(Cross-posted at M-Phi)
A few days ago Eric had a post about an insightful text that has been making the rounds on the internet, which narrates the story of a mathematical ‘proof’ that is for now sitting somewhere in a limbo between the world of proofs and the world of non-proofs. The ‘proof’ in question purports to establish the famous ABC conjecture, one of the (thus far) main open questions in number theory. (Luckily, a while back Dennis posted an extremely helpful and precise exposition of the ABC conjecture, so I need not rehearse the details here.) It has been proposed by the Japanese mathematician Shinichi Mochizuki, who is widely regarded as an extremely talented mathematician. This is important, as crackpot ‘proofs’ are proposed on a daily basis, but in many cases nobody bothers to check them; a modicum of credibility is required to get your peers to spend time checking your purported proof. (Whether this is fair or not is beside the point; it is a sociological fact about the practice of mathematics.) Now, Mochizuki most certainly does not lack credibility, but his ‘proof’ has been made public quite a few months ago, and yet so far there is no verdict as to whether it is indeed a proof of the ABC conjecture or not. How could this be?
As it turns out, Mochizuki has been working pretty much on his own for the last 10 years, developing new concepts and techniques by mixing-and-matching elements from different areas of mathematics. The result is that he created his own private mathematical world, so to speak, which no one else seems able (or willing) to venture into for now. So effectively, as it stands his ‘proof’ is not communicable, and thus cannot be surveyed by his peers.
Let us assume for a moment that the ‘proof’ is indeed correct in that every inferential step in the lengthy exposition is indeed necessarily truth-preserving, i.e. no counterexample can be found for any of the steps. In a quasi-metaphysical sense, the ‘proof’ is indeed a proof, which is a success term (a faulty proof is not a proof at all). However, in the sense that in fact matters for mathematicians, Mochizuki’s ‘proof’ is not (yet) a prof because it has not been able to convince the mathematical community of its correctness; for now, it remains impenetrable. To top it up, Mochizuki is a reclusive man who so far has made no efforts to reach out for his peers and explain the basic outline of the argument.
What does this all mean, from a philosophical point of view? Now, as some readers may recall, I am currently working on a dialogical conception of deductive proofs (see here and here). I submit that the dialogical perspective offers a fruitful vantage point to understand what is going on with the ‘Mochizuki affair’, as I will argue in the remainder of the post. (There are also interesting connections with the debate on computer-assisted proofs and the issue of surveyability, and also with Kenny Easwaran’s notion of the ‘transferability’ of mathematical profs, but for reasons of space I will leave them aside.)
Let me review some of the details of this dialogical conception of proofs. On this conception, a proof is understood as a semi-adversarial dialogue between two fictitious characters, proponent and opponent. The dialogue starts when both participants agree to grant certain statements, the premises; proponent then puts forward further statements, which she claims follow necessarily from what opponent has granted so far. Opponent’s job is to make sure that each inferential step indeed follows of necessity, and if it does not, to offer a counterexample to that particular step. The basic idea is that the concept of necessary truth-preservation is best understood in terms of the adversarial component of such dialogues: it is strategically in proponent’s interest to put forward only inferential steps that are indefeasible, i.e. which cannot be defeated by a countermove even from an ideal, omniscient opponent. In this way, a valid deductive proof corresponds to a winning strategy for proponent.
Now, when I started working on these ideas, my main focus was on the adversarial component of the game, and on how opponent would be compelled to grant proponent’s statements by the force of necessary truth-preservation. But as I started to present this material to numerous audiences, it became increasingly clear to me that adversariality was not the whole story. For starters, from a purely strategic, adversarial point of view, the best strategy for proponent would be to go directly from premises to the final conclusion of the proof; opponent would not be able to offer a counterexample and thus would be defeated. In other words, proponent has much to gain from large, obscure (but truth-preserving) inferential leaps. But this is simply not how mathematical proofs work; besides the requirement of necessary truth-preservation, proponent is also expected to put forward individually perspicuous inferential steps. Opponent would not only not be able to offer counterexamples, but he would also become persuaded of the cogency of the proof; the proof would thus have fulfilled an explanatory function. Opponent would thus be able to see not only that the conclusion follows from the premises, but also why the conclusion follows from the premises. To capture this general idea, in addition to the move of offering a counterexample, opponent also has available to him an inquisitive move: ‘why does this follow?’ It is a request for proponent to be more perspicuous in her argumentation.
This is why I now qualify the dialogue between proponent and opponent as semi-adversarial: besides adversariality, there is also a strong component of cooperation between proponent and opponent. They must of course agree on the premises and on the basic rules of the game, but more importantly, proponent’s goal is not only to force opponent to grant the conclusion by whatever means, but also to show to opponent why the conclusion follows from the premises. Thus understood, a proof has a crucial didactic component.
One way to conceptualize this interplay between adversariality and cooperation from a historical point of view is to view the emergence of the deductive method with Aristotle in the two Analytics as a somewhat strange marriage between the adversarial model of dialogical interaction of the Sophists – dialectic – with the didactic, Socratic method of helping interlocutors to find the truth by themselves by means of questions (as illustrated in Platos’s dialogues). This historical hypothesis requires further scrutiny, and is currently one of the topics of investigation of my Roots of Deduction project, in cooperation with the other members of the project.
Going back to Mochizuki, it is now easy to see why he is not being a good player in the game of deduction. He is not fulfilling his task as proponent to make his proof accessible and compelling to the numerous ‘opponents’ of the mathematical community; in other words, he is failing miserably on the cooperative dimension. As a result, no one is able or willing to play the game of deduction against and with him, i.e. to be his opponent. Now, a crucial feature of a mathematical proof is that it takes (at least) two to tango: a proponent must find an opponent willing to survey the purported proof so that it counts as a proof. (Naturally, this is not an infallible process: there are many cases in the history of mathematics of purported ‘proofs’ which had been surveyed and approved by members of the community, but which were later found to contain mistakes.)
Mochizuki’s tango is for now impossible to dance to/with, and as long as no one is willing to be his opponent, his ‘proof’ is properly speaking not a proof. It is to be hoped that this situation will change at some point, given the importance of the ABC conjecture for number theory. However, this will only happen if Mochizuki becomes a more cooperative proponent, or else if enough opponents are found who are willing and able to engage in this dialogue with him.