The week that changed maths for ever

0:00This is The Guardian.

0:12Did you miss what could be the biggest science story of 2026? Last week, OpenAI announced that it had solved a major problem in mathematics, one that comes with a million-dollar reward. The news was quickly buried under fears that AI will end humanity, but it's worth revisiting. The Navier-Stokes problem stood for 90 years, but it took 10 ,000 AI agents just 88 hours. That's less than four days to reach their solution. It left mathematicians reeling. And so today, the week that changed maths forever. From The Guardian, I'm Ian Sample and this is Science Weekly.

1:05Professor Culver-Roney Dougal, you're Head of Pure Mathematics at the University of St Andrews. First of all, what are the Millennium Prize problems and where do they fit into the work mathematicians do? So the Millennium Prize problems are a collection of problems released by the Clay Math Institute to celebrate the year 2000. And they were picked as a set of seven problems. They span mathematics. They're all very hard. They've all been open for a while. And the clay thought that focusing attention on them would lead to the development of lots of interesting mathematics. So it's not just the problem itself, it's the sense that working towards that would result in lots of other useful things.

1:51Until two weeks ago, only one had been solved. Now in the last few days, potentially, two have been solved. And why are the problems themselves important? I mean, do the solutions have practical value for society or what have you, or for mathematics? So I think it varies from problem to problem, and I think it depends on what kind of solution is found. So, for example, the Navier-Stokes one, which has been in the news just recently, is somewhere in between. I mean, it's about modelling fluid flow and hence also airflow. And that's very important for all sorts of engineering applications. But it's a question about whether our current model ever breaks down.

2:30So it's not clear that having a setup where it breaks down immediately is going to result in anything very practical. What was your reaction when you heard that OpenAI had solved the Navier-Stokes problem? I was very, very, very surprised. I thought we were years away from AI being able to do that kind of thing. And so, yeah, I got told it's standing in a common area just outside my office by a junior colleague. I was like, oh, yeah, and then went home to read about it. And what does that sort of trigger in your mind? I mean, are you wondering what's next? and how are you feeling about the fact that this is an AI company that's done this with an in-house model?

3:18I'm extremely unhappy that it's been sold for advertising purposes, to put it mildly. So, I mean, it's exciting that this kind of progress is occurring. It's terrifying thinking how much this is going to change how we day-to-day do our jobs, but at the moment we don't quite know how it's going to change, how they do our jobs. So the sort of models one can play with for free on the internet are not solving these kind of problems. And you're not going to solve them with the kind of access you could get for just a few hundred dollars. We're talking millions of dollars, which most mathematicians don't have in their grants.

3:56We tend to work on problems for very long periods of time. So it's made planning extremely difficult. I'm currently trying to finished a paper I've been working on for about six years. And that's not unusual. And so when I started it, CHAC GPT had barely been born. How do we know that the AI solution to that Navier-Stokes problem is correct? And does it matter? How important is it for mathematicians to understand the solution that these AI agents have come up with? My understanding is that they've released enough that some people are saying it's correct. It's really important that we understand why?

4:32So the question was, do these equations ever break down? And the answer seems to be yes, they can be made to break down in a particular situation. But then there's immediate follow-up of when do they break down? How do they break down? Are there other situations where they break down? And if you don't fully understand that proof, you can't immediately follow on with all of those things. So until something becomes initially something understandable only to a few experts and then to the broader community and eventually, presumably, into textbooks or something. It's not really part of the mathematical corpus.

5:13Some of your colleagues have called this a deep blue moment, and they're obviously referring to IBM's deep blue computer beating Garry Kasparov at chess in 1997. How big a deal then is this for maths and why is it that deep blue moment? Well, up to now, we thought that only humans could solve problems that hard. It's not that the machines have beaten us to it. I feel this quite strongly. What the machines have done has taken what is already in the literature and said that if you look at how sequences of words fit together, I can now make a sequence of words which is a proof of synabios soaks. That's how they work.

5:50This word is probably followed by that word and now it's got good enough that it can come up with a proof. So this is a new tool, but we tend to anthropomorphise them. So it's feeling like a new person has come along and been better at it than us. Maths has had to deal with sort of technological revolutions before. I know in your own work, you use computation and the introduction of computers must have been a massive deal for mathematics. But last week, we had an open letter signed by 25 Fields medalists. you know, these are the Nobel Prizes of mathematics, saying that they were worried that AI was going to sort of erode the foundations of mathematics.

6:28They had big concerns. Why are some of your colleagues, or why are many mathematicians, so worried? Because of the way in which this is being used. So it's not being used by people who are desperately keen to solve Navier Stokes, who want to understand everything about Navier Stokes. It's being used by an enormous company that wants to be in the headlines. And so they freely admit that they threw lots of resources at all of the Millennium Prize problems just trying to see what to do. They're trying to grab the medals without doing the work in some sense. So there will have been a lot of people who had been working on projects for years that were meant to be contributing bits towards Navier Stokes.

7:08Usually we all talk to each other about what we're doing and suddenly everything has disappeared from them and it's not clear how it's going to follow. So if the AI companies want to go and claim all seven millennium maths problems, well, fine, I guess we'll have to deal with that. But lots and lots of people's careers are going to be completely and utterly derailed by that. And we're then going to need to work out how everything gets reshaped. There's also a fear that with these generative AI, if we really start seeing as many automatically written papers as we're beginning to see now, we won't be able to cope with refereeing and absorbing the volume.

7:45Over the last few months, the number of papers in my field, which is group theory, has gone up by a factor of about four. My email inbox is like four times as long with things every morning that I should be looking at. And I can't look through that many anymore. So the net result is that we shrink in and we're not really trying to keep an eye even of our own narrow subfield anymore because of the volume. Do you think this could, in some sense, affect how humans understand mathematics or humans improving their understanding of mathematics or perhaps even impact on the sort of the joy and satisfaction that mathematicians get from doing these hard problems?

8:24Because mathematicians will still be working on problems, but maybe they may be doing more of sort of reading AI generated proofs than they were before. I have to say that would make it a lot less fun. Mostly by the time a human has really worked through something, they've thought hard about what concepts they're going to give names to. How are they going to break that up into chunks? How are they going to humanise it? And all of that disappears when you're reading AI stuff. It breaks it up very weirdly. It's immediately, you can smell it. and reading bad maths is depressing. It's very, very depressing because you're expending an awful lot of brain power and then going, oh, is that all they were trying to say?

9:05We're all trying to find, I think, not just the first proof or a proof of something, but usually there's this notion of the proof for the book, the most elegant proof, the thing that is clearly the way you should understand it. And when you find one of those, it really does spark joy. I remember the first time I solved an unexpected thing during my PhD. I showed it to my supervisor. He threw his head back laughing and slapped his thighs in delight at the thing I had just shown him. So for a mathematician, it really does give us this much pleasure to see the ideas being that beautiful. What is the impact, do you think, going to be on those up and coming mathematicians?

9:45I mean, you know, there are these, you know, your field medalists, your established professors of mathematics and so on. there are also people coming through this system who rely on solving problems to get their research papers and they rely on their research papers to get their jobs, to get their PhDs. Are you coming across any sort of instances where you can see AIs already kind of disrupting things? Yeah, massively. So this happened with me and a former student recently. So he was an incredibly strong student. I set him a problem that turned out to be harder than I thought. and so we worked on it and worked on it and we got most of it done but there was a gap which we really did try to fix anyway he had a postdoc already lined up because he's marvelous and so we were like okay well we'll just put this to bed with the gap in it you go off and do the next job the paper is currently submitted um with the gap in it even though we're a bit annoyed um and he wrote to me a few weeks ago saying he'd fed our preprint into chat gpt and using the results we already had, it filled the gap.

10:47And now what do we do? We already had it turned out all of the ideas you needed for this. We just hadn't quite put them together right. One solution, which is an immediate response, is to feel kind of devastated that the machines may be being smarter than you, because we really did try. And ChatGPT, free ChatGPT, did it in just a few seconds. Then there's the question of, well, do we now update the paper with the gap filled? Do we say just this one section has been done using generative AI? Do we incorporate it completely into everything and say this paper was written with the help of generative AI, even though all the rest of it wasn't because it's older than that?

11:26So it really is very troubling. So yeah, this is affecting the people at the beginning of their career much, much more. And it's terrifying for almost all the PhD students at the moment. So at the beginning of any PhD, your supervisor will typically give you a big chunky problem to work away at that will probably take you three or four years. And particularly for the current cohort of PhD students, when we picked their problems, this was not an issue. And so maybe you've got them working on a fairly famous conjecture, hoping to solve a little bit of it. And now there's a very real risk that before they finish their PhD, somebody's going to push a button and solve the conjecture.

12:05And there's not much we can do about that right now, other than sort of hope it doesn't happen, or if somebody does push the button, it's them.

12:17Coming up, how can mathematicians solve this new problem?

12:30It's really clear mathematics is going to have to change. I mean, what do you see as the sort of major hurdles that the community has to start finding a solution to, has to start working out how are we going to do maths now? Usually you'd like to know how a solution was reached. So the question of verifiability, reproducibility, equality of access is very important to us. So I'm sure as a community we are going to be working together with the computer scientists to try and at the least remove the financial impediment to using it. Then there's the question of how we form norms about how much to use it and when to use it and whether to check in with people working on problems before you dump their problem into it.

13:19So at the moment, we tend to do that. For example, if I have a new PhD student, I tell my broader research community what my new student's going to be working on so they know to stay clear. We're quite nice like that unless people are tackling really famous problems. So somehow we need to find a way to get back to that. And then the attribution question. So obviously AI doesn't give proper credit. And that's really important to us. Finally, Colbert, I mean, AI is obviously such a powerful tool. Do you think it's going to become the norm that mathematicians work with these AI tools, whatever they can get their hands on and that that becomes enormous and that it pushes mathematics to sort of greater heights rather than pushing humans out of the game i don't think it'll be uniform across all bits of maths so i suspect that some areas are going to use gen ai much more than others and you can see even now people are producing charts and graphs of how much it's being used to write preprints and some are being much more effective than others i think depending on how much energy, power, resources wind up being used to run these models.

14:28Some of us are going to be more comfortable than others. I think it will hopefully make us do better maths, but I think it's going to be a very rocky road together. It seems like there are perhaps a few years where it's going to be pretty dicey and then hopefully sort of a new sort of stable reality of mathematics on the other side. Yeah, I mean, I spent a chunk of this summer feeling really quite depressed about it all. And somehow as the autumn's come round, I've cheered up a bit and started feeling like we can manage this. So, I don't know, maybe ask me at Christmas and I'll be back to thinking it's all terrible again.

15:08Colver, thanks very much. You're very welcome. Thanks again to Professor Colver Roney-Dougall.

15:20and that's it for today this episode was produced by ellie sands the sound designer was gel cox and the executive producer is ellie burie we'll be back on tuesday see you then

15:38this is the guardian