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 a model weight. To me that's pretty unsatisfied. Comparing in-profit with opening eye. Do you detect any differences in how that is similar to human reasoning?
They definitely are not good at it, honestly. But with some hints you can kind of get them to do something interesting. A lot of progress in mathematics comes from letting, you know, thousands of different flowers bloom and people pursue their own curiosity and then the boundaries of knowledge expand in some kind of fairly uniform weight. What has been the most impressive result so far? My favorite fully autonomous result finding AI so far is the solution to the original distance problem.
There was some I wanted to prove, none of the French models could do it. 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. Once I had that statement, the models were able to very quickly prove that sort of better statement. And how should the mathematics community have best adapt and benefit from this? AI can increasingly solve math problems that would challenge professional mathematicians.
But solving a problem isn't necessarily the same thing as understanding it. In this episode, A16Z in for partner, Lisa Lee, sits down with the University of Toronto mathematician Daniel Lit to separate the headlines about AI and mathematics from what the models can actually do today. Daniel explains why recent results have changed his views of AI, where frontier models already resemble human mathematicians and where they still fall short, particularly when it comes to intuition, developing new theories, and even figuring out which questions are worth asking. They also explore what happens to mathematics when generating a proof becomes cheap, why academic incentives may need to change, and how mathematicians can use AI without outsourcing the understanding that makes the work valuable in the first place.
And more broadly, they ask what mathematics can teach us about working with AI, as increasingly capable models move into every knowledge profession. I am so excited to have you on Daniel. And Daniel Lit 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 natural practicing mathematician and in addition, he's been incredibly vocal about his evolving views of AI in math. And so I feel like if I just don't check in with you in two weeks, something different has been revealed and then you're very kind of like, what do you call it?
You have a lot of opinions. You have a lot of opinions, exactly. So I want to get into that. So I mean, one of the things that I'm most interested in is not just like a discussion of how the capabilities of advanced, I feel like in math, that's definitely the headline, et cetera. 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. It's not just like, hey, AI's been really making progress here, but delve into what actually mathematicians do. And so maybe like with that, arc in line, we can start with what has been the most impressive results so far, given all the recent progress for you. And then maybe, yeah, we'll kind of take it from there. Yeah, so there have now been a lot of results, some of them produced autonomously, some produced, like semi-harmously, some of them who's like the way the AI contributions are not at all clear.
There are a lot of different areas. So anything I say is kind of, I can only really comment on things that I have some expertise on. So it's quite possible that it be talking to different mathematicians. You'll get different answers here. So my favorite fully autonomous result by AI so far is the solution to the addition of distance problem, which I was announced in mid-May.
So at least what I liked about that is it seemed to me that 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 the techniques and apply them in maybe a clever way. And now they've kind of been as some of the results I've characterized as like last mile, like where it's recent work was done quite deep work with them by a group of human mathematicians. And then the AI took the final step. But yeah, with this sort of distance problem, I think it was something where the result was a little unexpected. So first of all, my sense was like that people working in the area thought it was true.
And then it was a counteresempel found. But then also it like brought in some techniques from another area. I think those techniques were not especially like deeper new. There's a class of ideas from the 60s, but they would new to this area of studying, 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 a bunch of mathematicians took those ideas and used them to find counteresempels to a bunch of other interesting open questions. So for example, like the song product conjecture for the real numbers. So that's what I like to think about how cool it is. Like you look at it post-talk and you see like, oh, well, we're whatever new ideas that were introduced, if any, like kind of useful to do other things, did they improve our understanding of something?
And I think that's maybe so far the main example of a result of that form. Yeah, I think it's really meaningful that you're commenting on this because that result came out to your point in May. And there's been 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 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 our all is so successful at this because you can kind of both verify it somewhat cheaply compared to other domains. And then also because the rules are quite illegible. And so you can kind of like, if you're super human 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 getting at truth, remaining confused and developing intuitions. And I mean, the tool to do that stuff is of course in having really strong abilities to push out logical implications. But maybe if you can kind of speak to like, when you say it's most impressive and creative, like decoupling just the inhumane maybe feats of just like logical implication from like, where is it being creative? What is it helping engender in terms of like mathematical activity as well? Yeah, I mean, so first of all, I mean, you kind of characterized it as like in
human in some way, I actually think the argument was very human. You know, I'd like to do that. I was opening I like, at least some chain of thought. Yeah, it was very recognizable. Like, if I try to imagine like my chain of thought and trying to solve a problem, like it might look kind of like that. Yeah, we haven't seen the raw chain of thought. Maybe that would maybe they cleanse it a
little bit. Yeah, maybe the model like to swear a lot in the middle of the chain of thought and make that up or something would know. But at least the summary seems pretty, 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 not so well written, if you just look at raw model output, it's not like there's some 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, definitely the models are still like, not very good 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 not seem to be doing, but certain things are very bad. So the ways they might be a little bit inhuman is like they don't
tired, they know a lot. But if you actually read the final output, it doesn't seem kind of inhuman. It's interesting because I mean, obviously we can't get too much of information from the labs we're producing and these models, why of what the training recipes are, how they're kind of advancing and reasoning. But at least one of the things we do know, and I think kind of opening eye-sear head of this is 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 polished, but textbook certainly 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 I'm kind of curious before maybe even going to the text on is an excuse, like when you're examining these models and their results comparing and fropping with opening eye, do you detect any differences in how that is similar to human reasoning? And then also if you have any comments on like insights on
perhaps why natural language scales so well that way, even though it's- Okay, so first of all, I really like your point though that they're mostly doing natural language reasoning rather than lean. I think that suggests to me that like these, 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, like probably the techniques are generalized other domains pretty well. That's just my guess. Okay, you asked a little bit about Claude versus JBT? My sense is that they're pretty similar in terms of capabilities.
I've played on a lot more with JBT than Claude Fable, but it seems like there's a lot of cases where OpenAI will drop a solution to some problems and we'll say, oh, we also do. Exactly. It actually seems like they're stopping a very similar collection of problems and it's like kind of a 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? Like there'll be things that- this isn't surprising. Like there'll be things that rely on the model strengths, like
their ability to round out a long computation or pull together kind of technical ideas from many areas or like maybe many papers that you know, even my mathematicians might not have read, but they're not like they seem like weaker and things like intuition or like having some big picture fun to be with a lot of what I do as a mathematician is like I have some kind of philosophy that's like very non-rigress. Like I think this thing is kind of analogous to this thing. And then like a lot of what I'm working out is like figuring out how to make that precise and like, you know, trying to measure the extent to which I've succeeded in understanding that by like can I solve a problem or whatever, like to find it interesting phenomenon which I don't understand it, but now I understand it. And like so far, I haven't, you know, you can see 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 non-techniques, which is to be clear that's like a very powerful thing to do to be very good at applying like all non-techniques. There are mathematicians who have had great careers doing very high quality work with that flavor and I think a lot of what the models are producing, it's 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 like signs of both, you know, all the frontier models starting to be able to do more fuzzy things. So like, I've tried to get both, both fable and Cenk, kp, 5.6 all, I guess, to do some kind of theory building. And it's like, they're not, 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. You know, when you do, when you do the models hints, it's always a little hard to tell
like what part is from the model and what part is new, 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. I do think there's signs that they're also kind of somehow picking up some of this, like implicit in 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 the small detail, which is that, you know, from the public, you've gleaned that anthropic and opening our probably neck and neck, but you personally are using a lot more chat GPT, like why is that or 5.6. Well, why is that? I know, I think it's just in our show.
Okay. I'm pretty sure have it. Yeah. One thing is that chat GPT got better at math earlier. So, like for a long time, the cloud model is not useful for research math. And I think we do it
on opus 4.5 or opus 4.6. They're like more or less caught up. But you know, some experimentation suggests to me they're pretty neck and neck. And so for my own work, except for my own just experimenting, I must have just stayed with one and started random. Yeah, as people outside of labs, like us, it's really interesting just to compare how they defer on the frontier. And I mean, you know, to your point, it might be a little bit of momentum.
I do think that at least from my anecdotal experience, my point six has been a little more clear in exposition. And it's like, it's a little bit, and this might not be true. I mean, obviously, the models are just incredibly jagged at the frontier. But like, it's 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 people might be explaining something very trivial, but then just like jump at like, well, you know, obviously, you should know these things and- Yeah, I find they're both pretty bad at theory. Okay, great. So then, seeing in your eyes, you're probably asking much deeper
questions. Okay, so the thread I really wanted to pull on was when you're talking about the models, maybe starting to get better in tuition or even theory building. So maybe before we even dive into that would be useful to talk through like, what is your primary, you know, activity as mathematician, especially in your area of like algebraic geometry, probably has a very different flavor than a combinatorialist or, you know, some other area. So if you can give a little maybe brief lay of the land and then kind of explain what your math mathematical activity was pre-AI and then maybe how it's kind of like changing your AI. Yeah, so I think there are a lot of different kinds of mathematicians. There's a lot of different spectra on which one can put
a mathematician. So definitely a lot of mathematicians like solving open problems. And then I think I'm one of those like I like to solve an open problem. That's a 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. At least for me, the point of an open problem is it's like supposed to measure your failure to understand something. So stand and look at benchmark. Right, like,
you know, one problem I really like is the grossed decaps, the curvature, what that is, it measures something that 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 conduction, we know we don't understand it. Okay, so in practice, like, how do you get 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 fill with it and then you stare at like what once you win, you stare at what you develop to win and try to turn that into some theory. So that's one thing you might do, you might try to solve a problem and so do a 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 then, you know, you might start building a table like this property A is related to property A prime, property B is related to property B prime and so on and so forth. So, for example, in my work, a lot of it is motivated by terminology between comology and algebraic varieties and representations of fundamental groups. So that's some fancy stuff. But it's just that this analogy is like very,
very fruitful. And like, there's any phenomenon that appears on one side, you can find it analog on the other side. And so, you know, trying to, to realize that dream has led to a lot of beautiful mathematics of less, like three or four years about people like Kyle Simpson and to grow much as a kid. 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. But 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, you know, you're trying to realize. And that philosophy is like super not rare, essentially. It's not like simple pushing at all. Yeah. So that's another kind
of activity that I like. Yeah, beyond that, you know, a lot of what you do when you try to, you know, a lot of the activities, like 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 like, figuring out what you don't know is like actually a very challenging thing to do. So, you know, there's like a list of, you know, you can go online and list it when conjectures or whatever. And this like doesn't really capture in a lot of ways. But we don't know, like often finding the
conjecture is like really, really hard. So I know a good example of this is like the version of the entire conjecture is one of the planning problems. It's this beautiful relationship between L functions of elliptic curves and the set of solutions to the course of the brain of the group of the group of solutions, what that means. It was discovered by like, it was like the first data conjecture. So version of the entire had like found all these statistics and elliptic curves in the 60s because it was one of the first ever concurated bits of mathematics. And they like
graphed these statistics and they noticed that, you know, some, so the slope of some line on the graph was related to some other algebraic and grand that was the source of this conjecture. So a lot of the time you're just like working at examples and like trying to split your doing kind of science, like you run an experiment and you try to grab an explanation for that experiment. And so I think going back to AI, I think like these are things where AI seems so far so you know, help a lot more in some things than other things. So like, you know, I think the more vague, of course, phenomenon is like the less you have a precise question in mind, the less useful 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. Like it's primarily kind of a substitute for Google or something. Like I might use it to like learn about some related topic or like something where I would have earlier Googled something and read a paper or maybe I'll just read it instead. So save some time for sure. It's like not really doing deep intellectual work for me.
But then, you know, because, you know, I know like, well, I suck at coding and so now I know, you know, my good friend who's really good at coding and now I have all these coding projects. It's like suddenly, if I had a question where coding would have been really useful, I would have gotten on it for six months until I tireless page to the student. Exactly. So yeah, now I picked up all these projects where coding is really useful. Like, 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 it would be 10 examples at a time, you know, 10 different sub-agents and like that's really useful. But these are like kind of different activities, which are like, on top of what I was doing before. I'd love to dig in. Yeah, go. Yeah, so because you were mentioning it, there's projects that you've been working on for like three, four, five years. And I don't know if it's correct to say like,
it was more of the theory building aspect of it because you did characterize yourself as like an open problem solver or like what is the thing that is it because like the deep thinking part, maybe, you know, kind of for this audience, it might also be useful to kind of say or explain, you know, most of pure math is it's not like it's being motivated by, you know, nothing on applied math, but apply math is like at least there's some external motivation for why a certain formal structure might be interesting to study whereas this one, it's like purely, it seems almost sociological until some extent, I think like thirsty made some comment, you know, point about this in the 70s, is that it's sociological phenomena with more and more mathematicians start examining something, then you'll maybe, you know, converge on some interesting structures. But it's just like, it's not, it's 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 maybe can make a more general comment about the profession? Yeah, it was definitely some people are like motivated by like, goody or or kind of aesthetic considerations. I try not to be motivated by that.
Oh, that's a bad way. Yeah. Like, well, one thing like that sort of limits you, right? Like, 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. And then like the proof feels really ugly and you decide. But like, okay, I mean, what if you're wrong? It's not ugly. Like,
why am I yourself too? Or the society? What is ugly? Because I have an information of what's ugly, but like, what is that for kind of stuff? I don't really know. I don't really have a aesthetic in this. But people sometimes feel this way.
Like, maybe it's a lot of grinding. Yeah. So in calculation, it's not illuminating or something like that. Exactly. But you should like win by any means necessary, in my opinion. Like, I like to think of what
I'm doing, like doing kind of physics, except with concepts. So, you know, instead of beauty, I try to think about maybe like, what is kind of fundamental, what's going to open up further understanding most? Yeah. Well, you know, am I introducing a new idea that will be like, broadly useful to understand this object? Yeah. I mean, I guess, you know, in some sense, there's
like some aesthetic consideration there, but it's, I think the orientation of like trying to do good science rather than trying to do art is a microver. But there's a huge, there's a huge variety of opinions here in my lots of mathematicians think of themselves as being like, closer to poets or something. Yeah. What I do think is broadly true is that like progress comes from people, like it seems to come in mathematics from people sort of pursuing their personal curiosity. And that it's sort of been crucial historically that there's like 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 know,
you have these opportunistic situations where a new idea has been introduced and you can kind of now find you now can suddenly cascade through a bunch of other things that we didn't understand before. 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? Like, when you're doing the deep thinking, like, is it just that it's just not clear that you formulated as a problem? And it's more that you're thinking about these, like, what are, you know, what are the fundamental kind of physics of, you know, if. So in some cases, I mean, but some cases, there is like a well stated problem here.
I was thinking about things for, you know, maybe 10 years now, it's one for certain problems. Sometimes there is just like 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 like the different truth. So like, I think that certain, you know, for example, you know, this, you know, distance can be a problem. Like the general belief in the community was that it was true and then after not to be false. And so what that means is that there's like a specific construction you
can do to, to, to, to refute it. On the other hand, I think a lot of the things I think about like, I don't know, maybe I'm about to, you know, this one will come up with a standard sample to the picture of a conductor who might have thought this is tomorrow and like, I'll look like a fool. But in general, like these conjectures fit into some very broad and theoretical framework, which means that we actually have a lot of evidence that they're true. And so, you know, yeah, so that's part of it. Like there's not like a construction you can do to refute it.
You need to somehow kind of, you know, there's this giant framework where certain, certain pieces of it are only conjectured or you probably need to resolve some of those conjectures to win. We also have like a pretty good sense, I think, that like very serious new ideas are needed to resolve those conjectures. So like, of course, you can't be sure like maybe there's some very clever construction that will let you, I don't know, avoid having a big new idea. I don't know. Maybe 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, there are, they're just not accessible to, you know, applying techniques very, very technically, strongly. So you need to develop a new technique. I'm just, like, I'm not saying the models won't be able to do this. It's like so far they seem not to. Actually, that's exactly the point that I went to, because it's some, you know, to your point is like, okay, we can try to calibrate and forecast like why they would get better at this. But it is true,
like, you know, that just providing construction of our counter example, they seem to be strong on. If you have to start developing either new theory or to your point techniques machinery to solve, to prove why conjectures are true, it struggles more. And it's probably because a lot of what it's drawing on is also just like techniques that have happened in other areas and reporting 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. I was like from another, at least it was like, right in something.
Exactly, exactly. And so like maybe then to the kind of ask the question, it's not that because now we're like, okay, fine, yeah, getting so good, so fast, we can't count it out. But like, why, what do you think it has to, yeah, I guess it's like you're smelling out what it has to get better at, but maybe some more kind of feelings on like why it's kind of particularly hard to then develop that new theory. And I think you just need a different like my guess actually is it's probably totally doable. It just hasn't been done yet. Like maybe you just need a different
ARL environment. I'm not, yeah, so I mean, this is what my expectation is just like that the trajectory will continue upwards. I'm not a skeptic of continued capabilities growth. Yeah. Yeah. Yeah. You know, I think what is definitely true is that like the seal of like developing a theory
or you're like building your understanding of some Portlander's job is like a fuzzier one. So it might be harder, you know, I guess you can try, you can tell it, you know, develop your understanding of data functions. And then once it proves that we want to pop this to see a better reward, but it's kind of hard, I think, to come up with some like intermediate things that you would reward. Yeah. I mean, that said, you know, I do think mathematics as a whole provides a lot of conjectures of varying levels of difficulty. So you know, maybe, maybe to explain better seems to
be a little bit of progress in these areas. Like presumably they are trying to, you know, get it to solve lots of problems and solve those problems to develop at least some of the skills of theory. Like new humans are able to develop these skills. You know, I guess sometimes they have rewards from their PhD advisors and their advice. That's a good idea or something based on some element of human taste or whatever. Yeah. And that might be something one can do too. But
yeah, my expectation is that just like as part of continued capabilities, we'll see growth in these areas too. Yeah. And I like the framing where you're casting these increasingly difficult conductors as a, you know, a form of curricula for both humans and of course, AI. And it, you know, maybe kind of getting a little bit too philosophical for some people's tastes. It's like, it gets up the question of like, why are we particularly good or maybe particularly bad at math too? Because it's, you know,
what is it that we're either struggling to do or some people are particularly good at when you develop new theory? Because it's not again, it's like not really, or maybe it's related to like, why can we formulate good structures for physics as well? It's just like, it's 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, you know, our minds because we need to, we can't understand anything beyond, you know, compressing stuff more finely. I don't know if that's also like the correct interpretation of like, yeah. I'm a little bit skeptical as like,
compression as a metric of interest, but it's definitely like, is an anthropological reason for like, why we do a certain theory building, I think it's true. And in fact, I think it's like, kind of like our inability to just grind is kind of important to our ability to make discoveries. So, and just like, as an example, so I have one paper out so far where the models were kind of useful. So they like, like proofable on us. So this is some situation where like, we had to kind of prove in the main result, and then there were like, some one was I was unhappy with, like, they seem not to be optimal. And so I like worked with, actually, this was with, with Gemini deep think,
which at the time was also on the frontier no longer. I, where I can work with it to improve the lemmas. And so what happened here was there was some lemma I wanted to prove, and the models couldn't do it. So none of the frontier models could do it. And 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.
Like, so I like a kind of better statement of one. And then, okay, once I had that statement, like, okay, probably I could have done it pretty fast, but also the models were able to very quickly prove this proof, prove that sort of better statement. So like, our inability to prove it led to an improvement in the result. Okay, now you can take the original lemma I had, but the models weren't able to prove put it into chat, GP 5.6 pro, and it will output like the worst proof you've ever seen, like 10 pages of like just brutal calculation with no insight whatsoever. And so, okay, this would have been a perfectly proof, but it would not have led to this discovery of like, I think a kind of beautiful conceptual explanation for why this thing is sort of true. So we found a better proof
because we couldn't do, and I say we couldn't do the calculation that could actually happen. And here I understand that I'm being hypocritical about my complaints of ugly proofs before. I realized that this horrible grind proof would work. I could not bring myself to do it. And so I mean, it's not, yeah, but now the model is going to do is very long, technical health calculations pretty reliably. Yeah, I mean, I don't know. I think I understand
why you're saying, like, don't, you know, just shy away from trying to prove something just because it seems ugly because it's like you have to take the first step, right? And eventually you all try to work towards insight. And so, you know, you might be shy about admitting it, but it is still maybe driven by whether it's like aesthetics or just like, you know, like, I want to understand. And if understanding just means it's a little bit simpler or, you know, more compressive than I've gotten understanding. I mean, that's just like a deep philosophical question too. It's like, what is it?
Right. Like, of course, any, you know, there's a reason we do informal mathematics rather than like writing out long and thermal strings and symbols of CFC or whatever. Yeah, yeah. Though, you know, besides length, somehow we're trying to put it in a way where we are actually getting some non-regarious understanding of. Yeah. I mean, I think somehow, yeah. And even how that
even relates to like why it helps. And again, it's like, is it because of our inability to grind or is it because like if we are pushing towards something that's more compressive, it also hopefully also ascends on the does it help understand other fields as well? And it's just somewhat magical that when we try to optimize for two, the both that it tends to coincide. I don't actually know if that's a fair, it's actually quantitative statement. But why, you know, that is, is kind of magical.
It's at least sometimes, yeah. Yeah, exactly. Or, you know, we certainly bias towards the cases of which it is. That's why we call that a good theory to build. But it's, it's, yeah, it's very much, I think, you know, it's like, 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 like, you know, pull on, which is like what given you've seeded that or not seeded, but you've believed that mathematical ability of AI is going to continue advancing and can do some of the things that you're, you know, doing nowadays, how should like the mathematics community kind of best adapt and benefit from this? I mean, as a, just kind of saying, as somebody who's, you know, doesn't have the time to kind of practice mathematics anymore, this is kind of 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 know, you've made the more precise point of we can not motivate the right kind of behavior of understanding and development. And so we'd love to kind of share more about your views there. Yeah. So first of all, like it is clearly really exciting. Like the very increasingly models that are like producing high-quality results, at least some high-quality results, also a lot of thought. Yeah. Yeah. Yeah. Some good stuff. And so, you know, as the models get really capable,
you know, my hope is that they will answer a lot of the questions that I've been like, you know, kept up at night thinking about them. Like I'll get to learn the answers. I've been back. I've been like, exciting. And you know, a lot of people got into math, like largely because they enjoyed learning math. Yeah. Right. Like the first thing you do is a math student is you like learn stuff
to other people. And you do that for like 20 years before you, before you start doing all maybe about 20 years. If you're lucky 20 years, yeah. You started early. Yeah. Yeah. Yeah. So, um,
that said, like the goal of mathematics is not to produce math and stapers, like it's to produce some kind of understanding. So maybe, okay, maybe that some of that understanding resides in model weights or something. To me, that's like pretty unsatisfying. Like my own, you know, my information pretty mathematics is like how to satisfy my own personal curiosity. I think people should be, have the capability to do that. And like that requires a pretty substantial apparatus. Like,
it's like simply not the case that you can like study the questions I think are fundamental, unless you've invested a huge amount of time and effort kind of getting to the point where you can meaningfully do so. And then moreover, like that, you know, even the people, like, you know, you have the small group of people doing like fancy research math on the frontier or whatever, that relies on like a huge apparatus of like, you know, thousands of millions or billions of people who are trying to learn to think mathematically, like you need to an entire mathematical community to support a small group of people who are on the frontier, like you just without the pipeline, that doesn't exist. So if you think that's important, like development, if you mean capital, like meaningfully engage with frontier mathematics, I think that, you know, 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. So right now, I think like the existing and instructors for math research do not do not do that. So, you know, right now, if you're like a postdoc on the market, you want to get a job, maybe for the next couple years before the community adapts. The best way to do that is like, you want to produce a lot of papers, which may be proof,
you know, open gestures or whatever, and you can do that by playing the slot machine until the model preces are hopefully correct proof of such a result. So you don't even have to pick the the theorem in advance. So like, here's an experiment you can do, you can take codecs, you can say, go online and find five recent conjectures now to break down and improve them. And 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, like bad papers, but correct papers, which okay, now I'm sitting on my hard drive waiting for me to email their element people, but this is not what people do so much time to know results. But yeah, you definitely see people doing this. So this, you know, been a huge object
in post archive, mostly not very interesting. Some of it is interesting. 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 this sort of no evidence that a human being is actually engaged with it, but there's no development of human capital 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 out and within a couple days of each other, which is clearly, you know, some situation where someone's playing the slot machine, yeah, check GPT is kind of consistently finding the same. But that's also, it's like, it's kind of mode collapsed on like certain paths of reasoning. Yeah. Yeah. And I think it's like not obvious that
this problem goes away as the models get better. What maybe it does, but maybe it doesn't. I think a lot of what I was saying earlier is that a lot of progress in mathematics comes from like letting you know, 1000 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 look at this really high dimensional space of mathematics. And then like opportunistically, you suddenly get some applications or like, as I told questions, we found, and it's like not clear to me that if, you know, we can subordinate mathematical exploration to what the model want to pursue, like if what you're getting is like one mathematician duplicated 1000 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 by changing center structures, I think this is actually a pretty, 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 capabilities, obviously incredibly powerful. And we've been, I mean, it's just created a lot of like great PR
headlines. But you know, to your point, and also like where my interest 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. And that's where they're strong out 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 with that other emergent, you know, stronger, diverse intuition might be coming from, I 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 you should be studying the math context, because like where it's good at and where it fails, and where human mathematicians are good, is exactly very kind of like, precise question to study it on. So there's like a long way of saying that I, it isn't clear to me that we'll maybe produce, if we don't incentivize enough people to interact with it, maybe unlike other domains, it might not actually continue producing a superior results without a human aid. Yeah, let me actually 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 time to the diversity. Okay, so I claim like still, actually, we still
want to be on the mathematician. So right, so we apply. So like, it's because, you know, it's like, it's like, it's like, it's a question of how we designing society, right? Like, you may be, the optimal situation is like, you have the models doing all sorts of math research, and like that leads to very, 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. So maybe that's the optimal thing to do. And like, in the end, that leads to lots of applications and lots of
improved understanding and so on. It's like, despite something being optimal, doesn't mean you do it, right? Like, there's no reason to think that, you know, if we hand over control of whatever, research to the models and just let them do their own thing, that it will do the optimal thing. 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, it might not be the case, like, that, what they decide to do that is like, is pursue a wide variety of interesting research, right? Like, they may just try to take the direct path. We don't know what's going to happen. So I
know if you believe that there is any value in sort of this broad based, like, fundamental research, which I do look at, that's not the most valuable thing that humans 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 broad interests, we're pushing the models to do it, like helping us to design a society, that's what we're pushing for. Like, I think at least, you know, my hopeful vision for the future is that humans are not like totally disempowered, like, we have some control over where we're going. Like, if that's the case, like, what we end up doing is going to be driven by human interests. And so you want to have people who have lots of different interests and also the capabilities to actually pursue them, which you want people who are like, smart and engaged and like, you know, well, do you not just like medical thinking and all sorts of thinking? Yeah, yeah. I mean, yeah, a great
fear is like, as advances, we don't develop the right ergonomic kind of interfaces to actually encourage us to also be good, continue to be good thinkers. And it's so easy to kind of relinquish that because you just awful and the models aren't even good at that level of, you know, thinking where it's like the top level structure, but despite that, it's so easy to. And so especially for math, I mean, just like, you know, another selfish reason is if you kind of math max, you know, I think it's actually great pedagogical excuse to actually get just really rigorous at thinking about various things. I mean, it does not why mathematicians do it. But as just somebody, it's part of my degree. Okay, great. Because I thought it just really helped give me a very
good framework to think about many things, not just mathematics. And as somebody who's also, you know, now a parent of a two year old, I kind of think about this a lot too, you know, it's not about kind of grinding, even though whatever it's still good, but like not to not to shit on grinding too much. But you know, just hearing, I used to collaborate a bunch with some Hungarian mathematicians, ballerjegadi among them. And I just heard that in in Budapest, they would just teach group theory when you're in primary school and like, well, we should definitely do that. We should continue doing that now that AI is so good at, you know, somewhat good at explaining, but it's far more accessible. We should actually, you know, probably proliferate that even
more. And so maybe that helps with bring more people to the frontier rather than, you know, just. Yeah, I mean, this is something I'm concerned about, right? I mean, of course, I mostly talk about math because that's like where I live. But like, I think, you know, one nice thing about thinking about this is that we're one of the first professions to kind of, you know, have a significant impact of high quality models. Although I think maybe one of the first professions for it to happen is
publicly, but you know, my sense is that there are plenty of professions that are totally floating. But also just like, and I think that, you know, anything you do at a computer, like probably a huge amount is being done by the models at this point, but I'm like, they're not in a public reckoning about it. But you know, because math capabilities are useful for the company for the model labs to talk about it, I think, but I think, you know, one reason to try to maintain, like, 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 experts and, you know, have like, talents and train skills and so on. Like, yeah, so it was, it's actually very convenient that the math profession is so involved with education here, because I think we're also seeing like, some amount of challenges among college and, like, education coming from there too. So as you said,
it's also an amazing tool to learn. So, you know, people have heard people start talking about a bi-modal distribution in their classes where there's some people who are really like figuring out how to take advantage of new tools and other people who are just like putting them in their homework and then bombing things. 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 it's just, you know, even if we talk about just the pure kind of optimization game that's better for us. So, but as, you know, just like a human being, like, that would be pretty inconvenient if we became worst thinkers, just as AI sons, 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 as well. Right. Yeah, I think it's like, it's sort of interesting to observe, like, the models, current level capabilities, that you do a lot more, they 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 fair to me that, like, in many cases, they're actually improving the quality of outputs. And I think this is common, like, you have a new technology that's doing something a little bit worse than what's previously done, but much cheaper. And so you get a lot of suddenly a lot of, like, low quality outputs
that are displacing high-preview side quality outputs. But I think it's possible to use the tools in a way that actually improves, you know, the quality of the whole, I have to mention it just requires some helpfulness and some redesign of institutions should actually incentivize that. Yeah. Well, hopefully capitalism works there. I do feel like the most high value things do require people to use it effectively. And right now, the models are not good enough without, like,
the human experts actually participate. But to your point, there's a vast majority, maybe, like more junior and entry level. And if, you know, it's harder for those rules to adapt as well. And so the thing that would be a mistake is to use the models in a way that doesn't, basically, you need to be ascending and using the models to pin 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 you kind of, you know, kind of forgive the very competitive language. But it really is that like, it's just so easy to kind of relinquish thinking to the models, the models can't really think. And so as things are setting so fast, it's like critical that you continue developing those facilities and actually leverage it to improve those facilities rather than relinquish. Yeah. Yeah. Yeah. Yeah. One thing I think has been nice
about sort of this vast increase in semi-aspirate attention or model attention on math problems. It's like, now there's been, you know, I've been complaining about soft papers or whatever, like, you know, people who are not, you know, producing the high quality stuff, often that's actually coming from professionals. So it's not saying, like, you know, people who, you know, like within academic mathematics, there are incentives to produce a lot of stuff. And that's, you know, that's one place to stop them from. So there's definitely also like, not coming from non-experts. But that I kind of actually don't see as a negative, like, okay, there's a lot of,
now there's a lot of documents on the internet one might have to come 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 like kind of a positive. So that's like a nice thing. I totally agree. I'm not even kind of positive. Obviously. Yeah. Yeah. Yeah. No, exactly.
It's like, suddenly there's a spotlight on it. I can nerd out about math more. I was actually curious. You've any comments on the, I guess this is another constructive result, but the elliptic curve of rank 30. That just came out yesterday. So if that's right. Yeah. Yeah.
Not a lot of people. We don't have any details about it yet. Yeah. There's rumors that we don't know how it, I mean, so it's, it's been a, I guess, clock table prompted by what EntelPosh and a letter whose name I unfortunately forget. Maybe you can post a letter. Yeah.
I just know the Twitter handle. Yeah. We can. Yeah. So yeah. So a lot of these nice
recent results have come from event with unclear amounts of autonomy. So in my sense, that's some of them are semi-communist residents from the autonomous. Yeah. With this, we don't know anything about the methods. So yeah, this is a fun construction without knowing what methods it's very hard to say how significant it is. Yeah. What I would say is that if you want to
understand kind of historically how such results have improved, or sorry, have been, have been understood by community, they're like cool, but I wouldn't say they're like a big deal. So like a typical place of result like this might go is like someone's website of records. It's not an annals level. Yeah. It's not an annals, but that's that is cool. I mean, you know,
it definitely, there were a few very, very, very, very talented mathematicians who like these kinds of questions. So no, not these being the most famous examples of Alfie St. Clag's, who kind of been pushing this record for a while. And then they recently kind of ran 29, example. One of them. The previous record. Yeah. Yeah. Yeah. 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, it's very, you know, one thing I always say about a model result is like you cannot evaluate it except in retrospect. 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 events. So it's always exciting when a
problem with that gets solved. But then like to figure out how significant is you start to look and like well, live in and cloud. And I think Eva, how old is the celebrity club? Okay, I have not yet told us how they did it. So have they been mostly more secretive? I think they
released some traces or stuff. But yeah. So for this one, I think they haven't yet. And also I missed it. Yeah. They, they, they, you know, live in like to tweet out his, his results. But yeah,
he has been, you know, slowly releasing some of the PDF right up to do it. I think it's, you know, he's just having fun on, on the internet. Yeah. Yeah. It's, it reminds me of when you're saying, you know, you have to evaluate how the results came. So had Mark Selke and
Mitab's 20 on from OpenAI recently. And they were saying how, you know, what's kind of in the most charming or delightful is 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. It was short. Or maybe do you find that on average,
stuff that you throw at on GPT, you know, Saul or Fable tends to be shorter and more legible to the human versus it might just go haywire and just grind it out. Yeah. So, so I mean, I, I think it is nice that we'll sometimes produce short cut or proofs. So that's, of course, everyone likes a short cut or proof. I think my sense is that the reason they're not producing long complicated proofs is that they cannot. Just like that, the ability to check correctness is not yet there.
So, you know, I mean, even actually, you know, if you ask the models to produce a short proof, you can then often ask them to discuss is that correct? And they will often say no. So they're much more reliable than they were six months ago for sure. But like, it's still, you know, they will still sometimes produce things that are just wrong and they know that. I think the problem with producing a very long thing is they might not know they're wrong.
And so what I wonder if, you know, presumably internally opening eye and probably solve a lot more problems than they released. And I imagine quite a few of them are just not sure if they're true. So for example, you know, with these recent list of 10 problems, it was by opening eye, those were all formalized in lean, which is of course a very good evidence that they're true. I've no doubt that there were a lot more that they could not formalize in lean, like the pre-rapis of options have not been put into mass, for example. There were probably more that were longer, which also makes a challenge in check.
So yeah, this is my guess. And we do actually see very long generated proofs on the archive. So for example, someone recently posted a claim to proof of resolution of singularities and other characteristics, which is 800 AI generated pages. It's definitely, I'm sorry, I haven't read it in an era, but there's no way it's correct. But this is the major result.
It's just not the capacities of current models, which are reasonably well-calibrated. And it's definitely no human has read it. Definitely, the models are not able to check this kind of thing. So this is my expectation. So the reason it's producing short-lover things is just that's what we can check. And if you get it to produce long things that are grindy or hard
to check, but then- 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 so good at these clever things. I think that's totally fair. Especially if you look at an adjacent domain code, remember reading something that Kurser put out about testing their long horizon harness, in this case they're trying to reproduce SQLite and Rust. And it was just so telling how far we are.
And that seems like a very comparable task. 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 actually need a harness there, in this case. It's not just the raw models. And it takes a while. And you can
actually compare with different frontier versus non-funteer models, who's the planner, etc., like differences in their capabilities as well. I was thinking in practice, to elicit a long proof, you need a harness. And 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. So it's a 10-2-3-5-6-4. It really tries not to say wrong stuff, for example. I know that it happens in the last few hours,
like, oh, is that correct? I don't say no. But when you're trying to get it to IDA, you try to get it to be creative. You try to get it out of this very rigorous rut in order to get it somewhere. I think if your interest is in, there are a lot of people who are arbitrage, the prestigious mechanics of the mathematics and trying to elicit a lot of proofs, which are not necessarily being checked. In order to do that, I think you just decrease the liability in order to get
a lot of stuff. So I think any harness that can elicit a 250-page paper is probably not being very careful about what it's producing. And you're saying that it's decreasing or it's not reliable just because the capability isn't there yet. And so it's just kind of forcing a longer horizon task on it. Yeah. I mean, in practice, like, a human checking a 250-page paper, either,
you can't reliably check it line by line. What you try to do to understand it is you try to understand the overall global structure of the argument and you express it in various ways. Like, would this even apply something else? I know to be false. Like, oh, does it work in a special case?
And also, the models seem not to be able to do that kind of like more, I don't know, fuzzy unit testing approved very well yet. It's not one of my favorite tests for the models, which they haven't succeeded at yet. There's a paper, I won't name it, that came out a couple years ago that was wrong. And it was very hard to find, like, in close enough area to myself, that I like to immediately download it and start 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. So it was like, you know, me and a bunch of other, he was proving something that was too strong to be true. If you can talk a little bit further. So me and a bunch of other experts immediately realized it was wrong and like, you know, email the author and like, we kind of went back and forth until someone figured out what the precise specific error was. And so far, the models seem not to have been able to do this, like the specific errors by subtle and they're also not able to do this kind of overall kind of big picture text checking, which is kind of what helps people in practice check papers. 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, still very weak. Maybe we'll send 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 how you, if you do any experiments with the harness there and how that improves, because it is a market kind of reasoning. Yeah. Yeah, I mean, I do, you know, I have my own sort of bad little
harness and codecs, but yeah, it's, you know, personally, I do not enjoy autonomous mathematics very much. Mostly do not use the harness. I must try to use it to help me understand stuff. Okay, no, it's totally fair. Yeah, exactly. You don't want to automate your driveway 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, this could be just something you haven't thought about, or actually got a lot about. I think you also have a toddler, right? Yeah. Yeah. So how
have you been three years old? Yeah, three years old. Great. Great. So you're one year, more advanced, probably, you know, have more thoughts on this. Like, how are you thinking about, um,
is it her education or his education? Or how are you thinking about her education in math and, you know, not to grind, but you don't really just like how you don't pass on the love of it and how to react to AI? Yeah. So she's, she's three. She's never used AI. She is starting to add. That's,
that's about as far as you're in math. That's more far as you can add single, single digit numbers, like, you know, like, on her fingers and count up. So it can be 30 reliably and 50 centi reliably. I'm very proud of that. Yeah. I definitely encourage that. We talked about shapes and stuff.
Actually, a couple of days ago, I will care up and she was like, hi, down to the blankets. Oh, I was like, oh, what are you up to under there? Sophia and she was like, oh, I'm doing some math. Oh, I think you're talking about that. That was great.
So, you know, I think she has like, sense that I like math and she's in two minutes of that. Yeah. I mean, I would say I don't know, you know, I think the world is probably gonna look pretty different in, you know, 20 years or whatever. She's kind of, you know, fully adult and doing her own thing. But you know, I think a lot of what we educate people for is like pretty robust changes in the nature of the world. Like, I think the reason to learn math has always been like to think clearly
and like better understand the world. And like, presumably that's something you want to do, even if there are some extremely capable ais. And it's 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. And I think like the actual values of the math profession and like education more broadly are like the things we definitely try to monitor, preserve, and think that I hope to install 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. But yeah, I mean, at least at a personal level, I'm definitely trying to, you know, convince my material that math is super cool. Oh, yeah, 100 words. One of her first words was like, oh, see, Adrian. So was what was what she has a she has a my parents gave her little like, was a kid, right? When she was one, she was very, very, not literally first word, but she learned
the platonic solids. Oh, very good. Very good. Next stop group theory. I mean, it's like very natural. Yeah, I mean, I mean, when I when I teach her about addition and subtraction for the whole,
we'll do it in the context of a general group. Oh, very good. Well, at least you give the motivation. I think a lot of probably skip that part. Yeah, yeah. And you know, maybe this is how
math grad students can also, you know, focus on like training the next much younger generation to use AI in service of actually getting better at math rather than just I can understand. Wonderful. Okay. Well, thank you so much, Daniel. Thank you. 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. So true. Thanks for listening to this episode of the A16Z podcast. If you like this episode, please share the like, comment, subscribe, leave us a rating, we're a view and share with your friends and family. One more episodes go to YouTube Apple podcast and Spotify,
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