*Beauty, explanation, and persuasion in mathematical proofs*, and to be honest at this point there is not much more to it than the title… However, the idea I will try to develop is that many, perhaps even most, of the features we associate with beauty in mathematical proofs can be subsumed to the ideal of

*explanatory persuasion*, which I take to be the essence of mathematical proofs.

*good*mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor

*that*the conclusion of the proof is true (given the truth of the premises) by showing

*why*that is the case. (See also this recent edited volume on argumentation in mathematics.) Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, beauty may well play an important role, but this role will be subsumed to the ideal of explanatory persuasion.

There is a small but very interesting literature on the aesthetics of mathematical proof – see for example this 2005 paper by my former colleague James McAllister, and a more recent paper on Kant’s conception of beauty in mathematics applied to proof by Angela Breitenbach, one of the organizers of the meeting in Norwich. (If readers have additional literature suggestions, please share them in comments.) But perhaps the locus classicus for the discussion of what makes a mathematical proof beautiful is G. H. Hardy’s splendid *A Mathematician’s Apology* (a text that is itself very beautiful!). In it, Hardy identifies and discusses a number of features that should be present for a proof to be considered beautiful: s*eriousness, generality, depth, unexpectedness, inevitability*, and *economy*. And so, one way for me to test my dialogical hypothesis would be to see whether it is possible to provide a dialogical rationale for each of these features that Hardy discusses. My prediction is that most of them can receive compelling dialogical explanations, but that there will be a residue of properties related to beauty in a mathematical proof that cannot be reduced to the ideal of explanatory persuasion. (What this residue will be I do not yet know).

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. (Detlefsen & Arana 2011, 1)

*aesthetic*concerns, in particular related to simplicity (which is one of the features widely associated with beauty).

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