I think of my goal as a mathematician as something like: to better understand the fundamental concepts “shape” and “number,” and to convey that understanding to others. I think this is a valuable goal for many reasons. The primary reason discussed on social media is the (admittedly slow, but I think very real) diffusion of mathematical ideas into technologies that better our lives. There are others, though: production of human capital (which plays some role in this diffusion), and diffusion of mathematical understanding into culture, for example. It is not obvious to me that even the bare technological need for mathematics can be filled without human experts; this seems to me to be a tricky question of institutional design. For me the most compelling reason is simply that we feel a pressing need to understand. While deep understanding is largely limited to professional mathematicians, the existence of mathematicians and of the institutions they make up is a huge part of what makes mathematical understanding available to those who want it. I believe that the capacity to understand mathematics (and the universe as a whole) is part of a good life, and we should try to provide people with a good life. We operationalize the goal of understanding in a number of ways, problem-solving and theorem-proving among them. My own research is oriented around certain open questions that I believe measure our lack of understanding (of numbers, shapes and so on), and I think resolving those is genuinely important. I would welcome a solution with any provenance. But I think proof is only a measure of progress—what actually matters (for technology, human capital, culture, a good life) is a bit harder to get at. Crucially it is very hard to measure progress—plausibly theorem-proving is used in large part because it is easy to measure rather than because it is a perfect proxy for what we care about. I worry a bit that we may automate many easy-to-measure things without automating the correlates we care about for which they are proxies. This presents a clear problem for the design of institutions aimed at achieving progress. In part because I think problem-solving is a proxy for something else, I don’t feel that my goal of understanding “shape” and “number” is particularly threatened by a machine that is good at problem solving. I expect that machine to be a substantial boon, instead. Ultimately I think we will operationalize mathematical progress in better ways. But how to do this is a difficult question and I think many on here are too quick to dismiss concerns about it.

Well worth reading (as with all of Frank’s writing): galoisrepresentations.org/20…

Without commenting on the correctness of this claim, I want to briefly say a few words about what it would mean in practice. Chess is supportable as a hobby in part because one can pick up the rules of chess in an afternoon. While one can't quickly play at a high level, it is relatively easy to appreciate the game, and have fun at it. Research math is not like this. Over the last decades we have made incredible progress understanding basic concepts (number, shape, etc.) but the edifice we have built to do so requires an incredible time investment. Learning the ideas that go into proving, or understanding the proof of, basic statements like Fermat's last theorem requires years of institutional support. It's possible that soon, AI will be able to do work at this level; despite recent successes, this isn't the case yet. But even if it can do this, why would it? Without people who care about these deep but fundamental ideas, who will ask it to? Perhaps it might on its own, or a person asking it to "do some cool math" might elicit a similarly deep idea, but who would understand it, if doing so requires years of unsupported work? In practice, turning pure math into a hobby means the end of pure math, and the end of people who understand basic mathematical concepts at a deep level. Whether this is something we want is something we'll have to decide together.

So there's a mathematical question I've been thinking about on and off for about 10 years. For the first 5 or so years I made very little progress, but learned a bit about the general area (while working on other things). In the last 5 years I've made some modest progress with collaborators, resulting in some (IMHO) nice results, though the question itself is still totally open. This has developed into some not-quite-rigorous research program which guides much of my current work (some of which is aimed at trying to make it rigorous). Of course I discuss this work with the models; for the first time, this week, I had a model produce, non-autonomously, a very nice partial result towards a question that came up in the course of this program, connecting it to various areas about which I know relatively little (microlocal sheaf theory, Lagrangian Floer homology). So now I get to learn about some of this stuff! Even in this (quite small) part of the project, there's lots for me to do--I've already found various improvements to and simplifications of the arguments it produced, and noticed some implications it seems to have that seem to me quite significant. An arguably underrated part of math is "discovering interesting new phenomena," and one function of mathematicians and the mathematical community is to certify this interest. That our open problems have been so significant to frontier labs is evidence we're pretty good at this! Of course I have thought about how I would react if GPT-8 Ultra Extreme or Claude Cthulhu 7 were to autonomously solve the entire problem. I think I would be excited--there would be a lot to learn, and surely a solution would raise many new questions. Maybe it would be a little bittersweet (I have an ego), but I view this problem as mostly an excuse to think about some fundamental objects, and I am quite convinced I'd continue to think about them.

I wonder how much of the divide among mathematicians on AI comes down to which of us consider ourselves scientists and which of us consider ourselves artists. (I'm very much in the former camp but for what it's worth I am sympathetic to the latter.)

I've been saying for some time that we should use AI systems to produce better work, as opposed to, say, more PDFs. One simple thing we can now do is identify and fix lots of extant errors in the literature. 1/n

Starting to think that the base rate at which well-known old conjectures are true might not be very high!