Daniel: The goal of mathematics is not to produce mathematics papers. It's to produce some kind of understanding. Maybe some of that understanding resides in model weights. To me, that's like pretty unsatisfying.

Host: Comparing Anthropic with OpenAI, do you detect any differences in how that is similar to human reasoning?

Daniel: They definitely are not good at it autonomously, but with some hints, you can kind of get them to do something interesting. A lot of progress in mathematics comes from like letting a thousand different flowers bloom and people pursue their own curiosity and then the boundaries of knowledge expand in some fairly uniform way.

Host: What has been the most impressive result so far?

Daniel: My favorite fully autonomous result by an AI so far is the solution to the Erdős unit distance problem. There was some lemma I wanted to prove none of the frontier models could do it. So I like worked out a ton of examples on my own and I realized oh well maybe like here's some reason why it could be true. Once I had that statement, the models were able to very quickly prove that sort of better statement.

Host: How should like the mathematics community kind of best adapt and benefit from this?

Daniel: Um...

Host: I am so excited to have you on, Daniel. And so Daniel is a professor of mathematics at the University of Toronto. Toronto is my hometown, so also very exciting. But the thing that is most special here is Daniel's an actual practicing mathematician and in addition he's been incredibly vocal about his evolving views of AI in math. And so I feel like every if I just don't check in with you, you know, like in two weeks something different has been revealed, and then you're very kind of like what do you call it? So you...

Daniel: I have a lot of opinions.

Host: You have a lot of opinions. Exactly. So I want to get into that. So, I mean, you know, one of the things that I'm most interested in is not just like a discussion of how the capabilities have advanced. I feel like in math, you know, that's definitely the headline, etc., but also, you've been very thoughtful about how practicing mathematicians should respond. And so that kind of gives us a chance and opportunity to talk about actually what is special about math, you know, like it's not just like, hey, AI has been really making progress here, but like delve into what actually mathematicians do. And so maybe like with that arc in mind we can start with what has been the most impressive result so far given all the recent progress for you, and then maybe yeah we'll kind of take it from there.

Daniel: Yeah. So okay, so there have you know now been a lot of results, some of them produced autonomously, some produced like semi-autonomously, some whose like where the AI contribution is just not at all clear. They're in a lot of different areas so anything I say is kind of you know I can only really comment on things that I you know have some expertise on, so it's quite possible that if you talk to a different mathematician you'll get like different answers here. So my favorite like fully autonomous result by an AI so far is the solution to the Erdős unit distance problem, which I think was announced in mid-May. So at least what I liked about that is it seemed to me that like it was in some ways a little bit creative. So I think some of the results we've seen have had kind of the flavor of like you know you kind of take some known techniques and apply them in maybe a clever way, or you I don't know they have kind of been some kind of results I would characterize as like last mile, like where some recent work was done like quite deep work done by a group of human mathematicians and then the AI kind of took the final step.

Host: Yeah.

Daniel: But yeah, with the sort of unit distance problem, I think it was something where the result was like a little unexpected. So first of all my sense was like that people working in the area thought it was true and then there was a counterexample found, but then also it like brought in some techniques from from another area. I think those techniques were like not especially uh you know like deep or new. They were sort of classical ideas from the 60s, but they were new to this area of studying you know point configurations in the plane, and so that was pretty cool and then afterwards we got to see like it was kind of fruitful so...

Host: A bunch of mathematicians took those ideas and used them to find counterexamples to a bunch of other interesting open questions, so for example like the sum-product conjecture over the real numbers.

Daniel: Um, so that's at least one way I like to think about how cool a result is, like you look at it post-hoc and you see like, oh well, you know, were whatever new ideas that were introduced, if any, like kind of useful to do other things, like did they improve our understanding of something? And I think that's maybe so far the main example I know of of a result of that form.

Host: Yeah, I think it's really meaningful that you're commenting on this because, you know, that result came out to your point in May and there's been so much kind of like so many headlines so far, and it's kind of probably hard for somebody who's not a practicing mathematician to appreciate the differences in these headlines. And so you kind of already started laying out sort of a taxonomy of like what is different in that proof, and so it'd be kind of interesting maybe to use as that as kind of both an excuse to talk about where you sense the model differences are and what like mathematicians actually do. So in this case, I mean the most kind of maybe naive understanding of what mathematicians do is that we're pushing around symbols in a logical manner. And this is why, you know, RL is so successful at this because you can kind of both verify it somewhat cheaply compared to other domains and then also, you know, because like the rules are quite legible, and so you can kind of like you know if you're if you're superhuman at that you might be good at math. But I think that of course betrays most of actually what is interesting about mathematics, which is perhaps I think you said this as well, but I think anybody who's tried to do math is sort of like it's about the understanding and sort of getting at truth, remaining confused, and developing intuitions. And very little—I mean the tool to do that stuff is of course in having really good, very strong abilities to push out logical applications. But maybe if you can kind of talk to speak to like, you know, when you say it's most impressive and creative, like decoupling the just the inhumane maybe feats of just like logical implication from like where is it being creative? What is kind of what is it helping engender in terms of like mathematical activity as well?

Daniel: Yeah. Okay. So, first of all, I mean you kind of characterized it as like inhuman in some way. I actually think the argument was very human-like. You know, OpenAI released some chain of thought, it was very recognizable. It was like kind of, you know, if if I tried to imagine like my chain of thought in trying to solve a problem, like it might look kind of like that.

Host: Yeah.

Daniel: Um you know we haven't seen the raw chain of thought, so maybe they...

Host: Maybe they cleansed it a little bit, yeah.

Daniel: Yeah. Or maybe, you know, maybe the model liked to swear a lot in the middle of the chain of thought and they cleaned that up or something. We don't know. But like at least the summary seemed pretty legible. And I would say that's actually like kind of typical of most of the results that I've studied. Like they don't seem inhuman at all. They seem absolutely like something a human mathematician could produce. And they're like typically understandable if like not so well written if you just like look at raw model output. It's not like there's some, you know, move 37 or whatever. It's like a human mathematician doing math. It's like a human mathematician doing certain types of math. So like they're definitely the models are still like not very good...

Speaker A: ...at some mathematical activities. And here I don't mean like by field, but just like certain things you do when you try to solve a problem the models don't seem to be doing, but certain things they're very good at. So, the ways they might be a little bit inhuman is like they don't get tired. They know a lot. But if you have to just read the final output, it doesn't seem kind of inhuman.

Speaker B: It's interesting because I mean obviously we can't get too much of information from the labs who are producing these models of like why, you know, how of what the training recipes are or how they're kind of advancing in reasoning. But at least one of the things we do know, and I think kind of OpenAI spearheaded this, is this like reasoning in natural language is actually what they happen to scale up, and it's not actually pushing a lot of Lean verified proofs as like the training corpus, and that's kind of amazing. On another side, what's kind of interesting is it's not clear that a lot of the mathematical training data, if they use that to a large extent at all, is reflective of how mathematicians think, if that's fair, because a lot of it is not like legible as traces, right? Like most of the papers are crisp and like polished. The textbooks certainly just show very little motivation of how something is developed, which is why it's like usually easier to kind of follow research direction by actually talking to the researchers and how they're thinking about it. So kind of curious before maybe even going to the taxonomy as an excuse, like when you're examining these models and their results comparing Anthropic with OpenAI, do you detect any differences in how that is similar to human reasoning? And then also if you have any comments or insights on perhaps why natural language scales so well that way even though it's...

Speaker A: Okay. So first of all, I really like your point by the way that they're mostly doing natural language reasoning rather than Lean. Like I think that suggests to me that—you know you hear a lot of people say like "math is a verifiable domain, like that explains the progress" whatever—like my sense is that because they're primarily scaling informal reasoning, probably the techniques are going to generalize to other domains pretty well. That's just my guess.

Speaker A: Okay. Um, you asked a little bit about like Claude versus ChatGPT.

Speaker A: My sense is that they're pretty similar in terms of capabilities. I've played around a lot more with ChatGPT than Claude Fable, but you know, it seems like there's a lot of cases where, you know, OpenAI will drop a solution to some problem and Anthropic will say, "Oh, you know, we can do that too." Exactly.

Speaker A: So, it actually seems like they're solving a very similar collection of problems and it's like kind of a small, you know, relatively small portion of what human mathematicians do. So yeah, one thing that is interesting is like we see the solutions have a certain flavor, right? There'll be things that—this isn't surprising—like there'll be things that rely on the model strengths, like their ability to like grind out a long computation or like you know pull together kind of technical ideas from many areas, or like maybe many papers that you know a human mathematician might not have read. But they seem like weaker in things like intuition or like having some big picture point of view. Like a lot of what I do as a mathematician is like I have some kind of philosophy that's like very non-rigorous, like maybe I think this thing is kind of analogous to that thing, and then like a lot of what I'm working out is like figuring out how to make that precise, and like trying to measure the extent to which I've succeeded in understanding that by like can I solve a problem or whatever, like can I find an interesting phenomenon which I don't understand and then now I understand it. And like so far, even the best results the models are producing, you don't see that much of this kind of reasoning. It's more like they're very, very good at applying some known techniques...

Speaker B: ...which is to be clear, that's like a very powerful thing to do, to be very good at applying like all known techniques.

Speaker A: There are mathematicians who have had great careers doing very high quality work of that flavor, and I think a lot of what the models are producing is high quality in that way, but it's like some kind of fairly narrow band of what mathematicians care about so far. I do think there's signs of all the frontier models starting to be able to do more fuzzy things. So like I've tried to get both Fable and ChatGPT 5.6 to do some kind of theory building. And they're not good. They definitely are not good at it autonomously, at least with like whatever scaffolding I've set up. But with some hints, you can kind of get them to do something interesting. When you give the models hints, it's always a little hard to tell like what part is from the model and what part is from you. But my experience is that like if they can do it with like a hundred bits of hints or whatever, in six months maybe they can do it without hints.

Speaker B: Yeah, you know, I do think there's signs that they're also kind of somehow picking up some of this like implicit and unwritten mathematical knowledge. Okay. I would love to go into the intuition part and where it sucks at, to put it in a very basic way, but you actually mentioned a small detail which is that you know from the public you've gleaned that Anthropic and OpenAI are probably neck and neck, but you personally are using a lot more ChatGPT—like why is that, or 5.6?

Speaker A: Well, why is... I don't know. I mean I just think it's just inertia, like I have...

Speaker B: Pretty sure of it.

Speaker A: One thing is that ChatGPT got better at math earlier. So...

Speaker A: ...like for a long time, the Claude models were just like not useful for research math, and then I think maybe around Opus 4.5 or Opus 4.6 they like more or less caught up.

Speaker B: Yeah.

Speaker A: But you know, some experimentation suggests to me they're pretty neck and neck, and so for my own work, you know, except when I'm just experimenting, I mostly just stick with one and sort of random.

Speaker B: As people outside of labs like us, it's really interesting just to compare how they differ on the frontier. And I mean, to your point it might be a little bit of momentum. I do think that at least from my anecdotal experience 5.6 has been a lot more clear in exposition, and there's a little bit—and this might not be true, obviously the models are just incredibly jagged at the frontier—but in the explanations of results to me, I always find that 5.6 is giving a more accurate theory of mind of what it assumes I know and don't know. Whereas Fable might be explaining something very trivial but then just like jump at like, "Well, you know, obviously you should know these things," and...

Speaker A: I find they're both pretty bad at theory of mind.

Speaker B: Okay, great, so from your eyes you're probably asking much deeper questions. Okay, so the thread that I really wanted to pull on was when you're talking about the models maybe starting to get better intuitions or even theory building. So maybe before we even dive into that, it would be useful to kind of talk through like what is your primary activity as a mathematician. Especially in your area of algebraic geometry, it probably has a very different flavor than a combinatorialist or some other areas. So if you can give a brief lay of the land and then kind of explain what your mathematical activity was pre-AI and then maybe how it's kind of changing with AI.

Speaker A: Yeah. So I mean I think there are a lot of different kinds of mathematicians. There's like a lot of different spectra on which one can put a mathematician. So definitely a lot of mathematicians like solving open problems.

Speaker B: Mhm.

Mathematician: Uh and then I think I'm one of those like I like to solve an open problem. That's uh I think of myself as a problem solver as opposed to like one you know one other taxonomy you could have is a problem solver versus a theory builder. Mhm.

Mathematician: Um at least for me, the point of an open problem is it's supposed to measure your failure to understand something. So it's kind of like a benchmark.

Mathematician: Um right like you know one problem I really like is the Grothendieck p-curvature [gross decap curp] whatever that is. It measures something about our failure to understand differential equations. So it's like there's some very basic object we would like to understand. If we can't answer this conjecture we know we don't understand it.

Mathematician: Mhm. Okay. So in practice like how do you get at a problem which is supposed to be measuring something you don't understand? Well like of course you try to understand the thing better. And in practice what that means is like well you try to find the smallest situation where you can't understand something and fiddle around with it, and then once you win you stare at like what you developed to win and try to turn that into some theory.

Mathematician: Um so that's one thing you might do. You might try to like solve a problem and in so doing develop some kind of new understanding of the situation. You might also just like have some feeling that like this thing is related to this other thing. And you know you might start building a table like oh this property A is related to property A prime, property B is related to property B prime, and so on and so forth.

Mathematician: So for example in my work a lot of it is motivated by some analogy between homology of algebraic varieties and representations of fundamental groups. Okay, so that's some fancy stuff. But this analogy is like very very fruitful and really any phenomenon that appears on one side, you can find an analog on the other side. And so trying to realize that dream has led to a lot of beautiful mathematics of the last like 30 or 40 years by people like Carlos Simpson and Takuro Mochizuki and others.

Mathematician: And so here it's really just like someone noticed like here's an analogy and then that analogy has led to a huge amount of developments. There's not really like an open problem at the end although of course like as you develop this you come up with lots of open problems. There's just like a philosophy that you're trying to realize and that philosophy is like super not rigorous actually. It's not like symbol pushing at all.

Mathematician: Um yeah so that's another kind of activity that I like. Um yeah, beyond that a lot of what you do, a lot of the activities, you're actually trying to figure out what the right question is even. Like here you have some object you feel like you don't understand it, and figuring out what you don't know is actually a very challenging thing to do. So there's a list of you can go online and list open conjectures or whatever, and this doesn't really capture in a lot of ways what we don't know, like often finding the conjecture is really really hard.

Mathematician: A good example of this is like the Birch and Swinnerton-Dyer conjecture which is one of Millennium Problems, is this beautiful relationship between L-functions of elliptic curves and the set of solutions to the corresponding equations of the rank of the group of solutions. It was discovered by, it was like the first big data conjecture. So Birch and Dyer had found all these statistics on elliptic curves in the 60s like this one of the first ever computer-aided bits of mathematics, and they graphed these statistics and they noticed that the slope of some line on a graph was related to some other algebraic invariant they knew, and that was the source of this conjecture. So a lot of the time you're just working out examples and doing kind of science like you run an experiment and you try to figure out an explanation for that experiment.

Mathematician: And so I think going back to AI, I think like these are things where AI seems so far to help a lot more in some things than other things. The more vague a phenomenon is or the less you have a precise question in mind, the less useful it happens to be. And so you were asking like how I use it in my daily life, like actually what I found is that the projects that I have that kind of predate AI, like the projects I've been thinking about for three or four or five years, it's just not that useful. It's primarily kind of a substitute for Google or something, like I might use it to learn about some related topic or something where I would have earlier Googled something and then read a paper, maybe I'll discuss it with AI instead. So it saves some time for sure. It's not really doing deep intellectual work for me.

Mathematician: But then you know, because I suck at coding, and so now I have you know my good friend who's really good at coding and now I have all these coding projects because suddenly, well if I had a question where coding would have been really useful I would have procrastinated on it for six months until I—

Host: —tireless PhD student.

Mathematician: Exactly. Yeah. So, so yeah, now I picked up all these projects where yeah, coding is really useful. Like—

Mathematician: —the models are like very good for kind of massively parallel things. Like if you want to find an example of something, you can just ask it to find, you know, work through a thousand examples in parallel or maybe 10 examples at a time, you know, 10 different subagents and like that's really useful. But these are like kind of different activities which are on top of what I was doing before.

Host: I'd love to dig in.

Mathematician: So yeah, go.

Host: Yeah. Sorry. Cuz you were mentioning there's projects that you've been working on for like 3 four five years and I don't know if it's correct to say like those are more of the theory building aspect of it cuz you did characterize yourself as like an open problem solver, or what is the thing that is because like the deep thinking part? Maybe kind of for this audience it might also be useful to kind of explain: most of pure math, it's not being motivated by—nothing on applied math, but applied math at least there's some external motivation for why a certain formal structure might be interesting to study. Whereas this one, it seems almost sociological, and to some extent I think Thurston made some comment or point about this in the 70s that it is a sociological phenomenon of more and more mathematicians start examining something then you'll maybe converge on some interesting structures. But there's some reason why people would prefer to study something or they think it's beautiful—like what is it that drives maybe you in particular and then maybe you can make a more general comment about the profession.

Mathematician: Yeah. I mean, so definitely some people are like motivated by like beauty or kind of aesthetic considerations. I try not to be motivated by that.

Host: I think that's a bad way like—Well, one thing like that sort of limits you, right?

Mathematician: One thing kind of a failure mode I see among young mathematicians sometimes is like you have something and you kind of think you know how to prove it—

Mathematician: —and then like the proof feels really ugly and you decide—but like okay, I mean what if you're wrong and it's not ugly? Why limit yourself, you should—

Host: —for this audience like what is ugly? Because I have an intuition of what's ugly, but like what is that, kind of spell out?

Mathematician: I don't really know, I don't really have aesthetic feelings, but people sometimes feel this way, like maybe it involves a lot of—

Speaker A: ...grinding calculation that's not illuminating or...

Speaker B: ...but you should win by any means necessary in my opinion. I like to think of what I'm doing like doing kind of physics except with concepts. So, instead of beauty, I try to think about maybe what is fundamental, what's going to open up further understanding most. Am I introducing a new idea that will be broadly useful to understand this object?

And I guess in some sense there's some aesthetic consideration there, but I think the orientation of trying to do good science rather than trying to do art is what I prefer. But there's a huge variety of opinions here and lots of mathematicians think of themselves as being closer to poets or something.

Speaker B: What I do think is broadly true is that progress comes in mathematics from people sort of pursuing their personal curiosity. And that it's been crucial historically that there's a lot of different people with different views of what's interesting, and then the frontier of knowledge expands and expands, and then suddenly you have these opportunistic situations where a new idea has been introduced and you now can suddenly cascade through a bunch of other things that we didn't understand before.

Speaker A: Yeah. I mean, so then going back to the three, four, five year problems and where are the models sort of not useful? Kind of ask it another way: when you're doing the deep thinking, is it just that it's just not clear that you formulate it as a problem, and it's more that you're thinking about what are the fundamental physics of...

Speaker B: Yeah, so in some cases, I mean, there is a well-stated problem here. Like, I've been thinking about things for maybe 10 years now at this point for certain problems, sometimes there is just a conjecture that I would like to prove that is there.

I think one reason the models might not be useful for some of these things is the conjectures are true. For example, this unit distance problem, the general belief in the community was that it was true, and then it turned out to be false. And so what that means is that there's a specific construction you can do to refute it.

On the other hand, I think a lot of the things I think about—maybe someone will come up with a counterexample to the curvature conjecture hypothesis tomorrow and I'll look like a fool—but in general, these conjectures fit into some very broad theoretical framework, which means that we actually have a lot of evidence that they're true. And so that's part of it: there's not a construction you can do to refute it. You need to somehow... we have this giant framework where certain pieces of it are only conjecturable, and you probably need to resolve some of those conjectures to win.

Speaker B: We also have a pretty good sense, I think, that very serious new ideas are needed to resolve those conjectures. So of course you can't be sure, like maybe there's some very clever construction that will let you avoid having a big new idea. I don't know, it's quite possible we'll find this out, but my sense is that for at least a lot of the things I've been thinking about, they're just not accessible to applying known techniques in a very technically strong way, so you need to develop a new technique. Just to be clear, I'm not saying the models won't be able to do this, it's just so far they seem not to.

Speaker A: Actually, that's exactly the point that I wanted to delve into because to your point, it's like, okay, we can try to calibrate and forecast why they would get better at this, but it is true that just providing a construction for a counterexample, they seem to be strong on. If you have to start developing either new theory or, to your point, techniques, machinery to prove why a conjecture is true, it struggles more. And it's probably because a lot of what it's drawing on is also just techniques that have happened in other areas and they're porting it over. And to your point, that's why maybe the unit distance problem was such a more creative result because it was maybe doing more of that on its own.

Speaker B: It was like from another, at least it was drawing in something from an unexpected area.

Speaker A: Yeah. Exactly. Exactly. And so maybe then to kind of ask the question—it's not that because now we're like, okay, fine, AI is getting so good so fast we can't count it out—but why, what do you think it has to... you're spelling out what it has to get better at, but maybe some more feelings on why it's particularly hard to then develop that new theory and technology?

Speaker B: Yeah, it's a good question. I think you just need a different... my guess actually is it's probably totally doable, and it just hasn't been done yet. Like maybe you just need a different RL environment. I don't know. Yeah. So at this point my expectation is just that the trajectory will continue upwards. I'm not a skeptic of continued capabilities growth.

Yeah, I think what is definitely true is that the skill of developing a theory or building your understanding of some poorly understood object is a fuzzier one. So it might be harder. I guess you can tell it, "develop your understanding of zeta functions, then once it proves the hypothesis you give it a reward," but it's kind of harder, I think, to come up with some intermediate things that you can reward.

Speaker B: That said, I do think mathematics as a whole provides a lot of conjectures of varying levels of difficulty, so maybe this explains why there seems to be a little bit of progress in these areas. Presumably they are trying to get it to solve lots of problems, and some of those problems develop at least some of the skills of theory building. And humans are able to develop these skills. Sometimes they get rewards from their PhD advisors, and their advisor says, "Oh, that's a good idea," or something based on some element of human taste or whatever. And that might be something one can do too. But yeah, my expectation is that just as part of continued capabilities growth, we'll see growth in these areas too.

Speaker A: Yeah. And I like the framing where you're sort of casting these increasingly difficult conjectures as a form of curricula for both humans and of course AI. And maybe kind of getting a little bit too philosophical for some people's tastes, it gets at the question of why are we particularly good or maybe particularly bad at math too. What is it that we're either struggling to do or some people are particularly good at when you develop new theory?

Maybe it's related to why can we formulate good structures for physics as well. It's just not obvious all parts of the world are kind of understandable and legible that way, but some parts are, and therefore we try to do it because there's maybe compressive pressures on our minds because we can't understand anything beyond compressing stuff more finely. I don't know if that's also the correct interpretation of...

Speaker B: I think that's... I mean, I'm a little bit skeptical as like compression as a metric of interest, but it's...

Speaker A: Definitely, like as an anthropological reason for why we do a certain theory building. I just think it's true, and in fact, I think our inability to just grind is kind of important to our ability to make discoveries. So, just as an example, I have one paper out so far where the models were kind of useful. They proved a couple of lemmas.

Speaker A: So this is some situation where we had kind of proven the main result and then there were some lemmas I was unhappy with, like they seemed not to be optimal. And so I worked with—actually this was with Gemini Deep Think, which at the time was also on the frontier, no longer man.

Speaker A: Where I kind of worked with it to prove the lemmas. And so what happened here was there was some lemma I wanted to prove, and the models couldn't do it. None of the frontier models could do it. And so I worked out a ton of examples on my own and I realized, "Oh, well, maybe here's some reason why it could be true." So I found a better statement of the lemma. And then, once I had that statement, probably I could have done it pretty fast, but also the models were able to very quickly prove that sort of better statement.

So our inability to prove it led to an improvement in the result. Okay, now you can take the original lemma I had that the models weren't able to prove, put it into ChatGPT-5.6 Pro, and it will output like the worst proof you've ever seen: like 10 pages of just brutal calculation with no insight whatsoever. And so, this would have been a perfectly fine proof, but it would not have led to this discovery of what I think is a kind of beautiful conceptual explanation for why this thing we discovered was true.

So we found a better proof because we couldn't do—I mean, I say we couldn't do the calculation, but what actually happened (and here I understand that I'm being hypocritical about my complaints about ugly proofs before) is like I realized this horrible grind proof would work and I just could not bring myself to do it.

Speaker A: And so I looked for another argument. And it's not—yeah, but you know, now the models can do very long technical calculations pretty reliably.

Speaker B: Yeah. Well, I mean, to your credit, I don't know. I think I understand why you're saying it's like: don't just shy away from trying to prove something just because it seems ugly, because you have to take the first step, right? And eventually you try to work towards insight. And so, you might be shy about admitting it, but it is still maybe driven by whether it's aesthetics or just pure, like "I want to understand." And if understanding just means it's a little bit simpler or more compressed, then I've gotten understanding. It's just like a deep philosophical question too, like what it is.

Speaker A: Right? Like I mean of course there's a reason we do informal mathematics rather than writing out long formal strings of symbols of ZFC or whatever, even though besides length, it's just like somehow we're trying to put it in a way where we are actually getting some non-rigorous understanding out. Yeah.

Speaker B: I think somehow...

Speaker B: Yeah. And even how that relates to why it helps. And again, is it because of our inability to grind, or is it because if we are pushing towards something that's more compressive, it also hopefully ascends on—does it help understand other fields as well? And it is somewhat magical that when we try to optimize for both, they tend to coincide. I don't know actually if that's a quantitative statement, but why that is is kind of magical.

Speaker A: It's at least sometimes true, yeah.

Speaker B: Yeah, exactly. Or we certainly bias towards the cases in which it is, that's why we call that a good theory to build. But it's very much, I think, this is what we can study and understand, and therefore it's also very convenient that it was rich in kind of mathematical results there.

Yeah, I mean, okay, so I think there's two things that we can pull on, which is: given you believe that mathematical ability of AI is going to continue advancing and can do some of the things that you're doing nowadays, how should the mathematics community best adapt and benefit from this?

I mean, as somebody who doesn't have the time to practice mathematics anymore, this is very great because I can maybe dabble more, there's a lot of results that can come out. But I can also see where you've made the more precise point of: we can not motivate the right kind of behavior of understanding and development. And so I would love to hear more about your views there.

Speaker A: Yeah. So, first of all, it is clearly really exciting that there are increasingly capable models that are producing high quality results. At least some high quality results, also a lot of slop.

Speaker A: Yeah, some good stuff. And so, as the models get really capable, my hope is that they will answer a lot of the questions that I've been kept up at night thinking about, and I'll get to learn the answers. I think that's really exciting. And a lot of people got into math largely because they enjoyed learning math.

Speaker B: Yeah.

Speaker A: Like the first thing you do as a math student is you learn stuff that other people did, and you do that for like 20 years before you start doing—maybe not quite 20 years, but...

Speaker B: 15 years.

Speaker A: If you're lucky, 20 years, yeah, if you started early. Yeah.

Speaker A: Yeah. So that said, the goal of mathematics is not to produce mathematics papers; it's to produce some kind of understanding. So maybe some of that understanding resides in model weights or something, but to me that's pretty unsatisfying.

My own motivation for doing mathematics is that I would like to satisfy my own personal curiosity. I think people should have the capability to do that, and that requires a pretty substantial apparatus. It's simply not the case that you can study the questions I think are fundamental unless you've invested a huge amount of time and effort getting to the point where you can meaningfully do so.

And then moreover, even the small group of people doing fancy research math on the frontier relies on a huge apparatus of thousands and millions or billions of people who are trying to learn to think mathematically. You need an entire mathematical community to support a small group of people who are on the frontier. Without the pipeline, the pipeline doesn't exist.

Speaker A: So if you think that's important—development of human capital that can meaningfully engage with frontier mathematics—I think you still have to incentivize those people to actually invest their time and effort getting to the point where they can engage, and then do so in a high quality, meaningful way.

Right now, I think the existing incentive structures for math research do not encourage people to do that. Right now, if you're a postdoc on the market and you want to get a job, maybe for the next couple of years before the community adapts, the best way to do that is you want to produce a lot of papers which maybe prove old conjectures or whatever. And you can do that by playing the slot machine until the model produces a hopefully correct proof of such a result. So you don't even have to pick...

Speaker A: the the the theorem in advance. So like here's an experiment you can do. You can take codex, you can say go online, find five recent conjectures in algebraic geometry and prove them. And uh, okay, I've run this experiment and with some back and forth, I was able to, you know, in an hour get like three, you know, quite bad papers,

Speaker B: but correct papers.

Speaker A: Um, which, okay, are now sitting on my hard drive waiting for me to email the relevant people, but this is not like a good use of my time to invest results. But yeah, you you definitely see people doing this. Um, so there's, you know, been a huge uptick in post to archive, mostly not very interesting. Um, some of it is interesting.

Speaker B: Um, but, you know, a lot of it is kind of clearly low quality, even if in so far as the like even if the result is something that would have been like kind of highly rewarded a year ago, just like the there's sort of no evidence that a human being is actually engaged with it. Like there's no development of human capital or understanding. Like sometimes, you know, we've seen examples where like three or four or five papers with the exact same proof of the exact same theorem have come out in within a couple days of each other, which is clearly, you know, some situation where someone's playing the slot machine. Yeah.

Speaker A: Chat GPT is kind of consistently finding the same.

Speaker B: But that's also interesting. It's like not it's kind of mode collapsed on like certain paths of reasoning.

Speaker A: Yeah.

Speaker A: Yeah. And I think it's like not obvious that this problem goes away as the models get better. Like maybe it does, but maybe it doesn't. Like I think a lot of what I was saying earlier is that a lot of progress in mathematics comes from like letting you know a thousand different flowers bloom and people pursue their own curiosity and then you know the boundaries of knowledge expand in some kind of fairly hopefully fairly uniform way and like this really highdimensional space of mathematics

Speaker A: and then like opportunistically you you suddenly get some applications or like answers to old questions we found. Uh and it's like not clear to me that if if you know we kind of subordinate mathematical exploration to what the model want to pursue like if what you're getting is like one mathematician duplicated a thousand times or like actually you know a million different mathematicians doing a million different things. You know, I think actually that's pretty I mean, you know, aside from like okay, math has to adapt um by changing incentive structures, I think this is actually a pretty um um maybe dangerous or something that the labs have to pay attention to because you know on one side what's been successful for them is that this emergent reasoning capability is obviously incredibly powerful and we've been I mean it's creating a lot of like great PR headlines. But

Speaker B: you know to your point and also like where my interests tend to is that like human mathematicians they are coming from all sorts of weird intuitions that like you know and that is why you end up developing you know to your point the frontier that is so diverse that you actually can draw from and then be able to connect and produce a lot more. And so it if it's true that most of these like proofs that are being pushed out by the labs are converging on very similar things because they are technically drawing on the same body of literature. Um and that's where they're strong at now. It's it's not clear that I mean one you know where are they developing their intuitions? It's from practicing mathematicians. It's not clear where that other emergent you know stronger diverse intuition might be coming from. It might come but I don't have a good theory for where it comes from. And it's also just not clear that you know just with like test time compute and various post-training scaling that you can even I mean it's a huge debate whether you could introduce new capabilities and it's like precisely I think it should be studied in the math context because like where it's good at and where it fails um and where human mathematicians are good is exactly a very kind of like precise um question to study it on. Um, so this is like a long way of saying that I it is unclear to me that we'll maybe produce um if we don't incentivize enough people to interact with it, maybe unlike other domains, it it might not actually uh continue producing uh a superior uh uh result uh without human aid.

Speaker A: Yeah, let me actually push push a bit further even. So like let's suppose the models become like really robustly superhuman like even like we're not even adding like meaningful cognitive capacity. Okay.

Speaker A: I claim like still actually we we still won.

Speaker B: Okay. Great.

Speaker A: Human mathematician. So, right. So, why? So, like it's because the you know there's like a it's like there's a question about how we've we're designing society, right? Like maybe uh

Speaker A: the optimal uh situation is like you have the models doing all sorts of math research and like that leads to very in you know in this like highly you know whatever non-uniform diverse way like maybe you don't need humans to add that kind of diversity. It's like maybe that's the optimal thing to do and like in the end that leads to lots of applications and lots of imp proofed understanding and so on. It's like despite like something being optimal doesn't mean you do it

Speaker A: right like there's no reason to think that you know if we hand over control of whatever math research to the models and just let them do their own thing that it will do the optimal thing. Um and in fact like if you you know if we are if we kind of instrumentalize what we want them to do like we want to say like oh you know make our life better or whatever

Speaker A: it might not be the case like be that that like what they decide to do that is like is is pursue a wide variety of interesting research right like they might just try to take the direct path we don't know what's going to happen

Speaker A: um so I don't know if you believe that there is any value in sort of this broad-based like uh fundamental research which I do look I think that's like one of the most valuable as human humans or or whatever or the models can be doing like I think the easiest way to guarantee it happens is like to keep a community with like broad interests who are like pushing the models to do and like helping us to design a society where like that's what we're what we're pushing for.

Speaker B: Yeah. Like I think at least you know my my hopeful vision for the future is that like humans are not like totally disempowered like we have some control over where we're going and like if that's the case like what we end up doing is going to be driven by human interests

Speaker A: and so you want to have people who have lots of different interests and also the capabilities to actually pursue them like you want people who are like smart and engaged and like you know well can do not just like mathematical thinking but all sorts of thinking.

Speaker B: Yeah. Yeah. I mean, yeah, a great fear is like as AI advances, we don't develop the right ergonomic um kind of interfaces to actually encourage us to also be good continue to be good thinkers and and it's so easy to kind of um relinquish that because you just off and and the models aren't even good at that level of like you know thinking where it's like the top level structure but despite that it's it's so easy to and so especially for math I mean just like you know another selfish reason is um if you kind of math max you know I think it's actually a great pedagogical um uh excuse to actually get just really rigorous at thinking about various things. I mean this is not why mathematicians do it. Um but as just somebody that's part of what we do

Speaker A: Okay, great. Because it was kind of, you know, I thought it just really helped give me a very um good framework to think about many things, not just mathematics. And as somebody who's also, you know, now a parent of a 2-year-old, I kind of think about this a lot, too. you know, it's it's not about kind of grinding, even though whatever it's still good, but like not to not to um on grinding too much, but you know, just hearing I I used to collaborate a bunch with some Hungarian mathematicians, um

Speaker A: ...among them, and I heard that in Budapest, they would just teach group theory when you're in primary school. And I'm like, well, we should definitely do that. We should continue doing that. Now that AI is somewhat good at explaining, but it's far more accessible, we should actually probably proliferate that even more. And so maybe that helps with bringing more people to the frontier rather than just...

Speaker B: Yeah. I mean, this is something I'm concerned about, right? And of course I mostly talk about math because that's like where I live. But I think one nice thing about thinking about this is that we're one of the first professions to have a significant impact of high quality models. Although I think maybe we're one of the first professions for it to happen so publicly. You know, my sense is that there are plenty of other professions that are...

Speaker A: 100%. Coding.

Speaker B: Coding. But also just like, I mean, I think that anything you do at a computer, like probably a huge amount is being done by the models at this point, and then like they're not having a public reckoning about it.

Speaker B: Because math capabilities are useful for the company, for the labs to talk about, I think probably a bit more publicly than everyone else. But yeah, I mean I think one reason to try to maintain human capital in this area is just like it's a model for all professions: like presumably we still want people who are like meaningfully engaging with the world and like experts and have talents and trained skills and so on. It's actually very convenient that the math profession is so entwined with education here, because I think we're also seeing some amount of challenges among college and secondary education coming from AI too.

Speaker A: So as you said, I mean it's also an amazing tool to learn.

Speaker A: You know, I've heard people start talking about a bimodal distribution in their classes, where there's some people who are really figuring out how to take advantage of new tools, and other people who are just letting them do their homework and then bombing everything else.

Speaker B: Unfortunately I don't think that adapts fast enough, but it's like we definitely want to be living in a world where we're producing better thinkers. I think even if we talk about just the pure kind of optimization game, I think that's better for us.

Speaker A: And so, as just a human being, I'm like that would be pretty inconvenient if we became worse thinkers just as AI ascends. And it's too easy to let that happen. So we should kind of be thinking hard on how to actually take advantage of this and harness it for our own improvement, right?

Speaker B: Well, yeah. I think it's sort of interesting to observe like the models at their current level of capabilities let you do a lot more. They let you do a lot of things you wouldn't have done more cheaply than—you know, cheaply enough to do them now. But it's not clear to me that like in many cases they're actually improving the quality of outputs.

Speaker B: Yeah. And I think this is common, like you have a new technology that's doing something a little bit worse than was previously done but much cheaper, and so you get a lot of suddenly a lot of low quality outputs that are displacing previous high quality outputs. But I think it's possible to use the tools in a way that actually improves the quality along every dimension. It just requires some thoughtfulness and some redesign of institutions to actually incentivize that.

Speaker A: Yeah. Well hopefully capitalism works there. I do feel like that the most high value things do require people to use it effectively and and right now the models are not good enough without like the human experts to actually participate. But to your point there's a vast majority of maybe like more junior and entry level kind of roles—it's harder for those roles to adapt as well. And so the thing that would be a mistake is to use the AI models in a way that doesn't...

Speaker A: Basically you need to be ascending and using the models to deepen your understanding. And it's so easy for human nature just to be lazy and you have to resist that because that is the moment that you will kind of lose basically. And so, forgive the very competitive language, but it really is that: it's just so easy to relinquish the thinking to the models. The models can't really think. And so as things are ascending so fast, it's like critical that you continue developing those facilities and actually leverage it to improve those facilities rather than relinquish. Yeah.

Speaker B: Yeah. Yeah. I mean, one thing I think has been nice about sort of this vast increase in semi-expert attention or model attention on math problems is like now there's been—okay, I've been complaining about slop papers or whatever, like people who are not producing really high quality stuff. Often that's actually coming from professionals; it's not—I'm not saying within academic mathematics there are incentives to produce a lot of stuff, and that's one place the slop comes from. So there's definitely also slop coming from non-experts, but that I kind of actually don't see as a net negative. Like okay, now there's a lot of documents on the internet one might have to comb through to figure out if a problem has been solved or not. But to me it seems like just the fact that there's lots of people excited about math is kind of a positive. So that's like a nice...

Speaker A: I totally agree. I know I get to talk...

Speaker B: And not even kind of a positive, obviously.

Speaker A: Yeah. Yeah. No, exactly. It's like suddenly there's a spotlight on it and I can nerd out about math more. I was actually kind of curious, you've got any comments on the—I guess this is another constructive result, but the elliptic curve of rank 30 that just came out yesterday.

Speaker B: That's right.

Speaker A: Yeah. Yeah.

Speaker B: Well, I mean, we don't have any details about it yet.

Speaker A: I know. There's rumors.

Speaker B: Yeah, we don't know how it—I mean, so it was due to I guess Claude Fable prompted by Levent Poge and... right.

Speaker B: A collaborator whose name I unfortunately forget. Maybe you can add it in post.

Speaker A: I just know the Twitter handle.

Speaker B: Yeah, we can. So yeah, I mean, a lot of these nice recent results have come from Levent with unclear amounts of autonomy. So my sense is that some of them are semi-autonomous rather than fully autonomous.

Speaker B: Yeah, with this we don't know anything about the methods. So yeah, this is a fun construction. Without knowing about the methods, it's very hard to say how significant it is.

Speaker A: Yeah.

Speaker B: What I would say is that if you want to understand kind of historically such results have been understood by the community, they're like cool, but I wouldn't say they're like a big deal. So like the typical place a result like this might go is like someone's website of records.

Speaker A: Like not...

Speaker B: It's not an Annals level result.

Speaker B: Yeah, it's not an Annals... but that said, it's cool, and it's like a—definitely there are a few very, very talented mathematicians who like these kinds of questions, Noam Elkies being maybe the most famous example. So Elkies and Klagsbrun are the ones who kind of have been pushing this record for a while, and they recently found a rank 29 example...

Speaker A: Example which was, you know, the previous record.

Speaker B: Yeah. You know, so those are mathematicians. And I think people like it, and it's cool now that the models can do this sort of thing. But yeah, one thing I always say about a model result is like you cannot evaluate it except in retrospect.

Speaker A: And like this is also true of human mathematics. Like sometimes a problem we thought was really important or would require really deep new ideas does not, and sometimes it does and you can't really know in advance. So it's always exciting when a problem like that gets solved, but then like to figure out how significant it is, you still have to look. And like well Levent and Claude and I think Ava Howell maybe is the third collaborator.

Speaker B: Okay.

Speaker A: Have not yet told us how they did it. So...

Speaker B: Have they been mostly more secretive? I think they've released some traces for stuff. But...

Speaker A: Yeah, so for this one, I think they haven't yet, unless I missed it. Yeah, Levent likes to tweet out his results, but yeah, he has been slowly releasing some kind of PDF writeups, too. I think he's just having fun on the internet.

Speaker B: Yeah. Yeah. It reminds me of when you're saying, you know, you have to evaluate how the results came. I had Mark Zelki and Metab Swani on from OpenAI recently, and they were saying how what's kind of been the most charming or delightful is just that the proofs have been relatively short. They're not like 200-page, and you know maybe corresponding to your grinding point as well. But maybe this is kind of optimized for in retrospect—it was picked that it was short, or do you find that on average...

Speaker A: ...just not sure if they're true. Um, so for example, uh, you know, with these this recent list of 10 problems uh, released by OpenAI, those were all formalized in Lean,

Speaker A: which is of course a very good evidence that they're true. I have no doubt that there were a lot more that they could not formalize in Lean, like because the prerequisite options have not been put into Mathlib yet for example.

Speaker A: Um there were probably more that were longer which also makes it challenging to check.

Speaker B: Yeah.

Speaker A: Um so yeah this is my guess and we do actually see very long generated um proofs on the arXiv. Um so for example someone recently posted a claimed proof of resolution of singularities in positive characteristic which was 800 AI generated pages. Um it's like definitely I mean I'm sorry I haven't read it I haven't got an error but there's no way it's correct like this will be a major result. It's just not within the capacities of the current models if you're reasonably well calibrated.

Speaker B: Yeah. Yeah.

Speaker A: Um and it's definitely no human has read it. Definitely the models are not able to check this kind of thing. Yeah. Um so well I mean um yeah. So this is my expectation is that the reason it's producing short clever things is just like that's what we can check.

Speaker B: Yep. And you can get it to produce long things that are grindy or hard to check. But then...

Speaker B: Yeah, yeah, we'll have to get there. This is actually more what it says about capabilities, frontier capabilities, rather than in a perhaps more negative way rather than, hey, it's just, you know, so good at these clever. No, I think that's totally fair. I mean, especially if you look at an adjacent domain like code, right? Um remember reading something Cursor put out about testing their long horizon you know, harness. In this case they're trying to reproduce SQLite in Rust and it was just so telling like how far we are from that. And that seems like a very comparable task of like it's very long, you have to make sure there's something to verify that it's a correct implementation by testing a suite of cases. But you know you actually need a harness there in this case, it's not just like the raw models and it takes a while, and you can actually compare with different frontier versus non-frontier models, who's the planner, etc. like differences in their capabilities as well.

Speaker A: Yeah. I also think in practice like the uh to elicit a long proof, you kind of need a harness.

Speaker B: Yeah.

Speaker A: And um when you make a harness whose goal is to elicit a proof, I think it often decreases reliability because you're just trying to produce output. Um so like ChatGPT Pro like really tries not to say wrong stuff for example and although you know it happens and then you'll ask it like oh was that correct and it'll say no. Um but uh you know when you're trying to get it to ideate like you try to get it to be creative. You try to get it out of this very like you know rigorous rut in order to like you know actually get it somewhere. And I think if your interest is in—I mean there are a lot of people who are you know trying to arbitrage the prestige mechanics of academic mathematics and trying to elicit a lot of proofs which are not necessarily being checked and in order to do that I think you just decrease the reliability in order to get a lot of stuff. Um, so you know, I think any harness that can elicit a 250 page paper is probably not being very careful about what it's producing.

Speaker B: And and you're saying that it's decreasing or it's not reliable just because it's the capability isn't there yet and so it's just kind of forcing a long longer horizon task on it.

Speaker A: Yeah.

Speaker A: Yeah. I mean in practice like how does like a human checking it like can't check a 250 page paper either like you know you can't reliably check it line by line. What you try to do to understand it is you try to like understand the overall global structure of the argument and you like stress test it in various ways like would this argument imply something else that I know to be false? Like oh does it work in this special case? Blah blah blah. And also the models seem not to be able to do that kind of like more I don't know fuzzy like unit testing a proof very well yet.

Speaker A: So actually like one of my favorite tests for the models which they haven't succeeded at yet is there's a paper, I won't name it, that came out a couple years ago that was wrong, and it was in close enough area to myself that was claiming some big result like I immediately downloaded and started reading it. And it was very hard to find the actual specific error, but it was also very clear from the structure of the argument that it couldn't work. It was like arguing something that was too strong to be true if you kind of took that a little bit further.

Speaker B: Yeah.

Speaker A: Um so me and a bunch of other experts like immediately realized it was wrong and we emailed the author and we kind of went back and forth until someone figured out what the precise specific error was.

Speaker B: Okay.

Speaker A: Um and so far the models seem not to have been able to do this. Like the specific error is quite subtle and but they're also not able to do this kind of overall uh kind of big picture test.

Interviewer: ...checking, which is kind of like how people in practice check papers.

Interviewer: Yeah. Yeah. Which is kind of it mirrors you know why in code it's like it's so clear it's good at the syntax, but you know higher level architectural stuff is still very weak. Maybe it'll ascend there, but probably need some harness help. Who knows? I mean, people have, you know, evolving opinions on how much the harness and the model have to co-evolve and which one is necessary, but the next model requires less. So, I feel like in math, it'd be very interesting to see if you do any experiments with the harness there and how that improves, because it is a mark of general reasoning. Yeah.

Daniel: Yeah. I mean, I do, you know, I have my own sort of bad little harness and Codex, but you know, I personally, I do not enjoy autonomous mathematics very much, so I mostly do not use the harness. I mostly try to use it to help me understand stuff.

Interviewer: Okay. No, it's totally fair. Yeah. Exactly. You don't want to automate your job away, because that does involve you being in the loop to understand it, which is, you know, necessary to participate. Maybe to finish off, I'd love to—and this could be just something you haven't thought about or actually thought a lot about—I think you also have a toddler, right?

Daniel: Yeah.

Interviewer: Yeah. A three-year-old?

Daniel: Yeah. Three-year-old.

Interviewer: Great. Right. So you're one year more advanced and probably, you know, have more thoughts on this. Like how are you thinking about her education in math? And you know, not to grind, but really just how to pass on the love of it, and how to react to AI.

Daniel: Yeah. So she's three. She's never used AI. She is starting to add, that's about as far as we are in math.

Interviewer: That's more than—

Daniel: She can add single-digit numbers by counting on her fingers, and count up to like maybe 30 reliably and 50 semi-reliably. So I'm very proud of that.

Interviewer: That's good. Yeah.

Daniel: Yeah, I definitely encourage that. We talk about shapes and stuff. Actually a couple days ago I woke her up and she was like hiding under the blankets, and I was like, "Oh, what are you up to under there, Sophia?" And she was, "Oh, I'm doing some math."

Interviewer: Oh, I did. I think you tweeted about that. That was horrible [funny].

Daniel: Yeah, it was great. So, you know, I think she has some sense that I like math and she's into it because of that. Um, yeah. I mean, I would say I think the world is probably going to look pretty different in, you know, 20 years or whenever she's fully adult and doing her own thing. But you know, I think a lot of what we educate people for is pretty robust to changes in the nature of the world. The reason to learn math has always been to think clearly and better understand the world, and presumably that's something you want to do even if there are extremely capable AIs. And this is also true, you know—I personally like math a lot, but also the reason to read a lot of books and do the humanities and so on. So I think the actual values of the math profession and education more broadly are things we definitely want to try to preserve, that I hope to instill in my daughter. You know, how much institutions have to change to make sure that happens is maybe an open question I think a lot about. But yeah, at least at a personal level, I'm definitely trying to convince my three-year-old that math is super cool.

Interviewer: Oh yeah, 100%.

Daniel: One of her first words was icosahedron.

Interviewer: Was what?

Daniel: My parents gave her a little icosahedron toy when she was one. Not literally a first word, but she learned the Platonic solids quite early.

Interviewer: Oh, very good. Very fun.

Interviewer: Next up, group theory. I mean, it's like very natural. That's right.

Daniel: Yeah. I'm actually, when I teach her about addition and subtraction further, we'll do it in the context of a general group.

Interviewer: Oh, very good. Well, at least you give the motivation. I think a lot of people probably skip that part. Yeah. And you know, maybe this is how math grad students can also focus, like training the next, much younger generation to use AI in service of actually getting better at math, rather than just lacking understanding. Wonderful. Okay. Well, thank you so much, Daniel.

Daniel: Thank you. It was a lot of fun.

Interviewer: Yeah. Yeah, it was a lot of fun. And yeah, I mean, I think there's going to be a lot more progress very soon. I'd love to maybe catch up and chat again.

Daniel: Sounds great.