Daniel Litt: The Mathematician's Guide to AI

12:29Daniel Litt:Creature of habit. Yeah.

12:31Lisha Li:It got, one thing is that ChatGPD got better at math earlier. So like for a long time, the cloud models were just like not useful for research math. And I think maybe around opus 4.5 or opus 4.6, they like more or less caught up.

12:47Daniel Litt:Yeah.

12:47Lisha Li:But, you know, some experimentations suggest 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.

12:56Daniel Litt:Yeah.

12:56Lisha Li:It's harder to random.

12:56Daniel Litt:As people outside the labs like us, it's really interesting just to compare how they differ 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, 5.6 has been like a lot more clear in exposition. And it just like it's there'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 Fable might be explaining something very trivial but then just like jump at like well you know obviously you should know these things and

13:40Lisha Li:Yeah I find they're both pretty bad at theory of mind

13:42Daniel Litt:Okay great so then seeing 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 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 little, maybe briefly of the land and then kind of explain what your mathematical activity was pre-AI and then maybe how it's kind of like changing with AI.

14:23Lisha Li:Yeah, so I think there are a lot of different kinds of mathematicians. There's like a lot of different, you know, 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. I think of myself as a problem solver as opposed to one other taxonomy you could have as a problem solver versus a theory builder. 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.

15:03Lisha Li:One problem I really like is the gross and decaps p-curperture concuture, 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. 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.

15:36Lisha Li:and then you stare at like what once you win, you stare at like what you developed to win and try to turn that into some theory. 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. So, for example, in my work, a lot of it is motivated by some analogy between cohomology of algebraic varieties and representations of fundamental groups.

16:13Lisha Li:So that's some fancy stuff. But it's just that this analogy is like very, very fruitful. And like really any phenomenon that appears on one side, you can find an analog on the other side. And so, you know, trying to realize that dream has led to a lot of beautiful mathematics over the last, like, 30 or 40 years by people like Carl Simpson and Jokuro Machizuki and others. And so here, it's really just, like, someone noticed, like, here's an analogy, and then that analogy has, like, 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.

16:51Lisha Li: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. 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 open conjectures or whatever.

17:35Lisha Li:And this like doesn't really capture in a lot of ways what we don't know. Like often finding the conjecture is like really, really hard. So I don't know, a good example of this is like the Bertrand-Swinner-Ton-Dyer conjecture, which is one of the millennium problems. It's this like beautiful relationship between L functions of elliptic curves and the set of solutions to the corresponding equations with the rank of the group of solutions, whatever that means. It was discovered by, like, it was like the first big data conjecture. So Bersh and Svindert and Dyer had, like, found all these statistics on elliptic curves in the 60s.

18:08Lisha Li:Like, it was one of the first ever computer-aided bits of mathematics. And they, like, graphed these statistics and they noticed that, you know, the slope of some line on the 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 like working out examples and like trying to, like you're doing kind of science, like you run an experiment and you try to figure out an explanation for that experiment. And so I think going back to AI, I think like these are things where AI seems so far to, you know, help a lot more in some things than other things.

18:39Lisha Li:So like, you know, I think the more vague phenomenon it is, or like 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've 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 then, like, read a paper.

19:12Lisha Li:Like, maybe I'll discuss it with AI instead. So it saves some time, for sure. It's, like, not really doing deep intellectual work for me. Right. But then, you know, because I, you know, I'm now like, well, 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 like suddenly, if I had a question where coding would have been really useful, I would have procrastinated on it for six months until I...

19:35Daniel Litt:Tireless PhD student.

19:37Lisha Li:Exactly. So, yeah, now I've picked up all these projects where, yeah, 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 maybe ten examples at a time. You know, ten different subagents. And, like, that's really useful. But these are, like, kind of different activities, which are, like, I don't know, on top of what I was doing before.

20:02Daniel Litt:I'd love to dig in.

20:04Lisha Li:Yeah, go on.

20:04Daniel Litt:Yeah, sorry, because you were mentioning, you know, there's projects that you've been working on for like three, you know, three, 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 because you did characterize yourself as like an open problem solver. Or like what is the thing that is, because like the deep thinking part, maybe, maybe like, 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 applied 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, and to some extent, I think Thurston made some comment, or a point about this in the 70s, that it is a sociological phenomenon.

20:47Daniel Litt:If 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, 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.

21:07Lisha Li:Yeah, I mean, so definitely some people are, like, motivated by, like, beauty or kind of aesthetic considerations. I'd try not to be motivated by that.

21:15Daniel Litt:Oh, that's controversial. Yeah, I think that's a bad way.

21:18Lisha Li: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 and it's not ugly? Like, why limit yourself?

21:38Daniel Litt:For this audience, like, what is ugly? Because I have an intuition of what's ugly, but like, what is that for kind of spelling?

21:43Lisha Li:Well, 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 grinding or some calculation that's not illuminating or something like that. 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.

22:10Daniel Litt:Yeah.

22:10Lisha Li:Well, you know, am I introducing a new idea that will be like broadly useful to understand this object? Yeah. I guess, you know, in some sense, there's like some aesthetic consideration there, but it's, I try to, I think the orientation of like trying to do good science rather than trying to do art is a biographer. but there's a huge variety of opinions here and like 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 now find, you now can suddenly cascade through a bunch of other things that we didn't understand before.

23:07Daniel Litt: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 the fundamental kind of physics of, you know.

23:32Lisha Li:So in some cases, I mean, some cases there is like a well-stated problem here. You know, I've been thinking about things for maybe 10 years now at this point 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 conjectures are true. So like, I think that certain, you know, for example, you know, this unit distance problem, like 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 like a specific construction you can do to refute it.

24:15Lisha Li: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, someone will come up with a found example to the P-curvature conjecture in order to do one hypothesis. tomorrow and like I'll look like a fool. But in general, like 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, 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, we have this giant framework where certain pieces of it are only conjectured and you probably need to resolve some of those conjectures to win.

24:54Lisha Li:We also have a pretty good sense, I think, that very serious new ideas are needed to resolve those contractures. So, of course, you can't be sure. 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. 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, you know, 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 like so far they seem not to.

25:36Daniel Litt:Actually, that's exactly the point that I wanted to delve into because it's, you know, to your point, it's like, okay, we can try to calibrate and forecast like why. They would get better at this, but it is true that just providing 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 solve, to prove why 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.

26:14Lisha Li:It was like from another, at least it was like drawing in something from an unexpected area.

26:17Daniel Litt:Exactly, exactly. And so like 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 like why, what do you think it has to, yeah, I guess it's, you're spelling 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 technique. Yeah, that's a good question.

26:40Lisha Li: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 RL environment. I don't know. Yeah. So at this point, my expectation is just like that the trajectory will continue upwards. I'm not a skeptic of continued capabilities growth. But yeah, you know, I think what is definitely true is that like the skill of like developing a theory or like building your understanding of some poorly understood object is like a fuzzier one. So it might be harder, you know, I guess you can tell it, you know, develop your understanding of zeta functions and then once it proves the real hypothesis, you give it a reward.

27:22Lisha Li:But it's kind of harder, I think, to come up with some like intermediate things that you can reward. Yeah, 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 this explains why there 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 some of those problems develop at least some of the skills of theory building. Humans are able to develop these skills. You know, I guess 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.

28:00Lisha Li:And that might be something one can do too. But yeah, my expectation is that just like as part of continued capabilities growth, we'll see growth in these areas too.

28:09Daniel Litt: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 it may be kind of getting a little bit too philosophical for some people's taste. It gets at 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.

28:51Daniel Litt: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.

29:13Lisha Li:I think that's, I'm a little bit skeptical as like compression as a metric of interest, but it's definitely like as 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 proved a couple of lemmas in the paper. So this is some situation where like we had kind of proven the main result and then there were like some lemmas I was unhappy with, like they seemed not to be optimal.

29:49Lisha Li:And so I like worked with, actually this was with Gemini DeepThink, which at the time was also on the frontier. no longer where I kind of worked 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 here's some reason why it could be true. So I found a better statement of the lemma. 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 that sort of better statement.

30:30Lisha Li:So like our inability to prove it like 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 chat GPT 5.6 Pro, and it will output like the worst proof you've ever seen, like 10 pages of like just brutal calculations with no insight whatsoever. And so, okay, this would have been a perfectly fine proof, but it would not have led to this discovery of like, I think a kind of beautiful conceptual explanation for why this thing discovered was true. So like we found a better proof because we couldn't do, I mean, okay, I say we couldn't do the calculation like what actually happened and here I understand that I'm being hypocritical about my complaints about ugly proofs before.

31:06Lisha Li:It's like I realized like this horrible grind proof would work. Yeah. And I like could not bring myself to do it. And so I went for another argument. And so, I mean, it's not, yeah, it's about noon and now the models can do very long and technical calculations pretty reliably.

31:23Daniel Litt:Yeah. Well, I mean, to your credit, I think I understand why you're saying it's 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 pure, you know, like I want to understand. And if understanding just means it's a little bit simpler or, you know, more compressed than I've gotten understanding. I mean, it's just like a deep philosophical question, too, is like what it is.

31:52Daniel Litt:Right.

31:52Lisha Li:Like, I mean, of course, any, you know, there's a reason we do informal mathematics rather than like writing out long informal strings of symbols of CFC or whatever. Yeah, yeah, yeah. Even though, you know, 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.

32:07Daniel Litt: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 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, it is somewhat magical that when we try to optimize for the both, that it tends to coincide. I don't know actually, you know, if that's a fair, it's actually a quantitative statement, but like why, you know, that is, is kind of magical.

32:39Lisha Li:It's at least sometimes true, yeah.

32:40Daniel Litt:Yeah, exactly. Or, you know, we certainly bias towards the cases in which it is. That's why we call that a good theory to build. But it's, yeah, it's very much, I think, you know, So 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 seated that, or not seated, but you've believed that mathematical ability of AI is going to continue advancing. it can do some of the things that you're, you know, doing nowadays.

33:24Daniel Litt: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 hear more about your views there.

33:56Lisha Li: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 stuff.

34:08Daniel Litt:Yeah.

34:09Lisha Li: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 like, you know, kept up at night thinking about. I'm like, I'll get to learn the answers. It's like, that's really exciting. And, you know, a lot of people got into math, like largely because they enjoyed learning math. Yeah. But like the first thing you do as a math student is you like learn stuff that other people did. And you do that for like 20 years before you start doing, oh, maybe not quite 20 years, but 15 years.

34:40Daniel Litt:If you're lucky, 20 years. Yeah, if you started early.

34:43Lisha Li:Yeah, yeah. Yeah, so that said, like, the goal of mathematics is not to produce mathematics papers. Like, it's to produce some kind of understanding. So maybe, okay, maybe some of that understanding resides in model weights or something. To me, that's, like, pretty unsatisfying. Like, my own, you know, my motivation for doing mathematics is, like, I would like 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.

35:27Lisha Li: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 and millions or billions of people who are trying to learn to think mathematically. Like you need an entire mathematical community to support a small group of people who are on the frontier. Like you just, without the pipeline, the pipeline doesn't exist. So if you think that's important, like development of human capital that can 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.

36:09Lisha Li:So right now, I think, like, the existing incentive structures for math research do not encourage people to 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 maybe prove, you know, 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 the theorem in advance. So like, here's an experiment you can do.

36:44Lisha Li:You can take codecs, you can say, go online and find five recent conjectures in algebraic geometry and prove 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, quite bad papers, but correct papers, which okay, are now sitting on my hard drive waiting. for me to email the relevant people, but this is not a good use of my time to invest in results. But yeah, you definitely see people doing this. So there's been a huge uptick in post-archive, mostly not very interesting. Some of it is interesting. But a lot of it is kind of clearly low quality, even if insofar as the, like even if the result is something that would have been like kind of highly rewarded a year ago, just like there's sort of no evidence that a human being is actually engaged with it.

37:33Lisha Li: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 within a couple days of each other, which is clearly, you know, some situation where someone's playing the slot machine. Yeah. ChatGPT is kind of consistently finding the same. Well, that's also interesting.

37:52Daniel Litt:It's like not, it's kind of mode collapsed on, like, certain paths of reasoning. Yeah.

37:56Lisha Li: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 it's really high dimensional space of mathematics. And then like opportunistically, you suddenly get some applications or like answers to old questions we found.

38:25Lisha Li:And it's like not clear to me that 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.

38:41Daniel Litt:Yeah, I think actually that's pretty, I mean, you know, aside from like, okay, math has to adapt by changing incentive 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 capability is 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 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.

39:27Daniel Litt:And so 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 at now, 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 of 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.

40:05Daniel Litt:And it's like precisely, I think, you should be studied in the math context because like where it's good at and where it fails and where human mathematicians are good is exactly a very kind of like precise question to study it on. So this is 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 result without a human aid. Yeah.

40:38Lisha Li:Let me actually push a bit further even. So like, let's suppose the models become like really robustly superhuman, like even and, like, we're not even adding, like, meaningful cognitive diversity.

40:47Daniel Litt:Okay.

40:47Lisha Li:Like, I claim, like, still, actually,

40:49Daniel Litt:we still want human mathematicians.

40:52Lisha Li:Yeah. So, right, so, like, why? So, like, it's because, you know, there's, like, a, it's, like, there's a question on how we're designing society, right? Like, maybe 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, like, 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, like, something being optimal doesn't mean you do it.

41:26Lisha Li: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. 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 don't know if you believe that there is any value in sort of this broad based like fundamental research which I do I think that's one of the most valuable things humans or whatever the models can be doing.

42:07Lisha Li:I think the easiest way to guarantee it happens is to keep a community with broad interests. We're pushing the models to do it and helping us to design a society. That's what we're pushing for. I think at least my hopeful vision for the future is that humans are not totally disempowered. We have some control over where we're going. If that's the case, 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. 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.

42:42Daniel Litt:Yeah, yeah. I mean, yeah, a great fear is like as AI 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 off. 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 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 it's part of what we do 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 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 um shit on grinding too much but you know just hearing i used to collaborate a bunch with some hungarian mathematicians um balas zegeri among them and i just It hurt that in Budapest, they would just teach group theory when you're in primary school.

43:58Daniel Litt: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...

44:15Lisha Li:Yeah, I mean, this is something I'm concerned about, right? Of course, I mostly talk about math because that's 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 we're one of the first professions for it to happen so publicly. But, you know, my sense is that there are plenty of professions that are totally impacted.

44:40Daniel Litt:Coding. 100%.

44:41Lisha Li:Coding. But also just like, I mean, 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. And then like there's not having a public reckoning about it. you know because math capabilities are useful for the labs to talk about I think more publicly than everyone else but yeah 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 you know have like talents and trained skills and so on yeah so it's actually very convenient that the math profession is so entwined with education here because I think we're also seeing like, you know, some amount of challenges, you know, among college and like secondary education coming from AI too.

45:35Lisha Li:So as you said, I mean, it's also an amazing tool to learn.

45:37Daniel Litt:Yeah.

45:38Lisha Li:People have, I've heard people start talking about a bimodal 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 letting them do their homework and then bombing everything else.

45:51Daniel Litt: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, I think that's better for us. But as, you know, just like 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 as well.

46:21Lisha Li:Yeah, I think it's sort of interesting to observe like the models at their current level 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.

46:39Daniel Litt:Yeah.

46:40Lisha Li: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 requires some thoughtfulness and some redesign of institutions should actually incentivize that.

47:04Daniel Litt: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 to actually participate. But to your point, there's a vast majority of maybe like more junior and entry level. And if, you know, 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, 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.

47:47Daniel Litt: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 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.

48:09Lisha Li:Yeah. Yeah, I mean, one thing I think has been nice about, you know, sort of this vast increase in like semi-expert attention or like model attention on math problems is like now there's been, you know, okay, I've been complaining about slot papers or whatever. Like, you know, people who are not, you know, producing really high quality stuff. Often that's actually coming from professionals. Like it's not, I'm not saying like, you know, people who, you know, like within academic mathematics, there are incentives to produce like a lot of stuff. And that's, you know, that's one place that's sloped from.

48:42Lisha Li:So there's definitely also like slop coming from non-experts. But that I kind of actually don't see as a net negative. Like, okay, there's a lot of, now there's a lot of like 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.

49:01Daniel Litt:I totally agree. I know I get to talk to you.

49:03Lisha Li:Not even kind of a positive, obviously.

49:05Daniel Litt:Yeah, yeah, no, exactly. It's like suddenly there's a spotlight on it. and I can nerd out about math more. Yeah. 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.

49:22Lisha Li:That's right.

49:23Daniel Litt:Yeah, yeah. That's cool.

49:25Lisha Li:We don't have any details about it yet. I know, there's rumors that it's... Yeah, we don't know how it... I mean, so it's due to, I guess, Claude Fable, prompted by Leventhal-Poge, and a collector whose name I unfortunately forget, but maybe you can edit in post.

49:41Daniel Litt:I just know the Twitter handle. Yeah, we can edit it.

49:45Lisha Li:So 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. 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. What I would say is that if you want to understand kind of historically such results have been proved, or sorry, 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.

50:25Daniel Litt:It's not an annals level result.

50:27Lisha Li:Yeah, it's not an annals, but that said it's cool and it's like you know, it definitely, there were a few very, very, or are a few very, very talented mathematicians who like these kinds of questions, so no Malky's being maybe the most famous example. So Alkes and Clagsbrun are the ones who kind of have been pushing this record for a while. And they recently found a rank 29 example. One up. Which was, you know, the previous record.

50:48Daniel Litt:Yeah, yeah.

50:48Lisha Li:You know, so those are mathematicians. And I think, you know, 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 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, have not yet told us how they did it.

51:25Daniel Litt:Have they been mostly more secretive? I think they released some traces for stuff.

51:30Lisha Li:Yeah, so for this one, I think they haven't yet, unless I missed it. But yeah, Levent likes to tweet out his results. But yeah, he has been solidly releasing some kind of PDF write-ups too. I think he's just having fun on the internet.

51:45Daniel Litt:Yeah, yeah. It reminds me of when you're saying you have to evaluate how the results came. So I had Mark Zelke and Metab Swani on from OpenAI recently. and they were saying how, you know, 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 maybe do you find that on average stuff that you throw at GPT, you know, Sol or Fable tends to be shorter and more kind of legible to the human versus it might just go haywire and just grind it out.

52:36Lisha Li:Yeah, so I mean, I think it is nice that they'll sometimes produce short cover proofs. Yeah. So that's, of course, everyone likes a short cover proof. I think my sense is that the reason they're not producing long, complicated proofs is that they cannot.

52:53Daniel Litt:Mm.

52:53Lisha Li:Just like 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 short proof you can then often ask them you also just ask is that correct and they will often say no so like 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 they're wrong 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 OpenAI and Anthropic have probably solved a lot more problems than they've released and I imagine quite a few of them are just not sure if they're true.

53:37Lisha Li:So for example you know with this recent list of 10 problems released by OpenAI those were all formalized in Lean 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 because the prerequisite options have not been put into MassLip yet for example. There were probably more that were longer, which also makes it challenging to check. So yeah, this is my guess. And we do actually see very long AI-generated proofs on the archive. So for example, someone recently posted a claimed proof of resolution of singularities and positive characteristic, which was 800 AI-generated pages.

54:16Lisha Li:It's definitely, I mean, I'm sorry, I haven't read it. I haven't done an error, but there's no way it's correct. Like this will be a major result. It's just not with the capacities of the 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 yet.

54:36Lisha Li:So this is my expectation, is that the reason it's producing short, clever things is just like that's what we can check. And you can get it to produce long things that are grindy or hard to check, but then...

54:47Daniel Litt:Yeah, we'll have to get there. This is actually more what it says about capabilities, these 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? I remember reading something that 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, you know, and that seems like a very, you know, comparable task of, like, it just, It's very long.

55:21Daniel Litt:You have to make sure there's something to verify that it's a correct implementation by testing a suite of, you know, 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, you know, non-frontier models, who's the planner, et cetera, like differences in their capabilities as well.

55:45Lisha Li:So, yeah, I was thinking in practice, like the, to elicit a long proof, you kind of need a harness. Yeah. 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, like, ChattGPT 5.6 Pro is like very, it 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. but 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 trying to arbitrage the prestige mechanics of academic mathematics and trying to elicit a lot

56:33Daniel Litt:of proofs which are not necessarily being

56:35Lisha Li:checked and in order to do that I think you just decrease the liability in order to get a lot of stuff 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.

56:47Daniel Litt: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.

56:56Lisha Li: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, like, 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-approved very well yet.

57:25Lisha Li: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 like very hard to find like in close enough area to myself that I like immediately like it was claiming some big result like I immediately downloaded it and like started reading it and it was like 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 well he was proving something that was too strong to be true if you kind of took that a little bit further so me and a bunch of other experts like immediately like 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.

58:13Lisha Li:And so far, the models seem not to have been able to do this. The specific error is quite subtle and they're also not able to do this kind of overall kind of big picture checking, which is kind of like how people in practice check papers.

58:27Daniel Litt: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 I'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 how you, if you do any experiments with the harness there and how that improves, because it is a marker of general reasoning.

58:59Lisha Li:Yeah, I mean, I do, you know, I have my own sort of bad little harnacks and codecs. Oh, yeah. But yeah, it's, but you know, 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.

59:15Daniel Litt: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, this could be just something you haven't thought about or actually thought a lot about. I think you also have a toddler, right?

59:34Lisha Li:Yeah.

59:34Daniel Litt:Yeah, yeah. And so how have you...

59:35Lisha Li:Yeah, I'm a three-year-old.

59:36Daniel Litt:Yeah, three-year-old. Great, great. So you're one year more advanced and probably, you know, have more thoughts on this. Like, how are you thinking about... Is it her education or his education? Her, yeah. How are you thinking about her education in math and, you know, not to grind, but, you know, really just like, you know, pass on the love of it and how to react to AI.

59:57Lisha Li: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.

1:00:04Daniel Litt:That's more far away than my theory.

1:00:06Lisha Li:She can add single-digit numbers by counting on her fingers. Nice. Count up to maybe 30 reliably and 50 semi-reliably. So I'm very proud of her.

1:00:15Daniel Litt:That's good.

1:00:16Lisha Li: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.

1:00:30Daniel Litt:Oh, I just told you, I think you tweeted about that. That was horrible.

1:00:34Lisha Li:Yeah, it was great. So, you know, I think she has like some sense that I like math and she's into it because of that. Yeah, I mean, I would say I don't know, you know, I think the world is probably going to look pretty different in, you know, 20 years or whenever 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 sort of extremely capable AIs.

1:01:09Lisha Li:And this is also true, you know, I personally like math a lot, but also the reason to, like, read a lot of books and do the humanities and so on. So, I mean, I think, like, the actual values of the math profession and, like, education more broadly are, like, things we definitely try to, 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. But yeah, at least at a personal level, I'm definitely trying to, you know, convince my three-year-old that math is super cool.

1:01:46Daniel Litt:Oh, yeah, 100 % of that.

1:01:47Lisha Li:One of her first words was icosahedron, so.

1:01:49Daniel Litt:Was what?

1:01:50Lisha Li:She has a, my parents gave her a little icosahedron toy. when she was one. That's the first word. Yeah, not literally the first word, but she learned the platonic solids quite early.

1:02:01Daniel Litt:Oh, very good, very good.

1:02:02Lisha Li:That was very, very fun.

1:02:03Daniel Litt:Next up, group theory. I mean, it's like very natural. That's right, yeah. I mean, it's active.

1:02:07Lisha Li:Yeah, I'm actually, when I teach her about addition and subtraction, for example, we'll do it in the context of a general group.

1:02:13Daniel Litt:Oh, very good. Well, at least you give the motivation. I think a lot of, but probably skip that part. Yeah, yeah. And, you know, Maybe this is how math grad students can also focus on training the next much younger generation to use AI in service of actually getting better at math rather than just understanding. Wonderful. Okay. Well, thank you so much, Daniel. Thank you. It was a lot of fun. 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.

1:02:46Lisha Li:Sounds great. Thanks for listening to this episode of the A60Z Podcast. If you liked this episode, be sure to like, comment, subscribe, leave us a rating or review and share it with your friends and family. For more episodes, go to YouTube, Apple Podcasts, and Spotify. Follow us on X and A16Z and subscribe to our Substack at a16z.substack.com. Thanks again for listening and I'll see you in the next episode. As a reminder, the content here is for informational purposes only. Should not be taken as legal business, tax, or investment advice or be used to evaluate any investment or security and is not directed at any investors or potential investors in any A16Z fund.

1:03:28Lisha Li:Please note that A16Z and its affiliates may also maintain investments in the companies discussed in this podcast. For more details, including a link to our investments, please see a16z.com forward slash disclosures.