Terence Tao – Kepler, Newton, and the true nature of mathematical discovery

Okay, today I'm chatting with Terrence Tao, who needs an introduction. Terrence, I want to begin by having you retail the story of how Kepler discovered the laws of planetary motion, because I think this will be great jumping off point to talk about AI from math. Okay, yeah, so I've always had amateur interest in astronomy, and so I've loved stories of how the earliest astronomers worked out the Nature of the Universe. So Kepler was building on the work of Copernicus, who was himself building on the work of Out of Starkis. So Copernicus very famously proposed the Hayley-O-Centric model, that instead of the planets and sun going around, the Earth, that the sun was at the center of the solar system and the other planets were going around the sun, and Copernicus proposed that the orbits of the planets were perfect circles. And his theory kind of fit the observations that the Greeks and the Arabs and the Indians had worked out over the centuries. I think Kepler got interested, like he learned about these theories in his studies, and he made this observation that the ratios of the size of the orbits that Copernicus predicted seem to have some geometric meaning. I think he started proposing that if you take, say, the orbit of the Earth and you enclose it in, I think, maybe a cube, the outer sphere that encloses the cube almost match perfectly the orbit of Mars and so forth. And there were six planets, none of the time five gaps between them, and there were five perfect electronic solids, the cube, the tetrahedron, isocrystion, octetion, and door-decadeion. And so he had this theory which he thought was absolutely beautiful that he could inscribe these tetan solids between the spheres of the planets, and it seemed to fit, and it seemed to him like, you know, God's design of the planets was matching this mathematical perfection of the tetanic solids. So he needed data to confirm this theory. And at the time, there was only one really high-quality data set. It almost had an existence, which was the so Tchaikovrache, the Danish astronomer, very wealthy, eccentric astronomer, had managed to convince the Danish government to fund this extremely expensive observatory, this, in fact, an entire island, where he had taken decades of observations of all the planets, Mars, Jupiter, every night, at least every night for which the weather was clear. With the naked eye, actually, it was the last of the naked eye astronomers. And so he had all this data which Kepler could use to confirm his theory. And so Kepler started working with Tchaikovrache, which was very jealous of the data. He only gave a little bit of bits a bit at a time. And I think Kepler eventually just stole the data, actually. He copied it and had to have a fight with Brachy's descendants. But he did work out, he'd dig in the data, and then he worked out to kind of his disappointment, that his beautiful theory didn't quite work. The data was sort of off from his patronic solid theory, about 10% or something. And he tried all kinds of fudges, moving the circles around and things. It didn't quite work. But he worked on this problem for years and years. And eventually, he figured out how to use the data to work out the actual orbits of the planets. And that was incredibly clever, genius amount of data analysis. And then he eventually worked out that the ellipses, not circles, which was shocking to him. And then he worked out the two also planetary motion, ellipses, also equal areas, super-at-equal times. And then 10 years later, after collecting a lot of data, the furthest planets, like Saturn and Jupiter, were the hardest for him to work out. But then he finally worked out this third law also, that the orbits, the time it takes for a planetary orbit, was proportional to some power of the distance to the Sun. And these are the three famous couples of laws of motion. And he had no explanation for them. It was just all driven by experiment. And it took Newton a century later to give a theory that explained all three laws at once. The take I want to try on you is that Kepler was a high-temperatured alum. Where Newton comes up with this explanation of why the three laws of planetary motion must be true. And of course, the way that Kepler discovers the laws of planetary motion, or figures out the relative orbits of the different planets as you say a work of genius. But then, you know, he's a through his career. He's just trying random relationships. And in fact, in the book in which he writes down the third law of planetary motion, it's sort of on the side on the harmonics of the world, which is this book about, you know, all these different planets have these different harmonies. And the reason there's so much famine and misery on Earth is because the Earth is me, for me, that's the note of Earth. And so all this random astrology, but in there is the cube square law, which tells you what relationship the period has to a planet's distance from the Sun, which is, as you're detailing, if you add that to Newton's f equals m a and then the equation for centripetal acceleration, you get the inverse square law. And so Newton works that out. But the reason I think this is an interesting story is, I feel like LLM's can do the kind of thing of like 20 years, let's try random relationships, some of which make no sense. As long as there's a verifiable data bank like Brahis data set, where, okay, I'm going to try out random things about like musical notes. I'm going to try out random things about platonic objects. I'm going to all these different geometries have this bias that is there's some important thing about the geometry of these orbits. And then one thing works and as long as you can verify it, it can then draw these empirical regularities can then drive actual deep scientific progress. Traditionally, when we talk about the history of science, I do generation has always been kind of the prestige part of science. So I mean, a scientific problem comes with there's many steps, you have to identify a problem and then you have to identify a good problem to work on the fruitful problem. And then you need to collect data, you need to figure out a strategy to analyze the data to make a hypothesis. And at this point, you need to propose a good hypothesis and then you need to validate. And then you need to write things up and explain that there's a dozen different components. But the ones we celebrate are the sort of eureka genius moments of ID generation. And so, so Kepler certainly had to, as I say, cycle through many ideas and several which didn't work. And I bet many that he didn't even publish at all. Because they didn't fit. And that's an important part of the process, trying all kinds of random things and seeing if they worked. But as you say, the, you know, the, if they have to match by an equal amount of verification, otherwise it's, it's slow. I mean, we celebrate Kepler, but we should also celebrate Brahe for his, his, his, his, his, his, a cityist data collection with which it was ten times more precise than than any previous observation. And that extra decimal point of accuracy was actually essential for Kepler to get his, his, his results. And, you know, and he was using, you know, Euclidean geometry and, and like, like, the most advanced mathematics he could use at the time to match his models with the data. So, like, all aspects had to be in play. You know, the data and the theory and the, the hypothesis generation. I'm not sure nowadays that hypothesis generation is the bottleneck anymore. Sciences has changed in the, in the century since. So, classically, sort of the two big paradigms for science for theory and experiment. Then in the 20th century, numerical simulation came along. And so he can also do computer simulations of, of, of, of, to test theories. But then finally, in the late 20th century, we had big data. No, we had the, the, the error of data analysis. And so a lot of new progress is actually driven now by analyzing massive data sets first, collecting large data sets and then drawing the patterns from them to, to, to do slow. Which is a little bit different from how science used to work where you make a few observations or you just have one out of the blue idea. And then you collect data to test your idea. That's the classic scientific method. Now it's almost reverse. You collect big data first. And then you, you try to get hypotheses from it. I mean, Kepler was maybe one of the first early data scientists, but, but even, even he didn't start with type of, and, and analyze it. He had, he had some preconceived theories first, but it's, it seems that this is less than this, the way we make progress in, in, in, yeah, just because, yeah, the data is just so much more massive. It's just so much more useful. Oh, interesting. I actually feel like the more the 20th century science that you're describing is actually very well described with Kepler, where he did have these ideas. 1595 and 96 is where he comes up with first polygons and then platonic objects theory, but they were wrong. And then a few years later, he gets brought his data. And it's only after 20 years of just trying random things that he gets this empirical regularity. And so it actually feels closer to brought his data as analogous to some massive data bank of simulations. And then we, he now, now that you got the data, you can keep trying random things. But if it was, and Kepler would be out there just writing books about harmonics and the platonic objects and there would be nothing to actually verify against. Yeah, yeah, yeah. So the data was extremely important. But the distinction I was trying to make was that sort of traditionally, you make a hypothesis and then you test it against data. Yeah. But now with machine learning and data analysis and statistics and some of you can, you can start with data and through statistics, work out, laws that were not personally. So, Kepler's third law was a little bit like this, except that for the third law, instead of having the thousand data points that Brahi had, Kepler had like six data points. Every planet, you knew the length of the orbit and the distance of the Sun, and there was like five or six data points, and he did what we would now call regression. You know, he could fit a curve to the six data points and he got a square coup d'etre, which was amazing. But actually, he was quite lucky. I mean, that the six data points gave him the right conclusion. You know, it's, that's not enough data to be really reliable. There was a later astronomer, Johannes Bordett, who took the same data, actually, the distances to the planets. And in spite of that, Kepler, I think he had a prediction that the distances to the planets formed basically a shifted geometric progression that he also fit a curve. Except there was one point missing. So there was a big gap between Mars and Jupiter. His low predicted that there was a missing planet. So it was a kind of a crank theory, except when Uranus was discovered by Herschel, the distance Uranus fit exactly this pattern. And then series was discovered this asteroid between, I think in the asteroid belt, and they're also fit the pattern. So people got really excited that that that board had discovered this amazing new law of nature. But then Neptune was discovered and was completely like way off. And, you know, and basically it was just a numerical fluke. You know, there were six data points. Yeah, so maybe one reason why Kepler didn't highlight his third law as much as the first two losses that maybe instinctively, even though we didn't have modern statistics, he kind of knew that with six data points, he had to be somewhat tentative with the conclusions. But maybe to ask the question about the analogy more explicitly. Does this analogy make sense to if we have, you know, the future will have smarter and smarter eyes and will have millions of them. And then they can go out and hunt for all these empirical regularities. It sounds like you don't think the bottleneck in science is finding more things that are for each given field, their equivalent of the third law planetary motion so that then later on somebody can say, oh, we need a way to explain this. Let's work out the math here. Here's the inverse square law of gravity. Right. So I think AI has basically driven the cost of idea generation down to almost zero. In a very similar way to the internet drove the cost of communication down to almost zero. Yeah. Which is an amazing thing. But it, you know, it doesn't make it doesn't create abundance by itself. Yeah. So now the bottleneck is different. So we're now in a situation where suddenly people can generate thousands of theories for a given scientific problem. And now we have to verify them, evaluate them. And this is something which we have to change our structures of science to actually sort this out. So, you know, in fact, traditionally we build walls, you know, so in the past, you know, before we hit AI slop, you know, we had sort of amateur scientists, you know, create, you know, have their own theories of universe. Many of which were basically a very little value. And so we've bought these like, you know, peer review publication systems and things to kind of filter out and try to isolate the high signal ideas to test. But now that we can generate these, these, these these possible explanations at massive scale. And some of them are good and a lot of terrible. I mean, human reviewers, we just, they're already being overwhelmed actually. I mean, many, many journals are reporting AI, during submissions, I just, I just, I just flooding their submissions. So it's great that we can generate all kinds of things now with AI. But it means that we have to, the rest of the rest of the aspects of science have to catch up. Rehabilitation, validation, and assessing what ideas actually move this up to forward and what, which ones are dead ends or, or, or red herrings. And that's, that's not something we've, we know how to do at scale. You know, for each individual paper, we can discuss it with, you know, if a debate among scientists and get consensus in a few years. But when we're generating, you know, 1000 of these every day, it's, yeah, this doesn't work. So I think there is this, a fairly interesting question of if you have billions of AI scientists, not only how do you gauge which ones are real progress, but how do you, I mean, this is actually a question that human sciences had to face and we've solved somehow and I'm, I actually am not sure how we solve this. But in any given field, less than in their 1940s and there's, if you're a bell, or if you're just generally trying to do these new technologies coming out of, post code modulation, basically how do you transfer signals, how do you digitize signals, how do you transfer more of our analog wires. And then there's like all these papers about the engineering constraints there and the details and then there's one which is like comes up with the idea of the bit, which has implications across many different fields and you need some system which can then look at that and say, OK, we need to apply this probability, we need to apply this to computer science, et cetera. And in the future, the AI is coming up with the next version of this kind of unifying concept and how would you identify it among millions of papers which might actually constitute progress, but which have much less general unifying ideas. A lot of us, the test of time, so many great ideas didn't actually get a great reception at the time that they were first proposed. It was only after some other scientists realized that they could take it further and apply them to their own deep learning itself was actually a niche area of AI for a long time that the idea of of getting answers entirely through training on data and not through first principles, you know, reasoning was was was very controversial and they would just took a long time before it actually started bearing fruit. You know, you mentioned the bit, you know, I mean, there were there were other proposals for computer architectures than the zero one that is universal today. I think there were there were tricks, you know, zero one, three valued logic and you know, in an alternate universe, maybe a different paradigm would have showed up. People argue that you know, the transformer, for example, is is the foundation of all modern language models and it was the first deep learning architecture that really was was sophisticated after capture language, but it didn't have to be that way they could have been some other architecture that was the first to do it and once that was adopted it would become the standard. So I think one reason why it's hard to assess whether a given idea is going to be fruitful is that it it depends on the future it depends on and it depends on also on the culture in society like like which ones get adopted, which ones don't. You know, the base 10 new system in mathematics extremely useful much better than the Roman human system for instance. But again, there's nothing special at 10 it's a system that we it's useful for us because everyone else uses it and we've standardized it and we've brought all our computers and our number of representation systems around it and so we're stuck with it now actually, you know, people are some people occasionally push for other systems than decimal but it's there's there's no this is no there's too much inertia. So you can't look at any given scientific achievement, puning isolation and give it an objective grade without being aware of the context both in the past and the future and so it may never be something that you can just reinforce and learn the same way that that you can for much sort of more localized problems. Yeah, it seems often in the history of science when what when the new theory comes up that in retrospect we realize is correct it seems to make implications that just either make no sense because they're wrong and we realize later on right or they're wrong or they're correct but you seem well thing plausible at the time so in as you talked about aristarchus had Helio centristism in the third century BC and then the ancient Athenians were like this can't be because it would if the earth is going around the sun we should see the relative position of the stars change as we're going around the sun and the only way that wouldn't be the case is if they're so far away that that you don't notice any parallax which is actually the correct implication but there's times when actually the implication isn't correct and we just need to graduate to a better level of understanding so a life nets would you know try Newton and disagree with you and see your gravity on the basis that it implied action at a distance and then there's we don't know the mechanism and in you and himself will sort of stand that inertial mass and gravitational mass were the same quantity so all these things which are resolved by Einstein yes yes but it was still progress and so the question for in system of pure euphoria would be even if you can falsify a theory how would you notice that it still constitutes progress relative to the thing before yeah so it often actually the ultimately correct theory initially is worse in many ways yeah so so it was less accurate than Tom Lee's theory so geocentrism had been developed for for you know I'm learning about that point and they had many many tweaks and very increasingly complicated ad hoc fixes to make it more and more accurate and simpler but much as accurate there's only a couple that made it more accurate than Tom Lee's theory I mean science is always a work in progress you know so yeah so when you only get part of of the solution it looks worse than then a theory which is incorrect but somehow you've been completed at clearly point where it kind of answers all the questions as you say you know Newton's theory had had big mysteries you know they had the equipment of mass and action at distance which were only resolved with a very conceptually different approach centuries afterwards often progress has to be made I can't not by adding more theories but by deleting some assumptions. that you have in your mind. So one reason why geocentrism held on for so long is we had this idea that objects naturally wanted to stay at rest. This is the Aristotle in the notion of physics. And so the idea that the Earth was moving, how can we also fall over? Once you have the notion of the motion, the object in motion remains in motion and so forth, then it makes sense. But you had to conceptually-- it's a very big conceptual leap to it to realize that the Earth is in motion. It doesn't feel like it's in motion. And the biggest advances Darwin's theory of evolution is the idea that species are not static. But it's not obvious because you don't see evolution in your lifetime. Well, now we can actually can, but it seems permanent and static. Right now we're going through a cognitive version of the Copernican Revolution, where we used to think that human intelligence is the central universe. And now we're actually seeing that there's very different types of intelligence that are out there with very different strengths and weaknesses. And so our assessment of which tasks require intelligence which ones don't has to be re-ordered quite a bit. And so it's trying to fit AI into our theories of scientific progress and what is hard and what is easy. We're struggling quite a lot. We have to ask questions that we've never been able to ask before. Or maybe the philosophers had, but now we all have to do with it. This actually brings up a topic I've been very curious about. So you mentioned Darwin's The Earth of Evolution. There's this book, The Clock of Universe by Edward Dalnack, which covers a lot of this era of history we're talking about. And he has this interesting observation in there that the origin of species is published in 1859. The Principia Mathematica is published in 1687. So the origin of species comes out basically two centuries after the Principia. And conceptually, it seems like Darwin's theory is simpler. There's a contemporaneous biologist to Darwin who reads the origin of species Thomas Huxley and says, how stupid not to have thought of that. And nobody ever says that about Principia. Then Shiding themselves are not having been immune to gravity. And so there's a question of, well, why did it take longer? It seems like a big part of the reason is that the evidence for natural selection is cumulative and retrospective, whereas Newton can just like, here's my equations. Let me see the Moon's orbital period and its distance. And if it lines up, then we've made progress. And so Lucretius actually had the idea, this idea that species adapted their environment in the first century BC. But nobody ever really talks about it until Darwin. Because Lucretius can't run some experiment and people are forced to pay attention. And so I wonder if wheel and retrospect end up seeing much more progress in domains, which are have this kind of tight data loop where you can verify them quite easily, even though they're conceptually much more difficult. I think one aspect of science is not just creating a new theory and validating it, but communicating it to others. So Darwin was actually an amazing science communicator. He wrote in English in natural language times people like it. So in Nolene-- [LAUGHTER] OK. I have to sort of get out of my technical mindset. Yeah, he spoke in plain English. Didn't use equations. And he synthesized a lot of disparate facts. So little pieces of evolution had been worked out in the past. But he had this very compelling vision. And again, still missing things. Like he didn't know that the mechanism for her editorate. He didn't have DNA. But his writing style was persuasive. And that helped a lot. Newton wrote in Latin. He invented a tie, new areas of mathematics, just to explain what he was doing. He was also from an era where scientists were much more secretive and competitive. So academia is still competitive. It was even worse back in Newton's day. So he held back some of his best insights because he didn't want his rivals to get any advantage. He was also, OK, someone I'm fuzz in person from what I gather. So it was actually only a couple decades after Newton where other scientists explained his work in much simpler terms that they became a widespread. So yeah, the art of exposition and making a case and creating a narrative is also a very important about a science. And if you have the data, it helps. But people need to be convinced otherwise they will not push it further. Or they want to take an initial investment to learn your theory and really explore it. And that's another thing which is really hard to reinforce and learn on. How can you score? How persuasive you are? OK, well, OK. There's the entire marketing departments we're trying to do. So maybe it's good that AI are not yet optimized to be persuasive. So yeah, there's a social aspect to science. Even though we've tried us also having an objective side to it where there's data and there's experiment and validation, we still have to tell stories and convince our fellow scientists. And that's a soft squishy thing. Like it's a combination of data and painting a narrative. And it's a very narrative of gaps. So even down at this, it's said, there are pieces of a theory he cannot explain. But he could still make a case that in the future, people would find transitional forms that they would find the mechanisms of inheritance. And they did. Yeah, I don't know how you can quantify that in such a precise way that you can start to reinforce something. Maybe that was to be forever the human side of science. One takeaway I had from reading and watching your stuff on the Cosmic Distance Latter. By the way, I highly, highly, highly recommend people watch your series with through the one round on the Cosmic Distance Latter. But one takeaway was that the deductive overhang in many fields could be so much bigger than people realize where if you just had the right insight about how to study a problem, you might be surprised at how much more you could learn about the world. And I wonder if you think that's sort of a product of astronomy at the particular times in history that you're studying? Or is it that based on the data that is incident and done the Earth right now, we could actually divine a lot more than we happen to know? Right. So astronomy was one of the first sciences to really embrace data analysis and squeezing every loss possible drop of information out of any information they had because data was the bottleneck. I mean, it still is the bottleneck. I mean, it's really hard to collect astronomical data. So astronomers are the best world class in extracting almost like Sherlock. So they're extracting all kinds of conclusions from the little traces of data. I hear that a lot of quantum hedge funds, they're preferred higher as an astronomy PhD. And they also are very interested for other reasons in extracting signals from various random bits of data. OK, speaking of clever ideas, one of my listeners, Sean, solved the puzzle that Jane Street made for my audience and posted a great walkthrough on X. For context, Jane Street trained a resume and then shuffled all 96 layers and then challenged people to put them back in the right order using only the model's outputs and training data. You can't brute force this. There's more possible orderings than atoms in the universe. So Sean broke the problem into two different parts. First, pair the layers into 48 different blocks and second, put those blocks in the right order. For pairing, Sean realized that in a well-trained resonant, the product of two weight matrices in a residual block should have a distinctive negative diagonal pattern. And this arises as a way for the model to keep the residual stream from growing out of control. From this insight, he was able to recover the right pairings. For ordering, Sean noticed that the model seemed to improve if he sorted the blocks by the size of the residual contributions. Starting with that rough approximation, he combined a clever ranking heuristic with local swaps to recover the exact right order. His full walkthrough is linked in the description. Don't worry if you didn't get to this puzzle in time, though. There's still one up about back door level ones that even Jane Street doesn't know how to solve. You can find it at jainstreet.com/thoracash. All right, back to Terence. We do under explore sort of how to extract extra information from various signals. Like I just put to pick one random study. I remember reading once that people had discovered-- we're trying to measure how often scientists actually read these citations that the papers are a site. So how do you measure this? You could try to survey different scientists. But they had some clever trick. So many citations have little typos. Like a number is wrong or a contribution is wrong. And they measured how often a type what it got copied from one reference to the next. And they could infer whether an author was actually just copying and pasting a reference without actually checking it. And so from that, they were able to infer some measure of how much attention people were paying. So there are also clever tricks to extract. So these questions you posed earlier, how can we assess whether a scientific development fruitful or interesting or. represents real progress. Maybe there are really useful metrics and footprints of this phenomenon in data-solving. We can examine citations and how often something is mentioned in the conference or something. And maybe there's a lot of social sociology of science research to be done. And that could actually detect these things. Yeah, maybe we usually get to most run with only case of the last few months. OK, so I think this brings us nicely to the progress that from the outside, it seems like AI for math is making. And I think you had opposed to recently we pointed out that over the last few months, AI programs have solved 50 out of the 1,100-odd, or those problems. But then I think-- I don't know if it's still correct, but as of a month ago, you said that there had been a pause because the low-hanging fruit had been picked. First of all, I'm curious if actually that is still the case that we have picked the low-hanging fruit and now we're at this plateau currently. It does seem so. I mean, there's so activity at the other-- yeah, so 50 odd problems have been solved with AI systems, which is great, but there's like 600 to go. And people are still chipping away at one or two of these right now. We're seeing a lot fewer sort of pure AI solutions now, where they are just one-shots the problem. So there was a month where that happened, and that has stopped. Not for lack of trying. I know three separate attempts to get frontier model AI to just attack every single one of the problems so much here and so. And they picked up some minor observations, or maybe they found some problems already solved in the literature. But there hasn't been any further AI purely powered solution yet. People are using AI a lot currently. So someone might use AI to generate a possible proof strategy, and then another person who is using a separate AI tool to critique it, or rewrite it, or generate some numerical data for it, or do a literature survey. And some problems have been solved by an ongoing conversation between lots of humans and lots of AI tools. But it does seem like it was this one-off thing. So maybe one analogy to complete problems is like, imagine like there's all these-- you're in some sort of mountain range of all kinds of cliffs and walls. And maybe there's a little wall, which is maybe like three feet high, and one of those six feet high, and there's 15 feet high, and then there's some mile high cliffs. And you're trying to climb as many of these cliffs as possible. But it's in the dark. We don't know which ones are tall, which ones are short. And so we try to lie some candles and make some maps. And slowly we kind of figure out some of them up, like climbable, some of them identify some partial track in the wall that you can reach first. And then these AI tools, they're kind of like these jumping machines that can kind of jump two meters in the air higher than any human. And sometimes they jump in the wrong direction, and sometimes they crash. But sometimes they can reach the tops of the lowest walls that we couldn't reach before. And so we basically have seven loos in this mountain range hopping around. And then there's this exciting period where they could actually find all the low ones, and they could reach them. But then there's been no-- I mean, maybe if the next time there's a big advance in the models, then they will try it again. And maybe a few more will be breached. But it's a different style of doing mathematics than sort of the-- so normally we would to hill climb, and we would make little markers and try to identify partial things. And these tools either succeed or they fail. And they'd been really bad at creating sort of partial progress or identifying intermediate stages that you should focus on first. Again, going back to the previous discussion, we don't have a way of evaluring partial progress. The thing where you can eval a one-shot success or failure of solving a problem. So there's two different ways to think through what you've just said. And one of them is more bearish on the air progress, and one of them is more bullish. And bearish on being, oh, they're only getting to a certain height of wall, which is not as high as humans are reaching. And the second is that, well, they have this powerful property that once they achieve a certain water line, they can fill every single problem that is available at that water line, which we simply can't do with humans where we can't make a million copies of view and give each of them a million dollars of inference compute and have you do 100 years of subjective time research on 100 different problems at the same time, or a million different problems at the same time. But once AI's reached Terence Tower level, they could do that. And once they reach intermediate levels, they could do the intermediate version of that. So the same reason that we should be bearish now is the reason we should be especially bullish, not even when they achieve superhuman intelligence, but just when they achieve human level intelligence, because their human level intelligence is qualitatively wider and more powerful than our human level intelligence. I agree. So they excel at breadth and humans excel at depth, like human experts, at least. So I think they're very complimentary. But our current way of doing math and science is focus on depth because that's where the human expertise, because humans can't do breadth. But we have to redesign the way we do science to take full advantage of this breadth capability that we now have. So as I said, we should have a lot more effort in creating very broad class of problems to work on, rather than one or two really deep important problems. I mean, we should still have the deep important problems. And humans should still be working on them. But now we have this other way of doing science. We can explore entire new fields of science by first getting these broad, moderately competent AI to sort of map it out and clear out all the easy-- make all the easy observations, OK? And then identify certain islands of difficulty, which then human experts can come and work on. So I see very much a future of very complimentary science. Eventually, you would hope to get both breadth and depth and somehow get the best of both worlds. But I think we need practice with the breadth side. It's too new. We don't even have the paradigms really to make full advantage of it. But we will. And then science should be unrecognizable after that. To this point about complementarity, programmers have noticed that they're way more productive as a result of these AI tools. And I don't know if you as a mathematician feel the same way, but it does seem like one big difference between vibe coding and vibe researching is that with software, the whole point of the thing is to have some effect on the world through your work. And if it leads to you better understanding a problem or are you coming up with some clean abstraction to embody in your code, that is instrumental to the end goal. Whereas maybe with research, the reason we care about solving the million and privacy problems is presumably that in the process of solving them, we discover new mathematical objects or better new techniques and those who understand our civilization's understanding of mathematics. And so the proof is sort of instrumental to the intermediate work. I don't know if you agree with that dichotomy or if that in any way will explain the relative uplift we'll see in software versus research. Right. Yeah. So certainly in math, the process is often more important than the problem itself. The problem is kind of a proxy for measuring the progress. And I think even in software, there's different types of software tasks. If you just kind of create a web page that does the same thing that 1000 other web pages do, there's sort of no skill to be learned. Well, there's sort of some skill maybe that the infrastructure program could pick up. But for kind of boilerplate type code, definitely, it's some of the things you could definitely offer to AI. But sometimes once you make the code, you still maintain it and this issue is for upgrading it and making compatible with other things. And I think I feel that that program is our reporting, even if an AI can create the first prototype of a tool, making it mesh with everything else and making it interactive with the real world in the way they want. That's an ongoing process. And if you didn't have the skills that you pick up from writing the code, that may impact your ability to maintain it down the road. So certainly, mathematicians, we've used problems to build intuition and to train people, to have a good idea as what's true, what to expect, what is proof of all, what is difficult. And so just getting the answers right away may actually inhibit that process. I made this thing to be a theory and experiment before. So in most sciences, there's an equal division between there's a theoretical side and experimental side. But math has been almost unique. It's almost entirely theoretical. We paint a premium on trying to have coherent, clean theories of why things are true in and false. And we haven't done much experiments as to maybe we have two different ways to solve a problem, which one is more effective. We have some intuition, but we haven't done large scale studies where we take a thousand problems and we just test them. But we can do that now. So I think AI type tools we really will actually revolutionize the experimental side of math where you don't care so much about individual problems and the process of solving them, but you wanna gather just large scale data about what things work, what things don't. Same way that if you want to, if you're a software company and you want to roll out a thousand pieces of software, you don't really wanna handcuff each one and learn lessons from each, you just wanna find what are the workflows that you scale. So we don't yet, the idea of doing mathematics at scale is at its infancy, but that's where AI is really gonna revolutionize the subject. Interesting. I feel like a big crux in these conversations about how good AI will be for science is, I think you said this, they're using existing techniques and modifying them. And it would be interesting to understand how much progress one can make simply from using existing techniques. Like how much of, if I looked at the top match journals, how many of them are, how many of the papers are coming up with, whatever, coming up with the technique means, doing that versus using existing techniques and new problems. And what the overhang is, where if you just applied every new technique to every open problem, would that just constitute a humongous uplift in our civilization's knowledge or would that not be that impressive and useful? - This is a great question. And we don't have the data to fully answer it yet. Certainly a lot of work that human mathematicians do, when you take a new problem, one of the first things we do is we just find, we look at all the standards things that have worked on similar problems in the past, and we try them one by one. And sometimes that works. And that's still worth publishing sometimes because the question was important. Sometimes they almost work and you have to add one more wrinkle to it and that's also interesting. But then the papers that go into the top journals are usually ones where you, you know, the existing methods can kind of solve 80% of the problem, but then there's this is 20% which is resistant and a new technique has to be invented to fill in the gaps. It's very, very rare now that a problem gets solved with sort of no reliance on past literature, where all the ideas come out of nowhere. That was more common in the past, but math is so mature now that it's so much of a handicap to not use the literature first. So AI tools are really good at getting really good at the first part of that, just trying all the static techniques on a problem, often now actually making fewer mistakes in implementing them than humans. They still make mistakes, but I've tested these tools on little tasks that I can do. And sometimes they pick up errors that I make, sometimes I pick up errors that they make. It's about a tie right now. But yeah, I haven't yet seen them take the next step. So when there are holes in the argument where none of those things are working, then what do you do? And then they can kind of suggest random things, and often I find that trying to chase them down and make them work and finding they don't work, it waste more time than it saves. So I think some fraction of problems that we currently think are hard will fall from this method. I mean, especially the ones that haven't received enough attention. So with the early problems, almost all of the 50 problems that were solved by AI's were ones for which basically there was no literature. I mean, the early should be put more so twice. I think maybe some people tried it casually and they couldn't do it, but they never wrote up anything. But there was a solution, and there was just, maybe combining this one obscure technique that not many people know about with some other result in the literature. And that's the kind of the median level of what AI can accomplish. And that's really great. It clears out 50 of these problems. So I think you will see some isolated successes. But what we found, so people have done large scale swoops of these early problems. And if you only focus on the success stories, the ones that get broadcast on social media, it looks amazing. Like all these problems that haven't been solved before for decades, now that they're falling. But whenever we do a systematic study, any given problem, an AI tool has a success rate of maybe 1% or 2%. It's just that they can buy a scale. And if you just pick the winners, it looks great. So I think there'll be a similar thing happening with-- there are hundreds of really prestigious, difficult math problems out there. A couple may make-- some AMA get lucky, and I keep solving them. And there was some back-door to solve the problem that everyone else missed. And that will get a lot of publicity. But then people will try these fancy tools on their own favorite problem. And they will, again, experience the 1 to 2% success rate. So there will be a lot of noise amongst the signal of when they're working, when they're not. We have to do-- it's increasingly important to collect these really standardized data sets. There are efforts now to create a standard challenge problems for AI to solve. And not just rely on the AI companies to only publish their wins and not disclose the negative results. So that will maybe give more clarity as to where we actually at. Well, I think it's worth mentioning how much progress in the AI constitutes already to have models that are capable of applying some technique that nobody had written down is applicable to this particular problem. The progress is simultaneously amazing and disappointing. It is a very strange feeling to see these tools in action. And also, a climate is really quickly. I remember when Google's web search came out 20 years ago. And it just blew all the others out of the water. You're just getting relevant hits on the front page, perfectly, almost exactly what you wanted. And it was amazing. And then after a few years, you just took a ground that you could just Google anything. And yeah, so a lot of-- I mean, 2026 level AI would be stunninged in 2021. And a lot of it, face recognition, natural speech, you had to do college level math problems. We just take for granted now. OK, so speaking of 2026, yeah, you made a prediction in 2023. I think by 2026, what was it that it would be like a colleague in mathematics or-- Yeah, I trust Worthy Kowal if used correctly. Which is looking pretty good in retrospect. Yeah, I'm pretty pleased. So let's see if we can continue this streak. You personally are too ex-mar productive as a result of AI. What year would you say that? Yeah, so productivity, I think, is not quite a one dimensional quantity. I'm definitely noticing that the style in which I do mathematics is changing quite a bit. And the type of things I do-- so for example, my papers now have a lot more code, a lot more pictures. Because it's so easy to generate these things now. So some plot, which I've taken me hours to do now, I can do in minutes. But in the past, I just wouldn't have put the plot in. In my paper in the first place, I would just talk about it in words. So it's hard to measure what 2x means. So on the one hand, I think the type of papers that I would write today, if I had to do them without AI assistance, they would definitely take five times longer. But interesting. But I would not write my papers that way. Five x. So yeah. But it's because these are sort of-- I mean, things like doing a much deeper literature search, supplying a lot more numerics. I mean, they enrich the paper. So the core of what I do, actually solving the most difficult part of a math problem, that hasn't changed too much. I still use pen and paper for that. But there's a lot of silly things. I use an AI agent now to reformat. Sometimes my parentheses are not quite the right size. I use them, I can actually change them by hand. I can get an AI agent to do all that quite nicely now in the background. So yeah, they really sped up lots of secondary tasks. They haven't yet sped up the core thing that I do. But it's allowed me to add more things to my papers. But by the same token, if I were to write a paper I wrote in 2020 again, and not add all these extra features, but just have something of the same level of functionality, then it actually doesn't save that much, to be honest. So it's made-- may the papers sort of richer and broader, but not necessarily deeper. You made this distinction between artificial cleverness and artificial intelligence. And I would like to better understand those concepts. What is an example of intelligence that is not just cleverness? Yeah, so intelligence is famously hard to define. It's one of these things that you can't know when you see it. But when I talk to someone, and we try to just collaboratively solve a math problem together, there's this conversation where we neither of us know how to solve a problem initially, but-- One of us has some idea and it looks promising. And so then we have some sort of prototype strategy and then we test it and then it doesn't work. But then we modify it. And there's some adaptivity and continue improvement of the idea over time. And eventually we sort of, we've sifted maped out what doesn't work, what does work, and we can kind of see a path forward. But it's evolving with our discussion. And this isn't not quite what the AI is. The AI is kind of mimicked this a little bit. So to go back to this analogy of these jumping robots, they can jump in fail and jump in fail. But what they can't do is they kind of jump a little bit and they reach some handhold. But then they sort of stay there and then they pull out people up and then they try to jump from there. There isn't this cumulative process, which is sort of built up interactively. It seems to be a lot more trial and error and just repetition brute force, which can scale. And it can work amazingly well in certain contexts. But yeah, this idea is sort of building up cumulatively from partial progress is kind of what's still not quite there yet. Interesting. You usually, if you're Gemini 3 or Cloud 4.5, whatever, solves a problem. It is not the case that it's owned and understanding of math as progressed or even if it works on a problem without solving it. It's not that it's owned and understanding of math as progressed. Yeah, you want a new session is forgotten what it just did. It has no new skills to attach to build on related problems. Maybe what you just did is part of 0.0% of the training data for the next generation. So maybe eventually, some of it gets absorbed. So Terrence talks about the importance of decomposing, particularly in hourly problems, into a series of easier chunks. Even if this doesn't result in a full solution, approaching problems in this way helps you build up the intuitions and practice the techniques that you'll need to keep making progress. But models today tend to struggle with these kinds of problem solving techniques. That's where the Bobox comes in. Liberalbox helps you train models not just to get the right answer, but to think the right way. They've operationalized these reasoning behaviors into rubrics, giving you the ability to evaluate every important dimension of a model's output. These rubrics go beyond simple correctness. Did the model reach for the right tools? Did it check its own work and explore alternative paths? How clear was its response? These skills are useful across domains-- math, physics, finance, psychology, and more. And the becoming increasingly important as models take on harder open-ended problems, some of which have multiple solutions, and some of which we don't even know the solutions too. Liberalbox can get you rubrics tailored to your domain, helping you systematically measure and shape how your models think. Learn more at labelbox.com/thorkech. [MUSIC PLAYING] One big question I have is, how plausible is it that if we just keep training AI as they get better and better at solving problems in lean, that they will continue to solve more and more impressive problems, and then we will in retrospect be surprised at how little insight be got from some lean solution to proving the rebound hypothesis or something? Or do you think it is a necessary condition also in the rebound hypothesis, even by an AI that is totally doing it in lean, that the constructions which are made, the definitions which are created, even in the lean program, have to advance our understanding of mathematics. Or do you think it could just be, assembly, could Google the Gook? Yeah, we don't know. I mean, some problems have been basically solved by pure brute force. A full color theorem is a famous example. We have still not found a conceptually elegant proof of this theorem. And maybe we're never wrong. I mean, some problems may only be solved all by just splitting into some enormous number of cases and doing brute force and insightful computer analysis on each case. I mean, part of the reason we fries problems like we know hypothesis, we're pretty sure that something amazing has to-- a new type of mathematics has to be created, or a new connection between two previous and unconnected areas of mathematics has to be discovered to make this work. We don't even know what the shape of the solution is. But it doesn't feel like a problem that will be solved just by exhaustively checking cases or something. I mean, it could be false, actually. We could actually-- there is an unlikely scenario that the hypothesis follows. And then it's just this, this is-- you can just compute-- oh, here's a 0 off the line. And a massive computer calculation verifies it. That would be very disappointing. I don't know. I do feel that, for the autonomous one-shot approaches, are not the right approach for these problems. I mean, I think you'll get a lot more mileage at the interplay between humans collaborating with these tools. And I can see one of these problems being solved by some smart humans assisted by some extremely powerful AI tools. But the exact dynamic may be very different from what we envision right now. It could be a collaboration of a type that we just doesn't exist yet. Yeah, I mean, there may be a way to generate a million variance of the human Zeta function and do some data analysis, AI assisted data analysis. And we'd discover some pattern between connecting them, which we didn't know about before. And this lets you transform the problem into a different area of mathematics. I mean, there could be all kinds of scenarios. So suppose the AI figures it out and latent in the lean is some brand new construction, which, if you realize a significant, we would be able to apply in all these different situations. How do you recognize it? Like if you just, again, a very naive question, but if you come over the equivalent of like, the card comes with this idea, oh, you can have the coordinate system where you can unify algebra and geometry. But the mean code would just look like R to R. And it would look that significant or something. Or similarly, I'm sure there's other constructions which have this kind of property. Well, the beauty of formalizing a proof and something like that is that you can take any piece of it and study it atomically. So when I read a paper with my humans, with which so some difficult problem, there's often some big sequence of lemmas and theorems and things. And so ideally, the author will talk their way through what's important, what's not. But sometimes they don't reveal what steps were the important ones in which ones are just kind of boilerplate standard steps. But you can study each lemma in isolation. And some of them I can say, oh, this looks very standard. This resembles something I'm familiar with. I'm pretty sure there's nothing interesting going on here. But this lemma-- oh, that's something I haven't seen before. And I could see why if you had this result, that would really help prove the men was up. You could assess whether some things are really sort of key to your argument or not. And lean really facilitated that. You can do this. Individual steps are really precisely. I think in the future there will be entire professions of mathematicians who might take a giant lean general proof and maybe do some ablation on it or something. I try to remove steps of parts of it. And try to find it in more elegant ways. Maybe some other AI's just sort of do some reinforcement learning. How can you make the proof more elegant? And maybe other AI's will grade whether this proof looks better or not. One thing that will change quite a bit in the near future is that until recently, writing papers was the most time-consuming and expensive part of the job. And so you did it very rarely. You only wrote up your results once everything was all the other parts of your argument were checked out and things. Because it's just rewriting it again, refactoring with this total pain. But that's one thing that's become a lot easier now with modern AI tools. So you don't have to have just one version of repayment. Once you have one, people can generate hundreds more. So one giant messy lean proof may not be very meaningful or understand one on its own. But other book can refactor it and do a kinds of things with it. We have seen with the Earthish problem website, people will generate a proof. And then he was 3,000 lines of code that verify the proof. But then people got other AI's to summarize the proof. And they will write their own proofs. There's actually post processing. Or once you actually have one proof, we actually have a lot of tools now to deconstruct it and interpret it. It's a very nascent area of science or mathematics. But I'm not as worried about-- so some people can say, what if the real hypothesis proof is a completely incomprehensible proof? I think once you have the artifact of a proof, we can do a lot of analysis on it. You posted recently that it would be helpful to have a former, semi-former language for mathematical strategies as opposed to just mathematical proofs, which is what lean specializes in. I would love to learn more about what that would involve or look like. We don't really know. I mean, we've been very lucky in mathematics that we have worked out the laws of logic and mathematics. But this actually a fairly recent accomplishment. I mean, it was started by Euclid millennia ago. But only in the early 20th century, did we finally-- this dark here, the axioms of mathematics are worth the standard axioms of what was the UFC. And the axioms are first-autologic. And this is what a proof is. and this we've managed to automate and have a formal language for. But there could be some way to assess plausibility of certain, you know, so you have a conjecture that something is true. You test a few examples and it works out like how does this increase your confidence that the conjecture is true. We have a few sort of mathematical ways to model this, like invasion probability, for example. But they're often, you have to set certain basic assumptions and there's a lot of subjectivity still in these tasks. So it is not clear. I mean, this is more of a wish than a plan to develop these languages. But just seeing how successful having a formal framework in place like Lean has made deductive proofs so much easier to automate and train AI on. If there was some similar framework, so the bottleneck for using AI to create strategies and make conjectures is we have to rely on human experts and the test of time to validate whether something's plausible or not. If there was some semi-formal framework where this could be done semi-automatically in a way that isn't sort of easily hackable to, you know, it is. So of course, yeah, it's really important with these formal proof assistance that there are just no there's no backdoors or exploits that you can do to somehow get your pure certified proof without actually proving it because reinforcement learning is just so, so good at finding these these backdoors. But yeah, if a strong framework that sort of mimics how scientists talk to each other in a semi-formal way, you know, using data and argument, but also, you know, contracting narratives and there's some subjective aspect of science that we don't know how to capture in a way that that we can insert AI into them in a useful way. So yeah, this is a future problem. I mean, there are research efforts to, you know, to try to create automated conjectures and and and and and maybe there are ways to benchmark these and and get some some way to simulate this, but this is it's all very, very new science. Can you help me get some intuition for I have two sub-questions. One, it would be very helpful to have a tangible sense of it would be helpful to have a specific example of what something like this would look like that the way scientists communicate that we can't formalize yet. And two, it seems almost definitionally paradoxical to say, building up some narrative or building up some natural language explanation and then also having something which you could have formalized. And I'm sure there's some intuition behind where that overlap is and I'd love to understand that better. Alright, so an example of a conjecture. So Gals was interested in the prime numbers and he computed, he created one of the first mathematical data sets. He just computed the first 100,000 prime numbers also hoping to find patents. And he did find a patent, but maybe not not the patent he was expecting, he found a statistical patent in the primes that he'd count how many primes there are up to 100, 100,000, 1 million and so forth. They get sparse and sparse, but the drop off in the density was inversely proportional to the natural logarithm of the range of numbers. So he conjectured what we're not going to know for the prime number theorem. The number of primes up to x is like x divided by the natural logarithm of x. And he had no way to prove this. It was data driven. So this was a conjecture. It was revolutionary for its time because it was maybe the first really important conjecture of math that was statistical in nature. So normally you talk about patent like maybe the spacing between the primes has a certain regularity or something, but this was really something which it didn't tell you exactly how many primes there were in any given range. It just gave you an approximate approximation that got better and better as you went further and further out. But it helped, so it started the field of Oracle and the number theory. But it was the first in many conjectures like the many of which got proved, which sort of started consolidating the idea that the prime numbers actually didn't really have a patent. That they behaved like random sets of numbers with a certain density. They had some patents like they're almost all odd. And they're not actually random. They're what's called pseudo random. There's no random number generation involved in creating the prime numbers. But over time it became more and more productive to think of the primes as if they were just generated by some god-roading dice all the time and creating this random set. And this allowed us to make all these other predictions. So there's still open conjecture in a number theory called the Trinprime Conjecture that there should be infinitely many pairs of primes that are twins. This is two apart, like 11 and 13. We can't prove that and there's actually good reasons why we can't prove it. But because of this statistical random model of primes, we are absolutely convinced this is true. We know that if the primes were sort of generated by flipping coins or something that we would just by random charts, just like infinite monkeys that are typewriter, we would see Trinprimes appear over and over again. And we have over time developed this very accurate conceptual model of what the primes should behave like based on statistics and probability. But it's all mostly heuristic and non-rigorous, but extremely accurate. So the few times when we actually can prove things about the primes, it has matched up with the predictions of what we call the random model of the primes. So we have this conjectural concept framework for understanding the primes that we, everyone believes in. And it's the same reason why we believe the re-enableces is true. Why we believe that cryptography based on the primes is basically is mathematically secure and so on. It's all part of this belief. In fact, one reason why we care about the re-enableces is that if the re-enableces failed, we knew it was false. It means it would be a serious blow to this model that this, it would mean there's a secret pattern to the primes that we were not aware of. And I think we would very rapidly abandon any cryptography based on primes because if there was one pattern that we didn't know, that was probably more. And these patterns can lead to exploits in crypto. And it's going to be a big, big shock. So we really want to make sure that that doesn't happen. So yeah, it's, so we've been convinced of things like the re-enableces and things of a time. But some of it is experimental evidence. Some is the few times we've been able to make theoretical results. They've always aligned. It is possible that the consensus is wrong and we've all just missed something very basic. They have been paradigm shifts in the past in scientific history. But we don't really have a way of measuring this. I think partly because we don't have enough data on how math and science develops. We have one timeline of history and we have like, you know, 100 stories of turning points in history. If we had access to a million alien civilizations and each of them, they're different development of history and science and different orders, then maybe we'd actually have a decent shot at at understanding of how do we measure what is progress and what is a good strategy. And we could maybe start formalizing it and having a framework. Maybe what we need to do is actually start creating lots of mini-universes, simulations of AI solving very basic problems, you know, in Earth, and take whatever. But coming over their own strategies for doing these things and having these little laboratories to test, I mean, there are people who investigate like trying to, what's the smallest, you know, newer network that can do tend to do an application and things like that. I think we could actually learn a lot from evolving small AI's on simple problems. We could learn a lot. I was super excited when Mercury reached out about sponsoring the ARCAST because I've been baking with them for years. I think I opened my first account with them in 2023. Something I've come to appreciate over the last three years is that Mercury is constantly updating things and adding new features. Take their newest feature, Insights. Insights summarizes your money in and out, showing you your biggest transactions and calling out anything that deserves extra attention. Like maybe you're revenue from a particular partner has gone down, or you've got a big, uncatarized purchase that needs to be investigated. It's a super low friction way for me to keep tabs on my business and make quick decisions. For example, I tried to invest any cash that I don't need on hand to keep running the business. With Insights, with just a couple of clicks, I was able to see exactly how much money I spent in each month of 2025. And that lets me know exactly how much cash I'll need for the next year or so of operations, and then I can go invest the rest. Mercury just keeps adding new features like this. Go to mercury.com to check it out. Mercury is a Fintech company, not an FDIC in shortbake. Faking services provided through choice financial group and column NA members FGIC. You have to learn about new fields, not only very rapidly, but deeply enough to contribute to the frontier. So in some sense, you're also one of the world's greatest auto-didacs. How does your process of learning about a new subfuel than math? What does that look like? >> Yeah. I certainly identified with the, we talked about depth and breadth before. It's not purely human AI distinction. Humans also split. I think it was Irving who split them into hedgehogs and foxes. The hedgehog knows one thing very, very well. A fox knows a little bit about everything. I definitely, I think of myself as a fox. I work with hedgehogs a lot and sometimes I can be a hedgehog if need be. Yeah, so I've always had a little bit of an obsessive streak. If there's something which I read about, which I feel like I should understand, I have the capability to understand this, but I don't understand why it works. There's some magic in it that, so someone was able to use a type of mathematics I'm not familiar with and get, which I would like to prove and I can't do it by myself, but they could do it by their method. Then I wanted to find out what was their trick. It bugs me that someone else can do something which I think I can do, but I can't. So I've always had that kind of obsessive completionist type streak. I've had to weed myself off computer games because I started game. I want to play it to completion several levels. So that's one way in which I learn new fields. I collaborate with a lot of people who have taught me other types of mathematics. I just make friends with other mathematicians who are working on another area of mathematics. I find their problems interesting, but they have to teach me some of the basic tricks and what's known, what's not known and learn a lot from that. I found that the writing about my, what I've learned, I have a blog where I sometimes record things that I've learned. Because in the past when I was younger, I would learn something and do this call track and say, okay, I'm going to remember this. And the sixth one's later I've forgotten. I remember remembering it, but I don't, but I can't reconstruct my arguments. And the first few times I was so frustrating to have understood something and then lost it. I sort of resolved, I should always write down anything cool that I've learned. And this is part of how this blog came about. I wanted to tell you the right about first. It's something I often do when I don't want to do other work. You know, like there's some referee report or something. There's something that I feel slightly unpleasant for me to do at the time. And so writing a blog, it feels creative and fun. Like it's something that I do for myself. So maybe depending on the topic, it could be a quick half an hour or several hours, but it doesn't, because it's something that I do sort of voluntarily, it doesn't feel like it doesn't feel time flies when I went out these things. As opposed to sort of doing something which I have to do for administrative reasons, but it's just it's it's it's it's drudgery. Okay, those are tasks by the AI is really helping with now. Exactly. Is it, um, if like civilization could from first principles to say how to use Terry Tows time? You know, it's like a limited resource. How what is the biggest difference between in the if the deal of ignorance got to decide how to use Terry Tows time versus what it does now? Okay, so podcasts wouldn't be happening. Yeah, so I get the as much of the complaint about certain tasks that I don't want to do, but I have to do it. So as you get more senior in academia, you get more responsibilities. I get some more committees and whatever. But I have also found that a lot of events that I kind of reluctantly went to because I was obliged to for one reason. Because there's outside my comfort zone, I often find interactions with people who I wouldn't normally talk to. I get like you, for instance. And I would learn interesting things and have interesting experiences. And I would have opportunities to to to to then network with other people that I would never have done before. So I do believe a lot in sensitivity. I mean, I do optimize my time and when I. So there's some portions of my other day where I do schedule very carefully. But I have been willing to sort of leave some some portions just okay, I'm going to do something which is which is not my usual thing. And then maybe it'll be always my time, but maybe I'll learn something. And more often than not, it's I've I feel like I've gotten a positive experience which is not something I would have planned for. And so I believe a lot on disability. And maybe there's a danger actually that in the modern society, it's not just AI, but we've become really good at optimizing everything. And and maybe we are optimizing. We're not optimizing other vocalization. That you know with with with covered for example, we switched like we switched a lot to remote meetings. And so everything was scheduled now. And so we kept busy at least in academia. You know, we met almost the same number of people that we met in person. But everything had to be planned. You had to schedule things in advance. And what we lost out on was sort of the casual like knocking on the hallway just meeting someone for you know while getting a coffee. And this there this. Yeah, so it's interactions that you may think are not optimal, but actually are really important. You know, when I was a grad student, I would go down to the library to look, I had to look for a journal article. Yeah, I had to physically go down the library, check out the journal and read your article. And sometimes the next article, you know, you can just browse through and the next article is also interesting. Sometimes it wasn't, but but you could accidentally find interesting things. Which is something which has basically been lost now because you can just type in, you know, if you if you want to access an article now, you just type it into to a search engine or even an AI and you can get instantly what you want. But you don't get so the accidental things that you might have have gotten if you've done it more inefficiently. So. Yeah, there've been times when I spent a year once at the Institute for Advanced Study, which is a great place to, you know, this no distractions. You know, there's just new research and like the first few weeks you're there, I could just great. You're getting all these papers written up that you've been wanting to do for a long time. You're thinking about problems for blocks of hours of a time. But I find if I stay there for more than several months, like I run out of inspiration somewhere like I get bored. I just, you know, the surface need a lot more. You actually do need a certain level of distraction in your life. It's somehow, uh, has enough randomness and that temperature, high temperature. Yeah. So yeah, I don't know the optimal way to schedule my life. It just seems to work. I'm very curious when you expect AI's that can like actually do frontier math better than the at least as good as the best human mathematicians. I mean, in some ways they're already doing frontier math that is super intelligent that humans can't do. But it's a different frontier from what we're used to. I mean, you could argue that Calcutta is doing frontier math that that humans could not accomplish. But it was, but it wasn't, you know, number crunching in. But, but replacing Terry Tao completely. I mean, what do you want me for? You was just going on the podcast after. I'm not sure we. It might not be the right question to ask. I think within a decade, a lot of things that mathematicians currently do. Where we spent a lot of the bulk of our time doing it and a lot of stuff we put in our papers today can be done by AI. But we will find that that actually wasn't the most important part of what we do. You know, 100 years ago, a lot of mathematicians were just solving differential equations. People needed and physicists needed some exact solution to to some system. And they were just they hired a mathematician to the laboratory to go through the calculus and work out the solution to this fluid equation, whatever. A lot of what 19th century mathematician would do, you could make a call to mathematical or with my alpha or computer algebra package or more now more recently and AI and it would just solve the problem in a few minutes. But we moved on with that. We worked on different types of problems after that. You know, once computers came along, you know, computers used to be human. People used to laboriously create log tables and work out primes that's goused it and that is all being outsourced to computers. But we moved on. In genetics, you know, to sequence at the genome of a single organism, that was an entire PhD of a geneticist. You know, it's so carefully, you know, separating all the chromosomes and one whatever. And now you can just spend a thousand dollars in center to a sequencer and get it done. But genetics is not dead as a subject. You move to a different scale. You know, maybe you study whole ecosystems rather than individuals. I take your point. But on the question of, well, when is most mathematical progress on almost all mathematical progress happening by AI? So if you find out, oh, this year, I'm going to lend him fries while the one has been solved. You would put, you know, and 95% odds that any I did it autonomously. Surely they're only such a year. I guess, I mean, I do believe that that hybrid human plus AI's will dominate my family. for a lot longer. It will require some additional breakthroughs beyond what we already have. So it's going to be sarcastic. I think AI currently are very good at certain things, but really tell what others. And why you can't sort of add more and more frameworks on top to kind of reduce the error rates and make them work with each other a bit more and so forth. It feels like we don't have all the ingredients to really have a truly satisfactory replacement for all intellectual tasks. It is complementary currently. It is not a replacement. But maybe, I mean, because current level AI is worse, accelerate science in so many ways, hopefully, you know, I mean, new discoveries, new breakthroughs will happen more quickly. I mean, it's possible that also by somewhat destroying some identity, we actually inhibit certain types of progress. Anything is possible, really at this point. I think the board is very, very unpredictable at this point in time. What is your advice to somebody who would consider a career in math, who is early in a career in math, especially in light of AI progress? How should they be thinking about the career different than the if and all as a result of the AI progress? Yeah. So we live in a time of change. It is, as I said, we live in a particularly unpredictable era. And I think, like, things that we've taken for granted for centuries may not hold anymore. So, yeah, the way we do everything, not just mathematics, will change. So I think, which is, you know, I mean, in many ways, I would prefer the much more boring quiet era where things are much the same as the 10 years ago, 20 years ago. So I think one just has to embrace this that this is going to be a lot of change. And that, you know, the things that you study, some of them may become obsolete or revolutionized, but some things will be retained. And so you somehow always have to keep an eye on, yeah, it could be a lot of opportunities for things that you wouldn't be able to do before. So, I mean, in math, you know, you previously had to basically build through years and years of education, you know, math PhD before you could contribute to the frontier of math research. But now it's quite possible at the high school level or whatever that you could get involved in math project and actually make a real contribution because of all these AI tools and lean and everything else. So there will be a lot of non-traditional opportunities to learn. So you need a very adaptable mindset. You know, there will be a lot of pursuing things as for curiosity, for playing around. And I mean, you still need to get your credentials for, I mean, I mean, I will thank you for a while, we'll still be important to sort of go through traditional education and learn math and science and so forth. That's the old-fashioned way for a while. But, yeah, but you should also be open to very, very different ways of doing science, some of which don't exist yet. Yeah, so it's a scary time, but also very exciting. Awesome, that's a great note to close on. Thanks so much. Yeah, thanks a lot, you're welcome.