Live from ICM 2026: What Is Math For in the Age of AI?

24:08And of course, producing a string of numbers is not human understanding. It's meaningless to us. So we don't consider the problem to be solved by the supercomputer. It's the human who adapts it in a way that, as you said, as human beings, we're able to appreciate it, understand it, and integrate it into the larger sphere of knowledge. And we're not computer scientists, of course, you're not computer scientists. But I just wonder if at any point do we think the AI will understand and will that understanding be completely different than the way in which human beings understand? Why are you guys looking at me?

24:45There's a number of ways to try to answer that question. You know, will the AI be conscious? I have no idea. Who knows? I'd like to think of, I think it was Richard Feynman who said a long time ago when speaking about technology and the way human beings harness technology that we see in nature and try to make it our own. And he pointed out that the way a bird flaps its wings is not at all how we figured out flight. So we have a completely different system that gets us from point A to point B in the air. Presumably we looked at birds and dreamed up such a thing. And maybe in the same way, depending on what our goals are, we invent technology that doesn't look exactly the way nature figured out to solve that problem, but solves it slightly differently.

25:30And so that's kind of what LLMs have become, is this alternative mechanism by which we can do a small part of what mathematical practice strives for. Does it mean that it understands? I don't know what the word understands means in this context. And it's all, you know, Turing never said what it means to understand language. He just said, does it look like language? Does the thing talk to me the way, and I can't tell if it's a human or a machine. So in the same way, maybe I'm even more pragmatic about it. Is the AI useful to me in what my goals are? My goals are to understand mathematics. I want to learn new math every single day.

26:05Is using AI helping that goal or not? To the extent that it's helping, I'm going to use it. And to the extent that it's not, I'm going to stop and go for a walk and, you know, play ball with my kids or something. I like your operational question. Can they be helpful to us? And I think the answer is already in that. Yes, of course, they're helpful. But the part that I feel is back to my list of taboo questions is what happens when they're not useful to us because they could do everything that we could do except much, much better. So I will stipulate, but maybe you won't agree, but I think it's obvious that not only will they be able to solve problems better than us, they'll be able to ask questions that we will find interesting.

26:44They'll be able to do exposition better than Martin Gardner and the whole thing. Will it still be fun for us to do, Matt? Well, I think you said two flavors of things, one of which was very reasonable, the other of which was very different. Go ahead. What were those? So on one hand, you could say, okay, once you postulate reaching the singularity, then I feel like all bets are off, and then it's not really worth having a discussion. Because it's so hard to know what will happen next, you mean? Or I just feel like the debates post-singularity tend not to have much value. You didn't use the phrase superintelligence.

27:21Anytime that gets used, I personally feel those are less useful discussions because of the lack of definitions. But then you said things that were very reasonable because they were precise. And there were statements about evidence of they will be able to write things that will be interesting. And those things were already here. That time has already come. And I think they are already telling us interesting things. I'm not hearing that they're already here. I hear people saying things, you know, again, because this is, I feel like, a sociological question. What are we talking about privately? and I hear that we can still have taste.

27:51We're better at knowing what's interesting. Ah, okay. So here I'm going to be going. And I'm claiming that that's only for the short term. So obviously there might be a new technological advance, but with the way things currently work, anything you can hill climb, it can hill climb. If there's anything that you can have enough examples of, it can improve. We train currently students to try to deal with the unknown, the situations that they have never seen anything like before, which is by definition, the thing you can't sort of train for. So is someone going to predict that a Pulitzer Prize winning novel will be written by an AI within three years?

28:26I'll put$20 against you on this. And people have made bets like this in the past, and they've lost those bets every single time. We were supposed to be on Mars by today, if you listen to certain very wealthy people 10 years ago. So absolutely large quantities of acceptable material, I expect to be very useful. And I'm not belittling it. This will change everything will happen. But I think the way things are currently happening is outstanding things are going to happen. And AI is involved and humans are involved. And I guess people are claiming that a time will come when there'll be no more humans, but they need some evidence.

29:03I very much agree. You have to be very careful about the premises, But let me give a, just let's suppose we're in this world where AIs can prove things, they can write questions that we find just as interesting. You know, like when I arrived at grad school, right, I met all these people who were just much better at math than I was. And at that time I was like, I'm never going to be as good at math as these people. But this is great. I get to study this field. It's a beautiful field. I'm not going to be the best at it. I'm happy to spend my life this way. So I think, you know, yes, we will still do math because it has some basic value to us as human beings.

29:38I think Akshay's point is crucial. And also to your point about students and the threat really to the young student is attitudinal. If they think there is no pleasure here, no joy here, no discovery, not just a career or being the best, but becoming a mathematician is no longer on offer and the hard work that it's going to require. and having to resist looking up the answers in the back of the book, which is to say to ask one of the models. And so how are you imagining addressing this to have the next generation of mathematicians survive? So I've been trying to explain to my students sort of three scenarios, three vignettes, as an analogy to exactly this question.

30:23Vignette number one, you have some heavy pallets and you need to move them onto a truck. So you get a forklift, you drive it over, you lift the pallets, you put them on the truck, end of story. Okay. Scenario number two, you're trying to learn how to use a forklift. So you get in the forklift and you move some pallets, not because they need to be moved from point A to point B, but because you're learning how to operate a forklift. Okay. Scenario three, you decide you want to get a little workout in, you go to the gym, you put 200 pounds on the bench press, you get in your forklift, you drive over and you do 10 reps with the forklift.

30:54One of these is the wrong use of technology. okay so if you just explain to students why are you here yeah what are you you know i don't need to know the answer to this real analysis homework i know how to do this it's in the back of the book before ai you could google the answers right okay so uh what is the purpose of this operation is it for you to learn how to use that technology to learn how to prompt ai to solve real analysis problem is it that i want the solution to the problem or is it you're supposed to sit there being frustrated and get the stamina for frustration that it takes to do any kind of difficult thing that human beings want to do.

31:31If that's what the goal of this exercise is, then either sign up voluntarily to do the exercise the way it's designed or drop this course because that's what I'm trying to teach. Yeah, I think the gym analogy is pretty good. And I do think it's very interesting. I once had to try to solve Einstein's equations for a black hole solution, not because it wasn't solved already or because I couldn't find the answer somewhere. And it was a great moment in my life, right, going through that process. But I think there's going to take some convincing for this generation that's growing up on AI. It's interesting your example, because this has always been a little bugaboo of my own, that there are people who talk about high school geometry is worthless, say the people who don't like math.

32:11But geometry will teach you to think. It will help you develop logic and patience and blah, blah, blah. I don't like that defense of geometry. Geometry is a magnificent creation of humanity. And this is like, you know, you could play soccer because you don't want to exercise your leg or because it's the beautiful game, you know? And so math to me, the content is beautiful. It's not just the calisthenics that I want to make myself stronger for solving a problem for a finance company, because that's what really matters. I want to do it for human flourishing, like Francis Sue would say, like what I think you've been pushing, Ravi.

32:46Not just for my own bulking up, because the content is inherently rich and makes me a happier person before I'm dead. Like, what's the meaning of life? I want to do math while I'm alive. And I like that we can have both. I like the fact that soccer in the United States is played in middle schools en masse, and you have fun. You're with people, and they're secretly, of course, getting healthier and stronger by enjoying it and building habits. So this seems a great model for similarly doing geometry to support similar things at the same ages. This might be very unpopular and I apologize to the mathematicians ahead of time.

33:23There's a belief that if all of human knowledge is wiped out tomorrow, not humanity, but human knowledge, that eventually will all be rediscovered. One plus one is two. If human knowledge was wiped out tomorrow, the AI stops. Kids got to start over. But there's a version of this question which you can ask right now, which is, okay, so there are these, we didn't get into formalization and interactive theorem provers, but if you take any interactive theorem prover, which has the logical axioms of inference, logical inference from one state to the next, and you seed it with nothing, like alpha zero, you've just given it the rules of chess, and you say, play a trillion games, and it comes up with, you know, its own theory and its own evaluation of states and so on.

34:02If you seed an interactive theorem prover with no library of what mathematics is, and you cut an AI loose, will it discover a proof of Fermat's last theorem? Right, without the reinforcement and... Yeah, without giving it any human knowledge, without anything. So that is, I think Peter Sarnak is popularizing this question. You know, is there like a math zero? Could someone train an AI to actually do that? That's a real question, and that's one that we don't have an answer to at the moment. But we know that we need, we know humanity could do it. I mean, we've done it once, we could do it again, right?

34:33But it also sort of harkens, and I think this was the part I was concerned mathematicians wouldn't like, to the discovered or invented question. are these discoverable truths that an entity that thinks entirely differently from human beings could discover. Sure. And you can imagine being the first human being to see a lake wandering in the woods in some new island that's discovered. Both can be true, that there's joy in human discovery, and also maybe a drone would fly over the island and discover the lake. Well, I think this is the place where we turn it over to the audience. You've been patient, and I appreciate our panel taking some pretty wild questions from us.

35:10They may get even wilder here. We're ready when you are. Please. Great. So far, AI is controlled by private companies, and access to the best models is increasingly being gated by financial resources. And I'm concerned about this divide between researchers who are going to have access and those who don't. Historically, our field has been known for all you need is pen and paper. To what extent will that no longer be true, and how can we make sure math is still accessible to everyone. There are a lot of groups that are working on open source analogs. Of course, they're behind, but they're catching up.

35:41I'm on a team that's working on open source models that solve math problems in particular or help mathematicians formalize mathematics and so on. So people are very concerned about this. They're working on it. It's hard. We need more resources. We should always say we need more resources. We always say that. My question is, I think there are some missing pieces now as AI begins churning out proofs, I think we are realizing that this is not what mathematicians want, just like the number of proofs. I think the missing piece, I feel, is aesthetics. And we haven't really defined that. Like, why is a problem interesting or why is a proof interesting?

36:21Where is the beauty? Where is depth? Where is interconnectedness? I think because these haven't been defined, when you're faced with hundreds of proof generated by AI, you have no idea what to look at. So is somebody working on measuring these? I'll go the other direction and say that you might be saying, can we mechanize aesthetics so we can measure what is the most aesthetic proof? And I don't think that's the direction to go. We already have a good human sense of what's aesthetically pleasing. We know what's interesting. We know most of the proofs that are produced by AIs are not interesting.

36:56We know some of them are fantastically interesting and beautiful. So I don't think it's mechanizable. And that pleases me a little bit, that it's a human subject, that it's a human understanding that matters. I also thought it was not a very happy, it's not a very conclusive answer, so maybe I should pass it to someone else. I was just going to say the various ways that other sciences try to mechanize quality by, you know, citation, you know, H-index and these kinds of things, which in mathematics are almost, you know, have zero correlation, basically. I'm really happy that our field is not looking at metrics and there are fields of medalists with an H-index of 4 or something.

37:29These things are completely meaningless. It's a problem when we have to go to the dean and say, no, no, this person really is very good. Please look at their papers, which the dean will not understand. Oh, go ahead, please. No, just as mathematicians, I think we should be very conscious of the use of numbers to stand in for another concept, and we remember it's really this other concept we're interested in. And we have interesting debates in the mathematical community, which I enjoy, about is this interesting or not. It's not like there's a single metric, this theorem deserves a 3.5 on the scale.

37:58but instead you know we'll argue we'll have interesting discussions over coffee that oh this was really something and actually it wasn't that impressed it's a very human measurement system some would argue then that it's open it's right for abuse which it absolutely is but on the other hand it also is most people are doing it in good faith and there's a sense of beauty in mathematics it is hard to i don't think i would want i don't think it's possible to quantify but it's something that you meet someone at the icm from another country brought up in a completely different school system and you meet and you hear about a new theorem and you both nod your head and you think that was amazing.

38:33And different people will have different evaluations of the value and even the same person at different times will have different evaluations of the same result. I know that there are results that I thought, oh, I don't think that's that interesting and then I started learning more about it and I realized, no, that's an amazing breakthrough. So true. Okay, please. Yeah, thanks for the great conversation. I think unfortunately, historically, the mathematical community has not appreciated exposition and teaching in the same way that it has solving really hard problems. I mean, this goes back to like Bertrand Russell describing people that describe math as being less than the people that solve math problems.

39:08Do you think this will shift in the age of AI towards an appreciation of pedagogy and of exposition? I think yes, for sure. I think the three of us here would have pushed before AI in exactly that direction. This is not new, but I think so, and I hope so, and I think we should have pushed before and yeah I think there's an example of where you know the stress that AI puts on the system will lead to something healthy something that was healthy quite without the issue of AI that's an optimistic reading that I like hearing actually that it may push us in a direction that we should have gone in before I mean Hardy explicitly wrote you know here I am in my old age yeah looking back and sort of surveying things and surveying is the work of lesser men.

39:50That's, I think, a literal quote. Yeah, he seemed to have had a lot of self-loathing. Some of it deserved, I think, maybe, but not all of it. No, that essay did a lot to damage our values. Yes, that is true. We'll save that for another time. Thank you for this conversation. I had a question for the entire panel. So the panelists laid out a vision of what AI use in education ideally looks like, right? You asked students not to use AI to train their minds. But this vision is sort of incompatible with the current infrastructure for like grad school applications or postdoc applications or tenure track applications or what have you, where, you know, it seems like the only thing that matters is how much you publish, how hard the problems you're solving are, and so on.

40:31How do you see this infrastructure changing? And what is your advice to a current graduate student worried about the fact that not wanting to use AI, say in order to train their minds, would leave them behind, especially given how slow the timelines of infrastructure changes tend to be. I'll begin, but I think what the other two have to say is going to be very important. I do think part of the system is currently quite robust and I feel happy that we can deal with it. Parts are going to have to change. I will say, for example, for applications for graduate school, our department now interviews in a way we didn't because we want to really see what people know and see the human being.

41:09And we ask questions not to test, but see their actual understanding. So I feel like that part of the system is robust. And what we're looking for at every single stage is, can you do the best mathematics? And I do think the best mathematics, some people will do very well augmented by AI. Other people will not. I will say to graduate students, if they wonder, do I need to use AI? I would say no currently. But I would also say don't be afraid of it if it helps. If you need to look for references, use it in the way you feel comfortable with. But the evidence currently out, don't go to Twitter. Don't follow what other people are saying.

41:44Don't be afraid. What really is happening on the ground is that the people who are using it effectively are using it in a very specific way. And I think the people who are best at it, who will give you the best advice, are fellow graduate students who are similarly struggling with it at the same level, where they will say, use it to perhaps track down the thing to do. How do I think about this? but then you go away and learn who among your peers is learning the fastest. And maybe they're using an AI to guide them what to think, or maybe not. And so I don't think you need to use AI. I don't think you should be afraid of it either.

42:18I was going to say the same thing about interviews. I'm also considering adding them for postdocs for exactly this reason. I think we want to understand who the person is, and language models are going to make that difficult if we just go on unwritten things. I think the broader question of how systems have to change is important. And I don't know, but I think the community is seriously thinking about it. And just to add, even before AI, it wasn't a number of publications or your GRE score, it was the letters. We want to know who the professors are that you've interacted with and what is their opinion.

42:54So, which, you know, is an interpersonal communication, and that's what a lot of the measure was. GRE scores are nice to look at, but I go straight to the letters. I think mathematics has always, I think it's made space for people who are independent minded and wanted to go and do their own thing. And I think we will continue to do that. And maybe doing your own thing means saying I refuse to use AI. But I think as a culture, we respect that kind of independence of mind. Nice. Okay. So one thing that we've heard a lot about is how AI is changing the way that we're attacking these problems the way we're thinking really.

43:32And I get a little pessimistic and wonder, you know, is it going to change the way we think in a bad way? I don't think students necessarily start using AI as a tool thinking, okay, I'm going to hurt my knowledge and the way I go about these problems, but slowly over time it does. And I worry that the same thing might happen to mathematicians. So my question is, are we really just like frogs boiling in a pot of water? Are we going to know when we're supposed to jump out?

44:01Who knows? Who knows? I think some people will make bad decisions. And the question is, if too many do, I think that is a real danger. And it's hard to know how warm the water is. And people will think their homework is due the next day. And just this one time, they'll do it. So I think that's a fair question. And it's tough when you're the person. It's easy to see it from the outside. It's tougher when you're the person deciding. I feel it a little bit when I learn a new subject with AI that sometimes I get the link to the paper and then maybe I'll get the summary. But in the old days, I would have read the paper and I would have fought with it for a few days.

44:39So there's a trade-off. And as I said earlier, we need to fight with things for a long period of time and be stuck. And I don't have much time right now. And so I sometimes make this trade-off, but I'm making potentially the same mistake the undergraduate is the day before the homework is due. I still can't understand anything without printing it out. I'm very old-fashioned in that way. If I'm looking at something on a screen, it's just going by. I tried annotating PDFs. It's just not the same as printing it out, laying seven pages across my desk, circling things, and I don't know. I'm a Luddite.

45:12Who loves AI? Yes, please. As a grad student, I feel low-hanging fruit is very important for grad students for getting started, getting confidence in math. And my concern is that AI will swoop and take all the low-hanging fruit, and then maybe it'll be hard for a lot of people to get started. And I'm concerned about the average Joe mathematician who doesn't work on things that are safe from AI. I was wondering what your thoughts are on that. I wonder about that, though. We didn't get a chance to talk about applied math versus pure math, and maybe you're in pure math, so this isn't relevant. But I think even within pure math, isn't it possible to start a new area, sort of?

45:55Like, there's this question of interpolating within math and extrapolating to do something brand new. And I realize it's hard to start a new subject, or even a new tiny piece of a subject. But I'm just thinking that might increase your chances that it won't be low-hanging fruit if it doesn't exist yet. You know what I mean? Like there may be tractable graduate student beginning level questions that you could get by talking to your friend who's a psychologist. I am going to go applied on you for a minute. We want you in applied math. There's a lot to do. And science may be more resistant than math.

46:28Well, this happened in particle physics. It was a victim of its own success. I mean, this is a different kind of a success. It's a technological success. But where the standard model seemed done. And it was no longer attracting tons of students who had lots of work to do. it was all kind of solved. And then, you know, luckily we realized we only know 5 % of what's out in the universe. So we're back, but you know, dark energy and dark matter kind of saved the fields, the unknowns. So it, it may be that there's just, as you're saying, there's a way of sort of pivoting, but it's very curious for mathematics.

47:00I think it is a curious question. So I want to express worry as well. I want to show your concern that this is, I think there are certain kinds of problems that would be excellent for training graduate student minds, such as you take a solution, a theorem that worked in a certain situation and I as an advisor know it's going to work. It's got to work in a slightly different situation. And that's perfect because you would have to really understand it to make it work. But now that really feels something that is accessible to models and rightly so that now this should be handed over to models. So the good thing is advisors are concretely thinking about how next to think about the right sources of problems.

47:40I don't feel like graduate students are the ones that are going to invent new fields, except for maybe three or four. But I do worry, that's one of the few things that I think are real concerns. But at least I share the concern, and we're thinking hard about it. So don't give up, because it's important to get the graduate students through this stage and continue to train, because they're the ones who are going to drive the field forward four years later. But I think even your example of I know this theorem works in situation X I want to you know make a slightly slight modification X prime the act of what we're asking the student to do to transfer from X to X prime we're not saying you have to come up necessarily with any new ideas the goal is for you to learn actually what theorem X is learn the innards of how it works because when you shake it just a little bit you you learn some more about oh actually the reason this is happening is because of this and not that and you thought it would work but it actually doesn't work and here's where I'm stuck and so on.

48:36There's still so much, even with the AI assisting you in that path, there's still so much for you to learn. The goal isn't necessarily just to arrive at X prime. Now, if you're saying, okay, but AI already knows X prime, who's pressing the button on the AI to get X prime, right? At the end of the day, someone has to ask the AI the question. And of course, it can ask its own questions, but even then someone's prompting it to say, go ask questions. So it's like, it's not turtles all the way down. Someone is paying the electric bill. What's being asked and that person is doing it for a reason at the end of the day What is being done has to be valued by human beings?

49:13Otherwise, it's not going to get done isn't it possible that we're in a peculiarly Anomalous time right now where the big AI companies are playing around with our field for their own purposes. Yes, but they may lose interest very quickly That's exactly what I expect right so so in fact what we're experiencing now may not last much longer and maybe we'll have it back to ourselves again for a while. And we'll have this great tool that they've created. And we'll have this great tool while they're solving something else. Well, okay, please, go ahead. So like many people in this room, I of course agree that math does amazing things to sharpen minds and to enrich lives.

49:47But at the same time, there's a lot of other disciplines that can lay similar claims. And much has been said even today about, you know, humanities. And that maybe math is going to start looking a little more like those. So my question is, do you think that as a profession, we need to be thinking about some sort of a value-added statement that would actually work to, in real life, nurture an ecosystem of people who would be engaged in this brave new math that's being shaped up? Or do you think that such a value-added statement will just kind of happen by itself? Okay, that was like softly lobbed across home plate.

50:23So this is something which is, so this is part of the mistakes that were made, the Hardy style. I only want to talk to those students who are future PhD mathematicians. That's decades ago. At many schools, math is one of the biggest majors. At our school, math is one of the biggest majors. Why? Not because people are going to become PhD mathematicians. It's because these habits of mind, this way of thinking, are empirically, more and more today, even more than ever, are incredibly useful. Our students, if you want a job where you know what you're going to be doing in 50 years, you should not be a math major.

50:57If you want to be a job in a career that doesn't even exist yet, where you're going to make the future, our students rule the world. In fact, I was just given from the Italian Mathematical Society, they did a study on the value added of mathematics to the economy. This is incredibly safe. And I think by hiding ourselves and pretending we're not useful, that's been really dangerous. I think we need to have an informed public. They want to know math. They want to understand math. We can easily convince them how useful it is if we would just make the case for it. So thank you for that very easy question.

51:28Well, you have been a fantastic audience. And so we thank you. We also thank our panel here. So we've been speaking today with Alex Kontorovich, Ravi Vakil, Akshay Venkatesh. And as always, Jana Levin is here co-hosting. We're signing off from the Joy of Y. Thanks again, Jana. Thank you, Steve. And thanks to our guests. Until next time. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. You turned on the recording? Run it again. If you're enjoying The Joy of Why and you're not already subscribed, hit the subscribe or follow button where you're listening.

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53:48Let's go.