Daniel Litt -- Human Mathematicians in the Age of AI

Yeah, quite a bit. Yeah, so I mean, maybe since I don't know, let me give you the history of my experiments with these things. So I started playing around with AI trying to get it to do math before ChatGPT-3 came out. So there was this, before it was available to the public because OpenAI felt it might be too dangerous to release. There was this video game AI Dungeon that used GPT-3 to tell Dungeons and Dragons stories. So I bought it and I would try to get it to do math inside of the middle of a Dungeons and Dragons story.

I think I probably started using the tools seriously to do math while trying around when o1 came out, which was in September 2024. And then maybe had my first successes around when o3 came out in summer of last year.

Yeah, so right now the primary way I use the tools is for literature search. Sometimes they can usefully prove a lemma or proposition that shows up in day-to-day research. And also it's very useful for finding examples of things or counterexamples, like when you're going down some, you know, you're trying to figure out if something could be true, so you look for a counterexample. It's very useful for that. And then also I've used them to code some scientific computing tools. So recently, for example, I used Codex to write some code to compute with parabolic bundles.

What do you mean by that?

Yeah, so far I have only had one paper come out in which a lemma was proven by AI, and we certainly noted that in the paper. I think that's kind of important to do right now, just at the level of, you know, attribution. Like, if part of your work was contributed by someone else, you should credit them, and I don't see why the same shouldn't be true for the tools. So yeah, I do and, you know, try to make sure my collaborators are okay with me using the tools and that kind of thing.

Yeah, I think from a scientific perspective, I'm really excited. Like, the tools are rapidly growing in capability, and I expect that, you know, in the next few years they'll be able to do some really interesting mathematics. So I think that's like an unalloyed good. I think doing interesting mathematics is important, and we should want more of it. From the perspective of the profession, I'm a little bit worried. Like, right now already it's possible to kind of generate papers that would have been publishable a year ago in a fully automated way, and I think the structure of our institutions is not prepared for that. And I think we really need to think about how we're going to adapt in terms of like publishing, training, and so on. And I don't know, my sense is that we are not yet adapting, and it would be good to start seriously considering that.

Yeah, I think it's changed it a little bit. So first of all, I think just thinking a lot about what skills are necessary to do math has been really useful, and that's something I've been doing is like trying to think about capability. But surprise, surprise, like, do they have the skills we have? An interesting thing I think I've observed, at least in my experience, is that despite growing capabilities, AI tools kind of have a very different skill set than I think strong mathematicians do. So a lot of what we do, I think because like an individual human doesn't know very much, is we reason about like, well, oh, could this be true? And then you kind of go to the literature and you try to understand, you know, has someone thought about this before? And there's a lot of sort of invisible reasoning that we do just to sort of rediscover things in the literature.

And I think that's actually sort of most of what we do. So there's a huge amount of creativity involved in being a human mathematician, but much of that creativity is focused on rediscovering what someone else knew. My experience is that the models are much less creative than humans right now, but because they know everything, they can substitute or know sort of all the existing literature, they can sort of substitute that knowledge for some amount of creativity.

There's a question as to how much math research creativity is really fundamental. My feeling is it's like a pretty small proportion of math research, in the sense that ultimately a lot of mathematical work comes down to recombining things in the literature, and the expenditure of creativity is in finding those things. But I think there is some portion of mathematical work which does require creativity in a way that isn't recombinatory.

I'm very interested to find out what portion of the math literature that is, and also, you know, to what extent AI tools will be useful for that, which presumably will come in the next few years.

Yeah, I mean, I think, right, I think I commented that like, you know, maybe one reason this solution was missed is that, you know, mathematicians are relatively specialized in what we know. And I think that's actually, you know, one of my hopes for progress coming from AI is that AI isn't as siloed in that way. It has like very broad knowledge, and so hopefully it can help people connect different areas.

You know, I think a lot of human work also is about kind of connecting different areas, and like, it's very common for me to like, you know, try to learn about a different field as part of a project. I think that's just part of being a mathematician. So it's exciting that AI can help with that. I think there's also something to be said about depth. Like, a lot of my favorite work is about, you know, not just kind of taking puzzle pieces together and, you know, from different areas and slotting them in, but also just, you know, really trying to deeply understand a mathematical object. My experience so far has been that AI tools are not as useful for doing that, but hopefully they will be useful for that eventually.

Yeah, that's a good question. I mean, maybe let's just briefly review the trajectory. So two years ago, AI tools could not reliably add numbers. A year ago, they got a gold on the IMO, and now they're resolving, you know, well, one major open question in combinatorial geometry, and I think, you know, broadly helpful with math research. I think a lot of mathematicians are really using the tools now.

So if you imagine that trajectory continuing, it's sort of hard to imagine, but I think it's natural to expect that in a year, you know, the tools will autonomously be proving much more interesting results, and, you know, broadly useful in all areas of mathematics.

I think there's a question about whether they will be equally useful for all mathematical activities. Whereas right now, there's been a huge amount of progress in like solving rigorously stated problems where the solution is relatively short, so like, you know, maybe five or six pages. I find it very plausible that they'll be able to in a year produce, you know, 30-page arguments or 40-page arguments that are correct.

There's also been, you know, some question about how useful they are for theory building or sort of more exploratory aspects of mathematics. And my sense is that so far the progress has been a little bit slower, but, you know, there has been progress. So maybe we'll see a lot more in year two. I would not be surprised if by, let's say, 2030, models available to the public could be prompted to produce, you know, in a single prompt, a math paper comparable to the best papers coming out this year.