Sunday, April 26, 2009

Tim Gowers on "correct" proofs

Catching up with Tim Gowers's blog, I notice he makes some interesting remarks in passing about the idea of a proof. He'd initiated/co-ordinated a "polymath" project -- a collaborative all-comers group effort trying to find a combinatorial proof of a result in Ramsey theory previously known from a proof in ergodic theory. After hundreds on contributions, but of course before any attempt to put the bits and pieces together into a conventionally written-up proof, Gowers says "I am basically sure that the problem is solved (though not in the way originally envisaged)."

Why do I feel so confident that what we have now is right, especially given that another attempt that seemed quite convincing ended up collapsing? Partly because it’s got what you want from a correct proof: not just some calculations that magically manage not to go wrong, but higher-level explanations backed up by fairly easy calculations, a new understanding of other situations where closely analogous arguments definitely work, and so on. And it seems that all the participants share the feeling that the argument is “robust” in the right way.
So, interestingly, getting a "correct" proof in his sense involves more than getting a proof in the austere logician's sense of a correct proof as just ("magically") getting out the desired result without logical gaps -- it goes with explanations and understanding.

Let's not fuss about terminology. It undoubtedly is the case that a notion of a correct proof in something like Gowers's sense functions centrally in mathematicians' thinking. But how good are the attempts by philosophers of mathematics to elucidate this notion?


Anonymous said...

I can't find the quote, but is that what he is saying? The quote sounded like the other proof was not correct -- just persuasive. In other words, it sounds like, in the first round, you have a sophistical argument that is very good looking at first but ultimately fallacious, but in this round, even though he has not completely worked out all the details, yet, it looks like this is going to do it and actually be correct this time.

Anonymous said...

Oh yeah... I found it. (Duh! LOL) I'm pretty sure he's not saying that it takes more than the correct proof. What he is saying is that it takes less than the correct proof. He is calling the shot without actually looking to make sure because he can tell this is probably going to be right. Ultimately what it takes is just the correct proof. I wasn't part of the project or antying, but I doubt the magic calculations are the correct logic sans intuitive motivation or some other additional component. They are not a logical argument at all. They are just suggestive outcomes that ended up not leading to a proof. Now, he thinks, even without dotting all the i's and corssing all the t's, that what they have now will actually lead to a correct proof.

Peter Smith said...

I think Anon is perhaps misreading my post. What I was noting was Tim Gowers's stressing that what we want from a proof is not just a (deductively ok!) argument with the right conclusion, but one which increases understanding, links in with explanations. Sure, Tim Gowers thinks that the there's enough, before dotting the i's and crossing the t's, to bet on being able to complete the job to everyone's satisfaction. But his confidence that the job can be completed, he says, is in part based on the thought that the ideas have fallen into place with the right explanatory ooomph. And I was wondering whether philosophers of maths have a good story to tell about what makes for being explanatory in the sort of way in question.

Karim Zahidi said...

Manin, in his "A course in Mathematical Logic", devotes a few pages on a discussion of proofs (and their relation with creating understanding of the subject at hand)in mathematics. He ends his discussion with the following sentence:

"The moral: a good proof is one which makes us wiser"

a.c. said...

*Goes off to look for a copy of Maddy's "Believing the axioms", parts 1 & 2, ... *

While her topic isn't quite the same, some of what she says might still be relevant.

I also wonder about Corfield's Towards a Philosophy of Real Mathematics (though it was lambasted on the FOM mailing list a while back).

Edward Dean said...

Erich Reck's recent paper "Dedekind, Structural Reasoning, and Mathematical Understanding", available from here, addresses the matter. This earlier paper by Jeremy Avigad bears on the topic as well, also using Dedekind as a case study. (You might also find this one interesting.)