Career Advice for Future Mathematicians

Today I'm chatting with Dran Anderson who runs S Bue one Brown and is now working on a new project documenting the progress AI is making in that And I wanted to talk to you about this because The I' been making the fastest progress in mathematics as of any other field So whatever is happening here and whatever we're seeing AF Mgress happen or not happen would tell us about what will happen to the rest of the world as A gets better and better. So I wanted to start with this question I asked you when I first interviewed you Th years ago And I asked you Once we have AIs, they can get gold in the international Math Olympic adad. 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, it hass gotten better in general ways since then, but there won't be some aha moment when this happens At first I think I'd be curious to get your characteristics 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 middleennium price problem Do you think it's so 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 But there's kind of a fractal nature to that spikiness because when you zoom into the specific progress within math, You have some things are a lot easier than others. So if we just think about IMO, which is old news at this point, it's kind of like two years ago that they're really like doing quite well. They would have gotten a gold in twenty twenty four if' not the following reason They were 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 It solves in like nineteen seconds in twenty twenty four 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 Comonatorx is the one that's the wild card of much more like playlayful puzzly saving problems. There were two commbonatorx 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 gold that year but it struggles on those comminatorics ones. and You know, someone who's trying to keep that torch of the last holdout of like math for humanity might say, well Those are the ones that require the more creativity E then though, I think The spirit of your question, like if they're solving know, 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 We could maybe like paint a couple different ways that like if we focus on, I don't know, remon hypothesis, like what would it look like to solve that Ums 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 like bizarre to have something with this superhuman breath that like knows all the field so well, that's not just finding those lightning bolts that connect them I think we're starting to see sparks of that of like actually finding connection between the things that it's an expert at. I'm sure we'll talk about it If the nature of the solution to the Reman 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 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 like This is a side tangnder, 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 statistical correlation between pairs of zeroos of the Reman Zeta funct. So the Remmann hypothesis is all about like, do all these zero sit on a straight line? And he's finding this like this quantitative question you could ask about and he writes down a formula that looks like one over sine squared or something like that Cream and dice and a physicist is like I 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 Remon' eta function. And I think it's a little bit of an open question, like is there fruit to be had there? But that kind of bridging together from two different fields. Like if it turned out that the solution to the Reman hypothesis was exploring an idea like that even further, that 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 Um, What's the right analogy? Maybe like if we think of Fair Mau's last theorem between the moment of Fair Mau phrising the question and then what the solution itself looks like where Ultimately, the solution involves such heavy machinery in math,? So theuty that problem is you can phrase it so simply. you ask about x to the n plus y to the end equals Z to the N. Do you have integer solutions for this when n is bigger than three? That' something 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, you know, maybe there's something simpler, but this might be the what it has to be There's such a complicated set of ideas that build on 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 Rman 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 h, 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 in the way that getting gold in the IMO did not have transformative effacts 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, you know, whatever does this and maybe that it's the same drug that can cure both things And yeah, I don't know, from an outsider's perspective, mathematics seems clearly like a field where Finding this counterxple to the unit distance problem conjecture was like an example of this kind of thing. tootal Gal plus moving, but then we can ask,kay, what is the next benchmark Now that AIs can do this thing that we should have thought they should they should be able to do. But 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 train these models to like we have these millennium price problems because Yeah, I know like mathematicians have noted like Reman came up with this idea of this like Reman's theetta function and because he thought that it would have some connection with like the density of prime numbers, or if the zeroos on this function would have some connection to prime numbers. And so like figuring out that there's Why 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 For 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 because all of these discussions, it causes people to reflect on like the process of doing math,? They're like, w, this thing can do these impressive stuff. Like what does that mean for us and he highlights this quote how good mathematicians prove theorems, G mathematicians come up with conjectures, and the greatest mathematicians come up with definitions. And that's more or less exactly your framing here on those two like we need the conjecture generator, and then like the definition generator, that's the preium tier of mathematician I don't understand 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 goalpost. The ball is through the goal or it's not. L you can clearly say like, yes, this is done, partly to be able to do things like R LVR, but also partly just to be able to like know that you haven't moved the goal post in answering. know Open A Ie can have their headline on disproving the unit distance conjecture because it's a s clear distinct, It's like it did it, right? Whereas imagine trying to have a headline onli PT five four came up with a really good conjecture, right? Like we promise everyone thinks it's a good conjecture. but just doesn't it doesn't land the same way 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 like 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 tone shift in conversations with mathematicians about the way that it's useful to work with Right And this series that you referenced that is not at all produced yet and probably won't be for a couple of months takes the form of us interviewing a lot of mathematicians. And what's interesting is we started doing this like 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 like mid twenty twenty five and where we are now in twenty twenty six You know, in the real world, that's a very short amount of time. In the AI world, that's eons, right? And like we're able to see over those eons like this tone shift I think the way that you'd measure conjecture generating ability is going to be more subjective unli that tone shift where They'll be mathematicians saying they're not just using it to like 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 like was genuinely helpful for that. I don't I don't think it's likely that you'd see it in the form of like a headline saying that like this was yet another benchmark knocked down. Right. And so it's very interesting. the kinds of things you can't make benchmarks for are 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. Y. I think it's very easy to come up with some dichotomy of like, here's a deep reason why A I can't do a certain thing and then it turns out well you're just thinking right it the wrong way and actually I can do it pretty soon thereafter I'm gonna come up with youre gonna come up with the temple anyway? And I think that this will probably It'll probably turn out that there's ways in which we can trainI to do these kinds of things in the relatively near term It seems like it would have to be different from current RVR 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 problem or the new way to understand the world. That then unifies 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 why light bends or white black holes exist. Th are phenomena he didn't know it needed to be explained in the first place But in mathematics, it often seem okay, 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'd be curious to hear you talk about whether Galwal coming with group theory and distinguishing his like solution to the The Quintic having no formula for the roots and Abble 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 looube on like is group theory an interesting concept that was like was something useful done here? Why is this proof better Potentially that verification loop is a hundred years long that involves the cryptography coming around and physics making progress and the ideas and group theory being relevant and understanding like symmetries in physics and all those kinds of things. There's like a one hundred year verification loop, but why is this a productive concept in the first place Boy, yeah, you struck a nerve because I had this like project about Go I was going to do in twenty twenty two that I put on the shelf, but I spent like a year of my life thinking a lot about what he did. So there's a risk of me accidentally talking too long on the specifics you can hold me back on. It's a perfect example for your case desescribing why it was a valuable insight does not come from immediate utility So certainly if you're thinking about RLVR environments, it's like, okay, this is going be really hard to do. But it's interesting to note how even with like human verifiers at the time, 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 that What makes the Gywa theory such an interesting example is you have Literally this one hundred 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 to back up a little bit I don I mean, do you want the background on the problem at all? All right. Well We all learn about the quadratic formula in school. I thought you' were going to say, we all learn about group theory in school. We We all miss that. We all learn 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 like 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 a natural open question for mathematicians is Can you find a formula that solves degree five equations Nowature the degree for, it's monsters. it would be in wild to write it down. You usually don't really write it down in full. You break it up as like a procedural thing So you might believe these things have this exponentially increasing complexity So many hundreds of years, nobody is like really answering that question. Usually we say Aba 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 Leag Grange where Leagrange found the right kind of question to ask about this As I can go into the details if you want, but I'll give it a very high level Cheap 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 like symmetric, more or less so. Like if I write down A plus b plus C plus d, just like adding four variables, if I permute those, it doesn't change the value of the expression Whereas if I write like A plus B multiplied by C plus D, some of the permutations don't change it, but some of them do And 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 So that's like you could put that in a puzzle book. You could put that in a brain teaser that like a twelve year old couldn't engage with U Ands it's not too hard to like find yourself feeling like that's an impossible task And so Lagrange is sitting here saying, Hmm, hereere's 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 But some kind of question about symmetry was the right way to be studying these polynomials. In his mind, it was just A way. It had yet to be discovered that actually there's a tighter connection. also Maybe rather than searching for the formula, we should be asking the opposite question, canan you prove that it's impossible? So he sort of planted that seed. Like Around fifty years later, Abel definitely read Lagrange and was influenced by it. Gaois, 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 pretty similar insights around that problem It's not like born from Lagrungee. But to your question, I'm like are you able to verify that this was a good idea? There wasn't any like result that LeGrange came to. There's never like he solved the problem and therefore we know that that was like the right question to ask. He asked it, there's some like intrinsically interesting thing. It also wasn't very important for math at the time. Like mostost people were more interested in the applications to physics. This is almost in that like. almost recreational hobbyist type thing. Like Abble, you know, he started working on intic stuff, but then he was advised to spend more of his efforts studying elliptic functions. and so more of his work was on that before he died young. He died at twenty six from tuberculosis And then Gawa U Kid. 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 while 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 complete proof. He wasn't giving a clear thought of what the theory actually was He wass just like a young sedgling mathematician getting his bearing. so it's like the verified reward there is like 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 he has 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, o, is the verified reward that he has solved a problem that other people haven't? It's like, well, Ab will proved that quQintics are unsolvable. And he said, what was Gawall doing? Well, In principle, the thing that Gawell 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, like X to the fifth minus one You know a solution is one, or to the fifth minus two, you can write down fifth root of two. So it's not that every quintic polynomia 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 He 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 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 it doesn' 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 Quintick. And so he asks his brother and his close friend like, get these notes to Gaus, get these notes to the important mathematicians of the day ' I think there's something here Even then, it didn't really take like His brother and his friend like tried to get them out. It wasn't another twenty 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 Gowwell getting it I mean even then, it was another twenty years or so until Jordan actually like puts together a hing like a modern treatment of group theory They attributeed to Gaois, you could easily imagine history turning differently where like these ideas were kind of coming about from other points in math and like Gais could have been forgotten in history if he was the less like florid character Between the time of Le Grange, like having this inkling of maybe symmetries of roots is the right way to go to where it all looks like modern group theory. you've got this long span. a lot of the time, it's like not even passing the verified reward of human reviewers,? Be it gets on someone's deskks, they say, I don't really know if there's anything here. gets on someone's desk, they don't. You have to have this 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 twentieth century before you have Gelan thinking, hm, maybe understanding the nature of how certain groups like breakdown has this relationship with what particles are made out of 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 to even predict the existence of quarkks is a group theoretically like question, that's so long after Lag Grange, 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 Gaois's mind when he says, I think there's something here. What's the instinct that's inside Le Gangee's mind when he says, like, I think this is the right way to think about it. What's the instinct inside Louisille's mind when he says 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 you know the whole compression is intelligence idea. And even though this isn't really the anle I'm taking, you know, there is something to the idea that The smaller expression that's more predictive like feels more intelligent. And so I wonder 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 Reman 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 Fermaaz'sl last theorem that's just 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 remon hypothesis that's like no one's really get anything out of it. And what you actually want is like what are the sucinct, like compressed versions of those ideas? then lend themselves to human understanding I don't know. Komagorv complexity, like maybe you throw that into your like 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 Gaois like instinct rather than just rewarding Have you solved a problem It's very hard to come up with a like the heuristic for science But it's clear like human humans have been doing this somehow and like Obviously guys will do it at some point. Well, it's relevant also not just in terms of verified reward, but like presumably, the end goal is understanding, human understanding. And so even if you do have some like thousand page proof of some math thing or some like grand new physical theory The goal is understanding. Yeah, right. Maybe if the goal is predictiveness, you can just have like automated engineers go off and like build rocket ships or something We're like, we have no idea how these work, but we can get between stars. But like there's going to be a lot of people want to understand. you're still going to want whatever the like concision function is that like distills down, hereere's this complicated way of thinking into like the right one, like the equivalent of the universal law of gravitation for Newton. Like you would still want train AIs to be able to do that and like find the the compressed representation. I grew up in Indac till I was eight, and so in addition to English, I also speak Gujarati. And since Google just released Gemini three point five Lve Translate, I thought it'd be fun to put it to the test in this midd role three point five Lve Translate automatically detects more than seventy different languages and translates them in almost real time into the target language. Live Tanslate your original speed and format while speaking, just like it's doing right now. I visited China back in twenty twenty four and I remember thinking at the time, 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. Now we have that technology. So if you're building an app that needs live translation, you should one hundred percent check out Gemini three point five live Tanslate It's available now via the Gemini live API and an AI stududio. Go to AI. studio slash live to get started So people have this worry about mathematics in particular that know the AIs will prove the everyement hypesis and our understanding of mathematics won't be any the better for it I have a couple of questions about this. The first one is whether this is like A thing you should expect Like isn't The reason humans come up with general Natural. Uh 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. So we can just think about like theoretically, would this even be a simpler way to solve the R hypothesis as opposed to just coming up with the natural abstractions that are relevant to thinking about the problem? And then two, empirically, is this what we observe when AIs do make progress on problems today? When the AI came up with that counterxample to the unit distance problem conjecture? You can just read its 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 for mathematicians, it was like understandable and it made it made use of like known conceptces mathematics and like proude relationships between them and all the natural language As a result, accelerated our understanding of the connection between this object and this conjecture Is this even a thing like empirically 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 Ryman hypothesis That one and the other like big one from this year is like a certain airdish problem numbered like one one, nine, six, but it's about these things called primitive sets, but basically it had that character of Bringing 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 use this like tryed to archive 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 Menold 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. 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 parible, right? Becauseuse all you have to do is just like show the start and endpoint of what those connections are. If the character of it is a mountain building, you do have to You have to put in a lot more time to understand that new mountain that was built because it's like a new thread. that's not just like lightning bolt between them And then if the nature of the progress was just like raw hustle, right? It's just like this just super long thing, no new theories, but it's just like long, long, long chain of reasoning answer, then you would have that word of 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 onn the mountain buildilding side It would actually be really interesting to see. like is it by default a very human understandable like the way that we like 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 we engage with. Right. Well, the closest example here would be like the, you know the attempted solution of the ABC conjecture that was we maybe shouldn't get into that one, but They felt it is not probably not a correct solution, but basically it's 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 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 just like long chain of reasoning A inter universal geometry or something. And so the fear that you would have is that like Yeah, it 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, hey 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 If we end up in this situation, David Besus had a really great blog post called follow the theorem of economy. We're just talking about this you know, historically there' as you're saying Mathematics is coming in up with these definitions and problems and it's about proving theorems about them And that really that they're improving stuff is what gets all the credit, but it's like really a parasite on the with the definition stuff. And historically, it's not been a problem in terms of credit apportionment because If you come up with a definition, you're probably going to be the guy who comes up with a theorem. But now we're in a situation where Um If the valuable work is the coming up with the insight and an AI just automates the ladatter part Okay, imagine a scenario where we have AI comes up with like the Able 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 diffusure AIs to then consolidate I 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 you can understand how this proof works that would make it easier to Come come up with the ideas behind group theory? Yeah I think it would It would be hugely helpful, right? Like Be I mean so much of like trying to discover new math is like mostly being wrong, right? You're like trying to solve a problem. like what it it doesn't feel like constantly 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. That feels like progress simply because It's providing like a sense of knowing that it leads to a solution. And there's plenty of like 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 I mean, 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 a physician named Timothy Cho who was trying to understand a concept called forcing. and so there's this problem called the continuum hypothesis. that more or less asks like you have a size of infinity for the natural numbers? you as of 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 it's just really, really hard to understand. It's the thing called forcing. And in the beginning of this paper, he writes like I want to like everyone knows the idea of an unsolved research problem. likeike I want to propose the idea of an unsolved expository problem where like I'm sure we've proven it, We don't really know why it's true. And so he proposes like a partial solution to that expository problem You can imagine why I loved that framing because this is my whole life.' like I don't do research math. it's just wholly about like what's the most clear way to understand this, even if it's proven just like there is a difference between proof and explanation. And so on that side, I think that You are basically like getting to the importance of that distinction. Yeah. And that will be the main incentive or or the incentive would have to change in not just mathematics, but in other areas of science from Um proroving things around the world to consolidating proofs into problems or higher level insights. but we having a discussion earlier at lunch about like Uh 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 forore an idea and the idea it's also You know, if you think about special relativity and like space timee diagrams, U and Mkvski spaceime. 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 Is like that is the reality So the 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 who are actually quite clear in their communication of it. L you might imagine given that the experience of a university student is often that the expert they are 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 Sannon 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.ike Feynman has this characteristic too, like very good expositor. And so maybe the same part of the brain that comes 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 AIs will become these automated theorem improvers, but like the role of the mathematicians is going to shift towards like my job, explain these things I kind of suspect that actually they'll also be quite good at doing that and 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 digesting and explaining what was going on, probably. the nature of how these things are going ision we can talk about like ways this might not be, 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. Yeah. That's my new like that's that's the way my I think beliefs have changed. What's the last thing you think you' willll be doing O they like both you and then also what would the mathematical community, the human mathematical community will be doing? I will probably be doing something like what I am until I die. Even so like, even And if the doomers'll be the exactly because It'll be for the same reason. Yeah, yeah, you know,' you like build a man a 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 someome of the Some of the like function of an explainer or a teacher is to like add clarity to a thing that someone's curious about. But some of it is like a little bit more relational and a little bit more providing like motivation, providing a sense of curation. Like 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 The I so all the things. so the art exists, right? They even know how to explain it really well, you know, on there. you still want someone to help you navigate in this like nearly infinite space of like what ideas are worth engaging with, like someone kind of doing that. that one, even if AIs were in some sense better at that, I think we would always still prefer like 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, you that might be different. you need to know there. think Like 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 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 of spending. And that's a lot of my job right now, even now,' basically I think people think a lot of the time for a video goes into the visuals. sure, a little it is,' not like immediate. but actually a lot of it is just deciding what's worth saying in the first place or what's worth putting there Um 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 aord. evenven if the AS are better than that. in the same way that likeike 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 objective quality of the MP three file coming out is like better from some model That's kind of what I see happening to my job. I want to go back to this question of Earlier, I was we were sort of Just as AA has crossed this threshold, this important benchmark of being able to connect existing ideas to come up with a new discovery your. rove or disprove something Just as cross the threshold, we're like, o, what's the next thing? I want to just There's a lot more to do on that one, by the way. Like just because a couple of lightning bolts have been I think there's like this flourishing future over the next couple years of like really connecting Yeah. And so in the limit, you could even say, I don't know if this is accurate to say, but potentially A lot of maybe the biggest breakthroughs like look like this at some level. It's just general relativity. Oh, like he's just connecting together like Romania geometry and special relativity, right? And so as they guys 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 I mean, a lot of the conversation focus has been on problem solving and that nature of math,, like taking off air dish problems or something. I would say it's not even a majority of mathematicians who would maybe characterize their work as like really targeting the next problem to take down Are you familiar with like the Langu's program? Okaykay. so this is like There's not even a field of math so much as it is like a like a research ethos where Thermaoss' theorem is one inkling of this and you had like these two different seemingly Dparate things and a connection between them like led to a solution So Lingus was a mathematician. yes, this like famous letter now, essentially spelling out how It seems likely that there's a lot more connections like that. and even got like a little bit more specific about the nature of the connections, such that you might imagine this like large map and you've got this valley over here and this mountain over here and this set of planains over there And there's a lot of mathematicians who would characterize their work as being part of like trying to understand the threads like 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 There's more that there's been enough time and time again cases where problems were knocked down by finding connections that it's almost preemptively finding the connections. And so you could im. Yeah. it's actually very interesting that like this anyytime you run into a mathematician, it like ask them whether you know the character of their work is more akin to like Langlland's program or if it's more akin to like targeting one particular problem, you get a certain like bifurcated split there. But the the possibility of AIs being supercharged connectors feels like it might be an amplifying tool in that pursuit It's It's hard to measure though, right? This cuts to what we were saying earlier How do you How do you assign a score to say like, 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 like PR move as the AI company to say we did it. Whereas like 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. But I think I think it will require a lot more like human in the loop to basically like say was it like the kind of connection that we're going for my guess on what most of the useful progress from these models will look like, in the next five years is just really filling in that landscape of connections that you can draw if you're an expert in multiple fields Like you've pointed out, it's kind of surprising weven't already had this. R. And what I'd be curious? I would be curious to know at a technical level causes the unlock there. because On the one end, 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 the method of reasoning is this auto reggressive chain of thought phenomenon. Auto regression 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 And 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 memory is wiped, right? And then you get like another slip of paper and you go Um, 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 them and be like, this is awful. That's not the essay that I would have written, right? Be like the process of like repeatedly like predicting something is just different from how you would think as a writer to like compose it and think it through and everything Um, And in particular, what would probably happen is you're sort of a slave to your context where U you might be answering some question about some particular field. and so you like drawwn all the context around that and you're going there. And the connection that actually is where all the substance is going to come from is like by its nature, a very like unlikely one and You know, you can do all the RL that you want to try to like get better in some way. but like what's the thing that's specifically upwighting 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 questioning 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 findind 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 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 productredictive to reason instead of architecture or even loss function to reason about data L a I don't know. we have diffusion models that do that do text and they're like not of a the kinds of things they produce are not of a wholly different character and they're just not been explored as much. I think the more relevant thing is what is the data on which whatever architecture or whatever loss function you have is incentivizing you to produce and u 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 they getting better at being autonomous Asians, just, I don't know, they have like they're in an environment we're auto regressively 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 Um progress in science or maybe in math is You have frontier math like problems, which require like mathematicians specifically design them because they require connecting together two different fields And there's all I'm guessing there's all kinds of clever like partially synthetic ways and which to make harder and harder problems like that that require these kinds of connections. For example, by eliminating assumptions and still requiring the EI to continue to get to the answer And then like it doesn't really end up mattering what the loss function is. It just like it's really about can you come up with an environment whichich incentivizes a cility. Yeah It feels like you should be able to can I certainly can't speak to the correct ways of doing that than 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, the key fact about them is that we can just paralyzed 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 I 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 of the 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 paralyization is like quite important property and I'd be curious about your predictions evenven if they're not as smart as human mathematicians The fact that they are just, you know, billions of because for PR reasons, the AI companies are are just dumping billions and billions of dollars at this would have a quantity has a quality of its own That seems in the right direction. I think I mean, if we take that, you know, that conversation between Montgomery and Dyson at the IAS that like suggests some connection between Reman hypothesis or remunza function zeroos and random matrices, that feels like the kind of thing that you could try to like automate and that you have you know, agents representing expertise in all these and basically having, okay We all know that an institute is smarter than an individual and that like the reason for having people all in the same geographic location is because you want those like serendipitous conversations to happen. What does it look like to sort of engineer those between agents? I mean, it's you sort of point out like you can sort of pool all your knowledge and so on. 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 like, I'll just like start again. Same do with humans, right? L sometometimes you 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 You know, there's 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 like prove the opposite? And that like unwinding your own context and going at it with a fresh mind, you could imagine systematizing that or like having multiple different agents deliberately given different pieces of context and try to like comparing and trust there. We don't have the same level of manipulation on our own context. in this like AI and Math series, the first episode will be about like 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 like leading someone in with without 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 international Mathylympiad 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 Ho Aye story is like for 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, like they'd probably answer it well. And you sort of want the same sometimes for like like human research in other contexts where like 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. like a little bit more of a systematic, what does it look like to like refresh your thinking, try answering two separate questions, like spin off two agents, onene who's trying to prove it, O one who's trying to disprove it. One who tries it like this way, one whoes. And they like deliberately have different contexts I would be curious to see if we're having this conversation three years from now, how many of the like significant results that make headlines have that character of basically like erasing the context previously, like trying a bunch of different things as opposed to merging the results of like a bunch of different. I this 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 go down the same path. and have similar patterns of speaking and so forth But Um, mayaybe 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 conjection 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 theS will have is actually to increase the entropy by systematically trying out both the negation and trying to prove the positive of any given statement 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 Iinst say just really motivated by this bias Like things should look the same in different reference frames. And then he had multiple other biases like these, but like that is actually 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 like systematically increasing entropy at the prompt level even though you have this like inevitable collapse at the like auto regression level.. Yeah. mean And I mean, Einstein would be an interesting example because it's like he's got this bias towards things should be able. He also has a bias towards like God should not play dice, right. 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 programs? Which actually goes to show you that there's not a correct heuristic. Exactly for science. Exactly you just need multiple independent research programs and their own heuristics. Yeah, ye. And that feels like old school software, right? As long as you're able to like describe that in some way. you have like old school software that amplifies that entropy in some way. and if you're able to like put a clear ontology to the distinct ways of thinking that you want to prompt explore that full ontology, and then each individual one runs off doing what it is But I You know, I think there's a certain design question there on like how exactly do you describe like the different approaches? The 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 suicient brereatth 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. For example, I started publishing my episodes on Billy Billy for a hopefully urgeoning Chinese audience 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 Kursor to do it instead and spare them. And 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 hodgepodatch 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 slash flore cash to get started. Obviously, AF forMath is making a lot faster progress than everything else. people to verifability 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 on. And u I'm outside the labs. I don't know what's actually going on, but there's totally naive theory Okay, a tangential question to why EI is making so much progress in math. Why has it been social a computer use which is what you, you know, computers is actually very verifiable like you know, is my EtsTy package coming or like is my event booked, you know, whatever. These are extremely verifiable things to survey. What computer use lacks is grind ability 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 of the same check offflow on Amazon because you'll get like shut down by Andy Jassse, right? And so you can personally presses the Red X onor cash button, Exactly. And so you could try to build clothes every single website. This is very labor intensive. slows you down And the reason, by the way, you need to do so many parallel ros in order to learn a skill currently with Deep learning is that we haven't solved sample efficiency. Sucking supervision to a strraw. E exactly what he says. Of course, people are working on many different techniques, but fundamentally, there's this big problem There's this big constraint in the way we training eyes that we just With code also, you can containerize a given a level of progress in repository and then just spin out thousands of pareallel containers or hundreds of parallel containers and say like try to implement this feature. It's totally deterministic. And because this iss deterministic, you can solve the Creditter Dsimond problem because you know that whatever caused this rollout to succeed and this one to fail, the diff is the thing that like worked. and this way you solve the Credittersimon problem. If you have situations that are starting off at different starting points, this credit design problem becomes much harder to solve But most of things in the real world are just very hard to containerize in the same way. likeike coding and math or U Exceptions to this rule, but if you're just trying toig out, how do I build a new business that succeeds? How do I like go trade in the markets for a day and like make money The chan Lager The fact that you had to interact with the real world and like 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 for progress in this domain and also in in in coding. It's not just verifiability. It has to be grindable The 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 united distance problem? Well, they or sorry Dprove the conjecture by the unit' problem They release the chain of thought or atleed the sub A rewrite of the shade of thought didn't have any lean in it I think it's just like the process based supervision that Lean provides where you know each step is correct It seems like less relevant than just having this grindable outcome that is verifiable It's an interesting point, like grind ability mattering more I guess I will say on the yeah, okay, so naively you might think Lean provides something unique for math because you're able to see if it can prove it. have oldld school software they can tell you yes or no, you use that as your VR. I mean, 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' to your point, not needed I think there is a um 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 That provides a certain bound on how like endlessly explorable things are. Like if you consider Alpha Go, alpha Zero Sle stuff, where they're just like off in their own universe, just like playing a bunch of G 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 Hey, you can do RL on that. It's also You basically never have to check in and you can just like poour compute at them like exploring the universe of Go What stands to be interesting, like maybe this won't pan out, but I think the The jury should still be out on like Um, whether this will yield anything withith Lean, you could imagine having a basically endlessly running program that's constantly trying to extend MathLib. So MathLib's this GitHub repository that's basically like all of math written 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 subcommunity 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 a certain taste for what they want to be in there. So you have like your fork of like the pure AI MathLib and it just goes and it just like doesn't stop. It doesn't need anybody to check in on it Right? It could just keep going It might come up with its own conjectures, it might come up with its own theories and like different definitions. Maybe many of them are useless, but it just has this infinite tree that it can like grow out That's a very unique thing that Math has that nothing else has where you could press go And then just like, compute edit and look away for ten years and then come back and say, like, what do you have? And there's going to be something, right? And then there's a question, is it useful or not? Like how do you sess that out That'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 Okaykay, there's There two different ways that like Lan 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. D did you see Carpathi's auto research ideia? He wrote this basically one Python file that does basic LON training And then just had a repo wherele agents would try to make modifications of the file. If it sped up the speedrun, the modification stays Eric Jang who came on to explain how Alago works did a similar thing when he was trying to build in a very strong uh Gbot, um And he had interesting observations about the kinds of like it's really going just go running experiment and going down that path, but it's bad at stopping at dead ends and just doing extremely parallel things. Anywways, 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 and interesting in USOAs. And this way, you don't have any outcome based supervision. No There'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 like if you were doing that, you don't want to completely go off the rails and like do a random walk to the space of logic. Youd probably want some like 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 like That's one of those like five years from now. I'd be curious to be able to get the future version of us like talking about whether like maybe that goes nowhere, Terry Tow was talking about onene like research project that's basically try to exhaustively search the space of possible like algebras. you could imagine different like axioms that you apply to algebraic systems. And so like when we come up with group theory, there's a certain axiom system it like has this flavor of they kind of look like arbitrary rules, unless you know the motivation But it's basically Have you tried all of them Do any of these yield useful things? And like the vast majority of them is just trash in some way. L it all collapses to like no interesting results. But like 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 like the number of theorems that can come out of it. And that's like bread and butter for what you would imagine like automated provers being good for, like exploring that space and seeing which one of them turns out to be something and 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. 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 su provpervision without lean. So Depseak had their deep seek math model and they released a paper on how they trained it No it's quite interesting. So they have The problem with ha natural language proofs is you don't know if it's correct or not. And so they have a verifier And then the verifier is trained by a metaverifier that makeakes sure that any all the problems that they're training this model to solve and like the aret of problem solving that the verifier is giving good feedback on that. And it works. And so it's just interesting. Natural language verification with some sort of metaverification 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 like writing clean code and refactoring code and stuff like that. And I'm sure that that there's process based like Alabama's 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 we like there arere there duplicates of the same kind of modular forms and so forth Um I feel like that should also work for mathematics, right? It's like it doesn't 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 you know, I was asking like why you can't just like they seem to be good judges if I give them two essays that like students write, they'd be able to say which one's more like accurate and insightful. So why can't you just have like a verifier saying, like, is this a good piece of writing or not Maybe the ultimate failure there is like even if they're good at discriminating between B essay and an A essay, they're not actually good at discriminating between like an A essay and like a thing you actually want to read that would be you know followable and substack and insightful and all of that. they actually end up preferring just un inssightful pieces 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 poof or not that lends itself to likeike 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 the previous way that things had been phrased before in the same way that You know, everyone talks about like moveo thirty seven and like Alphao and such. Like, what is the thing that lends itself to just going outside the prior heuristics. It seems productive to have a disconnection from the rest of the world in that exploration as like a complelmentary research pursuit to the natural language math front I mean, the other relevance of Le in there would be like, okay, let's say you have your pure natural language RL environments. and you have a pure natural language set of proofs. and people have the said like proceed AI mathematicians and they go and they generate like T papers a day that produce a bunch of stuff Um, If the error rate, if there's like any error rate to that at all So Alex Konorovich has talked about this. It becomes inufferable, 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 ninety nine out of one hundred 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 track mark that says Even 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, every field would love to have something like that. I think you are Right that Lan is maybe overrated on the side of the importance of it being used as a the R environment for any kind of like just progress in math generally But I definitely wouldn't write it out of the story. Yeah I also love this extension of MathLV as A metaphor for like, what's gonna to happen to our civilization pretty soon. sure. Yeah right? It' like for millennia, humanity is building this like corpose of knowledge and understanding and everything that we have now distilled into these models. And at some point the models will just like extend that arbitrarily U By the way, on the writing front, I actually have I have a theory of why writing is making worse progress than these other domains. So I think O one of them is what you said that they're bad at judging not only A versus B, but they get like distortally derailed by DSar which is this like 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 offhere else. I think the other important thing is that writing is not modular in the same way that code and math are, like 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 clear and stuff. like at the end of the day, it works, it works. sameame with like Lemmas in mathematics And then, you know, you can like have some end product that is different. from the way it is produced. So the code is the thing that produces an end product and you want a functional end product Um 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 any it's a You can't just be it can't like be slop. in the way that like code can be slopped and still produce an outcome that you want. you were just pointing out how actually we've gotten much better at agents writing, not just functional code, but clean code. Why is it not the case that the same progress that allows you to go from merely functional to like clean and like a mergeable PR doesn't also result in clearer writing? 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 For a lot of writing I consume, I find it's better to just copypaste it into an LOM 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 these are such terrible writers And also my real preference is just like, can I just have another le and 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, and 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' just going ask NLM how that works and then we can come back to your special piece of knowledge. It sounds, I mean, that's distillation. R, an explanation And so if I'm thinking like quality of youiew 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 better at wr L what is writing? It's not just distillation of pree existing ideas. It's not just like how to explain clearly because they are good explainers.'s What is the insight? And this is this is where it gets like But just auto regression is a very weird way to generate stuff because like when you're writing, you sort of 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 like generated it and you needed it some author who throughrew 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 coherent, well motivated narrative. It's like they put that all together in some way. And you know, 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 explore it 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 cite them being bad at writing. it's that element of unpredictability of being deliberately choosing something that's novel.'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 which I think is a very important skill in writing. So Annie Matuchak and another collaborator, whose name I'm getting 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 's just like people are talking about recursive self improvement in a year. and you can't get these things to write good flash cards. And what's going on there, right They tried many different kinds of techniques and they're like know, sophisticated people. like they tried 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, etcetera And the key constraint, it 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 passastage you're trying to make cards about. I think writing also is similar to this where if you're writing something, you're like the reason it's such a ennervating process that takes so long is each word you should be th or each sentence should be thinking What is happening in my reader's mind right now? Even if I flip the phrasing around so the end phrase goes to the beginning and like this is the first image that comes to your mind before you read the rest of the sentence that kind of maybe auto aggression is bad at that kind of there maybe a more diffusion like property of considering the whole rather than going sentence by sentence. but also I think that that requires a lot of mentalizing, which these models we really struggle at. Well, I mean, interesting question, like is it weird that they struggle at that So I might butcher 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? 'specially these really good tests online that'll have like a face and then four possible emotions. And it's like surprisingly hard to like describe exactly the correct emotion, but you also get the 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 like do really well on it and the ones who are a little bit more like left brain, like dumb. Okay. So that is a kind of test you can do. I vaguely remember experiment to this effect where they took people who had freshly gotten like Botox in some way And they did like a pret test and a post test. and like post tests, they were just much worse at like reading people's expressions. L that feels kind of weird. Wh 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 of partart of understanding like this emotion that you're looking at is doing it yourself. That's Like at a facial level, you, 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 like 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 like read what everyone wrote 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 not surprising at. They don't face muscles. their brain works completely different. It's just like is 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. it's like we've got the ready made hardware to just like place it in. And so are interesting. From that lens, it's not that surprising Okay, Grant, we are both partners with Jame Street. I'm sure over the years, you've interacted with a lot of Jame Streeters. What have you found that's unique about them or their culture? I mean 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' getting an inside view of that R 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 colleague's actual role is because everyone's doing a little bit of everything else. L even if you're officially a trader, you're doing a lot of research, evenven if you're officially a researcher, you're doing a lot of coding And I suspect maybe that's part of 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. All right, Grant, I'll do the vlog 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 threeb oneb d. co slash Jane Street Hi, Grant, let's talk more about AI and math What advice do you have about Us LMs. to learn. So as I was describing for a lot of well known concepts, I find them Very helpful And but often just a couple of further messages down and I'm trying to understand something And I just They'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 could clear up my confusion in three minutes I don't know. And then I feel like more and more we're going to want to use these things as somebody these I' talked a lot about education and you know, representation stuff, we're going to want to use these things to learn things. So Um Yeah, haveave you noticed the ways to use them more productively to understand concepts I'm curious to hear your take on this. I mean I'll give mine even pre LLM, I feel like a relevant insight in learning was recognizing that like who matters more than what. So like advice to any college student when they're choosing what courses to take Care a little bit less about your prexisting 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 someone 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 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. on And I'm getting to like LLMs on this. So like There's a 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 philosophyic and you go to the Stanford Encyclopia Philosophy Or if it's a math topic, you go to the like Princeton C compendium of math where 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 cororrectness on the way. but you can deliberately craft things that are a little bit wrong that you correct along the way. that it gets like edited out in a crowdsource environment LLM explanations feel to me at the moment a lot like Wikipedia, whichich is to say Amazing, right? Like imagine world before Wikipedia, like how long it would take to like find and like Susin 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 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. Often I like to just ask an LLM like who should I read, And maybe I can even give some specifics on ways I want to learn. I actually got gaslit by this once where I remember trying to learn about like, Like semiconductors or something, I was like, this feels very visual. This is all like text. I'm like, Is there any really good like well visualized math video? r not math, sorry, a well visualized video kind of like explaining the concepts that you're getting at. and Claude was like Yeah, H here's a couple in the top one. It was like Here's one from Three Blow and Brown. I'm like, I can guarantee that there's not. go it. And it was an actual video, an actual link, but it just had like misattributed to someone else's And it was good. And it 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. Um What about you, like you engage with these a lot? What's the best? I you put your finger on it. The most productive learning sessions I've had is when there's Some artifact that a human has produed, whether it's an article, a book, a video 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 LMens to just do a little bit pruding around this u this this branch that the book has identified. So I was um was actually I was going through I think you might have recommended Stehven Strogatz's textbook on the Chaos one? Yeah, ChaS and nonlinear dynamic. Yeah. I love that book. And so I was going through it. And u It was like bliss. It was like your videos in like a book form. He's so good. It was super fun And the way I was learning it is like I'd have on one third of the screen his lecturer 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 would just totally go over my head. Like these kids must be really smart. 'causeuse I'm like pausing and like reading the textb book and talking about the Ls and then restarting again. But with him curating what is the right order to understand the concepts? What is the right problem to motivate? understanding a concept. Oh also another thing Is are really bad at is, um 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 The correct way to organize these concepts is X. Yeah. And Een just can't really do that. Yeah, it's a little too placid. 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. Um That's a good point. And I think that cuts to theory of mind a little bit.. 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 teher, let's say you have like a middle school like math classroom or something. if a student like asks a question that suggest they're thinking about it in a different way It's actually really hard to like take seriously in the moment, hang on, could you get to a right answer with that before you say inststead of that, let's do this. Yeah. And like the really good teachers are able to like ju jitsu the like creative way that the student was thinking about it and bring it in I mean, LLMs aren't doing that, right when they are not reframing your question instead, they kind of like run off. But t the very least, it feels like there's three levels here. and so like Een is at one good explainers at another, but then like the A plus explainer is the one who can like Jjits to 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 LLMs will still be doing that, but in the better way What is your recommendation to U students who I'm sure email you this questession all the time Look, I was curious about doing mathematics. I'm really passionate about the subject, but seeing all the progress the RIs are making it doesn't 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 Um Yeah, what advice to other people M I wouldn't trust any advice that I give. It would maybe be how I'd like couch it Even pre AI, it feels very important for any job that you're going to go into U to really understand like if we're talking about a job, right? We're not talking about like you're a 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. specially students, they're in this environment where they They 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 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 salary is flowing in my direction? It's actually quite different in different cases. 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 we'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 correctly them predict that your progress will be in the spirit of that fundy. Sometimes it's just straight up teaching, right? It's like people like 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 being a teacher. reggardless of whether AIs are like proving theorems or not, or like whether we're talking in twenty sixteen or twenty twenty six, That is a 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 you know, I just like wasn't necessarily thinking about it and kind of stumbled into this career path where basically exploration can be monetized as entertainment 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 I 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 theory improving. 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 is best spent 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 Gant writer who like really understands algebraic number theory to understand that it's a good result And so there's going to be some inner culture of what constitutes like valuable contributions. Maybe it shifts away from theory improving, and maybe it shifts towards like good definition writing. Maybe it's that museum curator idea. You're going to have that same community. And as long as society as a whole is still like valuing like the premise of basic science. And if we're in the abundance world of what AI brings Hopefull there's more funding in that direction in some sense, right? On the side of prestige to institutions for like who their lecturers are, I 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. L even if LLMs are good explainers, the thing that a teacher is doing is such a social like coaching mentor type thing that probablyably the one of the most stable careers that's going to exist over the next fifty years. Um, And so insofar as what a lot of mathematicians role is like overlaps with that. you know, you as the perspective student going into it, you could lean into that. Actually think a lot more students should think about and give 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 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. Kn where the money is coming from, know where 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 perspective mathematicians. Yeah. in fact, I think In the crazy world, in the world where within five, ten years, the AIs are coming up with notot only solutions to the milleniing prize problems, but coming up with like just totally novel problems to be solving in the first place, novel mathematical fields and objects and stuff It is in that world where first of all, there's a ton of abundance and two the things that EI minds will have like gone furthest in wouldould they will have seen like Furth this beyond our horizons, will 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 AIs 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 insofar as there's any economic utility to it, you would imagine that the people who understand it and are able to make the decision of where it should point, like they actually have a lot more economic value by being able to make that judgment as curator and point this like behemoth of like, new math like pointed in a useful direction. like suddenly that's a much more levered move to make than it had been previously. Can I actually ask you about that? So obviously the One question for AI for math is not only can it do it? Isn't it any good? Yeah Or isn't any good for anything You were describing the ways in which group theory we were 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 place where mathematics is The field of human mathematics is accelerated at ten x or one hundred x that um some crazy shit happens or are we just actually going to be bottlenecked by Other fields or I think there's some fields probably will I mean, it's super spiky, right? I think like progress in algebraic number theory, it feels unlikely that that then like unlocks some. But I don't know I remember talking to this mathematician who does more like Like dynamics and like PDE solving type stuff And he was referencing Basically like his group had some ideas that At least 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 simulations 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 progress 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 Um, like engine design becomes just a little bit more fluid or like coming up with the right wing shape instead of running a whole bunch of complicated CFD, or maybe you're able to speed up your CFD simulations because of certain pure math insights that makes those more efficient. I bet you'd just see like a lot of great incremental improvement there Um It seems less likely that the massive breakthroughs and math immediately turn into like this massive economic breakthrough. L you solve the Navier Stokes like problems, and then that unlocks an ability to simulate more things. But you probably will see like at those fringes just some some meaningful like leakage outside of the pure math insights into other things. alsoso, I mean, there's a ton of people working on things like you AI engineers, like physical engineers, like material science and things like that that would be I have to imagine that like 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. so It's another one of these things where I'm not going to sit here and like put a flag in the sand like predicting that there will be. would be a little bit disappointing and a little bit surprising if there weren't over the next five years. economically valuable improvements that were made that were directly like referable to the AI progress in math. that just would be kind of disappointing if it was just ticaking down a bunch of air dish problems and like none of them actually you know, it wasn't doing any of the math that actually directly touches physical world. Yeah. too your point about, well, a lot of history and mathematics was about like building up these like piles of concepts and connections and whatever. And sometimes connect with each other or you discover an application somewhere else At the very least, you just build up this huge pile. and as you broader progress in society happens during the singularity, when like we get to the industrial part of the singularity, you just have all these different ideas that you can Hopefully are useful You know the front 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 revealing like, o, man, over the last like it's just become wholly useless. like the kind of questions being asked have become like so divorced from things that are physically applicable that that's one of the things mathematicians have to come to terms with. Everyone will look and be like, ye on a second. like whereere are' you guys supposed to? if there's so muchs like ten nexts progress there. Like why aren't we seeing it over here? And then M church is like,, everyvery time we wrote those grant proposals, it said, like trust us, the Ellipptic curve progress is gonna help with cryptography. Like it shines a light on the fact that like maybe it doesn't. So that's one possibility U Gant, this is super fun Thanks so much for doing it. Absolutely my pleas