The beauty (?) of mathematical proofs - methodological considerations

By Catarina Dutilh Novaes

This is the second installment in my series of posts on the beauty, function, and explanatoriness of mathematical proofs (Part I is here). I here discuss methodological issues on how to adjudicate the 'dispute' between the reductive and the literal accounts of the beauty of proofs, discussed in Part I.

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But what could possibly count as evidence to adjudicate the ‘dispute’ between the literal/non-reductive camp and the non-literal/reductive camp? We are now confronted with a rather serious methodological challenge, namely that of determining what counts as ‘data’ on this issue (and potentially other issues in the philosophy of mathematics). Both sides seem to have compelling arguments, but it is not clear that a top-down approach with conceptual, philosophical argumentation alone will be sufficient.[1] However, it seems that merely anecdotal evidence (“I am a mathematician and I use aesthetic terminology in a literal (or non-literal) sense”) will not suffice either. Firstly, there are of course limits to self-reflective knowledge. Secondly, what is to rule out that some mathematicians use aesthetic terminology as proxy for epistemic properties, while others use the terminology in a literal sense instead? It is not clear that a uniform account is what we are looking for.[2] Moreover, it may be a case of an is-ought gap: perhaps mathematicians do use aesthetic vocabulary in a particular way (either literal or non-literal), but should they use this vocabulary in this way and not in another way?

Ultimately, the question is: what is the explanandum in a philosophical account of the (presumed) aesthetic dimension of mathematical proofs? Are we (merely) offering an account of the aesthetic judgments of mathematicians? (Something that might be conceived as belonging to the sociology rather than the philosophy of mathematics.)[3] Or are we dealing with a crucial component of mathematical practice, one that fundamentally influences how mathematicians go about? Or perhaps the goal is to explain (purported) human-independent properties of proofs such as beauty and ugliness? What will count as data in this investigation will depend on what the theorist thinks is being investigated in the first place.[4]

Be that as it may, the recent work of Inglis and Aberdein (2015) presents a potentially novel methodological approach to the issue. They surveyed mathematicians’ own appraisals of proofs on a number of aspects, including aesthetic aspects. The instruction of the experiment (conducted online with 255 mathematicians at different places who volunteered to participate) was as follows:

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. […] Please read each word carefully, and then select the option that corresponds to how well you think it describes the proof. (Inglis & Aberdein 2015, 9/10)

Participants were then given a list of eighty adjectives in random order, such as: beautiful, enlightening, simple, profound, pleasant, explanatory, useful, charming, fruitful etc. The goal was to establish which (clusters of) adjectives were regularly used together to describe specific proofs (importantly, a different proof for each participant). Using a method borrowed from social psychology to study personality traits, from their data set Inglis and Aberdein established that the participants’ appraisals of proofs varied on four main dimensions: (in their terms) aesthetics, intricacy, utility, and precision.

Interestingly, they found no significant correlation between ascriptions of simplicity and ascriptions of beauty to proofs in the participants’ responses. This seems, at least prima facie, to count as evidence against McAllister’s (2005) account of beauty in terms of simplicity. Moreover, ‘beautiful’ correlated quite strongly with ‘enlightening’, which seems to lend some support to Rota’s account. However, Inglis and Aberdein (2015, 17) conclude:

One interpretation of these data would be to suggest that epistemic and aesthetic judgements of mathematical proofs are indeed different. Epistemic judgements concentrate largely on the utility dimension, whereas aesthetic judgements concentrate on the aesthetics dimension. Nevertheless, there do appear to be adjectives which reside at the conjunction of these two dimensions.

In other words, the non-literal account of ‘beauty as proxy for something else’ (for epistemic features in particular) does not seem to be fully corroborated by their analysis either. But the analysis does suggest an important connection between aesthetic and epistemic assessments, and (equally important for the present purposes, as will become clear) between epistemic judgments and utility.

But how much can this approach really tell us about the nature of aesthetic judgments in mathematics? A number of objections can be raised on the philosophical significance of this approach, for example: how reliable is data on colloquial uses of aesthetic terminology? Is the number of participants large enough to be representative? What about differences among different mathematical ‘cultures and traditions’ in function of geography, sub-areas within mathematics etc.? My own view is that, while this approach provides valuable data, it does not really cover all the aspects of the phenomena that we should be interested in (in particular the widespread phenomenon of proof predilection). But it remains unclear how else the dispute between the two camps described above could be settled, in particular what kind of data could be used as evidence in either direction.

In subsequent posts, I develop an alternative account of the beauty (or ugliness) of mathematical proofs, which seems to offer the possibility of ‘reconciliation’ for these apparently opposed accounts of aesthetic judgments in mathematics. The key notion will be that of functional beauty. In the final section of the paper I spell out some of the predictions emerging from the account, which could thus be used as empirical tests for its adequacy.

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

Gillies, D. [2014]: ‘Should philosophers of mathematics make use of sociology?’, Philosophia Mathematica (3) 22, 12–34.

Hafner, Johannes, and Paolo Mancosu [2005]: The Varieties of Mathematical Explanation. In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.

Mancosu, Paolo [2011]: Explanation in mathematics. In E. Zalta, ed., The Stanford Encyclopedia of Philosophy.

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.

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[1] Compare Mancosu (2011, section 7) on top-down approaches to the notion of mathematical explanation: “Previous theories of mathematical explanation proceeded top-down, that is by first providing a general model without much concern for describing the phenomenology from mathematical practice that the theory should account for. Recent work has shown that it might be more fruitful to proceed bottom-up, that is by first providing a good sample of case studies before proposing a single encompassing model of mathematical explanation.”

[2] The potential heterogeneity of the phenomenon being explained suggests that the case-study methodology advocated by Mancosu (see previous footnote) may also not be entirely satisfying; the range of case studies analyzed must be large enough so as to be representative of the potential variety of aesthetic judgments on proofs. Notice also that Mancosu himself (Hafner& Mancosu 2005) is sympathetic to the idea of non-homogeneous phenomena as the explananda for philosophical accounts of mathematics and mathematical practice.

[3] But see (Gillies 2014) on the relevance of sociology for the philosophy of mathematics.

[4] Personally, I am mostly interested in the second question, namely how (presumably) aesthetic values and aesthetic judgments influence and affect in fundamental ways the practices of mathematicians pertaining to proofs, in particular with respect to the fact that mathematicians typically prefer some proofs over others. For this question, surveys of mathematicians’ aesthetic judgments will represent valuable data, but will not be enough for an in-depth discussion of the phenomenon.