AI Just Solved One of the Hardest Problems in Mathematics

0:00Propel Fitness Water with Gatorade electrolytes, zero sugar and vitamins. Propel hydrates better than water to help you get the most out of your workout and get back to your best self. What propels you? Propel with Gatorade Electrolytes. AI has solved one of the most difficult problems in mathematics, the Navier-Stokes problem that has remained unsolved for 200 years. Yeah, I'm just going to read some quotes out about this. So here's one. This is a deep blue Kasparov moment. That was mathematician Tristan Buckmaster. and that's obviously referring to the first time a computer beat a human grandmaster in 1997.

0:37Massive landmark. Another one, this feels like one of those moments when the future arrives all at once. That's a German computational biologist called Fabian Thies. And then Jacob from our office. It's like imagining that an AI-generated film won an Oscar but that doesn't really work because solving Navier Stokes is much, much, much harder. So this is one of the biggest science stories of the year, probably the biggest. Yeah. And that's what we're getting all into today from new scientists. This is the world, the universe and us. And I'm Dr. Penny Sarsha. I'm Dr. Rowan Hooper and we are biologists, Penny.

1:11And so thank God that we've got some mathematically minded colleagues with us. Matt Sparks, Tim Revel, welcome to the pod. Matt, let's set the stage. What is the Navier-Stokes problem and why is it a big deal? Well, the Navier-Stokes equations model the flow of fluid. So that's air over a Formula One car or blood through your heart or anything like that. And they're very well known. They're very well used and they're very old. They were first written down in the 19th century. And when digital computers came along, it allowed us to do big simulations with them. So we've effectively made wind tunnels obsolete.

1:50We design new aircraft on a computer. or that sort of thing, all very reliable. But there was an unsolved question, a challenge that remained, and that was that we knew these equations could be sort of provoked in certain circumstances, in certain ways, to induce a bit of turbulence in these flows. And the question was, if you start with a 3D simulation of fluid with smooth flow, will that stay smooth forever, or will it blow up, turn into a singularity, break down, start producing nonsense essentially? And it was a very tricky problem to solve. And essentially OpenAI has now proved that yes, they can break down.

2:35These equations can produce a singularity, turn to nonsense in certain circumstances. Yeah, can I just say the thing with Navier Stokes is that like any maths undergrad will know these equations because they just kind of pop out magically from Newton's second law, F equals MA. You apply them to fluids and then they appear. What OpenAI has shown is that these seemingly simple equations that describe how fluids move suddenly break down into complete nonsense. It would be like if you were looking down in the bath and suddenly an infinite whirlpool arose where the water started moving at infinite speeds and shot out of nowhere.

3:08And that's what comes out of these equations. So this is not anything to do with chaos and a small difference leading to some crazy outcome. This is something different. This is to do with the equations themselves. So they are an approximation of the real world. The real world has molecules. Navier-Stokes doesn't. It thinks about fluids as a sort of homogenous lump. So the fact that the equations can break down and produce infinite jets of water in your bath is not a concern because they are an approximation. Yeah. So I think we'll get into this. But like your bath is going to be fine as a result of this.

3:44But mathematics may not. Okay. Okay. And so, Tim, Navier Stokes is one of the Millennium Prizes, these kind of key set of problems. Could you remind us what those are? Yeah, so these are mathematicians over the years have kind of got a history of listing out some of the most important problems to try to galvanise research effort around the most important challenges. And so at the turn of the millennium in the year 2000, the Clay Mathematics Institute, one of these big well-funded institutes, listed seven mathematical problems. that they said we will award whoever can solve these$1 million, which was a lot at the time and is increasingly less 26 years later.

4:21In that time, one had been solved before this. And then sitting on that list was also the Navier-Stokes problem. So now OpenAI and perhaps others, which I think we'll also get into, now have a potential claim for this$1 million prize. And there was a bit of a chat in the office this week about when we talk about these being the most important problems, they're not necessarily the most useful they're sort of the most mathematically interesting yeah that's right and especially with navier stokes because there was no one this wasn't like is this a true kind of thing wasn't just like can we prove this or not we know that these are some of the most basic and important equations we have yet we understand them incredibly poorly and so with what the millennium problem tried to do was list a property of them that we thought by solving it or working out whether it was true or not we would gain a better understanding of the equations themselves because of the nature of AI proofs the way that they kind of seem to just be able to create the answers without really giving much in the way of an explanation we can understand perhaps that's not quite been met but the the kind of pure statement in the prize that has been solved in open AI doing this Matt there has been a bit of a controversy about how exactly they managed to pull it off yeah it was a dramatic couple of days really and it starts with two researchers is Tristan Buckmaster and Levant Alpoge, they were sort of quietly working in the background on Navier Stokes, but also on a close cousin of it and a sort of stepping stone problem towards Navier Stokes, the Euler equations.

5:51Now, word of that had got out. OpenAI heard a rumor that progress was being made and decided to start their AIs on the same problem. They didn't know which of the Millennium Prizes these two were working on, so they decided to try all six. Subsequently, they found out that it was Euler, Navier-Stokes, and the two researchers found out that OpenAI was working on them. Everyone got together and had a talk. That's putting it lightly. There were discussions and hints of suggestions were made about various things in the process and who knew what and who saw what. everyone has denied everything and these two researchers rushed out their work the day before open ai right and it was important work it wasn't finished work yeah but uh very prominent mathematicians suggested they would they would have got there could have got there and perhaps even claim the credit for getting there that's just like a race to anything in science is it not it is but ai has the structure of dna for example yeah but ai has has lit a fuse underneath this I think maybe the structure of DNA is an interesting one because like you could make the claim that was there something improper done there when Franklin had done so much work and then did Crick and Watson just kind of nick the last bit.

7:11Yeah. That's arguably what's happened here. Or certainly what the what the conversation is around is that typically mathematicians will talk about problems they're working on thinking, well, nobody else is. I know so much at this stage. They can't just come along and do the final step. But what OpenAI is a is kind of the implication is what they have done is they've come along. They knew they'd made all of this progress and then they put$15 million worth of AI on compute on doing the last step, which is a real like kind of just pipping someone to the post. Yeah. Not academically brilliant. So perhaps that is on to work.

7:43But OK, let's talk about the AI. We're just saying AI did it. But this is an LLM, right? This is the thing that we've been all got used to over the last, you know, God knows how many years. but what sort of level just effectively a chatbot right has has somehow like this is what i can't get my head around because you might have i might have thought a few years ago oh this will be deep mind fashioning some sort of fancy ai that will crack these problems but this is really just a supercharged chat gpt right yeah yeah i mean they haven't named the model but there's every indication that it's the next chat gpt we don't know but yeah they're these models are built on increasing scale they're bigger and bigger and bigger trained on more and more data and there's lots of architectural tricks in the background which obviously not revealed so it's a scaling thing i think it's partly a scaling thing but there are also tricks being employed sebastian brubeck at um open ai told me that last year when there was a bit of a lull and there was this suggestion that we might have hit some sort of glass ceiling or a bottleneck no such luck no such luck he had on his plate 10 different ideas and he thought each one of these might nudge us along a bit and he told me recently they all worked so he was saying that maths used to be a difficult job for him you know he would sit there and there were problems that were really meaty and difficult to solve and now he's in ai he says he's just sort of pushing on an open door progress is just appearing in front of him.

9:16Wow. So that actually answers one of the questions we were thinking about, which is it has accelerated. It seems to accelerate now. We're in this real time of accelerated learning in AI. And if, you know, six ways to break the glass ceiling and they all worked, that might explain why, you know, we're suddenly seeing these leaps in discovery. I think an amazing, what an amazing time. Like, I mean, for us as science journalists to be kind of living through this moment where not only have we got probably the biggest math discovery in decades, we've got all this controversy and discussion points around it.

9:48Previous massive discoveries in maths gave us a new answer. This is perhaps rewriting what it means to be a mathematician and completely changing our view of these things. It's amazing. It feels like there's been this promise for a while that there's some, the fact that maths can be written down, it's all done on logical statements and you can work through it. It sort of always felt a little bit like solving a game which we know AIs have got incredibly good at it's sort of like the AlphaGo version of AI where it's the combinations are beyond what you could possibly compute but can you use AI to help navigate your way through them yeah that's kind of like what solving proving a theorem is about yeah it's you couldn't possibly try every method to get there but ultimately it's a game you work through the different steps and you've got to find the right path to it but now we've suddenly seemed to have hit that level where the AIs have got good enough that they can really make impressive progress.

10:39And over the last few months, we've seen increasingly impressive steps. And this one is like several rungs above the rest. This is a problem that is incredibly famous. It has had a lot of research effort on it for decades. And suddenly, AI has made the breakthrough. But it's not without effort. Yes, it was an LLM, but it was something like 10 ,000 AI agents were used. They worked for 88 hours, which is a relatively small amount of time but it was something like 15 dollars 15 million dollars worth of compute so real heavy resources going into that how many mathematicians for how many years could you have employed for 15 million dollars quite a few yeah i think that's the that's the crux of this is it may be doing something very very clever but it's also doing it on such a large scale if you employed that many world-class mathematicians for that long you may well have been able to do this if and with perfect communication, perfect memory, all the other advantages that computers have.

11:38So perhaps this is an AI story or perhaps it's just a story about scale. It's very hard to know, really. We're at a point now where most people are familiar with LLMs. They've tried them for various things. There's been a lot of hype, but actually they're not actually that good at quite a lot of things. When we're talking here about the huge breakthroughs, incremental improvements, the huge power are we talking about maths specifically and these other things that ai are very good at we're not talking about ai in general is now just brilliant at everything um it's it's hard to know i i think firstly mathematics is a is a wonderful problem for ai because you can sit there and just think about it you don't have to go off to a lab you don't have to have an idea test it come back tweak it go away test it you can just sit there and think through this stuff and computers can do that at the speed of light and in massive scale, in massively parallel scale.

12:31So firstly, maths is a wonderful problem for this new toy we have, but also this toy is getting very, very powerful. So we're not necessarily seeing the same progress in biology or chemistry. Well, no, but this is the thing. So as you said, they didn't name the model they used, right? But they said it was significantly more capable than the latest one, GPT-6 Astra, right? And then the president of OpenAI, Greg Brockman, he comes out and says, it's reasonable to view GPT-6 Astra as an early form of artificial general intelligence. This sort of mythical thing that we often talk about, super intelligence, basically.

13:09That seems like quite the claim, doesn't it? Yeah, this is where I put my sceptical hat back on. Okay, I mean, he would say that he's the president of OpenAI. Yeah. And we have to remember as well that we have no real definition of terms like intelligence or consciousness or AGI. So these are all very woolly claims to start with. And also, if you interact with a chatbot, you do get the sense that, OK, they can solve these amazing mathematical problems, but they can also fail to help you with very simple tasks. So this is not an intelligence that's universal. Well, I hear you, but I haven't interacted with the one after GPT-6 Astra.

13:51And, you know, we've seen people like Richard Dawkins, you know, say what you like about him, but he's not stupid. But he thought the chatbot he was interacting with was conscious. Many clever people are fooled into thinking this. So there's a lot going on under the lid. And I know there's a long, long history of people being fooled by things that seem like intelligence that goes a long way back to machines that are far more simple than this. So I think it's understandable that there are a lot of people who interact with this incredibly impressive technology and think, is there something underlying it that is artificial general intelligence?

14:27Someone said, is it conscious? All of these kind of things can certainly mimic that in a way that you can see why you can get confused. The question around, is it super intelligent? Well, certainly in some ways, but so is your calculator, right? Your calculator is better at making calculations than any human on the planet. Well, AGI, I meant to say, though, right? But yeah, so what is AGI? Is it being better, more intelligent than humans at everything? I think it's highly unlikely that what open AI have got now fits that category, not least because it's only sitting in a chatbot. There's so many forms of intelligence that we have, and many of them involve being able to go into the real world.

15:01But this question of like, where is this all going? What does it actually mean? What is it going to mean for mathematics as well? I just think that we are witnessing the kind of birth of a new kind of tool. And we don't know exactly what that tool is. But it seems like it's certainly for maths. There is this magic called formalization, which means that the AI can come up with many different versions of a proof. But then crucially, we can also check it with computers. We can work out whether it's something that just seems like it's true or whether it genuinely is true. And that's what makes the difference.

15:37And so mathematicians might soon find themselves with some sort of kind of magic truth machine that you can ask questions of. And maybe we can put huge amounts of compute on and we can suddenly find out whether theorems are true or not. But that's a long way. That's not the only job of a mathematician. A mathematician wants to know why things are true and work out the logical steps in between and gain an understanding of it. Like an example I've been thinking of recently is Pythagoras' theorem. You could find out that if you measured plenty of right-angled triangles, you would eventually work out that, okay, it's probably true that the sum of the two sides is equal to the square of the hypotheses.

16:13But what mathematicians want to do is they want to understand why that is true. And there are now something like 350 different proofs of Pythagoras' theorem, each of which gives you some insight into why this amazing relationship halts. and that's the job of the mathematician it's not just to work out is this statement true or not it's the why and so far the however amazing these machines are they are incredibly poor at that particular task so we may see an evolving of what an evolution of what a mathematician does but to me this just feels like an amazing part of their toolkit that is being underused so far we don't know how to use it but that could be an amazing game changer so on on the changing nature of maths.

16:56Matt, this week you spoke to Terence Tao. He's often said to be the finest mathematician of his generation. What did he have to say about all this? Well, it shouldn't surprise any of us that the finest mathematical mind of the current generation agrees with exactly what Tim just said, that these AIs are not, they're not doing the job of a mathematician. They're not assimilating this information. They're not consuming and absorbing it and adding it to the corpus of knowledge that we have which is the job of a mathematician they're just as terence tower put it just throwing a carcass of raw meat on the table of academia and saying have at it which is you know that's a phenomenally useful thing if they can develop these these chunks of knowledge but then they're not doing the rest of the work i think not yet right and they're not doing the perhaps the creative stuff yet well perhaps perhaps we don't know we don't know and And I've spoken to a lot of mathematicians this week, some on the theoretical side and some on the applied side.

17:54And it's the applied mathematicians who are telling me, I think I'm OK for a little while because my job is to absorb this and find a way to use it in the real world and help people. Whereas the theoretical people are a little bit more worried because AI can sort of seemingly swoop in and do their job. Another example from history is in a few hundred years ago, Descartes worked out that geometry and algebra go hand in hand. You can write equations down that describe perfectly geometrical shapes. And this now is just such a basic assumption of what we use mathematics for. But at the time, it was a major leap forward and one that gave a whole new toolkit to mathematicians.

18:34And mathematicians responded by going, I don't know about this. They thought that suddenly, at least some mathematicians thought, that the job of a mathematician might suddenly, rather than working in these beautiful geometric forms that the Greeks had handed down to us, would be pushing symbols on pieces of paper and would become a drone-like drudge who just went through these and no longer had any understanding. But instead, what happened with algebra and geometry is we got all of physics that we know today. Every bit of mathematics that has developed in the last 300 years has used these kind of relationships in one way or another.

19:07It completely changed the toolkit available to mathematics. And you couldn't have foreseen at that time what was coming but what came was an amazing new revelation in many ways or an amazing new view of what was to come maybe that's what we are looking at now we've got this amazing tool we can't possibly foresee how it is going to change things but i don't understand how an amazing new tool in mathematics can ultimately end up being a bad thing i think the way the ai companies are behaving is bad currently personally and i think that we are in a real like flux period where there's going to be upheaval but we're only adding to the toolkit we can still do maths on pen and paper but we've now got an extra tool where is that going to lead us i was at the royal society yesterday happened to walk past a portrait of rene descartes yeah and the the guy i was with as we walked past he swore at descartes and i wish i'd have i'm going to clip that little bit you've just said and send it to him.

20:06Well, there is this, he did. But look, just to go back to the thing you were saying about Pythagoras and right-angled triangles, and you're saying there are all these proofs as to why that relationship holds. When you say why to an evolutionary biologist, does it mean the same as it does to a mathematician? I would get, what does it mean to an evolutionary biologist? Well, Y to me means that I would say that that right angle and the relationship between the lengths of the triangle, there's some evolved reason for it. Like what there's a meaning to it. But that's not what you mean. I think it is in some ways.

20:44So like my best guess at putting my mind and putting myself in the mind of an evolutionary biologist, which may end up very poorly. But if you're looking at some sort of animal behavior, you want an explanation as to why they are acting that way. And in my mind, the behavior is the theorem, but the explanation here is the proof. What were the steps that led them to do that behavior? And in this case, why do right-angled triangles behave that way? Oh, amazing. Never thought of an evolutionary biology explanation for maths. I love it. That's amazing. That's the kind of thing you can end up from us, I reckon.

21:26To help you get the most out of your workout and get back to your best self. What propels you? Propel with Gatorade Electrolytes. Matt, you're due a day off really, but there are rumours that OpenAI has also solved another millennium prize, the Hodge conjecture. Yeah. Are we expecting them all to fall now? Do you know, I sort of am. I mentioned earlier that originally when OpenAI first got wind of this work, they set AI models off on all six of the millennium problems. Now, what would you do if one of them succeeded? What would you do with the other five? I'm pretty sure they're churning away on this problem right now.

22:05One thing I would say is we haven't spoken about negative results. AI companies never tell you when they couldn't solve something. So we've got every reason to believe that AI models can do amazing things and make amazing mathematical discoveries. But we don't know what sort of problem they can't solve. And that might be everything else they haven't announced yet. We will see. I think we are still getting a real, we really don't have a good sense of, like you say, which ones they can solve and which they can't. My guess is that it's still the case that there is a significant number or styles of problem that the AIs are not so good at.

22:42I personally hope the Hodge conjecture doesn't fault because it is by far the hardest one to explain. And we're going to have a tough podcast trying to talk about that one. I might be sick for that one. I think you've wrote it already. Matt, you've been dabbling yourself as well. can you tell us about your own work? Yeah I did a sort of quite quirky story about some mathematicians who had a toy problem that they'd sort of kicked around for over a decade. A very strange set of dice where if we were playing a board game now we could all throw one dice each and we'd be guaranteed that there wouldn't be a tie and one of us would throw the highest number so we could choose who went first.

23:16These are fair go first dice and what they were looking for was a set of five of these dice and it was a very tricky problem to solve and after over a decade they'd solved it and uh after i finished the story i sat there thinking i've done a lot about ai maths recently i wonder if chat gpt can solve this problem for a bigger problem and it took me 13 minutes to solve a six dice problem 13 minutes 13 minutes is there a million dollar prize unfortunately not but to be clear you're you're covering this but you're not actually a mathematician yourself no i'm a computer scientist but really my maths is pretty dusty so that's amazing i had to prompt it a couple of times but we're talking just plain english uh prompts nothing nothing technical i showed it to the researchers and admittedly it wasn't the most elegant solution but it was a working solution and it was one that doesn't exist on the internet so and this is the free version of chat solution yes it will become known as i do think a fascinating detail in this is the sort of prompts that end up with a solution that you do have to be able to explain the problem initially but after that it appears that really to get there you just have to do a lot of encouragement yeah you get back the answer and you say and it says i've not quite got there but here are some some steps yeah you can do it keep going and those steps happen again again and we've covered some of these really big impressive um results from ai has effectively been that just encouragement and encouragement and encouragement which is so strange i think It is.

24:43It hits these sort of halting points. I remember it saying, you know, I can't find a solution with dice that are all the same size. What if this? Shall I look for one that differs? Yep. Off you go. And then it comes back. Pass it on the head. That's a good idea. Off you go. It feels like that. It sounds considerably less annoying than like everyday uses of LLMs where there's all the sicker fancy and they're sort of pandering. This sounds more fun. My guess is you've got some of that too, right? But like you haven't come up with a solution yet. oh you're right I hadn't come up with a solution yet did you get some of that there was some pushback but I'll be honest I don't use AI very much I feel like I have to use it now and then to just sort of keep abreast of what's going on but I don't find it a particularly useful thing in my day-to-day life so but it's fun to play around with problems like this and look the other thing that happened this week is that um and I don't know what you make of this one of the a researcher from Anthropic, quit the company.

25:38And he said, the people building AI earnestly believed that it could kill us all by the end of the decade. And then two other engineers who are still at Anthropic haven't quit the company. They put out statements saying, yeah, we agree. There's a greater than 10 % chance that AI could make humans extinct by 2030. Which, as a reminder, is just four years' time. Yeah. What on earth is going on? well whether you believe this or not it's a tricky question but the the crux of it is it's not impossible so you you sort of have to lend some you have to worry about it to a certain degree there i mean i suppose there's two ways this could go if ai could either accidentally kill us which is the is the paperclip problem the famous paperclip problem if you if you run a factory and you're it's full of ai robots and you say to them make me as many paper clips as you can yes and And they use up all the raw materials and then they think, well, we'll break down the factory and turn that into paperclips.

26:38And then they go out into the street and they turn cars into paperclips. Yeah, okay. And then they turn us. Unintended destruction of everything. I think Prince, now King Charles, used to worry about this. It was a pet problem of his. The gray goo of nanobots. Yeah, okay, yeah. So that's the sort of accidental one. And then the other one is the sort of Hollywood malevolent AI wakes up and decides, I don't want these humans around and deliberately kills us off. We see those two threats, I think, are things that we've got worse things to worry about, right? We've got actual threats like climate change and nuclear war that are right there with a good chance of not making us all extinct, actually, but causing massive damage.

27:20And it does remind me of that book that was out last year. I don't know if you read that if anyone builds it, everyone dies. And this is really playing on this whole theme of if we build AGI, it's going to be bad for humanity. But I do think your point about there are also these other concerns is a valid one. Yeah. Around climate change, that is not something that is devoid from AI companies in the amount of compute and data centers and water and electricity they are using. And there's this term criti-hype, which is where you end up sort of hyping something up so that you can criticize it. And that's sort of a trap that we fall into a lot with AI where, yes, there is potentially this small possibility of complete existential destruction from AI.

28:06But by spending so much of our time talking about that and then criticizing it, we end up distracted from some of the clear problems that are definitely in front of us with AI around bias and energy use and all of these things that we know for sure. Yeah. Those are problems. It's funny you said that because it did pass through my mind that this was a kind of dead cat move by Anthropic. That being, if you're in trouble, you throw something unexpected on the table and everyone starts talking about that. I disagree. And I'm certainly not in the camp that believes AI is likely to do this. But it is possible.

28:42And you can't ignore an existential threat. You can't say, I'm going to worry about these existential threats, but not this one. I'm sorry but I was working with our reporters years ahead of the COVID pandemic everyone was ignoring the threat of a pandemic well okay like we knew we knew we were vulnerable we knew one would come we knew we would do one and people we knew it was a threat and no one was preparing the way I suppose I'm not saying it's impossible to ignore an existential threat but it's unwise to but you're quite right but that's that's exactly what we need we need people to worry about pandemics.

29:17We need climate researchers to worry about the climate. But people in AI companies, we need them to worry about AI rising up and killing humanity. I'm not saying it's a huge or likely problem, but it's not impossible. OK, so back to sort of science and maths. AI is brilliant at maths. It's a whole new tool. It could change the entire face of maths. But maths is this kind of, it's almost a computer language in a way, isn't it? So it's the first one. Where's this going? Is all science going to become AI? Is AI going to get out into the lab? What can we hope for or expect soon? I fully expect that AI will be a boost to every field of science to some degree.

29:58And I think that will vary and it will change over time. Some will be harder, some will be easier. We've already had examples of AI sort of making improvements in fusion power, for example. You know, more finely controlling the magnets and containing the plasma, that sort of thing. So if AI cracks fusion power, wonderful. You know, if AI makes hugely efficient solar panels, wonderful. So it could potentially have a big impact. We've had Alpha Fold from DeepMind helping us with protein folding, which could lead to better drugs, better treatments. So I don't expect soon to see the sort of mathematics-like progress that we've had in other fields, but I expect it to come eventually.