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

31:23So 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 AIs to just attack every single one of the problems simultaneously. And they picked up some minor observations or maybe they found that some problems are 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 will use a separate AI tool to critique it or rewrite it or generate some numerical data for it or do a literature survey.

32:04And 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 for these problems is like, imagine like there's all these, you're in some sort of mountain range with all kinds of cliffs and walls. And maybe there's a little wall, which is maybe like three feet high and one that's six feet high and then 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 uh we don't know which ones are tall which ones are short and um so you know we try to light some candles and make some maps and and slowly we kind of figure out uh some of them are climbable some of them we can identify some some partial um track in the wall that you can reach first um and then these these ai tools they're kind of like these jumping machines that can kind of jump you know two meters in the air you know higher than any human and sometimes they jump in the wrong direction and sometimes they crash but sometimes they they can reach um um the tops of of the lowest um you know um walls that we couldn't reach before and so we basically set them loose in this mountain range hopping around and you know and then there's this exciting period where they could actually find all the um all the low ones um and they could reach them um but then uh 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.

33:42But it's a different style of doing mathematics than sort of the, you know, so normally we would hill climb and, you know, we would make little markers and try to identify partial things. And, you know, these tools, they either succeed or they fail. and they've been really bad at creating sort of partial progress or identifying intermediate stages that you should focus on first. Again, going back to this previous discussion, we don't have a way of evaluating partial progress. The same way you can evaluate a one-shot success or failure of solving a problem. So there's two different ways to think through what you've just said.

34:22And one of them is more bearish on AI progress and one of them is more bullish. and bearish one 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 waterline, they can fill every single problem that is available at that waterline, which we simply can't do with humans, where we can't make a million copies of you and give each of them a million dollars of inference compute and have you do 100 years of subjective time research on a hundred different problems at the same time or a million different problems at the same time.

34:58But once AIs reach 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. yeah so they excel at breadth and humans excel at depth um human experts at least yeah so um i think they're very complementary um but our current uh way of doing math and science is focused on depth because that's where the human uh expertise is because humans can't do breath um but uh yeah so we have to redesign uh the way we do science to take full advantage of of this breadth capability that we now have um so as i said we do we should have a lot more effort in creating very broad classes of problems to work on rather than one or two um really um deep important problems i mean we should still have the deep important problems um and humans should still be working on them um but but now now we have this other way of of of doing um of doing science you know i mean we can explore entire new fields of science by by first getting the these broad um moderately competent AI to sort of map it out and clear out all the easy, make all the easy observations and then identify certain islands of difficulty, which, you know, then human experts can come and work on.

36:28So I see very much a future of very complementary science. Eventually you would hope to get both breadth and depth, you know, 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 will 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.

37:20And if it leads to you better understanding a problem or 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 Millennium Pipers 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.

37:58Right. 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 there's different types of software tasks I mean you know like if you just kind of create a web page that does the same thing that a thousand other web pages do um there's there's sort of no skill to be learned well um there's there's some skill maybe that the individual programmer could pick up um but you know for for kind of a boilerplate type code definitely um um you know it's it's it's something that you should definitely offload to AI.

38:34But, you know, sometimes once you make the code, you know, you still have to maintain it and there's issues with upgrading it and making it compatible with other things. And that, I think, I've heard that programmers are reporting, you know, that even if an AI can create the first prototype of a tool, making it mesh with everything else and making it interact with the real world in the way they want, I mean, 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 provable, what is difficult.

39:25and so just getting the answers right away may actually inhibit that process I mean so I made a distinction between 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 pay the premium on sort of trying to have coherent clean theories of why things are true and false. And we haven't done much experiments as to, like, you know, 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.

40:10But 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 want to gather just large scale data about what things work, what things don't. Same way that if you're a software company and you want to roll out a thousand pieces of software, you don't really want to handcraft each one and learn lessons from each. You just want to 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 going to revolutionize the subject.

40:52Interesting. I feel like a big crux in these conversations about how much how good AI will be for science is, I think you said this, it's like, oh, 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. If I looked at the top math journals, how many of the papers are coming up with whatever coming up with the technique means, doing that versus using existing techniques in new problems? And what the overhang is, where if you just applied every known 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?

41:35This 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 standard 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 the existing methodians methods can kind of solve, you know, 80 % of the problem, but then this is 20%, which is resistant.

42:16And 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. You know, that was more common in the past, but math is so mature now that it's just so much of a handicap to not use the literature first. So AI tools are getting really good at the first part of that, just trying all the standard 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 uh for um but um yeah i i haven't yet seen them take the next step you know so so when there are holes in in in the argument where none of the things are working to to how then what do you do um and then they can kind of suggest random things and it but it it um often i find that trying to chase them down make them work and finding they don't work it wastes more time than it saves yeah so um now so i think some fraction of problems that we currently think are hard will will fall from this this method um i mean especially the ones that haven't received enough attention um so like with the urdish problems you know like almost all of the 50 problems that were solved by ai's were ones for which basically there was no literature i mean irdish proposed from once or twice um i think maybe some people tried it casually and they couldn't do it, but they never wrote up anything.

44:03But there was a solution and it was just, you know, 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 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 sweeps of these early problems. If you only focus on the success stories that get broadcast on social media, it looks amazing. All these problems that haven't been solved before for decades, now they're falling.

44:38But 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, you know, there are hundreds of really prestigious difficult math problems out there a couple may make um you know some am may get lucky and actually solve them and there was there was some some backdoor to solve the problem that everyone else missed um and that will get a lot of publicity um but then people will try these fancy tools on their own favorite problem and they will again experience the one to two percent success rate right so um there will be a lot of noise and amongst the signal of sort of when they're working, when they're not.

45:26We have to do, yeah, it's increasingly important to collect these really standardized data sets. You know, there are efforts now to create a standard set of challenge problems for AI to solve and not just rely on the AI companies to only publish their wins and not disclose their negative results. So that will maybe give more clarity as to where we're actually at. Well, I think it's worth emphasizing how much progress in 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.

46:02It is a very strange feeling to see these tools in action. But also acclimatized really quickly. I remember when Google's web search came out 20 years ago and it just blew all the searches out of the water. You're just getting relevant hits on the front page like perfectly almost exactly what you wanted and it was amazing and then after a few years you just took for granted that you could you could just google anything um and yeah so a lot of yeah i mean 2026 level ai would be stunning in 2021 and a lot of it you know face recognition natural speech uh yeah doing you know college level math problems we just take for granted now right yeah okay so speaking of 2026 yeah you made a prediction in 2023 that i think by 2026 What was it?

46:49It would be like a colleague in mathematics or? Yeah, a trustworthy co-author if used correctly. Which is looking pretty good in retrospect. Yeah, I'm pretty pleased. So, you know, let's see if we can continue the streak. You personally are 2x more productive as a result of AI. What year would you say that? Yeah, so productivity, I think, is not quite a one-dimensional quantity. like 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 have taken me hours to do now I can I can do in minutes but in the past I just wouldn't have put the plot in my paper in the first place I would just talk about it in words so it's hard to measure what 2x means um so yeah on the one hand you know 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 5x so yeah that's but but it's it's because but these are sort of uh auxiliary i mean you know the um yeah so so things that yeah things like like um doing a much deeper literature search, supplying a lot more numerics.

48:13I mean, they enrich the paper. So, yeah, the core of what I do, like 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, you know, there's lots of silly things. I use an AI agent now to reformat. Like sometimes all my parentheses are not quite the right size. You know, I just manually change my hand and I can get an AI agent to sort of do all that quite nicely now in the background. So, yeah, they really sped up lots of secondary tasks. They haven't yet sort of sped up the core thing that I do, but it's allowed me to sort of add more things to my papers.

48:58yeah but by the same token like 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 sort of level functionality yeah then that doesn't have hasn't saved that that much uh to be honest uh yeah so it's made 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 um uh intelligence that is not just cleverness yeah so um it's intelligence is famously hard to define it's one of these things that you kind of know it when you see it um but when i when i talked to someone um and we're trying to collaboratively solve a math problem together um there's this conversation where you know we neither of us knows how to solve the problem um initially but um one of us has some idea and and it looks promising and and so then then we have some sort of prototype strategy and then we test it and then it doesn't work but then we we modify it and there's some adaptivity and and um and and uh continual improvement of of of the idea over time.

50:17And eventually, you know, we sort of, we've systematically mapped 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 is not quite what the AI is. The AI can kind of mimic this a little bit. So to go back to this analogy of these jumping robots, you know, so, you know, they can jump and fail and jump and fail and jump and 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 other people up and then they try to just jump from there.

50:55There 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 it scales 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 were saying if Gemini 3 or Claude 4.5, whatever, solves a problem, it is not the case that its own understanding of math has progressed. Or even if it works on a problem without solving it, it's not that its own understanding of math has progressed.

51:36Yeah. You run a new session and it's forgotten what it just did. Right. It has no new skills to attach to, to build on related problems. Maybe what you just did is part of 0.001 % of the training data for the next generation. So maybe eventually some of it gets absorbed. So Terence talks about the importance of decomposing particularly in Arlie problems into a series of easier chunks. Even if this doesn't result in the 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.

52:13That's where LabelBox comes in. LabelBox 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 they're 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 to.

52:48LabelBox can get you rubrics tailored to your domain, helping you systematically measure and shape how your models think. Learn more at labelbox.com slash Dwarakash. One big question I have is, how plausible is it that if we just keep training AI that 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 we got from some Lean solution to proving the rebound hypothesis or something. Or do you think it is a necessary condition of solving 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.

53:34or do you think it could just be assembly code gobbledygook? Oh, 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. It basically, and maybe we never will. I mean, some problems may only be solvable by just splitting into some enormous number of cases and doing a brute force unincitible computer analysis on each case. I mean part of the reason we prize problems like a hypothesis is that we're pretty sure that that something amazing has to a new type of mathematics has to be created or a new connection between two previously 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.

54:32So we could actually, there is an unlikely scenario that the hypothesis is false and there's this, you can just compute, oh, here's a zero off the line and a massive computer calculation verifies it. That would be very disappointing. I don't know. I do feel that, you know, fully autonomous one-shot approaches are not the right approach for these problems. I mean, I think you will get a lot more mileage out of the interplay between humans collaborating with these tools.

55:07And 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. I mean, it could be a collaboration of a type that just doesn't exist yet.

55:29yeah, I mean, there may be a way to generate, you know, a million variants of the human's data function and do some data analysis, AI-assisted data analysis, and we 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 the significance, we would be able to apply it in all these different situations. How do you even recognize it, right?

56:06Like if you just, again, a very naive question, but if you come up with the equivalent of like, Descartes comes up with this idea, oh, you can have this coordinate system where you can unify algebra and geometry, but in lean code, it would just look like R to R and it wouldn't 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 in something like Lean is that you can take any piece of it and study it atomically.

56:34So when I read a paper with my humans, which shows 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 and 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 fairly 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.

57:08And I could see why if you had this result, that would really help prove the main result. You can assess whether some things are really sort of key to your argument or not. and lean really facilitates that you know you can you can you can you know the individual steps are identified really precisely um i think in the future there'll be um you know there'll be entire professions of of mathematicians who might take a giant um lean generally proof and maybe you know do some ablation on it or something i try to remove steps of parts of it and um and try to find it find more elegant ways you know you know maybe some other ais to sort of do some reinforcement learning how can you make the proof more elegant and and and uh um maybe other ais will grade whether this proof looks better or not.

57:55One 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 just rewriting it again, refactoring was a 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 your paper. Once you have one, people can generate hundreds more. So yeah, one giant messy lean proof may not be very meaningful or understandable on its own, but other people can refactor it and do all kinds of things with it.

58:39We have seen with the Erdős problem website, an AI will generate a proof and then here's 3 ,000 lines of code that verify the proof. but then people got other AIs to summarize the proof and people write their own proofs. There's actually post-processing, 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 are concerned, what if the real hypothesis is proven with a completely incomprehensible proof? I think once you have the artifact of a proof, we can do a lot of analysis on it.

59:21You posted recently that it would be helpful to have a formal or semi-formal 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 is actually a fairly recent accomplishment. I mean, it was started by Euclid. millennia ago, but only in the early 20th century did we finally list our Q, the axioms of mathematics, or the standard axioms of what we call ZFC, and the axioms of first-order logic, and this is what a proof is, and this we've managed to automate and have a formal language for.

1:00:05But 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, Bayesian probability, for example, but you often have to set certain base assumptions and there's a lot of subjectivity still in these tasks. So it's 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.

1:01:01If 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. Of course, it's really important with these formal proof assistants that there's no backdoors or exploits that you can do to somehow get your certified proof without actually proving it because reinforcement learning is just so, so good at finding these backdoors.

1:01:51But yeah, if it's not a framework that sort of mimics how scientists talk to each other in a semi-formal way, using data and argument, but also constructing narratives and there's some subjective aspect of science that we don't know how to capture in a way that we can insert AI into them in any useful way. Interesting. So yeah, this is a future problem. I mean, there are research efforts to try to create automated conjectures and maybe there are ways to benchmark these and get 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 step questions.

1:02:43One, 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. all right so so an example of of a conjecture so um gauss um was interested in the prime numbers and he computed he created one of the first mathematical data sets he just computed the first hundred thousand prime numbers or so um hoping to find patterns um and he did find a pattern but maybe not the pattern he was expecting he found a statistical pattern in the primes that if you count how many primes there are up to 100 1 000 um um one million and so forth they get sparser and sparser, but the drop-off in the density was inversely proportional to the natural logarithm of the range of numbers.

1:03:57So he conjectured what we now call the prime number theorem. The number of primes up to x is like x divided by the natural log 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 you know so normally you talk about patterns like maybe the spacing between the primes has a certain regularity or something but yeah 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 it helped so it started the field of what we call analytic number theory but it was the first in many conjectures like this many of which got proved which sort of started consolidating the idea that the prime numbers actually didn't really have a pattern, that they behaved like random sets of numbers with a certain density.

1:05:02I mean they had some patterns, like they're almost all odd and they're not actually random they're what's called pseudo random i mean there's no random number generation involved in creating the prime numbers but um over time it became more and more productive to think of the primes as as if they were just generated by some some some god rolling dice all the time and just creating this this random set um and this allowed us to make all these other predictions um so there's a still open conjecture in in number three called the twin prime conjecture that there should be infinitely many pairs of primes that are twins distance two apart like 11 and 13.

1:05:37We can't prove that, and there's actually good reasons why we can't prove it, but because of this statistical random model of the primes, we are absolutely convinced it's 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 at a typewriter, we would see twin primes 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.

1:06:10So 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 everyone believes in. And it's the same reason why we believe the real hypothesis is true, why we believe that cryptography based on the primes is basically mathematically secure, things like that. It's all part of this belief. In fact, one reason why we care about the Riemann hypothesis is that if the Riemann hypothesis failed, we knew it was false. It means it would be a serious blow to this model that it would mean there's a secret pattern to the primes that we were not aware of.

1:06:53And I think we would very rapidly abandon any cryptography based on the primes because if there was one pattern that we didn't know about, there's probably more. and these patents can lead to exploits in crypto and it's going to be a big, big shock. So we really want to make sure that doesn't happen.

1:07:14So we've been convinced of things like agreement of offices and things over 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. You know, there have been paradigm shifts in the past in scientific history. Yeah, but we don't really have a way of measuring this. I think partly because we don't have enough data on how math or science develops. We have one timeline of history and we have like, you know, a hundred stories of turning points in history.

1:07:53If we had access to a million alien civilizations and each of the different development of history and of science in different orders, then maybe we actually have a decent shot at an understanding of how do we measure what is progress and what is a good strategy. And we could maybe start formalizing it and actually having a framework. Maybe if what we need to do is actually start creating lots of mini universes or simulations of AI solving very basic problems, in arithmetic or 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 what's the smallest neural network that can do 10-digit multiplication and things like that.

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1:09:48You 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 autodidacts. What is your process of learning about a new subfield in math? What does that look like? Yeah, so I certainly identify with kind of the, yeah, we talked about depth and breadth before. And it's not purely human AI distinction. I mean, humans also split, so I think it was Irving who split them into hedgehogs and foxes. And a hedgehog knows one thing very, very well, and a fox knows a little bit about everything. So I definitely, I think of myself as a fox.

1:10:31I work with hedgehogs a lot, and sometimes I can be a hedgehog if need be. But 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, you know, so someone was able to use a type of mathematics I'm not familiar with and get a result 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.

1:11:08So I've always had that kind of obsessive completionist type streak. I've had to wean myself off computer games because I start a game, I want to play it to completion, so both the levels.

1:11:23So 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 another mathematician who's working on another area of mathematics, and 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 I learned a lot from that. I found that writing about 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 cold trick. And I'd say, okay, I'm going to remember this.

1:12:02And then six months later, I'd forgotten. I remember remembering it, but I can't reconstruct my arguments. And the first few times it 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. How long does it take you to write a blog post? 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 feels slightly unpleasant for me to do at the time. And so writing a blog, it feels creative and fun.

1:12:37like it's something that i do for myself um so maybe depending on on the topic it could be a quick you know half an hour or several hours but i um it doesn't because it's something that i do sort of voluntarily it doesn't feel like it it it doesn't feel uh time flies when i write these things as opposed to sort of doing something which i have to do for administrative reasons but it's just that it's it's drudgery okay those are tasks that ai is really helping with nowadays actually Is it if like civilization could from first principles decide how to use Terry Tau's time? You know, it's like a limited resource.

1:13:17What is the biggest diff between if the veil of ignorance got to decide how to use Terry Tau's time versus what it does now? This podcast wouldn't be happening. Yeah. As much as I complain about certain tasks that I don't want to do, but I have to do. So as you get more senior in academia, you get more and more responsibilities. You 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 or another. Because it's outside my comfort zone, I often find interactions with people who I wouldn't normally talk to, like you, for instance.

1:13:56And I would learn interesting things and have interesting experiences. and I would have opportunities to then network with other people that I would never have done before. So I do believe a lot in serendipity. I mean, I do optimize my time when I... So there's some portions of a day where I do schedule very carefully. But I have been willing to sort of leave some portions just, okay, I'm going to do something which is not my usual thing and maybe it'll be a waste of my time, but maybe I will learn something. And more often than not, I feel like I've gotten a positive experience, which is not something I would have planned for.

1:14:41And yeah, so I believe a lot in serendipity. And maybe there's a danger actually that, you know, in modern societies, it's not just AI, but we've become really good at optimizing everything. And maybe we are optimizing, we're not optimizing a lot of optimization. that you know with with with covert for example um we we switched like we switched a lot to remote meetings um and so everything was scheduled now and so uh 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 um you had to schedule things in advance and what we lost out on was sort of the the casual like knocking on the hallway, just meeting someone while getting a coffee.

1:15:27And there's serendipitous interactions that you may think are not optimal, but actually are really important. When I was a grad student, I would go down to the library to look for a journal article. I had to physically go down to the library, check out the journal, and read your article. And sometimes the next article, you can just browse through, and the next article is also interesting. sometimes it wasn't but but you could accidentally find interesting things um 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'd done it more inefficiently um so um yeah there have been times when i'm um i spent a year once at the Institute for Advanced Study, which is a great place to, you know, there's no distractions.

1:16:27You're there to just do research. And like the first few weeks you're there, like it's great. You're getting all these papers written up that you've been wanting to do for a long time. You've been 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 somehow. Like I get bored. I actually surf the internet a lot more. You actually do need a certain level of distraction in your life. It somehow adds enough randomness and temperature, high temperature if you need. So yeah, I don't know the optimal way to schedule my life.

1:17:03It just seems to work. I'm very curious when you expect AIs that can actually do frontier math better than 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 calculus were doing frontier math that humans could not accomplish, but it wasn't, you know, number crunching. But replacing Terry Tao completely. I mean, what do you want me for? You'll just go on all the podcasts after.

1:17:47I'm not sure we've, It might not be the right question to ask.

1:17:56I think within a decade, a lot of things that mathematicians currently do, we spend 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 a hundred years ago a lot of mathematicians were just solving differential equations like people needed physicists needed some exact solution to some system and they were just they hired a mathematician to go through the calculus and work out the solution to this fluid equation or whatever a lot of what a 19th century mathematician would do you could make a call to Mathematica or Wolfram Alpha or a computer algebra package, or now more recently an AI, and it would just solve the problem in a few minutes.

1:18:50But we moved on. We worked on different types of problems after that. Once computers came along, computers used to be human. People used to liberally create log tables and work out primes as Gauss did, and that has all been outsourced to computers. But we moved on. in genetics, you know, to sequence the genome of a single organism, that was an entire PhD of a geneticist, you know, so carefully, you know, separating all the chromosomes and whatever. And now you can just spend$1 ,000 and send it to a sequencer and get it done. But genetics is not dead as a subject. You move to a different scale.

1:19:30You know, maybe you study whole ecosystems rather than individuals. I take your point. But on the question of, well, when is most mathematical progress, almost all mathematical progress happening by AI. So that if you find out, oh, this year, a millennium price problem has been solved, you would put, you know, a 95 % odds that an AI did it autonomously. Surely there will be such a year. I guess, I mean, I do believe that hybrid human plus AIs will dominate mathematics for a lot longer. It will depend, it will require some additional breakthroughs beyond what we already have. So it's going to be sarcastic.

1:20:10I think AI is currently very good at certain things, but they're really terrible at others. And while you can 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 sort of replacement for all. intellectual tasks. It is complementary currently. It's not a placement. But maybe, I mean, because current-level AIs will accelerate science in so many ways, hopefully, I mean, new discoveries, new breakthroughs will happen more quickly. I mean, it's possible that also by somehow destroying serendipity, we actually inhibit certain types of progress.

1:21:07Anything is possible, really, at this point. I think the world is very, very unpredictable at this point in time. What is your advice to somebody who would consider a career in math or is early in a career in math, especially in light of AI progress? How should they be thinking about their career differently, if at all, as a result of 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 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.

1:21:53And, you know, 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 they were 10 years ago, 20 years ago. But so I think one just has to embrace that there's 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, there'll be a lot of opportunities for things that you wouldn't be able to do before. So, I mean, in math, you previously had to basically go through years and years of education, be a math PhD, before you could contribute to the frontier of math research.

1:22:45But now it's quite possible at the high school level or whatever that you could get involved in a math project and actually make a real contribution because of all these AI tools and Lean and everything else. So there'll be a lot of non-traditional opportunities to learn. So you need a very adaptable mindset. You know, there'll be one for pursuing things just for curiosity, for playing around. And, I mean, you still need to get your credentials. I mean, I think for a while it'll still be important to sort of still go through traditional education and learn math and science the old-fashioned way for a while.

1:23:28But 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, pleasure.