Dwarkesh Podcast - Grant Sanderson _ AI and the future of math

0:00Today I'm chatting with Grant Sanderson who runs to Blue and Brown and is now working on a new project documenting the progress AI is making in math. And I wanted to talk to you about this because AI has been making the fastest progress in mathematics as of any other field. So whatever is happening here and whatever we were seeing AI progress happen or not happen would tell us about what will happen to the rest of the world as AI gets better and better. So I wanted to start with this question I asked you when I first interviewed you three years ago. And I asked you, Once we have AIs that can get gold in the International Math Olympiad, wouldn't that just be AGI? Wouldn't this just be able to do anything any human can do given how hard these problems are? And you had an answer, which in retrospect turned out to be very wise and correct, which is like, it'll be another benchmark, like all these other benchmarks that are passing. Obviously, AI has gotten better in general way since then, but there won't be some aha moment when this happens. First, I think I'd be curious to get your...

0:54heuristics on why that turned out to be true. And second, I'm curious how long you think this narrowness can continue to be true. So, by the point that AI has solved the millennium price problem, do you think it's still possible that at that point there's lots of tasks that humans are doing that AI still can't automate in the economy? It's an interesting question because it's hard to answer without knowing what the solution looks like ahead of time. I mean, if we take the IMO, that's something where I think the spirit of your question three years ago was in looking at how some of the solutions to these problems really seem to require creativity. And the designers of these problems, they'll try to have them come up with things that you can't train for as easily. I think the dirty secret with the IMO is that you really can train for a lot of them. And so with the whole AI and math project undergoing, I think, as you point out, one of the reasons it's interesting at all is that there's a spiky frontier to AI. Math is just right there in one of the spikes.

1:48But there's kind of a fractal nature to that spikiness because when you zoom into the specific progress within math, you have some things that are a lot easier than others. So if we just think about IMO, which is old news at this point, and it's kind of like two years ago that they're really like doing quite well. They would have gotten a gold in 2024 if or not the following reason. They're very good. They're just like cold solved geometry basically. And the IMO has these four categories of problems, this geometry, number theory, algebra, and combinatorics. So like geometry just solves in like 19 seconds in 2024 because it's kind of a brute force solver. And the dirty secret is for students, there's also sort of a brute force way that you kind of can go at it. Combinatorics is the one that's the wild card of much more like playful, puzzly, seeming problems. And there were two combinatorics problems on that year's test. There's not always. There's four categories, six different problems. So it's kind of a toss up which one is going to have two questions. Had it been more geometry questions, they would have gotten a goal that year.

2:47but it struggles on those combinatorics ones. And, you know, someone who's trying to keep that torch of the last holdout of like math for humanity might say, well, you know, those are the ones that require the more creativity. Even then though, I think the spirit of your question, like if they're solving, you know, a Millennium Prize problem, does that also service a lot of white collar work? It suggests that whatever the rate limiter is between where we are now and that is the same as the rate limiter for making things better at white collar work.

3:17We could maybe paint a couple different ways that we focus on, I don't know, Riemann hypothesis. What would it look like to solve that? One possibility would be these things are extremely good at a specific domain of knowledge and just knowing it very deeply and then knowing another domain and knowing another domain. And you've pointed this out. It's bizarre to have something with this superhuman breath that knows all the fields so well that's not just finding those lightning bolts that connect them.

3:45I think we're starting to see sparks of that, of like actually finding connection between the things that it's an exponent. I'm sure we'll talk about it. If the nature of the solution to the Riemann hypothesis was something like that, that feels pretty distinct to me then what's necessary to get good at white color work. And there's a reason to believe actually that that might be the nature of the solution. I don't know if you know the story of like Hugh Montgomery and Freeman Dyson at the IAS. This is a side tangent, but it's just kind of a fun story on how I don't know if it was over lunch or something like that. Basically, you have this number theorist who is pointing out just trying to understand the statistical correlation between pairs of zeros of the Riemann's data function. So the Riemann hypothesis is all about do all these zeros sit on a straight line. And he's finding this quantitative question you could ask about. And he writes down a formula that looks like 1 over sine squared or something like that. Freeman Dyson, a physicist, is like.

4:37I know that expression. That expression comes up in studying the eigenvalues for random Hermitian matrices, which was something that comes up in studying the energy levels of like a nucleus. And the idea that the statistics of those two seemingly different things were the same sort of prompted a potential exploration on, hey, are there aspects of random matrix theory that might be relevant to like Riemann's data function. And I think it's a little bit of an open question like is there food to be had there? But that kind of bridging together from two different fields, like if it turned out that the solution to the Riemann hypothesis was exploring an idea like that even further.

5:14that has this character of kind of how you expect LLMs to be good at math. It's like they're an expert at the quantum physics. They're an expert at the analytic number theory. They should be able to see that similarity in a way that doesn't require like Montgomery and Dyson to be having lunch and like happening to talk about that. That's totally different from white color work, right? In terms of like the extent to which you maybe have a hard time using an AI as an editor. It's not because they know everything and you just need them to find that lightning bolt in between. Different possibility would be

5:43What's the right analogy? Maybe if we think of Fermat's last theorem between the moment of Fermat phrasing the question and then what the solution itself looks like, where ultimately the solution involves such heavy machinery in math, right? So the beauty of that problem is you can phrase it so simply. You ask about x to the n plus y to the n equals z to the n. Do you have integer solutions for this when n is bigger than three?

6:10something you might expect there to be an elementary number theory approach to it. But just as far as we can tell, there's just not. Whereas the actual solution, maybe there is something simpler, but this might be what it has to be.

6:23There's such a complicated set of ideas that build on like centuries of work centered around elliptic curves. And then this other like mountain of ideas centered around these things called modular forms. And like both of those mountains have to be built before you can ask the right question that connects it. So if the solution to the Riemann hypothesis involved building a new mountain, like that's a kind of skill like the ability to like come up with the right new ideas that feels sufficiently different from like the character of how they're intelligent right now that It's not like that's what you need from your hired video editor per se, but that like if it's capable of building mountains that are, you know, the correct new theory that like crystallizes how we should be thinking about a subject. That's just such a level of intelligence that then it starts to feel like it would be surprising if that didn't permeate into other aspects of the economy besides like just the mountain building for math itself. Yeah. Or at the very least even if it couldn't like literally do every single thing white collar humans can do. Yeah. It would just have transformative effects.

7:21in the way that getting gold in the IMO did not have transformative effects on the world. First of all, I do want to point out that I'm totally moving the goalpost here, because when I interviewed Dario about two, three years ago, I asked this question about why haven't they been able to use their vast knowledge to connect ideas together and come up with a new discovery that way. That seems like the kind of thing even if a moderately intelligent person knew this much information, they'd be able to come up with a medical diagnosis from the fact that this drug causes migraines and this other thing, whatever does this, and maybe it's the same drug that can cure both things.

7:51And yeah, I don't know, from an outsider's perspective, mathematics seems clearly like a field where finding this counter example to the unit distance problem conjecture was like an example of this kind of thing. It's a total goal-plus moving. But then we can ask, okay, what is the next benchmark? Now that AI's can do this thing that we should have thought they should be able to do, what is the next thing that would be quite impressive? And there's a couple of candidate ideas here. So one could be coming up with interesting problems in the first place, and the other is, coming up with new kinds of objects or conceptualizations that create or unify fields. On the first one, right now we're just training these models to like we have these millennium price problems because, you know, the mathematicians have noted like Riemann came up with this idea of this like Riemann's data function and because he thought that it would have some connection with like the density of prime numbers or if the zeros on this function would have some connection to prime numbers. And so like figuring out that there's

8:47Why do we think this is an interesting thing to study in the first place? Why are we building this object and trying to answer questions about it and answer this particular question about it? Seems like the kind of thing that would be the next benchmark. I mean, you highlight two pretty good examples there.

9:02For anyone curious about the unit distance conjecture, there's this really nice video by a math channel called Polylog where they talk about it and one of the people in that. All of these discussions it causes people to reflect on the process of doing math. They're like, this thing can do this impressive stuff. What does that mean for us? And he highlights this quote.

9:21how good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions. That's more or less exactly your framing here. We need the conjecture generator, and then the definition generator. That's the premium tier of mathematician.

9:39understand how exactly you would make that a benchmark in the sense that usually when I think of the word benchmark I'm thinking something that you have like it's a goal post the ball is through the goal or it's not like you can clearly say like yes this is done partly to be able to do things like our LVR but also partly just to be able to like know that you haven't moved the goal post and answering you know openly I can have their headline on it disproving the unit distance conjecture because it's a clear distinct it's like it did it right or as imagine trying to have a headline on like Even D5.4 came up with a really good conjecture. We promise everyone thinks it's a good conjecture. It just doesn't land the same way. But maybe that doesn't negate the fact that that's the right thing to be thinking about. So I would be surprised if it ever took the form of looking like a benchmark and we have a score saying that it's past this benchmark because we can quantify how good a conjecture it is. But probably the nature of what it would take is that you would feel a.

10:34a tone shift in conversations with mathematicians about the way that it's useful to work with. And this series that you referenced that is not at all produced yet and probably won't be for a couple months takes the form of us interviewing a lot of mathematicians.

10:48And what's interesting is we started doing this over a year ago. And it's fun to see a little bit of a tone shift in the way that they talk about AI between mid 2025 and where we are now in 2026. In the real world, that's a very short amount of time. In the AI world, that's eons. And we're able to see over those eons this tone shift. I think the way that you'd measure.

11:09conjecture generating ability is going to be more subjective on that tone shift where it'll be mathematicians saying they're not just using it to solve their problems, but as they step back and decide what their research field should even be, that a conversation with such and such model was genuinely helpful for that. I don't think it's likely that you'd see it in the form of a headline saying that this was yet another benchmark knocked down. Right. And so it's very interesting. The kinds of things you can't make benchmarks for?

11:36are also the kinds of things, at least in the current paradigm, you can't easily train for, right? Because there's really no fundamental difference between a benchmark and a training environment. I think it's very easy to come up with some dichotomy of like, here's a deep reason why AI can't do a certain thing and then it turns out, well, you're just thinking about it the wrong way and actually I can do it pretty soon thereafter. But I'm gonna come up with- You're gonna come up with a couple anyway? And I think that this will probably- it'll probably turn out that there's ways in which we can train AIs to do these kinds of things in the relatively near term, but it seems like it would have to be different from current RLVR training. So the thing I'm curious about and the thing it seems to me that drives a lot of the big progress in mathematics and science generally is like coming up with a new way to think about a problem or the new way to understand the world that then

12:28unifies different fields, spawns entire new fields, solves problems we weren't even thinking we were trying to solve in the first place. Like the reason Einstein was thinking about GR is not because he wanted to explain white light bends or white black holes exist. These are phenomena he didn't even need to know and need it to be explained in the first place. But in mathematics, it often seems, okay, a total outsider, I don't even know the details of what I'm talking about here. From the outside, it seems like there's often ways to Say prove a specific problem that can motivate a new conceptualization One which results in a whole new field a whole new way of thinking which is immensely productive and one which doesn't I think I'm I'm curious to hear you talk about whether Galois coming with group theory and distinguishing his like solution to the

13:12the Quintic having no formula for the roots and Abel coming up with a different proof a few years earlier that didn't come up with group theory. But then if you wanted to do a verification loop on like is group theory a interesting concept that was like was something useful done here? Why is this proof better? Potentially that verification loop is 100 years long. Yeah, it involves.

13:30the cryptography coming around and physics making progress and the ideas in group theory being relevant and understanding like symmetries in physics and all those kinds of things. There's like a hundred year verification loop of why is this a productive concept in the first place? Yeah, boy. Yeah, you struck a nerve because I had this like project about Galois I was going to do in 2022 that I put on the shelf, but I spent like a year of my life like thinking a lot about what he did.

13:53There's a risk of me accidentally talking too long on the specifics. Hold me back on. It's a perfect example for your case because describing why it was a valuable insight.

14:06does not come from immediate utility. And so certainly if you're thinking about RLVR environments, it's like, okay, this is going to be really hard to do. But it's interesting to note how even with like human verifiers at the time, like it took a really long time to recognize it as being useful. Like I think Einstein with GR, people sort of felt, you can like feel this feels like a good theory right away. Like what makes the Galois theory such an interesting example is you have Literally, this 100-year segment of an idea that flows through many different people's heads before it settles into something that the math community agrees is good. So to back up a little bit, I mean, do you want the background on the problem at all?

14:48We all learn about the quadratic formula in school. I thought you were going to say, we all learn about group theory in school. We all know about group theory, quadratic formula. So this was known in some sense, like Greeks could solve quadratics, but they didn't really write things in algebra. And so it's really more like the Arabs that wrote down that formula. There's this delightful story around some like dueling Italian mathematicians, not real duels, just like intellectual challenges who like secretively found a formula for the cubic. And then very shortly thereafter found a formula for degree four polynomials. So natural open question for like mathematicians is can you find a formula that solves degree five equations? Now the degree four, it's monsters. It's like, it would be wild to write it down. You usually don't really write it down in full. You break it up as like a procedural thing.

15:39So you might believe these things have this exponentially increasing complexity. So many hundreds of years, nobody's like really answering that question. Usually we say, Abel was the first to prove it. He was this young precocious Norwegian mathematician, and he showed it's simply impossible. It's not that you can find a Quintic formula, he thought he found one, but he showed it's impossible. I think the real credit though, like you have to back up a little bit and talk about Lagrange, where Lagrange found the right kind of question to ask about this.

16:09I can go into the details if you want, but I'll give it a very high level. He was studying the question and he recognized being able to solve these polynomials is actually very related to understanding the way that certain algebraic expressions are symmetric, more or less so. If I write down A plus B plus C plus D, just adding four variables, if I permute those, it doesn't change the value of the expression. Whereas if I write A plus B multiplied by C plus D, some of the permutations don't change it, but some of them do.

16:37And he had this really, really nice insight about how if you can find expressions like this that have like four free variables, but all the permutations take on three distinct values that had this unexpected relationship with being able to reduce degree four into degree three. So he started approaching the like can we find a Quintic polynomial by saying I wonder if I can extend that. And to extend that method, you would have to have an expression that has five free variables such that as you permute them over all the five factorial permutations, it takes on only four values or fewer.

17:07So you could put that in a puzzle book. You could put that in a brain teaser that a 12-year-old couldn't engage with. And it's not too hard to find yourself feeling like that's an impossible task. And so Lagrange is sitting here saying, hmm, here is a strategy that I'm trying to solve this problem. Can I find a Quintic polynomial? This strategy doesn't. It seems like it might be impossible, at least from this strategy. But that was the first time in history that people had the instinct that some kind of question about symmetry was the right way to be studying these polynomials.

17:36In his mind, it was just a way. It had yet to be discovered that actually there's a tighter connection. And also maybe rather than searching for the formula, we should be asking the opposite question. Can you prove that it's impossible? So he sort of planted that seed.

17:48Like around 50 years later, Abul definitely read Lagrange and was influenced by it. Galois, we know that he loved Lagrange when he was like falling in love with math. And so it's very hard to imagine that like these two young geniuses, the fact that they both come up with like pretty similar insights around that problem, it's not like born from Lagrange. But to your question on like, are you able to verify that this was a good idea?

18:10There wasn't any result that Lagrange came to. He solved the problem, and therefore, we know that that was the right question to ask. He asked it. There's some intrinsically interesting thing. It also wasn't very important for math at the time. Most people were more interested in the applications to physics. This is almost in that side, almost recreational hobbyist type thing. Abel, he started working on Quintic stuff, but then he was...

18:34advised to spend more of his effort studying elliptic functions. And so more of his work was on that before he died young. He died at 26 from tuberculosis. And then Gawa, he pushed both of those ideas like in the right direction where he really understood the nature of abstraction. And so he had this really nice piece that he wrote.

18:55While he was in prison, actually, he was like, we could talk all about his life story. It's pretty wild. But he's like this teenager. He's in prison. He had tried to submit his math papers and they had been rejected. So again, it's like verifiable reward. The like verifier function that is the academy at that time is rejecting what he wrote. Because frankly, it was not very coherent. Like it wasn't a complete proof. He wasn't giving like a clear thought of like what the theory actually was. He was just like a young.

19:18fledgling mathematician getting his bearing. So it's like the verified reward there is like, eh, no good. But he has some instinct that there's something there. So he's writing this diatribe on like the nature of like math being something which is, it undergoes these like shifts over time. And he talks about like the advent of just algebra itself and going from just thinking in terms of numbers to like having a certain fluency just with like pure algebraic expressions where you're not tied to interpreting those expressions. And here's this instinct that like there is another layer of abstraction that seems like what we should be doing where rather than thinking about the formulas themselves, thinking about like what symmetries underlie those formulas. But it was still a pretty like ill-defined theory. So if you're trying to say, okay, is the verified reward that like he has solved a problem that other people haven't? It's like, well, Abel proved that quintics are unsolvable. And he said, what was Galois doing? Well,

20:08In principle, the thing that Gaoua theory will let you do is take a specific polynomial and it gives you the rules to say, does that specific polynomial have roots that you could write down? For example, x to the fifth minus one, you know that a solution is one, or x to the fifth minus two, you can write down fifth root of two. So it's not that every quintic polynomial you can't write down the solution, but could you find a specific one where you prove you can't write the solution using radicals?

20:31He also didn't even solve that exactly. Like he has a much more abstract. He didn't show for a specific example that he couldn't. So even describing like what problem did he solve is very tricky. So then he dies. It's this very like romantic story of he has this duel. We can get more into it. There's a lot of myth around like supposedly he writes up all his ideas the night before the duel. Really he tried to get them published. Working in the Quintic doesn't seem to be good for your health. It's very bad. Yeah, yeah, yeah. If you're a young genius, don't work on the Quintic. And so he he asks his brother and his close friend, like, get these notes to Gauss, get these notes to, like, the important mathematicians of the day, because I think there's something here. Even then, it didn't really take, like, so his brother and his friend, like, tried to get them out. It wasn't another 20 years until Louisville, like, sees these notes, sees that maybe there's something in them and tries to, like, clean it up and understand, like, what was Gauss while getting it. And then even then, it was another 20 years or so until Jordan actually, like, puts together a...

21:25something like a modern treatment of group theory that they attributed to Gaua. You could easily imagine history turning differently where like these ideas were kind of coming about from other points in math and like Gaua could have been forgotten in history if he was less like flurid character. But between the time of Lagrange, like having this inkling of maybe symmetries of roots is the right way to go to where it all looks like modern group theory. Like you've got this long span. A lot of the time it's like not even passing the like verified reward of human reviewers because it like gets on someone's desk. They say, I don't really know if there's anything here. It gets on someone's desk, they don't. You have to have this like one person sort of recognizes it. And then even then it's not really solving practical problems at that point. Like you point out cryptography and physics and things like that. You have to get into the 20th century before you have like Gell-Mann thinking.

22:13Hmm, maybe understanding the nature of like how certain groups like breakdown has this relationship with what particles are made out of and like he anticipates quarks based on a purely group theoretic question and like that's one of the more interesting applications of group theory is that like to even predict the existence of quarks is a group theoretic question. That's so long after Lagrange before you have anything like that. And so you have to ask like what is the way of measuring progress that's not based on solving a problem, right? And that's somehow capturing what is the instinct that's inside Galois's mind when he says, I think there's something here? What's the instinct that's inside Lagrange's mind when he says like, I think this is the right way to think about it? What's the instinct inside Louisville's mind when he says, hmm, these like scattered notes from this like long dead youngster like might have something to them? It's so hard to put a finger on that, but I mean a different like series of videos I'm making right now is about like...

23:10that the whole compression is intelligence idea. And even though this isn't really the angle I'm taking, there is something to the idea that the smaller expression that's more predictive feels more intelligent. And so I wondered the extent to which you can give some kind of verifiable reward around not just like did you solve it or what is it solving, but around the smallness of the concepts required to do it. I mean, going back to...

23:35Riemann hypothesis solutions, what would that look like if an AI solves it? I think a third way that it could happen is it just straight up works harder, right? In the same way that you could maybe have an elementary proof of Fermat's last theorem that's just like spelled out over like thousands of pages that would be incoherent. But like the cleaner way to view it is with elliptic curves and all that. Maybe there's some like thousand page proof of Riemann hypothesis that's like no one's really getting anything out of it. And what you actually want is like, what are the succinct like compressed versions of those ideas?

24:02would then lend themselves to human understanding. Like I don't know, comagorab complexity, like maybe you throw that into your attempt to quantify what you mean by elegance. But I don't think it's easy, but I do think it's something you would have to do in order to reward the Galois like instinct rather than just rewarding have you solved a problem. It's very hard to come up with the heuristic for science. But it's clear like humans have been doing this somehow and like Obviously, you guys will do it at some point. Well, it's relevant also not just in terms of verified reward, but presumably the end goal is understanding, human understanding. And so even if you do have some thousand page proof of some math thing or some grand new physical theory, the goal is understanding. Maybe if the goal is predictiveness, you can just have automated engineers go off and build rocket ships or something. We have no idea how these work, but we can get between stars. But there's going to be a lot of people who want to understand. You're still going to want

25:00Whatever the like concision function is that like distills down here's this complicated way of thinking into like the right one like the equivalent of the universal law of gravitation for Newton Yeah, like you would still want to train AI's to be able to do that and like find the the compressed representation I grew up in India till I was eight and so in addition to English I also speak Gujarati and since Google just released Gemini 3.5 live translate I thought it'd be fun to put it to the test in this withdrawal 3.5 live translate 3.5 live translate automatically detects more than 70 different languages and translates them in almost real time into the target language. Live translate your original speed and format while speaking, just like it's doing right now. I visited China back in 2024 and I remember thinking at the time that this trip would have been so much more productive if I could have been able to live translate the conversations I'm having with researchers and random people I meet on the street.

25:56Now we have that technology. So if you're building an app that needs live translation, you should 100% check out Gemini 3.5 live translate. It's available now via the Gemini Live API and in AI Studio. Go to ai.studio.live to get started. So people have this worry about mathematics in particular that, you know, AI's will prove their remand hypothesis and our understanding of mathematics won't be any better for it. I have a couple of questions about this. The first one is whether this is like I think you should expect like isn't the reason humans come up with general natural objects and sub goals and whatever when we're working on a big problem is that it's just like useful when you're trying to work on the complicated important problem. And so we can just think about like theoretically, would this even be a simpler way to solve the rebound hypothesis as opposed to just coming up with the natural abstractions that are relevant to thinking about the problem?

26:50And then two, empirically, is this what we observe when AIs do and make progress on problems today? When the AI came up with that counter example to the unit distance problem conjecture. You can just read, it's chain of thought, and it seems, it's not understandable to me, because I don't know anything about mathematics, but it seems to other mathematicians, it was understandable, and it made use of known concepts of mathematics and proved relationships between them in all the natural language. And as a result, accelerated our understanding of the connection between this object and this conjecture. So is this even, like, imperatively, is this a thing we should be worried about? I think it depends on the nature of, yeah. Like, again, if we sort of break down, like, the three possible ways of, like, solving the Riemann hypothesis, that one, and the other, like, big one from this year was, like, a certain Airdish problem numbered, like, 1196, but it's a, it, about these things called primitive sets. But basically, it had that character of...

27:43bringing an idea from a seemingly different field. As soon as you just present the basic idea to a mathematician, you say like, what if we like use this, like try the Markov chain process where we show that this thing is one from the bottom up probabilistically rather than the top down, and like use the Von Mangold function. If you like say that to someone in the know, they'd be like, they'd kind of know how to run with it. So this very like small idea that has the form of expertise in one field, expertise in another, draw a little lightning bolt between them. Like those are Those are going to be very human-parciple, right? Because all you have to do is just like show the start and end point of what those connections are. If the character of it is a mountain building, you do have to put in a lot more time to like understand that new mountain that was built because it's like a new thread that's not just like lineable between them. And if the nature of the progress was just like raw hustle, right? It's just like this just super long thing. There's no new theories, but it's just like long, long, long chain of reasoning answer.

28:36then you would have that word like, okay, there's this whole digestion process. So I don't think there's one clear answer. I think it depends on what the solution there would look like. And on the mountain building side, I would actually be really interesting to see like, is it by default, a very human understandable like the way that we see new theories from like great mathematicians? Or is it like an alien different kind of mountain being built where we even have to like reprocess the kinds of abstractions that we engage with? Well, the closest example here would be like the, attempted solution of the ABC conjecture that was, we maybe shouldn't get into that one, but it's probably not a correct solution, but basically it's like whole new way of thinking that this otherwise reputable mathematician in Japan had like come up with. And it just took mathematicians like a long, long time to even parse what he was saying, but it had the feeling of just like an alien bit of mathematics that's theory building. It's not just like long chain of reasoning.

29:33It's like, you call it like inter-universal geometry or something. And so the fear that you would have is that like, yeah, it like does that. The biggest fear would be that it does that. And then much like the ABC conjecture, like people work for years to go up the mountain and they're like, this just isn't right, right? And like, if it turns out to be wrong, but it like really looked right. But even if it was right, there's just a lot of effort to like hike up a new mountain. Yeah. If we end up in that situation, David Bessis had a really great blog post called, the fall of the theorem economy. We're just talking about this, you know, historically, as you were saying, mathematics is coming in with these definitions and problems and it's about proving theorems about them. And that really the theorem proving stuff is what gets all the credit, but it's like really a parasite on the definition stuff. And historically it's not even a problem in terms of credit apportionment, because if you come up with the definition, you're probably gonna be the guy who comes up with the theorem. But now we're in a situation where

30:29If the valuable work is the coming up with the insight, and then AI just automates the latter part. So, okay, imagine a scenario where we have AI comes up with like the obel like direct arguments about a bunch of important conjectures in the world, and then we just have these proofs. And now it's up to humans or to future AI's to then consolidate.

30:53I mean, I'm sure if you had access, again, having no object level understanding of this argument whatsoever, I'm sure if you had access to it, it would make it easier for you to then think about like, well, what is going on here? Is there some deeper way in which we can understand why this proof works that would make it easier to come up with the ideas behind group theory? Yeah. I think it would be hugely helpful, right? Because I mean, so much of trying to discover new math is mostly being wrong, right? You're trying to solve a problem. It doesn't feel like...

31:23Constantly taking the correct step up the mountain like mostly it feels like a random drunken walk where you're like doing a thing and then oh you're wrong and like constantly discovering So if at the very least you know that trying to digest what you know is ultimately leading to like a correct solution like that feels like progress simply because it's providing a sense of knowing that it leads to a solution. And there's plenty of instances in the recent history of math where it feels like the reach has sort of exceeded the grasp where there's things that are proven long before they're understood. And one of my favorite openings to a paper, it's not even like a research paper, it's more like an expository one is from this petition named Timothy Chao who was trying to understand a concept called forcing. And so there's this.

32:07problem called the continuum hypothesis that more or less asks, like, you have a size of infinity for the natural numbers, you have a size of infinity for the real numbers, is there something in between? And the answer is both yes and no. It depends on your axioms. Like, it's sort of outside the scope of our usual axiom systems. Which is an interesting answer. But the method to describe it is just really, really hard to understand. It's the thing called forcing. And in the beginning of this paper, he writes like, Everyone knows the idea of an unsolved research problem. I want to propose the idea of an unsolved expository problem. We're like, sure, we've proven it, but we don't really know why it's true. Instead, he proposes a partial solution to that expository problem. You can imagine why I loved that framing, because this is my whole life. It's like, I don't do research math. It's just wholly about what's the most clear way to understand this, even if it's proven. There is a difference between proof and explanation, and so on that side, I think that...

33:01you are basically getting to the importance of that distinction. Yeah, and that will be the main incentive for, or the incentive would have to change in not just mathematics but in other areas of science from proving things about the worlds to consolidating proofs into problems or higher level insights. But we have a discussion earlier at lunch about like a recent talk you were giving about you know, design and how it helps us understand things. And then in the limit, is there really a difference between the conceptualization for an idea and the idea itself? So, you know, if you think about special relativity and like space-time diagrams and Minskowski space-time, is it like, yeah, this is like a way in which we illustrate this idea of like why there's length contraction and time dilation. But is that like, like that is the reality. So,

33:55The exposition does seem to be like the explanation in some sense here. Yeah, I mean there's a couple of interesting things there. One is it seems like there's a really strong correlation between the people who come up with genuinely novel insights and also are actually quite clear in their communication of it. Like you might imagine given that the experience of a university student is often that the expert they're teaching them is not necessarily the best explainer of that topic because they are so spoiled by their expertise. But what seems at least in some cases to be the case is how The people who are really coming up with something quite novel, so you've got like Einstein or like Claude Shannon or something, you read their papers, they're really lucid papers, right? It doesn't feel like, oh, this is just for the experts and you have to chop through it with a machete to get, they're like very good expositors. Like Feynman has this characteristic too, like very good expositor. And so maybe the same part of the brain that...

34:47comes up with the correct new way of thinking about it at a research level, also has this knack for like good explanation. And I think this is pertinent to the AI one where I kind of used to think that AIs will become these automated theorem provers, but like the role of the mathematicians is going to shift towards like my job, like explain these things. I kind of suspect that actually they'll also be like quite good at doing that.

35:10and probably just like better than most humans are at like doing the explanation half and distilling half and that's actually not what's left for the mathematicians is like digesting and explaining what was going on probably the nature of how these things are going. I could have envisioned it we can talk about like ways this might not be it but like probably the same thing that is coming up with like the really good new idea that solves some new problem is just also good at explaining it. That's my new like that's a way my I think beliefs have changed. What's the last thing you think you'll be doing?

35:39Or they're like, both you and then also with the mathematical community, the human mathematical community, what are we doing? I will probably be doing something like what I am until I die. Even, so like, even... I have the doom for the right, maybe that'll be the same. Exactly. It'll be for the same reason. Yeah, yeah. You know, it's, you're like, build a man of fire and he's warm for one night, but set a man on fire and he's warm for the rest of his life. So that's where I am with AI. No, I, because...

36:09some of the function of an explainer or a teacher is to add clarity to a thing that someone's curious about. That's one thing. But some of it is a little bit more relational and a little bit more like providing motivation, providing a sense of curation. One interesting take that I've heard about what mathematicians will end up being is actually more analogous to art museum curators than anything else where...

36:34The A is all the things, so the art exists, right? They even know how to explain it really well, you know, all there. But like, you still... You still want someone to help you navigate in this nearly infinite space of what ideas are worth engaging with, like someone kind of doing that. And that one, even if AIs were in some sense better at that, I think we would always still prefer a human that we had a relationship with because the way that we get motivated to be interested in things is a social phenomenon. If you have some specific technology you're trying to build, that might be different. You need to know that. But I think...

37:05like the people listening to this podcast, they sort of trust your curation on like what's an interesting topic in the first place. It's not that they're landing on here because whatever your next topic is, that's like what they in a prior sense wanted to understand. They're trusting you as a curator. So my role and arguably that of like other mathematicians might actually just shift subtly into that curation direction of what ideas are worth displaying. And that's a lot of my job right now, even now. It's basically like, I think people think a lot of the time for a video goes into the visuals.

37:35Sure, a little bit. It is not like immediate, but like actually a lot of it is just deciding what's worth saying in the first place or what's worth putting there. And because that is, that's just I want to engage with that. And I think I have a trust with certain people and they are curious what I would choose to perform, even if they are better than that, in the same way that like human musicians are always going to have a role because of that like social function of the story behind them, even if they're like.

38:01objective quality of the MP3 file coming out is like better from some model. That's kind of what I see happening to my job. Yeah. I want to go back to this question of earlier, we were sort of just as AI has crossed this threshold, this important benchmark of being able to connect existing ideas to come up with a new discovery.

38:21prove or disprove something. Just as it's cross the threshold, we're like, okay, but what's the next thing? I wanna just... There's a lot more to do on that one by the way, just because a couple lightning bolts have been... I think there's this flourishing future over the next couple years of really connecting. And so in the limit, you could even say, I don't know if this is accurate to say it, but potentially, a lot of the maybe the biggest breakthroughs look like this at some level. It's just...

38:48General relativity. He's just connecting together like Riemannian geometry and special relativity, right? And so as AI's keep getting better and better at this connection thing, maybe a lot of big breakthroughs are not really of a different qualitative nature. I don't know if you have a take on that. Well, I mean, a lot of the conversation focus has been on problem solving and that nature of math, you know, like taking off Erdisch problems or something. I would say it's not even a majority of mathematicians who would maybe characterize their work as like...

39:16really targeting the next problem to TicTown. Are you familiar with like the Langlands program? No. Ah, okay. So this is like, it's not even a field of math, so much it is like a research ethos where Thermal's last theorem is one inkling of this on you had like these two different seemingly disparate things and a connection between them like led to a solution. So Langlands was a mathematician. He has this like famous letter now essentially spelling out how it seems likely that there's a lot more connections like that.

39:46and even got a little bit more specific about the nature of the connections, such that you might imagine this large map and you've got this valley over here and this mountain over here and this set of planes over there. And there's a lot of mathematicians who would characterize their work as being part of trying to understand the threads on this map. And the progress there, it's not even like, here's this one specific problem that we know will be solved by that connection. It's more that there's been enough time and time again cases where...

40:12big problems were knocked down by finding connections that it's almost preemptively finding the connections. It's actually very interesting. Anytime you run into a mathematician, ask them whether the character of their work is more akin to Langlin's program or if it's more akin to targeting one particular problem. You get a certain bifurcated split there. But the possibility of AIs being supercharged connectors feels like it might be an amplifying tool in that pursuit, it's hard to measure though, right? Because this cuts to what we were saying earlier. How do you assign a score to say, yes, you've done it? If it's knocking down a problem, you have a clear way of saying, yes, you've done it. You can write the headline. You can have your PR move as the AI company to say we did it. Whereas if it feels like that was the right connection drawn, you can write theorems around it. And this is the nature of what the papers in that field look like.

41:12I think it will require a lot more like human in the loop to basically like say what was it like the kind of connection that we're going for but that's my guess on what most of the useful progress from these models will look like like in the next five years is just really filling in that landscape of like connections that you can draw if you're an expert in multiple fields like you pointed out it's kind of surprising we haven't already had this right and what I'd be curious like I would be curious to know at a technical level what causes the unlock there because on the one hand, you can kind of paint an explanation in your head for why you could be an expert in all of these things and not be drawing those connections, which is when the thing is reasoning, like the method of reasoning is this autoregressive chain of thought phenomenon. Autoregression is actually like a really, really weird way to produce stuff, I think, if you think about it. Like, you're an intelligent person. Imagine I've walked you in a box, right?

42:10And then the only way that you have of interacting with the world is that you receive a slip of paper and then someone says, can you like predict what will come next, right? And then you predict what will come next and then your memories wiped, right? And then you get like another slip of paper and you go, imagine that was done a whole bunch and then what comes out on the other end, they're like, look at this essay that you wrote. You might look at that and be like, this is awful. That's not the essay that I would have written, right? Because like the process of like repeatedly like predicting something is just pretty different from how you would think as a writer to like compose it and think it through and everything.

42:39And in particular, what would probably happen is you're sort of a slave to your context where you might be answering some question about some particular field. And so you draw on all the context around that and you're going there. The connection that actually is where all the substance is going to come from is by its nature a very unlikely one. And you can do all the RL that you want to try to get better in some way. But what's the thing that's specifically up waiting and incentivizing making these unlikely connections when the vast majority of them aren't the predictable, you know, next token that would come in there. And so it's like, it might be the case that you just have this intelligence that's sort of locked in there inside that box, but it's just a weird way of interacting with it. So the thing I'm curious about is like, do you ever get any fruit by just like

43:25questioning the premise of how tokens are generated like every now and then in some way, right? And I don't think it would be as simple as you like manipulate the temperature or something like that. But like are there any things that you can do that take like the existing level of intelligence but like find the right ways of sparking those connections that like unlocks these sorts of things that we've seen or do you need just a little bit more intelligence such that.

43:50at the level of prediction, it's kind of predicting that it should be making that lightning bolt to another field. I think it's more productive to reason instead of architecture or even loss function to reason about data. Like, I don't know. We have diffusion models that do text and they're like, the kinds of things that produce are not of a wholly different character. They're just not being explored as much. I think the more relevant thing is what is the data on which whatever architecture whatever loss function you have is incentivizing you to produce and It does seem like they're getting better at like okay forget about math. I mean we did have this a couple of examples of this kind of thing But if you just look at why are you getting better at being autonomous Asians it just I don't know they have like

44:39they're in an environment where autoregressively producing the step that says, let's step back and do a search over the whole code base. Right. And then let's step back and like assess my mistake is like the thing that works. I assume what happened in the case of progress in science or maybe in math is you have frontier math like problems which require like mathematicians to specifically design them.

45:02because they require connecting together two different fields and there's all I'm guessing there's all kinds of clever like partially synthetic ways in which to make harder and harder problems like that that require these kinds of connections for example by like eliminating assumptions and still requiring the I to continue to get to the answer and then like it just doesn't really end up mattering with the loss function is it just like it's really about can you come up with an environment which incentivizes this ability yeah it feels like you should be able to I certainly can't speak to the correct ways of doing that that like unlock all this but it would just be pretty surprising. Like don't you think it would be kind of surprising if over the next three years there's not just like a lot more of those lightning bolts? So this I think is an important thing to think about, which is we often think about how smart a single system is. And we don't think about AIs having advantages that are more the result of other facts about them. So in this context,

45:59The key fact about them is that we can just paralyze and arbitrarily scale them so that whatever level of capability they have, it's not just like one idiosyncratic genius in the history of mathematics who makes a few connections and then dies in a duel. It's just universally applying that waterline across all problems that are accessible at the level of capability.

46:20I feel like this is among the many advantages that digital minds inherently have that we don't think enough about, the fact that you can, the other ones being the fact that you can like, they can merge all their knowledge together, at least there will be techniques that allow this to happen, that you can like, that you can spawn off copies with identical levels of knowledge. But yeah, I feel like this parallelization is like quite an important property. And I'd be curious about your predictions of even if they're not as smart as your mathematicians, the fact that they are just billions of, because for PR reasons that the AI companies are jumping billions and billions of dollars at this, would have a quantity as a quality all of its own. That seems in the right direction. I think, I mean, if we take that conversation between Montgomery and Dyson at the IAS that suggests some connection between Riemann hypothesis or Riemann's data function zeros and random matrices, that feels like the kind of thing that you could try to

47:15automate and that you have agents representing expertise in all these and basically having, okay, we all know that an institute is smarter than an individual and that the reason for having people all in the same geographic location is because you want those serendipitous conversations to happen. What does it look like to sort of...

47:34engineer those between agents. I mean, you can sort of point out like you can sort of pool all your knowledge. I actually wonder if one of the advantages is that you can do the opposite of that, where you have sometimes when an AI is failing, it's because it sort of gets into a bad chain of thought and it's really hard to get it out of it, right? So you're like, I'll just like start again. Same deal with humans, right? Like sometimes you like start thinking about it in a certain way. And actually what's required is to just like back up maybe sometimes the form of that.

48:02their stories about people trying to prove something for a long time. And then at some point they say, hang on a second, what if I tried to prove that it's impossible? They're like prove the opposite. And that unwinding your own context and going at it with a fresh mind, you could imagine systematizing that or having multiple different agents deliberately given different pieces of context and try to like comparing trust there. We don't have the same level of manipulation on our own context.

48:26in this AI and math series, the first episode will be about when they solve the IMO. And I want to focus on one specific IMO problem that they failed on, which is one that a lot of very smart students failed on. Terry Tao also failed on it. And the nature of it is basically that people were very mad at the problem because they called it a troll problem. I almost don't want to spoil it because I want to construct the episode around leading someone in with...

48:54without knowing that it turns out to have a simple solution because you like can really empathize with what it's like to be like a student solving this. Basically, there's a really elegant way of going down what you really feel like is going to be the solution based on the context of being the international math Olympiad problem positioned as it is. The like character of the solution is like really enticing but it's kind of hard to prove that it's the best. The reason is that it's not. There's like this almost brain dead solution that is the best. And so the like relevance of that to the whole AI story is like

49:23For a human, what's required to answer that question is to escape your context, escape the context that you're in the IMO, escape the context of the way you've been trained to solve these contest math problems. And if you just approached it like a brain teaser that I throw someone off the street, they'd probably answer it well. And you sort of want the same sometimes for human research in other contexts where sometimes just being able to say, refresh your thinking, come at it completely differently. So of all the advantages that digital minds have, that might actually be one of them. A little bit more of a systematic, what does it look like to refresh your thinking, try answering two separate questions, like spin off two agents, one who's trying to prove it, one who's trying to disprove it, one who tries it this way, one who tries it, and they deliberately have different contexts. I would be curious to see if we're having this conversation three years from now, how many of the significant results that make headlines have that character of basically,

50:19erasing the context previously like trying a bunch of different things as opposed to merging the results of like a bunch of different. It is incredibly interesting because a common concern people have about AIs is this entropy collapse where they all think the same way because they're trained in similar ways. This is why they're bad at writing. They kind of just like go down the same path and have similar patterns of speaking and so forth. But maybe actually the key advantage AIs have is that you can systematically, it sounded like one of the reasons the unit distance problem conjecture took so long to be disproven was because people assumed the conjecture was actually true. So mostly they were trying to figure out ways in which to prove it. And so maybe one of the key advantages they will have is actually to increase the entropy by systematically trying out both the negation.

51:09and trying to prove the positive of any given statement of like or being able to like systematically give different agents different biases. That's a good point. Like it seems like an important thing in the history of human science is that like Einstein is just really motivated by this bias that like things should look the same in different reference frames and then he had multiple other biases like this but like that is just a very formative in his thinking and you can just like systematically survey a bunch of heuristics and see which ones are being productive at a given problem. Yeah and so you would suggest basically

51:39like systematically increasing entropy at the prompt level, even though you have this like inevitable collapse at the like auto regression level. Yeah, and Einstein would be an interesting example because he's got this bias towards things you'd be able to do. He also has a bias towards God should not play dice. And it's almost like you want to make sure that you don't accidentally have all of your LLMs or Einstein because you might halt on quantum mechanics progress. Which actually goes to show you that there's not a correct heuristic for science. You actually just need multiple independent research programs with their own heuristics.

52:12And that feels like old school software, right? As long as you're able to describe that in some way, you have old school software that amplifies that entropy in some way. And if you're able to put a clear ontology to the distinct ways of thinking that you want to prompt, you explore that full ontology. And then each individual one runs off doing what it is. But I think there's a certain design question there on how exactly do you describe the different approaches?

52:38The easy one is are you trying to prove it or disprove it? The harder one would be to say what are all the tactics that you could take to prove this and make sure that you're like sufficiently applying sufficient breadth to exploring that. I don't think people appreciate the kinds of things that these models can just go handle for you when you equip them with a good harness like cursor.

52:58For example, I started publishing my episodes on Billy Billy for a hopefully burgeoning Chinese audience, but everything I upload there needs the sponsored segments cut out. Normally that would have meant that I would have to ask my editors to go back through all the old episodes, cut out the ads, and re-export everything. But in about just as much time as it would have taken me to send them that Slack message, I can just tell Cursor to do it instead and spare them.

53:20And for research for the podcast, I have a whole repo that I've set up where I've just put every single book and paper that's been relevant to prepping for any of the recent episodes. And I've been able to hodgepodge everything because the cursor harness is just extremely good at helping the model figure out exactly what information to pull, whether that's from my repo or from the web, in order to answer the questions I have while I'm doing research. So whatever you happen to be working on right now, just try pointing cursor at it. Go to cursor.com.com to get started.

53:48Obviously, AI for math is making a lot faster progress than everything else, and people point to verifiability of the domain as the key reason this is happening. I think that's one of the two important reasons, but I don't think, I think people really neglect the other one. And I'm outside the labs, I don't know what's actually going on. There's just a totally naive theory. Okay, a tangential question to why AI is making so much progress in math. Why has it been so slow at computer use?

54:18which is what you, you know, computers is actually very verifiable. It's like, you know, it's my Etsy package coming or like it's my event book, you know, whatever. These are extremely verifiable things to survey. What computer use lacks is grindability. So because websites have like bot detectors and also it takes a tremendous amount of compute to run parallel rollouts, it's very hard to just run like a thousand parallel rollouts at the same checkout flow on Amazon because you'll get like shut down by Andy Jesse, right?

54:47And so you can personally press the red X on door cash button. Exactly. And so you can try to build clothes every single website. This is very labor intensive and slows you down. And the reason, by the way, you need to do so many parallel worlds in order to learn a skill currently with deep learning is that we haven't solved sample efficiency. Sucking supervision to a straw like what he says. Of course, people are working on many different techniques. But fundamentally, there's this big problem in there's this big constraint in the way we train AI. With code also, you can containerize a given level of progress in a repository and then just.

55:25spin out thousands of parallel containers or hundreds of parallel containers and say like try to implement this feature and it's totally deterministic. And because it's deterministic, you can solve the credit assignment problem because you know that whenever cause this rule out to succeed and this one to fail, the diff is the thing that like worked. And this way you solve the credit assignment problem. If you have situations that are starting off at different starting points, this credit assignment problem becomes much harder to solve. But most of the things in the real world are just very hard to containerize in the same way, like coding and math or

55:54exceptions to this rule, but if you're just trying to figure out, how do I build a new business that succeeds? How do I go trade in the markets for a day and make money? The fact that you had to interact with the real world, and things change day after day, means that you can't keep replaying and grinding and farming the simulator. But the math, of course, is the exception, and I feel like this is actually an important driver of progress in this domain and also in coding. It's not just verifiable, it has to be grindable.

56:23The third reason that people point out that AI is making fast progress is they focus a lot on lean and formalization. Again, I have literally no idea what's going on in the labs. I feel like lean just doesn't matter that much for like the current level of progress in AI or like, why is AI able to solve the unit distance problem? Well, sorry, disprove the conjecture by the unit distance problem. They release the chain of thought or at least a rewrite of the chain of thought.

56:48didn't have any lean in it. I think it's just like the the process-based supervision that lean provides where you know each step is correct seems like less relevant to just having this grindable outcome that is verifiable. It's an interesting point like grindability mattering more. I guess I will say on the yeah okay so naively you might think lean provides um something unique for math because you're able to see if it can prove it you have old school software that can tell you yes or no you use that as your VR. I mean what so What would corroborate your point is the idea that like the initial attempts, again, I'll just circle back to IMO. It's like initially DeepMind basically does that. It's like everything in lean and then the next year it's all in natural language. So it's to your point not needed. I think there is a yet to be explored benefit of that formalization domain, which is at the moment you still need, you know, ultimately like a human is reviewing that counter example to the unit distance conjecture to say looks good and

57:47that provides a certain bound on how endlessly explorable things are. If you consider AlphaGo, AlphaZero style stuff, where they're just off in their own universe, just playing a bunch of Go and exploring themselves, just completely going potentially off the rails of what any human needs to look at, but they still have this automated verifiable reward. It's not just that, hey, you can do RL on that. You basically never have to check in and you can just poor compute at them exploring the universe of Go.

58:16What stands to be interesting, like maybe this won't pan out, but I think the jury should still be out on like whether this will yield anything. With Lean, you could imagine having a basically endlessly running program that's constantly trying to extend Mathlib. So Mathlib, it's this GitHub repository that's basically like all of math.

58:37written in code, it's very far from all of math, but they want it to be all of math, written in code that you can ask, like, is this proof correct? It's very labor-intensive to write these proofs. There's like a whole sub-community around it. But you could imagine, what if you just had an AI where you say, simply try to extend MathLib. Maybe it's a fork of it so that it doesn't have, you know, like, trash in it because people, you know, people have certain tastes for what they want to be in there. So you have like your fork of like the pure AI MathLib, and it just goes.

59:05And it just doesn't stop. It doesn't need anybody to check in on it. It could just keep going. It might come up with its own conjectures. It might come up with its own theories and different definitions. Maybe many of them are useless, but it just has this infinite tree that it can grow out. That's a very unique thing that math has that nothing else has where you could press go and then just pour compute at it and look away for 10 years and then come back and say, what do you have? And there's going to be something. And then there's a question, is it useful or not? How do you suss that out?

59:33that's just an interesting thing to be able to do. It would be very surprising if that didn't yield like some sort of interesting mathematical insight from it, right? So I think like that's the real case for, okay there's like two different ways that like Lean is important in this story. That's the first one of them basically is how it's like you could let go, not even check in, and progress will be made. You can do that with Go. I don't think you can do that with natural language math. This is very interesting. Did you see Carpathian's auto research idea?

1:00:03He wrote this basically one Python file that does basic LLM training and then just had a repo where LLM agents would try to make modifications to the file if it sped up the speed run, the modification stays. Eric Jang, who came on to explain how AlphaGo works, did a similar thing when he was trying to build in a very strong Go bot. He had interesting observations about the kinds of, it's really going to just go.

1:00:31running the experiment and going down that path, but it's bad at stopping at dead ends and just doing extremely parallel things. Anyways, this will probably be changed, this will change in the future. It's very interesting to think about what it looks like in the limit. I mean, this is fundamentally like what the human institution of mathematical research is, right? It's just like, this is a library, extended it and interesting in useful ways, and this way you don't have any outcome-based supervision.

1:00:57There's no outcome that you're trying to incentivize, but you have a process. You know the steps are correct. You just don't know if it's going in an interesting direction. But yeah, if you were doing that, you don't want to completely go off the rails and do a random walk through the space of logic. You probably want some supervisor model that's trying to provide heuristics on whether it's useful or not. But yeah, something of that character. I mean, you know people are working on it. And that's one of those five years from now. I'd be curious to be able to get the future version of us talking about whether.

1:01:26like maybe that goes nowhere but Terry Tao was talking about one like research project. It's basically try to exhaustively search the space of possible like algebras like you could imagine different like axioms that you applied algebraic systems and so like when we come up with group theory there's a certain axiom system that like has this flavor of they kind of look like arbitrary rules unless you know the motivation but it's basically like what have you tried all of them.

1:01:50Do any of these yield useful things? The vast majority of them is just trash in some way. It all collapses to no interesting results. But every now and then there would be this little island of a completely different type of axiom system that at the very least seems rich in terms of the number of theorems that can come out of it. And that's like bread and butter for what you would imagine automated provers being good for. Exploring that space and seeing which one of them turns out to be something.

1:02:13like maybe one of those islands actually turns out to be something you can retroactively put motivation on to say this is the kind of structure that's trying to get at in the same way that you could imagine looking at the axioms for a group not knowing that it's about symmetry but retroactively realizing like wow this is very relevant to studying symmetry. So you could imagine results of that flavor but instead of just exploring possible algebra systems it's like all possible like logical consequences of any kind of axiom. On the point about whether you can provide process-based supervision.

1:02:42without Lean. So DeepSeq had their DeepSeq math model and they released a paper on how they trained it. And it was quite interesting. So they have the problem with having natural language proofs as you don't know if it's correct or not. And so they have a verifier and then the verifier is trained by a meta-verifier that makes sure that all the problems that they're training this model to solve and like the other problem solving.

1:03:07that the verifier is giving good feedback on that. And it works. And so it's just interesting. Natural language verification with some sort of meta verification kind of work at least seems to work so far in the published literature. And also it seems to work in the published products that we're using. Like if you look at coding agents, they're getting better and better at writing clean code and refactoring code and stuff like that. And I'm sure that there's process-based like LLNS judge kinds of things which are saying, trying to provide taste and say, hey, is this like a clean way to write this function? Are there duplicates of the same kind of modular forms and so forth? I feel like that should also work for mathematics, right? It doesn't seem... It seems more plausible for math than anything else, even if you're only working in natural language that you could trust a verifier. I mean, you and I were talking earlier about why they're bad at writing, and I was asking why you can't just... They seem to be...

1:04:00good judges, if I give them two essays that students write, they'd be able to say which one's more accurate and insightful. So why can't you just have a verifier saying is this a good piece of writing or not? And maybe the ultimate failure there is even if they're good at discriminating between a BSA and an ASA, they're not actually good at discriminating between an ASA and a thing you actually want to read that would be followable on sub-stack and insightful and all of that. They actually end up preferring just un-insightful.

1:04:29pieces of writing. And so on the math front, I guess the question would be like that step to simply know like, is this a correct proof or not? That lends itself to like an automated verifier, even a natural language. You could probably still make a ton of the progress. It still doesn't like, I still like the sort of tree of logic out of lean front, just in that you can really go off the rails, right? Like there's just no constraint on like the previous way that things had been phrased before in the same way that everyone talks about like move 37 in like AlphaGo and such. Like what is the thing that lends itself to just going outside the prior heuristics? And it seems productive to have a disconnection from the rest of the world in that exploration as like a complementary research pursuit to the natural language math front. I mean, the other relevance of Lean there would be like, okay, let's say you have your pure natural language.

1:05:23RL environments and you have a pure natural language set of proofs and people have to said like proceed AI mathematicians and they go and they generate like 10 papers a day that produce a bunch of stuff. If the error rate, if there's like any error rate to that at all, so Alex Kontorovich has talked about this, it becomes inseparable like as a mathematician because you would basically be like Every single time I see one of these, I kind of don't know if it's worth my time, even if 99 out of 100 of them are right. I don't know if it's worth my time to even go through it because it's really labor intensive to find what that error would be. And it's like really frustrating if it turns out you spent all your time on a paper that was trash. And so having anything that's able to give you that green check mark that says.

1:06:08Even if this is going to be complicated to understand, even if it's going to be a pain, you at the very least know it is correct. Like every other field would kill for that, right? And like math has that. If the models are also able to take their natural language proofs and formalize them. And so that seems huge, right? The ability to have that. Like every field would love to have something like that. And so I think you are right that lean is maybe overrated on the side of the importance of it being used as a VR environment for any kind of like just progress in math generally. But I definitely wouldn't write it out of the story. I also love this extension of MathLive as a metaphor for like what's going to happen to our civilization for these students. For millennia humanity is building this like corpus of knowledge and understanding and everything that we have now distilled into these models. And at some point the models will just like

1:07:04extend that arbitrarily. By the way, on the writing front, I actually have a theory of why writing is making worse progress than these other domains. So I think one of them is what you said, that they're bad at judging not only A versus B, but they get like just totally derailed by B star, which is this like a shitty essay that just hits all the all the bells and whistles that like A is supposed to hit. And then so the reward hack thing just like totally goes off the rails. But I think the other important thing is that writing is not modular in the same way that code and math are. You know, you can write a function many different ways and they kind of do the same thing. And of course you want it to be very clean and stuff. But like at the end of the day, it works, it works. Same with like lemmas and mathematics. And then, you know, you can like have some end product that is different.

1:07:52from the way it is produced. So the code is the thing that produces some end product in New York. You want a functional end product. Whereas in writing, the end product is directly the thing the AI is producing. And each paragraph, sentence, word matters because that is a thing that is like, like that is the substance. It's not like some separate thing that is produced out of the writing. And so it any it's a It can't just be it can't like be slop it had the in the way that like code can be slop and still produce some outcome that you want But you were just pointing out how actually We've gotten much better at agents writing not just functional code But clean yeah, why is it not the case that the same progress that allows you to go from? merely functional to like clean and and like a mergeable PR doesn't also result in like clearer writing. Yeah, that's a good point. I mean also Has it not like I agree. There's many ways in which they're terrible writers, but

1:08:49For a lot of writing I consume, I find it's better to just copy paste it into an LLM and just say like, explain this to me. The explanation will be better than the thing that is produced by the human. So it's funny that we say like these are such terrible writers. And also my reveal preference is just like, can I just have an LLM explain it? Even when I'm talking to a human expert like live on a call. If it's a piece of knowledge they have that only they have that's not encoded in the distribution, I want them to explain it to me. But then if In order to understand that I need to understand a more basic concept, I would prefer if it was socially acceptable for me to just be able to say, let's pause there. I'm just going to ask NLM how that works, and then we can come back to your special piece of knowledge. Well, that's distillation, an explanation.

1:09:37If I'm thinking of quality of view as an essay writer, if it's that I give you a book to read and I want a book report, then I might believe that, okay, the LLM maybe gives me a better book report. But I think what people are really getting at when they say it's bad, what is writing? It's not just distillation of pre-existing ideas. It's not just like, how do you explain clearly? Because they are good explainers. It's like, what is the insight? And this is where it gets like...

1:10:03just auto regression is a very weird way to generate stuff because like when you're writing you sort of know in order for it to be good you have to have an element of the unpredictable and it's not just like increasing temperature in your mind or something right it's like knowing exactly the correct point when you want to make an unpredictable move and that that's going to be what's more insightful and so even if it's like better at explaining a pre-existing thing it's like What generated that book that you wanted distilled in the first place? It wasn't an LLM that generated it and you just needed it. It's like some author who threw a lot of exploration of ideas in the world and then deciding what aspects of it were interesting and which ways of presenting it were like the coherent, well-motivated narrative. It's like they put that all together in some way. And if they're a good author, it's probably one that actually you would err on the side of reading their book instead of the distillation. But so what makes it worthwhile to like

1:10:55explore at all in the first place and you're uploading it at all, I think it's all of that side of it that's the, like, when people will cite them being bad at writing. And it's that element of unpredictability, of being deliberately choosing something that's novel, that's like very directly contradictory to, like, the way that things are being produced. Yeah, that's a good point. I think they're also really bad at building really good mental models of people.

1:11:18which I think is a very important skill in writing. So Andy Matushak and another collaborator whose name I'm forgetting right now did an interesting report where they tried to teach LLMs to write good space repetition prompts. And I really like this because even though it seems like a really totally random skill, it's just like people are talking about recursive self-improvement in a year. And you can't get these things to write good flashcards. And what's going on there, right?

1:11:43They tried many different kinds of techniques and they're like sophisticated people. They tried to RL open source models, they tried all kinds of including chain of thought and the big prompt they sent to the best close source model, et cetera. And the key constraint that seemed to me was that writing a good card is about projecting somebody's mind in three months and what is the way in which they will associate the question, like what kind of answer we'll be thinking by the moment and is that is the elicitation that inspires the detail you actually want to take away from the passage you're trying to make cards about. I think writing also is similar to this, where if you're writing something, the reason it's such an innovating process that takes so long is each word you should be thinking, or each sentence you should be thinking, what is happening in my reader's mind right now? Even if I flip the phrasing around, the ending phrase goes to the beginning, and this is the first image that comes to your mind before you read the rest of the sentence.

1:12:42that kind of maybe auto regression is bad at that kind of maybe a more diffusion like property of considering the whole rather than going sentence by sentence but also I think that requires a lot of mentalizing which these models we're at least struggle at. Well I mean interesting question like is it weird that they struggle at that so I might butcher this this you know how when you like cite studies that you once read and it's like maybe the study wasn't real or something this is one very memorable one on Okay, so let's say you want to quiz people's EQ, like you show a flashcard of someone's like facial expression and someone's trying to describe like what's that emotion. So actually there's really good tests online that'll have like a face and then four possible emotions and it's like surprisingly hard to

1:13:29describe exactly the correct emotion, but you also get that since there really is a correct answer. And if you try this with people in your life, you'll notice that the ones who actually are pretty plugged in socially do really well on it, and the ones who are a little bit more left brain, don't. So that is a kind of test you can do. I vaguely remember an experiment to this effect where they took people who had freshly gotten Botox in some way, and they did a pre-test and a post-test.

1:13:53And like post-test, they were just much worse at like reading people's expressions. Like that feels kind of weird. But they got Botox. So the person taking the test, it's like, so you do the test and then you go and you get Botox and your face is all like frozen. And now you are worse at understanding the emotions of what you see, right? And the thought is that.

1:14:10part of understanding, like this emotion that you're looking at, is doing it yourself. That's crazy. Like at a facial level, like you're moving your facial muscles and it's like, you see that, you mimic that and you're like, oh yeah, that's anxiety, right? At some very subconscious level. So in that sense, if it is the case that models have bad theory of mind, sure, they know everything because they've read what everyone wrote. But at a level of like, actually able to put themselves in your shoes in the same way that like my face muscles are mimicking your face muscles. That's what helps me understand how you feel. Not surprising at all. They don't have face muscles. Their brain works completely different. It's just like, it's like an alien trying to empathize. Like, how could it have theory of mind? It would be like this very emergent thing to have theory of mind. Whereas we can just like plug it into our own minds. And it's like, we've got the ready-made hardware to just like place it in. And so. That's very interesting. It's not, that's, from that lens, it's not that surprising.

1:15:03Okay, Grant, we are both partners with James Street. I'm sure over the years you've interacted with a lot of James Streeters. What have you found that's unique about them or their culture? I did this interview with them this year that partly it was interesting because they don't usually have anything outward facing. I mean in the industry they're known as having like a pretty wild retention rate. Like people just stay there and I think getting an inside view of that. I remember one of the comments someone was saying Even though the people have role titles like you know researcher or trader or engineer they often don't know what their colleagues actual role is because everyone's doing a little bit of everything else like even if you're officially a trader you're doing a lot of research even if you're officially a researcher you're doing a lot of coding and I suspect maybe that's part of like why they have the insane retention that they do because anyone who wants to be growing they just have the chance to do a lot of different kinds of things. Alright Grant I'll do the plug for you this time if you want to watch this full sit down interview that Grant did with some of the folks there go to

1:15:563b1b.co slash Jane Street. All right, Grant, let's talk more about AI and math. What advice do you have about using LLMs to learn? So as I was describing, for a lot of well-known concepts, I find them very helpful, but often just a couple of further messages down, and I'm trying to understand something, and I just...

1:16:20They're so confused themselves, they're confusing me and they don't explain it the right way and then I'm just, I know that talking to the right human can clear up my confusion in three minutes. I don't know, and I feel like more and more we're gonna want to use these things as, somebody's just talked a lot about education and representation and stuff, we're gonna want to use these things to learn things. So yeah, have you noticed the ways to use them or productively to understand concepts?

1:16:46I'm curious to hear your take on this. I'll give mine. Even pre-LLM, I feel like a relevant insight in learning was recognizing that who matters more than what. So advice to any college student when they're choosing what courses to take, care a little bit less about your pre-existing interests because they're kind of arbitrary right now, and care a little bit more about whether the person teaching it is a good educator and so on you resonate with. I think in choosing what to read, like what books to read, who the author is maybe matters more than if it's a prior interest. So if there's a book you've liked before, read what else that author has written rather than reading another thing on that subject. And I'm getting to like LLMs on this. So like there's a difference in feel for trying to learn something if you look at a Wikipedia page of it versus if you look at let's say like it's a philosophy topic and you go to the Stanford Encyclopedia of Philosophy or if it's a math topic you go to the like Princeton Compendium of Math.

1:17:39where the difference there is like the articles are deliberately written by one individual who like tries to actually craft a motivation around it and everything whereas Wikipedia it's this like local minimum that's reached where basically every sentence has to be correct and I think a good exposition you care a little bit less about like correctness on the way but you can like deliberately craft things that are a little bit wrong that you correct along the way that gets like edited out in a crowdsource environment so like that LLM explanations feel to me at the moment a lot like Wikipedia, which is to say amazing, right? Like imagine world before Wikipedia, like how long it would take to like find in like Susan and everything, but nevertheless, what's the most useful part of a Wikipedia page? It's often just the references at the bottom, right? You look at the like key references and you go to them and you read them. It's like actually sometimes that gives a much like better overview of it. So often I like to just ask an LLM like who should I read, right?

1:18:39and maybe I can even give some specifics on the ways I want to learn. I actually got gas lit by this one, so I remember trying to learn about semiconductors or something. I was like, this feels very visual, this is all text. I'm like, is there any really good well-visualized math video? Or not math, sorry, a well-visualized video explaining the concepts that you're getting at in Claude. It was like, yeah, here's a couple in the top one.

1:19:01here's one from Three Blue and Brown. I'm like, I can guarantee that there's not. And it was an actual video, an actual link, but it just had like misattributed someone else's. And it was good. And I was like, I had a much better experience clicking over and watching that video to learn about the thing rather than like trying to proceed forward with questions there. So in that sense, basically using it like a very souped up version of Google on like zero in on the right human written resource.

1:19:27What about you? You engage with these a lot. What's the best way to do it? I think you push your finger on it. The most productive learning sessions I've had is when there's some artifact that a human is produced, whether it's an article, a book, a video that organizes the relevant concepts in the correct way and builds up the motivation of why building up the next idea would be relevant to solving the next problem you did encounter and the next idea and the next idea and using the LLMs to just do a little bit pruning around this branch that the book has identified. So I was actually, I was going through, I think you might have recommended Stephen Strogatz's textbook on- The chaos one? Yeah. Chaos and nonlinear dynamics. I love that book. And so I was going through it. And it was like bliss. It was like your videos in like a book form. He's so good. It was super fun.

1:20:20And the way I was learning it is like I'd have on one-third of the screen, his lecture from university, on one-third of the screen I'd have that part of the textbook and on one-third of the screen I have an LLM. Now I was actually thinking if I was back in college and watching this lecture live, we'd just totally go over my head. Like these kids must be really smart. Because I'm like pausing and like reading the textbook and talking to LLMs and then restarting again. But with him curating what is the right order to understand concepts, what is the right problem to motivate understanding a concept. Oh, also another thing Alan's are really bad at is a thing a really good human can do is when you ask a question they say like actually you're just like not really thinking about this topic the correct way. Yeah. Like the question you want to be asking. Yeah. The correct way to organize these concepts is X. Yeah. And Alan just can't really do that. Yeah. It's a little too placate. I mean this is ultimately like the very like the supplicants and you know that's very like oh what an insightful question you know that kind of thing. You want to you want to strip that down.

1:21:20That's a good point. And I think that cuts to theory of mind a little bit. Like recognizing that to ask a certain kind of question reveals that the mental structures are not, at least they're not the same as what the like explainer has. And sometimes people do this to a fault, right? Like I think a really good teacher, let's say you have like a middle school like math classroom or something, if a student like asks a question that suggests they're thinking about it in a different way.

1:21:46It's actually really hard to take seriously in the moment, hang on, could you get to a right answer with that? Before you say, oh, instead of that, let's do this. And the really good teachers are able to jujitsu the creative way that the student was thinking about it and bring it in. I mean, LLMs aren't doing that when they are not reframing your question. Instead, they kind of run off.

1:22:12The very least it feels like there's three levels here and so like LLM is at one, good explainer is at another but then like the A plus explainer is the one who can like jujitsu your way of thinking and say like oh that's where that's useful and so maybe there is a certain you know cycle all the way around where again five years from now the LLM's will still be doing that but in the better way. What is your recommendation to students who I'm sure email you this question all the time?

1:22:39Look, I was curious about doing mathematics. I'm really passionate about the subject. But seeing all the progress they are making, I don't know if it makes sense for me to pursue this as a career. And this is not relevant not only to people in mathematics, but I'm sure to people who are noticing that their field is more and more getting productivity gains or whatever from AI. So coding is very adjacent to this. Yeah, what advice do you have for people? I wouldn't trust any advice that I give. Would maybe be how I'd like But even pre-AI, it feels very important for any job that you're going to go into to really understand, like if we're talking about a job, right? We're not talking about like your gentleman scientist and you want to like engage with the math world or something. You should understand where the money's coming from and like what value you're actually adding and like the connection between those two. And I think often like a surprisingly small amount of thought is put towards that. Especially students, they're in this environment where they

1:23:37They probably want to go into math because they've always been good at it and they've just been rewarded in life for like proceeding through the next hoop correctly and next step. And when they think they want to be a mathematician, it's because it's a version of getting to continue to engage with that. It's like, well, I'll go like, where do people get to do this? Rather than thinking like, what value am I adding to other people? And to what extent is that like the reason that like salary is flowing in my direction? It's actually quite different in different cases. Like in some cases, It's a very prestigious mathematician and like their presence at a university lends a certain brand value and that's like why the university like wants them. In some cases it's like the NSF grant is given because you've got this like public good belief that we have that basic science has and like you've got this institution around that and there's going to be this whole bureaucracy around trying to act as a proxy for what we think that public good is and a whole song and dance around how to like correctly

1:24:30make them predict that your progress will be in the spirit of that funding. Sometimes it's just straight up teaching, right? It's like people like to send their kids to an institute that has experts teaching them and like that's what you're doing and you are providing the brand value by being an expert and then the direct value by like being a teacher. So regardless of whether AIs are like proving theorems or not or like whether we're talking in 2016 or 2026 like That is the thing that not enough students thinking I want to be a mathematician think about, but I think it's worth thinking about. Like for me, I think that, you know, it's, I just like wasn't necessarily thinking about it and kind of stumbled into this career path where basically math exploration can be monetized as entertainment, right? And I like stumbled into that. I'm like very grateful that I did, but it was an accident. It wasn't like this deliberate thing. And I think

1:25:15I could have avoided relying on serendipity and maybe done that a little bit more by design and had I been like thinking critically about it. So to your question, if it's the case that you have almost automated theorem proving, and then let's say it's the case, they're also really good explainers. So it's like even to get the human understanding, I think a lot of the like social role that mathematicians serve actually doesn't change that much, right? You still have a sense of as a public, we sort of feel like There's value to basic science and we're trusting in the judgment of mathematicians to determine where their time has best been. And the prestige comes from within that community. It's like other members saying that this was a really good result, more than it is like the grant writer who really understands algebraic number theory to understand that it's a good result. And so there's gonna be some inner culture of what constitutes valuable contributions. Maybe it shifts away from theorem proving and maybe it shifts towards good definition writing. Maybe it's that museum curator idea, but

1:26:14you're gonna have that same community. And as long as society as a whole is still valuing the premise of basic science. And if we're in the abundance world of what AI brings, probably there's more funding in that direction in some sense, right? On the side of prestige to institutions for who their lecturers are.

1:26:31I mean, I actually think teaching is one of the most stable like post-AGI jobs that there is because it's so relational. It's so like this is where parents want to spend their money if they have an abundance of wealth is like on good teaching and good educating and it goes so far beyond explanations. Like even if LLMs are good explainers, the thing that a teacher is doing is such a social like coaching mentor type thing that like that's probably one of the most stable careers that's gonna exist over the next 50 years. And so insofar as what a lot of mathematicians' role is like overlaps with that, you as the prospective student going into it, you could lean into that. Actually think a lot more students should like think about and give like pay credence to the idea of being like just a math educator and like the value that that can serve towards the next generation. So I'll couch again on, I don't think I'm the one to say

1:27:24Here, prospective young mathematician, like here's how you should think about the future, because I'm like a YouTuber, right? I'm someone who is not in the institution that they are thinking of going into. And so I'm speaking as an outsider looking in. But it feels like generally good universal advice, nowhere the money is coming from, nowhere you plug into that. And like if you're just asking those questions, you're actually already like steps ahead of all of the other like fledgling prospective mathematicians. Yeah. And in fact, I think in the crazy world, in the world where within five, 10 years, the AIs are coming up with not only solutions to the Millennium Prize problems, but coming up with like just totally novel problems to be solving in the first place, novel mathematical fields and object and stuff. It is in that world where first of all, there's a ton of abundance and two, the things that AI minds will have like gone furthest in, where they will have seen like furthest beyond our horizons.

1:28:22will be mathematics. And there will be so much demand of like, what have the AI seen? Can you explain it to us? Yeah, I feel like in that world, if there's any jobs whatsoever, surely distilling what the AI's have learned will be one of them. Also, it's funny because all of this sort of presumes that it's useless, right? Like we're not talking about the actual practical applications of what math is being done. So in so far as there's any economic utility to it.

1:28:48you would imagine that the people who understand it and are able to make the decision of where it should point, they actually have a lot more economic value by being able to make that judgment as curator and point this behemoth of new math pointed in a useful direction. Suddenly, that's a much more levered move to make than it had been previously. Can I ask you about that? Obviously, one question for AI for math is not only can it do it, but is it any good? Yeah.

1:29:19Or isn't any good for anything? You were describing all the ways in which group theory we're trying to solve this. We're trying to figure out random facts about the roots of different kinds of functions. And now it's all these different applications that are practical across many different fields. Do you have some sense of if we just totally get to a place where mathematics is. The field of human mathematics is accelerated at 10x or 100x. That we have some crazy shit happens or are we just actually going to be bottlenecked by.

1:29:48Other fields or I think there's some fields that probably will I mean, it's it's super spiky, right? I think like progress in algebraic number theory It feels unlikely that that then like unlocks something but I remember talking to this mathematician who does more like like dynamics and and and like PDE solving type stuff and he was referencing basically like his group had some ideas that Let me see if I summarize this right. It's like the way that Boeing would make planes is they would like make it and then they would do a bunch of tests and they had to like disassemble it and reassemble it based on those tests. And they essentially had some insights on how to like do more things in simulation such that you don't have to like deconstruct and rebuild it. And it saved Boeing just like billions of dollars or something. And then they just started funding that like group which is so that's it's like much more obviously application adjacent because like PDEs just sort of are that. So

1:30:45progress in that domain you would imagine like actually do unlock some things and I don't know if it's these like step changes but maybe it's more on the side of like engine design becomes just a little bit more fluid or you know like coming up with the right wing shape instead of running a whole bunch of complicated like CFD or maybe you're able to like speed up your like CFD simulations because of certain pure math insights like makes those more efficient. I bet you'd just see like a lot of like great incremental improvement there.

1:31:15it seems less likely that the massive breakthroughs in math immediately turn into this massive economic breakthrough. You solve the Navier-Stokes problems and then that unlocks an ability to simulate more things. But you probably will see at those fringes just some meaningful leakage outside of the pure math insights into other things.

1:31:39Also, I mean there's a ton of people working on things like AI engineers, like physical engineers, like material science and things like that that would be, you have to imagine that they would be in a good position to look at the AI math insights and decide if they're relevant in some way or not. And so it's another one of these things where I'm not gonna sit here and put a flag in the sand, predicting that there will be, it'll be a little bit disappointing and a little bit surprising if there weren't over the next five years.

1:32:07economically valuable improvements that were made that were directly like referable to the like AI progress in math. Like that just would be kind of disappointing if it was just taking down a bunch of Erich's problems and like none of them actually, you know, it wasn't doing any of the math that actually directly touches physical world. Yeah. I mean, to your point about, well, a lot of history and mathematics were like building up these like piles of concepts and connections and whatever. Yeah. And sometimes the piles connect with each other or you discover an application somewhere else.

1:32:37At the very least, you just build up this huge pile, and as a broader progress in society happens during the singularity, when we get to the industrial part of the singularity, you just have all these different ideas that you can hopefully are useful in other parts of the world. I mean, yeah. Like I said, one of the interesting things about what's happening is it causes people to step back and ask, like, what is math? And maybe one of the awkward conclusions of it will be, of revealing like, ah man over the last like it's just become wholly useless. Like the kind of questions being asked to become like so divorced from things that are physically applicable that like that's one of the things mathematicians have to come to terms with where everyone will look and be like.

1:33:16on a second, like, weren't you guys supposed to, like, if there's so much, that's like 10x progress there. Like, why aren't we seeing it over here? And then that church is like, every time we wrote those grant proposals and said, like, trust us, like, the elliptic curve progress is going to help with, like, cryptography. Like, it, like, shines a light on the fact that, like, maybe it doesn't. So that's one possibility. Grant, this is super fun. Thanks so much for doing it. Absolutely. My pleasure.