A flood from within the community
Gowers is a Fields medallist who did not sign the declaration the other Fields medallists signed on 11 September, and this is his explanation of why.
Perhaps the main reason I didn’t sign is that I don’t fully subscribe to this view. Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.
The declaration’s worry is that we are handed results faster than we can digest them. Gowers takes the volume itself much more calmly.
[…] even if the volume of AI output is too big for us to be able to digest it properly, that is not necessarily a bad thing.
In short, it seems to me that while a flood of “big” AI results would be likely to increase the amount of important mathematics that was not properly digested, it would also be likely to increase the amount that was properly digested, which seems like a pretty good bargain.
Very soon, the whole “credit system” will surely collapse, since finding an amazing proof will be no more of an intellectual achievement than when a citizen scientist spots through their telescope an object that turns out to be a new comet.
What he is not calm about is what happens around the mathematics.
Does all this mean that I am optimistic that mathematicians will end up digesting at least as much mathematics in a post-AI world as it would have if AI had not been able to prove major theorems? Not exactly. But my worry is not that we would be unable to do it, but rather that the social structures that currently support this digestion process will be destroyed and not adequately replaced.
My worry then is that we do not manage to transmit what we know to a new generation of mathematicians. Speaking for myself, my main motivation for becoming a mathematician was the dream that I would solve unsolved problems — the more famous the better. […] If the dream of solving a famous problem had not existed, I’m not sure whether I would have become a mathematician. […] I very much hope that there is a pool of young people for whom it will be a powerful motivation, because I think the survival of a human mathematical tradition may well depend on it.
A related risk is that the perception among policy-makers will be that mathematicians are no longer needed and that funding will become much harder to come by: we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.
[…] there will be a flood of new results, whether we like it or not, and it will no longer be the AI companies producing them, though perhaps the pattern will continue that the AI companies will have access to more powerful models and so will obtain more than their fair share of headline results. So I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly. In fact, it may well be that all that does is bring forward by a couple of months what was going to happen anyway […]
I think he’s predicting the future in a scaling law sense: that proving becomes much cheaper, quickly. Take the Navier–Stokes case as the example. Right now you spend multi-millions of compute to prove a problem worth a million, and that cost can come down fast enough for it to become economically feasible.
Can this be true? Conceptually there can be a plot of how hard a problem is, against the probability of solving it, against how expensive that is — basically, the more difficult the more expensive. Then there’s a “usefulness”, a.k.a. value, against the cost.1
If you spend the money proving something the community sees no value in, and you publish it, you perish.
My gut feeling is that the people doing the work — mathematicians — are the ones who define the value, right? So is the flood a problem, if it is coming from within the community?
Footnotes
The three-variable picture needs more care than a sentence, and I might come back to it in a future post.↩︎