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

Mathematicians are witnessing a profound shift in their field. AI systems are now producing proofs, spotting connections between distant fields, and, in a few cases, solving problems that had stumped mathematicians for decades. The pace of progress over the past several months has raised challenging questions: What can AI systems actually do? And what happens to the more human, creative aspects of the field once machines can match – or exceed – people at problem-solving? In this special live episode of The Joy of Why, recorded at the International Congress of Mathematicians in Philadelphia on July 26, 2026, hosts Janna Levin and Steven Strogatz are joined by three mathematicians responding to the rise of AI in math: Akshay Venkatesh at the Institute for Advanced Study; Ravi Vakil at Stanford University and president of the American Mathematical Society; and Alex Kontorovich at Rutgers University. Together they discuss what recent AI-generated proofs really demonstrate, what will be gained and lost as mathematics becomes more machine-assisted at the frontier, and what this means for the next generation of mathematicians. The conversation turns to a deeper question: What do mathematicians value about doing mathematics in the first place? The answer involves grappling with what proof and understanding really mean, and with the surprising importance of storytelling.

• 0:05Speaker 00

  Thank you so much for being here. I'm Janna Levin.

• 0:08Speaker 01

  And I'm Steve Strogatz.

• 0:10Speaker 00

  And this is the Joy of Why. Welcome to our first live.

• 0:15Speaker 01

  Yes, the Joy of Why, a podcast where we consider some of the biggest unanswered questions in math and science today. And we're very excited to be coming to you from the International Congress of Mathematicians 2026. It really does feel like a historic moment. The stakes are pretty high for the future of math. And we've had a good run for 3,000, 4,000 years doing this on our own. But there's a new kid in town, artificial intelligence. And it's starting to play with us and our field. And so we're excited to hear what our panelists have to say about the interaction between math and artificial intelligence now and in the short term and maybe in the longer term.

• 0:59Speaker 01

  So let me introduce our distinguished panel here. First up, I see Akshay Venkatesh from the Institute for Advanced Study. And next to him, we've got Ravi Vakil from Stanford University. And rounding out the panel, there's Alex Kontorovich from Rutgers. All right, Jana, do you want to get us launched in our discussion?

• 1:20Speaker 00

  I thought maybe we'd start with some of the recent advances that gave some mathematicians pause while others were very blasé and unimpressed. I'm going back to the 2024 International Mathematical Olympiads, when Alpha proof achieved a silver medal. And then later in 2025, a more advanced model with Deep Think achieved a gold medal. I first want to ask, before we get into exactly how it was done, how many of you participated in the International Mathematical Olympiads?

• 1:47Speaker 01

  Back in your day.

• 1:48Speaker 00

  You did? You did? And will you tell people about it? Because some of our audience might not know. It's six problems, nine hours, two days, and high school-aged competitors. Tell us what it was like.

• 2:00Speaker 01

  I've heard that Ravi was a terror at this. Jordan Ellenberg was talking about how you used to kick his you-know-what at that competition.

• 2:07Speaker 00

  People at the conference have been talking about how you terrorize them.

• 2:09Speaker 02

  So I first I first met Akshay at the event, I should say, although he may not remember.

• 2:14Speaker 03

  I don't. Sorry.

• 2:16Speaker 02

  Because he was quite young. But it's one of the many roads into mathematics. There's no royal road to mathematics, but this is one of the many ways by which we attract people into mathematics. And it's also a great way of advertising mathematics. And it helps to develop a certain way of mathematical thinking. The important thing about the competition isn't the six people or the problems. It's how it encourages millions and millions of kids to hone their minds. So it was wonderful when the models did well. It was really a proof of concept. I think it was extremely interesting. And I think it can both be an amazing milestone reached, while also, I think perhaps people might have misinterpreted it. So the first time a model answers a question, it's exciting. A milestone is reached. The first time a student answers a question, it's exciting for a different reason,

• 3:01Speaker 02

  because they've learned how to think. And the seventh time a model answers a question, it's far less interesting. And the seventh time a student answers the same question is far less interesting for a similar reason. So it was exciting, and I see it as only a good thing. But it did not represent superhuman intelligence or anything crazy like that.

• 3:18Speaker 03

  No, sure.

• 3:19Speaker 00

  Yeah, and it wasn't the highest score, even in the year that it reached.

• 3:22Speaker 02

  Sure, but the next year was, and the next year after it will be. And I'm happy about that, and I'm absolutely not alarmed.

• 3:28Speaker 00

  And Akshay, what's your experience?

• 3:30Speaker 03

  I think Ravi said it well. I think, you know, one unfortunate thing about this is it has contributed to the image, even among mathematicians, of mathematics as a competition, which is not really how I see it. And I think it's particularly unhelpful at the moment. There's some competitive drive. You know, when you're young, you get validation if people think you're good at something. I have mixed feelings about how healthy this thing is.

• 3:51Speaker 02

  As Akshay said, it could be a danger. Everything is a double-edged sword. People are driven out of mathematics because they think they're not good, because they didn't do well in a competition, whereas that just makes no sense.

• 4:00Speaker 01

  I'm very taken with your view of this. There's that old saying about it's the journey, not the destination. And you seem to be espousing that kind of philosophy. But it feels to me like that is also, in embryonic form, something we might get to by the end of our discussion today. Because all of us may have to start wondering, is mathematics about the journey and not the destination?

• 4:23Speaker 00

  I was going to say, for Alex, beyond that, that was 2025. We've recently had some major advances, recently published. One of the Erdosh problems was solved. And I'm just curious, one of you were impressed, surprised, and sort of your reaction to that? And if you could set the stage of what the problem was that was solved?

• 4:42Speaker 04

  Sure. Just to make one comment on the IMO. You know, two years ago, we had Alpha proof that, you know, it was one point off of a gold medal. And by the way, no one's handing medals to these companies. They're declaring themselves as having.

• 4:54Speaker 00

  Right, self-declared gold medalists.

• 4:56Speaker 04

  Yeah, that's right. And my prediction was that this summer, no AI models would get a gold medal, because none of them would bother competing anymore. So it's not interesting for them. The fact that I can push a button on GPT-5.6 and get perfect scores on the IMO. So, okay, next, what's the next interesting thing? That's not interesting anymore, exactly as you were saying. But let me go back to your question about these Erdosh problems. Okay, so we have problems that we're working on. And as Akshay said, solving problems is just one small component of what mathematicians do. We find definitions, we find new structures we're interested in, we create new questions and new problems all the time. One person who just loved to ask questions and didn't mind writing them down, whether he was right or wrong, was Erdosh, and he wrote down thousands of them.

• 5:41Speaker 04

  And thankfully, Tom Bloom, for whatever reason, several years ago, decided to collect these into a database. And what happens sometimes in mathematics, in fact, it happens with some frequency, is that problems that we thought were difficult end up being not nearly as difficult as we thought and doable by existing techniques if you only bothered to look at that problem and had the background necessary to apply the techniques that end up cracking it and making it easy. And so what AI is amazing at is you give it 1,000 problems, and maybe it has a 1% hit rate. And that's 10 good papers a year, which is a fantastic career in mathematics. So right now, we're seeing every day more and more of these Erdosh-style problems being announced.

• 6:26Speaker 04

  How many of them are actual solutions as opposed to maybe partial progress or complete abject nonsense, because it's just LLMs saying things, is one thing. I think what you're referring to is the Erdosh unit distance conjecture, which is something that quite a lot of people knew, quite a lot of people worked on. There was very significant progress trying to, you know, get this constant to the conjectured value. And it turns out that the conjectured value is just wrong.

• 6:49Speaker 00

  It's wrong.

• 6:49Speaker 04

  And autonomously, GPT found a counterexample, you know, constructed a solution where this exponent is not satisfied. What, to me, is most interesting about that, I mean, first of all, it's brilliant. It's fantastic. This would be a really great contribution if it was a human being that came up with this counterexample. The most fun thing about this is, a week later, a completely unrelated problem, called the sum product problem, was also solved in the negative by human beings who were applying the ideas, the techniques that were exposed to us from this AI solution. So this, to me, is sort of the, hopefully, golden age that we're going to enter. We're going to learn all kinds of amazing mathematics, thanks to these techniques. But what is it, if a theorem falls in the woods and no one cares,

• 7:33Speaker 04

  you know, does it have any value, kind of thing? So I don't think theorems have intrinsic value in the universe. Mathematicians value them. And right now, AI companies value them because they're great at marketing. They're a great way for them to say, hey, my tool is better than the previous tool and all the other tools out there. And it is very effective in that sense. And I think pretty soon, AI companies will need other, you know, things that actually bring value to the world, which theorems do not, except for to mathematicians. So but in their wake, they'll leave us a fantastic tool to learn lots of amazing facts.

• 8:04Speaker 00

  I do worry that the ability to generate so many proofs that could be wrong could be a sinkhole for mathematicians, that there is now an industry of just checking those proofs. Do any of you find yourself lured by the attraction of checking some of these proofs? I mean, some of our very prestigious colleagues have done that, right? Have offered their services. Would you ever do it, Akshay? To refer like Terry Tao did?

• 8:31Speaker 02

  100 AI slot proofs. Would you sign up for that?

• 8:34Speaker 03

  How much would someone have to pay?

• 8:35Speaker 01

  There's a lot of glamour. Come on. How can you resist?

• 8:39Speaker 03

  But this is, I think, adjacent to a very real problem in our community, which is about the functioning of journals.

• 8:44Speaker 01

  Yes.

• 8:44Speaker 03

  And how they will manage in this. So I think as a community, there are a lot of systems that we have to be looking at, which we're going to be placed under immense strain.

• 8:53Speaker 02

  Right. Part of the issue is that people are not going to be lining up to do this. The reason they're under strain is we're being flooded, where the flood is only beginning, of a large quantity of material, some of which really amazing things are happening. And also, I have in my inbox during these two hours, I'm not looking forward to seeing what kind of AI slot people are going to send me, what amazing things they think they proved. And to be honest, I'm not going to check them. It's not fun.

• 9:17Speaker 01

  It makes me think a little of this trifecta that Terry Tao has been mentioning about concepts like proof generation, proof verification, which is what we're talking about now, and also proof digestion, where we start to make sense of these proofs in human terms and can assess whether, how interesting they are. But I'm wondering if there might be another category of proof marketing.

• 9:43Speaker 03

  Well, I think I want to also even question the assumption that proof is the central activity.

• 9:48Speaker 01

  Okay, go on.

• 9:49Speaker 03

  Well, if I think about, you know, if I'm doing a piece of math, I probably think of it more akin to storytelling. And I'm trying to tell a story.

• 9:56Speaker 01

  Uh huh.

• 9:56Speaker 03

  Perhaps the proofs are part of the grammar of that. That's, you know, something that we should be thinking about. Like, should we define this discipline by proofs? And of course, we have done so. But I'm not sure it really matches with what we value and what we're doing.

• 10:11Speaker 02

  I'd also like to strongly endorse what Akshay says. I feel like I also think this is what we have always valued. This is not a change. This is something we've always valued and we've had various proxies. And we're forced to examine these proxies because of the change of technology. But human understanding and the storytelling, my graduate students, they're brilliant already. What I teach them, it's the storytelling. Learning how to do mathematics is when you write a paper, you have to understand the human understanding and the conveying of the understanding. I claim this is what we've always wanted. There have been brilliant mathematicians who've had ideas that went nowhere because they could not convince people of them. And there are other people who've really changed fields who've just been excellent. Erdich is a good example of someone who's changed mathematics. Or they're even on a very different level.

• 10:54Speaker 02

  Martin Gardner is someone who probably has advanced mathematics much more than most people because of what he's managed to do by storytelling.

• 11:02Speaker 01

  I think storytelling is not a term that would occur to most people. So I think you're on the hook now, both of you, to explain to us, what are you talking about?

• 11:11Speaker 03

  I think that that was the best word I could use for the process of, you know, I'm working on something. There's some landscape which I'm exploring. And there are all these things in that landscape. And I'm sort of thinking, how will I tell a story that will be interesting and engaging for my colleagues?

• 11:26Speaker 00

  Is that how you're drawn to problems? I mean, math is one of the last places people with a straight face talk about beauty and elegance. And in this case, storytelling. Can you give an example of that for somebody who's not familiar with the idea that this is beautiful or narratively powerful?

• 11:41Speaker 01

  Ravi, you look like you're ready to go.

• 11:43Speaker 02

  I would say I cannot think of a major advance in mathematics that does not fit this paradigm. Let me just take Fermat's Last Theorem as an example.

• 11:51Speaker 01

  Perfect.

• 11:51Speaker 02

  What makes Fermat interesting and Goldbach far less interesting? It's because Fermat is part of a long and rich story. People started to dig into it and in the 1800s, it led to the opening of lots of rich questions. And to be clear, it was beautiful, but it also was powerful. The thing about mathematics is when you have an amazing story that is beautiful empirically. I don't know why the universe works this way. But when you understand something and the story becomes simpler and simpler, you get something which is more and more powerful. And there's no question what the developments from the 19th century have led to in technology and science and in our standards of living. But it's because people were curiosity driven. And so Fermat led to the development of a good chunk, well, not alone, it's part of a grander story.

• 12:35Speaker 02

  And then the final proof, it was way better, the way it turned out, because it required the building of a far more interesting story with dramatic bridges that had no right to be there between one way of thinking and completely different way of thinking. And they were completely out of reach until they weren't, until you had dramatic leaps of thought. There are cliffhangers, there were examples where it looked like we were all done and then, uh oh, we weren't. And then the story is not over. Once you finish the book, you put the book down and the best theorems are the ones where you're waiting for the next volume. And everyone is talking about it around the world to try to think of how the story continues. It's hard to think of an example where things do not fit this.

• 13:12Speaker 01

  I would like to just try to sharpen up this idea of story. You're talking about the interconnectivity across time and across disciplines of math. That math is this amazing, often sort of subterranean thing that occasionally pops up above the surface and we can see peaks. But we didn't realize that there are these subterranean branches connecting parts that are invisible until... Okay, my metaphor is getting mixed up here. But it's something about that, that when you talk about a story, you mean something about the coherence of the subject, I think. Uh, but I don't know, you want to try to unpack that better than I just did, Alex?

• 13:51Speaker 04

  Maybe one way of saying it is that the question of Fermat's Last Theorem, you know, this x to the n plus y to the n equals z to the n, has no consequences whatsoever in the universe. No one really cares if it's true or not. You know, if there was some counter example, would it destroy something about the universe? Well, it turns out it would destroy something about the universe. It would destroy Langlands program. But, uh, but that, that was a theorem that had to be proved. But I, I think that question led to a series of other absolutely amazing breakthrough discoveries. From the very beginning of that question, it led to the understanding of non-uniqueness of factorization in number fields. And the fact that we have to understand number fields as their own entities and... So that one question, if all you think about is, I wonder if that's true or not, for no reason whatsoever.

• 14:36Speaker 04

  And you just follow the thread of that story from one breakthrough to another. You get all the way down to, you know, this massive Tanya Masumura and so on and Taylor Wiles breakthrough. So that's a beautiful arching, overarching story. And we have many of these in mathematics. Just to give one more, you know, the question of whether the parallel postulate, this fifth postulate in Euclid's Elements, can or can't be proved from the other four. Seems like the most esoteric, useless thing. Who the hell cares if the fifth postulate can or can't be proved from the other four? It's not like, we're not even arguing about whether the fifth postulate is true or not. We're saying, you know, should it be a theorem or should it be an axiom in the theory? And you followed the thread of that question 2,000 years, all the way to Einstein's General Theory of Relativity.

• 15:20Speaker 00

  And it very much matters to the universe.

• 15:22Speaker 04

  Exactly.

• 15:22Speaker 01

  In that case too. So Akshay, I feel like you really tapped into a rich vein with putting our attention on this concept of mathematical stories. I'm also wondering, since you seem to be saying that proofs are not the zenith of what we're after. Insofar as they can contribute to an unfolding story or a richer story, yes. But what about this thing that we say in math when we say something is morally true? Like before we have a proof, maybe even before we have a precise statement, we sort of know what's going on. Can you talk about that, what that feels like and what do we mean by that?

• 15:56Speaker 03

  Absolutely. You know, like just to give an example of my experience and probably of many other people in the room, right? When you write a paper, you know what's right long before you fill in all the details, right? You grasp an intuition, gaining that intuition is really the most enjoyable part. And it's sometimes mysterious how that intuition operates and how it fails sometimes. But yeah, in a way, it's, it's much more fun to know something without knowing how you know it.

• 16:23Speaker 00

  Well, isn't there a sense where choosing a problem invokes that intuition?

• 16:27Speaker 03

  Absolutely, right? You don't know what's there, but you have a sense. You have a sense that there's some richness there.

• 16:33Speaker 00

  I guess, Akshay, I would also ask similarly, are, do you care either way if a proof, if it's coming from a machine or if it's coming from an elder or if it's coming from a young upstart? Does it matter to you where it came from?

• 16:47Speaker 03

  You know, I certainly don't see why a machine couldn't produce something I found interesting. But I also think as we use machines, it will change those notions very rapidly. So it probably doesn't make a lot of sense to look at what we value now and ask, can machines do them? It may be more interesting to think how it'll change what we value. I mean, the fact that so many of us are using AI in different ways as part of our regular research. I, I think it shows you the practice of mathematics is in flux.

• 17:15Speaker 01

  So at this point, I'd like to ask a question that I feel is sensitive, but I think it could also be very interesting, which is, here at this meeting and in conversations I've been having with people before I came to the meeting, there seem to be certain taboo subjects. Things that we probably should be talking about that we're not as a community. And also things that I feel like a lot of us are worrying about, but we don't say out loud.

• 17:42Speaker 03

  Okay, Ravi has to answer this.

• 17:43Speaker 01

  Would anyone like to, as president of the American Math Society?

• 17:48Speaker 01

  No, you don't have to say anything if you're the president.

• 17:50Speaker 02

  No, no, no, no, I won't be circumcised at all.

• 17:52Speaker 01

  Go ahead, do it.

• 17:52Speaker 02

  But there are no taboos. So I think things people worry about in public or on social media are not the things that we should be worried about. I'm less worried about those. I do think there are things we should be very worried about. And I'm worried about younger mathematicians, I think. And the good thing is the community, at least uniformly, seems to realize the danger to the profession comes from the threat to younger mathematicians. But the threat isn't the obvious one. First, I need to say in practice, the evidence so far is that technology has always accelerated science in history. I can't think of a single time in history when it's been disastrous for science. And currently, the evidence is that it's been positive. We're in a period of great transition and the error bars are huge and I make no guarantees for the future. And so there's uncertainty and anxiety and also opportunity.

• 18:37Speaker 02

  But that's not what I'm worried about. What I'm worried about is, is this is being used as a pretext, as a Trojan horse for anti-intellectualism, which is that people are coming with pitchforks and torches to burn down the library. And we worry that maybe in a few years, what if this leads to disastrous cuts? You could imagine in five years, graduate programs might be cut by huge amounts. Funding for science might be cut, but that's actually happening right now.

• 19:01Speaker 01

  Yeah.

• 19:02Speaker 02

  And the pretext for it don't make sense. So when people say it's because of AI that we have to cut funding for graduate students, reduce jobs, cut REUs, that we don't no longer need to teach kids how to think mathematically because they can just Google it. They no longer need to understand science because they could just ask a model. So the danger is that it's a Trojan horse for anti-intellectualism and it's already happening. And we're fighting the cuts right now. So that's, that's my first real danger. And so, yeah, I mean, it makes no sense. So the second thing I'm worried about is students' learning. We talked about earlier, just fighting with a problem over weeks and maybe never solving it. And there's a reason why you don't want all the answers to be the back of the book. And now you have the temptation. The danger is that some of the tools, the problem sets to think over a week.

• 19:47Speaker 02

  Now we have to really trust students to work on their own. And the purpose of it is not to get the answer. It was never to get the answer on a test. It was to train your mind. The best students, I don't mean the smartest, but the wisest students are going to be accelerated. They will learn faster because it will give them clues as to how they should struggle with problems for longer. But I worry that a large number of people will think they understand something that will lose a large number of very bright students who will think less clearly because of the existence of this.

• 20:17Speaker 00

  There's an anti-correlation between homework scores and test scores.

• 20:20Speaker 02

  Exactly.

• 20:21Speaker 00

  And that's become very clear. Students are getting a hundred percent on all of their homework because they're using AI. So they go into a test. That's timed. They understand none of it.

• 20:30Speaker 02

  Wow.

• 20:30Speaker 00

  That's the way it's working.

• 20:31Speaker 04

  Scores are coming in bimodally. So there are the people who use AI to help themselves understand when there's something that they're confused about and they don't really know where to go to learn it.

• 20:40Speaker 02

  Right.

• 20:41Speaker 04

  And then there are the ones who use AI to do their homework. And you can see then on the tests, you know, it's no longer a single Gaussian. It's sort of a double peak. One peak is concentrated around the students who are really using that AI to accelerate their understanding.

• 20:53Speaker 02

  Uh huh.

• 20:54Speaker 04

  And train their brains in exactly the same way that we trained our brains, except, you know, when we would get stuck and I just even don't even know what this word means and have to wait five days to ask the professor in office hours. They can ask the AI right now. And so that's the way in which they're accelerating their education. As opposed to the students who just put the homework into the AI. They have the answers. Great. And then they're failing the test. But, you know, it's interesting. We mathematicians should have the longest time horizon over which to understand how education interacts with technology. Because pocket calculators have been around for seventy years or something. And we all know educators who tell their own students, you don't have to know your times tables because a calculator will do it.

• 21:33Speaker 02

  I mean, it seems a truism that, I mean, we've gone through this with calculators and there've been a lot of silly debates. But one thing which seems undeniable is the best person with a calculator is the best person without a calculator.

• 21:44Speaker 04

  That's a good aphorism. Yeah, I think that's right.

• 21:46Speaker 02

  So we're not afraid of calculators. Gauss made these huge, long tables and we no longer have to do these things. And that's good. But we still develop number sense. We change what we do. We don't do exactly the same things. And similarly, as Akshay had said, we're going to change what we value. Our long-term goals are the same, but we'll change these things.

• 22:06Speaker 04

  And it may not be good because if Gauss hadn't made those tables, he may not have noticed the relationship between the growth rate of the primes in a range versus the logarithm function. So there are all kinds of insights that I think we stand, we potentially.

• 22:20Speaker 02

  It's a trade-off.

• 22:21Speaker 04

  Yeah, it's definitely a trade-off. Yeah. And I think it's the trade-off that I will take.

• 22:24Speaker 02

  Yes.

• 22:24Speaker 04

  Uh, to have the technology, but, you know, there's no solutions. Everything is a trade-off.

• 22:29Speaker 01

  Well, let's focus a little bit on this question of human understanding. It's a theme in some of your talks. Actually, I've seen that you're concerned about this. What will be lost and what could maybe be gained as math interacts more and more with AI at the frontier and even just in everyday life?

• 22:48Speaker 03

  Right. I, I think something I would hope is it would lead to us, in fact, putting much more value on human understanding.

• 22:55Speaker 01

  Mm hmm.

• 22:56Speaker 03

  Human understanding is human, right? It's subjective. And I think math has this dual heritage of being a science and a humanity.

• 23:04Speaker 01

  Yes.

• 23:05Speaker 03

  And we have certainly in my lifetime, probably certainly since World War II, I think emphasized to scientific, right? We think of ourselves as a science because it's easier to get funding. That's where the resources are. No, I mean, right. But what I would like to see is really kind of embrace the humanistic aspect of math. That's a possible response to AI, which I think would be very nice.

• 23:29Speaker 01

  Mm hmm.

• 23:30Speaker 00

  Steve and I sometimes talk about this at the advent of supercomputers. There was a similar kind of hysteria that the supercomputer would replace the physicist or the scientist. And 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?

• 24:12Speaker 04

  Why are you guys looking at me?

• 24:15Speaker 04

  There's, there's a number of ways to try to answer that question. You know, will the AI be conscious? I have, I have no idea. Uh, who knows? I like to think of, um, 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.

• 24:39Speaker 00

  Flight.

• 24:40Speaker 04

  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, you know, 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. And 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.

• 25:14Speaker 01

  No.

• 25:15Speaker 04

  And it's all, you know, Turing never said, uh, 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, 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. Is 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.

• 25:44Speaker 01

  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, 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. They'll be able to do exposition better than Martin Gardner and the whole thing. Will it still be fun for us to do math?

• 26:24Speaker 02

  Well, well, I think you said two flavors of things, one of which was very reasonable, the other which was very different.

• 26:30Speaker 01

  Go ahead. What were those?

• 26:32Speaker 02

  So, yeah, 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.

• 26:42Speaker 04

  Because it's so hard to know what will happen next, you mean?

• 26:44Speaker 02

  Or I just feel like the debates post singularity tend not to have much value. You didn't use the phrase superintelligence. Anytime 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.

• 27:08Speaker 01

  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, I hear that we can still have taste. We're better at knowing what's interesting.

• 27:23Speaker 02

  Ah, okay. So here I'm.

• 27:25Speaker 01

  And I'm claiming that that's only for the short term.

• 27:27Speaker 02

  So, 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? I'll put twenty dollars 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 ten years ago.

• 28:09Speaker 02

  So absolutely large quantities of acceptable material, I expect to be very useful. And I'm not belittling it. This will change everything, uh, 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.

• 28:33Speaker 03

  I 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, I think, you know, yes, we will still do math because it has some basic value to us as human beings.

• 29:08Speaker 00

  I mean, I think Akshay's point is crucial. And also to your point about students and the, 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?

• 29:45Speaker 04

  So I've been trying to explain to my students sort of three scenarios, three vignettes as an analogy to exactly this question. Vignette number one, uh, 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 two hundred pounds on the bench press. You get in your forklift, you drive over and you do ten reps with the forklift. One of these is the wrong use of technology.

• 30:29Speaker 04

  So if you just explain to students, why are you here?

• 30:32Speaker 00

  Yeah.

• 30:33Speaker 04

  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.

• 30:41Speaker 00

  Right.

• 30:41Speaker 04

  Okay, so 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 a real analysis problem? Is it that I want the solution to the problem? Or is it that 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? If 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.

• 31:10Speaker 00

  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, going through that process. But I think there's going to take some convincing for this generation that's growing up on AI.

• 31:31Speaker 01

  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. But 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.

• 32:16Speaker 01

  Not 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.

• 32:26Speaker 02

  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 you, 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.

• 32:47Speaker 00

  This might be very unpopular, and I apologize to the mathematicians ahead of time.

• 32:51Speaker 04

  Okay, don't ask.

• 32:52Speaker 00

  There's a belief that if all of human knowledge is wiped out tomorrow, not humanity, but human knowledge, that eventually we'll all be rediscovered. One plus one is two. If human knowledge was wiped out tomorrow, the AI stops. Kids got to start over.

• 33:08Speaker 04

  But there's a version of this question which you can ask right now.

• 33:10Speaker 00

  Yeah.

• 33:11Speaker 04

  Which is, okay, so there are these, we didn't get into formalization, 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. If you seed an interactive theorem prover with no library of what mathematics is.

• 33:37Speaker 00

  Exactly.

• 33:38Speaker 04

  And you cut an AI loose, will it discover a proof of Fermat's Last Theorem?

• 33:42Speaker 00

  Right, without the reinforcement and...

• 33:43Speaker 04

  Yeah, without giving it any human knowledge, without, 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.

• 33:55Speaker 00

  That's a real question.

• 33:56Speaker 04

  And that's one that we don't have an answer to at the moment.

• 33:57Speaker 00

  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? But it also sort of harkens, and I think this was the part I was concerned mathematicians would like, to the discovered or invented question. Are these discoverable truths that an entity that thinks entirely differently from human beings could discover?

• 34:17Speaker 02

  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.

• 34:31Speaker 01

  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. They may get even wilder here. We're ready when you are. Please.

• 34:45Speaker 05

  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?

• 35:05Speaker 04

  There are a lot of groups that are working on open source analogues. Of course, they're behind, but they're catching up. I'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.

• 35:26Speaker 03

  We always say that.

• 35:27Speaker 04

  Yeah.

• 35:29Speaker 06

  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? Where is the beauty? Where is depth? Where is interconnectedness? I think because these haven't been defined, when you're faced with hundreds of proofs generated by AI, you have no idea what to look at. So is somebody working on, like, measuring these?

• 36:06Speaker 02

  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, I don't think that's the direction to go. We already have a good human sense as what's aesthetically pleasing. We know what's interesting. We know most of the proofs that are produced by AIs are not interesting. We 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 realize that's not a very happy, it's not a very conclusive answer, so maybe I should pass it to someone else.

• 36:41Speaker 04

  I was just going to say that 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 medalists with an H-index of four or something. These, these 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.

• 37:09Speaker 01

  Oh, go ahead, please.

• 37:10Speaker 03

  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.

• 37:19Speaker 02

  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. But 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 impressive. It's a very human measurement system. Some would argue then that it's open, it's ripe 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 this 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.

• 38:03Speaker 04

  And different people will have different evaluations of the value, and even the same person at different times.

• 38:08Speaker 00

  Yeah.

• 38:09Speaker 04

  Will have different evaluations of the same result. I know that there are results that, you know, I, I thought, oh, that, 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.

• 38:19Speaker 01

  So true. Okay, please.

• 38:21Speaker 07

  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. Do you think this will shift in the age of AI towards an appreciation of pedagogy and of exposition?

• 38:43Speaker 04

  I think yes, for sure.

• 38:45Speaker 02

  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...

• 38:54Speaker 04

  Yeah, I think this is 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.

• 39:04Speaker 01

  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.

• 39:11Speaker 04

  I mean, Hardy explicitly wrote, you know, here I am in my old age.

• 39:14Speaker 01

  Yeah.

• 39:14Speaker 04

  Looking back and sort of surveying things, and surveying is the work of lesser men that, I mean, that's, I think, a literal quote.

• 39:22Speaker 01

  Yeah, he seemed to have had a lot of self-loathing. Some of it deserved, I think, maybe, but not all of it.

• 39:29Speaker 01

  No, that essay did a lot to damage our values.

• 39:32Speaker 02

  Yes, that is true.

• 39:33Speaker 01

  We'll save that for another time.

• 39:35Speaker 07

  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 ask 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, so on. How 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.

• 40:19Speaker 02

  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. And 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? 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?

• 41:01Speaker 02

  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. Don't be afraid. What really is happening on the ground is that the people are using it effectively. You're 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 on what to think or maybe not.

• 41:43Speaker 02

  And so I don't think you need to use AI. I don't think you should be afraid of it either.

• 41:48Speaker 03

  I was going to say the same thing about interviews. You know, I'm also considering adding them for postdocs for exactly this reason. I think, you know, we want to understand who the person is. And language models are going to make that different, difficult if we just go on written things. And 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.

• 42:12Speaker 04

  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. So which, you know, is an interpersonal communication. And that's what a lot of the measure was. You know, GRE scores are nice to look at, but I go straight to the letters.

• 42:35Speaker 03

  I think, you know, 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.

• 42:52Speaker 01

  Nice. Okay.

• 42:54Speaker 07

  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. And 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?

• 43:31Speaker 02

  Who 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, 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 I get, 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. So 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.

• 44:15Speaker 02

  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.

• 44:23Speaker 04

  I still can't understand anything without printing it out. So I'm very old fashioned in that way, you know? 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, you know, I don't know, I'm a little, who loves AI?

• 44:46Speaker 02

  Yes, please.

• 44:47Speaker 07

  As a grad student, I feel low hanging fruit is very important for grad students for like getting started, getting confidence in math. And my concern is that AI will swoop and take all the low hanging fruits. And then maybe it'll be hard for a lot of people to get started. And I'm concerned about like the average Joe mathematician who doesn't work on things that are like safe from AI. But yeah, I was wondering what your thoughts are on that.

• 45:13Speaker 01

  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? Like 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 than math.

• 45:58Speaker 00

  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 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. I think it is a curious question.

• 46:32Speaker 02

  So I want to express worry as well. I want to share your concern that this, 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 eyes and 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 way, the right sources of problems. I don't feel like graduate students are the ones that are going to invent new fields unless, except for maybe three or four.

• 47:16Speaker 02

  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.

• 47:36Speaker 04

  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. There'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

• 48:20Speaker 04

  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 has, is paying the electric bill of what's being asked. And that person's doing it for a reason. At the end of the day, what is being done has to be valued by human beings. Otherwise it's not going to get done.

• 48:44Speaker 01

  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?

• 48:53Speaker 04

  Yes.

• 48:53Speaker 01

  But they may lose interest very quickly.

• 48:55Speaker 04

  That's exactly what I expect.

• 48:56Speaker 01

  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.

• 49:03Speaker 04

  And we'll have this great tool that they've created.

• 49:05Speaker 01

  And we'll have this great tool while they're solving something else. Well, okay, please, go ahead.

• 49:10Speaker 07

  So like many people in this room, I of course agree that math

• 49:13Speaker 02

  Does amazing things to sharpen minds and to enrich lives. But at the same time, there's a lot of other disciplines that can lay similar claims. And much has been said even today about 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. So this is something which is, so this is part of the mistakes that were made, the

• 49:57Speaker 02

  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. If 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.

• 50:41Speaker 02

  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.

• 50:57Speaker 01

  Well, 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, Janna Levin is here co-hosting. We're signing off from The Joy of Why. Thanks again, Jana.

• 51:15Speaker 00

  Thank you, Steve, and thanks to our guests. Until next time.

• 51:19Speaker 01

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• 51:51Speaker 00

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• 52:34Speaker 00

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• 52:49Speaker 01

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• 52:55Speaker 00

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