents, cats, or temperatures 

but have never been called upon or found it helpful to think or speak about hairpieces and firing 

ranges at one time. On this suggested view, the uniform conditions for determining which 

predicates and categories are acceptable or natural or well-entrenched would be given relative to 

a personal, historical, or cultural trajectory. Perhaps, then, the category which would allow us to 

decide whether ‗2 = Julius Caesar‘ is true or false is one which has not yet become (or will never 

become) relevant enough to become well-entrenched. Though at this point, after a century of 

arguing about Caesar‘s numerical status, it seems likely that we have dug a deep enough trench. 

Benacerraf, for his part, doesn‘t elaborate on this aspect of his view here, so we can only offer 

conjecture as to how we might capture just the categories he has in mind. Clearly, though, there 

will be significant difficulties in such a project. 

 

1.2.2 Empty generality 

 

As we‘ve seen, Frege‘s own solution to this problem (or his version of it) was to restrict 

meaningful statements to those whose terms referred to the elements of his ontological hierarchy 

and which obeyed the combinatoric logic of that hierarchy. This required him to employ very 

general notions like ‗object‘ and ‗function‘. Benacerraf finds the extremely general notions at the 

basis of Frege‘s ontological hierarchy so broad as to be useless—for him they simply do not 

form categories capable of providing the context for individuation between such radically 

disparate entities as, e.g., the number 27 and a particular radish. Such notions are not ad hoc in 

the way that custom-built predicates like ‗is a number or a general‘ are, and they seem to be 

rather ‗well-entrenched‘ on any charitable account of that phrase. So what, then, is the problem 

with these notions? Benacerraf is fairly brief in this objection, writing that 

 

To say that they are both ―entities‖ is to make no presuppositions at all—for everything 

purports to be at least that. ―Entity,‖ ―thing,‖ ―object‖ are words having a role in the 

language; they are place fillers whose function is analogous to that of pronouns (and, in 

more formalized contexts, to variables of quantification).
33

 

                                                 
32

 In some ways, the requirement of historical naturalness would mirror Hilbert‘s suggestion (cf. [Corry 2004a, 20]) 

that we ought only to axiomatize those parts of mathematics which have suitable historical pedigree. 
33

 [Benacerraf 1965, 65-66]. Interestingly, a key component of Benacerraf‘s own structuralist view of the natural 

numbers is that they are not objects.  



170 

 

 

Benacerraf suggests here that we cannot employ general notions like entity and object as 

appropriate categories enabling us to determine the truth value of identity claims because they 

are, in a sense, without real content. It is not that they are not well-entrenched notions—surely 

few notions have the historical pedigree of ‗object‘ and ‗entity‘—but, instead, that they offer no 

meaningful contrast through which a clear meaning can be assigned to an identity claim. They 

are ‗presuppositionless‘ insofar as we must already speak of entities or things if we are to speak 

logically about anything whatsoever. Part of Frege‘s problem, on this view, is that his 

ontological hierarchy was too general to capture Benacerraf‘s more restricted notion of 

semanticality. Indeed, for Benacerraf, it is precisely this generality which leads us into 

absurdities like dividing by Caesar. 

 

But here, too, we seem to run into significant difficulties with Benacerraf‘s attempt to side-step 

Frege‘s point of view and its related troubles. 

 

Is it true that there is no real content to notions like entity, object, and thing? In the brief passage 

just quoted, it seems that Benacerraf equates the roles of the words ‗entity‘ and ‗object,‘ while 

also holding that everything purports to be an entity. If the equation is to hold, then, whatever we 

mean when we use the word ‗entity‘ is what we mean when we use the word ‗object‘. On this 

view, neither of these terms is terribly useful because they don‘t pick out any particular category. 

Anything at all is an object or an entity or a thing. Indeed, the use of the term ‗anything‘ in the 

previous sentence seems to presuppose this fact. So, if we are trying to develop a category which 

delimits the semantical from the unsemantical, these notions will be powerless in their 

generality. They simply do not provide a meaningful contrast. 

 

But is this in fact the case? There are, I think, reasonable grounds for thinking otherwise. For 

one, it does seem that many mathematicians and philosophers assume some particular content for 

the notion of ‗object‘, at the very least, and probably ‗entity‘ as well. Even without the technical 

ontological notion of object possessed by Frege, it seems clear that most mathematicians 

(excluding, perhaps, set theorists)
34

 would differentiate between the objects of a given system 

and, say, the relations between the objects of that same system—this is particularly true of most 

branches of abstract algebra. On what basis is this distinction made if not on some presupposition 

about the nature of object-hood?  

 

Benacerraf‘s response to this query would likely be: indeed the category- or number-theorist 

does make a distinction between objects and relations, but this distinction is only made on the 

basis of a narrower, theory-specific notion of object-hood. On this view, orthographic devices 

could be developed to help maintain logical perspicuity (i.e., differentiation between objects of 

various types by use of subscripts: ‗objectcategory-theoretic,‘ ‗objectset-theoretic,‘ ‗objectFrege,‘ etc.). But 

                                                 
34

 The fact that ZF set theory a) was widely accepted as the best candidate for a foundation for mathematics when 

Benacerraf‘s paper was written and b) is purely extensional, containing no real differentiation between, e.g., objects 

and relations or concepts, might help to explain the origin of Benacerraf‘s view here.  Category theory, by contrast, 

operates on a basic distinction between the objects and the arrows (or morphisms) of a given category. We might 

also compare various axiomatizations of set theory to give some sense of the content of the notion of object; while 

ZF set theory deals exclusively with sets, [Von Neumann 1925] contains an interesting axiomatized ‗set‘ theory in 

which functions are logically prior to sets. 



171 

 

to attribute the content of the notion of object within various branches of mathematics 

exclusively to narrower theory-dependent decisions is a move which seems to me to avoid the 

fact that, prima facie, objects seem to play the same sort of role in many different mathematical 

domains. Indeed, Frege‘s distinction between objects and functions was meant to capture exactly 

this general feature of mathematical thinking. Without further argumentation, there is no obvious 

reason to reject the reasonable view that what can count as an object for a particular theory in the 

first place is dictated by a more general notion of object-hood, a notion which is often left tacit.  

 

1.2.3 The content of logic 

 

There also seems to be a conflict here between Benacerraf‘s claim that these notions are 

presuppositionless and other aspects of his own view. Note that Benacerraf shares with Frege 

and many others the belief that logic is a science of extraordinary generality, one which is 

applicable anywhere. Even with his demand that meaningful identity statements require a well-

defined category to gain purchase, he writes that  

 

Logic can then still be seen as the most general of disciplines, applicable in the same way 

to and within any given theory. It remains the tool applicable to all disciplines and theories, 

the difference being only that it is left to the discipline or theory to determine what shall 

count as an ―object‖ or ―individual.‖
35

 

 

Now, we‘ve seen in previous chapters why Frege was able to hold that logic was pan-

theoretically applicable. For him, it was the most general science of truth. But this did not at all 

mean that logic itself was in any way presuppositionless. One might even reasonably say that a 

large part of Frege‘s life‘s work was to bring the tacit presuppositions of logic into the light of 

day. This generality rested on logic‘s capacity to deal directly with those entities which were 

capable of being true or false, i.e., Fregean thoughts. Crucial to Frege‘s understanding of 

thoughts was his ontological hierarchy of objects and functions. Though Frege was quite aware 

that he could not define ‗function‘ and ‗object‘ in a purely precise manner, he nevertheless 

attempted to provide elucidations which were meant to guide the reader toward the content of 

these basic notions.
36

 The reason, then, why Frege believed logic to be universally applicable to 

any endeavour concerned with truth was because it dealt with the most general laws of truth, and 

truth itself is (partially) understood in terms of the ontological connections between entities in 

Frege‘s hierarchy. So, while Frege might have been willing to allow for narrower, theory-

specific notions of ‗object‘, ‗entity,‘ or ‗individual‘ he would likely have suggested that if these 

notions are going to be scientifically useful at all (i.e., if they are going to be employed in the 

search for truths), then they must already have built into them the more general assumptions of 

logic. For an individual theory to treat some class of entity as an object would, for Frege, already 

involve the assumption that such entities were self-standing, could act as arguments for first-

level functions, etc. While others might not share Frege‘s own assumptions about the nature of 

                                                 
35

 [Benacerraf 1965, 66]. This statement seems a particularly difficult one to maintain without argumentation in 

1965, well after the advent of intuitionism in the work of Brouwer and Heyting. For an interesting argument that at 

least some intuitionistic mathematics simply cannot share the same internal logic as classical mathematics, see 

[Shapiro 1989]. 
36

 Indeed, one of the three main guidelines Frege gives himself in the Grundlagen is ―never to lose sight of the 

distinction between concept and object‖ [Frege 1884b, xxii]. 



172 

 

objects or logic, it does seem clear that when the category or number theorist suggests that a 

given entity is an object rather than a function, they are tacitly employing a general notion of 

object which is not strictly determined by the definitions of category or number theory.  

 

One very basic assumption made by most forms of contemporary logic, for example, might be 

that the world is (at least partially) structured into entities at all, i.e., things which are particular, 

individuatable, and capable of possessing properties, et cetera. Even if this assumption turns out 

to be true—assuming that we can think of truth without already employing these notions—it is 

certainly not the case that everyone shares or has always shared this view. In philosophy and 

theology, for example, other ways of structuring the world have been developed extensively.
37

 

So, while the presuppositions implied by accepting a ‗world of objects‘ (or, using Rudolf 

Carnap‘s phrase, a ‗thing language‘)
38

 are extremely general in character, they are indeed 

presuppositions, and ones which not everyone might share. Certainly they seem necessary for 

many forms of contemporary scientific thinking, but this, it seems to me, is just another way of 

saying ‗We must employ the notions of logic to think logically‘. The language of logic and 

science is certainly not the only one possible. 

 

1.2.4 Which logic? 

 

But part of Benacerraf‘s goal in this paper is to distance himself from the rigidity of Frege‘s 

ontological hierarchy and its concomitant problems with fixed reference. So it would seem that 

his account of the pan-theoretic applicability of logic must resist this answer. So what is its basis 

for Benacerraf? 

 

I think part of the problem in answering this question for Benacerraf is that it is rather difficult to 

establish what is meant by ‗logic‘ here. What exactly is it that we are applying, what is this ‗most 

general of disciplines‘? Again, accepting Frege‘s detailed answer to this question would commit 

us to exactly the sort of profound difficulties with reference that Benacerraf is trying to avoid. 

But, while Frege‘s position is off-limits, it doesn‘t seem that there is a clearly superior 

alternative. 

 

Even more, while a few modern philosophers have argued in favour of the primacy of one form 

of logic over another—Frege and Quine being notable examples
39

—this has become an 

increasingly untenable position. The multiplicity of mutually incompatible logics developed 

throughout the 20
th

 and 21
st
 centuries is fairly good evidence that a conception of logic as the 

elaboration of a single intuitive notion of logical consequence might be outmoded or at least 

                                                 
37

 A fairly obvious example from twentieth-century philosophy can be found in the works of Martin Heidegger 

(1889-1976). In ―The Question Concerning Technology‖ (i.e., [Heidegger 1977]), for example, Heidegger suggests 

that the division of the world into objects occurs in the course of historical time, that the world need not be divided 

into objects, and that this division in fact obscures crucially important features of the world. A more ancient, though 

related rejection of the division of the world into objects occurs in the mysticism of the Neo-Platonist Plotinus 

(c.204/205 – 270), who writes ―Our way takes us beyond knowing; there may be no wandering from unity; knowing 

and knowable must all be left aside, every object of thought, even the highest, we must pass by, for all that is good is 

later than This and derives from This as from the sun all the light of the day‖ [Plotinus 2007, VI.9.4]. 
38

 [Carnap 1950]. 
39

 See, for example [Quine 1970], particularly chapter 6. 



173 

 

incapable of capturing much of what most of us would now happily call ‗logic‘.
40

 Examined in 

light of these developments, the history of logic from Frege to the present is similar in many 

respects to the confusion in geometry after the emergence of non-Euclidean geometry in the 19
th

 

century which we examined in the first chapter. We have moved from the informal investigation 

of a single consequence relation, to tinkering with its formal definition, to the realization that 

there are many (perhaps infinitely many) such relations. Instead of a single logic, we now have a 

huge variety of logics, all describing different consequence relations and with different possible 

applications. So, just as we have moved from geometry to geometries, we have (with less 

fanfare, perhaps) moved from logic to logics.
41

 Given the multiplicity of logics, then, the 

monolithic claim that logic is always and everywhere applicable must be slightly misleading or 

at least imprecise. Just as physical space might be describable in terms of a variety of quite 

different geometries, so too we can imagine a particular theory exhibiting many different logical 

structures, each with different consequence relations and (perhaps too) different assumptions 

about the nature of objects.
42

 Given these considerations, then, it seems that the uncritical 

acceptance of the pan-theoretic applicability of logic obscures the use of particular logical 

assumptions behind a veil of contentless generality which no seems longer tenable. 

 

1.2.5 Mathematical truth 

 

Another fairly serious difficulty arising from Benacerraf‘s attempt to restrict the meaningfulness 

of identity statements to specific contents lies in the area of mathematical truth. Assuming we 

have somehow overcome the foregoing difficulties in determining, in a uniform way, which 

contexts are semantically acceptable and which are not, it will become immediately obvious, 

given such a context, which identity statements are true, which are false, and which are 

unsemantical. It will be clearly true, for example, that within the context of ZF and the Zermelo 

reduction (assuming these can be made semantically acceptable), the statement ‗2 = {Ø, {Ø}}‘ is 

false. By contrast, if our context is the von Neumann account of the natural numbers, then this 

statement will be true. By changing the context we change the category which determines the 

content of the identity relation and the truth value of the relevant statements. For Benacerraf, 

there simply is no theory-independent account of the meaning or truth of these sorts of identity 

statements.  

 

As we saw in the second chapter of this thesis, Frege‘s emendations of Hilbert‘s view were 

similar in spirit to Benacerraf‘s demand for context, though they were intended to avoid 

precisely the sort of relativization of truth Benacerraf seems to suggest here. Frege‘s idea was to 

                                                 
40

 Note that this conception of logic as the elaboration of an intuitive notion of consequence, while common, is not 

shared by all philosophers. As Thomas Ricketts makes clear (in both [Ricketts 1997] and [Ricketts 2010]), it is 

rather difficult to characterize Frege in this manner, for instance. (For a contrasting opinion, see [Dummett 1973, 

81], which suggests that a consequence-centered view of logic is implicit even in Frege‘s work).  
41

 The locus classicus for logical pluralism is the work Rudolf Carnap (cf., [Carnap 1950] for one of the more 

provocative formulations of this view). More recently there has been renewed interest in the profound philosophical 

implications of the multiplicity of logics (see, e.g., [Beall and Restall 2000] and [Beall and Restall 2006]). 
42

 Indeed, some very interesting work has been done on the possibility of different sorts of objects within 

mathematics and logic. An early example can be found in the works of the Austrian philosopher and psychologist 

Alexius Meinong (1853-1920) (e.g., [Meinong 1904]). In more recent years, a number of philosophers of 

mathematics have entertained the possibility of ontologically ‗strange‘ objects, (e.g., [Akiba 2000] and [Balaguer 

1998]). 



174 

 

make explicit (what he thought to be) the implicitly conditional nature of Hilbert‘s uninterpreted 

statements. Thus, in place of stating that a claim like ‗2 = {{Ø}}‘ is true only within or relative 

to the Zermelo reduction of the natural numbers to the universe of ZF, we instead say something 

like: ―If the Zermelo reduction is true, and if the ZF axioms are true, then 2 = {{Ø}}‖. But, for 

Frege‘s conditionalization approach to work, some rigidly fixed ontological content has to be 

given to the background theory (in this case ZF and the Zermelo reduction)—something which 

Benacerraf is keen to avoid. 

 

Thus, we are again presented with a dilemma. On the one hand we can have fixed reference and 

a non-relative account of truth. With this choice, however, we also run into the problems 

associated with absolutely fixed reference (e.g., the Caesar problem and its modern cousin the 

non-uniqueness problem). On the other hand, we can relativize reference to a particular 

theoretical framework. This leads us again to two formidable problems. First: how are we to 

understand the relation of scientific truth to the world if that truth is relative to particular 

frameworks which are not always mutually compatible? Second: do these frameworks 

themselves have any connection to ontology or must they be free-floating in Hilbert‘s sense, 

mere scaffoldings of concepts? If the latter is the case, then we are back at the problems of 

meaninglessness faced by Hilbert in his correspondence with Frege. If these frameworks do have 

ontological implications, however, we need something more than Hilbert gave us. 

 

1.3 Structures 

 

For Benacerraf, as for Hilbert,
43

 the attempt to isolate some fixed collection of particular objects 

which are meant to serve as the natural numbers (or, in Hilbert‘s case, the elements of Euclidean 

geometry) is a fool‘s errand based on a serious philosophical misunderstanding of the nature of 

mathematical structures. This misunderstanding hinges, for Benacerraf, on the view that 

mathematics deals with various collections of objects (numbers, sets, points, groups, etc.) which 

can be picked out and which bear properties in the same way that objects like tables and chairs 

do. Benacerraf‘s view entails that mathematics isn‘t really about such objects at all, or, slightly 

more cautiously, that it only deals with isolable objects insofar as they are related to a given 

structure. The mathematically relevant aspect of an object is not one which can be discussed with 

indifference to the particular structure of which it is a part. This is a large part of the reason why 

questions like ‗Is Pompey Magnus identical to the number 2?‘ just don‘t make any sense. On this 

structure-focussed view, the typical set-theoretic reductionist‘s claim that the natural numbers 

just are sets is a mistaken one.  

 

In order to make all of this a bit more precise, Benacerraf makes a distinction between the 

ontological activity of (unrestricted) identification (‗x is y‘) and the ontologically neutral 

activities of reduction and explication (‗within context C, x can be treated as y‘). If we are to 

identify the natural numbers with some collection of sets, in the strong ontological sense, we 

mean quite literally that, when we speak about the natural numbers, we are really referring to 

                                                 
43

 Recall Hilbert‘s view that asking for referentially fixed terms prior to the application or interpretation of an 

axiomatic theory involves ― One […] looking for something that can never be found because there is nothing there, 

and everything gets lost, becomes confused and vague, and degenerates into a game of hide-and-seek‖ [Hilbert to 

Frege, in Frege 1971, 11-12]. 



175 

 

this particular collection of sets. Thus, if the set we identify with the number 3 has 847 members, 

then the number 3 has 847 members, regardless of the context within which we are speaking. 

But, as the non-uniqueness problem is meant to point out, we have been unable to provide any 

good reason for choosing one such reduction over another, as all of the distinguishing properties 

of individual reductions (set-theoretic or otherwise) will be arithmetically irrelevant. And, if the 

failure of Frege‘s best efforts is any indication, we are not likely to stumble on such reasons in 

the future. 

 

The activities of reduction and explication, by contrast, carry no such ontological commitment 

according to Benacerraf. If on a given reduction of the natural numbers to a collection of sets the 

number 3 has 847 members, we are not committed to the view that every single time we speak 

about the number 3 we are speaking about this particular 847-membered object. Rather, we are 

meant to understand that the number 3 is identified with an 847-membered set on this reduction 

and might very well be reduced to or explicated by a set possessing completely different non-

arithmetic properties on another reduction, or even with an object that isn‘t a set at all on some 

other reduction. Identification, in the strong sense, is, for Benacerraf, theory-independent, while 

reduction and explication are only possible within the context of some particular theory. 

 

Keeping these distinctions in mind, Benacerraf tries to overcome the dangers of Hilbert‘s 

relational view of structures by giving a more robust account of how abstract structures differ 

from the object-oriented accounts like those of Frege and the set-theoretic reductionists. 

 

The most obvious difference between Benacerraf‘s view and Frege‘s is that Benacerraf explicitly 

denies Frege‘s central claim that numbers are objects. Indeed, the central lesson that Benacerraf 

draws from the non-uniqueness problem is that, contra Frege, numbers simply cannot be objects 

(as the title of the paper is meant to suggest). The advantage he gains is fairly obvious at this 

point: he no longer has to address the Julius Caesar problem and others like it. By rejecting the 

broad claim that numbers are objects, Benacerraf can now look at ―the question of whether a 

particular ‗object‘ […] would do as a replacement for the number 3‖ as ―pointless in the 

extreme‖.
44

 

 

In place of the object-oriented picture of (at least) the natural numbers, Benacerraf suggests a 

structure-oriented view. The explanatory burden, then, is shifted. He no longer needs to explain 

how we pick out which particular objects the natural numbers are, but he must explain what the 

natural number structure is, and how its elements (the numbers) differ from objects. In Frege‘s 

attempted emendations of Hilbert‘s free-floating view, he sought to characterize Hilbert‘s 

uninterpreted formal systems by pinning them down within his ontological hierarchy of objects 

and functions. For Frege, then, the direction of best mathematical explanation was from the 

concrete world of objects and functions to the world of abstract formal structures built out of 

these entities. Benacerraf‘s central idea is to reverse this order of explanation, and suggest that 

mathematics is in fact about abstract structures, and that it only secondarily studies their concrete 

representations (e.g., representations within Frege‘s ontological hierarchy, or more familiar 

representations within the universe of set theory). 

 

                                                 
44

 [Benacerraf 1965, 69]. 



176 

 

Frege‘s objection, when Hilbert tried to make a similar explanatory shift, was that this seemed to 

leave mathematics contentless. In order to address this objection, and therefore to move beyond 

the Frege-Hilbert impasse, Benacerraf needs a more robust account of the relationship between 

ontology and the abstract structures which he supposes are the subject matter of mathematics. 

 

1.3.1 Concrete vs. Abstract Structures 

 

So, what exactly does Benacerraf mean when he speaks of structures? 

 

Given the tentative character of the paper‘s final few pages, it is perhaps not surprising that 

Benacerraf does not give us a thoroughgoing definition of the term ‗structure‘. Nevertheless, we 

can surmise that, for Benacerraf, at the very least a structure is composed of elements arranged 

under a certain relation (or relations).
45

 

 

So far as this goes, this account of structure is compatible with Frege‘s views or those of the set-

theoretic reductionist. On Frege‘s account, both the elements of and the relations which define a 

structure are entities with a fixed position in his ontological hierarchy, similarly for the set-

theorist and the universe of set theory, where everything (relations included) is a set. But 

Benacerraf wants to make a sharp distinction between concrete structures and abstract structures 

which will allow more careful differentiation between the ‗structuralism‘ which he wishes to 

espouse and the referentially-fixed reductionism of Frege and the set theorists. Let‘s look first at 

what this distinction says about the elements of each sort of structure.  

 

A concrete structure‘s elements will be entities
46

 which possess properties or stand in relations 

which are not solely determined simply by their being elements of that structure. For Benacerraf, 

 

That a system of objects exhibits the structure of the integers implies that the elements of 

that system have some properties not dependent on structure. It must be possible to 

individuate those objects independently of the role they play in that structure.
47

 

 

                                                 
45

 Cf. [Benacerraf 1965, 69-73], where he speaks repeatedly of the elements of the natural number structure, and also 

of structures as systems of relations. Some later structuralists, e.g., Stewart Shapiro, countenance structures which 

are composed of elements but no relations (cf., [Shapiro 1997, 115] and [Shapiro 2008] which discuss cardinal 

structures which have elements but no relations). Even further, Shapiro considers the degenerate case of the ―0 

pattern‖ which has neither elements nor relations [Shapiro 1997, 115].  
46

 Benacerraf seems to assume in this paper that the elements of a concrete structure will be objects, though it is not 

exactly clear what he means by ‗object‘ or even, as noted above, whether he thinks the term is a meaningful one at 

all outside of a well-defined theoretical context.  
47

 [Benacerraf 1965, 70; emphasis mine]. Here Benacerraf seems to tacitly view objects as independently existing 

entities. It should be noted here that a number of subsequent authors have suggested that there may indeed be objects 

(not merely ‗elements‘ or ‗roles‘) whose strange nature allows them to be elements of abstract structures in 

Benacerraf‘s sense. See, for instance, [Akiba 2000], which defends the position  ―that mathematical objects, along 

with some others, neither definitely possess nor definitely lack certain properties‖ [Akiba 2000, 26]; [McLarty 1993] 

suggests that ―the structuralist program [of Benacerraf] is already fulfilled or obviated, depending on how you look 

at it, by categorical set theory […]. Sets and functions in this theory have only structural properties. There is no need 

and no further place for a further theory of abstract structures‖ [McLarty 1993, 487]. Depending upon how we read 

the situation, these objections may amount to quibbles regarding the meaning of the term ‗object‘ as Benacerraf uses 

it. 



177 

 

Thus, for instance, even if Julius Caesar is to be an element of the natural number structure
48

 

(i.e., if he is arranged with other elements under an appropriate relation of recursion) he will 

have many properties which are not determined simply by his being an element of this 

structure. Simply being an element of such a structure will not tell us anything about the 

content of his Commentarii de Bello Gallico or which sort of olive he may have preferred, for 

instance. These latter properties are structure-independent (in relation to the natural number 

structure, at least). But, if we are to hold that he is an element of the natural number structure, 

then he will also have structure-dependent properties, like being prime or being the square 

root of 4 and so on. So, the elements of concrete structures will have both structure-dependent 

and structure-independent properties. 

 

It is specifically the possibility of extra-structural individuation (of the kind which might have 

solved the Caesar problem) and the possession of structure-independent properties which 

characterize concrete structures.  The structure-independence of their elements is also the feature 

which, according to Benacerraf, disqualifies them as the main focus of mathematical interest. For 

Benacerraf, the chief reason why mathematicians and philosophers have gotten into muddles like 

the Caesar problem or the non-uniqueness problem is because they mistake concrete structures 

for abstract ones, and they ignore the fact that what they are actually most interested in is abstract 

structures. The trouble is in understanding exactly what these abstract structures are—or, more 

pointedly, in distinguishing abstract structures from empty formal games. 

 

Benacerraf‘s abstract structures, like their concrete cousins, are composed of elements arranged 

under a relation (or relations). In contrast to concrete structures, however, the elements of 

abstract structures do not possess any structure-independent properties. Given what appears to be 

Benacerraf‘s tacitly Fregean notion of object, this means that the elements of abstract structures 

cannot be objects at all. 

 

What, then, are they? There are at least two ways to read this question. 

 

On the first, it asks us: what sorts of entities (other than objects) are abstract structures composed 

of? On Benacerraf‘s view, reading the question this way would simply be to recapitulate the 

object-oriented misunderstanding of mathematics. His point is that looking for the building 

blocks of structures, those isolable individuals which we might put together and eventually 

obtain a structure, is only ever going to get us concrete structures. 

 

There is, however, another way of reading this question which might help to shed some light on 

Benacerraf‘s view. Instead of asking ‗out of which sorts of things are structures composed‘ we 

can ask ‗what is the relationship between an abstract structure and its elements‘? And to this 

question we might give a very different answer. Here is how Benacerraf puts it: 

 

So what matters, really, is not any condition on the objects […] but rather a condition on 

the relation under which they form a progression. To put the point differently—and this is 

                                                 
48

 As we will see in what follows, for Benacerraf an object like Julius Caesar cannot be a member of the natural 

number structure (which is, for Benacerraf, an abstract structure) but only of a concrete structure which represents 

the natural number structure. 



178 

 

the crux of the matter—that any recursive sequence would do suggests that what is 

important is not the individuality of each element but the structure which they jointly 

exhibit. This is an extremely striking feature. One would be led to expect from this fact 

alone that the question of whether a particular ―object‖ […] would do as a replacement for 

the number 3 would be pointless in the extreme, as indeed it is. ―Objects‖ do not do the job 

of numbers singly; the whole system performs the job or nothing does. I therefore argue, 

extending the argument that led to the conclusion that numbers could not be sets, that 

numbers could not be objects at all; for there is no more reason to identify any individual 

number with any one particular object than with any other (not already known to be a 

member).
49

 

 

The relationship between the elements (which, again, are not objects for Benacerraf) of an 

abstract structure cannot be made sense of on a case by case basis. An entity is an element of a 

particular structure not because it possesses unique properties which are ideally suited to playing 

a peculiar role. No—Benacerraf‘s point is that while ―Any object can play the role of 3‖ no 

object is the role of 3.
50

 And it is the role played by 3 which is the element of the abstract 

structure, or, perhaps, the third place in the abstract natural number structure. Certainly this role 

can be represented by a particular object, or this place occupied by a particular object—indeed, 

any particular object, according to Benacerraf. But such representation can only occur within a 

concrete structure, i.e., a structure whose elements will have structure-independent properties 

like those of the set-theoretic representation of the natural numbers.  

 

So, for Benacerraf, the number 3 is not an object with a kind of structure-independent reality (as 

Frege suggested), out of which a structure is built, but is instead a role which can only be defined 

relative to the abstract structure of which it is an element. He writes that ―What is peculiar to 3 is 

that it defines that role – not by being a paradigm of any object which plays it, but by 

representing the relation that any third member of a progression bears to the rest of the 

progression.‖
51

 On this view, the elements of a structure are not prior to the structure. What is 

essential is the abstract structure itself, and this is defined by a particular relation or relations on 

elements. The relations which these elements stand in, and the properties which they might 

possess, are only those determined by the simple fact of their being elements of the structure. 

Thus, their nature is structure-dependent, or as Stewart Shapiro puts it, ―the essence of a natural 

number is its relations to other natural numbers.‖
52

 

 

This view, suggestive as it is, poses a number of difficult questions related to our central aim of 

clarifying what mathematics is about. If mathematics is chiefly concerned with these abstract 

structures,
53

 we‘ll want to know how (and perhaps which) relations define or otherwise relate to 

                                                 
49

 [Benacerraf 1965, 69]. 
50

 [Benacerraf 1965, 70]. 
51

 [Benacerraf 1965, 70]. 
52

 [Shapiro 1997, 72]. 
53

 Benacerraf himself is fairly limited in the scope of his claims in [Benacerraf 1965], sticking for the most part to a 

description of the arithmetic of the natural numbers. As we will see in the following section, it is later thinkers who 

sought to extend Benacerraf‘s ‗structuralist‘ suggestions about the natural numbers to the wider world of 

mathematics as a whole. Nevertheless, it does seem clear that, even for Benacerraf in 1965, his limited suggestions 



179 

 

abstract structures. We‘ll also want to know what ontological status—if any—such structures are 

meant to possess. In our pursuit of these questions, it is best to depart from the specificities of 

Benacerraf‘s account and begin an examination of more recent structuralist literature in the 

philosophy of mathematics. As Benacerraf himself notes, the positive conclusions of his paper 

are exploratory in character. He does not pretend to develop a fully fleshed-out account of the 

nature of abstract structures, but, instead, draws our attention away from the fool‘s errand of 

naïve set-theoretic reductionism toward the structural meat of mathematics. 

 

§2. Contemporary structuralism 

 

2.1 From sets to structures 

 

Historically, Benacerraf‘s paper was part of the impetus for a substantial reorientation in the 

philosophy of mathematics. As I‘ve mentioned at the beginning of this chapter, the philosophy of 

mathematics in the decades preceding Benacerraf‘s paper was dominated (with few exceptions) 

by the problems of set-theoretic reductionism. While set theory still figures prominently in the 

philosophy of mathematics even today, Benacerraf‘s paper helped to popularize a structuralist 

approach to all forms of reductionism, even set-theoretic ones. By the mid-1990s, to be sure, the 

philosophy of mathematics was dominated with questions concerning the viability of 

structuralism as a solution to some of the important foundational questions in mathematics. 

 

This philosophical movement toward structuralism was preceded by an increased focus on 

structure within mathematics itself. This focus is evident already in the work of Hilbert and 

others around the turn of the nineteenth century, but became increasingly more explicit and well-

defined in the work of the group of French mathematicians calling themselves Nicholas 

Bourbaki and in the (related) rise of abstract algebra from the 1930s onward.
54

 Bourbaki wrote 

and published an incredibly influential series of texts called the Éléments de Mathematique. The 

series involved the hierarchical and rigorous axiomatic development of mathematics from the 

ground up, ostensibly under the notion of structure. While Bourbaki‘s technical definition of the 

term ‗structure‘ was not terribly influential,
55

 their axiomatic-structural approach was immensely 

so. As usual, much of the philosophy of mathematics had lagged behind the actual development 

of mathematics for a few decades by the time Benacerraf‘s paper appeared. But, as is also often 

the case, very few mathematicians had a deep enough interest in the philosophical aspects of 

their discipline to develop a thorough-going account of these new structural developments and 

the nature of contemporary mathematics.
56

 

                                                                                                                                                             
implicitly contain much broader implications—for very little in the paper hinges on the peculiarities of the natural 

numbers or their set-theoretic representations. 
54

 [Corry 2004b] is the finest available historical account of these developments; [Wussing 1984] is an illuminating 

account of the ‗pre-history‘ of modern abstract algebra as found in the development of group theory; [Dieudonné 

1972] presents an insider‘s look at the somewhat more specific field of algebraic geometry.  
55

 Their technical definition is first advanced in [Bourbaki 1939]. Its lack of popularity is evident in the fact that it is 

rarely employed, neither by the members of Bourbaki nor those they influenced. In a survey of the notion of 

structure in mathematics, Mac Lane writes that ―as best I can determine [Bourbaki] never really made actual use of 

his definition, and I will not make any use here of my variant‖ [Mac Lane 1996, 179]. 
56

 This is not to say that mathematicians involved in these developments were disinterested in the philosophy of 

mathematics, only that there were (and continue to be) very few mathematicians whose interest extends beyond the 

publication of one or two brief papers on the subject; Mac Lane is a notable exception (cf. [Mac Lane 1986]). 



180 

 

 

2.11 Benacerraf‘s legacy 

 

In the 1965 paper, Benacerraf confines himself for the most part to a discussion of the arithmetic 

of the natural numbers. Since this paper, a more general position in the philosophy of 

mathematics called ‗structuralism‘ has emerged which extends these claims to include a much 

larger portion of mathematics. As Benacerraf himself puts it in a later paper, ―the structuralist‘s 

cri de guerre‖ is ―that mathematics is the study of (abstract) structures‖.
57

 My aim here is not to 

give a complete account of contemporary forms of structuralism which follow in the wake of 

Benacerraf‘s work, but to see whether they are capable of addressing the problems present in 

Benacerraf‘s 1965 account. 

 

As we saw, Benacerraf attempted to develop structuralism as a means of addressing both the 

Julius Caesar problem and its more contemporary manifestation, the non-uniqueness problem. 

His way of doing so involved denying Frege‘s claim that all identity statements must be 

meaningful. In effect, he sought to restrict meaningful identity claims to well-defined contexts. 

One way of reading his view is that, in mathematics at least, abstract structures serve to furnish 

these well-defined contexts. Thus, when we speak of identity, we speak of identity between the 

entities of a given structure, say the natural number structure. The benefit of this view is that we 

can eliminate the Julius Caesar problem by suggesting that, without further contextualization, 

there is no obvious structure which contains both Caesar and the natural number 2, so the 

question ‗Is Caesar identical to the number 2?‘ is rendered meaningless. When such a context is 

provided (say in a structure where components of the physical universe are employed as 

representatives of the elements of the natural number structure) the statement gains a truth-value. 

 

But there are clear difficulties with his attempt. 

 

First, Benacerraf has what appear to be inconsistent views of the role that objects play (or do not 

play) in this whole story. He famously denies that the natural numbers are objects – and, by 

extension, that the elements of any abstract mathematical structure are objects. He does so, it 

seems, because he takes ‗object‘ to mean ‗theory- or structure-independent entity‘, which is 

exactly what the natural numbers are not on his account. But he also maintains, in the same 

paper, that we can speak of theory-dependent objects and that the notion of ‗object‘ is too 

general to have any content whatsoever. Intermingled with these concerns are questions about 

the exact nature of the elements of abstract structures. As quite a bit seems to rely on how we 

understand his claims about objects and elements, these equivocations need to be addressed. 

 

Second, Benacerraf does not give us a thorough-going account of the exact nature of the abstract 

structures which he believes are the subject matter of mathematics. This is not terribly surprising, 

as his paper is intended to be (for the most part) diagnostic in character. If we are to accept his 

claim that arithmetic and mathematics concern these abstract structures, though, we need to 

develop a more precise understanding of these entities. Particularly pressing, from our point of 

view, are questions pertaining to the ontological status of such structures: do they exist or not? If 

they exist, are they to be understood as permanent, independent features of reality, or are they, 

                                                 
57

 [Benacerraf 1996a, 184]. 



181 

 

like their elements, dependent upon a particular context or theory for their existence? If they do 

not exist, does this leave us to treat mathematics as a contentless enterprise? 

 

Third, a great deal in Benacerraf‘s account seems to hinge on the status of logic within 

mathematics. Recall that a chief aim of his paper is to undermine the hold that set-theoretic 

reductionism has on the philosophy of mathematics. He does this by showing that what we are 

really interested in is not the ontological universe of set theory, but instead the abstract 

structures, most or all of which can be modelled or exemplified in this universe. His paper is 

meant to show that our representations of these structures within a single ontological universe are 

of secondary concern. But he also claims that logic (singular) is pan-theoretically applicable, and 

that it will operate the same in relation to any given abstract structure we might wish to study. 

The question we have to ask, especially given the almost universal understanding of logic in 

terms of the background ontology of model and set theory, is whether or not this ends up being 

circular. Does Benacerraf‘s view of the universality of logic commit him to the background 

ontology of set theory that the paper seeks to avoid? 

 

In order to further address these concerns let‘s turn to an examination of some more 

contemporary forms of structuralism. 

 

2.2 Two types of structuralism 

 

There are quite a number of distinct philosophical positions which can be placed under the 

heading ‗mathematical structuralism‘. Indeed, there are even a number of quite distinct 

typologies of such views.
58

 In what follows, I would like to provide a very general typology of 

my own, one which contains precisely two forms of structuralism. These I will call (loosely 

following a similar division made by Shapiro and Hellman)
59

 ‗assertoric structuralism‘ and 

‗algebraic structuralism‘. Very schematically, assertoric forms of structuralism hold that 

mathematics contains assertions about a particular content. Usually, but not always, this content 

is fleshed out in terms of some theory of abstract structures. Algebraic structuralism, by contrast, 

contends that mathematics does not have a single well-defined content, and that, to the extent 

that it contains assertions at all, these are always relative to a particular theory, framework, 

language, or structure. In the first, assertoric, case we might speak of a global content for 

mathematics, while in the latter, algebraic, case if mathematics is said to have content at all, it is 

only in a local sense. My intention is to investigate whether either of these forms of structuralism 

is capable of satisfactorily addressing the problems with Benacerraf‘s view and, thus, helping us 

to explain what it is that mathematics is about.  

 

                                                 
58

 [Hellman 1996] presents a typology containing four versions of structuralism (the model- or set-theoretic account, 

the category-theoretic account, the sui generis account, and his own modal account). Five years later, in [Hellman 

2001], the typology is reduced to three versions (set-theoretic, ante rem, modal). There are others besides (e.g., 

[Dummett 1991, chapter 23], [Horstein 2014]).  
59

 Cf., [Shapiro 2005]. For Shapiro, the division concerns the status of sentences within a given account – an account 

is assertory if its sentences ―are meant to express propositions with fixed truth values. Algebraic sentences are 

schematic, applying to any system of objects that meets certain conditions‖ [Shapiro 2005, 66]. While Shapiro‘s 

division is my inspiration here, my own may diverge from his in what follows. 



182 

 

Both of these forms of structuralism insist that mathematics is chiefly concerned with the study 

of abstract structures. They differ, however, in their understanding of the ontological status of 

these structures. For the assertoric structuralist, there is (or ought to be) what we might call a 

‗final theory‘ of structures, one in which the intricate network of relationships between structures 

(e.g., the sort of hierarchy of algebraic structures we find in Bourbaki‘s Éléments) is ultimately 

cashed out in terms of a theory which makes assertions about the world directly. The difference 

between this form of structuralism and the more straightforward, naïve version of set-theoretic 

reductionism, is that the assertoric structuralist is always concerned with abstract structures, in 

Benacerraf‘s sense, whereas the set-theoretic reductionist (or Frege) thinks in terms of a single 

concrete structure (the set-theoretic universe V, Frege‘s ontological hierarchy, etc.). 

 

The category ‗assertoric structuralism‘ is a very broad one, and it includes the work of 

structuralists who are, in many other respects, opposed. Thus, the (putatively) nominalist 

structuralism of Geoffrey Hellman
60

 and the realist structuralism of Stewart Shapiro
61

 both count 

as forms of assertoric structuralism. They differ markedly in what exactly they assert, of course. 

For Shapiro, structures are ante rem universals—about as far as one can get from nominalism. 

For Hellman, structures are ‗eliminated‘ ―in favor of sentences with modal operators‖.
62

 These in 

turn are cashed out in terms of ―a modal second-order logic with a restricted (extensional) 

comprehension scheme,‖
63

 with the end result being that Hellman makes assertions about 

possibilities, rather than actual universals. Aside from Shapiro and Hellman, there are a number 

of assertoric forms of structuralism which employ unique background ontologies to account for 

abstract structures for different reasons.
64

 What ties them together is that they claim that 

mathematics is best understood as making assertions whose truth value is ‗absolute‘ in the sense 

that there is a final theory of structures (or whatever) within which the relative truths of specific 

mathematical theories are embedded.  

 

Algebraic structuralism is so named because of its connections (conceptual and historical) to 

abstract algebra. A key component of abstract algebra involves detailed work on relations 

between various well-defined algebraic structures (e.g., rings, groups, monoids, etc.). Prior to the 

development of category theory in the 1940s and 1950s, these algebraic structures were 

ultimately cashed out in the background ontology of set-theory (usually ZF). But, in many ways, 

algebraists found—and still find—this connection to set theory and its ontology an encumbrance. 

Algebra has, historically, been concerned with operations at least as much as it has been 

concerned with particular objects (in most cases, much more so). To try to ‗pin down‘ algebraic 

work in the purely object-based ontology of set theory is, in many ways, foreign to the operation-

focussed character of abstract algebra. If, for example, your interest is in discussing certain 

homomorphisms between two rings, doing everything in set theory is both time-consuming and, 

for the most part, beside the point. In set theory, both morphisms and structures (as well as the 

elements of structures) will be understood in terms of sets.
65

 But, as many algebraists will note, 

                                                 
60

 Cf., e.g., [Hellman 1989], [Hellman 1996], and [Hellman 2001]. 
61

 Cf., e.g., [Shapiro 1997] and [Shapiro 2008]. 
62

 [Hellman 1996, 102]. 
63

 [Hellman 1996, 102]. 
64

 One influential such view is that of Michael Resnik, who has developed an account of structures based on the 

notion of patterns (cf. [Resnik 1980], [Resnik 1982], and [Resnik 1997]). 
65

 Or, possibly, urelements. 



183 

 

morphisms behave nothing like sets—so why understand them in these terms in the first place? 

Since its inception, category theory has been treated by many algebraists as a more ‗natural‘ 

mathematical language (or setting) for speaking about these structural interrelations. This is 

perhaps not surprising, given that the earliest incarnations of category theory, in collaborations 

between Saunders Mac Lane and Samuel Eilenberg, involved research into the surprising and 

profound morphisms between algebraic and topological structures, which were, at that time, 

pursued more or less independently.
66

 The impetus of category theory, then, was to make work 

on morphisms between structures easier, more flexible. To this end, category theory—which can 

be modelled in the universe of sets,
67

 but which need not be
68

—begins with a class of structures 

(i.e., categories) which are defined in terms of morphisms and objects.  

 

The emergence of category theory in the 1940s and 1950s thus pointed to a new direction for 

abstract algebra, where the idea of set theory having the last word no longer seemed necessary or 

even plausible. As John Bell eloquently puts it, 

 

Because the practice of mathematics has, for the past century, been officially founded on 

set theory, the objects of a category […] are normally constructed as sets of a certain kind, 

synthesized, as it were, from pure discreteness. As sets, these objects manifest set-theoretic 

relationships—memberships, inclusions, etc. However, these relationships are irrelevant—

and in many cases are actually undetectable—when the objects are considered as 

embodiments of a form, i.e., viewed through the lens of category theory. […] This fact 

constitutes one of the ‗philosophical‘ reasons why certain category theorists have felt set 

theory to be an unsatisfactory basis on which to build category theory—and mathematics 

generally. For categorists, set theory provides a kind of ladder leading from pure 

discreteness to the category-theoretic depiction of the real mathematical landscape. 

Categorists are no different from artists in finding the landscape (or its depiction, at least) 

more interesting than the ladder, which should, following Wittgenstein‘s advice, be 

jettisoned after ascent.
69

 

 

This is not to say that algebraists abandoned set theory, nor that category theory itself has not 

often been treated as a new ‗last word‘ in cashing out claims about relations between structures. 

But, for at least some algebraists, and some philosophers of mathematics as well, the emergence 

of category theory as a means of describing morphisms between structures has illustrated not 

only that set theory is not the privileged resting place of all structures, but also that we do not 

need such a privileged resting place and that, anyway, there can be none.
70

 This is the central 

claim of algebraic forms of structuralism: mathematical assertions can only be made relative to a 

particular background, and there can be only local reasons (reasons pertaining to one‘s particular 

problem, to a particular class of morphisms, etc.) for preferring one background or framework to 

another. Thus, for example, one might study the natural number structure in relation to the 

background ontology of set theory or category theory, or with indifference to both. The dream of 

                                                 
66

 Cf., [Mac Lane and Alexanderson 1989, 20] for a discussion of Mac Lane‘s initial collaboration with Eilenberg.  
67

 Cf., e.g., [Mac Lane 1998], though here Mac Lane is keen to point out that categories need not rely on set theory.  
68

 Cf., e.g., [Lawvere 1964], . 
69

 [Bell 2001, 152]. 
70

 [Awodey 2004] contains one of the most convincing defenses of this position. 



184 

 

a single universe within which all mathematical claims operate is, to the algebraic structuralist, 

just that: a dream. It is neither followed in practice (especially not in most algebraic practice), 

nor does it make sense philosophically (as Benacerraf‘s non-uniqueness problem can be said to 

show, at least partially). To make the move of the assertoric structuralist is, for the algebraic 

structuralist, to slip back into the game of hide-and-seek that Hilbert already saw to be 

impossible at the turn of the nineteenth century. 

 

In contrast to assertoric structuralism, the algebraic form of structuralism holds that there is not 

(and, often, there could not possibly be) any final background theory which rules over all of 

mathematics. For the algebraic structuralist, claims about relations between structures are 

themselves best thought of as embedded within a particular structure, or point of view. For 

different purposes, we might switch between a background theory of set theory or category 

theory or something else besides. The assertoric structuralist might suggest that switching 

between these theories implies that there is some sort of equivalence between the two, and, thus, 

that there is some broader, final abstract structure which encompasses both. The algebraic 

structuralist will hold that there may be such a structure, but that it, too, is subject to revision as a 

momentary background ontology, and not at all a final arbiter of what is. Though there are 

important differences, I hope the reader is struck by the similarity between the 

assertoric/algebraic divide and the debate between Frege and Hilbert. Frege, like the assertoric 

structuralist, sought to secure a determinate content for all of mathematics. In contrast to the 

assertoric structuralist, however, Frege‘s work endeavoured to embed mathematics within a 

single concrete structure. Similarly, the algebraic structuralist‘s unwillingness to pin 

mathematical claims down within a single ontological frame is strikingly similar to Hilbert‘s 

free-floating view. Neither the algebraic structuralist nor Hilbert believed that mathematical 

claims required extra-theoretical or extra-structural grounding. The question we must ask 

ourselves now is whether the divide between assertoric and algebraic structuralists holds more 

promise than the impasse of the Frege-Hilbert debate. 

 

2.3 Assertoric Structuralism 

 

Let‘s begin our investigation of contemporary structuralism by examining the capacity of 

assertoric structuralism to address the problems left by Benacerraf‘s account. In this section I‘ve 

chosen to focus chiefly on the work of Stewart Shapiro. Though the choice is partially arbitrary, 

there are a few motivating factors. First and foremost is the fact that Shapiro often frames his 

structuralism in direct response to Benacerraf, Frege, and Hilbert. This makes the task of 

situating his work in relation to our driving concerns much easier. Shapiro‘s work is also more 

directly and frequently engaged in the ontological aspects of structuralism with which I am 

chiefly interested here.
71

 Shapiro himself also provides a compelling argument which suggests 

that some of the key forms of assertoric structuralism are intertranslatable.
72

 So while the choice 

                                                 
71

 This is not to suggest that other structuralists are not interested in ontological questions, only that Shapiro‘s 

ontological focus is more often front-and-center. Michael Resnik‘s work on a pattern-theoretic account of structures 

strikes me as focussed on epistemological concerns to the same degree that Shapiro focuses on ontological ones. 
72

 Cf., [Shapiro 1997, 90-97]. There he writes that ―Anything that can be rendered in the modal structural system 

[…] can be rendered in either the set language or [Shapiro‘s own] structure language‖ [Shapiro 1997, 96] and, 

further, that ―when all is said and done, the different accounts are equivalent‖ [Shapiro 1997, 96]. The modal 

structural system he refers to here is Hellman‘s, as developed in [Hellman 1989] and [Hellman 1996].  



185 

 

is at least partially idiosyncratic, there are (I think) plausible reasons for engaging chiefly with 

Shapiro‘s form of structuralism.  

 

2.3.1 Shapiro on the nature of mathematical objects 

 

For Benacerraf, the elements of at least abstract structures (the chief focus of mathematics) 

cannot be objects. As we saw, his reasons for holding this belief are rather puzzling. On the one 

hand, he suggests that there is no real content to be given to any theory-independent notion of 

object, and, on the other hand, he presumes precisely that to be an object is to possess properties 

and stand in relations independently of any given theory, language, or structure. In any case, his 

view leads him to claim that whatever numbers might be, they simply cannot be objects (at least 

in the sense of entities possessing properties or standing in relations independent of theory or 

structure). As a result, the unrestricted form of identity which Frege employed, one which ranges 

over all objects, must be replaced. Thus, Benacerraf‘s strange account of objects is at least 

partially intertwined with his rejection of unrestricted identity. 

 

Shapiro finds this view as puzzling as we have. He writes that 

 

In mathematics, at least, the notions of ‗object‘ and ‗identity‘ are unequivocal but 

thoroughly relative. Objects are tied to the structures that contain them. It is thus strange 

that Benacerraf should eventually conclude that natural numbers are not objects. 

Arithmetic is surely a coherent theory, ‗natural number‘ is surely a legitimate category, and 

numbers are its objects.
73

 

 

Here I think we have a slightly more careful statement of what Benacerraf himself was trying to 

get at in his 1965 paper. The strangeness of Benacerraf‘s claim stems from the fact that there are 

two competing conceptions of ‗object‘ at work in his paper. The first is Frege‘s: objects—

mathematical or otherwise—are self-standing, independently existing entities. They exist prior to 

and irrespective of the relations they might stand in. This notion, both Benacerraf and Shapiro 

suggest, is not at all helpful when discussing the elements of abstract mathematical structures. To 

construe the elements of structures in terms of Frege-style objects is to become ensnared once 

again by the Caesar and non-uniqueness problems which structuralism seeks to avoid in the first 

place. 

 

But, in contrast with Benacerraf, Shapiro is not ready to abandon mathematical objects tout 

court. There is a second notion of object at work here, one which is dimly present in 

Benacerraf‘s paper, but which gets lost or muddled in his later equivocations. This notion gets 

rid of the theory-independence and self-standing character of the Fregean object and, instead, 

suggests that mathematical objects, at least, are essentially related to the structures of which they 

are components. Or, more simply put, Shapiro writes that ―mathematical objects are tied to the 

structures that constitute them‖.
74

 This can seem somewhat arcane if we limit ourselves to 

mathematics, but Shapiro marshals several down-to-earth examples to help explicate the 

position. Most of us are, presumably, quite happy saying that the game of chess exists, in some 

                                                 
73

 [Shapiro 1997, 81]. 
74

 [Shapiro 1997, 80]. 



186 

 

sense. Now, whatever metaphysical spin we want to give to this claim, it seems reasonable to 

admit that, if the game of chess did not exist, then the white queen‘s bishop could not exist 

either. Equally, most of us would probably agree that it doesn’t make any sense to say that the 

white queen‘s bishop exists unless we already accept that the game of chess exists. None of this, 

according to Shapiro, ought to convince us that the white queen‘s bishop is not an object. For 

him, it is indeed an object, but one whose existence depends upon the existence of a structure we 

call ‗the game of chess‘. So, in contrast to Benacerraf, Shapiro is quite happy to hold that the 

elements of even abstract structures can be understood as objects—only they are objects whose 

existence is dependent upon the structures which they constitute. 

 

But how exactly are objects related to the structures of which they are the elements? What does 

Shapiro mean when he uses the term ‗constitute‘? Shapiro‘s own position on this question has 

been refined in the last two decades. In his best-known work—his 1997 book Philosophy of 

Mathematics: Structure and Ontology—Shapiro gives some of the metaphysics behind his use of 

the language of constitution to describe the relationship between mathematical objects and 

abstract structures. There he writes that 

 

Each mathematical object is a place in a particular structure. There is thus a certain priority 

in the status of mathematical objects. The structure is prior to the mathematical objects it 

contains, just as any organization is prior to the offices that constitute it. The natural-

number structure is prior to the offices that constitute it [i.e., the natural numbers].
75

 

 

Here we see the core of his view. Mathematical objects are places in structures, and they depend 

for their existence on the prior existence of the structure. It is in this sense that they ‗constitute‘ 

the structure, while also being dependent upon it for their existence. The chess analogy is again 

helpful: while the chess-independent existence of the white queen‘s bishop makes no sense, 

similarly, the game of chess is constituted in part by the place of the white queen‘s bishop. 

Without this place, we have a significantly different structure (or, perhaps, no structure at all). 

 

Here we are reminded of Frege‘s criticisms of Hilbert‘s independence and consistency proofs. 

For Hilbert, these proofs involved the provision of models of the ‗same‘ content. For Frege, such 

proofs involved the equivocation between distinct contents. Here we see that this feature of 

Frege‘s critique has an analogue in Shapiro‘s system. Structures are constituted by their places 

and relations. If we change either of these, we do not alter the structure but begin speaking about 

a different structure. This seems to contradict the everyday intuition that we can still speak of the 

game of chess or baseball despite the significant alterations in their rules over time. To what 

extent this intuition can be recaptured by Shapiro via descriptions of interrelations among 

distinct structures is an open question. 

 

In a paper from 2008, Shapiro refines his understanding of the relation of constitution which 

holds between structures and their objects. He writes: 

 

Places are components of universals. Each ante rem structure consists of some places and 

some relations. If this makes metaphysical sense […] then the dependence relation in the 

                                                 
75

 [Shapiro 1997, 78]. 



187 

 

slogans for ante rem structuralism is that of constitution. A structure is constituted by its 

places and its relations, in the same way that any organization is constituted by its offices 

and the relations between them. The constitution is not that of mereology. It is not the case 

that a structure is just the sum of its places, since in general the places have to be related to 

each other via the relations of the structure. I think of an ante rem structure as a whole 

consisting of, or constituted by, its places and its relations.
76

 

 

And, further, that 

 

the metaphysical view is that a structure is constituted by its places and its relations. 

Neither the places nor the relations are prior to the other. Again the slogan that the objects 

depend on the structure can be understood in that spirit. The dependence is constitution.
77

 

 

Here we get a fuller sense of the view. A structure as a whole is constituted by both relations and 

places. The relations are not metaphysically prior to the places, nor are the places metaphysically 

prior to the relations (in fact we can‘t really understand either in isolation: what is the white 

queen‘s bishop without its allowable moves, what are the moves without the object?). It is only 

both together which constitute a complete structure, and, according to the last-quoted sentence, 

both are dependent for their existence on the structure which they constitute.
78

 

 

So, abstract and not just concrete structures contain objects as their elements. We must now ask: 

for Shapiro, what is the relationship between the objects of an abstract structure A and the objects 

of a system B which exemplifies A? This is a fairly difficult question and requires an 

examination of some of Shapiro‘s terminology. 

 

2.3.2 Places, positions, offices, and objects 

 

In the above quotations, we‘ve seen Shapiro refer to the elements of abstract structures by a 

number of terms: ‗places,‘ ‗offices,‘ and ‗objects‘. Elsewhere he also uses the term ‗position‘. 

For Shapiro, all these terms are, in a sense, inter-translatable. Nevertheless, there are some 

subtleties which will help us understand the relationship between the objects of abstract and 

concrete structures. Shapiro uses ‗place‘ and ‗position‘ as catchall terms which refer in general to 

components of abstract structures which are not themselves relations. He does not give precise 

definitions of these terms but, instead, provides elucidations through a wide variety of 

mathematical and non-mathematical examples. Much as Frege employs the primitive notion 

‗object‘ Shapiro suggests that ―‗places‘ are indeed simple, or atomic, in the sense that they do not 

themselves have places or other components. ‗Place‘ is a primitive of structure theory.‖
79

  

 

We‘ve already established that, for Shapiro, places are objects. There are a few other 

characteristics of places which are important. First, we must understand places both as objects (in 

                                                 
76

 [Shapiro 2008, 19] 
77

 [Shapiro 2008, 20]. 
78

 Though, again, note that [Shapiro 1997] countenances degenerate cases of structures which contain no relations 

(his ‗cardinal structures‘) or no elements (his ‗0 pattern‘). 
79

 [Shapiro 2008, 21]. 



188 

 

the above sense) and as ‗offices‘. What does it mean for a place to be understood as an office? 

Shapiro writes: 

 

The structures studied in mathematics are what I call free-standing. Anything at all can 

occupy their places. So perhaps it is not amiss to think of a place in an ante rem structure 

as itself a universal, albeit a rather trivial one. Each place of each structure can be 

instantiated by anything at all, in a system that exemplifies the structure. Perhaps a place in 

a structure is just the universal ‗thing‘ or ‗object‘ or ‗particular‘.
80

 

 

The language here deliberately echoes that of Hilbert: Shapiro begins by highlighting the fact 

that abstract structures are like mobile bits of scaffolding which can be laid over other parts of 

reality via exemplification, modelling, or instantiation. Here, too, Shapiro suggests, without 

using the term, exactly what it means for a place to be an office. If we are discussing the abstract 

natural number structure directly, then we can treat the number ‗2‘ as a structure-dependent 

object. This is the places-as-objects perspective, and it is the one that Shapiro believes is 

employed most often in pure mathematics. But the chief characteristic of abstract, versus 

concrete structures, is that their places possess no extra-structural properties, which allows these 

abstract structures, in turn, to be instantiated in or exemplified by other structures—Hilbert‘s 

‗scaffolding of concepts‘. When we note, for example, that Caesar, my right index finger, the set 

{{Ø}}, or any other object at all can represent the number 2 within a suitably defined concrete 

structure, we are treating the places of the abstract natural number structure as offices. That is to 

say, we recognize that the very character of the object we call ‗the number 2‘ involves the 

capacity to be represented or exemplified by any object whatsoever. This is fairly straightforward 

if we think of offices in the more familiar sense: the office of the registrar is a position with 

determinate relations to other positions within the university‘s administrative structure – in this 

sense it is a structure-dependent object. But it is of the essence of any bureaucratic office that it 

can be occupied or exemplified or represented by different people at different times.
81

  

 

The chief insight of Shapiro‘s position here is that places are always both offices and objects, 

though different perspectives may highlight one or the other aspect of their nature. From the 

                                                 
80

 [Shapiro 2008, 23]. In reading this passage, recall from above Shapiro‘s statement that ―In mathematics, at least, 

the notions of ‗object‘ and ‗identity‘ are unequivocal but thoroughly relative‖ [Shapiro 1997, 81]. [Shapiro 2008] 

speaks of places in structures as being capable of exemplification or representation by any object whatsoever (as 

long as the object is arranged in a suitable concrete structure, i.e., a system). It seems to me that Shapiro‘s remark 

here that we might treat places simply as the universal ‗object‘ implies a notion of object which is not in fact 

relative, but which encompasses all objects from whatever system we like. Despite the Hilbert-style approach to 

structures, this notion of places that accept any objects as occupants is strikingly reminiscent of Frege‘s 

understanding of first-level functions. As we will see in what follows, this similarity perhaps runs deeper than is 

obvious here. 
81

 Of course the office of registrar, unlike the office of the natural number 2, cannot be occupied by any object 

whatsoever. An air conditioner, for instance, cannot fulfill the duties of registrar, though it might very well act as a 

perfectly adequate number 2. Shapiro also attempts (at [Shapiro 1997, 98-99]) to distinguish between abstract and 

concrete structures on the basis of the sort of relations they contain. He does so using a version of Alfred Tarski‘s 

(cf., [Tarski 1986]) interesting distinction between the logical and the nonlogical. For Shapiro, the reason why an air 

conditioner or ‗any old object‘ is incapable of acting as a registrar is that there are implicit non-logical (or non-

formal) constraints on the allowable relations (e.g., the registrar must be a sentient, physical being). 



189 

 

perspective of a mathematician studying an abstract structure directly, the places of the structure 

appear to be (and are, according to Shapiro) bona fide objects. From the perspective of someone 

modelling an abstract structure, or applying it, the elements of the structure are best understood 

as offices, which can be occupied by any object whatsoever. The key thing to keep in mind is 

that the relationship of occupation or exemplification here is not the same as identity, even 

though a place might be said to exemplify itself.
82

 When I occupy or exemplify the number 2, I 

do so relative to a particular concrete structure. The difference between myself as an occupant of 

the 2-place in this structure and the number 2 of the abstract natural number structure is that I 

still possess extra-structural properties even when I exemplify the number 2. To be sure, we 

ignore these properties when I occupy the position, but, in the case of the actual number 2, the 2-

place of the abstract natural number structure, there are simply no properties to ignore. 

 

2.3.3 Cross-structural identity 

 

Part of Benacerraf‘s reason for rejecting the view that numbers can be objects is that he found 

claims which sought to identify wildly different types of objects to be ‗unsemantical‘. Thus he 

claimed that if we want to identify a particular number with a particular set, we must first 

provide a well-defined context within which to do so. The general (i.e., non-relative) notion of 

‗object‘ does not provide such a context, so we cannot discuss identity relations between entities 

on this basis alone. If we read Benacerraf as a progenitor of structuralism, we can understand 

well-defined structures as a means of providing the relevant context for identifying (structure-

dependent) objects with one another. On this view, then, Benacerraf‘s rejection of Caesar-type 

questions as unsemantical amounts to the claim that we cannot make cross-structural identity 

claims like those apparently made between the natural number structure and the set-theoretic 

structure unless the objects of these structures are (perhaps tacitly) understood to be elements of 

a single structure, a single well-defined context. But of course the traditional way of providing 

this single structure has been to reduce one structure (the natural number structure) to another 

(the set-theoretic hierarchy). 

 

It seems that Shapiro follows Benacerraf in much of this. He writes, for instance that ―a 

philosophy of mathematics should show why these questions [i.e., ‗does Julius Caesar = 2?‘] 

need no answers, even if the questions are intelligible‖
83

 and that ―Identity between natural 

numbers is determinate; identity between numbers and other sorts of objects is not, and neither is 

identity between numbers and the positions of other structures‖.
84

 And, as we‘ve already seen, he 

holds that ―In mathematics, at least, the notions of ‗object‘ and ‗identity‘ are unequivocal but 

thoroughly relative‖.
85

 

 

Given these views, one would expect that Shapiro would follow Benacerraf‘s conclusion as well, 

namely that we cannot make cross-structural identity claims. For, without a determinate context 

                                                 
82

 ―When we switch back to the places-are-objects perspective—and construe the places of a given structure as 

objects—we see that each structure exemplifies itself. That is, the relations of the structure hold of its places—when 

the latter are construed as objects‖ [Shapiro 2008, 24]. 
83

 [Shapiro 1997, 79]. 
84

 [Shapiro 1997, 79]. 
85

 [Shapiro 1997, 81]. 



190 

 

(i.e., a well-defined structure), it does not make any sense to speak of objects or an identity 

relation in the first place. But he does not follow Benacerraf‘s conclusion. Instead, he writes: 

 

I do not wish to go as far as Benacerraf in holding that identifying positions in different 

structures (or positions in a structure with other objects) is always meaningless. On the 

contrary, mathematicians sometimes find it convenient, and even compelling, to identify 

the positions of different structures. […] For [example], it is surely wise to identify the 

positions in the natural-number structure with their counterparts in the integer-, rational-, 

real-, and complex-number structures. […] The point here is that cross-identifications like 

these are matters of decision, based on convenience, not matters of discovery.
86

 

 

So, Shapiro wants to allow that we can make meaningful claims about identity between objects 

in distinct structures but that this is, in some sense, a product of a decision. I‘m not entirely sure 

what he can mean here, and what the contrast with Benacerraf‘s position is. Perhaps the 

‗decision‘ is to combine our distinct theory-bound notions of identity and object into another, by 

defining (at least tacitly) a more inclusive structure. For, if the claim of identity is to be a 

meaningful one, then it will have to be unequivocal and relative to a particular structure (on 

Shapiro‘s view). Whether or not we decide this to be the case seems to be beside the point, for 

irrespective of the question of decision or discovery, the identity claim will require relativization 

to a particular structure. Note, too, that Shapiro is committed to the independent existence of 

structures (i.e., to their existence regardless of our decisions about them).
87

 Without further 

elaboration on Shapiro‘s part, then, it seems slightly misleading to suggest that the objects of 

different structures can be identified with one another, as such identification requires a kind of 

equivocation between distinct notions of object and identity. I think Shapiro‘s refusal to accept 

the impossibility of cross-structural identification in fact belies a deeper commitment, on his 

part, to precisely the kind of absolute, universal background ontology which structuralism is (at 

least putatively) concerned to avoid. I think we can make this point most clearly if we turn now 

to an examination of Shapiro‘s account of abstract structures themselves. 

 

2.3.4 Shapiro on structures 

 

So, what are structures for Shapiro, exactly? 

 

In our discussion of objects, we‘ve already gotten a bit of insight into the nature of structures for 

Shapiro. They are, as we‘ve seen, constituted by their elements, i.e., their places, and relations on 

those objects. As far as abstract structures go, the nature of these objects does not matter at all, 

only that they stand in the appropriate relations with one another.  Shapiro gives a slightly more 

formal definition of structure along these lines: 

 

a structure has a collection of places and a finite collection of functions and relations on 

those places. The isomorphism relation among structures and the satisfaction relation 

                                                 
86

 [Shapiro 1997, 81]. 
87

He writes, for example, that ―Structuralists hold that a nonalgebraic field like arithmetic is about a realm of 

objects—numbers—that exist independently of the mathematician‖ [Shapiro 1997, 72] and, in a later paper, that 

―the structure itself is a chunk of reality‖ [Shapiro 2005, 67]. 



191 

 

between structures and formulas of an appropriate formal language are defined in the 

standard way. We could stipulate that the places of different structures are disjoint, but 

there is no reason to do so.
88

 

 

In relation to our earlier discussion of the possibility of cross-structural identity claims, the last 

sentence of the above quotation is quite revelatory. He claims that we could, if we so desired, 

stipulate that the places of each structure are distinct from the places of any other structure. This, 

of course, would be in keeping with Benacerraf‘s dictum that there can be no cross-structural 

identity claims. For Benacerraf, then, there is a very good reason to stipulate (or at least 

recognize) that the places of different structures are disjoint. But, as we see here, Shapiro holds 

that there is no reason to do so. Why? 

 

Though it is not made explicit here, I suggest (and will argue in what follows) that, for Shapiro, 

the reason that we need not explicitly state that the places of different structures are disjoint is 

because they are not in fact disjoint on Shapiro’s account. He has suggested that the notion of 

‗object‘ (and, thus, that of ‗place‘ as well) is relative to a particular structure. Or, more broadly, 

he argues that ontology is relative to a given structure. But here he seems to implicitly ignore this 

relativization by suggesting that places can be ‗shared‘ between distinct structures. How is this 

possible, given his previous account? It is possible, I will argue, because there is, ultimately, only 

one ontology for Shapiro, and the function of his structure theory is to describe this ontology. 

Indeed, the only point in having a theory of structures at all is to provide this background 

ontology. 

 

2.3.5 The ontology of structures 

 

As we‘ve seen, structuralism in the broadest terms involves the attempt to move beyond the 

uniquely fixed ontology of set-theoretic foundationalism, and, instead, to suggest that 

mathematics is the science of structure. Thus, it would be rather surprising if a dyed-in-the-wool 

structuralist like Shapiro simply recapitulated the traditional set- and model-theoretic foundations 

of mathematics.  

 

Yet, Shapiro takes his own axiomatic theory of structures to be loosely imitative of second-order 

Zermelo Fraenkel set theory,
89

 while also thinking of it as ―an axiomatization of the central 

framework of model theory‖.
90

 And, indeed, his few axioms mimic very clearly traditional 

formulations of axiomatic set theory.
91

 So, if Shapiro‘s theory is to differ significantly from the 

traditional set-theoretic foundationalism he wishes to combat, the difference cannot be solely in 

the form and content of his axioms. The difference will be in his philosophical account of the 

                                                 
88

 [Shapiro 1997, 93]. Presumably the ‗standard way‘ referred to here is that of model theory. In what follows, we 

will see that this implicit reliance on the model-theoretic approach is widespread in Shapiro‘s structure theory, as 

well as problematic for his goals. 
89

 [Shapiro 1997, 95]. 
90

 [Shapiro 1997, 93]. 
91

 For instance, in place of the usual set-theoretic axiom of infinite, which holds that there exists a set with an 

infinite number of members,, Shapiro‘s axiom of infinity holds ―There is at least on structure that has an infinite 

number of places‖ [Shapiro 1997, 93], in the place of the usual power set axiom, he has a power structure axiom, 

etc., etc. 



192 

 

point of these axioms, or of the nature of the items about which the axioms speak (e.g., 

structures). 

 

Like the traditional set-theoretic foundationalist, Shapiro accepts the necessity of a background 

ontology. In the traditional case, this ontology was the universe of set theory (and usually that of 

ZFC – Zermelo-Fraenkel set theory with the axiom of choice). On the traditional picture, this 

ontology is importantly non-structural: claims made about the ontology of ZFC are understood to 

be claims about absolute features of the world. This means that the fundamental notions of the 

ontology (notions like membership, for instance) are not ‗relative‘ but, instead, possess non-

structural properties just like an everyday object like my keyboard. This absolute ontology is 

exactly the sort of thing that structuralism is designed to oppose—as evidenced by the origins (of 

some forms) of structuralism in Benacerraf‘s critique of the viability of set-theoretic 

foundationalism. Benacerraf wanted to show that it was nonsensical to pin down mathematical 

structures like the natural number structure within the set-theoretic hierarchy. And, moreover, 

Skolem, Gödel, and Paul Cohen (among others) had by 1965 surely already shown that the very 

notions of set theory had to be relative themselves.
92

 

 

So in thinking about Shapiro‘s background ontology, we have a few questions to answer. 

 

(i) Why does Shapiro think it is necessary to presuppose a background ontology for 
structuralism? 

(ii) How does his background ontology differ from the absolute ontology of traditional 
set-theoretic foundationalism? 

 

Let‘s tackle these in order 

 

2.3.6 The necessity of a background ontology 

 

So, why do we need a background ontology in the first place? Why not eschew the universality 

so characteristic of set theory and instead allow for multiple distinct, disjoint ontologies? To 

answer this, we have to examine once more the role that the elements of structures play in 

Shapiro‘s account.  

 

Recall that Shapiro outlines two different (though ultimately inseparable) ways of understanding 

the places of structures: the places-as-objects view and the places-as-offices view. He suggests 

that, on the places-as-offices view, we must have enough objects to actually fill the offices of the 

structure. The structure, especially if it is to be ‗real‘ in Shapiro‘s sense, must be constituted by 

its objects and its relations. So we need objects to have a structure, and (for abstract structures at 

least) vice versa. This is why, in our discussion of any given structure, we need an ontology, 

however far in the background it might be when we actually get down to our mathematical work 

on the structure. 

                                                 
92

 Cf., [Skolem 1922], where the implications of Skolem‘s ‗paradox‘ are very clearly drawn. In two papers ([Cohen 

1963], [Cohen 1964]) Paul Cohen further dismantled the absolute picture of the set-theoretic hierarchy by producing 

models of ZFC within which the continuum hypothesis fails, alongside Gödel‘s models of ZFC wherein the 

continuum hypothesis is true. 



193 

 

 

We need a background ontology but, importantly, when we are dealing with mathematical (i.e., 

abstract) structures, this background ontology can contain any objects we please. Recall that the 

distinguishing feature of abstract structures is that the nature of their elements is a matter of 

indifference – any object at all can exemplify an element of the natural number structure (i.e., a 

number) as long as it stands in the appropriate relations to the other exemplars of the elements of 

the structure. So, the nature of the contents of the background ontology should not and does not 

matter for Shapiro‘s theory. Whether the objects are sets or hats or chimneysweeps or whatever, 

none of this matters for the background ontology. What does matter is that there are sufficiently 

many objects to fill all the places. And, since many of the interesting structures within 

mathematics are not only infinite but uncountably infinite, Shapiro‘s theory requires a 

background ontology which contains (at least) an uncountable infinity of objects as well. 

 

Now, as we‘ve seen, for Shapiro, the places of an abstract structure are also objects in their own 

right. This fact is embodied in what he calls the places-as-objects perspective. This means that 

we can simply populate the requisite background ontology of any given structure by the places of 

that very structure. The offices of the structure can be occupied by the objects of the structure 

(though, of course, they could be anything else as well). 

 

So far so good. We should now examine how this (necessary) background ontology differs from 

that of the traditional set-theoretic account which Shapiro‘s structuralism is meant to oppose. 

 

On the traditional set-theoretic view, the background ontology is populated by objects whose 

properties are not solely dependent upon a given structure. These can be either sets or urelements 

or both. In set theory (as in structure theory) we are interested in examining structures with 

infinitely many elements. Often, we are interested in structures with an uncountable infinity of 

elements. Thus, on this approach, our background ontology requires infinitely many structure-

independent objects. And so, in set theory, we have axioms which posit such entities. 

 

For Shapiro, since the elements of structures are not ‗structure-independent‘ in the way that sets 

and urelements are, this set-theoretic approach to the background ontology will not do. But, like 

the traditional set-theoretic foundationalist, Shapiro still wants to examine structures with 

infinitely many elements. Again, recall that, for Shapiro, structures are prior to their elements, 

and these elements are entirely dependent upon the structures they constitute. So, if we want 

structures with infinitely many elements, we cannot begin by postulating infinitely many objects 

to act as the elements of the structure. Instead, Shapiro suggests, we begin by postulating 

structures with infinitely many places. The ‗background ontology‘ is thus not filled with 

structure-independent objects like sets or urelements but with structures themselves. And, since 

the places of structures can occupy themselves, the existence of infinite structures guarantees the 

existence of infinite systems which exemplify those structures. 

  

On the traditional dispensation, set theory itself is not a structural theory, insofar at least some of 

its elements are understood to be ‗concrete‘. Shapiro‘s view ostensibly allows for a more 

thoroughly structural approach by insisting that the elements of his ontology (i.e., structures) are 

themselves understood in structural terms. 

 



194 

 

This approach to the background ontology is meant to improve not only on the traditional set-

theoretic approach to structures, but also to solve a problem to which eliminative forms of 

structuralism are prone.
93

 The problem, as Shapiro sees it, is one of infinite oscillation or regress 

between systems and structures. The eliminative structuralist qua structuralist wishes to study 

structures. In order to do so, a background ontology is employed which is itself non-structural. 

But, given the eliminative structuralist‘s general structural views, this background ontology itself 

exhibits a structure. This structure again requires a background ontology, which again will 

exhibit another structure, and on and on. Shapiro‘s solution aims to ―stop the regress of system 

and structure at a universe of structures‖.
94

  

 

Shapiro‘s putative solution to this regress is to identify structure and system one with the other. 

But does his solution actually stop the regress? There is good evidence that, in the final analysis, 

it does not. Let‘s take a look. 

 

2.3.7 Stop the regress I want to get off 

 

A good way into the problem is to examine Shapiro‘s criticisms of the views which he believes 

are subject to this regress. Of a set-theoretic approach to structuralism, he writes that  

  

The crucial feature of this version of eliminative structuralism is that the background 

ontology is not understood in structuralist terms. If the iterative hierarchy is the 

background, then set theory is not, after all, the theory of a particular structure.
95

 

 

In short, this approach is not a structural one, because it ends up being founded upon a theory 

whose very basis is ‗concrete‘, i.e., one whose elements have non-structural properties. The 

structural aspect of mathematics is actually founded upon a system. He makes much the same 

argument about the modal form of structuralism which is developed by Geoffrey Hellman. For 

Hellman, structures are not the final item on the ontological menu, but are instead eliminated in 

favour of the consideration of possible systems. The problem, as Shapiro sees it, is that the usual 

way of dealing with logical possibilities (particularly in the kind of modal language that Hellman 

employs) is to characterize them in terms of the set-theoretic hierarchy as employed in model 

theory. Shapiro writes that 

 

Hellman accepts this straightforward point, and so he demurs from the standard, model-

theoretic accounts of the logical modalities. Instead, he takes the logical notions as 

primitive, not to be reduced to set theory.
96

 

 

                                                 
93

The term ‗eliminative structuralism,‘ as Shapiro characterizes it, covers any form of structuralism which 

‗eliminates‘ structures in favour of some background ontology which does not itself contain structures. Thus, for 

instance, Hellman‘s modal eliminative structuralism employs a background ontology of possible systems. 
94

 [Shapiro 1997, 92]. The details of this regress are not made clear by Shapiro, nor does he give a clear account of 

why the eliminativist can‘t simply model the structure of the background ontological system in that system itself. 
95

 [Shapiro 1997, 87]. Shapiro calls any version of structuralism which eschews direct quantification over structures 

in favour of some other ontology ‗eliminative‘. Both the traditional set-theoretic approach to structures and Geoffrey 

Hellman‘s modal account of structures fall under this heading for him. 
96

 [Shapiro 1997, 89]. 



195 

 

But, Shapiro continues later in the work,  

 

because Hellman is out to drop the realist perspective, it is not clear why he is entitled to 

the traditional model-theoretic explications of the modal operators of logical necessity and 

logical possibility. For example, the usual way of establishing that a sentence is possible is 

to show that it has a model. For Hellman, presumably, a sentence is possible if it might 

have a model (or if, possibly, it has a model). It is not clear what this move brings us.
97

 

 

On the readings Shapiro offers here, his chief criticism of ‗eliminative‘ forms of structuralism is 

that they inevitably end up relying on a non-structural ontology of some kind. In the case of the 

set-theoretic approach to structuralism, this is very straightforwardly the ontology of set theory. 

In Hellman‘s case, rather more circuitously, he ends up relying (says Shapiro) on the very same 

ontology simply by his unexplained reliance on the usual model-theoretic semantics for 

modality. 

 

In response to these views, he writes that 

 

A structuralist might be tempted to step back from this competition of background theories 

and wonder if there is a structure common to all of them. However, on the ontological 

option, this temptation needs to be resisted. The structures studied in two theories can be 

compared only in terms of a more inclusive theory.
98

 

 

The ‗ontological option‘ here is the option of embedding abstract structures within a particular, 

determinate ontology. But, again, if the goal is to understand structures, and multiple available 

ontologies seem to ‗model‘ these structures equally well, we have no reason to choose between 

one or the other. And, moreover, since our interest is in the structures themselves, the use of an 

ontology whose elements possess non-structural properties is ultimately opposed to the 

structuralist programme. All of this simply recapitulates, perhaps on a grander scale, the non-

uniqueness problem we began with. So Shapiro needs an ontology which stops the regress, but 

which does not involve the reduction of structures to a concrete or non-structural background 

ontology. 

 

The obvious solution, and the one which we‘ve already mentioned, is to treat structures 

themselves as the background ontology by developing a theory of structures. The difficult part 

here will be in developing an account of structures which does not itself rely on non-structural 

notions. 

 

 

2.3.8 Shapiro‘s structure theory 

 

So Shapiro sets out to develop just such a theory. Note that he is quite happy to claim (and he 

gives compelling arguments supporting this claim) that his structure theory is intertranslatable 

                                                 
97

 [Shapiro 1997, 229]. 
98

 [Shapiro 1997, 87]. 



196 

 

with both the set-theoretic picture and Hellman‘s modal/eliminative approach. So what is the 

difference? 

 

Well, what he means by intertranslatable is not that they say the exact same thing (for he doesn‘t 

want to make any claims about non-structural entities like the traditional set-theorist‘s non-

structural sets). He writes 

 

there are several ways to render structuralism in a rigorous, carefully developed 

background theory, and there is very little to choose among the options. In a sense, they all 

say the same thing, using different primitives. The situation with structuralism is analogous 

to that of geometry. Points can be primitive, or lines can be primitive. It does not matter 

because, in either case, the same structure is delivered. The same goes for structuralism 

itself. Set theory and structure theory are equivalent in the sense defined above. To speak 

loosely, the same ‗structure of structures‘ is delivered. Modal structuralism also fits, once 

the notion of ‗equivalence‘ is modified for the modal language.
99

 

 

So, it isn‘t that they make exactly the same claims, but that they, ultimately, say the same things 

about structures. The key difference between them, according to Shapiro, is that his ontology 

never bothers to make the detour through a concrete system of sets or possible systems: it stays 

with structures the whole time. 

 

2.3.9 Quasi-concreteness 

 

But, near the close of this chapter on ontology, we find Shapiro making some rather remarkable 

claims about the relationship between his theory of structures and ‗concreteness‘ (or, what 

amounts to the same thing for our purposes, ‗quasi-concreteness‘). The term ‗quasi-concrete‘ 

Shapiro borrows from a 1990 paper by Charles Parsons. In that paper, Parsons presents a 

sustained argument against the viability of a purely structural account of mathematics like the 

one ostensibly offered here by Shapiro. On the basis of the importance he accords to the role of 

quasi-concrete objects and concepts within mathematics, Parsons concludes finally that 

 

if the structuralist view of mathematical objects is taken to mean that all mathematical 

objects are only structurally determined, it has to rest on legislation about what counts as a 

mathematical object. The explanatory and justificatory role of more concrete models 

implies, in my view, that it is not the right legislation even for the interpretation of modern 

mathematics.
100

 

 

Given this conclusion, it is somewhat surprising (as will be shown presently) that Shapiro seems 

to endorse much of Parsons‘ argument here, recognizing, in particular, that his own ante rem 

structuralism relies inescapably on the quasi-concrete. For Parsons, an entity is quasi-concrete if 

it is  

 

                                                 
99

 [Shapiro 1997, 97]. 
100

 [C. Parsons 1990, 338]. 



197 

 

directly ‗represented‘ or ‗instantiated‘ in the concrete. Examples might be geometric 

figures (as traditionally conceived), symbols whose tokens are physical utterances or 

inscriptions, and perhaps sets or sequences of concrete objects. The ‗concrete objects‘ that 

David Hilbert talked about in his accounts of intuitive, finitist mathematics are in my 

terminology quasi-concrete. A purely structuralist account does not seem appropriate for 

quasi-concrete objects, because the representation relation is something additional to 

intrastructural relations. Because they have a claim to be the most elementary mathematical 

objects, and also for other reasons, quasi-concrete objects are important in the foundation 

of mathematics.
101

 

 

This notion of the quasi-concrete is perhaps more widespread within mathematics than it might 

appear at first glance, for any proof or theorem which refers to or quantifies over, e.g., the 

formulas of a formal language will, on this dispensation, be quasi-concrete. In any case, the 

precise extension of the notion is not important for us here. What are important are the 

conclusions that Shapiro draws about his own theory, under the influence of Parsons‘ notion. 

 

After his discussion of Parsons, Shapiro gives three distinct reasons why his own theory of 

structures cannot but be understood, at least partially, in quasi-concrete terms. These reasons are: 

 

(i) His (crucial) account of the concept of coherence for structures is implicitly 
motivated/justified by quasi-concrete systems. 

(ii) His definition of ‗structure‘ itself entails reference to quasi-concrete 
entities/procedures. 

(iii) His use of a second-order language involves the use of quasi-concrete notions. 
 

Rather than enter into an examination of Shapiro‘s (admittedly problematic) notion of coherence 

for structures, I will examine the implications of the quasi-concreteness pertaining to the 

definition of ‗structure‘ and his use of a second-order language. Regarding his definition of 

‗structure,‘ he writes 

 

Another reason to think that the quasi-concrete cannot be eliminated is that I have appealed 

to quasi-concrete items in order to define the very notion of a structure. Recall that a 

structure is the form of a system, and a system is a collection of objects under various 

relations. The notion of ‗collection‘ is an intuitive one. There is something fishy about 

appealing to the set-theoretic hierarchy, a freestanding ante rem structure, in order to 

explicate the notion of ‗collection‘ in the characterization of ‗system‘ and thus ‗structure‘. 

Where did we get on this merry-go-round, and how do we get off?
102

 

 

Here we see that the problem of the quasi-concrete ultimately puts him back into the regress he 

sought to escape. His way of escaping the regress was to put structures directly into his ontology. 

But in his definition of structure, he employs the notion of ‗collection‘. This, according to his 

own rendering here, is a quasi-concrete notion which relies directly on our intuitive 

understanding of collecting (concrete) items together. So it seems that his attempt to characterize 

                                                 
101

 [C. Parsons 1990, 304]. 
102

 [Shapiro 1997, 105].  



198 

 

the set-theoretic hierarchy as a ‗freestanding ante rem structure‘ actually involves the more 

usual, intuitive notion of sets as concrete collections. Thus, the mechanism of his escape from the 

concrete account of mathematics itself involves an inescapable reference to the concrete. This, 

on its own, seems deeply troubling for his purely structuralist account. 

 

But he also notes that his use of a second order language involves the quasi-concrete. He writes 

 

A related point concerns the practice of characterizing specific structures using a second-

order language. Such languages make literal use of intuitive notions like ‗predication‘ or 

‗collection‘. A crucial step in the defense of second-order languages is that we have a 

serviceable, intuitive grasp of notions like ‗all subsets‘. This notion is also quasi-

concrete.
103

 

 

Thus, another deep problem emerges in the very language and logic used to describe his 

structuralist programme. In the face of the problem of ineliminable reference to quasi-concrete 

objects and notions, Shapiro writes 

 

In all cases, then, the conclusion is the same. We can try to hide the quasi-concrete, but 

there is no running away from it. […] However, the caveat does not undermine the main 

ontological thesis of ante rem structuralism, the idea that the subject matter of a branch of 

pure mathematics is well construed as a class of freestanding structures with formal 

relations. The role of concrete and quasi-concrete structures is the motivation of structures 

and the justification that structures with certain properties exist. The history of 

mathematics shows a trend from concrete and quasi-concrete systems to more formal, 

freestanding structures. There is no contradiction in the idea of a system of quasi-concrete 

objects‘ exemplifying a freestanding ante rem structure.
104

  

 

What are we to make of all this? 

 

It strikes me that Shapiro‘s final claim here, that there is no contradiction in his use of quasi-

concrete objects and freestanding systems might very well be true. But so what? The truth is that 

he has not given us freestanding structures, precisely because his definition of ‗structure,‘ as well 

as the language used to construct the theory of structures, both necessarily involve the use of 

quasi-concrete notions. Thus, the resulting structures are themselves quasi-concrete. If the whole 

endeavour of his particular form of structuralism was to avoid the pitfalls of an absolute 

ontology, and, instead, to develop an ontology of structures which contains no non-structural 

entities of any kind, then these admissions indicate that he has, rather straightforwardly, failed. I 

know of no other way to read them. If Shapiro was content to sweep the essential ontological 

role of these quasi-concrete objects under the rug, then the detour into ‗pure‘ structuralism was 

ill-motivated in the first place: we might just as well have ignored Benacerraf‘s qualms and stuck 

with traditional set-theoretic foundationalism from the beginning, ignoring ontological concerns 

along the way. 

 

                                                 
103

 [Shapiro 1997, 105]. 
104

 [Shapiro 1997, 105-106]. 



199 

 

In the end, then, what Shapiro‘s admission of the concreteness of his system shows is that the 

assertoric form of structuralism (exemplified in, at least, the work of Shapiro and Hellman) is, 

ultimately, another form of reductionism, which proposes to give us a new array of ontologies, 

but no good reason to choose between them. After all, they are intertranslatable on Shapiro‘s 

reading. If our statements must be about some particular non-structural ontology, but we can‘t 

choose which one, it seems as though we are not capable of picking out what mathematics is 

really about. At best, we have a variety of things that it might be about, and this, surely, is an 

insufficient response to Benacerraf‘s challenge. If we take the assertoric approach to 

structuralism, then, we still have no clear answer to the sort of non-uniqueness problems faced 

by Frege and the set-theoretic reductionists. 

 

2.4 Algebraic structuralism 

 

2.4.1 Categories 

 

Before we get to the philosophical content of algebraic structuralism, let‘s begin by briefly 

examining some of its mathematical underpinnings in category theory. What exactly is a 

category? In a moment I will provide a formal definition of the concept. Before this however, it 

will be useful to explain what categories are meant to do. Or, perhaps better, what sorts of 

mathematical activity they are intended to facilitate. 

 

In some ways, the notion of a category itself is a secondary matter in category theory. The chief 

aim of category theory is to provide a means of ‗translating‘ between different mathematical 

structures. The emphasis is often on the translations themselves, rather than on the categories 

(which can themselves be understood as object-like or morphism-like as we see fit). The point of 

category theory, at least in its initial incarnations, was to provide a theory of transformations 

between different types of structures. These transformations are called ‗functors‘ and category 

theory might have been better named ‗functor theory‘. In one of the founding documents of 

category theory—Eilenberg and Mac Lane‘s 1945 paper, ―General Theory of Natural 

Equivalences‖—the authors have this to say about the relative priority of categories and functors: 

 

the whole concept of a category is essentially an auxiliary one; our basic concepts are 

essentially those of a functor and of natural transformation […]. The idea of a category is 

required only by the precept that every function should have a definite class as domain and 

a definite class as range, for the categories are provided as the domains and ranges of 

functors. Thus one could drop the category concept altogether and adopt an even more 

intuitive standpoint, in which a functor such as ‗Hom‘ is not defined over the category of 

‗all‘ groups, but for each particular pair of groups which may be given.
105

  

 

And, further, that 

 

Perhaps the simplest precise device would be to speak not of the category of groups, but of 

a category of groups (meaning, any legitimate such category). A functor such as ‗Hom‘ is 

then a functor which can be defined for any two suitable categories of groups […]. Its 

                                                 
105

 [Eilenberg and Mac Lane 1945, 247]. 



200 

 

values lie in a third category of groups, which will in general include groups in neither [of 

the first two categories
106

 

 

I doubt that today many category-theorists would like to entirely dispense with the notion of 

category, but the emphasis of the above remarks is nevertheless illustrative of the general 

character of category theory today: the nature of the categories and their objects doesn‘t matter, 

it‘s the transformations between them (and the transformations between transformations, and so 

on) which reveal new and interesting mathematical structure. So, despite the importance of the 

notion of category, keep in mind in relation to the following definition that categories are best 

understood in relation to certain types of transformations.  

 

With that stage-setting out of the way, we can now ask again: what exactly is a category? 

 

I will give a definition of the term presented by Steve Awodey. Others could have been used,
107

 

but Awodey‘s is concise and manageable in that it does not involve any dependence on other 

mathematical terminology. He writes: 

 

A category by definition […] consists of objects A, B, C, … and morphisms f, g, h, … such 

that: (i) every f has a unique domain A and a unique codomain B, written f: A→B; (ii) given 

any g: B→C there is a unique composite g ∘ f: A→C, with composition being associative; 
(iii) each B has an identity 1B: B→B which is a unit for composition, i.e., 1B ∘ f = f and g ∘ 
1B for any f and g as stated. A category is anything satisfying these axioms.

108
 

 

For our purposes here, there are two relevant features of Awodey‘s characterization. First: a 

category contains both objects and morphisms (which are also called maps, mappings, or 

arrows). This already differentiates it immediately from the traditional account of set theory, 

which contains only objects, i.e., sets (or, possibly urelements as well) as basic elements. The 

second thing to note is that a category is anything at all that satisfies these axioms.  Implicitly, 

for instance, a category might contain Frege‘s functions as its objects, much to his horror. Since 

these axioms give very few constraints on what can count as morphisms and no direct constraints 

at all on objects, category theory begins with a fairly indeterminate ‗ontology‘ if it begins with 

one at all. The resulting generality means that category theory can describe an incredible array of 

structures. A few characteristic examples will help illustrate this point. 

 

1. The category Set, whose objects are sets and whose morphisms are functions between 
sets. 

2. The category Cat, whose objects are categories and whose morphisms are functors 
between categories. 

3. The category Top, whose objects are topological spaces and whose morphisms are 
continuous maps.  

4. The category K-Vect whose objects are vector spaces over a fixed field K and whose 
morphisms are K-linear transformations 

                                                 
106

 [Eilenberg and Mac Lane 1945, 247]. 
107

 E.g., [Mac Lane 1998, 7ff.] or [Lawvere and Schanuel 1997, 21]. 
108

 [Awodey 1996, 212-213]. 



201 

 

 

These are some of the better known examples of categories. We can also think of categories in 

relation to what we would normally call logical structures. Thus, for instance, any given 

deductive system can be understood as a category whose objects are the formulas of the system 

and whose morphisms are derivations or proofs of those formulas.
109

 Taken on their own, this 

definition and these examples don‘t yet tell us very much about what makes categories 

interesting. To see why many have thought of category theory as a new way of doing 

mathematics, we have to see categories in action.   

 

And for this, we need the crucial notion of a functor. 

 

2.4.2 Functors 

 

The term ‗functor‘ was taken by Eilenberg and Mac Lane from the linguistic work of Rudolf 

Carnap.
110

 Though Carnap‘s philosophical views seem (in some respects) in line with the 

pluralist approach of many category-theoretically minded philosophers of mathematics, his 

conception of a functor is somewhat different than the category-theoretic notion. In category 

theory, a functor is a homomorphism, or structure-preserving morphism, between two categories. 

More formally, a functor F: A → B has the following properties: 

 

1. To each object X of the category A it associates an object FX of the category B 
2. To each morphism f of the category A it associates a morphism Ff of the category B 
3. For each object X of category A, F1X = 1FX 

4. For any maps f and g of the category A such that f: X→Y and g: Y→Z, we have F(f ∘ g) = 
Ff ∘ Fg 

 

In other words, the functor is defined for all the objects and morphisms of the categories in 

question, and it preserves identity, composition, and the domain/codomain structure. Note here 

that the categories A and B needn‘t be distinct. Thus for instance we have ‗endofunctors‘ which 

simply map a category to itself—a specific example of which are identity functors, which 

associate every object with itself and every morphism with itself.  Identity functors are usually 

not of much interest, but there are other functors from categories to themselves which are of 

mathematical interest. For instance, if we take the category Set, we can define a ‗power set 

functor‘ P: Set→Set which maps each set to its power set and each function to its image. In fact, 

some of the most interesting functors are morphisms from a category into or onto itself.  

 

So, what is philosophically interesting about functors? Intuitively, functors provide us the means 

to ‗move between‘ different categories while examining what we might call their ‗shared 

structure‘. In a certain sense, they might be said to transform categories back and forth into one 

another. We might also say that they allow us to examine bits of shared mathematical structure 

without getting bogged down in conceptual concerns about the ‗nature‘ of a given category –that 

                                                 
109

 This basic insight lies at the heart of the field of categorical logic, first developed in [Lawvere 1963] and 

[Lawvere 1964]. 
110

 Though similar in many respects, Carnap‘s functors are defined only for linguistic expressions. See [Carnap 

2002, 14]. 



202 

 

is, they allow us a kind of indifference to the particular sorts of things which count as objects and 

morphisms in our categories. Understood in this way, functors can be a useful conceptual tool for 

a structuralist approach to mathematics. 

 

In addition to their capacity to isolate shared structure, functors can also be employed to reveal 

or construct new structure. This feature is perhaps best understood via the notion of a functor 

category, a concept which exhibits quite well the tendency towards algebraic abstraction so 

characteristic of category theory. 

 

Recall that a category consists of objects, morphisms, a domain/codomain structure, identity, and 

composition. The collection of all functors between two categories can themselves be thought of 

as the objects of another category, i.e., a functor category. In order to see how we might define 

such a category, we first need another crucial category-theoretic concept: that of the natural 

transformation. Intuitively, a natural transformation is a way of transforming one functor into 

another such that the compositionality of the morphisms of the underlying categories is 

respected.
111

 Now, with this notion of natural transformation in hand, we can give a clearer idea 

of what a functor category is. 

 

We start out with two categories, A and B. We can then define a category of functors from A to 

B (often written ‗Funct(A, B)‘). The objects of this category are the functors F: A→B. The 

morphisms of this category are the natural transformations between these functors.  I‘ve 

mentioned functor categories here not because they are particularly useful (though they certainly 

are useful) but because they represent exactly the sort of algebraic abstraction which is 

characteristic of the category-theoretic approach. I‘ll try to explain what I mean. 

 

Let‘s place ourselves in Frege‘s shoes for a moment. Perhaps the most essential core of Frege‘s 

‗background ontology‘ is the division between objects and functions. On his view, this is not a 

relative distinction but an absolute one. Any formal violation of this distinction would have 

disastrous results for his attempt to found mathematics. Let‘s see how the (linguistically similar) 

distinction between objects and morphisms plays out in the category-theoretic world. We might 

begin with Frege‘s universe as a category, let‘s call it Frege. The objects of Frege are just 

Frege‘s objects (i.e., thoughts, dogs, numbers, etc.) while the morphisms of this category are 

Frege‘s functions. We might be interested in some of the structure of this category, so we seek to 

define some functors from Frege to Frege. Thus, for instance, we might define the functor F: 

Frege→Frege which maps every object onto itself, and every function onto itself. We might 

then bundle together all such functors into a functor category Funct(Frege, Frege), whose 

objects are the functors between Frege and Frege and whose morphisms are the natural 

transformations between these functors. Then, if we are so inclined, we might define functors 

between the categories Frege and Funct(Frege, Frege), then another functor category, and so on, 

ad infinitum. This example isn‘t meant to be exciting or revelatory of any interesting 

mathematical content (and surely it isn‘t).  

 

What it shows is that the kind of absolute distinction that Frege wants to make between objects 

and functions just doesn‘t make any sense in the category-theoretic world. Nor does it make 

                                                 
111

 For a more precise technical definition, see [Mac Lane 1998, 16]. 



203 

 

much sense to ask which categories are the real ones, the ones out of which all other categories 

are ‗built‘. Instead, what the category-theoretic language of functors, functor categories, and 

natural transformation shows is that the questions of structure have no ‗final‘ or absolute answer: 

they must be given relative to a particular mathematical situation. On this model, to think of 

categories in the static terms of set theory might be possible but it certainly seems to miss the 

point. New structures can be built quite easily, and what might be a morphism of one category is 

an object of another category, or it may yet play the role of a functor between categories in 

another context. The question of the exact nature of the entities at issue is beside the point: what 

matters is the structure which is revealed by particular sorts of transformations. 

 

2.4.3 The philosophical aspects of algebraic structuralism 

 

With this rough conception of category theory, we will now have an easier time understanding 

the philosophical commitments of algebraic structuralists. Let‘s look at these now and see if this 

form of structuralism can move us in any significant way beyond the positions dictated by the 

Frege-Hilbert debate. Algebraic structuralism, inspired in large part by the methodology of 

category theory, holds that mathematics possesses no final or absolute ontology. On some 

dispensations, if it is meant to provide foundations at all, it is in a rather different sense than the 

ontological foundations we‘ve examined thus far. Take, for instance, the following comment by 

F. William Lawvere, one of the key proponents of the foundational importance of category 

theory: 

 

In my own education I was fortunate to have two teachers who used the term ‗foundations‘ 

in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-

Russell tradition). […] The orientation of [the works of his two teachers Samuel Eilenberg 

and Clifford Truesdell] seemed to be ‗concentrate the essence of practice and in turn use 

the result to guide practice‘. I propose to apply the tool of categorical logic to further 

develop this inspiration.
112

 

 

Of course Lawvere‘s remarks here are personal and idiosyncratic, but the inclination to oppose 

the traditional approaches to foundations as ‗speculative‘ and replace them with something more 

amenable to mathematical practice is a common theme amongst algebraic structuralists 

influenced by category theory and categorical logic.
113

 Like other forms of structuralism, 

algebraic structuralism holds that mathematics is, in the main, concerned with structures. In 

contrast to assertoric structuralism, however, algebraic structuralism does not seek a single 

theory of structures to account for the content of mathematics. Indeed, one way to describe 

algebraic structuralism is by highlighting its opposition to the idea that mathematics has a single 

determinate content, or that mathematics concerns itself with a global ontological background or 

framework.  

 

So, if mathematics is not about a single universe, either of sets or even of abstract structures, 

what is it about? There are a variety of different technical proposals in response to this question, 

                                                 
112

 [Lawvere 2003, 213]. 
113

 Cf., e.g., [Awodey 1996,]. 



204 

 

though most are framed, in some way, in category-theoretic terms. Thus, for instance, John Bell 

writes that 

 

The fundamental idea is to abandon the unique absolute universe of sets central to the 

orthodox set-theoretic account of the foundations of mathematics, replacing it by a 

plurality of local mathematical frameworks – elementary toposes – defined in category-

theoretic terms. Such frameworks will serve as local surrogates for the classical universe of 

sets.
114

 

 

Here we see the emphasis on local frameworks versus global foundations which is characteristic 

of the algebraic approach to structuralism. In place of a single theory of structures, the idea is to 

investigate a variety of different sorts of connections between different structures. There is not 

meant to be a final theory of structures, nor of the connections between them, but, instead, 

several distinct (perhaps incompatible, perhaps complementary) frameworks, each of which will 

highlight different sorts of connections, transformations, objects, etc. 

 

On Bell‘s particular view, the most natural way of organizing these frameworks is in terms of 

category-theoretically defined ‗elementary toposes‘.
115

 My interest here is not in the analysis of 

the precise details of topos and category theory which make them amenable to the structuralist 

programme,
116

 but instead in the general philosophical attitude towards foundations which is 

exhibited by those who suggest category theory can act as the basis for algebraic structuralism. 

For Bell, these elementary toposes are not meant to function as a kind of universal background 

ontology, in the manner of Hellman‘s modal theory of possible systems or Shapiro‘s structure-

theory. No, for Bell, these toposes themselves are ‗ambient‘ and variable, in the sense that 

neither their mutual interrelations nor their internal compositions (including their governing 

logic) are fixed outside of a given mathematical situation. 

 

An even more radical statement of the algebraic approach to structuralism can be found in a 2004 

paper by Steve Awodey. There Awodey writes (in response to some of Hellman‘s criticisms of 

the category-theoretic approach to structuralism) that 

 

As opposed to this one-universe, ‗global foundational‘ view, the ‗categorical-structural‘ 

one we advocate is based instead on the idea of specifying, for a given theorem or theory 

only the required or relevant degree of information or structure, the essential features of a 

given situation, for the purpose at hand, without assuming some ultimate knowledge, 

specification, or determination of the ‗objects‘ involved. The laws, rules, and axioms 

involved in a particular piece of reasoning, or a field of mathematics, may vary from one to 

the next, or even from one mathematician or epoch to another. The statement of the 

inferential machinery involved thus becomes a (tacit) part of the mathematics; functional 

analysis makes heavy use of abstract functions and the axiom of choice, some theorems in 

algebra rely on the continuum hypothesis; many arguments in homology theory are purely 

                                                 
114

 [Bell 1986, 409]. 
115

 Bell gives a formal definition of ‗ topos‘ at [Bell 1986, 412-413]. For a semi-technical elaboration on some of the 

interesting relations between toposes (or topoi), category theory, and set theory, see [Bell 1982]. 
116

 This is already accomplished in [Awodey 1996] and [McLarty 2004]. 



205 

 

algebraic, once given the non-algebraic objects that they deal with; theorems in 

constructive analysis avoid impredicative constructions; nineteenth-century analysis 

employed other methods than modern-day analysis, and so on. The methods of reasoning 

involved in different parts of mathematics are not ‗global‘ and uniform across fields or 

even between different theorems, but are themselves ‗local‘ or relative. Thus according to 

our view, there is neither a once-and-for-all universe of all mathematical objects, nor a 

once-and-for-all system of all mathematical inferences. Are there, then, various and 

changing universes and systems? How are they determined, and how are they related? Here 

I would rather say that there are no such universes or systems; or rather, that the question 

itself is still based on a ‗foundationalist‘ preconception about the nature of mathematical 

statements.
117

 

 

Let‘s try to unpack this a bit. For Awodey, the algebraic (or categorical) approach to 

mathematics involves accepting (embracing, really) certain limitations on one‘s mathematical 

activity. Instead of presuming to grasp the nature of the set-theoretic universe when proving a 

theorem about Abelian groups, for instance, his view suggests that in any given mathematical 

situation, we ought only to consider those ‗objects‘ and bits of ‗inferential machinery‘ which 

have actually been employed. On this view, we might say, a particular ‗structure‘ is not defined 

against anything other than the present mathematical situation: we form the structure out of the 

objects, properties, relations, and logic which we actually employ in the situation at hand and 

that is the end of the matter as far as ontology and reference-fixing goes. We state as much (and 

only as much) as we need to about the structures we are presently studying. From this point of 

view, the problem with the foundationalist approach to mathematics (within which both Awodey 

and myself would include Shapiro‘s and Hellman‘s assertoric forms of structuralism) is that its 

metaphysical principles require there to be some unifying element which collects together all 

forms of mathematical reasoning in a single universe (with its own particular collection of 

objects, relations, properties, and its own governing logic). The assertoric structuralist‘s 

complaint is that ―these constructions have to take place somewhere! They require some 

collection principles‖
118

 and some logic governing these principles. Ultimately, for Awodey, the 

assertoric structuralist‘s idea  

 

that one is ‗going up‘ in a hierarchy, and that this requires stronger and stronger collection 

principles and existence assumptions rests on the ‗foundationalist‘ conception that the 

‗objects‘ involved are fixed and determinate. From a categorical perspective, one is rather 

‗going down‘ by specifying more of the ambient structure to be taken into account […].
119

 

 

This contrast presented here gets at the heart of the algebraic structuralist view. Central to this 

view is the contrast between determinate collections and ambient categories or structures. What 

is the difference, exactly? 

 

2.4.4 Bottom-up vs. top-down 

 

                                                 
117

 [Awodey 2004, 56]. 
118

 [Awodey 2004, 62]. 
119

 [Awodey 2004, 62]. 



206 

 

The bottom-up approach is common in the set-theoretic construction of various number domains 

or algebraic structures. You begin with a very limited ontology—say the empty set or a finite 

collection of ur-elemente. From here, by applications of various ‗collection principles‘ (e.g., the 

power set axiom, the axiom schema of separation, etc.) you gain access to larger and larger 

collections, within which you can define more and more structures. This intuitive process is the 

reason why the set-theoretic universe is often called the iterative or cumulative hierarchy. 

Importantly, on this view, one‘s access to a particular structure is dependent upon there being 

some correlate to that structure in this cumulative hierarchy. So, if we are interested in 

investigating groups, we must first ‗build‘ them from the bottom up using the elements of the 

set-theoretic hierarchy in some way. Everything thus remains within the absolute ontology of set 

theory itself. Nothing in this is necessarily peculiar to set theory: we can approach other aspects 

of mathematics using this foundational attitude as well. Indeed, as we saw, Shapiro‘s approach to 

structure theory is bottom-up in exactly this way: he believes that, for his theory to get off the 

ground, he needs the basic building blocks which will guarantee that the more complex 

structures exist. If we wish to speak of categories, the category of sets, as understood by a 

traditional set-theoretic foundationalist, must be fixed, i.e., made non-algebraic,
120

 remaining the 

same for everyone in every situation. In this sense it is anything but ambient. From this 

determinate category, all the rest are built up. This is the ‗bottom-up‘ approach. 

 

Awodey and Bell, by contrast, wish to take a ‗top down‘ approach which does not even begin 

with a basic, universal ontology. And, in fact, the more usual approach to category theory is non-

foundational in precisely the required sense. Technical definitions of ‗category‘ (of which there 

are many) are not terribly important in the present context, what is important is the attitude taken 

towards such definitions. Thus, after giving his version of the category axioms, Awodey writes 

 

A category is anything satisfying these axioms. The objects need not have ‗elements‘, nor 

need the morphisms be ‗functions‘ although this is the case in some motivating examples 

[i.e., in the category Set]. But also, for example, associated with any formal system of logic 

is a category, the objects of which are formulas and the morphisms of which are deductions 

from premises. We do not really care what non-categorical properties the objects and 

morphisms of a given category may have; that is to say, we view it ‗abstractly‘ by 

restricting to the language of objects and morphisms, domains and codomains, 

composition, and identity morphisms.
121

 

 

Now, fairly obviously, this sort of language of abstraction closely mirrors Shapiro‘s (initial) 

approximation of the structuralist position. The difference between the two—and it is a crucial 

difference—is that, for the category-theoretic structuralist, such a language does not and should 

not be embedded within a single universe of discourse, a single logic, or a single ontology. Or, to 

                                                 
120

 This, in broad outline, is precisely the sort of claim that Hellman makes about category-theoretic approaches to 

structuralism: ―As its proponents have maintained, category theory does offer an interesting structuralist perspective 

on most mathematics as we know it. But it needs to be supplemented and set within a yet wider framework. […] 

[C]ategory theory—with its non-assertory, algebraico-structural axioms—depends on prior notions of structure 

(collection with relations) and satisfaction by structures to make sense of the very notions of ‗category‘ and ‗topos‘‖ 

[Hellman 2003, 154-155]. The objection clearly mirrors the problem of quasi-concreteness in relation to Shapiro‘s 

views. Similar claims are made in an earlier paper by Solomon Feferman (i.e., [Feferman 1977]). 
121

 [Awodey 1996, 213].  



207 

 

put it in terms familiar from the preceding section, the algebraic structuralist is not at all 

convinced of the necessity of a background ontology. 

 

The above examples of a category of sets and functions vs. a category codifying a system of 

formal logic are themselves illustrative of a key difference between the algebraic approach and 

the assertoric approach. On Shapiro‘s assertoric approach, he became embroiled in the problem 

of the quasi-concrete because his (non-structuralist) understanding of the logic animating his 

project seemed to commit him to sets. For the category theorist, this troubling situation might be 

easily resolved. On the one hand we have a category (let‘s just assume, for the sake of our 

example) that it is the category Set. The objects of this category are sets, and the morphisms are 

functions on those sets. The problem Shapiro has is that his logic seems to be understandable 

only in terms of this category of sets. But, from the category-theoretic perspective, here we have 

an interesting example of morphisms between categories. In addition to set we can define 

another category, let‘s say ShLog (for Shapiro-Logic), whose objects are the formulas of 

Shapiro‘s structure theory and whose morphisms are the allowable deductions among them. On 

this view, we needn‘t conflate the elements of Set and ShLog, but can instead describe the 

morphisms between them (perhaps there is a relation of isomorphism, as Shapiro seems to 

worry) as the elements of yet another category with a distinct structure. But we needn‘t take this 

to be revelatory of some secret or hidden quasi-concrete ontology in our original approach. 

Instead, it reveals new and interesting structure which emerges in this particular situation, or 

within this particular local framework. 

 

Perhaps the best way of characterizing the difference between this bottom-up approach and the 

top-down approach recommended by Awodey is to look at the components of each view which 

are allowed to vary. On the bottom-up approach, there are at least some parts which do not (and 

cannot) vary. These are the background ontology (usually some universe of sets) and the logic 

which governs the theory (whatever this might be). No matter what context you work in, if you 

build your structures out of a foundationally understood theory, then its ontology and logic 

remain static in every context. The key, on this approach, is to craft a subtle enough theory so 

that it covers all the desired instances without committing you to anything ‗extra‘. On the 

algebraic perspective, by contrast, both of these universally invariant things are allowed to vary; 

indeed, in Bell‘s proposal, their variability is an essential component of the notion of an 

elementary topos. 

 

 

 

 

2.4.5 No background ontology? No problem. 

 

As we saw in our examination of Shapiro‘s work, assertoric structuralism requires some 

background ontology in order to get off the ground. This background ontology was required in 

order to ensure that there were structures big enough to satisfy the usual mathematical demands. 

We also saw that commitment to such a background ontology involved Shapiro (and others) in 

an ultimately non-structural account of mathematics—exactly the thing he sought to avoid. The 

question we have to ask now is whether or not algebraic structuralism can really do without a 

commitment to such a background ontology. 



208 

 

 

Here is Awodey‘s statement on this question: 

 

the structural perspective [of topos theory] is at odds with the idea that all mathematical 

objects exist in a single, comprehensive universe of sets. The methods of topoi make it 

more natural to study the ‗local‘ logic of various kinds of structure than to force all kinds 

of structure into a ‗global‘ logic, say of the category of sets. Moreover, the very idea of 

‗foundations of mathematics‘ becomes less significant from a categorical perspective than, 

say, organizing and unifying the language and methods of mathematics. Such unification 

often points up related structural phenomena in disparate fields, suggesting fruitful 

directions for further research.
122

 

 

So the clear answer to the question ‗does algebraic structuralism require a single background 

ontology‘ is, for Awodey, a straightforward and emphatic ‗no‘. The point of the approach is to 

examine structures in situ, not to displace them and characterize them in the (usually) foreign 

terms of object-oriented set theory. Moreover, Awodey‘s remarks here suggest that, on topos-

theoretic approach to ‗foundations,‘ the troublesome connection to a non-structural account of 

logic (which Shapiro himself struggled with in relation to his ‗quasi-concrete‘ second order 

language) simply disappears. Logic itself is understood as a potentially variable component of a 

given topos. It is not pan-theoretically applicable, but a matter of specification in each case. 

 

The position as a whole is summarized by Awodey as follows: 

 

The structural perspective on mathematics codified by categorical methods might be 

summarized in the slogan: The subject matter of pure mathematics is invariant form, not a 

universe of mathematical objects consisting of logical atoms. This trivialization points to 

what may ultimately be an insight into the nature of mathematics.
123

 

 

The important point here, the point which Shapiro himself tried to get at in his form of 

structuralism, is that, for the algebraic structuralist, there is not and need not be any final 

accounting of ‗invariant form‘. The property of invariance is itself defined relative to a particular 

structure (i.e., a particular local framework or topos). There may be interesting relationships 

between different frameworks, which lead to the discussion of new and interesting sorts of 

invariant form (as happened, for instance, in the birth of category theory).
124

 Neither do these 

emergent forms of invariance require any final theory or background ontology. For the topos-

theoretic structuralist, ontology really is relative to a given framework. If we wish to remain 

‗foundationalists‘ we can seek to axiomatize a category of all categories,
125

 one which will 

encompass every other category. Category theory certainly allows for this possibility, but it is, in 

a fairly obvious sense, counter to the spirit of the algebraic structuralists endeavour. Even if we 

do construct such a category of all categories, there is no reason we can‘t treat this category (and 

                                                 
122

 [Awodey 1996, 235]. 
123

 [Awodey 1996, 235]. 
124

 Category theory, in its earliest incarnations, arose from the recognition that there were shared bits of invariant 

form between algebra and topology. See, e.g., [Eilenberg and Mac Lane, 1942] and [Eilenberg and Mac Lane 1945].  
125

 Cf., e.g., [Lawvere 1966]. 



209 

 

a concomitant logical structure) as one more local framework among others. The aim of category 

theory is to allow us to move between frameworks, while making it as easy as possible to 

highlight intrastructural morphisms (functors) which allow us to discern new forms of invariance 

and to construct new and interesting local frameworks all over again. 

 

2.4.6 Algebraic reference and truth 

 

Of course this shift from an absolute perspective to a radically relativized one does not come 

without a cost. As we‘ve seen in our examination of Frege, one of the key reasons why he sought 

fixed references for mathematical terms in the first place was to avoid the obvious relativization 

of truth which seemed to lie at the basis of Hilbert‘s view. Frege was deeply committed to the 

view that mathematics was a science, and that sciences had something to say about the world—

i.e., the single world shared by logic, mathematics, physics, chemistry, sociology, and all the 

rest. For him, the notion that there could be relative truths was simply a non-starter, and resulted 

in a meaningless, subjective morass from which no scientific findings could emerge. This is, at 

least partially, an explanation for his somewhat surprising (and perhaps misguided) rejection of 

the possibility that both Euclidean and non-Euclidean geometry could be ‗true‘.
126

 We saw, too, 

that Hilbert, in his later proof-theoretic years, was heavily invested in the production of direct 

consistency proofs, proofs which would stake direct claims about the world, and whose truth 

would be absolute. 

 

So it should be no surprise then that the only way to maintain the fully algebraic version of 

structuralism is to relativize not only truth but meaning and reference as well. These implications 

were already present in Frege‘s debate with Hilbert and Korselt, but are spelled out more 

thoroughly by Bell in a 1986 paper entitled ―From Absolute to Local Mathematics‖. There he 

writes  

 

With the relinquishment of the absolute universe of sets, mathematical concepts will in 

general no longer possess absolute meaning, nor mathematical assertions absolute truth 

values, but will instead possess such meanings or truth values only locally, i.e., relative to 

local frameworks.
127

 

 

And that 

 

Category theory […] suggests that mathematical concepts and assertions should be 

regarded as possessing meaning only in relation to a variety of more or less local 

frameworks.
128

 

 

These, framed in positive terms, are exactly the effects that Frege foresaw in Hilbert‘s turn of the 

century views on axiomatics. We are left, then, with the question of whether any significant 

advance has been made by algebraic structuralism toward answering our question ‗what is 

mathematics about?‘. Certainly great technical strides have been made since the Frege-Hilbert 

                                                 
126

 Cf. [Frege c. 1900, 169]. 
127

 [Bell 1986, 409]. 
128

 [Bell 1986, 410]. 



210 

 

correspondence, such that the variability of both ‗structure‘ and ‗logic‘ in a topos is possible in 

ways which were not even imagined at the close of the nineteenth century.  

 

2.5 Conclusion 

 

So, in the end, where do these two forms of structuralism leave us? Are we better off in our 

general understanding of the content of mathematics now than we were after the Frege-Hilbert 

debate?  

 

The goal of both the assertoric and algebraic forms of structuralism, as I have presented them 

here, is to present a clear and persuasive account of mathematical structures. Given such an 

account, we would be able to generate a satisfactory answer to our guiding question, ‗what is 

mathematics about?‘. The answer would be that mathematics is about structures and these are to 

be understood in the way dictated by this particular account.  

 

I think it is clear that there is no victor in the competition between the assertoric and algebraic 

forms of structuralism in terms of their respective accounts of structure. If we accept the 

assertoric account, we are pushed into a Frege-style dilemma which forces us to pick out our 

structures from a single background ontology. While the problems of assertoric structuralism are 

not identical to Frege‘s Caesar problem, and certainly need not lead to Russell-style 

contradictions, from a general philosophical point of view, they are unsatisfying in precisely the 

same way. Assertoric structuralism, like Frege, claims that mathematics has a specific content (in 

this case, some theory of structure) but it cannot give us good reasons for picking out this 

specific content from a variety of equally plausible options. Nor, contra Shapiro, can it rest 

satisfied with the possibility of a ‗shared structure‘ between these accounts because account of 

structure given is itself bound to a specific ontology. None of this ought to suggest that the 

assertoric form of structuralism is a fruitless endeavour
129

—only that it cannot provide an 

ultimately satisfactory answer to our particular question. 

 

Does algebraic structuralism fare any better? Like the assertoric structuralist, the algebraic 

structuralist also holds that mathematics is chiefly concerned with structure. Unlike the assertoric 

structuralist, however, the algebraic structuralist does not attempt to provide a single or final 

theory which pins down the nature of structures. Instead, following the lead of category theory, 

such structures are only ever determined locally. This has the benefit that the non-uniqueness 

problems which plagued Frege, set-theoretic foundationalism, and even assertoric structuralism 

are rather understood to be ‗features‘ of the view. But it also has the disquieting effect of 

introducing (or, perhaps, accepting an already-present) radical relativity into mathematics.  

 

The idea of a single ontology for mathematics is of course replaced with multiple distinct 

ontologies, defined relative to a given local framework or topos. This sort of view is already 

implicit in Benacerraf‘s work, and is accepted as a starting point for the algebraic approach. 

Alongside the relativization of ontology, however, we also have the more troubling relativization 

                                                 
129

 A feature of assertoric structuralism which has not at all been stressed in this chapter is its capacity to address the 

deep epistemological question in the philosophy of mathematics. Resnik, in particular, has made significant 

advances in these areas (cf., e.g., [Resnik 1999]). 



211 

 

of meaning, reference, and truth. Claims made within a local framework have no direct 

relationship with ‗the world‘ understood as a unified, final, or absolute ontological horizon. 

Frege, already in his correspondence with Hilbert, worried that the disconnection of axiomatics 

from any ontological background would have the effect of relativizing truth. The algebraic 

structuralist simply accepts this fact as an essential feature of mathematical discourse. Similarly, 

the idea that mathematical statements can be made meaningful only by their reference to a 

universal background ontology is rejected in favour of the view that both reference and meaning 

are relative to a given local framework. This fact is both supported by and helps to explain things 

like the existence of models of set theory within which the axiom of choice holds alongside 

models within which it does not hold. The implications of this kind of view are wide-reaching 

and essentially undermine any attempt to understand mathematics as a unified investigation of a 

single universe. 

 

Does the position of the algebraic structuralist offer us a significant advance beyond the impasse 

of the Frege-Hilbert debate? It seems that here, too, we have a recapitulation of an already-

familiar position. In the assertoric structuralist‘s case, the position was similar in the relevant 

respects to Frege‘s fixed-ontology view. In the algebraic case, the position closely resembles that 

of Hilbert (at least the Hilbert around the time of the publication of his Festschrift). For, recall 

Hilbert‘s position in his correspondence with Frege. 

 

His account of mathematical existence seems to be not much different than the ontological 

relativity presented by the algebraic structuralist. He writes that ―if the arbitrarily given axioms 

do not contradict one another with all their consequences, then they are true and the things 

defined by the axioms exist. This is for me the criterion of truth and existence.‖
130

 Of course 

Hilbert would later back away from this kind of relativity in his demand for absolute consistency 

proofs, but in this period he was for the most part happy to understand existence in terms of the 

consistency of a particular theory, and consistency in (admittedly) vague but not clearly 

absolutist terms. Certainly the category-theoretic/algebraic structuralist approach no longer relies 

uncritically on a theory-independent notion of consistency for its ontological relativity. But it 

seems reasonable to consider the newer position a technical refinement and expansion of 

Hilbert‘s attitude. 

 

Here, too, we see that, already at the close of the nineteenth century, Hilbert has accepted the fact 

that his purely relational (we might now say algebraic) approach to axiomatics commits him to 

the relativity of truth to a given formal theory. Without the definition or context provided by a 

given theory, there simply is no criterion for truth. The provision of the axiomatic theory (or the 

description of a structure) gives you the relative ontology, and it is only on the basis of this 

relative ontology that one can make (relatively) meaningful and truthful claims. Without such a 

context, there is no obvious way for Hilbert or the algebraic structuralist to speak of meaning, 

reference, or truth. 

 

In sum, then, it seems that the contemporary algebraic structuralist has been engaged in a project 

of refinement and increasing precision: the notions of category theory and categorical logic make 

more perspicuous the relativity of truth and reference, and they allow us to treat relations 

                                                 
130

 [Hilbert to Frege, 29 December 1899; excerpted by Frege in Frege 1980, 39-40]. 



212 

 

between various frameworks of truth and reference in a variety of systematic ways. These 

procedures were in their infancy in Hilbert‘s day, and it was, somewhat ironically, only through 

the development of formal logic (with the resources of model- and set-theory) which allowed 

them their fully algebraic form in contemporary approaches to structuralism. 

 

In any case, it seems that we are again left at a kind of impasse. Or, rather, we are left with two 

possible choices, both of which have significant implications for our understanding of the nature 

of mathematics. On the one hand, we can demand with Frege and the assertoric structuralists that 

mathematics possesses a unified content, and that we give a unique account of this content. This 

view faces the seemingly intractable non-uniqueness problem, but it has the benefit of 

maintaining non-relative conceptions of ontology, reference, and truth. On the other hand, we 

can insist that there is no final accounting of the content of mathematics, that there are different 

forms of mathematics with interesting relations with one another, but there is no final box within 

which they all live and breathe. This has the benefit of avoiding the non-uniqueness problem, but 

has the profound and far-reaching implication that mathematics does not speak directly about a 

single world, and that the truths of mathematics (and perhaps the rest of science as well) are 

merely relative to a given situation. 

 

How are we to choose between the two options? As I hope to have shown, neither view 

straightforwardly wins the day, as each position requires the acceptance of something which the 

other side imagines to be an obvious compromise or even a failing (e.g., ontological fixity, the 

relativity of truth). There seems to be no ultimately convincing argument to sway us either way. 

Either we believe in the non-relativity of truth, and reference, or we do not. The arguments that 

each side gives have most often been met with a shrug of the shoulders: either you demand a 

foundation for mathematics or you do not. A good example of this new impasse between forms 

of structuralism occurs in a debate between Hellman and Awodey regarding the possible 

usefulness of category theory for structuralism. Hellman suggests, following his assertoric 

scruples, that category theory requires ―a background logic of relations and substantive 

assumptions addressing mathematical existence of categories themselves‖
131

 That is, whatever 

approach is taken, there must be a final reckoning regarding the single logic (and, relatedly, the 

background ontology) which is taken up by the category-theoretic structuralist. Awodey, the 

arch-algebraist, responds to this ―problem of the home address‖
132

 by suggesting that it  

 

asks for something that only seems reasonable from a foundational perspective. This 

requirement exemplifies a general pattern in the discussion: Hellman sympathizes with the 

structural viewpoint […] but he thinks that there is still something missing in the overall 

category-theoretical position. The something is to be provided by his modal-structuralism. 

But I contend that what is missing is only a correct understanding of the categorical 

approach.
133

 

 

                                                 
131

 [Hellman 2003, 157]. 
132

 [Awodey 2004, 55]. 
133

 [Awodey 2004, 55]. 



213 

 

In effect, both Hellman and Awodey demand something that their opponent cannot give: 

Hellman cannot picture mathematics without foundations, and Awodey cannot understand why 

mathematics needs foundations at all. The two are at cross-purposes, and there does not seem to 

be a satisfying reason to choose one or the other. I do not see that this is a significant advance 

over the impasse of the Frege-Hilbert debate, though perhaps the battle lines have begun to be 

drawn more clearly when phrased in the terms of contemporary structuralism. 



214 

 

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