The beauty (?) of mathematical proofs - reductive vs. literal approaches

By Catarina Dutilh Novaes

I am currently working on a paper provisionally entitled 'Beauty, function, and explanation in mathematical proofs', and so this week I will post what I have so far as a series of blog posts. Here I start with a discussion on the current literature on the presumed beauty of some mathematical proofs. As always, comments very welcome!

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It is well known that mathematicians often employ aesthetic adjectives to describe mathematical entities, mathematical proofs in particular. Poincaré famously claimed that mathematical beauty is “a real aesthetic feeling that all true mathematicians recognize.” In a similar vein, Hardy remarked that “there is no permanent place in the world for ugly mathematics.” Indeed, in A Mathematician’s Apology Hardy offers a detailed discussion of what makes a mathematical proof beautiful in his view. More recently, corpus analysis of the laudatory texts on the occasion of mathematical prizes shows that they are filled with aesthetic terminology (Holden & Piene 2009, 2013). But it is not all about beauty; certain kinds of proofs that still encounter resistance among mathematicians, such as computer-assisted proofs or probabilistic proofs, are sometimes described as  ‘ugly’ (Montaño 2012). Indeed, mathematicians seem to often use aesthetic vocabulary to indicate their preferences for some proofs over others.

What exactly is going on? Even if we keep in mind that, in colloquial language, it is quite common to use aesthetic terminology in a rather loose sense (‘he has a beautiful mind’; ‘things got quite ugly at that point’), the robustness of uses of this terminology among mathematicians seems to call for a philosophical explanation. What are these judgments tracking? Are these judgments really tracking aesthetic properties of mathematical proofs? Or are these aesthetic terms being used as proxy for some other, non-aesthetic property or properties? Is it really the case that “all true mathematicians” recognize mathematical beauty when they see it? Do they indeed converge in their attributions of beauty (or ugliness) to mathematical proofs? And even assuming that there is a truly aesthetic dimension in these judgments, is beauty a property of the proofs themselves, or is it rather something ‘in the eyes of the beholder’? These and other issues are some of the explanatory challenges for the philosopher of mathematics seeking to understand why mathematicians systematically employ aesthetic terminology to talk about mathematical proofs (as well as other mathematical objects and entities).

In the existing literature, there seem to be two main kinds of accounts for this phenomenon: the literal, non-reductive accounts; and the non-literal, reductive accounts (literal vs. non-literal is Montaño’s (2014) terminology; non-reductive vs. reductive is Inglis & Aberdein’s (2015) terminology). According to the literal, non-reductive accounts, when mathematicians use aesthetic adjectives to talk about proofs, they truly mean what they say; they are talking about genuinely aesthetic features (which may be in the objects themselves or not, but are in any case genuinely aesthetic). Accounts of this kind may well attempt to explicate the idea of beauty in terms of other concepts,[1] but they attribute an irreducible aesthetic dimension to these judgments. In contrast, according to the non-literal, reductive accounts, when mathematicians use terms such as ‘beautiful’, ‘pleasing’, ‘ugly’ etc. to describe proofs, these terms are being used as proxy for non-aesthetic features. In other words, according to reductive accounts, these aesthetic adjectives are merely a façon de parler; they are not tracking truly aesthetic properties.

Hardy (1940) seems to belong to the literal, non-reductive camp; according to him, a mathematical proof is beautiful if it scores high on the following dimensions: seriousness, generality, depth, unexpectedness, inevitability, and economy (more on Hardy and these six dimensions shortly). McAllister (2005, 22) highlights brevity and simplicity as the key features of a beautiful mathematical proof:

Mathematicians have customarily regarded a proof as beautiful if it conformed to the classical ideals of brevity and simplicity. 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.

Another recent resolutely aesthetic approach to the presumed beauty (or ugliness) of a mathematical proof is developed by Montaño (2014), who draws, among others, from theories of musical aesthetics to develop a literal account of mathematical beauty (and ugliness).[2] These approaches seek to take what mathematicians say at face value, treating their uses of aesthetic terminology as suggesting that a truly aesthetic approach is required to understand this aspect of mathematics and mathematical practice. (See (Inglis & Aberdein 2014) for further examples of authors adopting the literal, non-reductive position.)

Reductive, non-literal approaches, in turn, posit that aesthetic terminology in the mouth of mathematicians is in fact tracking some non-aesthetic property or properties of proofs. One exponent of these views is Rota (1997), for whom the key property being tracked by these apparently aesthetic judgments is in fact a purely epistemic, non-aesthetic property: enlightenment. Rota’s understanding of enlightenment seems to come quite close to what is usually referred to as the explanatoriness of proofs, a property that will be discussed in more detail in the next section. But for now it is useful to note the reasons he gives for aesthetic terms being used as proxy for the epistemic property of enlightenment:

The term "mathematical beauty" [is a trick] that mathematicians have devised to avoid facing up to the messy phenomenon of enlightenment. … All talk of mathematical beauty is a copout from confronting the logic of enlightenment, a copout that is intended to keep our formal description of mathematics as close as possible to the description of a mechanism. (Rota 1997, 182)

In a similar vein, Todd (2008) contends that the prima facie aesthetic judgments of mathematicians are in fact epistemic judgments.[3] And so, aesthetic vocabulary is being used as proxy for epistemic assessment. The question then becomes, what kind(s) of epistemic assessment are we talking about? What exactly is being evaluated? In a future post, we will focus on the idea that mathematical proofs are to be explanatory, drawing on the literature on the explanatoriness of mathematical proofs.

FULL SERIES:
Part I is here
Part II is here
Part III is here
Part IV is here
Part V is here
Part VI is here
Part VII is here
Part VIII is here

References

Hardy, G.H. [1940]: A Mathematician’s Apology. Cambridge: Cambridge University Press.

Holden, H., and R. Piene [2009]: The Abel Prize 2003–2007: The First Five Years. Heidelberg: Springer.

Inglis, M., and A. Aberdein [2015]: Beauty Is Not Simplicity: An Analysis of Mathematicians' Proof Appraisals. Philosophia Mathematica 23 (1):87-109.

McAllister, J.W. [2005]: ‘Mathematical beauty and the evolution of the standards of mathematical proof’, in M. Emmer, ed., The Visual Mind II, pp. 15–34. Cambridge, Mass.: MIT Press.

Montaño, U. [2012]: ‘Ugly mathematics: Why do mathematicians dislike computer-assisted proofs?’, The Mathematical Intelligencer 34, No. 4, 21–28.

Montaño, U. [2014]: Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics. Dordrecht: Springer.

Rota, G.C. [1997]: ‘The phenomenology of mathematical beauty’, Synthese 111, 171–182.

Todd, C.S. [2008]: ‘Unmasking the truth beneath the beauty: Why the supposed aesthetic judgements made in science may not be aesthetic at all’, International Studies in the Philosophy of Science 11, 61–79.

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[1] E.g. simplicity for McAllister (2005).

[2] Recall that, in the traditional Liberal Arts curriculum, which dates back to later Antiquity and remained influential in the Middle Ages, music was part of the quadrivium together with arithmetic, geometry, and astronomy.

[3] “I argue that […] there are strong reasons for suspecting that many, and perhaps all, of the supposedly aesthetic claims are not genuinely aesthetic but are in fact ‘masked’ epistemic assessments.” (Todd 2008, 61)