By Catarina Dutilh Novaes
This is the final post in my series on beauty, function, and explanation in mathematical proofs (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). Here I tease out some empirical predictions of the account developed in the previous posts, according to which beauty and explanatoriness will largely (though not entirely) coincide in mathematical proofs. I also comment on how the account, based on a dialogical conception of mathematical proofs, could be made more palatable for those who would prefer a non-relative, absolute analysis of beauty and explanatoriness.
To summarize, the present account defends the thesis that when mathematicians employ aesthetic vocabulary to describe proofs, both positively (‘beautiful’, ‘elegant’) and negatively (‘ugly’, ‘clumsy’), they are by and large (though not exclusively) tracking the epistemic property of explanatoriness (or lack thereof) of a proof. Up to this point, the account is compatible with both subjective (agent-relative) and objective understandings of beauty and explanation, so long as the two dimensions go together (i.e. both understood as either subjective or as objective). However, on the basis of a dialogical conception of mathematical proofs, I’ve also argued that both explanation and beauty are essentially relative notions with respect to proofs: an explanation is not explanatory an sich, but rather explanatory for its intended audience; and if a proof is deemed beautiful to the extent that it fulfills this explanatory function, then beauty too emerges as a relative notion.
I’ve also suggested ways in which the present account can be made more palatable for those who strongly prefer objective accounts of explanatoriness and beauty. By maximally expanding the range of Skeptics who will deem a proof explanatory – and so aiming towards the notion of a universal audience – in the limit (idealized) case a proof may be deemed explanatory by all (i.e. those who have the required expertise to understand it in the first place). On this conception then, a proof may also be understood to be beautiful in an absolute sense, i.e. insofar it fulfills its explanatory function towards any potential (suitably qualified) audience. The conception of beauty as fit defended by Raman-Sundström (2012), which relies on an objectively conceived notion of fit, may be viewed as an example of such an account, and indeed her description of fit bears a number of similarities with concepts typically associated with explanatoriness.