### Wittgenstein and Thurston on Understanding

#### Posted by David Corfield

Two contributions to the Berlin Workshop I didn’t mention addressed the Jaffe-Quinn debate (original article, 15 mathematicians reply, Thurston replies, Jaffe and Quinn reply to the replies). Michael Stoeltzner spoke on The Ontology of Mathematics, while Oliver Petersen argued that Wittgenstein could offer us a way out of the bind at stake in the debate between Jaffe and Quinn, on the one hand, and Thurston on the other, by radicalising the former’s contentions about the importance to the health of mathematics of providing rigorous proofs for conjectures.

Petersen belongs to what I have called the generous interpreters of Wittgenstein, someone who deflects the charge that Wittgenstein is evidently wrong when he argues that propositions which we don’t know how to prove are meaningless, or that different proofs of the ‘same’ proposition must be proving different things, by saying that Wittgenstein didn’t quite mean these things. Furthermore, he argues that we can go beyond Jaffe and Quinn when we understand the real reason that proofs are so vital is that they add to the stock of proof techniques or argument patterns, augmenting the allowable moves of the language game. Knowledge of the moves of the language game is all we need mean by ‘understanding’, and it allows us to avoid ‘Platonic’ talk of things, like the infinite, that we’re coming to understand better in some mysterious way. I don’t believe that Wittgenstein was brought to make this move through any distaste for the mystical itself, being, after all, a close reader of Tolstoy, but rather through a dislike of people speaking about the mystical - “Whereof one cannot speak, thereof one must remain silent”. So Cantor’s theological dressing of his set theory would have offended.

However generous one chooses to be about Wittgenstein, his vision of mathematics remains a skeletal affair, where Thurston’s is much fleshier, going far beyond the proving of propositions. Now, what brought about Thurston’s response? As I listened to Petersen’s talk, I came to realise what was really at stake in Thurston’s contribution to the debate. In essence, he and a few others were being accused by Jaffe and Quinn of acting very irresponsibly. By tossing out a conjecture and sketching how a proof might go, they were not acting for the good of mathematics. Big names shouldn’t behave like this, as it causes confusion, misleads the young, and discourages people from sorting out the field. Thurston’s response was to agree that responsibility is precisely what is at stake, but he goes on to say that responsibility involves so much more than maintaining standards of rigour. To keep his field alive, he says, he has had to leave the frontiers of his explorations to return to show others the way and to allow them to make their own discoveries. In his accompanying account, he gives us a very rich picture of mathematical understanding, one which rings many bells for those involved in mathematical research. Should it worry Wittgenstein?

I think it’s important to separate two ‘realist’ issues: the kind of thing mathematics is about and the notion of the ‘correctness’ of a concept, construction, definition. Let’s start with the first, where much of the philosophy of mathematics aims itself. One way to understand Wittgenstein’s motivation to translate *understanding* as an ability to make some moves within, or add to the allowable moves of, a mathematical language game, is to see how it deflates ‘Platonist’ talk. (I use scare quotes to indicate that it’s a long way off from what Plato thought.). My saying that a mathematician has a better understanding than I have is then not so very different from my saying Kasparov understands chess better than I do. The fact that chess can be conducted by moving some physical objects about a board to achieve a certain end is not important. Even if we invented a game that couldn’t be played by moving pieces on a board within 3-space, but required at least 27 dimensions, we could still have world champions, who would understand the game better than us, and who might well be better able to think up similar gripping games. ‘Ontological commitment’ about mathematics, on this view, is no more frightening than that.

But, as Petersen detected, there is something more in Thurston’s idea of understanding. For me the key here is to realise the right relationship between the goal of a discipline, an improved understanding, and the proving of conjectures. For Thurston, the primary goal is understanding. The making and proving of a conjecture may or may not aid the understanding of the field. The former is a subsiduary means to the proper end of understanding. In a 1982 paper, he suggests that the Poincaré conjecture has proved not to be a great guide and hopes that his Geometrization conjecture (now apparently the Geometrization theorem) will prove a better guide (Thurston, W. 1982, ‘Three Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry’, Bulletin of the American Mathematical Society 6: 357-81). I think one would have to be an extraordinary generous interpreter of Wittgenstein not to see the difference between Thurston and Wittgenstein here.

I think that our strong communal emphasis on theorem-credits has a negative effect on mathematical progress. If what we are accomplishing is advancing human understanding of mathematics, then we would be much better off recognizing and valuing a far broader range of activity.

…the entire mathematical community would become much more productive if we open our eyes to the real values in what we are doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation” and “proving”. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this. (Thurston 1994: 171-2)

The question then is whether this something extra should worry us. In other words, when I say Thurston understands 3-manifolds better than I do, if I forego Petersen’s advice to follow Wittgenstein, and say that this is more than his ability to make some moves within, or add to the allowable moves of, a mathematical language game, have I now committed myself to some entities in a Platonic realm about which we cannot say how we have access?

What is this extra? It involves the question of the rightness of the language game, and how it can be improved. I’m glad Thurston has his say in the way research into 3-manifolds is conducted. I trust his judgment to introduce relevant new concepts, and to formulate new guiding conjectures. I think he is far more likely to lead the game in the right direction than I ever could. At this point we cannot avoid the use of the word ‘story’. It is crucial to realise that mathematics, as with any intellectual discipline, is scored through with stories, is constituted by dramatic narratives. In Sir Michael Atiyah’s *Mathematics in the Twentieth Century* (Bulletin of the London Mathematical Society 34(1), 1-15, 2002), the words ‘story’, ‘stories’ or ‘history’ appear 23 times in just 15 pages. Many of these concern the story of a part of mathematics.

So, the best way to phrase Thurston’s understanding of 3-manifolds is to say that he is someone to whom we can entrust the story of this part of geometry. He is better able to tell the story so far, see how earlier viewpoints were partial, and better able to sketch out how the next chapters might go, how future viewpoints might see ours as partial. He is likely to be part of the story when told centuries hence, and for good reason. In effect, Jaffe and Quinn are charging him with jeopardising his part of the story of mathematics, and he robustly rebuts this charge. He more than most has helped the next generation: “I do think that my actions have done well in stimulating mathematics”.

To end, I don’t see that there’s anything here to worry Wittgenstein.

## Re: Wittgenstein and Thurston on Understanding

Thank you - a nice article, and one that is particularly pertinent to n categories. After all, we are still not sure what the

definitionof an n category should be. Of course, most (all?) of the candidate definitions have some theorems, so it could be argued that there is a theory of n categories in the sense of (miserly read) Wittgenstein. But much of the real work I would suggest here is still searching for the right conceptualisation - and that is very valuable mathematics. Indeed, for my taste, it is more valuable than the proof of theorems ‘everyone’ ‘knows’ are true anyway. To pick a concrete example, isn’t the original Lawvere Pavia paper on enriched categories a great example of Thurston style mathematics?