Grant Sanderson – AI and the future of math

23:12And even though this isn't really the angle I'm taking, there is something to the idea that the smaller expression that's more predictive feels more intelligent. And so I 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 Riemann hypothesis solutions, what would that look like if an AI solves it? I think a third way that it could happen is it just straight up works harder, right? In the same way that you could maybe have an elementary proof of Fermat's last theorem that's just like spelled out over like thousands of pages that would be incoherent.

23:48But like the cleaner way to view it is with elliptic curves and all that. Maybe there's some like thousand page proof of Riemann hypothesis that's like no one's really getting anything out of it. And what you actually want is like, what are the succinct, like compressed versions of those ideas like that would then lend themselves to human understanding? Like, I don't know, Komogorov complexity. Like maybe you throw that into your, like your attempt to quantify what you mean by elegance. But I don't think it's easy, but I do think it's something you would have to do in order to reward the Galois-like instinct rather than just rewarding, have you solved a problem?

24:22Yeah, it's very hard to come up with the heuristic for science. But it's clear humans have been doing this somehow, and obviously AI's will do it at some point. Well, it's relevant also not just in terms of verified reward, but presumably the end goal is understanding, human understanding. And so even if you do have some thousand-page proof of some math thing or some grand new physical theory, the goal is understanding. 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.

24:56But 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. Here's this complicated way of thinking into like the right one, like the equivalent of the universal law of gravitation for Newton. Yeah. Like you would still want to train AIs to be able to do that and like find the compressed representation. I grew up in India till I was eight. And so in addition to English, I also speak Gujarati. And since Google just released Gemini 3.5 Live Translate, I thought it'd be fun to put it to the test in this mid-roll.

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26:03It's available now via the Gemini Live API and in AI Studio. Go to ai.studio slash live to get started. So people have this worry about mathematics in particular, that the AIs will prove the human hypothesis 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 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 a complicated, important problem.

26:40And so we can just think about like theoretically, would this even be a simpler way to solve the Rayman 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 counterexample 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 mathematicians, it was like understandable.

27:08And it made use of like known concepts of mathematics and like proved relationships between them and all in natural language. And as a result, accelerated our understanding of the connection between this object and this conjecture. So is this even 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 Riemann hypothesis, that one and the other like big one from this year was like a certain erdish problem numbered like 1196 but it's a 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 like use this uh like try the markov chain process where we uh show that this thing is one from the bottom up probabilistically rather than the top down and like use the von bengold 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.

28:02So you have this very like small idea that has the form of expertise in one field, expertise in another, draw a little lightning bolt between them. Like those are, those are going to be very human parsable, right? Because all you have to do is just like show the start and end point of what those connections are. If the character of it is mountain building, you do have to, you have to put in a lot more time to like understand that new mountain that was built because it's like a new thread that's not just like lightning bolt between them. And if the nature of the progress was just like raw hustle, right?

28:29It's just like this super long thing, no new theories, but it's just like long, long, long chain of reasoning answer. Then you would have that where it's like, okay, there's this whole digestion process. So I don't think there's one clear answer. I think it depends on what the solution there would look like. And on the mountain building side, I would actually be really interesting to see. Like is it by default a very human understandable, like the way that we see new theories from great mathematicians? or is it like an alien, different kind of mountain being built where we even have to reprocess the kinds of abstractions that we engage with?

29:02Right. Well, the closest example here would be the attempted solution of the ABC conjecture. We maybe shouldn't get into that one, but it's probably not a correct solution, but basically it's this whole new way of thinking that this otherwise reputable mathematician in Japan had come up with. and it just took mathematicians like a long, long time to even parse what he was saying, but it had the feeling of just like an alien bit of mathematics that's theory building. It's not just like long chain of reasoning. It's like, he called it like inter-universal geometry or something. And so the fear that you would have is that like, yeah, it like does that.

29:39The biggest fear would be that it does that. And then much like the ABC conjecture, like people work for years to go up the mountain and they're like, this just isn't right. Right. And like, if there, if it turns out to be wrong, but it like really looked right. But even if it was right, there's, There's just a lot of effort to hike up a new mountain. Yeah. If we end up in that situation, David Bessis had a really great blog post called The Fall of the Theorem Economy, where he's talking about this, you know, historically, as you were saying, mathematics is coming in about these definitions and problems, and it's about proving theorems about them.

30:12And that really the theorem proving stuff is what gets all the credit, but it's like really a parasite on the definition stuff. And historically, it's not even a problem in terms of credit apportionment because if you come up with a definition, you're probably going to be the guy who comes up with a theorem. But now we're in a situation where if the valuable work is coming up with the insight and then AI just automates the latter part. So, okay, imagine a scenario where we have AI comes up with like the ABBA-like direct arguments about a bunch of important conjectures in the world. And then we just have these proofs.

30:46and now it's up to humans or to future AIs to then consolidate. I mean, I'm sure if you had access, again, I'm 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 why this proof works that would make it easier to come up with the ideas behind group theory? Yeah, I think it would be hugely helpful, right? Because I mean, so much of like trying to discover new math is like mostly being wrong, right? You're like trying to solve a problem.

31:21It 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, like 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 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.

31:52And 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 mathematician named Timothy Chow, 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 have a size of infinity for the real numbers. Is there something in between? And the answer is both yes and no. It depends on your axioms. Like, it's sort of outside the scope of our usual axiom systems, which is an interesting answer.

32:23But the method to describe it is just really, really hard to understand. It's the thing called forcing. And in the beginning of this paper, he writes, like, I want to, like, everyone knows the idea of an unsolved research problem. Like, I want to propose the idea of an unsolved expository problem. Where like, sure, we've proven it, but we don't really know why it's true. And so then he proposes like a partial solution to that expository problem. You can imagine why I loved that framing, because like this is my whole life. It's like, I don't do research math. It's just, it's just wholly about like, what's the most clear way to understand this, even if it's proven?

32:55Just 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 for, or the incentive would have to change in not just mathematics, but in other areas of science from proving things about the world to consolidating proofs into problems or higher level insights. But we were having a discussion earlier at lunch about a recent talk you were giving about design and how it helps us understand things. And then in The Limit, is there really a difference between the conceptualization for an idea and the idea itself?

33:37So, you know, if you think about special relativity and like space-time diagrams and Minkowski's space-time, is it like, yeah, this is like a way in which we illustrate this idea of like why there's length contraction and time dilation. But is that like, like that is the reality. So 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 are actually quite clear in their communication of it. Like, you might imagine, given that the experience of a university student is often that the expert they're teaching them is not necessarily the best explainer of that topic because they are so spoiled by their expertise.

34:21But what seems, at least in some cases to be the case, is how the people who are really coming up with something quite novel. So you've got like Einstein or like Claude Shannon, something there. You read their papers. They're really lucid papers, right? It doesn't feel like, oh, this is just for the experts and you have to chop through it with a machete to get. They're like very good expositors. Like Feynman has this characteristic too, like very good expositor. And so maybe the same part of the brain that comes up with the correct new way of thinking about it at a research level also has this knack for like good explanation.

34:51And I think this is pertinent to the AI one, where I kind of used to think that AIs will become these automated theorem provers, but like the role of the mathematicians is going to shift towards like my job, like explain these things. I kind of suspect that actually they'll also be like quite good at doing that 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 is like digesting and explaining what was going on. Probably the nature of how these things are going. I could have envisioned it.

35:24We can talk about like ways this might not be it, but like probably the same thing that is coming up with like the really good new idea that solves some new problem is just also good at explaining it. That's my new, like that's a way my, I think, beliefs have changed. What's the last thing you think you'll be doing? Both you and then also with the mathematical community, the human mathematical community. what we do i will probably be doing something like what i am until i die even so like even i have the doomers right maybe that'll be the same exactly it'll be for the same reason yeah yeah you know it's um you like uh 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 um so that's where i am with ai no i because some 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.

36:15That's one thing. But some of it is like a little bit more relational and a little bit more like providing like motivation, providing a sense of curation. Like one interesting take that I've heard about like what mathematicians will end up being is actually more analogous to art museum curators than anything else where the AI solve the thing. So the art exists, right? They even know how to explain it really well, you know, there. But like, 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. And 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.

37:00If you have some specific technology you're trying to build, you know, that might be different. You need to know there. But I 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 what they in a prior sense wanted to understand. They're trusting you as a curator. So my role, and arguably that of like other mathematicians might actually just shift subtly into that curation direction of what ideas are worth displaying. And that's a lot of my job right now, even now it's basically like i think people think a lot of the time for a video goes into the visuals like sure a little it is not like immediate but like actually a lot of it is just deciding what's worth saying in the first place or what's 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 perform even if the ais are better than that in the same way that like human musicians are always going to have a role because of that like social function of the story behind them, even if the like objective quality of the MP3 file coming out is like better from some model.

38:05That's kind of what I see happening to my job. Yeah. I want to go back to this question of earlier I was, we were sort of, just as AI has crossed this threshold, this important benchmark of being able to connect existing ideas to come up with a new discovery or prove or disprove something, just as it's crossed the threshold we're like, okay, but 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 of years of, like, really connecting.

38:35And so, in The Limit, you could even say... I don't know if this is accurate to say it, but potentially... A lot of maybe the biggest breakthroughs, like, look like this at some level. It's just general relativity. Oh, you're just connecting together, like, Ramanian geometry and special relativity, right? And so as AIs keep getting better and better at this connection thing, maybe a lot of big breakthroughs are not really of a different qualitative nature. I don't know if you have a take on that. Well, I mean, a lot of the conversation focus has been on problem solving and that nature of math, you know, like taking off Erdős problems or something.

39:11I would say it's not even a majority of mathematicians who would maybe characterize their work as really targeting the next problem to take down. Are you familiar with the Langlands program? No. Ah, okay, so this is like, it's not even a field of math so much as it is like a research ethos where Fermat's last theorem is one inkling of this. You had like these two different seemingly disparate things and a connection between them like led to a solution. So Langlands was a mathematician. He has this like famous letter now essentially spelling out how it seems likely that there's a lot more connections like that.

39:45and 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 like valley over here and this mountain over here and this like set of plains 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. It's more that there's been enough time and time again cases where big problems were knocked down by finding connections that it's almost preemptively finding the connections.

40:17And so you could imagine. Yeah, it's actually very interesting. Like this, anytime you run into a mathematician, it's like to ask them whether, you know, the character of their work is more akin to like Langland's program or if it's more akin to like targeting one particular problem, right? And you get a certain like bifurcated split there. But the possibility of AIs being supercharged connectors feels like it might be, you know, an amplifying tool in that pursuit. It's hard to measure, though, right? Like, because this cuts to what we were saying earlier. How do you assign a score to say, like, yes, you've done it?

40:55If 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. This is the nature of what the papers in that field look like. But 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? But that's my guess on what most of the useful progress from these models will look like, like in the next five years is just really filling in that landscape of like connections that you can draw if you're an expert in multiple fields.

41:32Like you pointed out, it's kind of surprising we haven't already had this. And what I'd be curious, like I would be curious to know at a technical level, what causes the unlock there. Because on the one 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, like the method of reasoning is this autoregressive chain of thought phenomenon. Autoregression is actually like a really, really weird way to produce stuff, I think, if you think about it. Like, you're an intelligent person.

42:07Imagine I've locked 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. Imagine that was done a whole bunch. And then what comes out on the other end, they're like, look at this essay that you wrote. You might look at that and be like, this is awful. That's not the essay that I would have written. Right. Because like the process of like repeatedly like predicting something is just pretty different from how you would think as a writer to like compose it and think it through and everything.

42:39And in particular, what would probably happen is you're sort of a slave to your context, where you might be answering some question about some particular field. And so you, like, draw on all the context around that, and you're going there. The connection that actually is where all the substance is going to come from is, 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 upweighing and incentivizing, making these unlikely connections when the vast majority of them like aren't the predictable, you know, next token that would come in there.

43:12And so it's like, it might be the case that you just have this intelligence that 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 find the right ways of sparking those connections that like unlocks these sorts of things that we're seeing?

43:46Or do you need just a little bit more intelligence such that at the level of prediction, it's kind of predicting that it should be making that lightning bolt to another field? I think it's more productive to reason instead of architecture or even loss function to reason about data. Like, I don't know, we have diffusion models that do text and they're like, the kinds of things they produce are not of a wholly different character. They're just not being explored as much. I think the more relevant thing is what is the data on which whatever architecture, whatever loss function you have is incentivizing you to produce.

44:24And 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 agents, it just, I don't know. They have like, they're in an environment where auto-regressively producing the step that says, let's step back and do a search over the whole code base. And then let's step back and like assess my mistake. Is like the thing that works. I assume what happened in the case of progress in science or maybe in math is you have frontier math-like problems, which require, like, mathematicians specifically designed them because they require connecting together two different fields.

45:05And I'm guessing there's all kinds of clever, like, partially synthetic ways in which to make harder and harder problems like that that require these kinds of connections, for example, by, like, eliminating assumptions and still requiring the AI to continue to get to the answer. and then like it just doesn't really end up mattering what the loss function is. It's just like it's really about can you come up with an environment which incentivizes this ability. Yeah, it feels like you should be able to. Yeah. I certainly can't speak to the correct ways of doing that that like unlock all this, but it would just be pretty surprising.

45:38Like 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 paralyze and arbitrarily scale them so that whatever level of capability they have, it's not just like one idiosyncratic genius in the history of mathematics who makes a few connections and then dies in a duel.

46:13It'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 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 parallelization is like quite an important property. And I'd be curious about your predictions of, even if they're not as smart as your mathematicians, the fact that they are just, you know, billions of, because for PR reasons, the AI companies are just dumping billions and billions of dollars at this, would have a, quantity has a quality all of its own.

46:56That 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 Riemann hypothesis or Riemann zeta function zeros 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.

47:32What does it look like to sort of engineer those between agents? I mean, it's interesting because you sort of point out like you can sort of pool all your knowledge. So I actually wonder if one of the advantages is that you can do the opposite of that, where you have sometimes when an AI is failing, it's because it sort of gets into a bad chain of thought, and it's really hard to get it out of it, right? So you're like, I'll just like start again. Same deal with humans, right? Like sometimes you like start thinking about it in a certain way. And actually, what's required is to just like back up maybe sometimes the form of that, you know, there's stories about people trying to prove something for a long time.

48:04And then at some point, they say, hang on a second, what if I tried to prove that it's impossible. They prove the opposite. And that unwinding your own context and going at it with a fresh mind, you could imagine systematizing that. Or having multiple different agents deliberately given different pieces of context and try to compare and trust there. We don't have the same level of manipulation on our own context. In this AI and math series, the first episode will be about when they solved 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.

48:37Terry Tao also failed on it. And the nature of it is basically that people were very mad at the problem because they called it a troll problem. I almost don't want to spoil it because I want to construct the episode around leading someone in without knowing that it turns out to have a simple solution because you can really empathize with what it's like to be a student solving this. basically there's a really elegant way of going down what you really feel like is going to be the solution based on the context of being the international math olympiad problem positioned as it is the like character of the solution is like really enticing but it's kind of hard to prove that it's the best the reason is that it's not there's like this almost brain dead solution that is the best and so the like relevance of that to the whole ai story is like for a human what's required to answer that question is to like escape your context escape the context that you're in the IMO.

49:29Escape the context of the way you've been trained to solve these like 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 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, Like, what does it look like to, like, refresh your thinking, try answering two separate questions, like, spin off two agents, one who's trying to prove it, one who's trying to disprove it, one who tries it like this way, one who tries it.

50:06And 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. It is incredibly interesting because a common concern people have about AIs is this entropy collapse, where they all think the same way because they're trained in similar ways. This is why they're bad at writing. They kind of just like go down the same path and have similar patterns of speaking and so forth.

50:39But maybe actually the key advantage AIs have is that you can systematically, it sounded like one of the reasons the unit distance problem conjecture took so long to be disproven, which because people assumed the conjecture was actually true. So mostly they were trying to figure out ways in which to prove it. And so maybe one of the key advantages they guys 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 systematically give different agents different biases. That's a good point.

51:17Like it seems like an important thing in the history of human science is that like Einstein is just really motivated by this bias that like things should look the same in different reference frames. And then he had multiple other biases like this. But like that is just a very formative in his thinking. And you can just like systematically survey a bunch of heuristics and see which ones are being productive at a given problem. Yeah. And so you would suggest basically like systematically increasing entropy at the prompt level, even though you have this like inevitable collapse at the like auto regression level.

51:47Yeah. Yeah. And I mean, Einstein would be an interesting example because it's like he's got this bias towards things should be able to. 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 progress. Right. Which actually goes to show you that there's not a correct heuristic. Exactly. For science. Exactly. You actually just need multiple independent research programs with their own heuristics. Yeah. Yeah. And that feels like old school software, right?

52:13As long as you're able to like describe that in some way, you have like old school software that like 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, you like explore that full ontology. And then each individual one, you know, 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?

52:44And make sure that you're like sufficiently, applying sufficient breadth to exploring that. I don't think people appreciate the kinds of things that these models can just go handle for you when you equip them with a good harness, like Cursor. For example, I started publishing my episodes on Bilibili for a hopefully burgeoning Chinese audience, but everything I upload there needs the sponsored segments cut out. Normally, that would have meant that I would have to ask my editors to go back through all the old episodes, cut out the ads, and re-export everything. But in about just as much time as it would have taken me to send them that Slack message, I can just tell Cursor to do it instead and spare them.

53:19And for research for the podcast, I have a whole repo that I've set up where I've just put every single book and paper that's been relevant to prepping for any of the recent episodes. And I've been able to hodgepodge everything because the cursor harness is just extremely good at helping the model figure out exactly what information to pull, whether that's from my repo or from the web, in order to answer the questions I have while I'm doing research. So whatever you happen to be working on right now, just try pointing cursor at it. Go to cursor.com slash thwarkash to get started. Obviously, AF or math is making a lot faster progress than everything else, and people point of verifiability of the domain as the key reason this is happening.

53:56I think that's one of the two important reasons, but I don't think, I think people really neglect the other one. And I'm outside the labs. I don't know what's actually going on, but this is a totally naive theory. Okay, a tangential question to why AI is making so much progress in math. Why has it been so slow to computer use? Which is what you, you know, computers is actually very verifiable. It's like, you know, is my Etsy package coming? Or like, is my event booked? You know, whatever. These are extremely verifiable things to survey. What computer use lacks is grindability. So because websites have like bot detectors and also it takes a tremendous amount of compute to run parallel rollouts, it's very hard to just run like a thousand parallel rollouts at the same checkout flow on Amazon because you'll get like shut down by Andy Jassy, right?

54:46And so you can personally... Presses the red X on Dworkesh button. Exactly. And so you can try to build clothes on every single website. This is very labor intensive and slows you down. And the reason, by the way, you need to do so many parallel roles in order to learn a skill currently with deep learning is that we haven't solved sample efficiency. Sucking supervision to a straw, like that's 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 you're training AI. that we just, with code also, you can containerize a given level of progress in a repository and then just spin out thousands of parallel containers or hundreds of parallel containers and say like, try to implement this feature.

55:29And it's totally deterministic. And because it's deterministic, you can solve the credit assignment 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 credit assignment problem. If you have situations that are starting off at different starting points, this credit assignment problem becomes much harder to solve. But most of the things in the real world are just very hard to containerize in the same way. Like coding and math are exceptions to this rule. But if you're just trying to figure out, how do I build a new business that succeeds?

55:57How do I like go trade in the markets for a day and like make money? You can't like, 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 of progress in this domain and also 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 lab.

56:33I feel like lean just doesn't matter that much for the current level of progress in AI. Or why is AI able to solve the unit distance problem? Well, they—or sorry, disprove the conjecture by the unit distance problem. they released the chain of thought or at least a rewrite of the chain of thought. Didn't have any lean in it. I think it just like the process-based supervision that lean provides where you know each step is correct seems like less relevant than just having this grindable outcome that is verifiable. It's an interesting point, like grindability mattering more. I guess I will say on the, yeah.

57:05Okay, so naively you might think lean provides something unique for math because you're able to see if it can prove it. you have old school software that can tell you yes or no you use that as your vr i mean what so what would corroborate your point is the idea that like the initial attempts again i'll just circle back to imo it's like initially deep mind basically does that it's like everything in lean and then the next year it's all in natural language so to your point not needed i do i think there is a um a yet to be explored benefit of that formalization domain which is at the moment you still need you know ultimately like a human is reviewing that um counter example to the unit distance conjecture to say looks good.

57:45And that provides a certain bound on how like endlessly explorable things are. Like if you consider like AlphaGo, AlphaZero style stuff where they're just like off in their own universe, just like playing a bunch of Go and exploring themselves, just completely going potentially off the rails of what any human needs to look at, but they still have this automated verifiable reward. It's not just that, hey, you can do RL on that. But it's also, you basically never have to check in and you can just like poor 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 jury should still be out on like whether this will yield anything.

58:24With Lean, you could imagine having a basically endlessly running program that's constantly trying to extend Mathlib. So Mathlib, it's this GitHub repository that's basically like all of math 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 sub community around it. But you could imagine what if you just had an AI where you say simply try to extend Mathlib. Maybe it's a fork of it so it's that it doesn't have, you know, like trash in it because people, you know, people have certain taste for what they want to be in there.

59:00So you have like your fork of like the pure AI Mathlib and it just goes and 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 different definitions. Maybe many of them are useless, but it just has this infinite tree that it can grow out. That's a very unique thing that math has that nothing else has, where you could press go and then just pour compute at it and look away for 10 years and then come back and say, what do you have? And there's going to be something, right?

59:30And then there's a question, is it useful or not? How do you suss 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, okay, there's like two different ways that like Lean is important in this story. That's the first one of them basically is how it's like you could let go, not even check in, and progress will be made. You can do that with Go. I don't think you can do that with natural language math. That's very interesting. Did you see Karpathy's auto-research idea?

1:00:02He wrote this basically one Python file that does basic LLM training and then just had a repo where agents would try to make modifications to the file if it sped up the speed run, the modification stays. Eric Jang, who came on to explain how AlphaGo works, did a similar thing when he was trying to build in a very strong Go bot. And he had interesting observations about the kinds of, like, it's really good at just running an experiment and going down that path, but it's bad at stopping at dead ends and just doing extremely parallel things. Anyways, this will probably be changed. This will change in the future.

1:00:41It'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 in interesting and useful ways. And this way you don't have any outcome-based supervision. 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 through the space of logic.

1:01:09You'd 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 like be able to get the future version of us like talking about whether, like maybe that goes nowhere. But Terry Tao was talking about one like research project. It's basically try to exhaustively search the space of possible like algebras. Like you could imagine different like axioms that you apply to algebraic systems. and so like when we come up with group theory there's a certain axiom system that like has this flavor of they kind of look like arbitrary rules unless you know the motivation but it's basically like what have you tried all of them do any of these yield useful things and like the vast majority of them is just trash in some way like it all collapses to like no interesting results but like every now and then there would be this little island of like 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 is like exploring that space and seeing which one of them turns out to be something and like maybe one of those islands actually turns out to be something you can retroactively put motivation on to say this is the kind of structure that's trying to get at in the same way that you could imagine looking at the axioms for a group not knowing that it's about symmetry but retroactively realizing like wow this is very relevant to studying symmetry so you could imagine results of that flavor but instead of just exploring possible algebra systems.

1:02:34It's like all possible like logical consequences of any kind of axiom. On the point about whether you can provide process-based supervision without Lean, so DeepSeq had their DeepSeq math model and they released a paper on how they trained it. And it was quite interesting. So they have, the problem with having natural language proofs 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 makes sure that all the problems that they're training this model to solve in like the art of problem solving, that the verifier is giving good feedback on that.

1:03:08And it like, it works. And so it's just interesting, natural language verification with some sort of meta-verification kind of work, at least seems to work so far in the published literature. And also it seems to work in the published products that we're using. Like if you look at coding agents, they're getting better and better at like writing clean code and refactoring code and stuff like that. And I'm sure that there's process-based like LLMS 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, are there duplicates of the same kind of modular forms and so forth?

1:03:43I feel like that should also work for mathematics, right? It's like, it doesn't seem... It seems more plausible for math than anything else, even if you're only working in natural language that you could trust a verifier. I mean, you and I were talking earlier about why they're bad at writing. And, you know, I was asking like why you can't just have... they seem to be good judges. If I give them two essays that students write, they'd be able to say which one's more accurate and insightful. So why can't you just have a verifier saying, is this a good piece of writing or not? And maybe the ultimate failure there is even if they're good at discriminating between a B essay and an A essay, they're not actually good at discriminating between an A essay and a thing you actually want to read that would be followable on Substack and insightful and all of that, like they actually end up preferring just uninsightful pieces of writing.

1:04:30And so on the math front, I guess the question would be like that step to simply know like, is this a correct proof or not? That lends itself to like an automated verifier, even a natural language. You could probably still make a ton of the progress. It still doesn't like, I still like the sort of tree of logic out of lean front, just in that you can really go off the rails, right? Like there's just no constraint on like the previous way that things had been phrased before in the same way that, you know, everyone talks about like move 37 in like AlphaGo and such. Like what is the thing that lends itself to just going outside the prior heuristics?

1:05:05And it seems productive to have a disconnection from the rest of the world in that exploration as like a complementary research pursuit to the natural language math front. I mean the other relevance of Lean there would be like okay let's say you have your pure natural language RL environments and you have a pure natural language set of proofs and people have to said like proceed AI mathematicians and they go and they generate like 10 papers a day that produce a bunch of stuff if the error rate if there's like any error rate to that at all so Alex Konturovich has talked about this it becomes insufferable 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.

1:05:51Even if 99 out of 100 of them are right, I don't know if it's worth my time to even go through it because it's really labor intensive to find what that error would be. And it's like really frustrating if it turns out you spent all your time on a paper that was trash. And so having anything that's able to give you that green 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.

1:06:21And so that seems huge, right? The ability to have that. Like every field would love to have something like that. And so I think you are right that Lean is maybe overrated on the side of the importance of it being used as a VR environment for any kind of like just progress in math generally. But I definitely wouldn't write it out of the story. Yeah. I also love this extension of MathLiv as a metaphor for what's going to happen to our civilization pretty soon. For millennia, humanity is building this corpus of knowledge and understanding and everything that we have now distilled into these models.

1:07:01And at some point, the models will just extend that arbitrarily. 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 one of them is what you said that they're bad at judging not only A versus B but they get like just totally derailed by B star which is this like shitty essay that just hits all the all the bells and whistles that like A is supposed to hit and then so the reward hack thing just like totally goes off the rails but I think the other important thing is that writing is not modular in the same way that code and math are.

1:07:39You can write a function many different ways, and they kind of do the same thing. And of course, you want it to be very clean and stuff. But at the end of the day, it works, it works. Same with lemmas and mathematics. And then you can have some end product that is different from the way it is produced. So the code is the thing that produces some end product. And you want a functional end product. Whereas in writing, the end product is directly the thing the AI is producing. and each paragraph, sentence, word matters because that is a thing that is like, like that is the substance. It's not like some separate thing that is produced out of the writing.

1:08:16And so it, any, it's a, it can't just be, it can't like be slop. It had, in the way that like code can be slop and still produce some outcome that you want. But you were just pointing out how actually we've gotten much better at agents writing not just functional code, but clean 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 like clearer writing. Yeah, that's a good point. I mean, also, has it not? Like, I agree there's many ways in which they're terrible writers, but for a lot of writing I consume, I find it's better to just copy paste it into an LLM and just say like, explain this to me.

1:08:56The explanation will be better than the thing that is produced by the human. So it's funny that we say like, these are such terrible writers. and also my reveal preference is just like, can I just have an LLM explain it? Even when I'm talking to a human expert like live on a call, if it's a piece of knowledge they have that only they have that's not encoded in the distribution, I want them to explain it to me. But then if in order to understand that, I need to understand a more basic concept. I would prefer if it was socially acceptable for me to just be able to say, let's pause there. I'm just going to ask an LLM how that works and then we can come back to your special piece of knowledge.

1:09:30Well, it sounds, I mean, that's distillation, right, and explanation. And so if I'm thinking of like quality of you as an essay writer, if it's that I give you a book to read and I want a book report, right, then I might believe that, okay, the LLM maybe gives me a better book report. But I think what people are really getting at when they say it's bad at writing, like what is writing? It's not just distillation of preexisting ideas. It's not just like how do you explain clearly because they are good explainers it's like what is the insight and and this is this is where it gets like just auto regression is a very weird way to generate stuff because um 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.

1:10:25It's like, what generated that book that you wanted distilled in the first place, right? It wasn't an LLM that like generated it and you just needed it. It's like some author who threw a lot of exploration of ideas in the world and then deciding what aspects of it were interesting and which ways of presenting it were like the coherent, well-motivated narrative. It's like, they put that all together in some way. And, 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 still, what makes it worthwhile to, like, explore at all in the first place, and you're uploading it at all, I think it's all of that side of it that's the, like, when people will cite them being bad at writing.

1:11:03And it's that element of unpredictability, of being deliberately choosing something that's novel. That's, like, very directly contradictory to, like, the way that things are being produced. Yeah, that's a good point. I think they're also really bad at building really good mental models of people, which I think is a very important skill in writing. So, Andy Matushak and another collaborator, whose name I'm forgetting right now, did an interesting report where they tried to teach LLMs to write good space repetition prompts. And I really like this because, even though it seems like a really totally random skill, it's just like people are talking about recursive self-improvement in a year.

1:11:37And you can't get these things to write good flashcards. And what's going on there, right? And they tried many different kinds of techniques and they're like, you know, sophisticated people. like they tried to rl open source models they tried all kinds of including chain of thought and the big prompt they sent to the best close source model etc and um 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 what kind of answer we'll be thinking by the moment and is that is the is the uh elicitation that inspires the detail you actually want to take away from the passage you're trying to make cards about i think writing also is similar to this where if you're writing something you're like the reason it's such a innervating process that takes so long is each word you should be thinking or each sentence you should be thinking what is happening in my reader's mind right now even if i flip the phrasing around where 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 regression is bad at that kind of um this It's maybe a more diffusion-like property of considering the whole rather than going sentence by sentence.

1:12:51But also I think that requires a lot of mentalizing, which these models weirdly 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. you show a flashcard of someone's facial expression, and someone's trying to describe what's that emotion. So I think there's really good tests online that'll have a face and then four possible emotions.

1:13:25And it's surprisingly hard to describe exactly the correct emotion, but you also get the sense there really is a correct answer. And if you try this with people in your life, you'll notice that the ones who actually are pretty plugged in socially do really well on it, and the ones who are a little bit more left brain don't. Okay. So that is the kind of test you can do. I vaguely remember an experiment to this effect where they took people who had freshly gotten like Botox in some way. And they did like a pre-test and a post-test. And like post-test, they were just much worse at like reading people's expressions.

1:13:56Like that feels kind of weird. 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 part of understanding like this emotion that you're looking at is doing it yourself. Like at a facial level, like you're moving your face 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.

1:14:32But at a level of like actually able to put themselves in your shoes in the same way that like my face muscles are mimicking your face muscles. That's what helps me understand how you feel. Not surprising at all. They don't have face muscles. Their brain works completely different. It's just like, it's like an alien trying to empathize. Like how could it have theory of mind? It would be like this very emergent thing to have theory of mind. Whereas we can just like plug it into our own minds. And it's like, we've got the ready-made hardware to just like place it in. And so. That's very interesting.

1:15:00It's not that, from that lens, it's not that surprising. Okay, Grant, we are both partners with James Street. I'm sure over the years you've interacted with a lot of James Streeters. What have you found that's unique about them or their culture? I did this interview with them this year that partly was interesting because they don't usually have anything outward facing. I mean, in the industry they're known as having a pretty wild retention rate. Like people just stay there and getting an inside view of that. I remember one of the comments someone was saying, even though the people have role titles, like you know, researcher or trader or engineer, they often don't know what their colleagues' actual role is because everyone's doing a little bit of everything else.

1:15:34Like even if you're officially a trader, you're doing a lot of research. Even if you're officially a researcher, you're doing a lot of coding. And I suspect maybe that's part of 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 plug for you this time. If you want to watch this full sit-down interview that Grant did with some of the folks there, go to 3b1b.co slash Janestreet. All right, Grant, let's talk more about AI and math. What advice do you have about using LLMs to learn?

1:16:07So as I was describing, for a lot of well-known concepts, I find them very helpful. But often, just a couple of further messages down, and I'm trying to understand something. And I just, 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 can clear up my confusion in three minutes. It's, I don't know. And I feel like more and more, we're going to want to use these things as, somebody who's taught a lot about education and representation and stuff, we're going to want to use these things to learn things.

1:16:40So, yeah, have 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 pre-existing interests because they're kind of arbitrary right now and care a little bit more about whether like 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, like who the author is maybe matters more than if it's a prior interest.

1:17:16So if there's a book you've liked before, read what else that author has written rather than reading another thing on that subject. And I'm getting to like LLMs on this. So like there's a difference in feel for trying to learn something if you look at a Wikipedia page of it versus if you look at, let's say, it's a philosophy topic and you go to the Stanford Encyclopedia of Philosophy. Or if it's a math topic, you go to the Princeton Compendium of Math. The difference there is the articles are deliberately written by one individual who tries to actually craft a motivation around it and everything.

1:17:48Whereas Wikipedia, it's this local minimum that's reached where basically every sentence has to be correct. And I think a good exposition, you care a little bit less about like correctness on the way, but you can like deliberately craft things that are a little bit wrong that you correct along the way that gets like edited out in a crowdsourced environment. So like that LLM explanations feel to me at the moment a lot like Wikipedia, which is to say amazing, right? Like imagine world before Wikipedia, like how long it would take to like find and like susan and everything. But nevertheless, what's the most useful part of a Wikipedia page?

1:18:24It'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 better overview of it. So 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 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? Or not math, sorry. A well-visualized video kind of like explaining the concepts that you're getting at.

1:18:56And Claude was like, yeah, here's a couple. And the top one, it was like, here's one from Three Blue and Brown. I'm like, I can guarantee that there's not. And it was an actual video, an actual link, but it just had like misattributed someone else's. And it was good. And 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. What about you? Like what you engage with these a lot.

1:19:29What's the best way to do it? I think you put your finger on it. The most productive learning sessions I've had is when there's some artifact that a human has produced, whether it's an article, a book, a video that organizes the relevant concepts in the correct way and builds up the motivation of why building up the next idea would be relevant to solving the next problem you didn't encounter. and the next idea and the next idea. And then using the LLMs to just do a little bit pruning around this branch that the book has identified. So I was actually, I was going through, I think you might have recommended Steven Strogatz's textbook on - The chaos one?

1:20:09Yeah. Chaos and Nonlinear Dynamics. I love that book. And so I was going through it and it was like bliss. It was like your video is 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 like lecture from university. On one third of the screen, I'd have that part of the textbook. And on one third of the screen, I have an LLM. And I was actually thinking if I was back in college and watching this lecture live, it would just totally go over my head. Like these kids must be really smart. Because I'm like pausing and like reading the textbook and talking about LLMs and then restarting again.

1:20:43but with him curating what is the right order to understand concepts what is the right problem to motivate understanding a concept oh also another thing LLMs are really bad at is a thing a really good human can do is when you ask a question they say like actually you're just like not really thinking about this topic the correct way yeah like the question you want to be asking yeah the correct way to organize these concepts is X yeah and LLMs just can't really do that yeah it's a little too placated 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 that's a good point and I think that cuts to theory of mind a little bit yeah like recognizing that to ask a certain kind of question reveals that the mental structures are not at least they're not the same as what the like explainer has and sometimes people do this to a fault right like I think a really good teacher Let's say you have like a middle school like math classroom or something.

1:21:40If a student like asks a question that suggests 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, oh, instead of that, let's do this. And like the really good teachers are able to like jujitsu 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 at the very least, it feels like there's three levels here. And so like LLMs at one, good explainers at another.

1:22:17But then like the A-plus explainer is the one who can like jujitsu your way of thinking and say like, oh, that's where that's useful. And so maybe there is a certain, you know, cycle all the way around where, again, five years from now, the LLMs will still be doing that, but in the better way. What is your recommendation to students who I'm sure email you this question all the time? Look, I was curious about doing mathematics. I'm really passionate about the subject. But seeing all the progress the AI is making, I don't know if it makes sense for me to pursue this as a career. And this is 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.

1:22:59So coding is very adjacent to this. Yeah, what advice do you have for people? I wouldn't trust any advice that I give. It would maybe be how I'd like couch it. But even pre-AI, it feels very important for any job that you're going to go into to really understand. Like if we're talking about a job, right? We're not talking about like you're a gentleman scientist and you want to like engage with the math world or something. You should understand where the money is coming from and like what value you're actually adding. and like the connection between those two. And I think often like a surprisingly small amount of thought is put towards that, especially students, they're in this environment where they probably want to go into math because they've always been good at it.

1:23:40And they've just been rewarded in life for like proceeding through the next hoop correctly, a next step. And when they think they want to be a mathematician, it's because it's a version of getting to continue to engage with that. It's like, well, I'll go like, where do people get to do this? Rather than thinking like, what value am I adding to other people? and to what extent is that like the reason that like salary is flowing in my direction? It's actually quite different in different cases. Like in some cases, it's a very prestigious mathematician and like their presence at a university lends a certain brand value and that's like why the university like wants them.

1:24:10In some cases, it's like the NSF grant is given because you've got this like public good belief that we have that basic science has and like you've got this institution around that and there's going to be this whole bureaucracy around trying to act as a proxy for what we think that public good is and a whole song and dance around how to correctly make them predict that your progress will be in the spirit of that funding. Sometimes it's just straight up teaching, right? It's like people like to send their kids to an institute that has experts teaching them. And like, that's what you're doing. And you are providing the brand value by being an expert and then the direct value by being a teacher.

1:24:46So regardless of whether AIs are like proving theorems or not, or like whether we're talking in 2016 or 2026, Like, 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 that, you know, I just, like, wasn't necessarily thinking about it and kind of stumbled into this career path where basically math exploration can be monetized as entertainment, right? And I, like, stumbled into that. I'm, like, very grateful that I did, but it was an accident. It wasn't, like, this deliberate thing. And I think I could have avoided relying on serendipity and maybe done that a little bit more by design had I been, like, thinking critically about it.

1:25:21So to your question, if it's the case that you have almost automated theorem proving, and then let's say it's the case they're also really good explainers, so it's like even to get the human understanding, I think a lot of the 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 grant writer who like really understands algebraic number theory to understand that it's a good result.

1:26:01And so there's gonna be some inner culture of what constitutes like valuable contributions. Maybe it shifts away from theorem proving and maybe it shifts towards like good definition writing. Maybe it's that like museum curator idea, but you're gonna 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, probably there's more funding in that direction in some sense, right? On the side of prestige to institutions for who their lecturers are, I mean, I actually think teaching is one of the most stable post-AGI jobs that there is because it's so relational.

1:26:37It's so, like, this is where parents want to spend their money if they have an abundance of wealth is on good teaching and good educating. And it goes so far beyond explanations. Like, even if LLMs are good explainers, the thing that a teacher is doing is such a social, like, coaching mentor type thing that, like, that's probably the most, one of the most stable careers that's going to exist over the next 50 years. and so insofar as what a lot of mathematicians role is like overlaps with that you know you as the prospective student going into it you could lean into that actually think a lot more students should like think about and and give uh like pay credence to the idea of being like just a math educator and like the value that that can serve towards the next generation so i'll couch again on i don't think i'm the one to say here prospective young mathematician uh like here's how you should think about the future because I'm like a YouTuber, right?

1:27:30I'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, know 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 prospective mathematicians. Yeah. And in fact, I think in the crazy world, in the world where within five, 10 years the AIs are coming up with not only solutions to the the Millennium 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 the things that AI minds will have like gone furthest in where they will have seen like furthest beyond our horizons will be mathematics and there will be so much demand of like, what have the AI seen?

1:28:27Can you explain it to us? Yeah. 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 like make the decision of where it should point, like they actually have a lot more economic value by like 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 um one question for ai for math is not only can it do it but is it any good yeah or isn't any good for anything you were describing all the ways in which group theory we're trying to solve this, we're trying to figure out random facts about the roots of different kinds of functions.

1:29:28And now it's all these different applications that are practical across many different fields. Do you have some sense of if we just totally get to a place where mathematics is the field of human mathematics is accelerated 10x or 100x that some crazy shit happens? Are we just actually going to be bottlenecked by other fields or? I think there's some fields that 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 thing. But I don't know, I remember talking to this mathematician who does more like dynamics and like PDE solving type stuff, and he was referencing basically like his group had some ideas that let me see if I summarize this right it's like the way that Boeing would make planes is they would like make it and then they would do a bunch of tests and they had to like disassemble it and reassemble it based on those tests and they essentially had some insights on how to like do more things in simulation such that you don't have to like deconstruct and rebuild it and it saved Boeing just like billions of dollars or something and then they just started funding that like group which is so that's it's like much more obviously application adjacent because like PDEs just sort of are that So progress in that domain, you would imagine like actually do unlock some things.

1:30:50And I don't know if it's these like step changes, but maybe it's more on the side of like engine design becomes just a little bit more fluid. Or like coming up with the right wing shape instead of running a whole bunch of complicated like CFD. Or maybe you're able to like speed up your like CFD simulations because of certain pure math insights like makes those more efficient. I bet you'd just see like a lot of like great incremental improvement there. it seems less likely that like the massive breakthroughs in math immediately turn into like this massive economic breakthrough like you solve the navier stokes uh like problems and then that unlocks like an ability to simulate more things but you probably will see like at those fringes just some some meaningful um like leakage outside of the pure math insights into into other things but also i mean there's a ton of people working on things like you know ai engineers like physical engineers like material science and things like that that would be you 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 and 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 it would be a little bit disappointing and a little bit surprising if there weren't over the next five years like economically valuable improvements that were made that were directly like referable to the like AI progress in math.

1:32:14Like that just would be kind of disappointing if it was just taking down a bunch of Airtish problems and like none of them actually, you know, it wasn't doing any of the math that actually directly touches physical world. Yeah. I mean, to your point about, well, a lot of history and mathematics is about like building up these like piles of concepts and connections and whatever. Yeah. And sometimes the piles connect with each other or you discover an application somewhere else. At the very least, you just build up this huge pile. and as a you know broader progress in society happens during the singularity when like the we get to the industrial part of the singularity you just have all these different ideas that you can hopefully are useful in other parts of the world i mean yeah it 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 oh 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 like that's one of the things mathematicians have to come to terms with where everyone will look and be like on a second like where are you guys supposed to like if there's so much that's like 10x progress there like why aren't we seeing it over here and then that church is like every time we wrote those grant proposals and said like trust us like the elliptic curve progress is going to help with like cryptography like it like shines a light on the fact that like maybe it doesn't so that's that's one possibility um grant this is super fun thanks so much for doing it absolutely my pleasure