Rieger, A. (2018) The beautiful art of mathematics. Philosophia 
Mathematica, 26(2), pp. 234-250. (doi:10.1093/philmat/nkx006). 
 
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Accepted final version. Published paper in Philosophia Mathematica.

The Beautiful Art of Mathematics∗

Adam Rieger

School of Humanities

University of Glasgow

Glasgow G12 8QQ

U.K.

Adam.Rieger@glasgow.ac.uk

Abstract

Mathematicians frequently use aesthetic vocabulary and sometimes

even describe themselves as engaged in producing art. Yet aestheticians,

in so far as they have discussed this at all, have often downplayed the as-

criptions of aesthetic properties as metaphorical. In this paper I argue

firstly that the aesthetic talk should be taken literally, and secondly that it

is at least reasonable to classify some mathematics as art.

∗Ancestors of this paper were presented some years ago at the Universities of Edinburgh and
Nottingham; I thank audiences there, and Nick Zangwill for discussions at that time. A later
version was presented at a conference on Aesthetics in Mathematics held at the University of
East Anglia in December 2014; I thank the organizers, Angela Breitenbach and Davide Rizza,
and other participants, for feedback and enjoyable discussion.



1 Introduction

One might think it strange if there were a field of human activity in which the

practitioners regularly describe themselves as motivated by aesthetic consider-

ations, appraise each others’ work using apparently aesthetic vocabulary, and

even explicitly describe what they are doing as ‘art’; yet aestheticians show no

interest in the field.

But such is, or was until recently, the peculiar position of mathematics.

What literature that does exist on this topic — and it is rather little — has con-

sisted mostly of scattered remarks made by mathematicians reflecting on their

subject, with not much written in a systematic way by philosophers. The situ-

ation is starting to change, but the subject of the aesthetics of mathematics is

still in its infancy.

My aim here is primarily to argue that there are indeed genuinely interest-

ing aesthetic issues here, and that mathematics is a perfectly good topic for

aestheticians to discuss.

For the most part I will give an overview of the issues; many of them deserve

a far more lengthy treatment than I have space for here. But I shall be arguing in

a little more detail for two specific theses. Firstly, that the aesthetic vocabulary

used in discussing mathematics should be taken literally. If that isn’t so, the

aesthetics of mathematics is a pseudo-subject, and attempts to nurture it into

maturity are misguided. Secondly, and more tentatively, I shall outline how it

could be argued that mathematics is sometimes an art.

2



2 The use of aesthetic vocabulary

Here are some quotations from mathematicians:

I think it is correct to say that [the mathematician’s] criteria of se-

lection, and also those of success, are mainly aesthetical. [von Neu-

mann, 1947]

A mathematician, like a painter or poet, is a maker of patterns . . . The

mathematician’s patterns, like the painter’s or poets, must be beau-

tiful . . . Beauty is the first test: there is no permanent place in the

world for ugly mathematics. [Hardy, 1941, pp. 84–5]

I am interested in mathematics only as a creative art [Hardy, 1941,

p. 115].

The mathematician’s best work is art, a high perfect art, as daring

as the most secret dreams of imagination, clear and limpid. Math-

ematical genius and artistic genius touch one another. (Mittag-

Leffler, quoted in Rose and De Pillis [1988])

I like to look at mathematics almost more as an art than as a sci-

ence; for the activity of the mathematician, constantly creating as

he is, guided though not controlled by the external world of senses,

bears a resemblance, not fanciful I believe, but real, to the activities

of the artist, of a painter, let us say. [Bôcher, 1904, p. 133]

Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth

Symphony beautiful. If you don’t see why, someone can’t tell you.

3



I know numbers are beautiful. If they aren’t beautiful, nothing is.

(Erdös, quoted in Devlin [2000, p. 140])

Such instances are not at all exceptional. Indeed, as Rota [1997, p. 171] ob-

serves, ‘whereas painters and musicians are likely to be embarrassed by ref-

erences to the beauty of their work, mathematicians instead like to engage in

discussions of the beauty of mathematics’.

A little statistical evidence can be found in the empirical study by Zeki et al.

[2014]. Of the 14 mathematicians in the study, all but one reported emotional

responses to equations. (They were not asked explicitly whether they thought

equations could be beautiful, but the data is suggestive.)

3 Examples

It might be useful to have some examples of (putative) mathematical beauty in

front of us. Here are three of my own favourites.

Euler’s famous formula

1

12
+ 1

22
+ 1

32
+··· = π

2

6

has appeared high up in some polls, coming 5th out of 24 in the survey Wells

[1990] of readers of the Mathematical Intelligencer and 9th out of 60 in the study

Zeki et al. [2014] of postgraduate and postdoctoral mathematicians.1

1Instructions for each survey were respectively ‘give each theorem a score for beauty be-
tween 0 and 10, inclusive’ and ‘rate the beauty of each equation on a scale from -5 (ugly),
through zero (neutral) up to +5 (beautiful)’. The formula e iπ = −1 (or a re-arrangement of it)
came top in both surveys. Wells reports, though (p. 38), that some awarded it a low score; he
speculates that, remarkable though this equation is, those who have a reasonable familiarity
with complex analysis may find it too obvious to score it highly.

4



For a second example, here is a proof that
p

2 is irrational, remarkably not

discovered until quite recently [Apostol, 2000]. The usual proof2 is algebraic;

this is a geometric variation. Suppose
p

2 is rational, say it is M /N where the

fraction is in its lowest terms. Then there is an isosceles right-angled triangle

with integer sides which is the smallest one possible:

M

M
N

We can show, for a contradiction, that there is another, smaller similar tri-

angle, also with integer sides. Construct the circle with centre and radius as

shown; we claim the shaded triangle is isosceles and right-angled with integer

sides. It is clearly right-angled; it is isosceles since it shares an angle of 45° with

the larger triangle; and its hypotenuse is of integer length since it equals M mi-

nus the length of one of the shorter sides, N − M (tangents from a point to a
circle are equal).

2Which, incidentally, is actually cited as a paradigm of mathematical beauty by Hardy [1941,
p. 94].

5



M

M

N − M

Here is a third example. The equation z 4 = 1 has 4 roots (±1 and ±i ). The
Newton-Raphson method is an elementary iterative technique for finding the

roots of an equation; given an approximation to a root, it (almost always) re-

turns a better approximation, converging on the root if applied repeatedly.

Suppose we colour each point of the complex plane according to which root

it converges to — what do we get? A reasonable guess would be that each point

converges to the nearest root, and indeed that’s what happens if we start near

to a root. But the full behaviour is quite extraordinary:

6



Of course, the mathematical beauty here is distinct from (though perhaps

related to) the beauty of the picture.3

Some whole areas of mathematics are sometimes cited as particularly beau-

tiful: for example number theory and complex analysis (an area that stands out

in my own memory of studying mathematics as an undergraduate). In con-

trast, the theory of differential equations, which has the appearance of a ragbag

of disparate techniques, has been cited as particularly ugly: ‘this is botany, not

3Multiple coloured pictures of this sort may be found by typing ‘Newton fractal’ into a search
engine.

7



mathematics’, Sawyer [1961, p. 145].4

Finding necessary and sufficient conditions for beauty is not something

many aestheticians think is possible.5 However, in the mathematical case, a

number of features have come up quite frequently in discussion (for example

Wells [1990], Hardy [1941], Rota [1997]).

Many of these features appear in Hutcheson [1726]. Hutcheson considers

that the key to beauty is uniformity amidst variety (I.II.III). He devotes an entire

section (I.III) to ‘the Beauty of Theorems’, claiming there is no kind of beauty

‘in which we shall see such an amazing variety with uniformity’ (I.III.I). How

widely this idea is applicable to beauty in general may be debatable, but in my

view Hutcheson is definitely on to something in the case of mathematics. An

interplay between simplicity and complexity is typical of paradigmatic exam-

ples of mathematical beauty. In the Newton-Raphson example, a very simple

equation generates a very complex pattern. In the theory of numbers, the sim-

plest building blocks exhibit endlessly intricate behaviour. Another example of

the same phenomenon is Conway’s Game of Life, where there is a striking con-

trast between the simplicity of the rules of generation and the complexity of the

pattens they create.

Another aspect picked out by Hutcheson is surprise, though he is careful to

note it is not a sufficient condition for beauty. This seems to be playing a part

in all three examples I have given; in no case would one, on seeing them for the

first time, anticipate what is coming.

4Sawyer rightly observes that the comparison is unfair to botany, which of course aims at
more than a collection of specimens.

5For example, McMahon [2007, p. 40] writes: ‘. . . there are no necessary or sufficient con-
ditions for beauty when these conditions are construed as properties an object must have in
order to be beautiful.’ See also Sircello [1975, p. 44].

8



Hardy [1941, §§14–18] mentions seriousness,6 which he analyses as combin-

ing generality and depth; the best theorems are not isolated facts, but concern,

or are generalizable, to a variety of cases, and have far-reaching consequences.

(Hutcheson also draws attention (I.III.V ) to cases where ‘one theorem contains

a great multitude of corollaries deducible from it’, and gives an example from

Euclid; this seems also related to the ‘uniformity amidst variety’ idea.) The best

arguments are economical; for example, a proof which argued by considering

many similar cases could not be beautiful.7

A final aspect concerns a certain kind of understanding. For example, Rota

[1997, p. 180] talks about enlightenment, contrasting it with cases where one

merely follows the steps of a proof without grasping its ‘sense’.8 The geometric

proof of the irrationality of
p

2 above is an example of this; it makes it clear, al-

most obvious, why
p

2 is irrational, by making visible the method of infinite de-

scent. (These criteria can sometimes pull in different directions; Barker [2009,

p. 66] gives the example of the (second) recursion theorem, which has a short,

elegant proof which, however, makes it hard to see to why the theorem is true.9)

4 The objects of mathematical beauty

If there is beauty in mathematics, what exactly is beautiful?

6Hardy writes of beauty and seriousness as the two criteria by which mathematics is to be
judged, but he is quite explicit (§11) that they are not independent: ‘. . . the beauty of a mathe-
matical theorem depends a great deal on its seriousness’. He apparently regards seriousness as
either a component, or at least a necessary condition, of beauty in mathematics.

7An extreme case is the first published proof of the four-colour theorem [Appel et al.,
1977a,b] which required the checking of 1936 different cases by computer.

8Rota’s view (p. 181) is that talk of mathematical beauty is really indirect talk about enlight-
enment, a concept he (somewhat implausibly) claims mathematicians dislike and avoid dis-
cussing directly because it admits of degrees.

9There are slick proofs of the incompleteness theorems which have the same property.

9



Taking the examples above, in the first case we have an equation or theo-

rem. The beauty does not seem to depend on the exact syntactic formulation

(for example, it would not matter greatly if the left hand side were replaced by

a verbal description of the sum). This suggests that it is what the equation ex-

presses, rather than the syntactic equation itself, that is really what is beautiful

here.

In the second case, although the theorem itself may also be beautiful, it is

the proof which is the main focus. Again, what seems important is not the exact

words and pictures used, but the ideas they express.

The third case is more difficult to pinpoint — perhaps it is a fact which is

beautiful? What seems to be beautiful is that such a richly complex pattern can

be generated by such a simple equation.

In fact most of the cases cited in the literature are either theorems or proofs.

The surveys carried out by Wells [1990] and Zeki et al. [2014] concentrate on

theorems; Zangwill’s objections discussed below focus on proofs, as do the col-

lectors of beautiful proofs ‘from the book’ in Aigner et al. [2010]. (An entire

area, such as Galois theory or complex analysis, is a collection or sequence of

theorems and their proofs.)

Interestingly, there are not many claims in the literature that mathemati-

cal objects are themselves beautiful. Although the Erdös quotation above sug-

gest that numbers are literally beautiful, mathematicians do not usually refer

to particular integers, or π, as beautiful. Perhaps Erdös should be interpreted

as meaning that the totality of numbers, or the number structure, is beautiful,

but even that would be contrary to the way most mathematicians talk. Some

geometrical figures are cited as beautiful, but this is perhaps visual rather than

10



mathematical beauty.

An unusual suggestion in Rota [1997, p. 171] is that a definition can be beau-

tiful. He gives the example of the notion of category, which facilitates the study

of mathematical structure at an extreme level of abstraction. The definition of

compactness in topology might provide another example. When one first en-

counters this, one is puzzled as to why such an apparently complex property

deserves a label; but doing so makes possible beautifully simple proofs of vari-

ous theorems.

Rota (pp. 173–4) is careful to distinguish the beauty of a theory from the

beauty of any particular exposition of it, and cites Galois theory as an exam-

ple of a beautiful theory, none of the expositions of which have succeeded in

matching its beauty. He is surely right to make the distinction; moreover, of the

two, it is again the beauty of the theory itself, of the content, that seems by far

the more significant. (It is notable that the word ‘elegant’, rather than ‘beauti-

ful’, is often used when discussing particular presentations; for example Rota

(p. 74) uses it when describing expositions of the Lebesgue integral.)

The paradigmatic cases, then, all seem to be, roughly speaking at least, propo-

sitional (a proof being naturally thought of as a sequence of propositions, though

this may be arguable in the case of a visual proof ).

5 Is this genuinely aesthetic?

If the subject of the aesthetics of mathematics is to get off the ground, it had

better be that case that the judgments referred to above are genuinely aesthetic.

Yet some have denied this. Thus Rom Harré has written

11



. . . quasi-aesthetic appraisals are not a queer sort of aesthetic ap-

praisal but simply not aesthetic appraisals at all . . . the satisfaction

that we call peculiarly aesthetic is absent from the mathematical

situation. [Harre, 1958, p. 136]

Other writers to express hostility to, or scepticism about, the use of aesthetic

vocabulary in this context being literal are Zangwill [2001] and Todd [2008].

The burden of proof, it seems to me, is really on the deniers. The testimony

of a large number of mathematicians, who are using this vocabulary without

irony, is itself a prima facie case in favour of their experiences being genuinely

aesthetic. Perhaps it is the case that only a small proportion of the population

talk in this way,10 but this hardly seems relevant. The Zeki et al. [2014] study re-

vealed that the same areas of the brain fire when mathematicians contemplate

equations they find beautiful as when they appreciate beautiful pieces of music

or art, though this is suggestive rather than conclusive.

There is a sense in which nothing is more convincing than one’s own in-

trospection. If I reflect on my own experience in contemplating the examples

above, it seems to belong to the same distinctive class as that involved in appre-

ciating art and music. But this is rather unsatisfactory as a means of collectively

reaching a conclusion on the matter. What actual arguments can be adduced

pro and con?

Harré notes (p. 135) the relative paucity of the vocabulary used about math-

ematics, compared with other areas — we use ‘beautiful’ and ‘elegant’, but not

10In contrast to the 14 mathematicians in the Zeki et al. [2014] survey mentioned above, 9 of
the 12 non-mathematicians questioned denied having an emotional response to beautiful the-
orems; on the other hand, Hardy [1941, p. 87] cites the popularity of chess, bridge and puzzles of
various sorts as evidence that the ability to appreciate mathematics is in fact quite widespread.

12



‘charming’, ‘delightful’, ‘lovely’ or ‘handsome’. (Incidentally I think Harré is

wrong about ‘lovely’.) Todd [2008, p. 71] also appeals to this feature. I do not

think a huge amount can be drawn from this alone, however. The range of

aesthetic experiences obtainable from mathematics is no doubt less wide than

those to be obtained from painting, music or literature, but this hardly shows

that the experiences themselves, or the resulting appraisals, are not aesthetic.

On the pro side, it is a perfectly sensible activity for a mathematician to

search for better and better proofs of a result already known to be true. This

seems to show that mathematicians aim at more than the pursuit of truth. What

else is it? One answer is that they seek proofs that are explanatory; that give un-

derstanding as to why a theorem holds, with promise perhaps of further devel-

opments and applications. But another answer, consistent with the first, that

has been given is that the motivation is aesthetic:

Much research for new proofs of theorems already correctly estab-

lished is undertaken simply because the existing proofs have no

aesthetic appeal.11 [Kline, 1964, p. 470]

Sensory dependence

A direct challenge to the idea of aesthetics in mathematics comes from the

idea that aesthetic qualities are tied up with perception. For example Levinson

writes

My proposal is that aesthetic properties . . . are higher order ways of

appearing, dependent in systematic fashion on lower-order ways of

11Hutcheson [1726] seems to be making a similar point in the second paragraph of I.III.V.

13



appearing . . . [Levinson, 2005, p. 218]

Similarly Zangwill [2001] has argued that ‘sensory properties are necessary

for aesthetic properties’, which entails that no abstract objects have any aes-

thetic properties, and hence (assuming for the moment a platonistic concep-

tion) that no mathematical proofs, theorems or objects can be beautiful.

Actual arguments for the sensory dependence thesis seem hard to come by.

It is a natural view perhaps, given the historical concentration of aestheticians

on the visual arts and, to a lesser extent, music. Indeed the etymology of ‘aes-

thetic’ suggests dependence on perceptual properties. For Zangwill the thesis

fits into a wider project of aesthetic formalism.

Here, maybe, is an unfortunate by-product of the neglect of mathematics as

a topic for aestheticians! Had mathematics had the discussion it deserved, per-

haps no-one would have been tempted by this thesis. But the position seems

implausible independently of any mathematical considerations. For example,

on literature: Zangwill believes that the content of a literary work — that is,

‘what the work means, the story it tells, the characters it portrays, the emotions

it evokes, the ideas it involves, and so on’ (p. 135) — have no aesthetic value.

And so, for example, if we read War and Peace in English, virtually all12 of its

aesthetic properties are literally ‘lost in translation’. This seems to me to be

close to a reductio ad absurdum of the position.13

Zangwill is one of the few aestheticians to have discussed the case of math-

ematics explicitly. In his view, ascriptions of beauty etc. to mathematics are

metaphorical. In support of this, he notes that a proof has a purpose, and ‘our

12A few may remain, for example names may be sonically well-chosen for their characters.
13Zangwill himself does express some doubts (p. 137) as to the correctness of the sensory

dependence theory in the case of literature.

14



admiration of a good proof. . . turns solely on its effectiveness in attaining this

end, or else its having properties which make attaining the end likely. Could a

proof be elegant if it was invalid, or did not possess properties which tend to

make proofs valid?. . . Surely not’ (p. 141).

Perhaps, though, Zangwill imagines an opponent replying, this very pur-

pose can give rise to dependent beauty, that is, a proof might be ‘beautiful in

the way it fulfilled its function’, like the way ‘a building is beautiful as a thing

with a certain function or a painting is beautiful as a representation of some-

thing’ (p. 141).

However, he rejects this on the grounds that in genuine aesthetic cases,

beauty and function can ‘come apart’. Thus, ‘a building might aesthetically ex-

press the function of being a library but not actually function well as a library.

It might even have no potential to function well as a library (p. 141).’ In con-

trast ‘there cannot be proofs. . . which which are disfunctional yet beautiful or

elegant’ (p. 142).

Even if we grant him the possibility that the library may have dependent

beauty of the sort described without actually functioning well as a library, Zang-

will seems to have overlooked that some dependent beauty may depend on ac-

tual success in fulfilling the function.

His argument goes through, therefore, only if (i) the only possible mathe-

matical beauty is dependent beauty to be cashed out in terms of effectiveness

of a proof in fulfilling its function, (ii) genuine cases of dependent beauty arise

entirely from expressing a function, rather than actually fulfilling it, and (iii) in

the mathematical case there is no possibility of ‘expressing’ the function com-

ing apart from fulfilling it. But all of these seem highly contentious.

15



With regard to (i), while proofs have purposes, it’s less clear that theorems

or mathematical objects do (although perhaps a theorem has some kind of rep-

resentational purpose, as I discuss below). In addition, it seems misconceived

to set things up in this way: there is surely more to the (purported) beauty of a

proof than its simple effectiveness, or else any two correct proofs of the same

theorem would be on a par. (ii) seems false; a library could have dependent

beauty in virtue of the way it actually functioned as library, and a painting in

virtue of accurately depicting its subject. And (iii) is also dubious; a proof might

perhaps be strictly invalid but still contain valuable ideas which made it beau-

tiful.14 Overall, therefore, Zangwill’s remarks are unconvincing. (For a more

detailed critique of Zangwill’s view, see Barker [2009].)

Beauty and truth

The most serious threat to the literal interpretation of the aesthetic vocabulary

arises from the observation that mathematicians are ultimately concerned with

producing truths; hence, even if they describe themselves as pursuing beauty, it

is dubious that they really mean it. Thus Todd [2008] is sceptical that aesthetic

appraisals should be taken literally. What appear to be aesthetic judgments are,

he suggests, really disguised epistemic ones.

He presents (p. 68) a dilemma for the literalist: either there is ‘some im-

portant connection between epistemic and aesthetic factors in theory assess-

ment’ (a conjunctive view), in which case ‘it is difficult to see what independent

role aesthetic factors could play in theory assessment, or indeed what the dif-

14Euler’s original proof of the π
2

6 formula, in which he lacked the relevant results on infinite
products, might provide an example.

16



ference between aesthetic and empirical15 criteria of assessment actually is’;

or else ‘they are essentially unconnected’ (the disjunctive view), in which case

‘problems also surround the mysterious role that aesthetic factors could play in

theory assessment, particularly in respect of the problematic idea that theories

could somehow be beautiful but not true’.

Surely some version of a conjunctive view is correct; beauty and truth are

importantly connected (though the relationship is certainly more complex than

identity, as Keats would have us believe). It does seem at least roughly right that

truth (or validity in the case of a proof ) is a necessary condition for beauty. The

Euler proof mentioned earlier is invalid as it stands, but can be made rigorous

by filling in some gaps. There could perhaps be a ‘near miss’ theorem that was

untrue as stated but possessed some beauty — but it would be flawed, like a

cracked vase, and the falsity certainly sharply reduces the aesthetic value.

Kivy [1991] suggests that the beauty of theories16 should be thought of by

analogy with representational painting:

The scientist does not admire a theory’s beauty and then admire it

or not admire it for its truth. It is admired for how beautifully it is

true; for how beautifully it represents nature. (p. 192)

If we take ‘nature’ in the case of mathematics to be ‘mathematical reality’,

we have here, I think, a promising way to make sense of mathematical beauty.

And one that is perfectly consistent with its relation to truth. For it seems the

aesthetic value of a representational painting depends on the the success, the

15Todd’s argument is framed in terms of empirical science, but is intended to apply to math-
ematics as well.

16Kivy’s main focus is also on the empirical sciences, but again can be applied also to the
mathematical case for most purposes.

17



truthfulness of the representation (of course, this should not be understood

in a crude way as ‘the more like a photograph, the better the painting’; but a

successful painting says something illuminating and true about how its subject

appears, or how one experiences seeing it).

Kivy goes too far in his ‘conjunctive account’; he says that beauty and truth

cannot be ‘prised apart’ (p. 193), and comes close to endorsing Keats at the

end of his paper. But whether or not we can have beauty without truth, we

can certainly, in mathematics, have truth without beauty.17 Todd’s charge that

Kivy’s conjunctive account does not keep the aesthetic sufficiently distinct from

the epistemic is just.

It is not true though, as Todd claims (p. 66) that ‘science just aims to get it

right’. Truth isn’t all there is. There is a position which avoids both the horns

of Todd’s dilemma: beauty and truth are neither independent, nor to be iden-

tified. Indeed, in the latter, more concessive, part of his paper, Todd counte-

nances the possibility of ‘explaining the aesthetic value of proofs and theories

in terms of the way in which their epistemic content is conveyed’ (p. 77), which

suggests a position not far from Kivy’s, though without the near-identification

of the true and the beautiful.

Exactly what is the connection between beauty and truth is a large question!

If, for example, seeking beauty is somehow to be a guide to finding the truth, it

is an urgent matter to explain why.18 I shall have a little more to say about it in

17Paradigmatic examples might be the kind of theorems we find in combinatorics, giving
complex formulae for such things as the number of ways of tiling polygons with other polygons;
in addition valid but unattractive proofs such as the one of the four-theorem cited earlier. This
perhaps also serves as an example of quite a beautiful theorem (9th on Wells’ survey) without
a beautiful proof; Rota (p. 172) cites the prime number theorem, giving the asymptotic density
of the primes, as another such example.

18James McAllister has developed, over a series of publications, an elaborate theory which

18



Section 7. But that mathematical beauty is enmeshed with truth does not seem

a good reason to think that it is not really beauty at all.

6 Aesthetics and the metaphysics of mathematics

I have suggested above a way in which thinking about mathematics might have

consequence for aesthetics, in telling against the sensory dependence thesis.

An interesting questions is whether we might have interaction in the other di-

rection: might aesthetic considerations have implications for more mainstream

philosophy of mathematics?

If propositions are the locus of beauty, then this suggests that is no easy

route from aesthetical considerations to conclusions about the ontology of math-

ematics. It seems Euler’s π
2

6 theorem can have beauty whether one platonisti-

cally regards it as being about an externally existing realm of mind-independent

mathematical objects or, alternatively, about a world of fictional objects created

by human activity.

The Croce-Collingwood theory, according to which artworks are mental,

has interesting parallels with the intuitionism of Brouwer, according to which

mathematics consists of ‘mental constructions’. And the theories run, I think,

into parallel difficulties, that in each case what goes on in one’s head seems to

be answerable to something external, in a way that the theories struggle to ac-

count for; there are wrong ways to interpret a picture, and there are false math-

ematical propositions.

connect beauty and truth in science and mathematics, via what he calls the ‘aesthetic induc-
tion’ (see for example [1996] and [2005]). I do not have space to discuss McAllister’s work here,
but it is addressing exactly the questions I think need exploration.

19



Breitenbach [2015] expands some brief remarks of Kant into a worked out

account of the beauty of mathematical proofs within a Kantian framework. Ac-

cording to Breitenbach’s account, our experience of mathematical beauty is

grounded in ‘our felt awareness of the imaginative processes that lead to math-

ematical knowledge’ (p. 2).

I cannot here discuss Breitenbach’s intricate account in the detail it de-

serves. By focussing on mathematical demonstration as a human activity, Bre-

itenbach is able to go some way towards accounting for the roles of surprising-

ness and understanding in mathematical beauty. An account focussing only on

objects and their properties will surely struggle to do this.

On the other hand, the Kantian framework explicitly allows only proofs19 to

have beauty, and not theorems.20 This is because, for Kant, a cognitive judg-

ment, such as is involved in contemplating a theorem, differs essentially from

an aesthetic one (in the first, but not the second, a ‘synthesis of the sensory

manifold’ is ‘subsumed under concepts’ — see p. 6). This seems a serious weak-

ness of the Kantian account, since the position that proofs but not theorems

can be beautiful does not accord well with the experience and testimony of

mathematicians. (It is not simply that we need to seek a different account for

the beauty of theorems — such an account is ruled out in the Kantian system.)

It would be an interesting project to see to what extent Breitenbach’s insights

could be preserved in a way that was less embedded in the Kantian approach.

19It is informal proofs that are intended here, and in particular in geometry. The question
as to how far the account is extendible to other areas of mathematics is raised by Breitenbach
herself (pp. 15–6).

20Breitenbach describes the contrast as existing between proofs, on the one hand, and math-
ematical objects and their properties, on the other, but that does not seem quite right. It is
presumably not the (universal) properties such as primeness, straightness etc. which might be
beautiful, but rather particular objects’ having them, that is, something propositional.

20



What of someone who wanted to defend the beauty of mathematical the-

orems and proofs, but rejected propositions? One way out might perhaps be

to argue that one can have an aesthetic experience without an object, analo-

gously to adverbialist theories of perception. Perhaps it can simply be beautiful

that something is the case without there being any object which is beautiful. I

do not know whether a plausible theory could be developed along these lines,

but it would be interesting if so and might have applications in other areas of

aesthetics.21

7 Mathematics as art

Let us take stock. In arguing for the literalness of aesthetic appraisals in math-

ematics I made use of the analogy with representational painting, pointed out

by Kivy, and the kinship with literature since the bearers of beauty are, in most

cases, propositional. But one can now ask a further question: is mathematics,

like painting and literature, an art? This question, of course, is separate from

the question of whether mathematics has aesthetic properties. It could be that

mathematical beauty exists but is natural beauty, like the beauty of a landscape

or a flower.

Though several mathematicians, including some quoted above, have talked

about mathematics as an art, as far as I know, no-one has explicitly defended

this thesis philosophically. It certainly seems implausible that all mathematics

should be art; in particular, a lot of applied mathematics will not be. But in this

21A referee suggests another route for the rejecter of propositions: to find beauty in the lin-
guistic expressions. But as argued above (Section 4) mathematical beauty seems primarily lo-
cated in the content of theorems and proofs, rather than the particular way that content is ex-
pressed.

21



mathematics is like several other activities (not all writing or drawing is art, for

example).

How might one argue for the thesis that some mathematics, of the pure sort

which its practitioners say is pursued for aesthetic reasons, is an art? In contrast

to the case of beauty, a considerable amount of philosophical work has gone in

to attempts to define art, without any great agreement (in this, of course, art is

hardly exceptional). It would be an interesting task to assess whether mathe-

matics counts as art according to some of the main theories that have been put

forward, but that would hardly give us a conclusive answer, and is not some-

thing I shall attempt here.

In any case, perhaps there are no necessary and sufficient conditions for

art. Wittgenstein’s ‘family resemblance’ idea may be helpful here: I am cau-

tiously inclined to think that the parallels, noted above, between mathemat-

ics and both representational painting and literature, combined with the gen-

uinely aesthetic elements in mathematics for which I have already argued, sug-

gest that mathematics is sometimes an art.

The main consideration on the other side seems once again to be that truth

plays too great a role in mathematics. But as in the discussion of beauty and

truth above, the case of representational painting suggests this is hardly deci-

sive. Not only is the representational accuracy of a painting no obstacle to its

being art, it is (understood, as previously mentioned, in a non-simplistic way)

essential to the aesthetic value of a painting.

The case of literature is more complicated. With prose at least, paradig-

matic examples of literature-as-art tend to be fictional. (On some non-platonist

views, mathematics itself is a kind of fiction and the objection loses its bite; for

22



an explicit defence of a such a view, see Bueno [2009].) We are more likely to

regard a novel as art than a work of biography, history or travel-writing.

There are several points to make here. Poetry always seems to be art, whether

or not its subject matter is fictional. A landscape painting does not count as art

only if the landscape depicted is fictional. Sometimes we do regard works of

history or biography as art; and here, as in the case of representational paint-

ing, not only is the constraint to be truthful no obstacle to their being art, but

its violation would be a serious flaw. Finally, even fiction is related to truth in

an indirect way; the faithfulness of a novel to human nature in particular, and

also to such things as the era in which it is set, is of direct relevance its aesthetic

value.

Mathematics, then, is one of a family of activities which tell us how things

are, in a way that is aesthetically valuable. It seems no travesty to call such a

practice ‘art’.

Hardy, who sees no contradiction between his platonism [1941, pp. 123–4]

and explicitly stating that mathematics is an art (p. 115), raises an interesting

issue which points to a difference between mathematics and the other arts I

have been discussing. He quotes (p. 84) with approval Houseman’s comment

that ‘poetry is not the thing said but a way of saying it’, and of the lines from

Richard II ‘Not all the water in the rough rude sea/Can wash the balm from an

anointed King’ comments ‘Could lines be better, and could ideas be at once

more trite and more false?’. Mathematics works only with ideas, thinks Hardy,

and is hence ‘more permanent’.

Hardy does bring to light an important contrast here. In mathematics, the

main aesthetic value lies with the thing represented, not the representation.

23



(Sometimes a particular presentation or formulation may be notable for its ele-

gance, but it is mainly the theorems or proofs themselves that are seen to have

the aesthetic value.) This seems to mark a distinction between mathematics

and literature, and also representational art, where we talk of the beauty of the

painting, not its subject.

This disanalogy might be seen to threaten the status of mathematics as art.

But even if correct, it does not seem a conclusive reason why mathematics can-

not be art. There can be art in selecting which pieces of (mathematical) reality

to display, as du Sautoy discusses in a recent popular piece in which he is com-

paring mathematics and music:

Most people’s impression is that a mathematician’s job is to estab-

lish proofs of all true statements about numbers and geometry . . . What

is not appreciated is that mathematicians are actually engaged in

making choices about what is being elevated to the mathematics

that deserves performance in the seminar room or conference hall.

The proof of Fermat’s Last Theorem is considered one of the great

mathematical opuses of the last century, while an equally compli-

cated calculation is regarded as mundane and uninteresting. It is

the narrative journey that the first proof takes you on that makes

this proof worth telling. [du Sautoy, 2015, p. 50]

In any case, the contrast does not seem sharp. Arguably the most valued

paintings have beautiful subjects, as well as being themselves beautiful repre-

sentations; part of the what the artist is commended for is having successfully

conveyed a beautiful part of reality. And it seems to me Hardy is wrong about

24



the lines of Shakespeare; the lines would be finer, were the thought expressed

by them true and insightful.22 I argued above that part of the aesthetic value

of literature is genuinely propositional, survives translation, and contra House-

man, is the ‘thing said’, rather than ‘the way of saying it’. So the disanalogy with

mathematics is less than Hardy suggests.23

8 Conclusion

In the course of this survey, I have argued firstly that aesthetic appraisals of

mathematics should be taken literally. That the practioners of mathematics use

aesthetic vocabulary apparently intending it to be understood non-metaphorically

suggests the burden of proof is on those who deny the genuineness of the aes-

thetic appraisals. But neither an argument from sensory dependence, nor one

maintaining that mathematics is too concerned with the pursuit of truth to be

an aesthetic activity, seem convincing.

Moreover, mathematics seems to have enough in common with paradig-

matic arts such as painting and literature that there is a case for counting at

least some mathematics as itself an art.

I have only sketched how one might argue in more detail for these claims,

but if I am correct, then mathematics is an area of human activity which de-

serves a lot more attention from aestheticians than it has so far had. And even

if the claims are false, articulating exactly why promises to be illuminating in

22Indeed, Hardy himself comes close to recanting a few pages later (at the end of §11).
23The form of literature closest in analogy to a mathematical theorem is perhaps the Wildean

epigram: for example, ‘A man cannot be too careful in the choice of his enemies’, from The Pic-
ture of Dorian Gray. We might note the element of surprise; the truth of the thought expressed;
and that it survives translation.

25



clarifying our concepts of art and the aesthetic. Either way, there are plenty of

interesting avenues for further exploration in the area.

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