Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics

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Tao does not propose a definite answer. Instead, he offers an expressly non-exhaustive list of twenty-one features which could result in a piece of mathematics being positively assessed. These include its beauty, elegance, creativity, depth, strength, intuitiveness, and definitiveness. But although the problem may appear hopelessly complicated, it cannot be truly intractable, unless the assessments of mathematical quality that seem ubiquitous in mathematical practice lack all content.

Tao's approach to resolving this conundrum begins with the observation that good mathematics in any of his senses tends to beget further pieces of good mathematics in the same or other senses. How then does a mathematician judge the quality of a novel piece of mathematics, which has not yet had time to beget any further mathematics, good or bad?

But there are empirical findings that pose a challenge for this expertise-based account of mathematical evaluation. Notably, students, who presumably have scant experience of observing mathematical development, and therefore no obvious means of determining whether or not proofs are likely to fit into a larger picture, are apparently nevertheless capable of appreciating the aesthetics of at least some mathematical arguments [ Koichu et al.

So, Tao does not seem to have resolved his conundrum. However, if the dimensionality of mathematical qualities were substantially smaller than Tao suggests, the evaluation of mathematics need not be intractable; so the conundrum would not arise. Our goal in this paper is to investigate this question empirically. Specifically, we ask: on how many broad dimensions can perceptions of mathematical proofs be said to vary? Prior to describing our strategy, method, and findings, we briefly review existing accounts of what is sometimes considered the most valuable quality of mathematical proofs, that of beauty.

Although mathematical theorems and definitions are sometimes perceived as being beautiful [ Wells, ], it seems that the mathematical objects most commonly described in this way are proofs. But the notion of beautiful proofs raises a number of serious questions. Different authors have taken different positions on this latter question.

For example, G. A classic view of mathematical beauty is to relate the notion to simplicity. James McAllister [ ] epitomises this stance: Mathematicians have customarily regarded a proof as beautiful if it conformed to the classical ideals of brevity and simplicity.

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The most important determinant of a proof's perceived beauty is thus the degree to which it lends itself to being grasped in a single act of mental apprehension. Simplicity has also been associated with the beauty of aspects of mathematics other than proofs. David Wells [ ] surveyed sixty-eight mathematicians to determine which of twenty-four theorems they found to be the most beautiful.

He found that, even though his survey stated the theorems without proof, simplicity of proof impacted upon the perceived beauty of the theorems. In sum, one attempt to provide a characterisation of which proofs mathematicians find beautiful is to suggest that the perceived beauty of a proof is identical to, or at least highly correlated with, its perceived simplicity. Some have questioned whether aesthetic judgements in science and mathematics are really related to aesthetics at all. Many scientists seem to subscribe to the conjunctive view, believing that the perceived beauty of a scientific theory is somehow a sign of its truth [ Engler, ; Kivy, ; McAllister, ].

Todd [ ] rejects this idea, on the grounds that it is very difficult to see how aesthetic factors could be systematically related to a theory's empirical adequacy. In contrast, disjunctive theories suppose that aesthetic judgements are independent of epistemic judgements. This view seems to imply the existence of, or at least the possibility of, scientific theories and mathematical proofs which are true but not beautiful, and beautiful but not true. While this may be true for scientific theories, it is not clear that this claim is meaningful for mathematical proofs.

Nevertheless, Todd rejects both conjunctive and disjunctive views. He concludes that the most parsimonious account involves rejecting the notion of scientific and mathematical aesthetic judgements entirely. Instead he contends that when mathematicians and scientists talk of beautiful proofs or theories they are actually making epistemic not aesthetic claims. Therefore if the dimensionality of mathematical quality could be appropriately assessed, as we attempt to do in this paper, it would go some way to challenging Todd's argument for a reductive account of aesthetics in mathematics.

Todd is not alone in proposing a reductive account: Gian-Carlo Rota's [ ] mathematical aesthetics is a particularly clear instantiation. So presumably mathematicians perceive simplicity to be a characteristic of beautiful mathematics simply because it is easier to gain enlightenment from a simple proof compared to a complex proof.

Rota also has an idiosyncratic account of mathematical elegance. In the sections below we empirically investigate the relationship between perceived beauty, enlightenment, and definitiveness. There is general agreement that the notion of mathematical beauty is widespread in mathematical practice.

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Mathematicians characterise one another's work as beautiful or not , and strive to produce beautiful proofs. But there is disagreement about what beauty is in a mathematical context. Two broad classes of theory have been proposed. Non-reductive accounts of beauty in mathematics suppose that when mathematicians talk of beautiful proofs they are genuinely making an aesthetic judgement e. A classic view is that simplicity is central to these assessments. In contrast, reductive accounts, such as those proposed by Todd [] and Rota [] , suggest that when mathematicians talk of beauty they are using the term as a proxy for an epistemic assessment.

In particular, Rota argues that beautiful proofs are those which provide enlightenment. And Todd argues that, until the dimensions along which aesthetic and epistemic appreciation operate can be fully determined, reductive accounts offer the most parsimonious approach to mathematical beauty. Our goal in this paper is to investigate directly the dimensions along which mathematical proofs are typically evaluated. In particular, is Tao [] correct to argue that the dimensionality of mathematical quality is high?

On how many dimensions do assessments of mathematical proofs vary? And does the answer to this question offer insights into the nature of mathematical beauty? Our approach to addressing the dimensionality of mathematical assessment relies upon a statistical procedure known as a factor analysis. Factor analyses attempt to model the covariation among a set of observed variables in terms of functions of a small number of latent constructs, or factors, which are themselves unobservable.

The goal is to explain as much of the original variance as possible using a small set of factors, and to express each of the original variables as a function of these new factors. The technique works by looking for patterns in a matrix of the correlations between the original variables: if a set of variables are all strongly inter-correlated they can, in some sense, be said to represent the same underlying construct.

Indeed, exactly this study was performed by Burt and Banks [] , who looked at nine different measurements taken from adult males, and successfully identified a general size factor. Factor analyses are widely used in a wide variety of research domains, including the study of intelligence e. Human personalities are clearly more complex than human body measurements, but an approach methodologically analogous to Burt's [ ] study can be, and has been, adopted for their investigation. There are a great many natural-language terms which can be used to describe someone's personality in fact, Allport and Odbert [] identified almost 18, English words which can be used to describe a person's traits.

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These ratings can then be subjected to a factor analysis to determine how many factors, or dimensions, emerge. Thus an individual's personality can, as a rough approximation, be seen as a point within five-dimensional space. There are now various methods of estimating where a given person lies within this space e. For example Big Five profiles predict, among other things, life satisfaction [ Boyce et al. Our conjecture is that classifying mathematicians' perceptions of the qualities of mathematical proofs is an analogous problem to the challenge of characterising human personalities.

Thus we followed a research strategy analogous to that used by social psychologists interested in personalities.

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First, we produced a list of adjectives which have often been used to describe mathematical proofs. Second, we asked a large number of mathematicians to think of a proof that they had recently read, and to state how accurately each adjective described that proof. Finally, we subjected these ratings to an Exploratory Factor Analysis EFA , to determine on how many broad dimensions mathematicians' perceptions of proofs vary. We were particularly interested in using these findings to interrogate the various accounts of mathematical beauty described above. Our research strategy does not, of course, allow us to draw any conclusions about objective qualities of proofs or indeed, to say whether or not proofs have objective qualities , and neither does it allow us to understand which proofs have which qualities, or whether there are between-mathematician differences in the assessment of mathematical quality.

It does, however, allow us to investigate the structure of the language with which mathematicians characterise the qualities of mathematical proofs. What is distinctive about Naess's approach is his emphasis on empirically investigating the use of philosophically significant terms as a guide to their meaning, an approach we share.

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We differ from Naess in the more quantitative character of our methodology and our focus on the practice of a specific community, research mathematicians, rather than the population as a whole. Of course, the strategy of looking for meaning in use has a much broader philosophical pedigree. It may be traced back even further to the American pragmatists of the nineteenth century, and especially to Peirce. Our approach could be seen as an attempt systematically to study the language game which takes place when mathematicians evaluate proofs. Our first task was to select a long list of adjectives which have been used to describe mathematical proofs.

Using Tao's [ ] list of mathematical qualities as a starting point, we formed a list of eighty adjectives that we conjectured may be used by mathematicians to describe the traits of mathematical proofs. These are shown in Table 1. Like earlier researchers interested in studying empirically research mathematicians' behaviour [ Heinze, ; Inglis et al.

All mathematics departments with graduate programmes ranked by U. If the department agreed, they forwarded an email invitation to participate to all research-active mathematicians in their departments. As with all research which requires participants to give informed consent prior to participation, our participants were self-selected and so cannot be said to be a truly random sample.

The email gave a brief outline of the purpose of the research, and provided a web link to the location of the study. Participants who decided to take part first saw an introductory page which again explained the purpose and nature of the research. On the second page participants were asked to select their research area applied mathematics, pure mathematics, or statistics , and state their level of experience PhD student, postdoc, or faculty.

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On the third page participants were given the following instructions: Please think of a particular proof in a paper or book which you have recently refereed or read. Keeping this specific proof in mind, please use the rating scale below to describe how accurately each word in the table below describes the proof. Describe the proof as it was written, not how it could be written if improved or adapted.

So that you can describe the proof in an honest manner, you will not be asked to identify it or its author, and your responses will be kept in absolute confidence.

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Please read each word carefully, and then select the option that corresponds to how well you think it describes the proof. Emphasis in the original. Participants were then shown the list of eighty adjectives, presented in a random order, and asked to select how well each described their chosen proof using a five-point Likert scale very inaccurate, inaccurate, neither inaccurate nor accurate, accurate, very accurate. Finally participants were thanked for their time, and invited to contact the research team if they wanted further information.

A total of mathematicians participated in the study, consisting of PhD students, 23 postdocs, and 86 faculty. Sixteen participants did not respond to one or more adjectives, resulting in a total of 20 missing values 0.

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Prior to conducting the main analysis, the suitability of participants' ratings for factor analysis was investigated. Thus both tests supported the use of an EFA. Participants' ratings for the 80 adjectives were entered into an EFA, using the maximum likelihood method. Both the Scree Test and Parallel Analysis are approaches which attempt to find a balance between extracting sufficient factors to explain a high proportion of the original variance, and extracting so many that closely related latent constructs are represented. Loadings for the five extracted factors are given in Tables 2 and 3.

These numbers describe how well each adjective describes each factor: so if an adjective has a high positive loading then it is very representative of that factor, if it has a zero loading then it is independent of that factor, and if it has a high negative loading then it is very unrepresentative of that factor. Adjectives which loaded strongly onto Factors 1 aesthetics and 2 non-proofs.

Adjectives which loaded strongly onto Factors 3 intricacy , 4 utility , and 5 precision. Before describing the factors in detail we first note that Factor 2 appeared to be somewhat different from the other factors. We therefore concluded that Factor 2 was not a true dimension upon which proofs vary or at least, proofs which mathematicians have recently read or refereed and chose to think about do not vary substantially on this dimension.

Consequently we do not discuss Factor 2 in the remainder of the paper. Repeating the EFA on the 60 adjectives which had mean ratings significantly greater than 2 on the 1 to 5 Likert scale resulted in four factors which were essentially identical to Factors 1, 3, 4 and 5 discussed below. The table of factor loadings for this analysis is given in Appendix Table A1. We refer to this factor as the aesthetics dimension.

We refer to this factor as the intricacy dimension. We refer to this factor as the utility dimension. We refer to this factor as the precision dimension. We structure our discussion of these findings in five sections. First we summarise our main results, then we discuss the implications of these findings for the three accounts of mathematical beauty discussed in the introduction.

Finally, we discuss the role of empirical data in discussions of mathematical practice, and argue that reflective reports from mathematicians must be treated with appropriate caution. Our analysis indicated that mathematicians' appreciation of the qualities of mathematical proofs can be reasonably well understood using four-dimensional space. Looking directly at the correlates of beauty revealed that there appears to be no relationship between a proof's perceived beauty and its perceived simplicity, contrary to the classical view espoused by many mathematicians and philosophers.

As noted in the introduction, the classical idea that mathematical proofs tend to be regarded as beautiful if they are simple has been supported by many notable mathematicians and philosophers [ Engler, ; Wells, ; McAllister, ; Tappenden, ; Chandrasekhar, ; Cherniwchan et al. We found no evidence for this view.

How plausible is this latter suggestion? We do not believe that this is the case. Because our EFA indicated that aesthetics and intricacy are orthogonal factors, it should be at least in principle possible to write simple proofs which have positive, zero, or negative values on the aesthetics dimension. If simplicity and beauty are independent, why would so many mathematicians and philosophers link the two notions?

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