But Mochizuki didn't actually prove anything, whereas the labs have already done so.
The pessimistic scenario is the AI labs will continuously hoover up new developments in mathematics and gazump everyone in their respective fields. It's clear they have no desire to participate in any silly academic niceties, like properly assigning credit or expository work for normal humans. What incentive then do humans have to do this work ?
Generally the mood amongst research mathematicians is pretty dire, and I don't really blame them.
To be clear I think AI is super useful for mathematical research, the problem is the methods of the big labs are massively disincentivising mathematicians from engaging in research, and other associated tasks like giving seminar, teaching writing books etc. These arguably have much more value than finding an obscure counter example to Navier-Stokes.
i find this and other efforts from anthropic somewhat antisocial. technically they have achieved their goal, but in a way which does not benefit mathematics or humanity. Kevin Buzzards headline goal was to formalise FLT, but i’m sure the real aim was to create a formalised library of mathematics which is comprehensible to humans. By solving these famous problems by brute force, they are disincentivising the important work of making it digestible for everyone else, and so in my view this work in particular has negative societal value.
You can say that American society made OpenAI and Anthropic possible. No other current society would have. Suddenly, formalisation of math is becoming cheap. That's not a problem, that's the goal, and it is here much earlier than expected. That's not antisocial. That is scientific progress.
You're stating it's not a problem- but I'm giving you a reason why it is. This is serendipitously mirrored by a recent post from Terence Tao on Mastodon (https://mathstodon.xyz/@tao/117207856734787448)
In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field through the efforts to solve such problems, and then to digest any partial or complete solutions that emerge for further insights. Prematurely solving the problem by purely AI-powered methods - particularly without full transparency into the solution process - can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.
Yes. But that is a problem for pure mathematics, not for society. I think that pure mathematics is over. At the same time, applied mathematics will probably subsume most of pure mathematics. Fermat's theorem now is applied mathematics! It will be used to improve implementations of proof assistants for a long time.
Yes, it has been. And in the form of applied mathematics it will continue to be. It is not so much that pure mathematics disappears, but that in the future there will be just mathematics, and of course it is applied. You would be surprised what kind of mathematics appears when you actually try to formalise your applications properly. It pretty much includes everything that is thought of as pure mathematics today, and much more.
The pessimistic scenario is the AI labs will continuously hoover up new developments in mathematics and gazump everyone in their respective fields. It's clear they have no desire to participate in any silly academic niceties, like properly assigning credit or expository work for normal humans. What incentive then do humans have to do this work ?
Generally the mood amongst research mathematicians is pretty dire, and I don't really blame them.
To be clear I think AI is super useful for mathematical research, the problem is the methods of the big labs are massively disincentivising mathematicians from engaging in research, and other associated tasks like giving seminar, teaching writing books etc. These arguably have much more value than finding an obscure counter example to Navier-Stokes.