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 THE NOTION OF NUMBER AND 
 THE NOTION OF GLASS 
 
 BY 
 
 RICHARD A. ARMS 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL IN 
 
 PARTIAL FULFILLMENT OF TFE REQUIREMENTS FOR 
 
 THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 PHILADELPHIA 
 1917 
 
1 
 
UNIVERSITY OF PENNSYLVANIA 
 
 THE NOTION OF NUMBER AND 
 THE NOTION OF CLASS 
 
 BY 
 
 RICHARD A. ARMS 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL IN 
 
 PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR 
 
 THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 PHILADELPHIA 
 1917 
 
Press of 
 
 Steinman & Foltz. 
 
 Lancaster. Pa. 
 
CONTENTS 
 
 Page 
 
 A. The Notion of Number 5 
 
 B. The Psychological Aspect 19 
 
 C. The Notion of Class 39 
 
 D. Mathematical Usage 55 
 

 The Notion of Number and the Notion 
 
 of Class 
 
 A. The Notion of Number 
 
 §i. Introductory 
 
 Discussions about the nature of number may be readily 
 divided into various types. It is one of the striking character- 
 istics of the true mathematician that anything in the way of a 
 vague statement causes him acute discomfort. He desires a 
 change from the suggestive impressionistic expression of the 
 subjectively self-evident to an objective explicit derivation. 
 A striking example of this is Frege's rebellion against psycholog- 
 ical theories of number. "States of mind have no place in 
 mathematics," says the mathematician, and, so far from differing 
 with him, we may well inquire whether, as such, they have any 
 place in philosophy. 
 
 To the post-Kantian metaphysician no such ideal ever pre- 
 sented itself. For him the boasted Copernican revolution had 
 indeed taken place, and like our pragmatists, he could think 
 of no problem except in connection with some mind. Number 
 depends upon the way we think of it and the logically simple 
 is what we can think of most readily. The method used is 
 introspection. Opposed to these is the empiricist who demands 
 that all be reduced to physical terms. Crudities such as this 
 represent J. S. Mill's rash intrusion into the philosophy of arith- 
 metic. 
 
 Corresponding to these three classes of thinkers we find widely 
 different conceptions of number and three variants of Aris- 
 totelian logic. The empiricist defines number as a physical 
 property of external objects and emphasizes induction; the 
 Kantian talks about thought-content, introducing all manner 
 of the psychology of reasoning into his so-called logic. It 
 remains for the formalistic mathematician to regard number 
 as definable in terms of class and that bewildering maze of 
 svmbolic ramifications which has gained the title of logistic 
 as the pinnacle of mathematical method. 
 
 5 
 
 
6 The Notion of Number and the Notion of Class 
 
 The present discussion will not be confined to a purely mathe- 
 matical treatment of the notion of number. The objects of 
 arithmetic are not entities met with only on the pages of text- 
 books; number is something which we use in every variety of 
 present day experience, and which even primitive man was 
 forced to recognize. Some introduction of mental states seems 
 unavoidable. However, we may demand that this introduction 
 be made in terms of actual or virtual behavior so that any descrip- 
 tion of man's relation to number can be verified. The result 
 must be fixed and accessible. 
 
 We shall first be concerned with the notion of number as 
 it appeared to Mill and the Kantians and shall attempt to show 
 that the various historical positions are liable to one or both 
 of the following objections: (a) subjective mental states have 
 been introduced, (b) violence has been done to the facts. Frege's 
 distinction here between a posteriori and a priori methods is 
 important. He holds that a priori is deductive, a posteriori, 
 inductive; synthetic means that which uses conceptions belong- 
 ing to a special science while an analytic proposition is derived 
 from pure logic alone. Frege held that the concepts and laws 
 of arithmetic are analytic a priori, that they are derived from 
 the fundamental definitions and axioms of logic. We shall 
 find Bertrand Russell espousing the same cause although inde- 
 pendently and from slightly different motives. 
 
 §2. Empirical and Kantian Views 
 
 John Stuart Mill is the classic exponent of the inductive 
 theory of number and his conception of the basic truths of 
 arithmetic is narrow in the extreme. We find him contending 
 that number is the expression of an act, a physical property 
 of physical objects on the same plane as weight, color, and 
 extension; that the laws of arithmetic are merely inductions 
 of a very high order. Addition is physical combination of 
 objects, and calculation in general does not depend upon rule 
 and definition but upon observation of actual objects. 
 
 Leibnitz, Locke, and commonsense agree, however, that all 
 entities whatever can be numbered. To render his doctrine 
 consistent the empiricist must hold that all existents are physical 
 beings. Here we may reply that if this is the case, we are 
 confined to the mechanical world order and that where the 
 
The Notion of Number 7 
 
 empiricist desired to put number in a concrete world, rich with 
 individuality, he either contradicted himself, or placed number 
 in the most abstract of all worlds, — a world from which life, 
 mind, and purpose had been perforce abstracted. 
 
 Kant's theory of the "Schema" of Quantity is noteworthy 
 because of the manifest belief that number is the science of 
 time, while geometry is the science of space and the emphasis 
 on succession, which is due to an introspective analysis of count- 
 ing. It must always remain a matter of doubt whether Kant 
 believed that arithmetic was the science of time from the merits 
 of the case or because of the symmetry of his system. We may 
 not without grounds suspect him of falling victim to the archi- 
 tectonic passion. 
 
 Among the closer followers of Kant, in their emphasis on the 
 intimate connection of number and time are Bain, Hamilton, 
 and Helmholtz. The most interesting of these theories is found 
 in Helmholtz' "Zahlen und Messen." It offers the paradox 
 of combining in its attitude the extremes of both the objective 
 and subjective standpoints and is a fine example of nominalism, 
 regarding numbers as mere words or symbols which have no 
 significance except when used. Berkeley was among the first 
 to regard number as a mere tag. This standpoint must be 
 sharply differentiated from that of the formalists in mathe- 
 matics. The mathematician at the close of his manipulations 
 is concerned with the various interpretations of his symbols; 
 the nominalist says in effect that there is no interpretation, 
 that mathematics deals with words alone. This sweeping 
 conclusion cannot be accepted without more reason than has 
 yet appeared. 
 
 Lange, Baumann, and Brix cling to the Kantian formula of 
 a "Schematismus" but are inclined to hold that number is more 
 closely connected with the intuitions of space than those of 
 time. Lange in his "Logische Studien" blends in a curious 
 way the views of Kant and Mill. He agrees with Mill in deriv- 
 ing number from the contemplation of spatial objects and follows 
 Kant in his reiteration of the magic word "Synthesis." He even 
 goes so far as to say that all logical and mathematical thinking 
 is in terms of spatial syntheses which are thus not only the 
 basis of the number concept, but also of everything else. How- 
 ever, such terms as "Schema" and "Synthesis" tend to obscure 
 
8 The Notion of Number and the Notion of Class 
 
 more than they explain and are more puzzling than the notion 
 of number itself. 
 
 Let us pause for a moment to estimate the value of the intro- 
 spective method. The theories above cited have been gained 
 by its use. As a result we see one party holding that number 
 depends upon time and in no way involves space; another school, 
 whose members happen to be of the visual type, contending 
 for the opposite theory. Such a divergence casts no doubt 
 upon the validity of the theorems of arithmetic. It is rather 
 upon the method that these inconsistent conclusions reflect. 
 Is it not by this time clear that whatever an investigator dis- 
 covers by delving within the recesses of his "inner experience" 
 cannot be safely offered as a truth unless it is solidly verified by 
 observable fact? Otherwise we have the mad scramble of each 
 man to his taste, and as many philosophies of number as there 
 are philosophers. 
 
 §3. Number as Derived from Identity and Difference 
 
 Jevons preferred to assume a higher point of view. He con- 
 tends that every means of differentiation can be a source of 
 plurality, that number is only another name for difference. 
 However, it is clear that a certain likeness as well as difference 
 must be possessed by the counted units. Now if the units of 
 arithmetic are radically different, the fundamental laws of the 
 subject must be stated in terms of any unit whatever, or we will 
 be quite unable to calculate. However, it is manifest that when 
 we say "any unit" it is the quality of being a unit which is essen- 
 tial and their likeness is more important than their difference. 
 
 Hegel's dialectic treatment of number in the " Logik" leaves 
 much to be desired from a standpoint of clarity but is extremely 
 suggestive. He shows that the contradictions in the nature 
 of One make One and not-One qualitatively identical and thus 
 engender a Plurality. The synthesis of unity and plurality 
 is number. This synthesis denotes the collective unity of a 
 class whose reality lies in its parts. The units composing the 
 number, although distinct, are qualitatively identical. Thus 
 from the discontinuity or isolation of the units we pass to their 
 continuity or connection. In realizing this fact, Hegel has 
 advanced beyond Jevons. 
 
 Schuppe in his "Grundriss der Erkenntnis theorie und Logik" 
 sets forth the theory that plurality is already presupposed in 
 
The Notion of Number 9 
 
 the principle of identity, for the concepts of identity and differ- 
 ence presuppose at least two distinct impressions. For number 
 the principal matter is the comprehension ("Zusammenfassung") 
 of the units, by which a pre-existing difference is made explicit. 
 Number also asserts likeness and needs another concept under 
 which the differentiated objects are to be put, for only the 
 similar can be counted. So far, Schuppe's analysis is accurate 
 and subtle. However, he goes on to say that difference in 
 spatial position is essential to number and that when we seem 
 to count non-spatial objects we really localize them in imaginary 
 space, — being apparently convinced that because he, Schuppe, 
 happens to visualize counted objects, every one else does so. 
 
 Bergson's views, as elaborated in the second chapter of "Time 
 and Free Will," are not unlike those of Schuppe. He insists 
 that psychic states are continuous and non-spatial, whereas 
 number is discrete and spatial. It is not enough to say that 
 number deals with units. We also require that these units be 
 identical with one another, so far as the counting is concerned. 
 Nevertheless, the units must be distinct. We conclude, there- 
 fore, that they are differentiated by space. 
 
 Everything is not counted in the same way, for there are 
 two different kinds of multiplicity. Material objects are given 
 as separated by space. Psychic states, however, cannot be 
 retained in time; they must be differentiated with respect to 
 some homogeneous medium. Their continuity is not numerical 
 but a succession of qualitative changes which melt into one 
 another. Between the discrete and the continuous there is a 
 wide gulf. 
 
 A correspondingly sharp line of demarcation between number 
 and the continuum is found in Sigward's "Logik." Number, 
 to him, is only a development of the notions of Identity and 
 Difference. Everything we take to be one is cut out of the 
 continuum by a definite limited act of perception, whether that 
 continuum be time or space. A number is not a bare plurality 
 but a plurality definitely comprehended and bounded, and its 
 possibility lies in the fact that we are conscious of our progress 
 from one unit to another, making use of formal activities alone. 
 Number is a free creation of our self-conscious thought and is 
 independent of what is given by sense. 
 
 Now the introduction of any desired group of fractions and 
 irrationals between two consecutive integers seems to reduce 
 
io The Notion of Number and the Notion of Class 
 
 number, originally discrete, to a continuum. Notwithstanding 
 the superficial plausibility of this conclusion, argues Sigwart, 
 number can never be anything but discrete; since, however far 
 we push our interpolations, we can never succeed in reaching 
 more than a finite number of intervening members. The con- 
 tinuous process which intuition gives us in space and time can 
 never be expressed in the forms of number. 
 
 Here is evidently a point of view -closely akin to that of Berg- 
 son. Number and the continuum are placed as far asunder 
 as the poles and "never the twain shall meet." Sigwart's spirit 
 in this discussion is not far from that of Bergson, when the 
 latter accuses physics of transforming continuous movement 
 into mathematical smoke. The continuum is an insurmountable 
 obstacle and barrier to logical progress. But was it not Hegel 
 himself who said "Knowledge of a limit is knowledge beyond"? 
 
 Bosanquet is rather a failure as a philosopher of the continuum. 
 He makes continuity an invariable concomitant of classification 
 and accordingly must hold that a dozen eggs form just as con- 
 tinuous a group as the points of a line. The problem as it arose 
 is not attacked. As a matter of fact the only consistent answer 
 Bosanquet could have given is one like Sigwart's, for their 
 theory of counting is similar in all essentials. 
 
 The results of these two self-styled logicians are important 
 for two reasons. They continually emphasize the abstract 
 nature of numbering and make it subordinate to the logical 
 process of classification; and they are aware that order is a funda- 
 mental notion. While their methods are those of subjective 
 psychology, these conclusions are not necessarily so. They lead 
 to the notion that number is derived from formal logic, a position 
 most prominently represented by Frege and Russell. 
 
 §4. Number as Derived from Formal Logic 
 
 To gain his -definition of number Frege introduces a new 
 and widely applicable logical method. Just as we say that 
 lines parallel to a given line have the same direction, so we lay 
 it down that concepts having the same number are those which 
 are equivalent ("gleichzahlig") to a given concept A and the 
 particular number is defined as the "Umfang" of the concept 
 "equivalent to A." The relation of equivalence is that of one- 
 one correspondence, and Frege gives the following definition of 
 
The Notion of Number n 
 
 it in purely logical terms: Two concepts F and G bear to each 
 other the relation of one- one correspondence when there is a 
 relation R such that if a,e belong to F and b,d to G, then, 
 (i) IfdRa and d R e then a and e are identical. 
 (2) If d R a and b R a then d and b are identical. 
 The word "one" is dexterously avoided. 
 
 The number Zero offers some peculiar difficulties. It is 
 defined as the number which pertains to the concept "non- 
 identical with itself." The question naturally arises, — how can 
 two null-concepts be placed in a one-one relation if there is 
 nothing there to correspond? Frege's device is the clever but 
 unconvincing one of cutting up. "Every F has the relation R 
 to some G" into the components "a is an F" and "a is related 
 by R to no G" and postulating that one must be false. In the 
 case where either F or G are null, it is the first component which 
 fails of acceptance. 
 
 Frege has passed over some logical difficulties here which need 
 elucidation. It is sufficiently obvious that this analysis is an 
 afterthought and was brought in by main strength for the express 
 purpose of dealing with Zero. Moreover, a delicate issue is 
 involved here, — what can we conclude about the truth or false- 
 hood of a proposition whose subject does not exist? Frege's 
 method is to chop up the supposedly complex proposition by 
 analyzing the subject into its logical parts. For example, 
 "The green horse is in the field" becomes equivalent to the 
 falsehood of one of "The horse is green" and "The horse is 
 not in the field." As a consequence, Frege would hold that 
 we may pasture our horse, in as much as he does not exist. Now 
 this is not obvious to common sense and needs more argument 
 than he gives it. 
 
 Granted the definition of Zero, Frege finds it easy to define 
 One and then proceeds to discuss the law of the natural number 
 series. A number n is said to be the immediate successor of m 
 if "there is a concept F and an object x coming under it such 
 that the number which pertains to F is n and to "objects under 
 F not identical with x," m. It is important to observe that an 
 assumption has been made here without explicit mention. This 
 x which is to be extracted from the members of F must be held 
 fixed. It is not a variable but must be so definite that the expres- 
 sion "under F not identical with x" is unambiguous. In other 
 
12 The Notion of Number and the Notion of Class 
 
 words, the result is independent of what choice we make of x, 
 but once having made a choice, we must abide by it. Now this 
 fact, unless we admit subjective caprice, involves the ability 
 to define x in terms of known properties of F. It does not 
 appear obvious that this is always possible. 
 
 Upon such a basis Frege claims in the "Grundlage," his early 
 work, to be able to establish upon the basis of logic alone, the 
 following theorems : 
 
 (i) If a is the immediate successor of Zero, a is One. 
 
 (2) If F has the number One, then a is an F is not always 
 false. 
 
 (3) If F has the number One, a is an F, b is an F, then a and 
 b are identical. 
 
 (4) The relation of immediate succession is one-one. 
 
 (5) Every number has a successor. 
 
 From our present standpoint the validity of (5) may well 
 appear doubtful. It insures the existence of just as many con- 
 cepts and distinct objects as there are numbers, — in other words 
 infinitely many. If (5) is derived from the assumptions of 
 pure logic alone, which contain no ontological or existential 
 elements, its truth should be independent of how many distinct 
 beings there are. If we limit our widest universe of discourse 
 to a small finite number of terms it is hard to see how the exis- 
 tence of any infinite class can be deduced without regarding 
 terms and collections of terms on the same level, and this is 
 not a process to be unhesitatingly accepted. 
 
 Husserl in his "Philosophic der Arithmetik" criticized the 
 "Grundlage" and one of the points he makes deserves special 
 mention. He recalls Frege's definition by abstraction and 
 points out that this method does not define the content of the 
 number concept but its Umfang. Now Umfang, says Husserl, 
 if it is to mean anything, can only mean the collection of objects 
 falling under thq concept. Thus where Frege's definition pur- 
 poses to make number a definite single object, it really makes 
 it a plurality. 
 
 Husserl has put his finger upon the weakest link in Frege's 
 chain of argument. Even Russell, admirer of the definition 
 as he was, admitted that at first sight it must appear a "wholly 
 indefensible paradox." Clearly the definition of number as an 
 "Umfang" loses all claims to plausibility, if we regard this to 
 
The Notion of Number 13 
 
 mean a mere plurality of concepts, and like Husserl we may well 
 wonder how any one could have thought of number in such a 
 strange way. However, this does not by any means show that 
 Frege's definition is as faulty as Husserl would have us believe. 
 It is a convincing proof that Frege did not identify the "Um- 
 fang" of a concept with the mere enumeration of the objects 
 falling under it. 
 
 Kerry also criticized the position of Frege in some detail but 
 only one of his objections has any point to it, for his work is 
 based upon a misunderstanding of the aims and methods of 
 the "Grundlage." In attempting to show that Frege has mis- 
 takenly identified concept with its Umfang, Kerry remarks 
 that there are concepts which do not have any definite Umfang 
 at all, — such as "heap" which is clear as a concept, but whose 
 Umfang is obscure, since we do not know whether we are going 
 to call two objects a heap or not. This amounts to a demand 
 for an analysis of the process of definition. 
 
 Let us enumerate the difficulties we have found. (1) Given 
 any "Umfang," can we determine upon a definite member of it? 
 (2) When is a concept adequately defined? (3) Can the exist- 
 ence of infinite classes be deduced from pure logic? (4) The 
 notion of Umfang is so indefinite as to be misunderstood by both 
 Husserl and Kerry. It cannot be identified with a mere collec- 
 tion from the extensional point of view or Frege's definition 
 is invalid. What is the logical theory of an Umfang, then, which 
 will support the definition? (5) Is the inclusion of the null 
 class in every class justified? 
 
 Russell's initial discussion of number occurs in the eleventh 
 chapter of the "Principles." He lays down two fundamental 
 principles, akin to Frege's, — (a) numbers are to be regarded 
 as properties of classes, (b) "Two classes have the same number 
 when, and only when, there is a one-one relation whose domain 
 includes the one class, and which is such that the class of corre- 
 lates of the terms of the one class is identical with the other 
 class." This relation of equivalence is easily shown to be 
 reflexive, symmetrical and transitive. Russell gives his reasons 
 for dissenting from Peano's definition which postulates the 
 existence of a common property whenever we have a field of 
 such an isoid (reflexive, symmetrical, transitive) relation. 
 
 Peano defines number as this common property, to which 
 the equivalent classes have an identical relation. Russell objects 
 
14 The Notion of Number and the Notion of Class 
 
 to definition by abstraction in general, and to this actual defini- 
 tion in particular. He points out that while the existence of 
 this common property is guaranteed by the principle, its unique- 
 ness is not. Two ways are open, the definition of number as 
 the whole class of entities to which all equivalent classes have 
 a many-one relation, — which is practically useless, and the other, 
 "the" number of a class as the class of classes equivalent to 
 the given class. This last, after some argument, is finally 
 adopted. Moreover, Russell announces his intention of hold- 
 ing to the following: "Whenever Mathematics derives a common 
 property from a reflexive, symmetrical, transitive relation, all 
 mathematical purposes of the supposed common property are 
 completely served when it is replaced by the class of terms 
 having the given relation to a given term." 
 
 The tone of this passage is noteworthy. The desirability 
 of a definition different from the "class of classes" is not denied; 
 but its successful accomplishment is despaired of. We are not 
 able to define number as a common property because of defects 
 in the logical relations of predicates and classes and are forced 
 to the apparently "indefensible paradox" as the only way out of 
 the difficulty. Now why should Russell see paradox where 
 Frege saw none? Clearly Russell's notion of a class is more 
 extensional than Frege's and he sees the risk of making one 
 side of the definition (the number) singular where the other side 
 (the class) is plural. Substituting the class of objects having 
 the isoid relation for the desired common property is only legiti- 
 mate if we arc given a clear notion of what is meant by class, 
 and then not its identification with a mere plurality of terms. 
 Support from a satisfactory theory of classes is not only desir- 
 able, but absolutely necessary. 
 
 In his definition of multiplication, in purely logical terms, 
 the haste with which the "multiplicative class"and the very 
 delicate issues involved in the assumption of its existence are 
 passed over, is striking. In the "Principles" no doubt seems 
 to be felt as to the validity of the process. We are uncondi- 
 tionally commanded to choose and to keep on choosing. Now 
 here is not only Frege's assumption of the selecting of a definite x 
 from a class but this same assumption carried to an infinite power. 
 
 Russell bases his theory of the natural number series on a 
 set of postulates and indefinables due to Peano. The indefin- 
 ables are zero, integer, and successor. The postulates: 
 
The Notion of Number 15 
 
 (1) Zero is an integer. 
 
 (2) If a is an integer, the successor of a is an integer. 
 
 (3) If a is an integer and b is an integer, and the successor 
 of a is identical with the successor of b, then a and b are identical. 
 
 (4) Zero is not the successor of an integer. 
 
 (5) If s is a class to which zero and the successor of every 
 integer belonging to s belong, then every integer belongs to s. 
 
 Russell correctly remarks that another assumption is neces- 
 sary, i. e., 
 
 (6) Every integer has one and only one successor. 
 
 It is very important that this should be explicitly stated, for 
 it formed one of the dubious points in Frege's theory. 
 
 Russell objects to Peano's treatment on the ground that it 
 does not define a unique system. This deficiency is to be rem- 
 edied by reducing the whole to purely logical terms. Zero is 
 the class of classes whose only member is the null class. Num- 
 ber is defined as before. One is the class of classes not null 
 such that if x and y belong to the class, they are identical. " Hav- 
 ing shown that if two classes are equivalent and a class of one 
 term be added to each, the sums are equivalent," the successor 
 of n is defined as the number resulting from the addition of a 
 unit to a class of n terms. As a result Peano's postulates 
 are satisfied and are shown to possess the categoricity and 
 consistency of logic itself. "There is, therefore, from the mathe- 
 matical standpoint, no need whatever of new indefinables or 
 indemonstrables in the whole of Arithmetic or Analysis." 
 
 This is, of course, a very laudable conclusion, but what has 
 become of our added postulate (6)? It is not self-evident that 
 this is satisfied. The point at issue is the logical status of infinite 
 classes. Here we may refer to a later passage in the text (p. 
 357) where three arguments are given to show the existence 
 of infinite classes. These so-called proofs require careful scru- 
 tiny. The first necessitates the gratuitous assumption that 
 One and Being are distinct, which pure logic does not seem 
 to be concerned with. The third argument derives its plausi- 
 bility from a strange and perverted use of the term "idea." 
 If an idea is not a psychological or physiological reaction, there 
 is no reason to accept the premise, "of every term there is an 
 idea." Consequently we must reject both of these, and are 
 thrown back upon the second proof, which is essentially depen- 
 
1 6 The Notion of Number and the Notion of Class 
 
 dent upon the proposition that zero, the class of classes equiva- 
 lent to the null-class, is an integer which has a successor but 
 not a predecessor. Hence so far as the "Principles" is con- 
 cerned, Russell's doctrine of number stands or falls with the 
 notion of zero. 
 
 Here we have our old difficulties with the null-class, and we 
 may well ask, has Russell avoided the troubles of Frege? In 
 his first treatment of the null-class Russell does not depart from 
 the customary attitude of symbolic logic. However, he shows, 
 in his criticism of Peano, that all is not such smooth sailing 
 as we might suspect from the passage referred to above. The 
 definition of the null-class as included in every class is objected 
 to, because "there are no such terms x; and there is a grave 
 logical difficulty in trying to interpret extcnsionally a class 
 which has no extension." On p. 68 we find that, rejecting 
 Peano's identification of class and class concept, "there is no 
 such thing as the null-class, although there are null-class con- 
 cepts." 
 
 However, a new problem immediately offers itself. Null 
 class concepts possess in their equality an isoid relation, conse- 
 quently the principle of abstraction operates. For these the 
 required common term to which they have the same relation 
 must be taken to be the class of null concepts. "The null-class, 
 in fact, in some ways is analogous to an irrational in Arithmetic; 
 it cannot be interpreted on the same principles as other classes." 
 Now the null class seems to be not a fiction, a class with no 
 terms, but a class of infinitely many terms, namely all null- 
 concepts. 
 
 Passing on, immediately after a discussion of the paradoxes 
 we meet with another change. Now the class with no terms 
 is admitted, inasmuch "in the present chapter we decided that 
 it is necessary to distinguish a single term from the class, whose 
 only member it is." Hegelian dialectic in its most intricate 
 stages was not more baffling than this. At one moment the 
 null class has no members and is rejected, at another an infinity 
 of members and is accepted. After such bewildering changes 
 we should feel no surprise when our latest report shows that 
 it is again without members and accepted. Worse than this, 
 in Appendix B, we find the following: "This renders the very 
 definition of Zero erroneous; for every type of range will have 
 
The Notion of Number 17 
 
 its own null-range, which will be a member of Zero considered 
 as a range of ranges, so that we cannot say that Zero is the range, 
 whose only member is the null-range." Finally, in the Preface 
 occurs the following very significant remark: "On questions 
 discussed in these sections I discovered errors after passing the 
 sheets for the press; these errors, of which the chief is the denial 
 of the null class . . . are rectified in the Appendices. The 
 subjects treated are so difficult that I feel little confidence in 
 my present opinions." 
 
 Where the propounder of the theory feels little confidence, 
 we may not be blamed for feeling less. Russell, however, does 
 not seem to realize than an irreproachable definition of Zero is 
 indispensable for his proof that Arithmetic and Analysis are 
 derived from logic. Let us recall the difficulties we found in 
 the work of Frege. Selections, infinite classes, the null-class, 
 have all been discussed in the " Principles" but our doubts have 
 not been dispelled. On the problem of definition Russell's 
 position may be imagined from his remarks on formulas. Can- 
 tor and Dedekind thought that a class c was well defined if, 
 given x, it was determinate whether x belonged to c or not. 
 Applied to the theory of functions, the functional relation is 
 only specific correspondence and any particular function, to be 
 well-defined, must be given by a set of rules, which if mutually 
 independent, must be finite in number. 
 
 Russell's view is quite different. "The usual meaning of 
 formula in mathematics involves another element which may also 
 be expressed by the word law. It is difficult to say precisely 
 what this element is, but it seems to consist in a certain degree 
 of intensional simplicity. . . . It is therefore essential to 
 the correlation of infinite classes should be one in which . . . 
 the formula should be one which we can discover. I am unable 
 to give an account of this condition, and I suspect it of being 
 purely psychological. There is, however, a logical connection. . 
 This amounts to saying that the defining relation of a function 
 must not be infinitely complex. . . . This condition, though 
 it is itself logical, has, I think, only psychological necessity." 
 Under such opinions it is difficult to see how we can be guaran- 
 teed the possibility of choosing a definite member out of an 
 arbitrary class, whose members are not identifiable as individuals 
 and therefore cannot be explicitly distinguished. Without this 
 
1 8 The Notion of Number and the Notion of Class 
 
 possibility being granted, the multiplicative class fails to exist 
 for even a finite number of classes. To be told that these diffi- 
 culties are "purely psychological" is poor consolation. 
 
 The logical foundations of Arithmetic have been discovered to 
 rest upon the notions of Zero, Number, and Succession. To 
 each of this trio, so far as the "Principles" goes to show, dam- 
 aging objections may be raised, and to sum up our consideration 
 of Russell's philosophy of arithmetic, we will review them briefly. 
 
 (a) Zero is admitted at the close of the work to have received 
 no proper logical interpretation. The doctrine of the null- 
 class, although symbolically convenient, is not as yet justified. 
 
 (b) Number is applicable to classes, if they are many. If 
 a number is a class, it must be one or the definition is invalid. 
 Hence a class cannot be exclusively many, nor exclusively one, 
 if Russell's definition is to hold. The early proposed identifi- 
 cation of class with numerical conjunction invalidates the 
 definition of number. 
 
 (c) Succession has not been shown to be always possible 
 unless Zero be granted a correct definition. The notion of 
 Predecessor is not explained without a theory of Selections in 
 some way contraverting the notion of formula given above. 
 
 What conclusions are we to draw from this hasty review? 
 In the first place, it is clear that Russell has not answered the 
 great majority of our objections to Frege if, indeed, he has 
 answered any of them. For, in order to preserve Mathematics 
 as a derivative of Logic, he has been forced to depart from the 
 conception of Logic as laid down early in the book and, to change 
 this science almost beyond recognition by varying the use of 
 the term "class." To keep Arithmetic, therefore, he has done 
 violence to Logic. 
 
 In the second place, if we survey the work carefully, we see 
 that in establishing the indefinite extension of the natural num- 
 ber series, our author has resorted to extremely questionable 
 means and we may doubt whether, after all, theoretical arith- 
 metic has been deduced. Even granted the changes in Logic, 
 has Arithmetic been gained? We are forced to conclude that 
 in this case Frege's methods have not been bettered. Moreover, 
 by making numbers classes and by throwing doubts upon the 
 nature of classes not only the deduction of Arithmetic but its 
 very content is imperiled and the foundation of mathematics 
 
The Notion of Number 19 
 
 is shaken. To deduce Arithmetic and save Logic, violence has 
 been done to Arithmetic. Both sciences are common sufferers. 
 
 B. The Psychological Aspect 
 
 §1. Introductory 
 
 The problem of number is now to be considered from a slightly 
 different point of view. In our survey of the philosophy of 
 arithmetic we have seen that those who looked upon psychology 
 as a science based entirely and fundamentally upon introspec- 
 tion (Frege must be included here, since his contempt for psy- 
 chology is due to his belief in its subjectivity), have disagreed 
 radically and have left us with a difficult and delicate logical 
 problem, — the nature of classes. The question now naturally 
 is suggested, — would the result have been entirely different, if 
 another method had been used? What conclusions might be 
 drawn, if observation is substituted for introspection? 
 
 Such an investigation has to do with the observable behavior 
 of men toward number, and, it must be confessed, is beset with 
 as yet unsurmountable technical obstructions. Psychology as 
 an exact science is still in its infancy, and many of its conclusions 
 are not only debated by skeptical critics, but are questionable 
 in themselves. It will be our method, therefore, to avoid par- 
 ticular assumptions of a technical character and to confine our- 
 selves to observations of a general nature, not confirmed, perhaps, 
 by concrete instances, but, at any rate, not definitely refuted. 
 In brief, we shall endeavor to state not a solution, but a hypo- 
 thesis. 
 
 The customary behavior of adults is so complicated, and so 
 entangled with extraneous considerations that an investigation 
 does not promise any great results. What we shall do, therefore, 
 is to confine ourselves to the more elementar.y types of behavior 
 as found in (a) primitive man, (b) children. We must attempt 
 to find whether the notion of number is a particular act, a mode 
 of behavior, or whether it is not reached until we have a still 
 higher level of action. If the consciousness of number is a 
 definite concrete act, we shall say that it is gained by an indi- 
 vidual quantitative discrimination; if it corresponds to a type of 
 such actions, it will be termed a percept, and if it is not attained 
 in any of these ways it will be assumed to be conceptual. Thus 
 
20 The Notion of Number and the Notion of Class 
 
 the possible conclusions are parallel in a way to the theories of 
 Mill. Kant and Frege, as representative of the three great classes 
 of the philosophy of mathematics. 
 
 §2. The Implications of Anthropology 
 
 Just how primitive man perceived number is a matter of some 
 doubt. The problem has been considered in a careful way by 
 several anthropologists, among them Conant, McGee, Tylor and 
 Levy-IB ruhl. It is necessary to take into account the scanty 
 available historical material, as well as existing stages of primi- 
 tive culture. 
 
 The main opportunities in this field are afforded by the savage 
 number-words — since these, being a part of language, indicate 
 easily observed behavior. It might be imagined that a con- 
 sideration of the primitive methods of counting (as by fingers 
 or toes) would be fundamental, but investigators concur in the 
 opinion that these are comparatively late developments. As 
 a matter of fact numerals among the lower types of savages are 
 disappointingly few. McGee gives examples of Brazilian tribes, 
 which have words for one, two, three, and four and another 
 much used term meaning "many," "heap," or "multitude." 
 The behavior indicated by this last term is especially significant 
 since in effect it is equivalent to "non-denumerable" and sug- 
 gests a way of attacking a quantitative problem where the limit 
 of number has already been reached. 
 
 Conant establishes clearly that even where savages were 
 deficient in number-language, they were not deficient in the 
 conception of number. Serial arrangements give rise to types 
 of act, which have no explicit number words corresponding to 
 them. In fact, almost all of the primitive tribes have a 
 definite limit to their counting, which corresponds in some 
 measure to their facility in their perception of number. 
 
 Two outstanding factors in their attitude toward number 
 seem to be order and correspondence. In the district of Adelaide, 
 Australia, Moorhous?, the explorer, found that the children as 
 they come into the world receive in the order of birth numerical 
 names, as follows: 
 
 first child, if boy kertameru, if a girl kertanya 
 second, icarritya warriarto 
 
 third. kudnutya kudnarto 
 
 fourth, monaitva monarto 
 
 
The Notion of Number 21 
 
 and so on, up to nine. Now these same Australians possess 
 numerals only up to three, so we can imagine the savage parent 
 calling the roll and being convinced that the roster of his children 
 is complete without having the necessary knowledge of abstract 
 number, or, it may be, without any notion whatever of abstract 
 number. He does not count up to nine but calls names up to 
 the ninth. He has a comparatively adequate and successful 
 type of behavior toward a particular group, — his children, but 
 his numerical acts in other directions are complete failures. 
 It appears, therefore, that number is not gained by the perception 
 of order, or by mere perception of any kind. 
 
 The Murray Islands in Torres Straits offer an interesting 
 example of the further development of number-perception by 
 means of correspondence, and show that this process of tallying 
 off is of fundamental importance psychologically. The natives 
 of these regions have only two numbers, netat (one) and neis 
 (two). Beyond this, they proceed by repetition, such as neis neis 
 for four, or, more generally by correspondence, — a fixed method 
 of reference to the parts of the body. By this last device they 
 can count up to thirty-one. They commence with the little 
 finger of the left hand and go from the fingers to the arms, 
 shoulders, etc., ending with the little finger of the right hand. 
 If these are not sufficient, the islander has recourse to the toes 
 and the lower part of the anatomy. The number of any desired 
 group is indicated by the requisite part of the body, and then 
 starting from the little finger of the left hand, the embryo expon- 
 ent of one-one correspondence ascertains the relative position 
 of the particular part desired. In such behavior as this we 
 have the extension of successful behavior toward a particular 
 group, namely the parts of the human organism over into any 
 other arbitrary field by reference to this given particular group. 
 Even on this higher level, however, we have not yet found the 
 abstract numbers of arithmetic. 
 
 The transition to the formation of a number system may be 
 made in many ways, a discussion of which would be interesting 
 but hardly important for our purposes. Whatever base be 
 chosen, be it two, five, ten or what not, the fact that there is a 
 base is for us the fundamental issue. It gives us, so to speak, 
 the time for our numerical melody, the rhythm of the scale 
 helping us to remember and to group while counting. The 
 
22 The Notion of Number and the Notion of Class 
 
 distinction between the possessors of a system based on the 
 fingers and such more primitive numberers as the Murray 
 Islanders is well illustrated by the Baikaris. A member of this 
 tribe touches the counted objects with the right hand, holding 
 up corresponding fingers of the left as he does so, then for groups 
 above five he reverses the process. He does not count the 
 fingers themselves but counts with the fingers and may continue 
 enlarging this process. 
 
 We have seen that the different tribes have varying limits 
 to their counting which correspond to their mental rank. Be- 
 sides the gradual extension of this limit to include larger groups, 
 an extension of numbers in generality is necessary. For instance, 
 a Canadian tribe has the following words for groups of two 
 objects depending on the qualitative character of the objects: 
 Maalck, Masem, Mats'ak, Matlgsa, Matlouth, Matsemala, 
 Maalis, etc. The extraction of the common root is the indis- 
 pensable prerequisite to the gaining of the number concept. 
 
 The number-perception of the savage may far outrun his 
 words to describe it and he may have an intuitive sense of num- 
 ber which defies clear perception and which demands for its 
 expression the words for "multitude" or "heap." This sig- 
 nifies not only his inability to apply a numeral to the presented 
 group, or his failure to refer it successfully to the parts of the 
 body, but his lack of conviction and assurance. There seems 
 to be a fundamental factor in human nature which demands 
 a vague and doubting attitude toward that which cannot be 
 counted or numbered. We do not mean to apply the intro- 
 spective method here by the use of such terms as "conviction" 
 and "assurance." They are to be defined in terms of the time 
 elapsed between the presenting of a situation and the action 
 which aims at the successful handling of that situation. 
 
 Any attempt to go back of the actually observed data of the 
 anthropologists is, of course, pure speculation. Speculation, 
 however, although hazardous, is not always unprofitable and 
 the result may justify the means. Can we not conclude, with 
 some measure of plausibility, that before explicit counting, two 
 species of number-behavior existed; — a clear cut, swift and 
 assured reaction to a presented group, and a type of act some- 
 what similar to the first but lacking in precision and decision, 
 doubtless accompanied by doubt and uncertainty? The pro- 
 
The Notion of Number 23 
 
 gressive development of number and general number words 
 represents from this point of view a gradual and slow, but none 
 the less persistent conquest of the residuum of hesitation and 
 ignorance. Might we not indeed go further and trace the origin 
 of number to this very type of doubtful act? First we might 
 have, let us say, a specific act of discrimination such as the crow 
 performs when it is able to behave differently when one of a 
 group of four is absent, or like the nightingale's selection of 
 three worms. This is repeated until a type of successful dis- 
 crimination is present which by continual practice becomes 
 habitual under a certain situation. Here we have swift, con- 
 fident action. 
 
 If we concede to man, who is defined as the possessor of free, 
 rational behavior the ability not to do different things at the same 
 time but the ability to do the same thing under different circum- 
 stances, we yield to him the power of extending a type of success- 
 ful behavior into other fields. Granted this, which is the very 
 root and foundation of abstract thought, we can clearly see 
 how elementary particular quantitative discriminations, the 
 mere perception of change or invariance in magnitude, leads 
 by its very vagueness to an attempt to differentiate explicitly 
 and to act in conformity with some established custom or habit. 
 From the mere perception of such change or invariance, which 
 is undoubtedly possessed in some degree by the lower animals, 
 we pass to the recognition of different changes, and from that 
 to a type of constant behavior toward varying groups of the 
 same changes. Under this drying out process of extension 
 and abstraction, qualitative differences gradually evaporate, 
 until, at last, when abstract number is reached, the complexity 
 of the preceding process is elided and forgotten. 
 
 The number words do not evolve from this struggle for mastery 
 of quantitative change, but seem to traverse parallel lines. In 
 their rudimentary forms, they are bestowed with all manner 
 of religious and imaginative significance and are not applied 
 abstractly to presented groups. McGee says, 1 "These number 
 systems are distinct from Aryan arithmetic. . . . They are 
 devices for binding the real world to the supernal, and it is only 
 in an ancillary way that they are prostituted to practical uses 
 . . . yet by reason of the extraordinary potency imputed 
 
 1 American Anthropologist, 1899. 
 
24 The Notion of Number and the Notion of Class 
 
 to them they dominate thought and action in the culture stages 
 to which they belong." 
 
 The actual employment of numerals seems to be the meeting 
 point of two tendencies, therefore, — (i) A demand from prac- 
 tical sources for a universal term applicable to the similarity 
 observed among groups of qualitatively different objects. (2) 
 An emotional and personal reaction toward an ordered series, 
 in which rhythm plays an important part — any one who has 
 heard children count while skipping rope can appreciate this. 
 From this sense of rhythm we can see the development of the 
 systems of numeration. The binary scale comes first; with 
 more fully developed mentality we have the other bases. The 
 decimal scale is a surprisingly late development, showing clearly 
 that the finger- toe method is not fundamental and that number 
 is not on the perceptual level. The so-called mystic properties 
 of numbers are probably derived from the affective reaction 
 toward the ordered scale, the numbers seven and thirteen being 
 especially interesting in this respect. Such emotional factors 
 are markedly present in the beliefs of the Pythagoreans, where 
 they ascribed mystic virtues to all the simple numbers. It is 
 clear that no mere practical demand leads to the construction 
 of a mathematical cosmology; a certain esthetic craving must 
 be super-added. 
 
 We seem to be led to the conclusion that primitively number 
 is both logical and esthetic — not only a universal, but a uni- 
 versal with an emotional value of varying degree, that 1 " (1) 
 the origin of number names is at the bottom of the scale of 
 human development; (2) primeval man does not cognize decimal 
 systems; (3) does not use his fingers and toes as mechanical 
 adjuncts to nascent notation." 
 
 With the passing of time and the banishment of animism 
 the mystic powers of numbers tend to become disregarded and 
 number becomes more and more an abstract means of adjustment. 
 The fingers and toes appear as practical aids; the abacus, marks 
 in the sand, large and small pebbles, etc., are used. Order 
 and rhythm on the one hand, correspondence on the other make 
 these modes of behavior habitual and successful. Finally the 
 concrete aids to tallying become replaced by symbols and reck- 
 oning develops into arithmetic and that into algebra. If this 
 is correct, the numbers of pure mathematics must be not only 
 
 1 McGee, op. cit. 
 
The Notion of Number 25 
 
 universals, but concepts of the very highest generality and order 
 of abstraction, — and anthropology fails to confirm the psy- 
 chological views of Mill and Kant. 
 
 §3. The Results of Experiment 
 
 There is a striking parallel between the child and the primitive 
 man in the ability of each to grasp concrete number before 
 acquiring a knowledge of numerals, or even the ability to count. 
 Children can reproduce a group numerically even when giving 
 it the wrong numeral, after learning the names; for the associa- 
 tion of the mechanically learned series of names with the def- 
 initely given group may be made erroneously or not at all. 
 Phillips 1 cites a case where a child who knew the words for num- 
 bers and was able to recite them in order, on counting sticks 
 was as apt to call the third one "six" as "three." There seems 
 to be a strong element of rhythm in the motor activities of 
 counting, which is not always successfully transferred. 
 
 It is not until the process becomes mechanical by frequent 
 repetition that the names are applied correctly and the connec- 
 tion between name and group is perceived. However, the 
 motor tendencies are still apparent. The child is likely to count 
 in sing-song and will be much more accurate if allowed to observe 
 a definite rhythm. All experience agrees that counting of con- 
 crete groups must become familiar and habitual, before the 
 abstract character of number is understood. It is the accepted 
 procedure in elementary arithmetic to introduce many concrete 
 problems such as "3 apples and 5 apples make how many ap- 
 ples?" This is not without significance. It shows that the 
 gradual transition of number from a perceptional reaction cannot 
 be eliminated. The too hasty progress of our common schools 
 through the mechanical routine of methodological devices fre- 
 quently leads to a surprising knowledge of mathematical machin- 
 ery with an equally surprising ignorance of what it is all about. 
 This can be directly traced to the abrupt transition from names 
 of numbers to a parrot-like memorization of rules for their 
 manipulation without enough concrete illustrations being given 
 to induce any realization of the meaning of these rules. Having 
 once started taking rules for juggling symbols on faith, the 
 student is prepared for the easy descent to that mathematical 
 Avernus, of which the type of pupil who can solve any literal 
 
 1 " Number and its Applications," Ped. Sem., Vol. V, p. 262. 
 
26 The Notion of Number and the Notion of Class 
 
 problem in Algebra, while helpless before one couched in con- 
 crete terms, is an all too familiar denizen. No matter how 
 abstract and formal a science Mathematics has become, the 
 way to its comprehension is by the gradual ascent from percept 
 to concept, from act to type of act. 
 
 The perception of number has been carefully studied by a 
 number of experimenters, notably Messenger, Burnett, Dietze, 
 Arnett, Nanu, Lay, and Howell. 
 
 It is important from a psychological as well as a philosophical 
 point of view to decide whether the apprehension of number is 
 gained by counting (or some equivalent process), or is due to 
 an immediate perception, and in addition, to determine how 
 far accurate numbering can be carried without actual counting. 
 
 Cattell found that, where certain lines were exposed, up to 
 four or five the estimation was correct and that in case of 
 failure the number was likely to be underestimated. Dietze 
 experimented with metronome beats, and discovered that correct 
 judgment depended on the rapidity of the beats and their 
 rhythmical grouping. On the most favorable conditions eight 
 groups of five, forty beats in all, were accurately perceived. 
 How far Dietze succeeded in eliminating counting is dubious. 
 Warren used the reaction method and had his subjects open 
 their mouths when they apprehended the group (in this case 
 one to eight circles). Three simultaneous objects and five 
 presented successively were correctly estimated. 
 
 Nanu repeated the experiment of Dietze's, in a more careful 
 way. The indicated results were that in every case the limit 
 reached as high as eleven beats and in some instances up to 
 forty-nine. All the subjects involuntarily subjected the beats 
 to a kind of rhythm. When sets of points were presented, the 
 greatest number of simultaneously given points was five, of suc- 
 cessive, six. The experiments also showed that linear arrange- 
 ments were not as favorable as symmetric and that circular and 
 polygonal arrangements are undesirable. The number, more- 
 over, of simultaneously given elements successfully numbered 
 is greater than Warren thought, being eight to ten under the 
 best conditions. Counting was eliminated as far as possible. 
 
 An exceedingly valuable account of these researches has been 
 given by Howell in his "Foundational Study in the Pedagogy 
 of Arithmetic," to whom the following observations are in large 
 
The Notion of Number 27 
 
 part due. It seems to be the consensus of opinion among investi- 
 gators that the apprehension of number varies with the way the 
 group is presented, a theory which has enough superficial plausi- 
 bility about it to render it acceptable. For instance, Messenger 
 found that larger objects in a given space give the idea of a greater 
 number; that the perception of number is influenced partly 
 by spatial qualities and partly by other considerations. In the 
 majority of cases the perception is due to the association of 
 qualitative differences in the unit with numbers of parts derived 
 from actual count. With the establishment of the association 
 it becomes mechanical and number-perception seems immediate. 
 The apprehension of large groups is due to either rapid mechani- 
 cal counting or the association of number and form. 
 
 Arnett's experiments with counting showed: "Any lack of 
 rhythm caused the count to be very difficult and inaccurate. 
 . . . . Irregular counting may be made easier and more 
 accurate by practice." Here we have again the intimate con- 
 nection of rhythm and the number scale. The actual corres- 
 pondence between objects and numerals was found to necessitate 
 some motor reaction, as touching the objects, nodding the head, 
 etc. From the mere citing of names as a motor response, to the 
 mechanical tagging of objects with these names, the child passes 
 to a realization of the immense range of application of the method 
 and the subsequent abstraction. 
 
 Lay opposed the experimenters such as Messenger, who held 
 counting to be implicitly present in numerical judgments. His 
 work aims at showing that an immediate apprehension of number 
 is possible. He made a laborious and careful series of experi- 
 ments with school children in which various types of number- 
 pictures were used as stimuli, and came to the following con- 
 clusions: 
 
 (1) The faculty of number-perception varies with (a) the 
 power of clear imagery, (b) the homogeneity, distance, size, 
 and color of the objects. 
 
 (2) It is improved when the sense of touch is used along with 
 the sense of sight. The number image unites objects spatially 
 and temporally. 
 
 (3) The particular number-perception is independent of any 
 arrangement in a spatio-temporal series. The row form is not 
 necessary for the apprehension of number. 
 
28 The Notion of Number and the Notion of Class 
 
 (4) The images of the elementary numbers are immediately 
 perceived; those of the higher numbers (above ten) are gained 
 by the visualization of the elementary cardinal numbers in 
 connection with collections of tens. 
 
 (5) Those who hold that perception of number depends upon 
 counting have no valid psychological basis for their argument. 
 
 Such a position as Lay's is extremely radical. He over- 
 emphasizes perception almost as much as his opponents over- 
 rate counting. His conclusions have been challenged and 
 various attempts made to check them up by parallel series of 
 experiments. Of these the most careful have been those of 
 Howell. He comes to the conclusion that Lay's theories are 
 not corroborated, although experimental results tend to cast 
 increasing doubt on the theory that number is derived purely 
 and simply from the process of counting. He arrives at one 
 illuminating result: "The number pictures themselves consti- 
 tute for the child a tremendous step toward complete abstrac- 
 tion, in that he is taken away from the specific numbered objects 
 of various kinds to a representation that may stand for any of 
 them." The discovery of Lay that images of successions lead 
 to particularity much more than images of groups is substanti- 
 ated. It is also found that number concepts arise rather late 
 in children, and Howell recommends that the introduction of 
 abstract number be postponed until the third or fourth year of 
 school. 
 
 Counting is found to play the largest part in numbering 
 among those of an auditory type, and for this reason it is unjust 
 to neglect the process, as Lay does. His view that the percep- 
 tion of a group "becomes, through a Kantian reaction, numerical 
 apprehension" is not confirmed, although it is not proved that 
 counting is an indispensable prerequisite. The exact deter- 
 mination of this issue is a matter of great delicacy, and, pending 
 its unlikely settlement, an attitude of compromise is necessary. 
 "The instantaneous grasp of a group of objects visually pre- 
 sented is not intuitive. . . . It is made possible by repeated 
 prior experiences of association of certain qualities differing 
 in the unit expression. . . . Lay's notion that the appre- 
 hension is immediate cannot be sustained. . . . The burden 
 is on him to show that these children did not practise a primitive 
 method of numeration." 
 
The Notion of Number 29 
 
 Howell is certainly right in rejecting the hasty conclusion 
 that the immediate perception of number is possible. Proof 
 of such a statement involves not only great logical difficulties, 
 but must also surmount formidable technical obstructions. 
 Moreover, on the face of it, it is unlikely that any abstraction 
 of the nature of pure number should be intuitively perceived 
 by a child. Such a faculty would involve a discontinuity in 
 intellectual development which would be hard indeed to explain. 
 Transition from the repeated concrete to the abstract is the 
 only power we have any right to postulate in the child. That a 
 sharp break in the process occurs in the special case of number 
 is an assumption to be made only in the case of dire necessity, 
 and such necessity by no means exists. 
 
 The exponents of counting are not, however, to be completely 
 agreed with. They claim that the visual apprehension of a 
 group numerically contradicts the psychological principle that 
 the field of consciousness is limited, that only a relatively small 
 number of objects can be grasped simultaneously. This view, 
 however, neglects the power of grouping, by which the visualiza- 
 tion of the number-pictures is widely extended and which is 
 greatly facilitated by practice. Even after a group is counted, 
 the simultaneous grasp of it soon follows, and this can be so 
 quickened by practice that counting becomes unnecessary. The 
 image replaces the serial count. 
 
 Another charge of the advocates of counting is that children 
 who would be taught to visualize number by grouping instead 
 of counting would be restricted to their images and would never 
 attain pure number in its abstraction. Although it is no doubt 
 true that counting makes this abstraction easier, that does not 
 show by any means that it could not take place without it. More- 
 over, the visualized images aid in calculation as a check on the 
 work. Speaking generally, it is quite easy to gain a false mathe- 
 matical result by plausible symbolical consideration, where 
 concrete illustrations (say by a geometrical figure), if introduced, 
 provide less rapid, but far safer means of progress. The child 
 who has distinct recollection of five as applied to all the fingers 
 of one hand and six as adding one finger of the other hand is 
 far less likely to say one, two, three, four, six, than the child 
 who is merely learning words by rote. 
 
30 The Notion of Number and the Notion of Class 
 
 Thus we see that there is unquestionably ground for the asser- 
 tion that number is not essentially dependent upon order and 
 counting although there is no reason to go to the extremes of 
 Lay. Children manifestly possess the ability to make quanti- 
 tative differentiations independent of order, this ability varying 
 with their ability to visualize and the color, shape, etc., of the 
 objects. This ability may be developed until accurate observa- 
 tions of absolute number are made with confidence. A bank 
 cashier, for example, does not need to count a small number of 
 coins lying before him; he groups them and behaves just as if 
 he had counted. 
 
 The experimental results show, therefore, a striking parallel 
 between the attitude toward number of the child and the behav- 
 ior of primitive man. There is the same elementary quantitative 
 distinction developing by order and correspondence through prac- 
 tical application to the pure concept; there is also the esthetic 
 factor of rhythm which makes for the homogeneity of the number 
 scale. The numbers of theoretical arithmetic are show in attain- 
 ment and abstractly conceptual in character. 
 
 §4. The Philosophers Again 
 
 We may now resume our discussion of the metaphysical the- 
 ories, this time, with a view as to their psychological implica- 
 tions, looked at abstractly. The problem we have now before 
 us is: What is the nature of the experience which leads to the notion 
 of abstract number? Hoiv is it that man is capable of mathematical 
 behavior? We may neglect the crudities of Mill and the em- 
 piricists who would reduce mathematics to the level of quan- 
 titative discrimination. If our genetic investigations have told 
 us anything, they have taught us that the notion of number is 
 far above the level of immediacy. 
 
 It seems to be an implicit presupposition among philosophers 
 that whatever is to be counted must first be collected; and, with 
 this in mind, we can see plainly that the differences between 
 the various theories are due to disagreement about the psy- 
 chological character of collecting. Individual elements com- 
 pared as such do not give either the notion of plurality or of 
 definite number. To the elements must be added the property 
 of their being a whole and the question which is thus raised is, — 
 
The Notion of Number 31 
 
 what is this collective principle? What is the psychological 
 basis for our combining elements? 
 
 A possible answer which might be advanced is that the required 
 common property is simply co-presence in consciousness which 
 is, abstractly considered, equivalent to having a common relation 
 to a single observer. To this we may reply that many things 
 are present in consciousness, which we do not treat as collections. 
 The combining process is essentially teleological ; to unite the 
 many into a single whole our interest must be elicited and our 
 attention thus far focused. If the theory we are considering 
 were tenable, in our experience at a given moment there would 
 be the material for one and only one collection; while on the 
 contrary, it is a familiar fact that we have the material for many 
 concepts constantly before us. 
 
 There is a latent confusion here which it would be well to 
 bring to light, since in the sequel we shall find ourselves frequently 
 beset by it. The distinction which has been omitted is an 
 essential one. In no distinction of this kind can it be safely over- 
 looked that there is a difference between a subject behaving toward 
 an object, experiencing it; and that same subject reflecting on his 
 experience. The relation in the first case is comparatively 
 simple; from the reflective standpoint we have a set of relations 
 which are mediated and complex. When we hastily write 
 down subject-relatcd-to-objects it seems as a direct consequence 
 of strict logic that a collection of objects-&«0w«-&y-subject must 
 exist, which would of course prove what is required. There is 
 no trouble in the logic; it is with the premise that we must 
 quarrel. We have no right to assume subjcct-knoiving-objects, 
 for by so doing we are putting ourselves in the reflective stand- 
 point and any conclusion we arrive at exists only for its reviewing 
 the process. The subject knows the objects as such and not 
 as known. Consequently the common quality of being known 
 they do not possess for him. 
 
 The temporal theory is not essentially different from this. 
 We may dismiss rather summarily the notion that to collect 
 elements, they must be simultaneously presented. To grasp 
 them as a collection, we must in some fashion know them at the 
 same time; but this by no means shows that we know them to be 
 temporally coexistent. The necessary similarity is hardly the 
 possession of the same time abscissa. 
 
32 The Notion of Number and the Notion of Class 
 
 The view of Kant is more subtle. Instead of a cross-section 
 of time a longitudinal view is taken. The contention is that 
 succession is the important factor, thus placing the required 
 similarity among the mutual time relations of the objects. In 
 defense of this view it is alleged that we can attend to but one 
 thing at a time. Granted this, which is at best extremely 
 dubious, we cannot concede that the situation of the temporal 
 theory is a whit more promising. If we attend to a succession 
 and know it to be a succession we recall all the past objects and 
 in a given instant have the whole collection present before us. 
 We do, of course, know and attend to the group as one but we 
 just as much attend to it as many; and the whole positive force 
 of the argument we have been considering lies in the tacit elision 
 of half the process and that half wherein, as a matter of fact, 
 the crux of the difficulty lies. It may very well be questioned 
 whether temporal conditions are even a necessary basis for the 
 required common property. To be sure, the events we do 
 unite have a temporal beginning and passing away, but we 
 cannot be sure that what is ever present is never irrelevant. 
 Here again we must emphasize the difference in standpoints; 
 that while for us reviewing the process the time relations stand 
 out clear and distinct, for the actual experient they were slurred 
 over and deduced later, giving a logical and not a psychological 
 condition. 
 
 The spatial theory is no better. To Schuppe's ingenious 
 theory that the counting of non-spatial objects is only possible, 
 if we locate them in imaginary space, we may reply: (a) any 
 particular spatial relations are quite irrelevant. Two men are 
 two men, whether walking arm in arm on Broadway or separated 
 by the width of the Pacific; (b) if it is asserted that spatial 
 relations in general are necessary — that is, that the objects must 
 possess some spatial relations, — to this we may reply that this 
 fact si seen, if at all, only after reflection, the object of immediate 
 perception being the particular space relation. Thus in either 
 case the theory falls. 
 
 The opinion of Jevons that number is the form of abstract 
 difference betrays a lack of analysis of the notion of differentia- 
 tion. The psychological process of conscious distinctive is a 
 type of act resulting from repeated comparisons. Difference 
 is a concept which derives its force from denying a certain kind 
 
The Notion of Number 33 
 
 of similarity. Before this similarity can be denied, the objects 
 must be brought together for comparison, collected, and thus a 
 more fundamental kind of similarity asserted. Thus the plur- 
 ality is perceived before the difference. We may say with 
 Husserl, that differentiation is a process which follows upon 
 analysis. We may know A and we may know B, but it is not 
 until later that we know that A is different from B. Perception 
 of differences is not the same as different perceptions. 
 
 We have now examined enough of these psychological the- 
 ories to make it fully apparent that a confusion of standpoints 
 pervades the great majority of them. Logic is so entangled 
 with psychology and psychology with logic that the results are 
 untenable in either domain. The question naturally arises, — 
 can we do any better? Before we go on, it might be well to 
 re-iterate what we are trying to do. We want an account of 
 the type of act which by repetition leads to the concrete numbers 
 of practical life and thence to the abstract concept of mathe- 
 matics. 
 
 §5. Number in Terms of Behavior 
 
 The clue to our number-behavior must be found in the fact 
 that man is subject, not only to mechanical but also to teleologi- 
 cal classification. That a mere collection of electrons whose 
 behavior can be exactly calculated by a set of equations could 
 draw arithmetical conclusions is a theory which may surely 
 be disregarded. It is with the possession of mind that we rise 
 above the mechanical level. 
 
 It is one of the chief characteristics of man that, as we have 
 indicated before, he is able to behave selectively, that is, a certain 
 part of the environment is singled out and attended to. This 
 process is essentially teleological and governed by the prevailing 
 interests of the person in question. A striking example of this 
 is found in the varying comments different persons make upon 
 the same presented scene. It is a familiar fact, that no matter 
 how similar physical conditions may be, one person will observe 
 objects which are entirely unnoticed by another. This ability 
 to concentrate attention is essential for self-preservation; and 
 we will call it, the "/Hs" type of behavior. Now in cases of 
 erroneous perception, hallucination and the like, the selected 
 "this" cannot be said to exist, for our very discovery that it is 
 erroneous is due to the attribution of contradictory predicates 
 
34 The Notion of Number and the Number of Class 
 
 to the object. Consequently existence is not a necessary con- 
 sequence of this-ness. That it is behavior toward something, 
 however, is undeniable; we shall say, accordingly, that each 
 this has being, although not necessarily existence. 
 
 As another condition for the success of self-preservation, man 
 must possess a kind of preparation for the future which envelops 
 him like a fringe. It is the extension of the past into the future, 
 the result of the fundamental tendency of each act to become typical 
 and habitual. This casing about us of the accustomed response 
 to the well-known environment enables us to make needed 
 adjustments rapidly and easily, conserving our strength for 
 situations demanding concentrated attention. Every manifes- 
 tation of hope, belief, or fear is an illustration of this forward- 
 looking characteristic of human behavior. We are wound up, 
 so to speak, like a spring, and always charged with potential 
 action. Now this tendency to assume the attitude of tension, 
 of expectancy, is a distinct mode of behavior, quite different from 
 what we called the "this" type. This new element we will call 
 the and behavior. 
 
 This relation is implicitly present in every act and has some 
 properties which need explaining. The connection between 
 the situation A in which the "and" element is present, and the 
 situation B to which it is connected, is, of course, the "and" 
 relation itself. Now if B is extremely novel and striking, if A 
 and B are non-homogeneous, the connection breaks down and a 
 new type of act presents itself. The old accustomed reactions 
 fail to work; an attitude of doubt and hesitation is assumed, 
 until when the pressure becomes great enough, man exerts his 
 inherited power of carrying over one of the old types. This 
 doubtful behavior will be called the "stop" type. The condition 
 that it should arise is that A and B should be non-homogeneous. 
 
 Let us pause for a moment and recall the qualities with which 
 we have found man to be endowed. We have seen that in 
 ordinary experience, our behavior is of the type "this" and "that" 
 and "the other" and so on, until a "this" is reached which is 
 sharply distinct from the preceding with reference to the modes 
 of action already possessed by the subject. Here we come to 
 the "stop." We may state the typical a priori form of cognitive 
 experience thus: this and this and this . . . stop, this and 
 . . . stop, etc. 
 
The Notion of Number 35 
 
 Here we have in the most rudimentary form of cognition the 
 basis for numerical behavior. Let us suppose a man gazing at 
 a group of trees and his behavior of the type "this tree A and 
 this tree B and this tree C, stop" and let us also suppose the act 
 repeated until the type of act "A tree and A tree and A tree" 
 replaces the concrete individual act. This type of act when 
 further abstract and carried into other fields becomes "An X 
 and an X and an X," finally emerging as the recognition of 
 the abstract number three. At first we have the concrete con- 
 tents filling the places between the "and"s; next they are replaced 
 by a typical term representative of the similarity or homogeneity 
 of the objects; finally the particular kind of similarity is ab- 
 stracted and we are left with a bare set of homogeneous "this"-es 
 or units. The next stage is the explicit number, obviously 
 abstract and conceptual in character. 
 
 This is enough to generate a number system such as a Bra- 
 zilian tribe might possess, but does not suffice to give us our own 
 scale. We may now introduce the rhythmical sense, which we 
 found so prominent in the behavior of children and primitive 
 men. Manifestly in experience abstractly considered the stop 
 or break may occur at different places in successive situations; 
 man as the possessor of rhythmical behavior is able to interpolate 
 the discontinuities at uniform places. In this way a collection 
 can be grasped in groups of two's, five's, etc. In our ordinary 
 decimal scale the interpolated stop comes at every tenth element. 
 
 Not only does this principle of rhythm enable us to put in 
 the breaks but it provides for the extension to the general num- 
 ber system. This result is accomplished by the change from 
 the stop externally imposed as when the man surveying a herd 
 of cows comes quite suddenly upon a bear to the arbitrary stop 
 of the bank teller who stacks his coins. In the first case the 
 homogeneity of the known objects breaks down completely; 
 in the second the self-imposed rhythm gives a motive power 
 which bridges, so to speak, the gap — very much as dancers 
 keep step during a rest in the music. We come then to the 
 possibility of an experience in which all the breaks are inter- 
 polated and any non-homogeneity can be neglected. Put 
 abstractly, this would be "this arid this and this, & this and this 
 and this, & this and this and this. . . . After any given 
 trio, the rhythm of the process carries us on. Here we have only 
 
36 The Notion of Number and the Notion of Class 
 
 two main principles at work, "this" with "and" and the proviso 
 that no real "stop" interfere. "This is not the same as if we 
 should attempt to count a homogeneous set with an inadequate 
 supply of numerals and should be compelled to^ apply the term 
 "heap" or "many" to it. That act was essentially of the stop 
 variety, whereas the assumed proviso in the second case directly 
 forbids such hazy behavior and gives the clear conception of a 
 never-ceasing array of isolated objects, which are necessarily 
 homogeneous. This new conception involved a new principle, 
 the abstraction of the external stop, and is therefore essentially 
 different from the experiences which gave the ordinary numbers. 
 
 Two other abstractions are possible. The question now 
 arises, what is to be done with the potentiality of abstracting 
 the "and"} If this is granted to be conceivable we have be- 
 tween the initial "this" and the final "stop" a collection of 
 "this"-es not disconnected by "stop"s and not connected by 
 "and"s. Although they are distinct from one another, with 
 the exception of the first and last, they are not distinguishable 
 one from another and are not separated. Of no individual 
 "this" can we say that it has a next following or a next preceding; 
 the isolation has been abstracted. It follows, therefore, that 
 if we consider the two elements which we can distinguish, namely 
 the first and last, some connection must replace the "and"; 
 otherwise they could not belong to the same part of experience. 
 This connection can only be the collection of elements between 
 the first and last. If by the principle of rhythm we interpolate 
 a "stop" in the set and are thereby enabled to distinguish a 
 third "this" between this new element and each end there must 
 be a similar connecting collection. The group of "this"-es is 
 consequently compact, i. e., between any two distinguishable 
 elements there is always a third. It is moreover cohesive; there 
 are no finite gaps which might be filled by "and." It has, how- 
 ever, one further property which is of essential importance. 
 Let us consider the possibilities of interpolation. They are 
 manifestly of the form "this" and "this" and "this" . . with 
 nothing to destroy their homogeneity, consequently can be 
 considered a collection of the kind where the "stop" is abstracted 
 and the "and" left in. What then is the difference between the 
 whole set and its subset? 
 
 The distinction seems to be this — that no matter how far 
 our interpolations be carried, there is always a remainder which 
 
The Notion of Number 37 
 
 at any given step is perfectly determinate; that there is a re- 
 mainder totally outside all interpolations, indeterminate and 
 containing no distinguishable element, and around each inter- 
 polated element on either side cluster the elements of the re- 
 mainder. Any subset of the interpolated elements is deter- 
 minate; this is not true for the whole collection, for the remainder 
 is not determinate. We shall call such a set where the "and" 
 is abstracted, a continuous segment. If the "stop" now is ab- 
 stracted we get the notion of a continuous manifold or, in brief, 
 a continuum. This, however, comes later. 
 
 If we let the "and" in and abstract the "this" we get nothing 
 in particular — vague awareness but no actual behavior. This 
 represents the numerical zero which is far from being absolutely 
 nothing. Bergson devotes himself to some length in his "Crea- 
 tive Evolution" in showing that the image of zero is a psycho- 
 logical impossibility. From our point of view this is quite 
 needless for apprehension of Zero does not represent the absence 
 of experience, but the absence of purposive, selective behavior. 
 Bergson would put the Zero in the object; our view places it 
 in the subject. 
 
 So far, therefore, we have gained: 
 
 (1) A group of types of indefinite extent, corresponding to the 
 different possible positions of the "stop." 
 
 (2) The type with the "stop" abstracted — definite infinity. 
 
 (3) The type with the "and" abstracted — continuous seg- 
 ment. 
 
 (4) The type with the "stop"-"and" abstracted — continuum. 
 
 (5) The type with the "this" abstracted — zero. 
 
 So far as can be seen, the possibilities of abstraction are ex- 
 hausted. Now (2) (3) (4) and (5) represent a different level of 
 behavior from the different types of (1) and it is possible for a 
 man to possess a great variety of such modes of behavior with- 
 out having gained, let us say, such an abstraction as zero or 
 infinity. Infinity, for example, only comes with the feeling of 
 the inherent rhythm of the number series. 
 
 So much for the psychological origin of number. Now where 
 does mathematics come into the process? We might say that 
 a group of objects possess a certain number when our behavior 
 towards the group is of such and such a type. This is, however, 
 only Husserl's fitting of forms over again and one-one corre- 
 spondence a little disguised. The goal of separating number and 
 
38 The Notion of Number and the Notion of Class 
 
 correspondence is illusory and offers small advantages if gained. 
 Cantor, for example, defined the number of an aggregate as the 
 concept which is attained by abstraction of the nature of the 
 elements and the order in which they are given. If unity is 
 substituted for each element, this new aggregate is regarded by 
 Cantor as a symbolical representation of the number. This 
 view is substantially one which would make the different num- 
 bers our abstract forms "this" and "this" and "this," etc. We 
 however regarded the actual number as gained on the next 
 higher level of abstraction; having the number being in fact 
 that qualify in the group which renders the application of the 
 form possible. 
 
 Such a theory might seem natural enough and an easy con- 
 sequence of the psychology involved — yet one question has 
 been completely begged. The very fundamental assumption 
 made in our discussion of the strata of number-perception 
 (from one "stop" to the next) was that the objects should be 
 homogeneous. What, now, is the significance of the term 
 "homogeneous"? If we say a new object is homogeneous with 
 the old when it is similar to them, the problem is merely re- 
 moved, not solved. If we define the homogeneous as that which 
 occurs between two "stop" acts we fall into a vicious circle. 
 The real state of affairs is that to define the psychological nature 
 of number we have been obliged to assume, in some measure 
 at least, a grouping of the objects of experience independent 
 of and prior to an observing mind. Now such an "independent 
 grouping" is nothing more nor less than the familiar fact that 
 the objects of the universe are arranged in classes. The term 
 homogeneous refers, therefore, to membership of the same class. 
 
 Here again as with the philosophers as Bosanquet and Sig- 
 wart and the logisticians Frege and Russell, from an attempt 
 to explain the notion of number we are brought face to face 
 with the notion of class. Now this notion is not to be accepted 
 blindly as an indefinable, if our short experience of Russell's 
 bewildering treatment of the null-class is to serve as a sample. 
 To sum up our discussion thus far, we can define number neither 
 philosophically, logically, nor psychologically without presup- 
 posing the notion of class. To presuppose that notion blindly 
 is the extreme of rashness, consequently we are forced from 
 the question "What are numbers" to the allied problem, "What 
 are classes?" 
 
PART TWO 
 
 C. The Notion of Class 
 §i. The Viewpoint of the Mathematician 
 
 The ordinary theory of the unreflective mathematician in 
 regard to classes is that of pure extension. When he mentions 
 a point-set, he means just the several individual points, which 
 seem to group themselves into a set by a sort of pre-established 
 harmony. The class is simply its elements; if it has only one 
 member, it is identical with that member and most mathe- 
 maticians would view a class with no members as fictitious, 
 if not absolutely absurd. This position is rather naive and if 
 adhered to entails all manner of complications. For example, 
 on such a theory, how can any statement be made about infinite 
 classes? The very essence of an infinite group is that it cannot 
 be given in its extension. Moreover, the mathematician is 
 constantly using the determining law of the group, although 
 unconscious, perhaps, of its full importance. Couterat at one 
 time asserted boldly that mathematical logic could exist only 
 if the standpoint of extension were used; but this would allow 
 a class to be defined only by enumeration of its terms and would 
 fail to account for an infinite series constituting a whole, in 
 that the n-th term is given by a definite formula. It is no 
 doubt true that the part of mathematics known as the theory 
 of aggregates (" Mengenlehre") regards an "aggregate," — which 
 is only a logical class under another name, — as constituted by 
 the actual catalog of the elements. Here we have, let us say, 
 a case of objects, or terms connected by "and." As plural, not 
 considered as forming a single whole, we shall call such an enum- 
 eration of term, a collection. 
 
 As we have indicated, it is impossible to rest with collections 
 if the entire range of mathematics is to be traversed, for infinite 
 classes do occur most frequently. Moreover, we may as well 
 state once and for all that any such view as that of Russell's 
 in the following passage: "Particular classes, except when they 
 happen to be finite, can only be defined intensionally, i. e., as 
 the objects denoted by such and such concepts. I believe this 
 distinction to be purely psychological ; logically, the extensional 
 definition appears to be equally applicable to infinite classes, 
 
 39 
 
4-0 The Notion of Number and the Notion of Class 
 
 but practically, if we were to attempt it, death would cut short 
 our laudable endeavor before it had attained its goal," — is not 
 that of the present discussion. The complete enumeration of 
 an infinite collection is to be regarded by us not only as repug- 
 nant to psychology, habit and commonsense but absolutely and 
 finally as a flat logical contradiction, to be rejected by both 
 mathematics and philosophy. Any treatment of infinite classes 
 must necessarily involve an intensional element. What is more 
 than this, infinite classes enter into mathematics in so many 
 ways, that the point of view of extension is clearly inadequate 
 and to insist upon it would be pure affectation. 
 
 §2. Frege's Theory of Ranges 
 
 It is one of the chief merits of Frege to have called attention 
 to this breakdown of the theory of extension, and to have advo- 
 cated most earnestly the rival doctrine. He was, however, not 
 so much influenced by the consideration of infinite classes as 
 by desire to define One and Zero. The null-class manifestly 
 does not occur in extension, and it is obvious that if the null- 
 class is absurd, Zero is no better. 
 
 His most important contribution to the refutation of the 
 extensional theory, however, is in the case of the solitary ele- 
 ment, which, viewed from the extensional point of view is identi- 
 cal with its containing class. This has been put in the following 
 form: " If a is a class of more than one term, and if a is identical 
 with the class whose only member is a, then to belong to a, is 
 to belong to a class whose only member is a and hence is to be 
 identical with a. Hence a is the only member of a, which con- 
 tradicts the hypothesis that a is a class of more than one term." 
 This discussion, if correct, absolutely disposes of the theory of 
 extension, in its general aspect. 
 
 Frege's positive theory of classes is one of great complexity 
 and difficulty. It depends mainly upon the primitive ideas 
 function and truth-value. A function is, roughly speaking, the 
 invariant part in some expression, where we regard some sign 
 or set of signs as being replaceable by something else. In 
 x 2 -|-x we do not have a function; the function is: ( ) 2 -f( ) 
 and the x is the argument. When we have a logical function 
 (Russell's propositional function), it can only have the values 
 truth or falsehood and is the same as a BegrifT or concept. If 
 
The Notion of Class 41 
 
 a satisfies the function — if (J) (a) is true — then a falls under the 
 concept. Objects (Gegenstande) are contrasted with functions 
 — anything which is not a function is an object. Examples 
 of objects are numbers and truth-values. The name of an 
 object is a proper name. 
 
 We are now ready for the introduction of ranges (Wertver- 
 laiife). Two functions are said to have the same range when 
 they always have the same value for the same argument. In 
 the case of truth-or-falsehood4unctions we can replace function 
 by concept and range by our old friend "Umfang" for example, 
 "the concept 'square root of four' has the same Umfang as the 
 concept 'that whose square multiplied by three is twelve' " means 
 the same thing as "'x 2 = 4' has the same range as '3X 2 =I2.'" 
 
 Suppose we have now the proposition "For all values of x, 
 (J)(x)=4^(x)." The notion of ranges enables us to change the 
 universality of the equation into a range-equality. Thus 
 x(<J)(x))=x(<{/(x)) in Frege's notation. How important this 
 step is may be judged from his own language. "This substitu- 
 tion may be regarded as unimportant. . . . To such I may 
 recall that in my "Grundlage der Arithmetik" I have defined 
 number as the Umfang of a concept." "The introduction of 
 the symbol for ranges seems to me one of the most notable exten- 
 sions of my symbolism." 
 
 The notion of range, however, still leaves something to be 
 wished. The complete "Grundgesetze" gives us some infor- 
 mation which it might be well to summarize: 
 
 (1) Numbers are ranges and are applicable to ranges. 
 
 (2) Ranges are chiefly useful in reducing the order of a func- 
 tion. 
 
 (3) Falsehood is the range of |(<ce=e) 
 
 (4) Truth is the range of — x. 
 
 (5) "The" a where a is a concept with more than one par- 
 ticular under it indicates the range of a. "The horse is an intelli- 
 gent animal" gives information about the range of the concept 
 "horse." "The present King of France" denotes the range of 
 e(t£=e) or falsehood. 
 
 (6) If a range has only one member, it is to be distinguished 
 from that member. 
 
 We may now pass on to the "Nachwort" of the "Grundge- 
 setze" (pub. 1903) where ranges come again to the foreground 
 
42 The Notion of Number and the Notion of Class 
 
 in a most perverse and disturbing fashion. It seems that Russell 
 communicated to Frege (while the second volume of the " Grund- 
 gesetze" was on the press) the results of his well-known contra- 
 diction that the class of classes not members of themselves both 
 is and is not a member of itself. The " Xachwort" is the result 
 of this communication. It opens with the rather dramatic 
 words: "Einem wissenschaftlichen Schriftsteller kann kaum 
 etwas Unerwunsch teres begegnen, als dass ihm nach Vollendung 
 einer Arbeit eine der Grundlage seines Baues erschuttert wird." 
 
 Frege proceeds to give an account of Russell's discovery, 
 saying that something belongs to a class ("Klasse") if it falls 
 under the concept whose Umfang is the class. The paradox 
 is readily elicited and various ways out of the difficulty sug- 
 gested. Since all these mean substantial modifications in the 
 original notion of range we must consider them: 
 
 (i) It is possible to hold that the law of Excluded Middle 
 does not hold for classes. This would deny that a class was an 
 object while about it there is nothing unsatisfied (argument 
 place) which would destroy its objectivity. If we suppose 
 classes to be pseudo-objects ("uneigentliche"), it must be speci- 
 fied just what functions can take these pseudo-objects for argu- 
 ments; for there certainly is at least one property which classes 
 have, viz., Identity. An interminable vista of distinctions 
 opens up, and we may well despair of giving a complete law, 
 for determining what objects are permissible for any arbitrary 
 function. 
 
 (2) Another method would be to hold that classes are only 
 fictions (Scheineigennamen) which do not have an indication. 
 They would then be parts of symbols, which would only indicate 
 something in connection with the whole — incomplete symbols. 
 Such a notion is nothing surprising or new. The very sim- 
 plicity of every undefined symbol requires that the parts to be 
 detected in it do not mean anything by themselves. However, 
 what does the theory that classes are such parts lead to? We 
 would have to admit that what we are accustomed to regard as 
 numerals would not be symbols, but only meaningless parts 
 of symbols. An explanation of the symbol "2" would be impos- 
 sible; instead of this we would have to explain many symbols 
 which would contain "2" as a part, but are not logically the 
 sum of 2 and another part. It would then be illegitimate to 
 
The Notion of Class 4^ 
 
 represent such a part by a single letter, since by itself it is 
 meaningless. With this the universality of the laws of arith- 
 metic disappears, along with such expressions as ''number of 
 classes," "number of numbers." 
 
 (3) Since these paths seem to lead nowhere, Frege is forced 
 to retrace his steps and investigate the matter on the basis of 
 his own postulates. He gives a symbolic statement of the paradox 
 entirely without the use of the epsilon relation and uses postulates 
 V(b) and 11(b), 1(g), 111(a). It is in one of these that the 
 inconsistency must lurk therefore, 1(g) is to the effect that a 
 and not-a imply not-a. 11(b) is to the effect that what is true 
 of all first order functions with one argument is true of any 
 particular one of them. 111(a) says that if a = b, and b has 
 any property f, a also has this property. 
 
 If any of these foregoing postulates be denied, it is hard to 
 see how any formal reasoning whatever could survive; how 
 Contradiction, Identity, or the Syllogism could be ever legiti- 
 mately employed as logical terms. The trouble therefore must 
 lie in Yb. Postulate V, we recall, assumed the equivalence of 
 equality of ranges with universal equality of functions and was 
 the basis for the argument that even* range has an indication. 
 However Yb, which is suspected, is not so strong; it is merely 
 half of the equivalence, i.e., that the equality of ranges implies 
 the equality of functions. With its downfall, comes the possi- 
 bility that there are concepts which possess no Umfang and 
 that the expression iij>(e) has no indication — "und doch ist 
 eine solche fur die Begrtindung der Arithmetik unentbehrlich." 
 Without Y(b) ranges seem to have vanished from the substantial 
 world and have become trifles light as air. 
 
 With praiseworthy courage Frege seeks to repair the damage. 
 He attempts to generalize the argument, so that instead of going 
 out from Yb and arriving at a contradiction, the falsehood of 
 \~b will be the end attained. 
 
 This effectually disposes of (Y)b at one fell swoop, and proves 
 moreover from the very generality of the argument that it is 
 impossible to give "range" or "Umfang" such a meaning that 
 that half of postulate V could be accepted. What is then to be 
 done? By using a different method of approach, Frege is able 
 to show that there are two concepts which give the same value 
 when used as arguments for the general second-order function 
 
44 The Notion of Number and the Notion of Class 
 
 M ((}>), and under one of these this value falls and under the 
 other it does not. The exception which makes all the trouble 
 is clearly the Umfang itself. 
 
 Since all this analysis was made on the basis of unquestioned 
 postulates, the conclusion may be accepted. It suggests clearly 
 that the criterion for equality of ranges must be revised and 
 indicates a method for evading the contradiction. Frege states 
 the revised criterion as follows: "The Umfang of a concept f 
 is equal to that of a concept g if every object with the exception 
 of the Umfang of f, which falls under f, falls under g and also 
 if every object except the Umfang of g which falls under g also 
 falls under f." This leaves V(a) unaltered, but changes the 
 suspected Vb by the addition of two exceptions. It is now 
 easy to show that the contradiction is avoided. 
 
 We have now plunged through Frege's doctrine of classes 
 and if all has not been transparently clear, we have company in 
 our misery. Bertrand Russell himself confesses quite frankly: 
 "The chief difficulty which arises in the above theory of classes 
 is as to the kind of entity that a range is to be. . . . It would 
 certainly be a very great simplification to admit, as Frege does, 
 a range which is something other than the whole composed of 
 the terms satisfying the propositional function in question ; but 
 for my part inspection reveals to me no such entity. . . 
 Frege's notion of a range may be identified with the collection 
 as one, and all will then go well. But it is very hard to see any 
 entity such as Frege's range, and the argument that there must 
 be such an entity gives us little help. ... It would seem 
 necessary, therefore, to accept ranges by an act of faith, without 
 waiting to see if there are such things." 
 
 The status of ranges seems to be this — we have here a symbol 
 which may be significantly said to have properties (in Frege's 
 usage, possess an indication). It is further determined by 
 certain postulates. The justification of this indefinable seems 
 to rest upon an interpretation of the word range (as limited by 
 postulates) in some sense consistent with experience, which may 
 be limited to some previous objective doctrine of logic, mathe- 
 matics, or philosophy. Frege himself offers no such interpre- 
 tation, which fact proves one of two conclusions: either (a) he 
 could hit upon no justification of the symbol or (b) the notion 
 had become so familiar to him through long use that he could 
 not imagine failure to grasp it. 
 
The Notion of Class 45 
 
 Even if ranges were interpreted, which they are not, other 
 difficulties arise in connection with Frege's logical system. The 
 first of these was noticed by Kerry and arises in connection with 
 the theory of Concepts (Begriffe), Objects (Gegenstande), and 
 Proper Names. Frege's distinctions are so delicate in regard 
 to these matters that he is extremely liable to misconstruction. 
 
 Frege holds that a concept cannot be made a logical subject 
 while an object can be so used; in his own words, "Begriff ist 
 ein mogliches Pradicat eines singularen beurteilbaren Inhalts: 
 Gegenstand ein mogliches Subject eines solchen. " The names 
 of objects are used with the definite, of concepts with the indef- 
 inite article. Kerry thinks that this position is untenable. If 
 we mean by subject grammatical subject such an example as: 
 "The concept of which I was just now speaking, etc.," is suf- 
 ficient to refute it Here we have a concept used as the subject. 
 He goes on to argue that concepts can be shown to be objects 
 in general, for we find no difficulty in speaking of a concept of 
 concepts or of the subordination of one concept to another. 
 Take for instance, "The concept horse is an easily attained 
 concept." Here we have both subject and predicate concepts 
 with no object entering at all. 
 
 Kerry's objections were answered by Frege in an article 
 "Uber Begriff und Gegenstand." He contends that even if 
 Kerry's criticisms were themselves correct, the possibility of 
 the existence of a Begriff which could not be a subject, and a 
 Gegenstand which could not be an object, must be recognized, 
 the distinction being available as needed. What is of more 
 weight, however, is his claim that Kerry's argument is mistaken. 
 While it is of course true that we may have a concept falling 
 under another concept, this must not be taken to mean that 
 the concept itself is used as the subject. Here we have the proper 
 name of the concept not the function itself. In "the concept 
 horse is an easily gained concept" we are talking about the name 
 of the concept, it is possible to indicate this usage by appropriate 
 quotation marks. 
 
 To make plain what Frege means by this, we shall have to 
 revert to his theory of proper names, where he and Russell are 
 in flat disagreement. He distinguishes between the meaning 
 (Sinn) and the Bedeutung or Indication of a term; this distinc- 
 tion is interpreted by Russell as being roughly equivalent to 
 
46 The Notion of Number and the Notion of Class 
 
 the difference between a concept as such and its denotation. 
 Thus "the first President of the United States" and "the father 
 of his country" have the same indication but not the same mean- 
 ing. If we wish to speak of the meaning of a term we must 
 signify this by notation, otherwise the indication is considered 
 to be meant. "A proper name expresses its meaning and indi- 
 cates its indication." In Frege's theory, every concept has a 
 proper name which indicates the denotation of the concept 
 but, as name, expresses the meaning of the concept. In the 
 subsumption of one concept under another, it is not the indi- 
 cation of the former but its meaning that we are concerned with, 
 hence it is the proper name itself which has the relation. In a 
 footnote we find the following significant observation ". . . 
 und nicht 'der Begriff $(£).' Die letzten Worte bezeichnen 
 also eigentlich nicht einen Begriff in unserem Sinne, obwohl 
 es nach der sprachlichen Form so aussieht." Every proper 
 name has the two sides, can be used predicatively as well as 
 in the role of subject. Truth and Falsehood which are assumed 
 to be objects do not have proper names as such, so we arbitrarily 
 give them the proper names ( — x) and (x = term not identical 
 with itself). This is in direct conflict with Russell's theory 
 of denotation, as we shall see later. 
 
 We can see now why Frege does not use a function quad- 
 ratically. If we have cj>[4>(x)], the argument is not the indi- 
 cation of the function but its proper name; consequently the 
 revision of postulate V appears to be justified, as a translation 
 into symbolism of the distinction between meaning and indi- 
 cation. The significance and sweeping importance of this 
 distinction seems to have passed over Russell's head. 
 
 Russell goes on to give what he thinks a destructive example 
 of Frege's notion of function, as independent of an expressed 
 variable. Consider the identical function of x, namely x, if 
 this were indicated by no variable, when we abstract, there 
 would be nothing left, hence no function. Frege, however, 
 did not admit x as a function of x; its place is filled by ( — x) 
 (x assumed) which is always a function whether x is itself or 
 not. 
 
 We have now gone far enough to appreciate the profundity 
 and breadth of Frege's analysis. His deficiencies are largely 
 due to failure to see the wide-reaching implications of his own 
 
The Notion of Class 47 
 
 distinctions. He has re-introduced meaning into logic and 
 sees clearly the double side of concepts, and proper names, 
 although the treatment given is rather formless. Ranges remain 
 mysterious, but necessary and their interpretation a problem 
 for our own undertaking. All in all, Frege seems to have walked 
 unscathed among the pitfalls into which Russell has fallen and 
 to have maintained an unshaken serenity in the face of tre- 
 mendous complications. His solution of the paradox, moreover, 
 at least on the face of it, is consistent with his general logical 
 doctrine. The neglect of his contemporaries may have been 
 due to his cumbrous symbolism and the obscurities of his deli- 
 cate analysis but we may look to the future to avenge him, for 
 it would be scarcely too much to say that no more brilliant, 
 subtle, or original work on the foundations of logic has appeared 
 since the days of Aristotle. 
 
 §3. Russell's Infinite Variety 
 
 When we turn from Frege to Bertrand Russell, it is very 
 much as if we should glance away from the regular symmetry 
 of some Greek temple and contemplate the myriad ramifications 
 of a Gothic cathedral. The thought of Frege is not without a 
 certain majestic sweep as it continually rises above its former 
 incompleteness and develops by virtue of its own imperfections. 
 With Russell, on the other hand, a bewildering variety of the- 
 ories follow upon one another in such dazzling profusion that 
 the result is not so much to convince the reader as to hypnotize 
 him. No sooner does a conclusion receive its full and persuasive 
 enunciation than the possible alternatives and objections crop 
 up; to be repudiated or sympathized with, as the case may be. 
 The greater part of the "Principles of Mathematics," therefore, 
 does not resemble a constructive theory so much as a Parlia- 
 mentary debate, and gives the impression that its author is a 
 brilliant, but unsafe guide to mathematical philosophy. 
 
 We caught some hint of this erratic tendency in Russell while 
 we were considering his theory of the null-class. His general 
 theory of classes suffers the same changes as the null-class. 
 We start out in the "Principles" with the dictum that the exten- 
 sional view of classes is necessary for symbolism and that the 
 failure of this view in regard to infinite classes is "purely psy- 
 chological." However, there is a distinction between the class 
 
48 The Notion of Number and the Notion of Class 
 
 as many and the class as one, and this distinction may be enough 
 to solve the paradox, although it does not distinguish a unit 
 class from its only member. The notion of propositional func- 
 tion is declared to be the genesis of classes. Since it is, as a 
 function, neither true nor false, but indeterminate and may be 
 true for certain values and false for others. Thus a class is 
 determined — a collection of values for which it is true. Equiva- 
 lent propositional functions, i. e., true for the same value, deter- 
 mine the same class. Russell calls this point of view modified 
 extension. The many variations of this theme in the "Prin- 
 ciples" we have already found in the treatment of the null class. 
 
 Under the pressure of his paradox and Frege's seemingly 
 irrefutable argument that the unit class is not to be identified 
 with its member, Russell is compelled to change his theory, 
 of classes in the same volume, while considering Frege's ranges. 
 He says: "Nevertheless the non-identification of the class with 
 the class as one . . appears unavoidable, and by a process of 
 exclusion the class as many is left as the only object which can 
 play the part of a class. By a modification of the logic hitherto 
 advocated in the present work, we shall, I think, be able at 
 once to satisfy the requirements of the Contradiction and to 
 keep in harmony with common sense." Back in the sixth 
 chapter occurs the following: "A plurality of terms is not the 
 logical subject when a number is asserted of it; such propositions 
 have not one subject, but many subjects." This seems to be 
 directly contradicted by the identification of class with class 
 as many. 
 
 Russell goes on to recapitulate the possible theories of classes 
 which present themselves. His enumeration is sufficiently 
 important for our purposes to justify repetition. A class, he 
 says, may be identified with: (i) the predicate; (2) the class 
 concept; (3) the concept of the class; (4) Frege's range; (5) 
 the collection of terms; (6) the whole composed of the terms 
 of the class. The first three are inadmissible, since they do not 
 permit a class to be defined by the enumeration of its members 
 Frege's range is declared to be incomprehensible. A collection 
 cannot be a class because it is grammatically plural, and (6) 
 is refuted by the argument that the singular class is not to be 
 identified with its only member. 
 
 Having thus demolished all previous theories, Russell finds 
 himself in a quandary. After considering and rejecting various 
 
The Notion of Class 49 
 
 alternatives, he declares himself thus: "The logical doctrine 
 which is thus forced on us is this: the subject of a proposition 
 may be not a single term but essentially many terms . . but 
 the predicates or class concepts or relations which can occur 
 in propositions having plural subjects are different from those 
 having single terms as subjects. Although a class is many and 
 not one . . classes can be counted as though each were a 
 genuine unity; and in this sense we can speak of one class. It 
 will now be necessary to distinguish (1) terms; (2) classes; (3) 
 classes of classes and soon. We shall hold . . that no mem- 
 ber of one set is a member of another and that x s u requires 
 that x should be of a set a degree lower than the set to which 
 u belongs." 
 
 He asserts that it will be necessary to indicate whether the 
 field of every variable is terms, classes, or classes of classes, etc., 
 arriving finally at a more extensional view than before, finding, 
 "that the class as many is the only object defined by a preposi- 
 tional function . . that the class as one is probably a genuine 
 entity except when the class is defined by a quadratic function." 
 
 Later, in Appendix B, a slightly different point of view is set 
 forth, which elaborates the theory of logical types previously 
 hinted at. This is substantially a rigid distinction between 
 terms, classes of terms, etc. The different possibilities are rather 
 carefully examined and all goes well until §500 is reached, when 
 the dread bugbear, the Contradiction, crops out again, this 
 time in considering propositions as a type. Thus, although the 
 contradiction of classes is solved by the doctrine of types, a 
 closely analagous difficulty is raised in the doctrine itself. "It 
 seems possible, of course, to hold that propositions themselves 
 are of various types and that logical products must have propo- 
 sitions of only one type as factors." This alternative is de- 
 nounced as "harsh and artificial," consequently the notion of 
 class remains paradoxical at the end of the very last Appendix 
 of the "Principles." 
 
 The next important sign of Russell's increasing passion for 
 extension and growing distrust of the notion of class is found 
 in his article, "On Some Difficulties in the Theory of Transfinite 
 Numbers and Order-Types." Here he again contemplates the 
 paradoxes and is able to develop a great budget of similar ones 
 by a common recipe. 
 
5<3 The Notion of Number and the Notion of Class 
 
 Three ways are open, he finds, for the logician who wishes to 
 avoid paradox and contradiction. They are, following his 
 designation: (i) the zigzag theory; (2) the theory of limitation 
 of size; (3) the no-classes theory. The zigzag theory presumes 
 that logical functions determine classes when they are simple 
 "and only fail to do so when they are complicated and recondite." 
 Its great disadvantage is that the postulates determining which 
 functions are to determine classes and which shall not, are 
 necessarily complicated, difficult and without much plausibility. 
 Indeed the great guiding principle in the adopting of any definite 
 axiom of this kind is that it avoids a well known paradox. This, 
 our author admits, "is a very insufficient principle, since it 
 leaves us always exposed to the risk that further deductions 
 will elicit contradictions." 
 
 The theory of limitation of size is based on the idea that it is 
 "size which makes classes go wrong." This theory becomes 
 specialized into the assumption "that a proper class must always 
 be capable of being arranged in a well-ordered series." The 
 great drawback is that we are led to a sort of trial and error 
 process. The No-classes theory remains to be considered, and 
 although it is not definitely adopted at this time, is evidently 
 viewed with greatest favor. Its procedure is brief, stringent, 
 and "drastic." Classes and relations are to be straightaway 
 banished from logic; everything is to be expressed in terms of 
 proposition, function and truth or falsehood. This process is 
 admitted to be complicated and difficult, not to say somewhat 
 startling to common sense. A degree of safety is claimed for 
 it, however; although at the time of writing (before Nov. 24, 
 1905) it is not espoused. Russell is still disposed to cling doubt- 
 fully to the ordinary logic. 
 
 A footnote dated February 5th, 1906, reveals the decided 
 break: "From further investigation I now feel hardly any doubt 
 that the no-classes theory affords the complete solution of all the 
 difficulties stated in the first part of the paper." The decision 
 has now been made. Classes are banished, and Russell has 
 from that celebrated "lair" "midway between extension and 
 intension" come out unqualifiedly for extension. 
 
 The character of these investigations was revealed in articles 
 in the "Revue de Metaphysique et de Morale" and the "Ameri- 
 can Journal of Mathematics." In these the theory of types of 
 
The Notion of Class 51 
 
 Appendix B is united with the no-class theory and a formal 
 disproof of the contradictions attempted. This treatment 
 appears in very nearly unaltered form in the "Principia," and 
 its abolition of classes needs careful consideration. Russell's 
 point is that classes are incomplete symbols. He neglects 
 absolutely to reply to Frege's objections to this supposition. 
 
 On p. 69 of the "Principia" we find an incomplete symbol 
 defined as a symbol which is not supposed to have any meaning 
 in isolation but is only defined in certain contexts. Such incom- 
 plete symbols are to be used differently from proper names. 
 Here we may object. Proper names by no means have a con- 
 stant meaning independent of context. We can agree with 
 Russell when he says that "Socrates" is a proper name which 
 constantly denotes an individual, only if we suppose "Socrates" 
 to denote the physical man; if we discuss Greek history and 
 mention the Socrates of the "Memorabilia" and the Socrates 
 of the Platonic dialogs and the youthful Socrates and so on, 
 here "Socrates" has anything but a constant meaning. 
 
 Russell goes on to exhibit a kind of proposition in which the 
 subject, while purporting to be a proper name, because preceded 
 by the definite article "the" seems to be masquerading under 
 false colors. The cited case is: "The round square does not 
 exist" and we are told that we can not regard this proposition 
 as a denial of the existence of the object "round square." "For 
 if there were such an object it would exist; we cannot first assume 
 that there is a certain object and proceed to deny that there is 
 such an object." Where the asserted proposition — such an 
 object is implies that such an object exists" — comes from, is 
 left to the reader's imagination. The two words are confused 
 throughout the passage in question. To disentangle ourselves 
 from the meshes of this equivocation we shall have to make use 
 of a distinction, which Russell himself was once very scrupulous 
 in making. 
 
 On p. 449 of the " Principles" we find this illuminating passage. 
 "Being is that which belongs to every conceivable term, — to 
 everything that can possibly occur in any proposition, true or 
 false, and to such propositions themselves. . . . Being 
 belongs to whatever can be counted. To mention something 
 shows that it has being. Existence, on the contrary, is the 
 prerogative of some only among beings. To exist is to have a 
 
52 The Notion of Number and the Notion of Class 
 
 specific relation to existence." We see from this that the class 
 of existents is a sub-class of beings. A round square is denied 
 by the proposition to fall in this sub-class. The proposition 
 "The round square is not" is an incomplete symbol; "The round 
 square does not exist" is not such. 
 
 If we accept the distinction between being and existence (and 
 surely no good reason has been advanced for discarding it) 
 the situation presents no difficulty and there is no need for 
 analyzing out the grammatical subject as our author proceeds 
 to do. The argument goes on to show that such "the" phrases 
 are always incomplete symbols, but a like confusion infests 
 the proof. Here the illustration is "Scott is the author of 
 Waverley" and the argument is as follows. "Scott" is a genuine 
 proper name, and "the author of Waverley" if a complete symbol 
 denotes "Sir Walter Scott" also, giving the proposition "Scott 
 is Scott" which is denounced as trivial. Hence it is concluded 
 summarily that "the author of Waverley" must be an incom- 
 plete symbol. 
 
 We may object in two ways, (a) The principle of Identity 
 is not so trivial as is commonly supposed. Boyce Gibson says 
 in his "Problem of Logic" — "To state a proposition we must, 
 to put the matter quite generally, specify our meaning. If we 
 wish to make a definite statement of fact we must first specify 
 that aspect of the total topic which we wish particularly to 
 speak about; this will give us the subject of our statement. We 
 have then to specify this subject by predicating something about 
 it that is other than itself. The Principle of Identity will be a 
 Principle of Identity in relation to difference." Thus what we 
 are doing in a statement of identity is connecting contexts, 
 (b) Moreover in "the author of Waverley" and "Scott" we 
 do not have a proper name and a denoting phrase but two denot- 
 ing phrases and our proposition merely asserts their equivalence, 
 that is, says that they have the same denotation. Why our 
 customary proper names should be given an exalted rank is 
 hard to see; they do not individuate in the true philosophical 
 sense any more than denoting phrases. 
 
 Most of the difficulty is due to Russell's doctrine of individuals, 
 which is in some respect necessitated by the theory of types. 
 We can only make a distinction between types as n and n+i if 
 there is a first type; this must be the type of individuals. We 
 shall return to this later. 
 
The Notion of Class 53 
 
 Another supposed puzzle is exhibited in order to justify the 
 peculiar theory of denotation. This is "The present King of 
 France is bald." The present King of France not existing, 
 the statement is paradoxical. However, we may reduce this 
 to the problem, whether we are allowed to introduce non-ex- 
 istents into arbitrary existent classes, and this current logical 
 usage appears to allow. What Russell desires to do is to convert 
 the proposition into "The King of France exists and is bald," 
 claiming to find his justification in ordinary usage. Ordinary 
 usage is not a safe guide in connection with existence, however, 
 for it is prone to identify existence with the physical world 
 order, and the circle, qua perfect circle, would not exist any 
 more than the present King of France. 
 
 We may, for example, interpret the phrase as meaning "the 
 rightful King of France" and thus have an existent subject. 
 This shows that a phrase may not denote an existent when we 
 consider a certain limited context or interpretation, but often 
 the sphere may be widened enough for the proposition to be 
 definitely interpreted. No statement of existence can legiti- 
 mately be imported into the proposition. What we have is the 
 statement of membership in a class. If we hold that an existent 
 class can only have existents as members, this would settle the 
 question, but that is another problem. 
 
 Classes, we. are told, are incomplete symbols; they themselves 
 do not mean anything at all. "Thus classes, so far as we intro- 
 duce them, are merely symbolic or linguistic conveniences, not 
 genuine objects, as their members are, if they are individuals." 
 Russell admits that no proof is forthcoming that classes are, as 
 stated, incomplete symbols although "arguments of more or less 
 cogency" are adduced, all of which are equivalent to the assump- 
 tion that the same object cannot be both one and many, which, 
 observes our author, "seems impossible." If it is impossible, 
 a formal proof should be available; if it is mere disbelief and 
 want of faith, such matters of psychology should not be allowed 
 to intrude. 
 
 Given two propositional functions which have the same truth 
 value for the same argument, i. e., are formally equivalent, it 
 seems obvious that they have something in common, — namely 
 that which we would naturally suppose to be the class, — but 
 we are forbidden to rush to the conclusion that this is the case. 
 
54 The Notion of Number and the Notion of Class 
 
 "We do not assume that there is such a thing as an extension; 
 we merely define the whole phrase 'having the same extension." ' 
 What Russell does is to say that say that any function f of a 
 class is equivalent to the corresponding function of a predicative 
 function formally equivalent to the one which defined the class. 
 In this way classes are regarded as having been avoided. How- 
 ever, this process of defining any function of x without defining 
 x appears to be invalid. Let us recall that in criticism of Frege, 
 Russell used the so-called identical function of x, namely x 
 itself; this function overthrows his evasion of the notion of class. 
 For regarding the class as a function of the class we see it defined 
 as the predicative function, which is, in fact, gained by the axiom 
 of reducibility. Here is really an extremely intensional view 
 of classes, for which there appears little philosophical defense. 
 
 The existence of Individuals is deemed of fundamental impor- 
 tance. That the axiom which insures it, is convincing, is hardly 
 to be conceded. For if any term is an individual, it cannot be 
 a function or class and in ordinary usage no such "principium 
 individuationis" appears. The facts of the case seem to be 
 that the term individual is essentially relevant to the universe 
 of discourse. As we have seen, Socrates is in one "sense" or 
 universe an individuating term, in another, a universal. As for 
 the absolute individual, which is to be a universal in no universe 
 of discourse — this is a concept which appears to involve insuper- 
 able difficulties. 
 
 As far as considering the theory of types to be a disguised 
 theory of the universe of discourse, we have seen that this theory 
 necessitates a first type, — individuals — and individuals only 
 exist in reference to a universe of discourse. How can the 
 theory of types explain something which it presupposes? This 
 problem has been dexterously avoided. 
 
 Russell then has, in effect, while ostentatiously rejecting 
 classes, regarded them as equivalent to certain propositional 
 functions. Frege has regarded them as ranges, vague and 
 unexplained notions. We can be satisfied with neither. Rus- 
 sell's ideas of types and individuals infect his theory of classes, 
 and Frege's notions are lacking in justification. We shall turn, 
 therefore, to current mathematical usage, for clues by which 
 to investigate the notion of class, with which the number con- 
 cept seems to be inseparably connected. 
 
The Notion of Class 55 
 
 D. Mathematical Usage 
 
 We must now examine the most formal and rigorous procedure 
 of the mathematicians. They customarily hold a mathematical 
 science to be dependent upon (1) the undefined symbols, (2) 
 the general axioms, (3) the existence postulates. The axioms 
 provide schemes of classification of the undefined symbols, and 
 suggest means for defining unwieldy combinations in terms of 
 these undefined symbols. After investigating the various pos- 
 sible classifications which interest him, these being tabulated 
 under the name of theorems, the mathematician becomes con- 
 cerned with a critical survey of his initial suppositions. It is 
 important to note that so long as he is merely establishing theorems, 
 the mathematician is only connecting arbitrary symbols (the inde- 
 finables) by the means of arbitrary rules — the postulates. 
 
 From this new and later point of view, however, he wishes 
 to investigate his postulates with a view as to their consistency, 
 categoricity, and mutual independence. Consistency is the 
 consideration of first and foremost importance. If a set of 
 postulates is inconsistent, it must be revised or discarded. What 
 is the test of consistency? Two methods are generally accepted: 
 
 (1) A set of postulates is said to be inconsistent if contra- 
 dictory statements can be made about the undefined symbols, 
 these statements being correctly deduced from the whole or a 
 subset of the assumptions in question. 
 
 (2) Such a set is said to be consistent if some interpretation 
 of the undefined symbols can be found which satisfies the postu- 
 lates. 
 
 It is important to remember that these are the only criteria 
 of consistency which are accepted. If we look at them more 
 closely we can see that both (1) and (2) directly depend upon 
 the ingenuity and perseverance of the mathematician who is 
 working upon the system. For example, a contradiction may 
 lurk hidden among the implications of some obscure theorem, 
 whose possibilities no one has ever thought of investigating. 
 From this point of view the mathematician is always walking 
 upon the brink of a precipice, for, no matter how many theorems 
 he deduces, he cannot tell that some contradiction will not await 
 him in the infinity of consequences. Let us suppose, then, that 
 as far as he has gone, no inconsistency has been come upon, and 
 being dissatisfied with the indefinite conclusion that the postu- 
 
56 The Notion of Number and the Notion of Class 
 
 lates may or may not be consistent, he grasps the other possi- 
 bility, and undertakes to demonstrate their consistency. 
 
 Now it devolves upon him to find an interpretation for his 
 arbitrary words such that the postulates still hold. The only 
 way of doing this is by recourse to the accepted conclusions of 
 some other set of postulates. The problem is thus shifted from 
 one domain to another, and the only thing the mathematician 
 is justified in saying is that his set is just as consistent as the set 
 in which the interpretations occur. This process can go on 
 until he can establish, it may be, that his set is just as consistent 
 as the axioms for the natural number system. Back of this 
 the non-philosophic mathematician does not care to go, nor 
 is it evident that continuing the search would lead to any profit- 
 able results. 
 
 Now our survey of this attempt of the mathematician to 
 interpret his "words" makes it plain that he is reducing his 
 system to a system involving defined combinations of the unde- 
 fined symbols of some other branch of the science. He is assum- 
 ing the formal equivalence of his symbols to these particular 
 combinations; that is, he feels himself justified in substituting 
 the "interpretation" for the undefined symbol wherever it 
 occurs and vice versa. Suppose now that no such scheme of 
 equivalences occurs to the mathematician. It is entirely con- 
 ceivable that, for some length of time, the science may contain 
 no inconsistency and yet receive no interpretation, being, so 
 to speak, in a delicate position. Moreover, even when inter- 
 preted to its fullest extent, it is just as consistent, neither more 
 nor less, as the system of positive integers. That the number 
 system contains no contradiction is the fundamental rock of 
 mathematical faith. 
 
 The question of categoricity is closely allied. We call a set 
 of postulates categorical if there is only one interpretation to 
 be given to the undefined symbols or if any two such systems 
 of interpretations can be made simply isomorphic. Now the 
 task of examining all possible interpretations and rejecting 
 some as not satisfying the postulates, and others as being essen- 
 tially and abstractly the same as the system of interpretations 
 we wish to consider categorical, is not a process which can be 
 completed in any finite length of time. As in the case of con- 
 sistency, no matter how far and how favorably the investigation 
 
The Notion of Class 57 
 
 progresses, some residuum of doubt must always remain. Such 
 being the situation, the mathematician is regarded as having 
 demonstrated the categoricity of his set of postulates when he 
 has proved that they possess the categoricity of the axioms for 
 the number system. The consideration of independence forces 
 us to a somewhat different point of view. Here we wish to show 
 that each assumption in turn cannot be deduced from all the 
 rest; in other words, that no one of the axioms is redundant. 
 This is, in effect, the opposite of categoricity. The more postu- 
 lates used, the more they will tend to be categorical and not 
 independent. The happy medium is the goal of the mathe- 
 matician — a set which contains no redundancies, and is abso- 
 lutely categorical. An assumption Pi, let us say, is said to be 
 independent of the other assumptions Pa . . P n when an 
 interpretation can be given to the undefined symbols which 
 satisfies P 2 . . P n but not Pi. This particular interpretation 
 constitutes an independence example for Pi. 
 
 A delicate issue is raised here. How can we be sure that 
 P 2 . . P n are really satisfied by the independence example? 
 It is clearly necessary that this example be contained within 
 the scope of a science determined by postulates which are both 
 consistent and categorical. Here again we are driven from one 
 domain to another, and if we wish to show unqualifiedly the 
 validity of an independence example, we must be led, as before, 
 back to the number system. 
 
 We must make special mention of the method of so-called 
 vacuous satisfaction, which is often used in setting up inde- 
 pendence examples. Suppose we have an axiom to the effect 
 that "two distinct points determine a straight line" and we 
 are considering the geometry of a single point. Here there is no 
 distinct second point, but we say that this unit space has the 
 property that every two points determine a straight line. The 
 axiom is vacuously satisfied by a non-existent point and a non- 
 existent line. Now the curious part of the reasoning involved 
 is this, — if the contradictory axiom "two points never determine 
 a straight line" were under consideration, this would also be 
 vacuously satisfied by the space of a single point. Now this is 
 somewhat of a departure from the attitude of naive common 
 sense and we must be prepared to admit that accepted mathe- 
 matical method involves to some extent the logical doctrine 
 
58 The Notion of Number and the Notion of Class 
 
 that a false proposition implies any arbitrary proposition, and 
 its analogues for classes. 
 
 The denoting terms used in mathematics merit especial atten- 
 tion. Let us take a familiar example from elementary analytic 
 geometry. Suppose we wish to deduce the equation of a curve. 
 We must first know how that curve is generated by conditions 
 on a representative point; we say the curve is the locus of points 
 such that this condition is satisfied. The coordinates of this 
 general point which typifies every point on the locus are taken 
 to be (x,y) and it is required to connect x and y by a functional 
 relation determined by a given condition on the general point. 
 First we have the condition, second the general element and 
 then the equation, the collection of particular points on the 
 curve. We derive this equation by a significant procedure. 
 We single out a constant point P which is any definite point 
 on the locus and assume its coordinates to be (xi,yi). From 
 the condition on the general point, we arrive at a functional 
 relation connecting Xi and yi ; and then, since P was any definite 
 point on the curve, we replace xi and yi by x and y, giving the 
 equation. Now it might be said that the point (x,y) is really 
 any point on the locus, and stands for each particular one; so, 
 since (xi,yi) also was any point, it must be identical with (x,y). 
 This contention has only superficial and verbal plausibility. 
 The two expressions, considered as denoting terms, must be 
 carefully differentiated. In the first place, (x,y) is not a con- 
 crete, particular point at all. If the expression is permitted, 
 it is an abstract point which satisfies the common condition, 
 generates the points of the curve but is not one of them. 
 P(xi,yi), on the other hand, is one of the points of the curve, 
 although not a specifically designated one. In the case of P, 
 a set of points exists which are on the curve and which do not 
 include P. Where the typical or generating point is considered, 
 no such set exists; a collection of points independent of its gener- 
 ator would be self-destructive. Whatever can be proved of 
 the general term is certainly true of P; but the converse does 
 not follow. We distinguish therefore between the indefinite 
 any (in the sense of the variable), any definite member, the 
 condition given at the start, which may be regarded as the predi- 
 cate common to the points on the locus and the curve as a single 
 whole regarded as the range of the variable point. We have 
 
The Notion of Class 59 
 
 also concluded that mathematical reasoning is concerned primar- 
 ily with words considered as mere symbols or marks and their 
 possibilities of interpretation. 
 
 §1. The Mathematical Universe 
 
 If we adhere to this semi-verbal character of accepted mathe- 
 matical reasoning, it is possible to take a somewhat broader 
 view of the method and content of mathematics than has been 
 the fashion among the logisticians. Ordinary experience pro- 
 vides us with a vast array of propositions (taken in the verbal 
 sense). With how these propositions are acquired we are not 
 concerned ; suffice it that they are there, — the objective, common 
 property of those with whom logical argument is possible. Where 
 there is no such common store of propositions, no logic, no code 
 of agreement or disagreement is possible. With a native of the 
 South Sea Islands, for example, no logical disputation is possible. 
 
 This array of propositions as presented by individual experi- 
 ence, common sense, and inductive science is undeniably loose 
 and rambling in its structure. We have the same word serving 
 a variety of uses and we feel that between such propositions as 
 "2+2 = 5" an d "the present King of France is bald" there is 
 absolutely no connection. Now from our point of view, the 
 function of logical method is just this, — to arrange the heterogen- 
 eous concatenation of propositions in a systematic order; to 
 separate the disconnected propositions into different universes 
 of discourse. 
 
 The postulates of any particular code of logic constitute a 
 specific definition of a systematic universe of discourse; and the 
 failure of any part of an agreed-upon code is taken to be an 
 evidence that the limits of the universe in question have been 
 overpassed. For example, the propositions "A is blue," "A 
 is red," "A exists," where red and blue are taken to be contra- 
 dictory, cannot be contained in the same systematic universe. 
 We conclude that the first and second must be taken in discon- 
 nected universes, psychologically speaking, from different points 
 of view and revise them thus, to put them in the same universe 
 "A is blue at time x," "A is red at time y." The goal of logical 
 method, in general, is an exact and accurate statement and 
 analysis of our rudimentary and vague notion of a systematic 
 universe of discourse. 
 
60 The Notion of Number and the Notion of Class 
 
 Some such theory of the universe of discourse is implicitly 
 presupposed by any logic and may be summed up: "Any prop- 
 osition which violates the logical canons can be split up into a 
 complex of propositions to be taken in different universes." 
 The code must at all costs be preserved. Puns must be handled 
 in this way, for instance. We are not allowed to attribute 
 existence to that which has contradictory predicates (as in the 
 case of blue and red) but are forced to divorce the predicate 
 if we wish the existence to remain. Now existence is not a 
 quality, like extension, magnitude, or color. The size or color 
 of an object may well differ in different universes, but it has 
 or has not existence in a given universe. In the sphere of a 
 lecturer on Modern Drama, Hedda Gabler has just as real and 
 objective an existence as anything you please, while to a chemist 
 she is relegated to the domain of the round-squares and the 
 chimaeras. Any word which violates the law of contradiction 
 when the two contradictory propositions are united in a common 
 universe cannot have existence attributed to it. 
 
 From the mere absence of contradiction, however, existence 
 is not to be inferred. A nonsense word can be given existence 
 in a universe, but does not possess existence of its own accord 
 unless this is implied by accepted postulates. Hence the pres- 
 ence of the existence postulates in a mathematical system. 
 They give a beginning to the collection of known existents, 
 which is added to by use of the other postulates. Roughly 
 speaking, existence is nothing more nor less than membership 
 in a certain class. Following Russell (in 1903), we will hold 
 that words which do not exist in a given universe have being 
 with regard to that universe. Any definite word has being with 
 reference to any definite universe we choose to add it to, and, it 
 may be, existence in some. The round-square has being in 
 the universe of metrical geometry; it may have existence where 
 "square" is interpreted in the sense of "public square" as the 
 space between intersecting streets. Thus a public square which 
 is round is not a paradox if we find the right universe. Obviously 
 this new universe and geometry are disconnected, at least so 
 far as the term "square" is concerned. 
 
 We can see now just what mathematicians mean when they 
 say that along with their particular axioms they assume the 
 postulates of logic. They mean that the universe of discourse 
 
The Notion of Class 61 
 
 with which they are about to deal is a systematic universe, subject 
 to the law of contradiction. 
 
 Assertion is to propositions what existence is to terms not 
 a quality or attribute but containing an essential reference to 
 the universe of discourse. An asserted proposition occurs as a 
 unit factor in an implication and can always be dropped. Axioms 
 of a science are the fundamental asserted propositions of the 
 universe they define, and their assertion is not absolute, but 
 relative. Assertion is not the same as truth, which in fact is a 
 highly controversial idea and seems to involve notions extraneous 
 to pure logic. When we consider "A is B" as a proposition, 
 it may imply its contradictory in one universe and be implied 
 by it in another. In the first case "A is not B" is asserted, in 
 the other "A is B." Propositions which are implied by their 
 contradictories are unit propositions with respect to the particu- 
 lar universe; those which imply their contradictories are null 
 propositions. 
 
 One of our most fundamental demands that we make upon 
 a systematic universe is that with regard to it the same prop- 
 osition cannot be both unity and null. When such a state of 
 affairs occurs we have the choice of two alternatives: (i) either 
 we assume that somehow the unit part and the null part of the 
 contradiction occur in different universes and the proposition 
 cannot be asserted without some change which expresses this 
 fact, or (2) the axioms of our logic are insufficient to define a 
 truly systematic universe and must be revised to meet the 
 situation. One or both of these methods must be adopted in 
 dealing with Russell's paradoxes. 
 
 §2. The Internal Structure of a Universe 
 
 We have in any universe words which play the part of con- 
 ditions or predicates. Other symbols satisfy these conditions 
 Conditions which are satisfied by one term are concepts; by 
 couples with sense, relations; by trios with betweenness, operations 
 or triadic relations. The collection of symbols which satisfy 
 a condition P, considered as forming a single whole, constitute the 
 range of P. 
 
 Suppose that within some systematic universe we have a well- 
 defined concept C. Its range will be said to constitute a connected 
 universe of discourse if any definite symbol satisfying C occupies 
 
62 The Notion of Number and the Notion of Class 
 
 only one place with regard to the relation of satisfaction. Thus 
 if A satisfies C, A satisfies B, C satisfies A; the A of the first two 
 propositions is not in the same connected universe as the A of 
 the last. A term therefore cannot satisfy itself, unless we specify 
 by some notation that the first term of the relation is in a different 
 universe from the second. The same restriction holds upon the 
 range of A; it cannot satisfy A without some specification. "Soc- 
 rates, the philosopher, is human" and "Human is a word having 
 five letters" are propositions occuring in different connected 
 universes. 
 
 Russell's conception of the absolutely unrestricted variable 
 does not appear to be needed. He says that in " 'x is a man' 
 implies 'x is a mortal,'" it is only a vulgar prejudice in favor 
 of true propositions which restrains us from substituting bicycles 
 and teaspoons for x. This may be doubted; and from our 
 (intensional) point of view what we assert in the "formal impli- 
 cation" in question is a relation between the concepts "man" 
 and "mortality" which we may call the relation of inclusion. 
 The so-called "apparent" variable appears to be a confusion 
 in terms. 
 
 Where we would say "for all values of x, x satisfies C," we 
 say "C is a unit concept"; "for no values of x, x satisfies C" or 
 "for all values of x, x satisfies not-C," — "C is a null concept." 
 If C is a unit concept, not-C is null, and vice versa. " For some 
 values of x, x satisfies C" becomes "C is not null" and "for 
 some values of x, x satisfies C," — "C is not unity." 
 
 To express relations between concepts we make use of denot- 
 ing terms, which together with concepts form denoting com- 
 plexes. Where C is a condition and x is the variable term which 
 stands for any term of the range of C, for this variable term 
 we use the denoting complex "every-C." If x is a fixed term 
 but not specified by particular properties, we replace it by 
 "any-fixed C." 
 
 Where "every-B satisfies A" we say that the range of B is 
 included in the range of A; similarly for "any fixed-B satisfies 
 A." 
 
 In the term "any-fixed" no particular element is understood 
 to be specified; we have something analogous to the constant 
 in indefinite integration which can be said to vary, but is none 
 the less a constant and not a variable. There is an ambiguity 
 
The Notion of Class 63 
 
 about a parameter which is distinct from the essential ambiguity 
 of the variable and which is essential to mathematical usage. 
 Consider ax+by+c=o. For fixed values of a,b,c we have a 
 definite equation; taking into account the ambiguity of the 
 parameters we see that we have a triply infinite set of equations. 
 In this discussion x and y are regarded to maintain a placid 
 and invariant state of variability if we may be pardoned the 
 paradox; a,b,c are seen to vary in different ways, though always 
 remaining constants. In the case of a concept C with only a 
 finite number of terms Si . . . S n , "every-C" denotes Si 
 and denotes S2 . . . and denotes S n , "any-fixed-C" denotes 
 Si or S2 or . . . S n . The case of "a selected-C" is quite 
 different. Here we have a true constant independently defined. 
 
 The range of C is to be taken very much as "all-C's" consid- 
 ered collectively. Following Frege we arbitrarily assert that 
 the denoting complex "the-C" is equivalent to the range of C. 
 Thus "the positive integer between one and three" is "two," 
 considered, not as individual, but as sole member of the range. 
 "The square root of four" is, without previous restriction, the 
 collected unity of such roots; "the-lion" expresses the range of 
 the concept lion. "The round-square" (in a geometrical uni- 
 verse) denotes a null-range. Thus every description is given 
 an equivalence. 
 
 Any symbol which satisfies a null condition can be imported 
 into the range of any condition we please; a null range or any 
 part of it is included in any fixed range, null or not. Thus if we 
 have one point P and one null point P 1 , P and P 1 can be regarded 
 as determining a straight line, or not, at pleasure. In actual 
 practice the activity of the logician contributes something. 
 The existent members of a range are considered to be there, 
 given; these non-existents are only potentially present and are, 
 in effect, put there by the voluntary choice of the logician who 
 chooses to make them satisfy this rather than the contradictory 
 condition. Logic actually contributes something; it is not a 
 mere passive spectator in its own development. 
 
 The calculus of ranges is precisely that of Peano's calculus 
 of class concepts (which he looks at from the standpoint of 
 extension), if we allow the term inclusion to be used in such a 
 way that the concept C is included in C l , if every C satis- 
 fies O. 
 
64 The Notion of Number and the Notion of Class 
 
 §3. Definition, Selection, and Critical Questions 
 
 We will say that a condition C is defined with reference to a 
 systematic universe, if every symbol which satisfies C and not-C 
 is a non-existent. If the universe is also connected, C is said to 
 be well-defined. Any well-defined condition not-null is an exis- 
 tent condition. If C is an existent, the range of C, every-C, and 
 any-fixed-C exist. 
 
 Any existent in a systematic universe is said to possess an 
 indication with respect to that universe; if the universe is con- 
 nected, it possesses a unique indication. Any non-existent 
 possesses a unique indication (since it indicates nothing at all). 
 Limiting assertions to a connected universe is no more nor less 
 than an objective rendering of the traditional dogma that in 
 every argument the terms used must have a constant "mean- 
 ing." The intrusion of subjective elements in "meaning" seems 
 unavoidable, but by making mathematical reasoning purely 
 verbal, the psychological factors are reduced to a minimum. 
 Looked at from the mathematical point of view what we have 
 said amounts to this, — given a set of postulates, general and 
 existential, then for every demonstrated existent in the universe 
 determined by the postulates, there is one and only one inter- 
 pretation for a given interpretation of the undefined symbols. 
 For any demonstrated non-existent we can substitute nothing. 
 "Meaning," therefore, becomes a term relative to the universe. 
 The more systematic the universe, the less intrusion of the 
 subjective. In a hap-hazard universe where only a few funda- 
 mental laws hold, psychology may indeed run riot. The prog- 
 ress of objective science and mathematics is essentially the 
 analysis of such loosely put together universes into several 
 systematic ones. Consider the proposition, "Walt Whitman 
 is a poet." In a conversational universe, individual opinion of 
 necessity intrudes and we assent or dissent according to our 
 individual notions of what "true poetry" may be or not be. 
 "Poet" is a term which does not possess a unique indication 
 since the universe is unsystematic. However, if we go about 
 the matter logically and scientifically, we define poetry in objec- 
 tive terms and give the term a unique indication. 
 
 We must now take up our postponed discussion of the denot- 
 ing complex "a-selected" which is roughly equivalent to "an 
 independently defined." What we mean by this is — given a 
 
The Notion of Class 65 
 
 concept (not null) is there any means of defining a symbol which 
 satisfies it? Any concept whose range is included in the range 
 of the concept "positive integer" does provide such a means. 
 Here the selective function is "the first," since every collection 
 of positive integers must have a first term. Consequently if 
 = "positive integer" and C2 satisfies Ci, a-selected-Ca exists. 
 (C = "sub-condition of C.) 
 
 Any condition C where a-selected-C exists is said to be selec- 
 tive. Any condition C whose associated condition C (of included 
 conditions) is uniformly selective is said to be a Zermelo con- 
 dition. 
 
 We must now consider two logical axioms which have to do 
 with the notion of selection. 
 
 (1) Every condition which is well-defined and not-null is 
 selective. This will be called the selective axiom. 
 
 (2) Every well-defined condition not null is a Zermelo con- 
 dition. This is the so-called Zermelo postulate. 
 
 First we may ask, can it be proved directly that being selec- 
 tive follows from being well-defined? Let us suppose a concept 
 C contained in an infinite universe, where we can name as many 
 terms as we please. Is there a method by which we can name 
 a definite symbol satisfying C? If we take term after term 
 of the universe, it is determinate in each and every case whether 
 it satisfies C or not-C, but we may have the ill luck to hit upon 
 those which satisfy not-C. Since we are unable to specify 
 all the terms of the universe, no proof seems possible on the 
 basis of the bare supposition that C is well-defined. 
 
 The truth of the axiom may, in point of fact, be sincerely 
 doubted and Russell actually cites "all products of two integers 
 which never have been and never will be thought of by any 
 human being" as a well-defined condition not null of which no 
 instances can be given, — hence non-selective. Physical objects 
 and other people's minds are other such instances, he contends. 
 
 Let us consider abstractly what must be done to demonstrate 
 that a given well-defined condition P should be non-selective, 
 i. e., a-selected-P does not exist. Since from the fact that P 
 is well-defined it follows that any fixed symbol we mention 
 determinately satisfies P or not-P, then any such symbol must 
 satisfy not-P if we are to show P to be non-selective. Any 
 demonstrated existent in the universe must be shown to satisfy 
 
66 The Notion of Number and the Notion of Class 
 
 not-P. Therefore, if P is a known existent, we must show that 
 P satisfies not-P and this cannot be done without departing 
 from our connected universe, which cannot be allowed since P 
 was supposed to be well-defined. Such examples as Russell's con- 
 tain a latent vicious circle fallacy. Take for example, the familiar 
 "proof" of idealism that all entities are ideas. Everything that 
 can be mentioned is an idea, and you cannot show an entity to 
 exist without mentioning it, so all entities are ideas. The prop- 
 osition to be proved, however, is certainly an entity and to 
 show that it is mentioned, it is necessary to change from one 
 point of view to another and so to use "mentioned" in a different 
 sense. Such notions as mentioning, thinking of, prove nothing 
 about all entities. — The other examples such as other people's 
 minds and physical objects are more philosophical scandals 
 than anything else, and in the sense in which Russell uses the 
 terms, their existence may be denied without any great display 
 of radicalism. 
 
 Can we conclude from this that we are justified in assuming 
 the selective axiom? Here is a cross road where two schools 
 part company. It is possible to steer a safe and cautious course 
 not denying the axiom but not assuming it and when necessary 
 actually determining the needed selective function. The great 
 majority of mathematicians, however, implicitly assume that 
 every well-defined condition is selective. A point which appears 
 to justify this assumption is the nature of indirect proof (reductio 
 ad absurdum). We start with a condition P and we seek to 
 show not-P to be null. We take (let us suppose) a-definite 
 not-P and show that it has contradictory properties. From this 
 we conclude that not-P is null since a-selected not-P does not 
 exist. If the selective axiom is false nothing can be concluded 
 and all indirect proof fails. 
 
 Although the necessary distinctions are very difficult and 
 cannot be made with any great degree of confidence, it appears 
 that a case may be made in rebuttal. In all but a very small 
 percentage of proofs by reductio ad absurdum, the member of 
 not-P being definite has nothing to do with the argument. 
 Consequently for ordinary mathematical purposes instead of 
 a-selected not-P we take any-fixed-not-P and show that it does 
 not exist. This is, of course, a valid proof since we assumed 
 that if P is well-defined and not null, any-fixed-P exists. For 
 
The Notion of Class 67 
 
 the cases where definiteness is needed we either reject the con- 
 clusion, seek a direct proof or take the conclusion as a postulate 
 
 The Zermelo postulate has gained much greater notice in 
 mathematical circles. Cantor, Bernstein, Whitehead, Schonfiies, 
 and many other prominent mathematicians have accepted it 
 either explicitly or implicitly and it is thought that a refusal 
 to adopt it is paradoxical and involves a sweeping away of 
 "much interesting mathematics." The more philosophic writers 
 on the foundations of Analysis, such as Russell, Poincare, Hob- 
 son, and Borel, have expressed doubts as to its truth. It is 
 contended that it is the equivalent of an infinite number of 
 acts of choice and involves all the difficulties of the selective 
 axiom (which is a particular case) with additional ones of its 
 own. The postulate, itself, is equivalent to the selective axiom 
 and either of (a) every condition can be well-ordered or (b) of 
 any two cardinal numbers one is greater than, less than, or 
 equal to, the other. It is preferable to prove (a) and (b), how- 
 ever, if the postulate is justified, since they are essentially mathe- 
 matical in character. Before discussing the postulate critically, 
 it is well to say a few words about its suppossed self-evidence. 
 This is brought about by assuming that "any-fixed" can be 
 regarded as a selective function which applied to any definite 
 collection of conditions yields an equivalent collection, definite, 
 of members of the conditions. While superficially plausible 
 there is no reason to suppose that "any-fixed" yields anything 
 definite except in the case of singular conditions (with only one 
 term). This "self-evidence," therefore, is illusory. 
 
 A certain uniformity of selection is what is guaranteed by 
 the postulate. The conditions from which we are to choose, 
 unless finite in number, must possess a similarity of structure 
 which permits the condition gained by the selection to be well- 
 defined. Consider the interiors of circles. The interiors of 
 mutually exclusive circles obey the Zermelo postulate, since 
 they are not homogeneous but contain an element bearing a 
 unique relation to the concept — the center. The selective 
 axiom assumes this non-homogeneity with reference to each 
 individual condition. 
 
 We will now discuss some critical points with especial refer- 
 ence to the paradoxes and the interpretation of ranges. The 
 paradoxes are, we hope, successfully avoided by the notion 
 
68 The Notion of Number and the Notion of Class 
 
 of unique indication; to have a unique indication, we recall, a 
 symbol must be contained in a connected universe. Now 
 Russell has shown that the paradoxes result as particular cases 
 of the following: "Given a property § and a function f such that 
 if (J> belongs to all the members of u, f(u) always exists, has the 
 property <j> and is not a member of u ; then the supposition that 
 there is a class w of all terms having the property <J> and that 
 f(w) exists leads to the conclusion that f(w) both has and has 
 not the property <J>." Here f(u) occurs as function (condition) 
 and as symbol satisfying (J>, which cannot occur in one connected 
 universe, without some notation to express the difference. How- 
 ever, if the notation is introduced, the paradox vanishes. 
 
 For a particular example, take the predicate "not predicable 
 of itself." In our language this would be the condition "con- 
 dition which does not satisfy itself." Let this be W. Then 
 if A satisfies W, A satisfies not-A and conversely. So if W 
 satisfies W, W satisfies not-W and conversely, which is paradox. 
 To keep the argument in one connected universe, however, we 
 must give our terms when conditions the subscript c and as 
 symbols, s. All we get now by our substitution, in A s sat. W c = 
 A 9 sat. not-A c is, if W s satisfies W c , W s satisfies not-A c which 
 has nothing to do with the case, or if A satisfies W c , A satisfies 
 not-W c which shows that W c is not well-defined. The supposed 
 paradox results from ambiguity. 
 
 Burali-Forti's interesting ordinal contradiction merely shows 
 that the ordinal number of all ordinal numbers is not an ordinal 
 number in the same sense (or connected universe) as its mem- 
 bers — thus if B is the suspected ordinal, B>B is consistent 
 because in the one case it is condition and in the other symbol. 
 The inequality is not asserted in a connected universe because 
 there the set of ordinal numbers does not have an ordinal number 
 which is comparable with its members. The class of all classes 
 and cardinal number of all cardinal numbers involve similar 
 ambiguities, where ambiguity is understood to mean not any- 
 thing subjective but lack of a unique indication, as we have 
 defined the term. 
 
 Classes are, as might be supposed, dependent upon ranges 
 and our original query "what is a class?" must wait for the 
 answer to "what is a range?" We must now meet Russell's 
 objection that no such entity as a Frege range is comprehensible 
 
The Notion of Class 69 
 
 and to do this we must interpret the term "range" in some way 
 consistent with experience. We recall that if every-C satisfies 
 D and every-D satisfies C, the range of C is equivalent to the 
 range of D; if A satisfies C, it belongs to the range of C. 
 
 The clue to the interpretation of ranges must be found in 
 Frege's all too brief remarks in the opening paragraph of the 
 "Grundgesetze." Here in introducing the notion of a function, 
 he comments by way of explanation that the essence of a func- 
 tion is not in the mere collection of values which satisfy it, but 
 rather in their "Zusammengehorigkeit." Correspondingly we 
 may say that a range possesses a coherency, a continuity and 
 a homogeneity which a loosely put together heterogeneous collec- 
 tion does not. Each member of a range is an example of the 
 typical, representative general term; they are all held together 
 by the fact that they satisfy the same condition. 
 
 The difference between a range and a collection may be illus- 
 trated by the classical example of the man who could not see 
 the forest for the trees. The trees represent (from his point of 
 view) a collection and unless known to be finite do not constiutte 
 a single whole; the forest represents a synthesis and each par- 
 ticular tree is indicated by "the tree of the forest." The analytic 
 person who was a victim of this strange blindness failed to see 
 in each separate tree the residuum of identity which it has in 
 common with the other different trees. With the perception 
 of this community of quality comes the recognition that the 
 collection is a range — a forest. When we have a range and 
 perform an operation on a general term, that self same operation 
 is performed on the different terms of the range. If x is the 
 variable tree of the forest, then for x to be leafless is to imply a 
 forest denuded of its foliage. The unity of a range is not the 
 unity of an additive whole but more that of an organism and is 
 based upon the common possession of an identical property. 
 A finite collection can be turned into a range by the addition 
 of singular ranges. An infinite collection cannot. Hence we 
 must insist that mathematics deal only with ranges, and tolerate 
 collections only as potential ranges. 
 
 §4. Classes, Numbers, and the Principle of Abstraction 
 
 Let us assume that the use of ranges has been justified; we 
 have still not attained the goal of this whole discussion, — the 
 notion of class. It must be concluded that whereas a range is 
 
yo The Notion of Number and the Notion of Class 
 
 something concrete, dependent upon its particular condition, 
 a class is independent of any particular condition and has just 
 as much to do with any fixed one of a set of formally equivalent 
 conditions as any other; it is, therefore, not concrete in the sense 
 that a range is concrete. Whereas we found a range to depend 
 upon the internal homogeneity of the symbols satisfying a con- 
 dition, a class depends upon the external homogeneity of ranges 
 themselves, in short, upon the principle of abstraction. 
 
 Russell is enabled to dispense with the principle of abstraction 
 as a postulate by using as his underlying identity, belonging 
 to the class of terms which bear to one another the isoid (reflex- 
 ive, symmetrical and transitive) relation. When he comes to 
 the notion of class itself he finds himself at a loss, for here where 
 the use of the principle of abstraction would be most advantage- 
 ous, its employment is impossible. To hold that the definition 
 of class is that a class is the class of formally equivalent conditions 
 which determine it would be a vicious circle obvious at a glance. 
 The principle as used by Russell fails utterly to take care of 
 the very fundamental notion of class. 
 
 Against his use of it in general, we must contend that although 
 he avoids the introduction of a new axiom, this advantage 
 is superficial and illusory, (i) If the existence of classes is 
 doubtful, the existence of the entities gained by the abstraction 
 is also doubtful, since they are classes. Numbers, since defined 
 by this process become meaningless, and arithmetic falls peril- 
 ously near to nominalism. (2) The principle is not consistently 
 employed, as has been very clearly pointed out by Jourdain. 
 The definitions of cardinal number and ordinal number go well 
 enough, but when we come to real number — instead of defining 
 a real number as a class of coherent classes of rationals, as the 
 principle clearly indicates, Russell arbitrarily defines the real 
 number as one particular class, a segment. This inconsistency 
 moves Jourdain to inquire whether cardinal numbers could not 
 have been defined in a similar way, and he in fact, proposes 
 another method based upon this new principle of abstraction. 
 Me believes that it is possible to define Zero as the null-class, 
 One as the class of classes whose only member is the null class, 
 and so-on, — substituting for the Frege-Russell "class of classes 
 equivalent to the given class" one particular member of this 
 set. Now this definition in no way helps us with the notion of 
 
The Notion of Class 71 
 
 class and has little advantage of any kind to recommend it. 
 It does, however, tend to cast considerable doubt upon Russell's 
 use of the principle of abstraction. 
 
 To maintain the validity of the principle along with the 
 unreality of classes seems mere verbiage, and to dispense with 
 its use altogether, an unnecessary hardship. We propose, there- 
 fore, the following logical postulate which, if we are correct, is 
 sufficient foundation for the existence of classes and numbers 
 and in fact is somewhat analogous to Russell's axiom of reduci- 
 bility: Given a non-singular collection of entities, between any 
 fixed pair of which, there subsists an isoid relation R, then there 
 exists a unique well-defined symbol A R to which every term of the 
 collection bears the relation R, but which is distinct from any 
 definite term of the collection. This abstract entity is not one 
 of any collection of entities having the relation; it is exceptional 
 and not on a par with them, since they are concrete. 
 
 A comparison with Cantor's conception of abstraction will 
 make this postulate clearer and more plausible. Cantor, we 
 recall, abstracted order and quality from a group and gained 
 the cardinal number. Obviously this new abstract entity he 
 got in this way is not identical with any of the concrete groups, 
 although it bears to them a very curious relation, very much 
 as the general term of a condition bears to any particular one. 
 This abstract group does not have the number; it is the number. 
 It bears to any of its concrete groups the relation of one-one 
 correspondence, and is a new entity, gained by a neiv principle. 
 
 Let us apply our postulate to the problem of the definition of 
 classes and numbers. Ranges may have the relation of formal 
 equivalence (mutual inclusion), which is isoid. By our postulate, 
 therefore, given say the range of Ci formally equivalent to the 
 range of C 2 , etc., then there exists an abstract range or class 
 formally equivalent to O and to C 2 , etc., but not identical with 
 any definite one of these ranges. Hence if A satisfies Ci, it 
 belongs to this Class, which we will speak of as the class of sym- 
 bols satisfying any fixed one of the conditions whose ranges are 
 formally equivalent. The necessary and sufficient condition for 
 the existence of a class is the existence of two well-defined con- 
 ditions whose ranges are formally equivalent. One is not enough 
 because the symmetry and transitivity of the relation do not 
 appear; in fact from one range, there does not seem to be any 
 
72 The Notion of Number and the Notion of Class 
 
 motive for abstracting. We require, therefore, that before the 
 abstraction occur, at least two entities must have the required 
 relation to each other. 
 
 Classes as we have defined them possess the limitations of 
 ranges. They are in the same connected universe as the ranges 
 they are equivalent to and cannot satisfy the conditions of 
 these ranges and at the same time possess members without 
 this difference being symbolically expressed. The paradoxes 
 of classes are analogous to those we have considered for con- 
 ditions. 
 
 We come now to the definition of abstract cardinal number. 
 Let us suppose numerical equivalence defined as Frege has 
 defined it, an isoid relation. Then given a collection of classes 
 (not ranges) which are numerically equivalent, our postulate 
 gives us a new entity more abstract than the classes in question 
 but equivalent to them — this is the cardinal number. 
 
 Given a definite class A and the relation of numerical equiva- 
 lence, then providing A has been abstracted from a selective 
 condition, then the cardinal number of A exists. For the selec- 
 tive member of A can be replaced by something else at pleasure 
 and by this method we get a new class A 1 , which is in (1,1) 
 correspondence with A, satisfying the hypothesis of the prin- 
 ciple of abstraction. If Po and Po 1 are null conditions, their 
 ranges are formally equivalent and determine the null-class 
 which has no existent members. 
 
 A singular condition can be defined in logical terms and One 
 is the cardinal number gained by abstracting from singular 
 conditions. Now if One exists, a singular condition C to which 
 it applies also exists and by considering these as symbols we 
 get an additive range, selective, which yields by abstraction 
 the cardinal Two. A cardinal R is said to be the immediate 
 successor of another cardinal S provided R applies to a condition 
 P which contains a singular condition Pi such that the cardinal 
 number of P-Pi is S. Two is therefore the immediate successor 
 of One. Three exists, by considering Two, One, and C as sym- 
 bols and, in fact, we can proceed as far as we like by this method, 
 changing the connected universe at each step. 
 
 It does not appear, however, that in any given connected 
 universe every number has a successor, for the universe has 
 been changed at each step. Can this gap be filled by Frege's 
 
The Notion of Class 73 
 
 method of introducing Zero? We must remember that the 
 null class contains no existent members and that it is unique, — 
 there may be many null ranges and null conditions but there is 
 only one null class. The necessary and sufficient condition 
 that this null class should give rise to a cardinal number Zero 
 in the way that a singular class yields One is that it be selective. 
 Now can we possibly hold that the null-class is selective? It is 
 hard to see how such a contention can be maintained with any 
 show of plausibility, — yet, if we deny it, Zero is not a cardinal 
 number in the sense that One, Two, Three, etc., are cardinal 
 numbers. There are, however, certain psychological reasons 
 for believing that this is the case. 
 
 We seem to be confronted with a choice of alternatives: (a) 
 we assume the selective character of the null class and by the 
 method of logical activity (including the null class in particular 
 classes) show that every number has a successor, or (b) we start 
 the number series with One and substitute for the use of Zero, 
 an axiom to the effect that every number has a successor. 
 
 It is possible to hold that in either case an assumption has 
 been made which differentiates Cardinal Arithmetic from Pure 
 Logic. The notion of logical activity enables us to hold that 
 Logic is the class of all propositions gained without placing 
 non-existents in arbitrary classes. However (a) seems to depend 
 upon the null-class in an objectionable way and there seems 
 to be much better reason for adopting the second of the two 
 possibilities and holding that the axiom (every number has a 
 successor) marks the dividing line between Logic and Arith- 
 metic. 
 
 In this we are to some extent confirmed by the verdict of 
 others. This was one of Kerry's objections to Frege, and it 
 was, in germ, the keynote of Poincare's vigorous assaults upon 
 logistic. He believed that a new type of reasoning was involved 
 in theoretical arithmetic, that mathematical induction repre- 
 sented a distinct type of intuition, a separate faculty of the mind. 
 These results he gained more by introspection than by abstract 
 reasoning, and we cannot accept subjective reasons for such 
 theories. Couterat and Russell were undoubtedly in the right 
 when they emphasized the objective character of logistic. Trans- 
 lated into objective terms, however, Poincare's theory is quite 
 different and is equivalent to: A new indefinable or a new axiom 
 
74 The Notion of Number and the Notion of Class 
 
 must be introduced into logic before its indefinables and axioms 
 are sufficient to yield cardinal arithmetic. This is substantially 
 our own conclusion. 
 
 Russell has himself implicitly given up the battle although 
 the admission of defeat is not so clear as might be wished. He 
 says in the "Principia" — "Some of the properties which we 
 expect inductive cardinals to possess . . can only be proved 
 by assuming that no inductive cardinal is null. . . This 
 amounts to the assumption that in any fixed type, a class can 
 be found having any assigned inductive number of terms. . . 
 This assumption . . will be adduced as a hypothesis when- 
 ever it is relevant. It seems plain that there is nothing in logic 
 to necessitate its truth or falsehood and that it can only be 
 legitimately believed or disbelieved on empirical grounds. 
 
 Since this, in effect, concedes the point at issue, we need not 
 discuss the matter further. Poincar6 must be adjudged the 
 victor, and the sweeping assertion of 1903 — "The fact that all 
 mathematics is symbolic logic is one of the greatest discoveries 
 of our age; and when this fact has been established, the remainder 
 of the principles of mathematics consists in the analysis of sym- 
 bolic logic itself" appears to stand in need of considerable revi- 
 sion. The dividing line between Logic and Mathematics is 
 the axiom that every number has a successor; roughly speaking 
 a new notion is introduced — "aw d so on." 
 
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