(Cross-posted at NewAPPS)
In December, I will be presenting at the Aesthetics in Mathematics conference in Norwich. The title of my talk is 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.
As some readers may recall, in my current research I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? The general hypothesis is that most of the defining criteria for what counts as a mathematical proof – and in particular, a 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 its 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: seriousness, 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).
As I mentioned, this is still very much work in progress, but for now I would like to sketch what a dialogical account of beauty in a mathematical demonstration might look like for a specific feature. Now, a fascinating desideratum for a mathematical proof, which has been discussed in detail recently by Detlefsen and Arana, is the ideal of purity:
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)
A mathematical proof is said to be pure if it does not rely on concepts that are not present in the statement of the conclusion of the proof (the theorem). Many famous mathematical proofs are not pure in this sense, such as Wiles’ proof of Fermat’s Last Theorem, which utilizes incredibly sophisticated and complex mathematical machinery to prove a theorem the statement of which can be understood with knowledge of standard high school level mathematics. (The impurity of Wiles’ proof is one of the motivations often given to seek for alternative proofs of FLT, as described in this guest post by Colin McLarty.) Now, I take it to be fairly obvious that purity concerns can be readily understood as aesthetic concerns, in particular related to simplicity (which is one of the features widely associated with beauty).
What would a dialogical account of the purity desideratum look like? Going back to the idea that the function of a proof is that of eliciting persuasion by means of understanding in an interlocutor (hence the stress on the explanatory dimension), it is clear that, in general, the less complex the mathematical machinery of a proof, the less it will demand of the interlocutor being persuaded in terms of cognitive investment. Moreover, if it relies on simpler machinery, the proof will most likely reach a larger audience, i.e. be persuasive for a larger number of people (those possessing mastery of the concepts used in it). In particular, a proof that only uses concepts already contained in the formulation of the theorem will be at least in theory comprehensible to anyone who can understand the statement of the conclusion. Thus, a pure proof maximizes its penetration among potential audiences, as it only excludes those who do not even grasp the statement of the theorem in the first place. In other words, purity sets the lower bound of cognitive sophistication required from an interlocutor precisely at the right place. (Naturally, I can also be convinced of the truth of a theorem even if I do not understand the proof myself, i.e. by relying on the expertise of the mathematical community as a whole.)
As I said, these are only tentative ideas at this point, so I look forward to feedback from readers. In particular, I would like to hear from practicing mathematicians their answers to the question in the title: what makes a mathematical proof beautiful? Do you agree with Hardy's list? (I could definitely use some input so as to render my investigation more in sync with actual practices!)