Chief Scientist: Timothy Gowers, GOSIM 2026 Paris

Hello everybody. I'm Alexa Proball, the Hello everybody. I'm Alexa Proball, the head of commutate Lake sale and here we head of commutate Lake sale and here we are at the Go Sim Paris and with us we are at the Go Sim Paris and with us we have Sir Timothy Gowers who is a have Sir Timothy Gowers who is a professor at College de France and professor at College de France and University of Cambridge and he can order University of Cambridge and he can order this conference and he's a field medal this conference and he's a field medal winner winner in mathematics and in combinatorics in mathematics and in combinatorics and and it's really exciting. I went to math it's really exciting. I went to math school in school in Moscow. One of my classmates Igor Park Moscow. One of my classmates Igor Park is a well-known combinatorics scientist. is a well-known combinatorics scientist. So I So I you know, I'm not you know, I love you know, I'm not you know, I love mathematics but never you know, work at mathematics but never you know, work at this this amazing levels. It's really exciting and amazing levels. It's really exciting and all all the eyes math, right? It's all all all the eyes math, right? It's all matrices all the way down. So it's matrices all the way down. So it's really exciting to have really exciting to have Timothy with us and can you tell us a Timothy with us and can you tell us a little bit about how you came here and little bit about how you came here and what you talked about in your keynote? what you talked about in your keynote? Yes, I mean I came here because uh Yes, I mean I came here because uh I've got an invitation and uh I've got an invitation and uh I'm although I'm a mathematician, I'm I'm although I'm a mathematician, I'm also very interested in automatic also very interested in automatic theorem proving and I have a group in theorem proving and I have a group in Cambridge Cambridge >> Mhm. devoted to automatic theorem >> Mhm. devoted to automatic theorem proving. That's to say um proving. That's to say um just trying to get computers to do just trying to get computers to do mathematics. mathematics. But uh our focus is a little bit But uh our focus is a little bit different from um different from um a lot of what's going on in that uh a lot of what's going on in that uh domain. So a lot of what's happening is domain. So a lot of what's happening is what you might just call prompt what you might just call prompt engineering. You're just you're giving engineering. You're just you're giving problems to uh problems to uh language models to solve and if you get language models to solve and if you get the prompts right, they're getting the prompts right, they're getting better and better at solving them. But better and better at solving them. But we are interested in really we are interested in really understanding in a more fundamental way understanding in a more fundamental way how how human beings solve problems. It's a very human beings solve problems. It's a very mysterious mysterious uh question because um uh question because um we know that uh we know that uh it shouldn't in some sense it shouldn't it shouldn't in some sense it shouldn't be possible. If you have a general be possible. If you have a general mathematical statement Mhm. and uh you mathematical statement Mhm. and uh you ask does this statement have a proof? ask does this statement have a proof? That's actually uh halting complete. That's actually uh halting complete. There's no algorithm that can tell There's no algorithm that can tell whether a whether a in a sense there's no algorithm that can in a sense there's no algorithm that can do mathematics in general. Mhm. do mathematics in general. Mhm. But, uh we can do mathematics, and But, uh we can do mathematics, and that's because we don't do general that's because we don't do general mathematics, we do sort of interesting mathematics, we do sort of interesting mathematics. mathematics. >> Right. >> Right. So, we're trying to understand what it So, we're trying to understand what it is that about the interesting is that about the interesting mathematics that makes it possible for mathematics that makes it possible for human mathematicians with their rather human mathematicians with their rather limited resources limited resources to be able to come up with complicated to be able to come up with complicated proofs. Right. And we hope that by proofs. Right. And we hope that by understanding that, we'll be able to understanding that, we'll be able to teach teach AI to do it better. Right. And so, AI to do it better. Right. And so, as a result of that, I have as a result of that, I have an interest in what's going on in AI, an interest in what's going on in AI, and so coming to somewhere like Ghost and so coming to somewhere like Ghost Server is a good way to Server is a good way to meet people and find out what other meet people and find out what other people are thinking about in the people are thinking about in the more AI domain, so. more AI domain, so. Thank you. I I mean, it really struck Thank you. I I mean, it really struck me. One thing in your keynote, right? me. One thing in your keynote, right? So, I went to this best math school in So, I went to this best math school in Soviet Union because and I should have Soviet Union because and I should have studied history. I like history. I studied history. I like history. I really like humanities. I'm not very really like humanities. I'm not very good at math. good at math. I'm okay at math, but now I'm a better I'm okay at math, but now I'm a better programmer. So, but there was no other programmer. So, but there was no other path for people to do something real path for people to do something real because, you know, all the humanities because, you know, all the humanities were Marxist-Leninist. were Marxist-Leninist. So, So, and I thought that math is very and I thought that math is very antisocial. When I read math books, they antisocial. When I read math books, they rubbed me the wrong way because, rubbed me the wrong way because, effectively, they they provide very like effectively, they they provide very like very dense statements, and then there is very dense statements, and then there is an answer, and in between there is a an answer, and in between there is a phrase like as an exercise to the phrase like as an exercise to the reader. Yeah, yeah, yeah. Or it easily reader. Yeah, yeah, yeah. Or it easily follows that and there is nothing like follows that and there is nothing like like it's followed easily, right? And like it's followed easily, right? And so, I thought that I'm dumb. I don't so, I thought that I'm dumb. I don't understand it. But then you said that understand it. But then you said that this is this is basically a wrong way for people to basically a wrong way for people to understand math, and you said that LLMs understand math, and you said that LLMs they should do math they should do math basically became antisocial and mean basically became antisocial and mean this way because mathematicians write this way because mathematicians write books this way from which we shall learn books this way from which we shall learn to read. And so, suddenly it just to read. And so, suddenly it just occurred to me that math could could occurred to me that math could could have been different. It could have been have been different. It could have been more convenient. And the And you also more convenient. And the And you also said that you're interested not in the said that you're interested not in the achieving the result, but you want to achieving the result, but you want to see how a mathematician arrived at that see how a mathematician arrived at that result. This was very interesting to me. result. This was very interesting to me. Can you explain a little bit like how do Can you explain a little bit like how do you go about it? you go about it? Yes, I mean just before I Yes, I mean just before I get on to that, I mean I think something get on to that, I mean I think something very important happened in the beginning very important happened in the beginning of the 20th century, which was putting of the 20th century, which was putting mathematics on a rigorous footing, mathematics on a rigorous footing, developing the axioms developing the axioms uh ZF axioms and uh ZF axioms and uh uh formalizing what we meant by formalizing what we meant by rigorously what we meant by a rigorously what we meant by a mathematical proof mathematical proof and so on. But unfortunately, I think and so on. But unfortunately, I think that development, though extremely that development, though extremely important for mathematics and very important for mathematics and very positive development in many ways, positive development in many ways, had a slightly unfortunate stylistic had a slightly unfortunate stylistic consequence that people then tried to consequence that people then tried to write in this kind of write in this kind of formal logical style. formal logical style. >> Dry. Yeah, which um >> Dry. Yeah, which um sort of lost sort of lost Something was lost in that process or Something was lost in that process or something to do with the human presence. something to do with the human presence. They look like computers. They They want They look like computers. They They want to be machines. to be machines. >> Yeah, exactly. >> Yeah, exactly. Um Um So, how does one counteract that? I So, how does one counteract that? I mean, actually I've been trying to mean, actually I've been trying to counteract that throughout my career, counteract that throughout my career, uh long before anything to do with AI or uh long before anything to do with AI or automatic theorem proving, automatic theorem proving, just by just by really examining carefully how really examining carefully how I myself come up with proofs or thinking I myself come up with proofs or thinking about things that I was taught when I about things that I was taught when I was a student. Mhm. Um that I just was a student. Mhm. Um that I just accepted accepted the proofs of them. So, I would think, the proofs of them. So, I would think, "Actually, how could somebody have "Actually, how could somebody have thought of that argument?" And thought of that argument?" And sometimes if I thought hard enough about sometimes if I thought hard enough about it, I realized actually often after it, I realized actually often after teaching it myself, Mhm. I would teaching it myself, Mhm. I would suddenly see something that would make suddenly see something that would make me realize, "Ah, yes, that's how to have me realize, "Ah, yes, that's how to have probably how that was or how could that probably how that was or how could that have that how how that could have been have that how how that could have been discovered." discovered." Um Um and so and so for me there's always been quite a for me there's always been quite a strong link strong link between between a really good way of explaining a really good way of explaining mathematics to humans Mhm. and a really mathematics to humans Mhm. and a really good way of explaining mathematics to good way of explaining mathematics to computers. Mhm. Um computers. Mhm. Um and um what I hope is that uh and um what I hope is that uh Well, I don't think we have this at the Well, I don't think we have this at the moment, but I'm hoping that moment, but I'm hoping that sometime in the future sometime in the future we will be able to teach computers we will be able to teach computers to think about mathematics in a more to think about mathematics in a more human way Mhm. so that they will then be human way Mhm. so that they will then be able to explain to us in a better way able to explain to us in a better way uh what they've been doing. So, at the uh what they've been doing. So, at the moment, you know, they they've copied moment, you know, they they've copied the style that they see in textbooks. the style that they see in textbooks. They just give us They just give us give us the answers, basically. give us the answers, basically. >> Right. And uh >> Right. And uh sometimes that's not what you want. sometimes that's not what you want. Sometimes you want This is fascinating, Sometimes you want This is fascinating, right? Because I I vividly remember So, right? Because I I vividly remember So, when I uh you know, went to school 57, when I uh you know, went to school 57, which is the most famous math school in which is the most famous math school in Soviet Union. So, a lot of our graduates Soviet Union. So, a lot of our graduates basically, like Igor, like the basically, like Igor, like the professors of math in in famous places. professors of math in in famous places. And the other half became oligarchs and And the other half became oligarchs and they're all retired. Uh because they're they're all retired. Uh because they're very good at computers. Uh but you know, very good at computers. Uh but you know, So, we have this series of exams. So, So, we have this series of exams. So, basically, the the class basically, the the class of the math class was filled of the math class was filled progressively. First, the half of the progressively. First, the half of the people who attended the evening courses, people who attended the evening courses, and then a quarter, and then and so, you and then a quarter, and then and so, you have to pass exams. So, and um right? have to pass exams. So, and um right? And so, And so, uh and I remember that uh and I remember that uh So, I read some book on the number uh So, I read some book on the number theory. So, Soviet Union had these theory. So, Soviet Union had these amazing math books, right? And so, amazing math books, right? And so, basically, I was supposed to explain basically, I was supposed to explain something, and I thought that I know I something, and I thought that I know I kind of mechanically memorized a lot of kind of mechanically memorized a lot of mathematical formulas. And so, the guy mathematical formulas. And so, the guy uh was looking at me. Uh I was uh was looking at me. Uh I was explaining some some inference, but I explaining some some inference, but I mechanically memorized it. I didn't mechanically memorized it. I didn't understand the underlying math. I just understand the underlying math. I just kind of memorized how the the proof kind of memorized how the the proof goes. And then, I forgot midway. Uh goes. And then, I forgot midway. Uh and and basically, I did all the steps and and basically, I did all the steps except the final step. And then, the guy except the final step. And then, the guy looked at me smiling as like expecting looked at me smiling as like expecting me to just normally explain what me to just normally explain what follows. So, because I followed follows. So, because I followed mechanically, I looked at him with mechanically, I looked at him with horror, and like I realized that I horror, and like I realized that I forgot the next step. And then because I forgot the next step. And then because I never understood the whole thing, uh I never understood the whole thing, uh I couldn't do it. And so they it just couldn't do it. And so they it just struck me that, you know, some people struck me that, you know, some people really understand it, and some people really understand it, and some people mechanically follow this, and I couldn't mechanically follow this, and I couldn't do it. And so and then basically it made do it. And so and then basically it made me think like I'm not really I don't me think like I'm not really I don't really get mathematics, but that's really get mathematics, but that's probably because the the way it was probably because the the way it was explained was very mechanical and not explained was very mechanical and not explaining it to me myself. Yeah, it's explaining it to me myself. Yeah, it's an interesting measure of an interesting measure of the level of understanding of say a a the level of understanding of say a a proof in mathematics. proof in mathematics. >> Mhm. How much you can actually compress >> Mhm. How much you can actually compress it. it. >> Yes. So uh >> Yes. So uh the lowest level of understanding, you the lowest level of understanding, you just have to learn each line and how just have to learn each line and how what comes in what comes in But if if you have a better level of But if if you have a better level of understanding, you might have sort of understanding, you might have sort of five ideas that if you first you do this five ideas that if you first you do this and getting to that idea is a fairly and getting to that idea is a fairly standard calculation, and then you get standard calculation, and then you get to this by a fairly standard to this by a fairly standard calculation. But then you have to calculation. But then you have to remember those five ideas. Yes. And remember those five ideas. Yes. And maybe you understand it better than you maybe you understand it better than you actually see there's only one idea, and actually see there's only one idea, and those five ideas were natural. those five ideas were natural. Yes. And then maybe at some point you Yes. And then maybe at some point you reach the point where reach the point where just the whole thing just you're doing just the whole thing just you're doing what feels like the obvious thing, and what feels like the obvious thing, and that's when you've reached the sort of that's when you've reached the sort of ultimate understanding of the proof. ultimate understanding of the proof. >> Right. But but to me it was, you know, a >> Right. But but to me it was, you know, a lot of it is like when you do lot of it is like when you do integration, a lot of it is mechanical integration, a lot of it is mechanical moving of parts of of formulas around. moving of parts of of formulas around. And so you can remember the mechanics of And so you can remember the mechanics of doing it, but not the underlying doing it, but not the underlying principle. And so I think a lot of principle. And so I think a lot of people will do mathematical people will do mathematical transformations as just some kind of transformations as just some kind of mechanical movements, right? And they mechanical movements, right? And they will not And I think a lot of school will not And I think a lot of school children do this. Yeah, I've seen that children do this. Yeah, I've seen that with my own children, actually. Uh-huh. with my own children, actually. Uh-huh. Everything seems to be fine, and then Everything seems to be fine, and then they suddenly make some bizarre mistake, they suddenly make some bizarre mistake, and then I realize that There's no and then I realize that There's no understanding. Yeah, that the reason it understanding. Yeah, that the reason it seemed fine was just because they'd seemed fine was just because they'd learned the manipulations very well, and learned the manipulations very well, and uh uh but uh if they got sort of out of but uh if they got sort of out of distribution, so to speak, distribution, so to speak, >> Right. Right. >> Right. Right. >> then suddenly uh it fell apart. >> then suddenly uh it fell apart. I want to kind of uh I want to kind of uh zoom in on the formal methods. So I had zoom in on the formal methods. So I had a friend who was in formal methods in a friend who was in formal methods in 1993 working with a computer company, 1993 working with a computer company, right? And so it seems like a very very right? And so it seems like a very very kind of kind of stagnant little like academic niche, stagnant little like academic niche, right? And so, there were people they right? And so, there were people they had the formal method conferences, and had the formal method conferences, and they went there, but they were not they went there, but they were not connected to the real world, right? No, connected to the real world, right? No, so the first time I've seen a talk about so the first time I've seen a talk about formal methods actually being practical formal methods actually being practical was Leslie Lamport, who gave a talk at was Leslie Lamport, who gave a talk at the computer conference, and TLA+ and he the computer conference, and TLA+ and he mentioned how the Amazon Dyna DynamoDB mentioned how the Amazon Dyna DynamoDB team used TLA+ to basically formulate team used TLA+ to basically formulate that it's a correct implementation, that it's a correct implementation, right? And and but now then I've seen right? And and but now then I've seen this whole rise of Lean in conjunction this whole rise of Lean in conjunction with and people basically want to model with and people basically want to model the world now with Lean and prove, you the world now with Lean and prove, you know, things about it and apparently it know, things about it and apparently it does work in math. So, I'm just curious does work in math. So, I'm just curious like do you see this is this a like do you see this is this a breakthrough for formal methods? Um I breakthrough for formal methods? Um I think it's certainly a breakthrough of think it's certainly a breakthrough of one kind. I mean, it's uh one kind. I mean, it's uh we're seeing we're seeing the the the the story always used to be the the the the story always used to be that that one one by and large mathematicians were by and large mathematicians were they didn't really need this extra they didn't really need this extra guarantee of correctness because if guarantee of correctness because if something was important something was important um it would be read by a lot of people. um it would be read by a lot of people. If it was if it was If it was if it was incorrect, you know, usually there would incorrect, you know, usually there would be small mistakes, but if there were be small mistakes, but if there were serious mistakes, somebody would at some serious mistakes, somebody would at some point realize that. Mhm. point realize that. Mhm. And if it wasn't important, then who And if it wasn't important, then who cares? cares? Um but two things have changed. And also Um but two things have changed. And also the other the other side of the story the other the other side of the story was that uh was that uh the effort needed to write down a the effort needed to write down a completely formal proof in a proof completely formal proof in a proof assistant such as Isabelle or Coq or assistant such as Isabelle or Coq or Mizar or Mizar or or Lean which came in a bit later or Lean which came in a bit later was so great and so much greater than was so great and so much greater than just writing out the usual style of just writing out the usual style of proof Mhm. that it just wasn't worth it. proof Mhm. that it just wasn't worth it. So, two things have changed, I think. So, two things have changed, I think. One is that One is that conventional proofs have become more and conventional proofs have become more and more complicated Mhm. to the point where more complicated Mhm. to the point where there is genuine worry about their there is genuine worry about their correctness and uh Right. if the too correctness and uh Right. if the too many proofs come out many proofs come out >> Fermat's theorem it took a long time to >> Fermat's theorem it took a long time to show that it's correct. So sorry. And um on the other side on the other side as Lean has developed as Lean has developed um and people have built more and more um and people have built more and more tools and added more and more to math tools and added more and more to math Lean Lean that you can then use that you can then use um barrier to entry to formalizing has um barrier to entry to formalizing has perhaps gone down and down and we find perhaps gone down and down and we find that uh that uh undergraduates can formalize deep undergraduates can formalize deep bits of mathematics Mhm. which they bits of mathematics Mhm. which they don't even necessarily have to don't even necessarily have to understand very well as long as they can understand very well as long as they can just cover it line by line. just cover it line by line. >> Right. Um so >> Right. Um so things have changed a lot and um things have changed a lot and um there's now a big sort of thriving Lean there's now a big sort of thriving Lean community. So I think I could call that community. So I think I could call that a a maybe not a breakthrough exactly because maybe not a breakthrough exactly because it wasn't really a conceptual change. it wasn't really a conceptual change. It's just more like a sort of phase It's just more like a sort of phase transition that's taken place. transition that's taken place. And um And um a further phase transition seems to be a further phase transition seems to be about to take place which will be about to take place which will be that auto formalization will get that auto formalization will get sufficiently good sufficiently good that people will be able to formalize that people will be able to formalize without even bothering to learn Lean. without even bothering to learn Lean. They would just write formalizations and They would just write formalizations and >> Right. Write formalizations. >> Right. Write formalizations. So that means that I mean one immediate So that means that I mean one immediate change will be I think that uh change will be I think that uh instead of journals sending articles out instead of journals sending articles out to referees to check the proofs we'll to referees to check the proofs we'll just get computers to check proofs just get computers to check proofs and so then the only remaining job of and so then the only remaining job of journals will be journals will be judging whether something's interesting judging whether something's interesting or not. Do you think it will ever happen or not. Do you think it will ever happen that you don't need a human in the loop? that you don't need a human in the loop? That this vibe proofing will actually be That this vibe proofing will actually be good enough? good enough? I do think it will happen. I don't know I do think it will happen. I don't know how long it'll take but I think that how long it'll take but I think that that is the end that is the end >> automatic proof Yeah yeah. that you will >> automatic proof Yeah yeah. that you will be comfortable with. be comfortable with. >> Oh that's slightly different question >> Oh that's slightly different question whether I'm comfortable with [laughter] whether I'm comfortable with [laughter] Will you ever accept something that Will you ever accept something that computer said is correct and then computer said is correct and then Yeah, because you you like you're an Yeah, because you you like you're an editor of the journal, you send a editor of the journal, you send a submission to a lean verifier and it submission to a lean verifier and it came back saying this is correct. Will came back saying this is correct. Will you be now comfortable publishing this? you be now comfortable publishing this? Uh if I judge it to be interesting Uh if I judge it to be interesting enough then I'd be enough then I'd be I mean I mean I'd be at least as comfortable in fact I'd be at least as comfortable in fact probably a lot more comfortable probably a lot more comfortable judging it to be correct judging it to be correct if it had been verified in lean. Yes. if it had been verified in lean. Yes. Than if I just sent it out to a referee Than if I just sent it out to a referee who probably didn't read it very who probably didn't read it very carefully. carefully. >> Right, that's right. I mean what we have >> Right, that's right. I mean what we have the system and one has to remember that the system and one has to remember that the system we have now the system we have now is very far from perfect. Right, right, is very far from perfect. Right, right, right. Unless you know and trust the right. Unless you know and trust the referee to do a good job. referee to do a good job. >> a lot on those sort of >> a lot on those sort of webs of webs of trust and acquaintance with people and trust and acquaintance with people and so on. So so on. So So I'm not thinking what you described So I'm not thinking what you described is very interesting because and you is very interesting because and you basically mentioned in the keynote that basically mentioned in the keynote that the models in the recent months got so the models in the recent months got so good that the best models operated at good that the best models operated at the level of graduate students, right? the level of graduate students, right? At the graduate student PhD candidate. At the graduate student PhD candidate. >> I may have to qualify that but at least >> I may have to qualify that but at least uh uh some of the time. Some of the some of some of the time. Some of the some of the time. the time. >> And I think there may be parts of >> And I think there may be parts of mathematics where they haven't reached mathematics where they haven't reached that level but Right. that level but Right. Um Um and there will be plenty of problems and there will be plenty of problems where yeah, but where yeah, but they can operate at that level if you they can operate at that level if you give them the right give them the right questions to do right at least. And and questions to do right at least. And and and basically you said that um and basically you said that um a lot of so you're concerned that a lot a lot of so you're concerned that a lot of low-level work uh will you know, of low-level work uh will you know, students will now delegate to students will now delegate to LLMs instead of doing this manually, LLMs instead of doing this manually, right? Like a lot of steps that normally right? Like a lot of steps that normally they would be doing manually, they would they would be doing manually, they would delegate it delegate it and and um and and um So So you you're basically worried about that. you you're basically worried about that. But I'm just thinking that like in in in But I'm just thinking that like in in in coding now everybody uses coding now everybody uses code code and similar tools and nobody code code and similar tools and nobody There was a period of time in ancient There was a period of time in ancient history 6 months ago when people history 6 months ago when people questioned that. Now, the consensus is questioned that. Now, the consensus is that the best developers use AI. And it that the best developers use AI. And it just makes them even better. The rich just makes them even better. The rich gets richer, right? Like this is gets richer, right? Like this is Marxism. So, the 10x developers, the Marxism. So, the 10x developers, the magical creatures who are super magical creatures who are super productive, become even more productive, productive, become even more productive, and juniors are get getting eliminated and juniors are get getting eliminated because they cannot compete with even a because they cannot compete with even a basic LM. So, uh, so the question is basic LM. So, uh, so the question is now, I wonder if if Lean now becomes now, I wonder if if Lean now becomes approachable because machines can do a approachable because machines can do a lot, right? Like it was impractical for lot, right? Like it was impractical for people to do a lot of formalization. people to do a lot of formalization. Now, this will be by formalization. Will Now, this will be by formalization. Will it now become a tool in a similar it now become a tool in a similar fashion for mathematicians, for fashion for mathematicians, for students? So, you will always work students? So, you will always work alongside Lean or something like this, alongside Lean or something like this, right? So, so it will not compete it right? So, so it will not compete it will be like a coding tool. It will help will be like a coding tool. It will help you, but human is still in charge you, but human is still in charge because a smart our computer architect because a smart our computer architect developer, he knows what to ask, and developer, he knows what to ask, and that really makes a difference. And that really makes a difference. And like, is this like, is this can this be the course of mathematics can this be the course of mathematics education? So, now you will teach with education? So, now you will teach with Lean from see, you know, math 101. Lean from see, you know, math 101. Do you think that's that's going to Do you think that's that's going to happen? happen? I don't know whether we'll teach with I don't know whether we'll teach with Lean because [clears throat] I'm Lean because [clears throat] I'm learning Lean learning Lean presents an extra barrier somehow. And presents an extra barrier somehow. And the the way that you have to do the the way that you have to do mathematics if you want to do Lean mathematics if you want to do Lean is a little bit different from the way is a little bit different from the way you do normal mathematics. A little bit you do normal mathematics. A little bit less intuitive in various ways. Okay. less intuitive in various ways. Okay. Um, or at least that's how it seems Um, or at least that's how it seems because I've grown up the other way because I've grown up the other way maybe. Maybe maybe. Maybe >> Maybe it will be some something better >> Maybe it will be some something better than Lean. But something maybe a than Lean. But something maybe a platform that has Lean running in the platform that has Lean running in the background background >> Mhm. that >> Mhm. that is a bit more sort of um, is a bit more sort of um, user-friendly than Lean. Maybe Lean is user-friendly than Lean. Maybe Lean is integrated into LaTeX and as you write a integrated into LaTeX and as you write a paper, like it gets verified. Yeah, paper, like it gets verified. Yeah, exactly. I think if you've got some exactly. I think if you've got some interface sort of a higher level interface sort of a higher level language that you're using yourself Mhm. language that you're using yourself Mhm. but Lean is um, but Lean is um, verifying what you write as you write. verifying what you write as you write. Or Lean combined with uh, automatic Or Lean combined with uh, automatic theorem proving tools. Mhm. That could theorem proving tools. Mhm. That could be be Yeah, I I think we will get to that Yeah, I I think we will get to that stage. But then there's a sort of race stage. But then there's a sort of race between between people developing a platform like that people developing a platform like that and just AI getting better and better at and just AI getting better and better at maths. Right. On its own. maths. Right. On its own. Um so I mean a future that it would be Um so I mean a future that it would be very I think very nice would be if um very I think very nice would be if um we had platforms that made it really we had platforms that made it really easy to do mathematics and have it easy to do mathematics and have it formalized as you do it and uh formalized as you do it and uh um um in other words, sort of tools that just in other words, sort of tools that just help you think at your screen. help you think at your screen. Um Um and I I know a lot of people are and I I know a lot of people are interested in um developing such tools, interested in um developing such tools, including my group in Cambridge, including my group in Cambridge, actually. actually. Um Um but uh but uh if you have a really good platform like if you have a really good platform like that, then you can also train an AI that, then you can also train an AI system to use the platform, so Right. Uh system to use the platform, so Right. Uh So that will be our ask for developers. So that will be our ask for developers. If you want to build such a platform, it If you want to build such a platform, it will have an impact. So maybe, you know, will have an impact. So maybe, you know, I'll close with the kind of general I'll close with the kind of general question because, you know, I come from question because, you know, I come from Soviet Union, which had very strong math Soviet Union, which had very strong math and physics, and a lot of people here and physics, and a lot of people here come from China or India or France, come from China or India or France, where they have very strong STEM where they have very strong STEM education, right? And so I think that's education, right? And so I think that's why uh you know, for instance, like, you why uh you know, for instance, like, you know, there is a question, you know, know, there is a question, you know, women in tech, women in science. In women in tech, women in science. In America, it's very, you know, much America, it's very, you know, much dramatized, but in India, Russia, China, dramatized, but in India, Russia, China, France, there is more balance women in France, there is more balance women in tech because everybody gets strong tech because everybody gets strong education. There is no question, you education. There is no question, you know, of somebody not getting math. This know, of somebody not getting math. This is I think we're of the last generation is I think we're of the last generation which very educated without the AI. And which very educated without the AI. And now, of course, everybody says, "You now, of course, everybody says, "You don't need to know anything. AI will don't need to know anything. AI will solve it for you." So I wonder, and you solve it for you." So I wonder, and you are at Cambridge, right? Like the are at Cambridge, right? Like the citadel of uh kind of uh academia and citadel of uh kind of uh academia and tradition. And so, how do you see the tradition. And so, how do you see the role of math? role of math? Uh so math uh so AI is all math, as we Uh so math uh so AI is all math, as we know, is all math. This is all the way know, is all math. This is all the way down, but down, but in practical application, it's so much in practical application, it's so much removed now from math that very few removed now from math that very few people can do the underlying math, people can do the underlying math, right? The actual gradient descent, uh right? The actual gradient descent, uh uh uh understanding matrices, it's uh uh understanding matrices, it's actually it used to be a part of AI actually it used to be a part of AI education. Now, nobody really needs education. Now, nobody really needs this. People just can like ask this. People just can like ask right? So, right? So, I think like every year more and more I think like every year more and more people in the AI have no idea what is people in the AI have no idea what is the underlying math, how it work How do the underlying math, how it work How do you think like it becomes specialized to you think like it becomes specialized to like PyTorch people optimizing for GPUs, like PyTorch people optimizing for GPUs, they need to know both the math and how they need to know both the math and how the computation becomes a very the computation becomes a very specialized tool. So, I wonder as a specialized tool. So, I wonder as a professor of mathematics, how do you see professor of mathematics, how do you see uh uh math math kind of as a foundation of science? kind of as a foundation of science? Are we going to be able to preserve it Are we going to be able to preserve it that way? Or is it going to become a that way? Or is it going to become a very niche very niche uh area or like how do you see the role uh area or like how do you see the role of math in in education, right? Uh to of math in in education, right? Uh to make the people of tomorrow? I don't know what's going to happen. I'm I don't know what's going to happen. I'm I'm a bit worried about that. Uh as I I'm a bit worried about that. Uh as I like I said in my uh not in the keynote like I said in my uh not in the keynote panel yesterday, there was a discussion panel yesterday, there was a discussion about about AI and education. AI and education. >> Right. >> Right. And I'm worried about this perception And I'm worried about this perception that we don't need to know anything that we don't need to know anything because we can outsource all our because we can outsource all our So, there was somebody uh Bill Wren who So, there was somebody uh Bill Wren who said that uh said that uh we can outsource we can outsource knowledge, but we can't outsource knowledge, but we can't outsource understanding or something like that. I understanding or something like that. I think that's think that's I think that's quite true and I think if I think that's quite true and I think if we don't have we don't have um a significant percentage of people in um a significant percentage of people in a society who really understand a society who really understand mathematics and uh mathematics and uh at some level, at some level, uh then we will operate uh then we will operate we'll be somehow under the control of we'll be somehow under the control of these systems. We won't we'll be passive these systems. We won't we'll be passive consumers of them and um So, that worries me and I I think that So, that worries me and I I think that maybe what happens. On the other hand, I maybe what happens. On the other hand, I think it will remain the case Mhm. that think it will remain the case Mhm. that uh people who do make the effort to have uh people who do make the effort to have a good understanding of mathematics a good understanding of mathematics will have a big advantage. will have a big advantage. Um Um bit like what you were saying earlier bit like what you were saying earlier about the rich getting richer. If you about the rich getting richer. If you Right. If you know how to Right. If you know how to If you If you know maths well, then If you If you know maths well, then you'll be able to you'll be better at um you'll be able to you'll be better at um using AI and using it to do mathematics using AI and using it to do mathematics and and having an understanding of what you can having an understanding of what you can what you can use the tools for and so what you can use the tools for and so on. on. Um so I hope that uh that message will Um so I hope that uh that message will get through. get through. Do you see young people, so you're Do you see young people, so you're teaching in France and then in teaching in France and then in Cambridge. Do you see Cambridge. Do you see young people coming into maths who will young people coming into maths who will make great mathematicians despite of make great mathematicians despite of everything else happening? everything else happening? Like do you see like is there supply of Like do you see like is there supply of smart young students Well, I'm very smart young students Well, I'm very lucky to be um at Trinity College, lucky to be um at Trinity College, Cambridge and we get regularly Cambridge and we get regularly incredibly good mathematicians coming incredibly good mathematicians coming >> Mhm. >> Mhm. >> there. So they're still being produced >> there. So they're still being produced somewhere and get to Trinity. And for somewhere and get to Trinity. And for the moment, I think they are still the moment, I think they are still noticeably ahead of what AI can noticeably ahead of what AI can Mhm. Mhm. Um the very best students anyway. Um the very best students anyway. Um Um So there is hope. There is hope for the So there is hope. There is hope for the future of maths. Yes, there is. future of maths. Yes, there is. But uh But uh Yeah, it's it's Yeah, it's it's It could go either way, I think. It's It could go either way, I think. It's We We really don't know what's going to We We really don't know what's going to happen. That's great. I mean, it's it's happen. That's great. I mean, it's it's it's really great to see that you're it's really great to see that you're doing things with Lean. So what is your doing things with Lean. So what is your goal for the next year? So, you know, if goal for the next year? So, you know, if we meet here next year, what do you want we meet here next year, what do you want to have achieved by this to have achieved by this like in a year with your lab? like in a year with your lab? Well, Well, I was talking in my keynote a little bit I was talking in my keynote a little bit about a platform that my group is about a platform that my group is developing developing um Mhm. that would in sort of encourage um Mhm. that would in sort of encourage people to people to produce not just proofs, but proofs that produce not just proofs, but proofs that are transparent, so you can see where are transparent, so you can see where the ideas come from. the ideas come from. So my sort of main short-term hope So my sort of main short-term hope Maybe short What I think of as Maybe short What I think of as short-term, but in this world, a year short-term, but in this world, a year feels like long-term as well. feels like long-term as well. Is just platform will be much more Is just platform will be much more developed and we'll be able to developed and we'll be able to demonstrate that it really is uh demonstrate that it really is uh a good way of thinking about how to do a good way of thinking about how to do mathematics. This sounds great. So, if mathematics. This sounds great. So, if you are ever in San Francisco Bay Area, you are ever in San Francisco Bay Area, we're on the uh meet up there. Just come we're on the uh meet up there. Just come and give us a talk and do a demo of the and give us a talk and do a demo of the platform. We'd like We'd like our center platform. We'd like We'd like our center student to show it. We'd love to play student to show it. We'd love to play with it. with it. Well, thank you so much. We appreciate Well, thank you so much. We appreciate it. Thanks a lot. Yeah.