The beauty (?) of mathematical proofs -- beauty and explanatoriness

By Catarina Dutilh Novaes

This is the third installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here). In this post I start drawing connections (later to be discussed in more detail) between beauty and explanatoriness.

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A hypothesis to be investigated in more detail in what follows is that there seems to be an intimate connection between attributions of beauty to mathematical proofs and the idea that mathematical proofs should be explanatory. Indeed, the reductive account of Rota in terms of enlightenment immediately brings to mind the ideal of explanatoriness:

We acknowledge a theorem's beauty when we see how the theorem "fits" in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods. We say that a proof is beautiful when such a proof finally gives away the secret of the theorem, when it leads us to perceive the actual, not the logical inevitability of the statement that is being proved. (Rota 1997, 182).

It is not surprising that there should be such a connection for non-literal, reductive accounts such as Rota’s; explanatoriness is a very plausible candidate as the epistemic property that is actually being tracked by these apparently aesthetic judgments. However, the connection is arguably present both in reductive and in non-reductive accounts. Indeed, it is striking to notice that many of the properties that Hardy (1940) attributes to beautiful proofs are in fact properties typically associated with explanatoriness in the literature (to be discussed in an upcoming post). According to Hardy, a beautiful mathematical proof is:

• ·      Serious: connected to other mathematical ideas

• ·      General: idea used in proofs of different kinds

• ·      Deep: pertaining to deeper ‘strata’ of mathematical ideas

• ·      Unexpected: argument takes a surprising form

• ·      Inevitable: there is no escape from the conclusion

• ·      Economical (simple): no complications of detail, one line of attack

(Notice that the depth criterion is closely related to his realist, Platonist conception of mathematics: some concepts are inherently more fundamental, i.e. ‘deeper’, than others according to a ‘strata’ organization.) As we will see shortly, (what Hardy describes as) seriousness, generality, depth, inevitability, and economy are properties associated with explanatory proofs.

Do judgments of explanatoriness play a role in the practices of mathematicians similar to the role of aesthetic judgments? Prima facie, it seems that aesthetic vocabulary is more often present in how mathematicians talk about proofs than explanation-related vocabulary.[1] So one may wonder whether mathematicians themselves do (or do not) view the categories of ‘explanatory’ and ‘non-explanatory’ as relevant for their practices. Colyvan (2012, 79) recognizes that judgments of explanatoriness are typically left out of the published work of mathematicians; he also notes that “it is difficult to find a great deal of agreement in the mathematical literature on which proofs are explanatory and which are not”.[2] Nevertheless, he maintains that these categories are in fact relevant for the practices of mathematicians, and thus that explanatoriness is an important topic for the philosopher of mathematics.[3] Hafner and Mancosu (2005) reach a similar conclusion on the relevance of explanation for mathematical practice on the basis of some case studies. (It is to the credit of these authors that they raise the question at all; most philosophers writing on explanation in mathematics seem to take the relevance of the topic as a given.)

While brief, these considerations should in any case be sufficient to offer initial motivation for the pertinence of the concept of explanatoriness in mathematical proofs for the present purposes (i.e. a discussion of the presumed beauty of mathematical proofs). With this in mind, in the next post we will take a brief look at what has been said on explanation and mathematical proofs in the philosophical literature.

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

Colyvan, Mark [2012]: Introduction to the Philosophy of Mathematics. Cambridge: CUP.

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.

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

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

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[1] If Rota is right, then this is not surprising: for some reason, mathematicians would eschew vocabulary related to explanation and enlightenment and use instead aesthetic vocabulary to talk about these epistemic properties instead.

[2] Again, whether mathematicians converge in their attributions of beauty and of explanatoriness to proofs is still essentially an open question. Much of the literature on both topics seems to presuppose that they do, but this is not to be taken for granted (Inglis and Aberdein have been conducting some pilot studies which seem to suggest significant variations in proof appraisal). The philosophical significance of consensus or lack thereof on the beauty and/or explanatoriness of proofs will be discussed in more detail shortly.

[3] Some philosophers may want to maintain that whether mathematicians themselves view explanation as an important concept is irrelevant to determine the philosophical significance of the topic. But it seems to me to be a mistaken attitude to disregard mathematical practice in this way.