(Cross-posted at M-Phi)
“That's the problem with false proofs of true theorems: it's not easy to produce a counterexample.”
This is a comment by Jeffrey Shallit in a post on a purported proof of Fermat’s Last Theorem. (Incidentally, the author of the purported proof comments at M-Phi occasionally.) In all its apparent simplicity, this remark raises a number of interesting philosophical questions. (Being the pedantic philosopher that I am, I'll change a bit the terminology and use the phrase 'incorrect proof' instead of 'false proof', which I take to be a category mistake.)
First of all, the remark refers to a pervasive but prima facie slightly puzzling feature of mathematical practice: mathematicians often formulate alternative proofs of theorems that have already been proved. This may appear somewhat surprising on the assumption that mathematicians are (solely) in the business of establishing (mathematical) truths; now, if a given truth, a theorem, has already been established, what is the point of going down the same road again? (Or more precisely, going to the same place by taking a different road.) This of course shows that the assumption in question is false: mathematicians are not only interested in theorems, in fact they are mostly interested in proofs. (This is one of the points of Rav’s thought-provoking paper ‘Why do we prove theorems?’)
There are several reasons why mathematicians look for new proofs of previously established theorems, and John Dawson Jr.’s excellent ‘Why do mathematicians re-prove theorems?’ discusses a number of these reasons. The original proof may be seen as too convoluted or not sufficient explanatory – ideally, a proof shows not only that P is the case, but also why P is the case (more on this below). Alternatively, the proof may rely on notions and concepts alien to the formulation and understanding of the theorem itself, giving rise to concerns of purity. Indeed, recall that Colin McLarty motivates his search for a new proof of Fermat’s Last Theorem in these terms: “Fermat’s Last Theorem is just about numbers, so it seems like we ought to be able to prove it by just talking about numbers”. This is not the case of the currently available proof by Wiles, which relies on much heavier machinery.
From the point of view of the dialogical conception of proofs that I’ve been developing, as involving a proponent who wants to establish the conclusion and an opponent who seeks to block the derivation of the conclusion (see here and here), an important reason to re-prove theorems would be related to the persuasive function of proofs. Dawson does mention persuasion in his paper, but he does not adopt an explicit dialogical, multi-agent perspective:
[W]e shall take a proof to be an informal argument whose purpose is to convince those who endeavor to follow it that a certain mathematical statement is true (and, ideally, to explain why it is true). (p. 270)
(In my opinion, this is a fabulous definition of mathematical proofs, except for the fact that it is not explicitly multi-agent.) That is, a given proof, while correct, may still fail to be sufficiently convincing in this sense. I am here reminded of Smale’s original proof of the possibility of eversion of the sphere, which however did not exhibit the process through which the eversion would take place. It was only when Morin built clay models of the stages of the process that it became clear not only that a sphere can be eversed, but also how it can be eversed. (In mathematics, whys often become hows, i.e. how to construct a given entity, how to realize a given process etc.) In fact, it is now known that there are different ways of eversing the sphere.
Still within the dialogical framework, another reason to formulate alternative proofs of theorems are the different commitments and tastes of various audiences. A mathematical proof is a discourse, and even though there is an absolute sense in which a proof is or is not correct, different proofs will be more or less persuasive to different audiences. For example, this observation would explain the search for constructive as well as classical proofs of the same theorems, thus catering for different groups of potential addressees. More generally, different preferences in argumentative styles (not only in theoretical commitments as the ones separating classical and constructivist mathematicians) may also create the space for several proofs of the same theorems.
And here is a final, less ‘noble’ reason for re-proving previously established theorems: such proofs are harder to refute. The mathematician’s preferred approach to refuting a proof is to provide a counterexample, i.e. a situation or construction where the premises hold but the conclusion (the theorem) does not. Now, if providing such a counterexample were the only move available to opponent to block the inference of the conclusion by proponent (a thought that I confess to have entertained for a while), then every proof of a true theorem would be a valid proof, no matter how absurd and defective (such as the one motivating Shallit’s comment above).
This is exactly why a proof cannot be a one-step argument going directly from premises to conclusion (which is, in effect, a necessarily truth-preserving move in the case of true theorems): a proof spells out the intermediate steps, which must be individually perspicuous and explanatory – and yes, also necessarily truth-preserving. So incorrect proofs of true theorems require the additional work of delving into the details of the proof in its different steps in order to reveal where the mistake(s) lie(s) – more work, and often tedious work.
Nobody said it was easy being a mathematical opponent.