You have a new tool. It's a lot of fun to use, but you know, I think a lot of people are talking about accelerating science. I'm not necessarily skeptical that that's going to happen in some sense, but I think they're talking about it without trying to rigorously run tests as to how much the tools are accelerating things. Anytime I think about an open problem, the first thing I do is ask models for some ideas. They're almost always sort of nonsense. Never gotten an idea that sort of passes the sniff test for sort of a deep open problem. It's weird that we haven't seen a lot of mildly interesting conjectures resolved by AI. I guess we're now starting to see those arguably. So I think right now, at least where I expect AI to have a significant impact is that the marginal cost of like trying something is getting very small.
Hello everyone. I'm Greg Bernham. I'm a researcher at Epoch AI and this is my colleague.
I'm Anson. I'm also a researcher at Epoch AI.
And we're joined today by Daniel Litt. Hi Daniel.
Hey, nice to see you guys and meet you in person. I'm Daniel Litt. I'm a professor at the University of Toronto in mathematics.
I wanted to start with sort of a fun one. Could you characterize like what's the hardest math problem that AI systems today can solve? However you might think about that.
That's a good question. Yeah. So I think, okay, so of course there are these examples where now every frontier model basically has gotten a gold on the last IMO. And I think that's a pretty good baseline for what you should expect. So I think probably later in the discussion we'll discuss a few open problems that have been resolved either with the help of AI or autonomously by AI, and I think it's probably accurate to say that those are about at the level of some kind of a mid-tier, low-tier IMO problem.
Gotcha. I think there's some evidence that systems can do a little bit better than that right now.
And I think, you know, with some work one can probably even elicit better performance than that from current gen models. But I think that's overall like about the level of a contest problem. Something you would expect a strong high school student or undergrad to solve in a couple hours seems to be about where the systems are at.
And one thing you've, you know, stuck your neck out on is that within the next year we might see like mildly interesting conjectures resolved. Maybe you can say what that means.
Yeah. So for me, I think what I mean by a mildly interesting conjecture is something that someone has stated in print. So at least one person was really interested in it and hopefully someone has, you know, spent at least a few hours thinking about it. I think there are a good number of problems at that level. I would expect that current gen systems can resolve some of these things with sort of the, you know, pass at 1 million or something like that.
I think arguably some recent examples actually fit that bill. So maybe that prediction where I stuck my neck out has actually come true. Although there's some questions about exactly, you know, how interesting such problems are, right? Yeah.
Very nice. You've actually referred a couple of times already now to sort of time frame for human solve. I was going to get around to that eventually. Is that a metric you like in general? I'm curious what you think about that.
To be honest, not really. I think that's not a great way to think about difficulty. So right, so an IMO problem, you know, a strong high school student, a very strong high school student, the best students in the world have about an hour and a half to solve the problem. That definitely gives you some upper bound on difficulty.
You know, I think that for a lot of those problems, if you gave them to a, you know, a professional mathematician, they would actually take longer.
And that's just because, you know, they'd fiddle around. They wouldn't be motivated in the same way as in a contest. And also just solving a contest problem, there's a certain very constrained set of techniques you can use, and, you know, in research mathematics you're not constrained at all. So you don't just try the tricks and see what works. You try everything and you fiddle around and you maybe break out a computer and do some, you know, work out some examples or whatever. So yeah, I think difficulty, it's sort of a funny thing, like how do you judge it?
And a lot of the times maybe the best way to judge it is after the fact. So you look at the structure of the proof and you say, 'Oh, well actually this wasn't that hard.' I think that's a little dangerous, right? Of course, especially when we're talking about like evaluating a model. I think that leads to goalpost moving. Like the model comes up with a proof and you say, 'Oh, well that wasn't that hard.' You know, I didn't have to do anything. I just pressed a button.
One thing you've said often a lot on Twitter is that the amount of uplift you get from these systems is actually quite limited. But then I think there are also some other people who feel like the uplift is quite a lot bigger. I'm curious like how you would characterize like what the difference is, like what's going on.
Yeah. I mean I do think there's some areas where the models are just better. So, you know, if you are working in optimization for example, my sense is that there's a lot of people, for example at OpenAI, who are experts in that area and they've generated a lot of data and probably use their human expertise to guide the models. And so I would not be surprised if people, for example in that area, are getting more out of the models. Comparatively, my sense is that in algebraic geometry, number theory, the models are just not that good.
I think also there are areas which are more amenable to sort of the tool use the models have access to. So, you know, if you're trying to come up with a counterexample to some inequality or whatever, you know, writing code is a very natural thing to do. If you're trying to come up with a counterexample to a conjecture about the intermediate Jacobian of a cubic fourfold, probably there isn't really any code you can write that will help you.
That's interesting. That said, I also think probably people are just misreporting the amount the models are helping them. You know, you have a new tool. It's a lot of fun to use. But, you know, I think a lot of people are talking about accelerating science. I'm not necessarily skeptical that that's going to happen in some sense, but I think they're talking about it without trying to rigorously run tests as to how much the tools are accelerating things. I'm very happy to believe that some frictions are being removed, but there are also a lot of bottlenecks in research that I think the models just don't touch. And, you know, so are you really accelerating if you, you know, have removed a bottleneck to, you know, opening a paper and finding Lemma 3.7, but not to, you know, having a good idea.
Yeah. This is why we're glad to have you tell us what's right and what's wrong.
Yeah. One thing I find interesting about that is like the typical way I would characterize differences in capabilities or the spikiness of AI capabilities across domains is through something like Moravec's paradox. But then it seems like for things within math, like we also get this kind of spikiness within math. And I guess we were kind of alluding to some of this coming from what training data there is. Some of it is coming from like, you know, what is more amenable to AI. Do you think like that's like explaining most of it? Do you think there's anything else like, or how would you characterize like the relative importance of these factors?
Yeah, I mean what I do think is like doing research math is a skill that's a little different from contest math in that it's a very high dimensional skill. So it's not, you know, there's not some more or less finite set of known techniques that are useful. I mean, okay, sometimes contest math requires a little bit more creativity, but okay, to be honest, the models have not really succeeded in solving problems where I think one sure can argue that that's required. So yeah, I think, you know, a lot of the jaggedness we see in math, I think is just the same jaggedness we see everywhere. I don't think there's anything special. Like in my view, the biggest obstacle to the models like autonomously doing high quality math research is just the same as the biggest obstacle to automating anything, which is like they can't do sort of long context tasks. You know, something that would take a human six months. There's just no task that takes a human six months that the models can do. So yeah, I think once we start seeing the models performing like software engineering tasks at that scale, I would not be surprised if they also start doing high quality math research.
So I just actually don't think there's anything special about math in this regard.
Makes sense. Yeah. Well, one sort of galaxy-brained take I have that I probably, that I will hold lightly is that models are weak in spatial reasoning, visual spatial intuition. That's not the galaxy brain part. The galaxy brain part is that that is a big latent factor explaining what they're good and bad at. I do know from hearing narratives of various mathematicians about their thought process that often there is something quite spatial or geometrically intuitive about a reasoning process. And I do wonder a little if the AI models are especially good when there's a more symbol manipulation approach to solving something, coding being the most obvious, but even just working through algebra that might be.
To be honest, I'm a little bit skeptical of that kind of explanation. Like first of all, there's a lot of diversity in how mathematicians think about mathematics. So some of, you know, some of us are whatever, shape rotators, some of us are more like word cells or... I was curious actually, trying to question this belief, whether there are, what's the word, aphantasic mathematicians. Actually quite famously, some even geometers who are. Right now there's a huge number of mathematicians working on problems and we have lots of different approaches and that's partly why, you know, there are mathematicians who I think are overall, if you tried to put all mathematicians in a line, way, way better than me at math, nonetheless I'm proving some theorems which they haven't proved or probably couldn't prove in the same amount of time. It's just because we have a different approach. So I think also, you know, now there are whatever, like three or four models that can reasonably well solve math problems. They have slightly different approaches, a much smaller diversity of approaches than humans.
And I think we actually see that reflected in the benchmarks. It seems like the set of problems that are being solved, there's actually quite a lot of commonality between the models, is my understanding from, you know, hearing what you guys have been doing analysis. Whereas, you know, of course, all the problems in the benchmarks have been solved by at least one human.
Yeah, so I think, yeah, my sense is that you should think of the model as like a mathematician, and so there's certain problems that they'll be quite good at and certain problems that we'll be quite bad at, but maybe we shouldn't read too much into what those problems are. It's just an artifact of the fact that there's, you know, only two or three models to look at to begin with.
Since you mentioned it, it's a big question that is very hard to answer, but I'm curious. How much transfer do you think we're seeing in capabilities across different subfields of math? So if you get this edge, surely you get an edge from generating synthetic data. How big an edge do you, as capabilities grow, do you not really need that, or do you have any sense of this?
Yeah, it's a good question. I mean, I think it's sort of hard to say. My sense is that most of what happens when you try to get a model to prove a statement in algebraic geometry is that it tries to find it in literature, or like, you know, you could find something very close in literature and, you know, just, you know, try to make one or two reasoning steps beyond that. Like it's really, compared to what happens when you ask it sort of a combinatorics question, it's sort of not, at least in my observation, doing the same kind of like, you know, real attempt to solve a problem.
Interesting. And I think, you know, compared to a graduate student, you know, who maybe knows, like a graduate student who knows all the stuff the model knows about algebraic geometry, number theory, or sort of fancy math topics, would be able to do a lot more reasoning and like really try to prove theorems. So it seems to me like, you know, the models are sort of superhuman in some mathematical subjects in terms of knowledge. But they don't somehow, yeah, whatever reasoning capabilities they have or whatever tech, like they haven't necessarily learned the same techniques that a student who has that same knowledge base would know.
This is just based on vibes.
Totally. Of course. But I don't know, we've chosen you to hear your vibes. What areas do you have the sense that the models are stronger in, like native reasoning?
Yeah, I think, I mean, okay, at least I don't know about superhuman, but they're definitely super me at like proving an inequality, for example, that kind of thing. My guess is just it's sort of easier to generate data and there's probably a lot more data in that area than in algebraic geometry.
And when you say an inequality, are we talking like contest style inequalities? Not like something a little more interesting or important from analysis where everything's inequality.
Yeah. I mean, yeah. Something, yeah. And I know, yeah, so something where coding is useful. They're very, you know, typically very strong. Like, you know, every once in a while I'll need to prove an inequality and like, you know, now my first step is just to sort of explore what the space looks like by writing some code using a model.
I think we'll get into this a fair bit more, but you did mention two things I thought were maybe a little in tension. One is a thing that the models are missing is having a good idea, but another was if they can just do things for six months in many domains, maybe they'll be good in math as well. I say these are in tension because I think there are six-month projects humans do that don't require having like brilliant flashes of insight or anything particularly, planning a wedding or something, which models could not do today and takes months. Like do you think creativity comes just with time and so, or I don't know, how do you...
Yeah, that's the question. I know, I agree that those two statements were kind of in tension. What I would say is I think there's some kind of continuum between like just applying a technique that's well known and developing a new technique. Arguably a new technique is, you know, you take a hundred different things and put them together in some way. And yeah, I presumably at least one ingredient to doing that is just time.
Sure, fair enough. It's just not clear to me whether it's the only ingredient.
I would say my experience of doing math is very rarely that I have a, you know, a brilliant idea that just solves the problem. Of course, sometimes, you know, you wake up in the middle of the night and like the problem is solved. But typically for that to happen, you have to marinate in the problem for four months. So like there is some secret ingredient of time. I don't know that my introspection is well-developed enough or trustworthy enough to say whether that's the only real ingredient. I mean, I think there are... Okay, let me amend that slightly. You know, there are other things that happen, like you develop philosophies or analogies, and there's some kind of mystical aspect to doing mathematics that I think we haven't seen the models do, but it's also sort of BS, right? Like that mystical aspect of doing mathematics is maybe just kind of compressing a lot of sort of ideas that you've read or absorbed into some kind of package that's digestible to humans. So I don't know, maybe it's close to context compaction or something like that.
Interesting. Yeah, I guess there are these big analogies for like intelligence is search or intelligence is compression or something like that, and we're just better at that for now, and models have been getting better at it, and...
Yeah, generally speaking, I'm also skeptical of those kinds of analogies, but like...
Oh, I just think, you know, my sense is that there are a lot of ways to be good at math. Like, you know, if you just look at what different people are able to do, there's not that much overlap in capabilities. Like I don't know that there's any mathematician who like can prove the same theorems I'm proving, and there's plenty of mathematicians doing stuff where I think their way of thinking is just quite different from mine.
Oh, but I suppose, I mean, maybe I said this poorly, but if I think of... maybe it's not falsifiable. If I think of intelligence as search, then what that sounds to me like is you and a different mathematician are pursuing different heuristic search algorithms or something like that.
Sure. I think that there's maybe a way of making sense of it, but I would argue that's sort of not very contentful, right?
Like if intelligence is some hugely high dimensional space and you just make a name for that space, doing a good job at that is, yeah, I don't know, you can't tell if it's enlightening or not.
Could you contextualize the utility you're finding in the context of previous generations of tools? Like literature search is better, but Google Scholar existed, you know, and that was presumably an improvement over card catalogs and conference proceedings.
Yeah, I mean, I think right now the tools are on a continuum with previous generations. So maybe in two ways. So first of all, literature search. Yeah, definitely the models are now, at least for some literature search tasks, better than Google or better than Google Scholar. That probably saves some time. Is it a lot of time? Well, I don't know. I mean, how much does it, how much time does it save compared to going to the library and probably a couple hours?
A bit every once in a, you know, that kind of 2% productivity, long run productivity improvement, right, on trend.
In general, I think, yeah, those improvements seem to typically be, as you're saying, like fairly small. I would be skeptical that this is more than a percent or two.
Like if AI progress stalled today, you wouldn't expect that we already have baked in an explosion in quality of mathematics compared to what was developed 10 years ago.
Yeah, I would expect, yeah, a sort of similar productivity growth to what we've seen, which is, you know, maybe attributable to some extent to technology, probably mostly attributable to population growth. You can ask just the same question at Google. Like how much did Google or email, that kind of thing?
Did you live through that? Did that feel like, you know...
So I was born in 1988, so I got my PhD in 2015. So yeah, Google was already around by the time I started thinking about math. So I have actually asked older mathematicians this question. I think the general consensus, just self-reported, is that like Google did increase mathematical productivity. But it's pretty hard to see if you just like try to look. I mean, it's hard to come up with a proxy that lets you measure this. But I don't think it's obvious just from vibes, for example, that like the advent of Google led to like really, you know, a remarkable growth in good new mathematical ideas. I just think literature search is not really where the main bottleneck is.
Makes sense.
So there's another precursor to the development of these sort of AI tools, which is just the development of computing.
Sure. So we saw a lot of progress in a lot of different areas, like in the, you know, already in the '60s, '70s, '80s, just with the advent of computers. So for example, maybe a famous example is Euler had this famous conjecture, the sum of powers conjecture, which is, you know, when is there always a solution to a sum of kth powers being another kth power for some number of kth powers. I don't remember the exact number. So the first counterexample was found just via computer search. And then maybe more famously, the case of fourth powers was resolved by Elkies in 1988 using a sort of very clever computer search.
Got it, but like that method would have been dead in the water without computer search, even though there's a lot of cleverness.
Yeah, yeah, that's right. Yeah. So he sort of found a way to make these questions accessible to a 1988 computer. I think probably now still they would not be accessible to just naive force. But yeah.
But yeah, so this was a huge development, and I think in a lot of ways, like if we just stopped with existing models, like we would see some kind of natural continuation of that trend.
What would that look like? I guess if right now we mostly use them for literature search and coding, maybe it's the coding aspect, like AlphaEvolve style. What do you think would be the developments you see?
What I'm imagining here is, you know, like sometimes to make progress in mathematics you have to do a search.
So like this Elkies example, this conjecture of Euler. And a lot of the time that search is sort of, there's some kind of art to it. Like maybe you're working through a thousand different examples and you don't have an algorithm to work through each example. So each example requires some little idea or executing some standard technique, but you know, it's hard to write a computer program to do it. So maybe algebraic geometry, you need to work through some, there's parts of it you can automate with a Python program, parts of it that require some real idea.
So I think, you know, probably where the models are now, you could imagine automating with relatively high reliability some of those kind of example searches.
That's cool. I see. So these are cases where it would have had to be, much of it would have had to be, or at least the amount of manual work scaled linearly with the size of problems, and now you cut that down.
Yeah. So I think that's something that I'm really looking forward to. I think like, you know, sometimes I'll write a paper which is just like, here's a beautiful construction, right? And like to get some, to find it, you know, I need to do some search and like kind of think about where the right place to look is. And, you know, we already see AlphaEvolve is maybe some baby version of this, where we kind of see some kind of automated search aided by a sort of clever LLM. I think that's something where I can imagine that having a really significant impact on mathematics, but I think it would be like sort of in the same spirit.
Yeah. It wouldn't be an automation of mathematics. It would be a continuation of figuring out how to use computers to reduce labor, open up new...
Yeah. If you think about, you know, maybe the proof of the four color theorem or the conjecture, that's a similar thing where you've many cases that one needs to check and you have a computer, you figure out how to get a computer to do that.
And I suppose you could get lots more of this if you just way amped up the compute. Like one question we ask about AI, because it's true all the time, is what's compute constrained? Do you have a sense of are some areas of math you could really get some fascinating things if you just gave them, you know, a thousand times as much compute as they have now, or are these kind of hard to find, the little ones? I don't know.
Yeah, I mean, I think a thousand... Yeah, so people do this. So if you talk to computational number theorists, I've had actually really fun conversations where someone takes a problem and say, 'Oh, well, this problem will be solved in this year,' just by taking Moore's law and saying, you know, when will it no longer be computable? And those predictions are, you know, reasonably accurate.
Maybe you know about this example. Can you write an integer as a sum of three cubes?
So here we sort of know exactly how hard, at least conjecturally, each instance of this problem is. So you fix an integer, try to write it as a sum of three cubes. We kind of know about how big to expect those cubes to be.
But, you know, I think it's cool to do that for numbers where it's hard, but, you know, you double the computer, triple the computer, or multiply... yeah, you get a few more, get a few more interesting integers. But yeah, there's a question as to what extent that constitutes progress in my view. I think there are definitely people who are more excited about it than I am.
Makes sense.
Yeah, I'm imagining things where, you know, you're maybe looking for an example of some construction where there's not just a known algorithm that computes it, and you instead ask, you know, GPT to cleverly search and let it come up with ideas. And if you give it enough test time compute, yeah, maybe eventually it will come up with something. I'm just here, I'm not imagining it coming up with better and better ideas as the test time continues, but just trying more and more things.
Sure. And certainly if you had something verifiable, like some way to verify, even if it's just more test computing, does this idea look good or something like that.
Okay. And then I did want to just cover this. Are there problems that AI is causing right now? I mean, I think we know like cheating in college is clearly an issue. It's more tempting, lower bar to cheating sort of thing. Like what junk papers? I don't... what's the state of this?
So I mean, there's definitely junk papers. So I think I started counting in maybe September of this year, papers on the arXiv with the phrase Hodge conjecture in the title. The Hodge conjecture is one of the six remaining open millennium problems. It's the one that's probably, maybe the statement is the hardest to understand for a layperson. So for a long time it was safe from cranks because you just like couldn't write anything comprehensible about it. So now that's no longer true. Of course, of course, you know, the frontier models can write, you know, reasonable looking text about the Hodge conjecture. So yeah, so I think in September and October of this year, there were something like 12 or 13 papers posted to arXiv math, math.AG, with Hodge conjecture in the title or abstract, and all but one of them were nonsense.
And of course I can't prove that they were generated by an LLM, but based on the writing style it was quite clear. So to be fair, this was... there were maybe, you know, 12 or so papers of this form that came from a smaller... the number of authors was not 12, it was maybe like six or... I see, that's some repeated authors, but yeah.
So I mean, to what extent is this causing a problem? Well, you know, it wasted several minutes of my time.
As the LLMs get better at writing coherent looking text, you know, before you would, you know, have to spend 10 seconds to find nonsense. Now you have to spend a few minutes actually. So, like I think the most serious offender here, the argument really did not make sense. But I, you know, I really had to go to the middle of the paper and see that some statements were just nonsense. Like the introduction was totally reasonable and...
Yeah. And making some quite dramatic claims, which is why I was motivated to check that it was BS, but yeah, it wasn't a totally trivial job to do, and I think that problem will get harder.
So in particular in areas where like, you know, formalization, like for example using Lean or other formalization software is not really practical right now, I think that's going to be an issue.
And you can imagine way worse versions of that, which I'm sure are really happening, where a serious person, like maybe a graduate student who's stuck on a problem, you know, uses one of these models to generate a nonsense proof of a lemma in the middle of their paper where they, you know, where maybe it doesn't make sense. There, you know, 99% of the proof of the paper is probably correct, has no value because there's some nonsense.
And that's very hard to catch. I mean, that of course already happens without... people have been bullshitting since time immemorial.
There's lots of wrong papers out there, but, you know, a lot of this is about the marginal cost of doing something, and the marginal cost of lying and cheating is getting a lot lower. Again, if capabilities froze, is this sort of at the level of society manages, because society generally, you know, muddles through? Are there any things you'd be particularly worried about?
I mean, I think it's just contributing already to a... like there's right now in mathematics, I think a refereeing crisis. There's way more papers that are being generated than can be refereed carefully, and I think it would contribute to that probably. It would continue to worsen, right? A lot of that is about the incentives in math academia and not the models themselves. I think we'd manage the way we've been managing, but it's not...
Yeah. I mean, and I do think there's some hope here, like of course the models can also help check papers, and I mean there's already sort of nice tools that are being developed.
Yeah. So building on top of that, I want to think of it longer term now.
Great. A big part of all of this, like progress in AI and math, is about like compute and scaling. And one of the things that we discussed earlier was being able to, you know, get the AIs to run lots of examples, help you work through lots of examples and do this kind of stuff. So I'm kind of curious like how we should expect like the fields of math to evolve as we get more of this like ability to run lots of experiments at scale. So I'm kind of thinking like, you know, with previous numerical simulations maybe that enabled say lots of simulations in meteorology and economics. What should I expect in the case of math where we have like systems that say are able to solve frontier math and we're able to run lots of these models at the same time.
Yeah. So here you're asking, just to make sure I understand, you're asking about like the sort of thing I was proposing earlier where you're just sort of working out lots of examples, not like trying to get the models to solve through one hypothesis.
Yeah. So I think we'll see kind of a continuation of previous trends, like, you know, coming with the advent of computers, like we'll be able to work through larger volumes of interesting examples. Like maybe, I don't know, I want to find an algebraic variety with properties X, Y, and Z. I can like just have the model start trying things. Like the main benefit I think you get is like the cost of trying, like the first dumb thing you would think of, gets very low. Like, you know, historically, if I want to, you know, find a construction, I have to sit down and try a few things, even if doing that requires very little cleverness, you know, it still takes a few days of my time maybe, and, you know, I'm a busy guy. And oh, moreover, you know, this might be just an idle question and like I have other things I'm more excited about. Okay, so there's some opportunity cost to doing these things. That opportunity cost and also like just the monetary cost of like the marginal cost of trying something will get very low, even if you're asking a model that's like not that capable to do it. That's very valuable.
So yeah, I think, you know, one way mathematics moves forward is like you just like look for examples of cool stuff and then like you just every once in a while you discover such examples. You don't... that doesn't necessarily require like deep insight or brilliance. It just requires spending some time, and so having automated search for cool stuff would be a really big deal. So here I'm thinking about like, I don't know, there's some kind of sporadic examples of cool things like the sporadic simple groups or the exceptional Lie groups, where, you know, of course people are searching for them in some kind of fairly principled way, but, you know, ultimately at some point you have to make a discovery, and that a lot of the time you just make that discovery by like, oh, noticing someone worked out a cool example and then you like observe some interesting property of that example. That's happened to me. I mean, I think some of the projects I'm most proud of are like, you know, I noticed something cool in the literature and like sort of drew some consequences out of it.
That's interesting. So that can be very productive.
Yeah, I think that can be very productive. I think it's a big part of how math moves forward. Like it's not just that like, you know, the most brilliant people are proving amazing theorems. Like there's a huge background of, you know, people like me who are, you know, maybe more workmanlike, doing work and like just thinking about lots of cool stuff, and every once in a while they'll find something important. And I think, you know, that's something where just the ability to automate is probably a huge deal, even if the automation is not at the level of like, you know, even the median professional mathematician.
So yeah, that's I think the primary way I expect near-term sort of mathematically capable models to make a big difference.
How quickly did the returns to running these kinds of experiments diminish? Like, you know, I kind of imagine if I had loads and loads of these like AIs that are solving, like working through these examples, like maybe like a thousand examples isn't really that much more helpful than a hundred. So like would it be bounded?
I think it depends a little bit on how you're searching, right? So the most interesting examples are like, at least for me, is there's an infinite collection of objects and you're looking for 26 of them, like they're really very special exceptional things. So, you know, a brute... you know, it's not like you're doing a brute force, you're doing some cleverness, and, you know, the models can automate a little bit of cleverness. So I think that's, yeah, I can imagine situations where being able to search through a million things is way more valuable than being able to search through a hundred things.
And does that imply to like, or which particular fields would be most amenable to this?
I think everything. I think there's lots of... you know, I guess I'm thinking of algebraic geometry specifically where there's certain sort of exceptional constructions I like, and, you know, the famous example is this 27 lines on a cubic surface and 28 bitangents to a plane quartic and so on. And these examples have some interesting properties, like they turn out to be connected in various ways to the exceptional Lie groups, for example. I mean, this is all known.
classically in 1900s, but you know it would be amazing to find. I think this is sort of part of a series where there provably cannot be more examples, but it would be amazing.
Yeah, I think that's and that's not something that seems impossible to me. You know, people do regularly find sort of cool examples of the same nature, like weird weird phenomena.
Yeah. So, it sounds like compute is like really important. Then it sort of raises a question to me of like why aren't all the mathematicians going to try to go into like a big lab and working with like, you know, OpenAI or DeepMind, or why aren't they going to places where there's a huge ton of compute and trying to like ride the wave and exploit?
Yeah, I think I mean to be honest I think model capabilities are not really there yet, right? Like right now the way you would try to automate this maybe is you just like run a model in a loop saying like look for a cool example of this sort of phenomena, like here are the examples you've looked at so far. You know, when you try to ask the model to do a question like this it will basically fail 100% of the time. So I mean I think at least in algebraic geometry just like the capabilities are not there yet, like they just can't work through an interesting example.
So maybe that'll happen soon. I don't know.
But we also think like if we extrapolate the trend on FrontierMath and like maybe by the end of like next year and it's probably going to be saturated. So like should we be...
I think saturating FrontierMath is somehow not relevant to...
So let's maybe ask about timelines more just on the skills you care about. You made this bet with former... Tami Baseru. I think it was, we looked this up so I'm hoping I'm getting this right, three to one odds that by early 2030, call it four or five years from now, time has progressed so it's not five, it was five years at the time of the bet, that an AI system would not be able to more or less autonomously produce a paper that would be publishable at today's standards in one of the top...
...journal in your field. And so that was, if I'm not getting the arithmetic wrong, that's like you giving that a 25% chance of that happening.
Yeah, I think there's some chance I was overconfident. I think I probably revised my estimate more towards Tami's side since then.
It sounds like, if I recall, Tami though, maybe I'm misreading from your perspective, the evidence was saying he was less confident, like he was more confident.
Yeah, I think actually so Tami has moved towards me and I've moved towards him since then.
That's interesting. Yeah.
Yeah, it's also clear to me. I mean I think right so this is something like and yeah there were a few more constraints maybe they're interesting. So first of all it has to be a repeatable capability, although you know of course if the model proof the room hypothesis probably...
We're past that, yeah, yeah.
Repeatable for the sake of there's not like a quirk of...
Yeah, it's not like it finds one weird example that's a counterexample to some major open problem and everyone cares just enough, the stars align just right.
Regularly producing stuff. And then there was a cost constraint: the marginal cost had to be about $100,000, which was some kind of estimate of the marginal cost of getting a human mathematician to produce an Annals paper.
So you know I've produced an Annals paper, many other mathematicians produce Annals papers, and you know...
You don't make millions either. So you know.
Yeah. And I think that's a little bit hard to evaluate like in the sense like probably we're not going to spend $100,000 compute trying to do this. Although maybe open, but you know maybe we expect the cost to drop. So if in 2031 it's doing it for $1,000 then...
Great. Yeah. So I... Yeah. So sorry. What was the question?
Oh, just sort of, well one good question here is, so first of all, 25, like just acknowledging 25% of this happening baseline in 5 years. Like that's a lot of progress for where we are today.
I think that's very plausible as a forecast.
I mean we could do a lot with that alone. Ask about what does math look like then. But I'm curious your updates since then, like why you might have moved in.
I think 25% to me feels a little bit low. Not really based on anything besides vibes. I actually think it's reasonably likely that models will be able to autonomously produce, you know, very high quality research in that time.
I think in the... so that bet was supposed to be a proxy for will models have absolute advantage over humans in doing math research, and I think I've come to the view that it is a very poor proxy.
And so one reason is the following. So right now there are mathematicians who I think are much more capable of, you know, the marginal cost of having them produce an Annals paper is much lower than the marginal cost of me doing it. You know, they're doing one a year. You know, I do it, I've produced one Annals paper so far and then a few other papers in comparable venues. So yeah, so nonetheless like you know they're not like they don't have absolute advantage over me in some sense, right? Like we have different points of view and we're doing mathematics in different ways. So like there are areas where I have a substantial advantage in proving an interesting theorem or you know understanding interesting mathematics, which I view as the actual goal.
So if I hear you correctly, it's like if models have 5 years from now the same characteristic they sort of seem to have today of they're all kind of doing the same thing. It's like a mathematician.
That's right. Yeah. So I think if you imagine, you can absolutely imagine models that are very strong at certain, you know, even doing certain types of mathematics or proving theorems of a certain shape, or even proving theorems of many many different shapes. Yeah. But you would still maybe expect, you know, the fact that Maxim Kontsevich is around doesn't mean that there's no role for other mathematicians.
The cost criteria is relevant there a little because let's just imagine that the cost was zero for this and that you know we weren't at 200 tokens a second, which feels like a bizarre coincidence that we are sort of close to human speed of thought for models. I think of computer, they're much faster than us at arithmetic. So putting those aside, which are totally real practical, let's imagine we can instantly get some Annals papers. And I guess my question is just, well, more almost from the human analogy though, tell me if you think this isn't fruitful, if you put your own brain in a data center and could you know run 100 subjective years in a minute, like can you give every mathematician in the world a run for their money? Like maybe it takes you longer to do the sorts of things you're not so good at actually as a human, but that's a hidebound 2025 human perspective. What's the electronic version?
That's a good question. I think it's hard to say. So, right now, because I'm limited in various ways, I mostly work in areas where I'm very good. Yes. And so, you know, every once in a while that will involve like learning a new technique, and very frequently it will involve learning a new technique and or you know developing a new technique, which is something I think the models have not yet really showed capability of doing, which takes a lot of effort. I think if I was not so constrained like I could start trying to learn other people's techniques more easily and you know so that's a good question, but clearly the answer depends a little bit on my capability of doing some kind of continual learning.
So I think when we're comparing to a model maybe it depends a little bit on how things develop in that direction. That said my expectation is that no, I would probably do a lot of cool mathematics. I hope I'd be very embarrassing if I was given so much resources and was not able to do it. But no, I think it's quite clear that other mathematicians would still be very valuable and there's certain sort of I would say styles of thinking or modes of thought within mathematics that I do not feel suited for, like maybe that's some kind of comparative advantage calculation I'm...
Yeah, I mean this whole question just to summarize it is does AI break down comparative advantage dynamics that we're so familiar with.
Yeah, and I think I don't see a reason to think that. I think it's, you know, I mean of course we also asked the question without resource...
the key analyzing comparative... just happens not to be good at, humans gravitate toward that, and then a few humans are guiding the math that the AI is good at and building on...
Yeah, and I imagine one would also try to automate systems and you know try to develop some kind of taste evaluation for the AI and try to get it to do interesting things on its own too. So I think that's a possibility. 5 years. Yeah. Okay. Maybe 25% is still around where I'm at. I would go a little higher. I think it's very likely, you know, within 15 or 20 years that we'll have, you know, there will be a lot of areas of mathematics where automated systems have advantages over humans. I would be surprised if that doesn't happen.
And this is a little our area so I don't mean to unfairly ask you to do our job for us, but what is your, when you're saying 5, 15 to 20, like what's, what are you using to draw the trend line? Like what's vibes? Is it probably finance?
Primarily vibes. I mean I do think I'm participating in the age-old tradition of AI forecasting where 5 years means you think it will happen soon and more than 5 years means anything from you know 5 years...
[laughter] So I think you know, okay, there were a lot of people forecasting AI progress in the 60s and 70s who I think were saying 5 to 20 years in the same way, whatever, there's sort of today there's soonish probably maybe I don't know and there's like oh come on who knows what.
What I'm saying is that I would be, I think it's reasonably likely but far from certain that if current trends continue we'll see kind of high quality automated research.
So maybe we should see what happens when we go actually 15 or 20 years or perhaps even more into the future. So one thing I'm especially curious to get your take on is like the returns to scaling and there's kind of like multiple dimensions that we could do this. So one dimension is kind of like what we were discussing of like thinking longer. Yeah. Another one is like having many more researchers and the last one is kind of like the scaling to intelligence. So maybe let's start off with the first one, is like how do we compare like the scaling of like maybe one person thinking for 100 years as opposed to 100 people thinking for one year, and like how much is there a way to characterize like how much better a person gets over the course of this like 100 years.
That's a good question. I mean so maybe let me kind of connect this to something we were discussing earlier which is like how to decide how hard a problem is. So I think there are a lot of problems which clearly have the flavor of like if you had a Manhattan project to solve them they would be solved pretty fast. Or you know like if you like the primary thing missing is attention and if you somehow even got just a few mathematicians to really devote time to them like it's almost certain they would be solved. Like I think if you want me to name such problems I probably can.
Sure. I mean I'm curious. So I have in mind, you know, for example, we've discussed privately the Enri problem for the less sporadic simple group, well known the Mathieu group M23. I think that's probably a hard problem, a bunch of people have thought about it. I absolutely think that if you know if we had the US government devote, you know, whatever the US government spends, $200 million on mathematics annually historically, although maybe less this year we'll see, that if they devoted a substantial portion of that to solving that problem, that would get solved.
I think yeah, that's a good, yeah that's helpful. Yeah. So that's maybe talking about just devoting some kind of resources broadly speaking.
So if you ask a single person to think for a long time, I think there are also a lot of problems where you sort of expect a solution. So maybe just as an example, there's a problem I started thinking about in maybe 2016. And like it sort of came up in a talk I saw and I really liked it and I tried to solve it and I didn't really get anywhere at the time but I kind of kept it in the back of my head and then you know solved it in 2024. And so I think you know you learn stuff, you connect it to the sort of things you're thinking about, you know there's just some problem of like matching your knowledge to a conjecture, like knowing a conjecture even exists. And you know so as you kind of just as time passes and you just try various things sometimes you eventually hit on a solution. And here in fact part of the issue was like one of the ingredients of the solution just did not exist until 2023. So you have to sometimes you just have to wait for someone else to do some work and you realize it's relevant and then you...
That makes the individual mathematician thinking actually sound a little bit more like a group of mathematicians because someone else does other piece.
Yeah. So right, so some of it is just time passing. You learn things, you try different things. You become a better mathematician various ways. You learn techniques also, you know, a big part of mathematics, and I think this is something where humans still have a very substantial edge over the models, is like you have a new object and you try to make friends with it, like you play around with it, work out various, you know, ask questions about it, work out some examples and relevant special cases of the thing you're interested in. And I think this is something where you know you learn about an object that way and develop some theory for it. And this is something where I really worry about model capabilities in the current paradigm and like trying to understand sort of marginal cost of progress. Right, like right now for a model to learn about a new subject or like to have a tool that has learned about a new subject, you have to train a new model. Right, like I don't know, so Peter Scholze is developing this theory of condensed mathematics which is some kind of new kind of space and none of the models know about it, you know it's in some I guess they can go online and look on paper but they're you know they don't understand it sort of level.
Yeah, they can't like take the object and like try to work with it and make friends with it.
And yeah, so this is something that a human can do over time and I think that's somehow the main way that time helps in solving a problem is you kind of become friends with the object and develop some intuitions and the models just can't do that without training an entirely new model. So there the marginal cost of solving a problem that involves some kind of new object or new way of thinking is the cost of training a model is huge, like way way worse than asking a human to try to understand the object.
So continual learning seems very key for this. Yeah.
And it's not just, I mean it's somehow not just continually learning in the sense that I can read a paper and understand the object but like you know build up.
Yeah. If you expect you know this the models to behave like a person it's sort of yeah playing playing. And I guess we have seen some attempts maybe at getting models to do some sort of self-play with mathematics but not clear to me what kind of success that has had.
I mean it's interesting that you can yeah I don't know you see the chains of thought and I feel like you certainly see glimmers of this where they will try different things and play around a little and eventually hit on something probably because it's what they've been trained to do for the sake of solving a concrete problem rather than this more taste oriented thing of like okay I've got a better angle or a better grasp on this. It does feel like maybe that scales.
Yeah, I'm not necessarily skeptical. I just... that's what it will take. Yeah, makes sense.
Yeah, I feel like the other part that I found really interesting about that was like kind of like the limits to how much you can parallelize this because like you know there was like this dependence from like another result in 2023 was it? And so like if we're comparing like the Daniel thinks for 100 years versus like 100 Daniels, like maybe like the 100 years thing is like probably more productive in the long run with the continual learning but maybe there's also like a parallelization bottleneck and like or you need something else to happen broadly throughout the math economy or something.
Yeah, I think I can I think that's accurate, right? So like for example in this case, this result took input from sort of two deep theories.
So one was non-abelian Hodge theory and the other was the Langlands program for function fields. And so the sort of main thing that happened that allowed this project to go through was someone else. So Anouar and Groenewald had proven something using the Langlands taking input from the Langlands program for function fields and the companions whatever that means. And I think that's not something I would have thought of. It's you know okay it's close enough to my area that I knew about it and there's some chance yeah some chance I would have like thought about the question at least but it's really very creative and important development that was kind of different from you know different from the things I was thinking about so I think it's really not clear like I would have even known it was relevant in some sense like if I was just thinking about this problem for hundred years.
I also think I mean historically like asking a human to think about just one problem for a long time without you know in isolation is probably not such a productive thing to do like you try a bunch of things and at some point you start hitting diminishing marginal return.
Yeah. Mhm. I guess the parallelization it sounds like this is often the case, diversity diversification is important there even. And I guess today's models, have you tried to get them to do something surprising or more original even if wrong?
Like to yeah demonstrate creativity for sure. I mean I would say right now anytime I think about an open problem the first thing I do is ask the models for some ideas. They're almost always sort of nonsense. Yeah so yeah I would say I've never maybe never gotten an idea that sort of passes the sniff test for sort of a deep open problem like that.
And so I mean it feels like again it feels like random initialization in some abstract way should solve this problem but we're just not seeing...
Yeah. Yeah. You can also try to you know every day I wake up with a different random initialization right? I think there's still you know some kind of persistence in who I am limits my ability to really really kind of search the full space of mathematics and... and so yeah.
I guess the other aspect of this was the returns to intelligence. Yeah. So there was like comparing you know I think if you had a million Ansons trying to do the stuff that you do I think I would not make very much progress. And then if we're comparing even if I like spend a lot of time trying to learn like full-time just learn that. And then maybe like the next scale of this is like there's a million Daniels compared to super intelligent mathematicians. Like would we expect this to have like the same kind of dynamic where like the super intelligence is just like so much better than even like no matter how many times you scale up the number or like the duration of our researchers like does this just keep going up like all the way up.
Yeah. So let me first push back on the claim about you versus me and then we'll talk about me versus [laughter]. I actually think I find it plausible that if you really devoted a long time to like learning cool mathematics, you would be able to do cool mathematics. Like probably it would be different math from what I'm doing just because we have different preferences and you know different capabilities and you know we're stronger in different things. But I don't you know you seem like a sharp guy like yeah. Oh, thank you. Yeah. Well, like I don't see a reason to think you wouldn't do interesting mathematics if you were sort of had the motivation and resources you know devoted to it. So similarly I think with a super intelligent AI if whatever that means like well intelligence and even just within a fairly narrow domain of math research is exceedingly high-dimensional thing like we just see comparing different human mathematicians that we're very good at very different things. Yeah. And so I think even if you imagine AI which is very very capable of solving interesting research problems, it's not clear that it's very very capable at solving all interesting research problems that all humans are good at.
But I suppose we could try to sharpen the question a little by saying of so whatever Terence Tao is famous but surely but you've mentioned others as perhaps in other fields sorry whose name Schulz I guess... Schultz and Convitch or something. Yeah. Yeah. Exactly. So within whatever net you want to cast for people who are well suited by interest or disposition to a particular field like what do we observe in...
There's still huge huge differences right so like you know... so high dimensional. Yeah maybe an example is there's a sort of classical dichotomy even within any field is that of theory builders versus problem solvers. And I think that's you know in large part about interest but also about capability. I think of myself maybe as more of a problem solver than a theory builder. And I think part of that is just like I'm not as good at theory building. Sometimes I have to do it in order to solve a problem. But like my you know this part partly what I'm good at and part like I enjoy trying to solve interesting problems. But I think that's also even that shade is like some kind of continuum or maybe high dimensional continuum like like you know some problems are like develop a theory to do this. Yeah.
Well, I still feel like we could sharpen this because you just say, 'Okay, so add whatever dimensions you want and now we see a spectrum of...'
Yeah, you can absolutely of course imagine an AI which exceeds human capability in every dimension.
Yeah. I guess what I'm curious about is whether just from human evidence do we see, I have in the back of my mind I should just say it, this kind of you know light-hearted quote by I think Fefferman on when Tao got his Fields Medal saying like everyone you know if you get stuck on a problem the best way out is to interest Terence Tao suggesting sort of that there was this sense of... yeah you know like if you get a more capable human on a problem that you know that they're a good fit for like that isn't you know a theory building problem or whatever you know project for a theory builder in an area that they're interested in or something like that like do you have I don't know...
I think that they're a good fit for is the key there though right? Like I think this may be a matter of expertise rather than capability but like you know I think if I got Tao interested in one of my problems it probably wouldn't be that helpful in the near term like maybe he would load up a huge amount of algebraic geometry or arithmetic geometry into his brain and that would be helpful. You know, that would take a while.
And I guess the question is just for whoever is most um is smartest subject to being a good fit for lack of you know more nuanced terms. Whether that like just what that scale of productivity looks like and are there people who you avoid working on that problem if you know they're working on it cuz you're like oh surely they're going to crack this before I do.
I would say that happens in some sense but I don't think of it as about capabilities. It's like you know for example there's a very active area now is p-adic theory. I you know adjacent to it in various ways consume the output of p-adic theory. I don't really work in the area but it's just because it's a very fast-moving area and like if I wanted to get into it I would have to learn a lot and then catch up to these people who are working very quickly.
Got it. So it's more about opportunity cost and capability. So we don't see so much like the return to whatever we call intelligence in humans. It's not...
You know I have a two and a half year old who I think is very sweet and very sharp but you know she's not doing math research right now. She's two and a half. So yeah there are absolutely differences in capability between people. But yeah, I think among professional mathematicians, it seems to me mostly about the fact that it's a high-dimensional space and people are interested in and specialize in different parts of that space even within sort of fields you feel are kind of narrow.
Yeah. Maybe a remark on that is like I think you know so if models kind of continue in the current paradigm where they are kind of very jagged and have in some sense sort of narrow capabilities compared to a person. Even if that happens, you still imagine some kind of very substantial value. Just because you know the fact that we specialize in these sort of tiny little zones in a very high dimensional space means that basically all problems are kind of attention bottleneck. You know the number of people who even know what the words in the problem statement mean is like 10. And you can imagine that in some sense those problems are not very difficult because of more paradox or just like no one has really had time to think about them.
I guess like another way of kind of thinking about this in the way that like say to people who care about super intelligence and like fast AI takeoffs and whatever is like suppose we increase the rate of math progress by a certain amount based on some measure that we think most people agree is reasonable. And then we try to break this down into like because we have these like you know 15 to 20 years in the future AI that are doing the math research. And we break it down into like the three factors that we were talking about. It's like more AI, AI thinking for longer, and like smarter AI. Like what would the do we have like a sense of like what the breakdown would be?
That's a good question. I mean my sense is that whatever okay it's not clear to me to what extent like progress in math research has increased over the last 40 years or whatever let's say prior to AI but maybe my sense is that productivity growth like total productivity growth comes down largely to the increased number of mathematicians. So you know maybe once you expect AI to reach a certain level of capability like at par with humans if this happens maybe you expect it to come down to number. A little bit of a concern there is like human you know why are there returns to numbers of humans? I think a part of it is just attention, part of it is like there's diversity of modes of thought and maybe you don't have as much of that with AIs depending on how systems develop. So I don't know but I do think you know probably a lot of the returns to increased number of mathematicians just comes from increased time spent thinking about different problems, increased attention and that I expect you could see substantial returns to even with sort of not very capable AI.
So yeah, so what about returns to intelligence or whatever? Again, I think this is not such a well-posed problem. Like I think it's a very high dimensional like intelligence very high dimensional object in my view. But yeah, I mean so okay maybe let's call it something like the ability to you know search through some you know high-dimensional space of proofs in a clever way. I guess you can imagine that being very valuable like you know presumably the proof of the Riemann hypothesis will develop you know involve developing various objects and theories that don't yet exist and that would you know take a lot of you know you'd have to search for them in some very high dimensional search space. But I don't know this it feels sort of like too much speculation like I don't know.
Yeah like in fact I think a lot of the way that search happens in practice is like that people who are you know just normal mathematicians are sort of just thinking about objects and observing like doing this sort of low-level search I was discussing like working through lots of examples and notice patterns and then people start to sort of encapsulate those patterns into relevant objects and like I think somehow that sort of high level theory building which we attribute to maybe the most brilliant mathematicians really depends on sort of a background of just like hard work and example analysis.
Mhm. So you know arguably some of that is also attributable to just scale and the number of mathematicians and the number of sort of questions people have thought about. As a funny fun little just what are your solve probabilities over time for the Riemann hypothesis or pick your millennium problem if that's not your...
Yes. Okay. I mean the ones I'm probably most familiar with are Riemann hypothesis, Birch and Swinnerton-Dyer, and the Hodge conjecture. I think of those the Riemann hypothesis actually is the one where in some sense we know what a proof should look like. Uh-huh. And the reason why is that there's an analogous conjecture the Weil conjecture which is the Riemann hypothesis for varieties. Well part of it is the Riemann hypothesis for varieties over finite fields that was proven by Deligne in the 70s. And maybe the part that's most closely analogous to the Riemann hypothesis is the Weil conjectures for curves which were proven by Weil much earlier.
So we really have some sense of like what the shape of a proof should look like. Should look like Weil's proof of he proofs but one of Weil's proofs of the Riemann hypothesis for curves. And then the problem is that there's various maneuvers one makes in that proof that don't make sense for like the integers than a curve over a finite field. And so one has to figure out how to make those maneuvers work. I mean this is sort of science fiction but I think this is at least some kind of plausible you know mystical way of thinking about a proof. So yeah of course many mathematicians have thought about how to try to make these maneuvers work and I think there's some chance that you know one of those attempts will work out in the next 10 years. I don't know. I mean it's the number I put on it is like 15% or something.
But that's just meant to be like a fairly small but not small but not zero.
Yeah. And in general I think if you look at like sort of solutions to major open problems like the time difference between the sort of last big idea you need and like completing the proof has been fairly small like you know for Mordell theorem like the distance between the Tamagawa number conjecture sort of a very hard conjecture related to the Langlands program and the proof of the Riemann hypothesis not not the... temporally not so long yeah sorry for theorem it's not so long [snorts] that does make it sound a little like maybe because vague conjectures for curves is you know decades old that maybe...
Yeah so in fact people expected I think when Deligne proved the Weil conjectures that the proof of the Riemann hypothesis would come soon. That's well before I was born but that's my understanding of the history and then it wasn't.
So maybe some people don't know yeah so I think people don't really know what the missing new ideas are you know we have some sort of very vague senses what their shape should be. And then I guess the obvious follow on is what has your time prediction time frame changed with AI progress at all.
That's a good question. I mean I think I am right now I haven't really like I don't feel like I've so part of the reason of my bet with for bet with my earlier is I don't feel like I've seen some aspects of what I think is necessary for high quality research. Got it. I don't think I've seen sparks of it from AI. You know the distance between solving an IMO problem and writing doing high quality math research is larger in my view than I think some people some other people seem to think it is. Yeah I would say it has not moved so much. I mean right like right now the tasks that AI can help with and the tasks it seems like it will soon be able to help with don't seem to me to be the primary bottlenecks for resolving major open conjectures. That said, I do think, you know, there's I gave a 25% chance that AI these problems will be solved and AI will be able to you know resolve do high quality research. I think depending on the there's you know maybe some 5% of that there's some depending on how that works that could be could be so good that it's on track to yeah that to do something you know crazy.
So just for fun, Millennium Prize problems, these big targets in math, I can't tell where are you on, how likely you think those will be solved and whether AI will contribute substantially to those solutions.
Yeah, I mean I think my median prediction is that zero will be solved autonomously by an AI in the next 10 years.
Yeah, I see. My at all I would say maybe zero to one I would expect to be solved in the next 10 years. You know, there's arguably been some progress on Navier-Stokes, which is far from my area, but I don't find it totally implausible that that will be resolved. And I, you know, the current news about it is about a team working with jointly with DeepMind's.
Yeah, that would be with some kind of maybe more traditional deep learning techniques involved, not LLMs or reasoning models. And I guess our I mean, one of the seven was resolved not too long after they were codified, the Poincaré conjecture. So our base rate is not zero. We shouldn't expect like these are impossible. They're just very hard.
Yeah. But I think with a couple of them like Hodge conjecture seems like there's just no ideas. BSD similar.
Yep. With yeah with the Riemann hypothesis I guess we discussed there's some kind of ideas but it's really kind of unclear how far we are away. I think it could happen in the next 10 years. It could happen in the next hundred years. And so relating this to your other AI timeline senses the 25% chance that we do hit Annals level papers in five years some part of that goes toward you know...
Yeah. Yeah. I think it's but my expectation is that it seems very possible for that to happen without... without starting to knock out more... basically not much of that and then you're sort of like 20 years you're you know it's beyond the when when this all has had a lot more time to brew and develop.
Yeah. And I think it depends a little bit on what kind of you know we talked a little bit about acceleration. And I'm not an expert in this, but you know, presumably there's some question about acceleration of AI progress itself or self-improvement.
And that I'm generally pretty skeptical of, in the same way I'm skeptical of acceleration in general.
But yeah, we'll see, I guess.
So, and this might slowly transition us into measurement benchmarking. But first, with sort of a fun speculative version of that. You've said you're not seeing so many glimmers of some of the things you think are critical. Well, let me ask the boring version first. What are some of those things? You've talked about them plenty on Twitter, but just for our sake here, like what are some of those?
You know, so let me imagine, like, what are some things that would be signs for me that...
Like for example, you know, making a new interesting definition would be important.
So showing some kind of research taste would be important. So like asking a question or like discovering even just like conjecturally some new phenomenon. Like I think a lot of the most important mathematics is actually just making conjectures, which seems sort of hard to get current systems to do. I think it would be interesting to get to do experiments there.
Yeah, developing, you know, this is sort of related to making a definition, but some kind of theory building. I don't think we've really seen, like somehow it's already been very surprising to me that the models are able to learn during training. I don't mean during use or during deployment and like techniques that, you know, apply them. So that I think that's somehow the thing that really changed with the reasoning models, like they were able to take some sort of well-known techniques and now start to really apply them with high reliability to do math. Yeah, if they were to develop, this is sort of a fuzzy thing, but if they were to develop a new technique...
Even in, I mean, this is, you know, there's some kind of continuum between old techniques and new techniques, like put a little twist on a new technique on an old technique, is that a new technique? But yeah, if I were to recognize something like that, that would be a sign. I don't think any of those things have happened. Will they happen soon? Maybe. No.
And just in case it catches anything else in the net, if I were to say of the tasks that constitute doing math, which are the hardest for AI systems, this kind of theory building, conjecturing category seems critical. Are there others you might put?
Yeah, I mean, right. So like for me, I try to have all of my papers contain a new idea, and again, this is a fuzzy thing, new idea, new technique, whatever.
It's not true for all of them. Sometimes you find a trick to resolve some old conjecture, and of course down...
I think that is, you know, most people don't have a lot of new ideas. Like that's mathematicians, I think typically, right, okay, depending a little on the area, but you know, one to two papers a year. So compared to other subjects, we're not very productive in terms of total output.
And I think that thing where you develop a new technique, that's really the, or a new sort of general new idea. I mean, you know, what does that even mean? But when you see it, that's the hard part.
Yeah. Yeah.
And so I just think we've practically answered it already, but the...
With something people like to point at for the game playing systems in Go in particular, AlphaGo was move 37. I suppose some, like I think part of what was exciting about move 37 is it was described as an inhuman move, like a move no human would have thought of. And so the AI had really stepped out of the bounds of, I mean it wasn't trained on much human data, but it had stepped outside the search space that humans had looked at.
It sounds like it would be enough of a move 37 to just have a new idea, even if it was something a human could think of. I don't know.
It's kind of funny. I mean, in some sense that's happened prior to the deployment of these AI systems. If you think about, you know, the proof of the Kepler conjecture or proof of the four color theorem, like those are inhuman proofs. Like they're, you know, of course a human was in charge, but the sort of the majority of the work was some kind of horrible casework done by a computer, or beautiful casework, sorry, no such thing as...
Does something horrible. Yeah. No, I'm not, I like to say in mathematics that everything true is beautiful, and you've said you encourage your grad students to adapt their way of thinking to this...
Not, yeah, you should prove things by any means necessary. Totally. Yeah. Why tie your hands behind your back? But yeah, so that's some kind of move 37 arguably, right? Which had nothing to do with an AI system.
Would I consider an AI system generating that kind of proof, so like automating a huge amount of casework itself, to be a move 37? No, because we've already seen humans do it. I guess you can also imagine some kind of weird like Lean proof of some hard theorem where, you know, with no comments, that itself is very hard to extract a human argument from. Like if you've ever looked at Lean code, it's not easy.
Tough. So yeah, but yeah, I think if I were to see something that I considered to be like a new technique, like something where I couldn't really find a precursor in literature, I mean, well, whatever, again, it's sort of a judgment call as to what a precursor is, that would be very exciting.
And I mean, this is part of why I pay attention to your commentaries. I think that like goalpost moving post hoc judgment is very hard. But you seem to be trying to do an honest job.
Yeah, I mean, I should say there's a huge temptation to both post here. Like, you know, maybe if I do think, to some extent it's fair, like right now the signal you get from AI solving a problem is partly the AI is capable and partly the problem is easy.
And so, you know, mathematicians have a habit of saying, oh, this problem was solved by AI, but it was actually easy, so maybe we shouldn't update on that, right?
To some extent, that's... I think it should be a mix, yeah.
Yeah, but yeah, I think, you know, we should try to evaluate what the models are doing and try to see like, oh, well, if a human had written this, like, would I be excited? And I think that's what I try to keep in the back of my head here. Like, so for example, for these recent Erdős problems, like if a human told me they solved this Erdős problem with this proof, which has happened, I would be like, 'Oh, that's cool.' And that's also how I feel about the AI. 'Oh, that's cool.'
Yeah. Yeah. And like life goes on, but you know, don't diminish it in the moment. Yeah. Good. You did a thing. Yeah.
Yeah. Curious if you think it's like feasible to kind of...
Well, actually, let me step back a bit. So, one thing you've mentioned on Twitter is like that whether these problems are, like solving these problems is impressive depends a lot on whether or not there was a lot of human effort that was put into it previously. And do you think it's possible for us to kind of like say go through these like Erdős problems and then kind of like just, or has it already been done? Like should I try to like say...
Yeah. How much effort put into...
You can look at, you can kind of look at, you know, they were proposed in some paper, you can look at how many citations that paper has. I think with the ones that were solved, I actually don't know. So I think the hard version of 124 that was not solved, the paper has 14 citations. So for a 1996 paper, that's, you know, that's math, is not a lot of citations. So yeah, the baseline 14 is not a lot, but it's not, you know, there are plenty of papers from 1996 with zero citations.
