296 – Navier–Stokes and What AI Means for Mathematics

0:00Hello and welcome to the Idols podcast. I'm George Simmons and I'm joined today by David. How are you David? I'm Daniel Lloyd George. I'm excited by today's discussion topic. Yeah so this discussion is coming on the back of OpenAI's announcement that, or from some of its research is that they have solved a portion of the Navier Stokes Millennium problem. So this news came out on Tuesday, we're recording on Friday and yeah we're going to give our discussion on what we think about it, what we think the implications of this are and some of the wider discussions around it. And it is of course a trending topic right now. Lots of mathematicians or maths people are wondering what the implications of this are. So it's a really important topic, it's a really important moment potentially. Absolutely. So I guess we should start with a couple

1:02of caveat at the time of recording. This solution has been claimed, it has not been fully verified, it has not been independently reviewed, all those things as far as the claims used in the Millennium problem to concern this is still an open problem for those caveats. And the second caveats we give is that neither of us are experts in fluid mechanics or Navier Stokes, that kind of thing. We are mathematicians. So yeah. In this context this is something where of course as mathematicians the Millennium problems are things that we're aware of and deeply aware of. This is not a problem which I haven't spent time looking at. This is something which is on my consciousness. And more importantly, this is something where the way it has happened is really I guess it ties in with what's happening all over the place. And this is where I think it's a wonderful representation

2:06of where these AI systems are and just how disruptive they are becoming in ways that could be very positive or could be very negative. So it was about a month ago that Karen Stow gave a talk on exactly this kind of thing. So he was reflecting on the role that AI already is playing in mathematics and the role that it should play. And the part of what he said was that in a way the regime that has been used to generate this solution is not a new train of thought. It is a continuation of through building solutions of more simplified Navier Stokes problems by the equations, unforeseen solutions with walls in them or the kind of stuff. So the train of thought is not new and therefore in a way it is not a surprising result that a massive brute force effort

3:07like this building on existing knowledge has come about. So can I say a few things about that? Yes. So as I understand it, deep mind which is Google's big effort, they've been working on this specific problem for a long time. And more generally we know they've been working on mathematical proof for a long time and they've cared deeply about it. So it would not have been a surprise in some sense if deep mind had solved it. It is a surprise and I think it's a surprise to many that open AI's latest models have solved it. And what you've stated I think is the key point. And this is where I want to refer back to the AI Empire's book from Karen Houd and AI's approach.

4:13So Google's approach have a lot of respect for. As you've read it, open AI's approach was more than ever more compute brute force. Yeah. So I read that this is a, so I can eight eight hour coordinated effort and was contributed by 300 billion output tokens on the AI which I know it's an internal model so you can't quite use public facing pricing but on the new after six pricing which is $50 per million output tokens is cost 150 million dollar effort which by any academic research standard is an enormous amount of expenditure. So just to put into context the amount of effort that's gone into these 10,000 agents coordinating here. But this is the key thing. What you're talking about there

5:13is still 88 hours. So they weren't interested in this before. They just said okay can we test what can we achieve with our latest big model. And if we put in as you say enough effort to it. And 88 hours is nothing. This is less than four days. Compared to the 200 years that these problems have existed for. So the question is wait a second. Why? Has it been that it just needed four days? And why if it just needed four days is the effort that Google's gone about on this has not born fruit. Now some people are thinking this is okay, this is because open AI's model is so much better now. I don't believe that. Now other people are mentioning the fact that maybe open AI's model had access to some unpublished research which had to belong the way. Whether that's true or not it's just another piece of the puzzle. As you said

6:19this is building on the research that was done over many years by many people. The key point is even if you think about that in terms of human effort actually this would have been a huge amount of human effort to go through and do this which wouldn't have been done. And the approach that Google has been taken has been to try and actually train the systems to do things in different ways which would be more legitimate. As far as I know this brute force approach just using and setting in what it can do is not the approach that the deep mind team are doing. They're doing much more interesting research. So to me there's three key things I would like to sort of lay out on this. One is that is mathematical proof of something like the Navier Stokes equation, the right thing to be judging against. Don't get me wrong this is impressive. This is important if it is verified this is really an important moment. But it is important to discuss. Is this

7:25even the right indicator of different things? What does this say about mathematical proof itself as I'm going to say the word pastime because that's what it is in some sense. It is a pastime. This is a way people spend time. So let's come back to that. And then the second thing I want to draw out is that actually this is only possible because of the power of the latest AI models. It is absolutely possible to do an equivalent amount of work that previous models would have taken years and years maybe more to do this effort and maybe never arrived there. And so the latest models there are some things. What does this mean in terms of what we expect in terms of the business model of the likes of open AI and Claude in the future? Are they going to continue to release their latest models? Or one of the things that I've heard from colleagues, boost bassitists, a colleague of mine

8:27Ames Dase, and he had this wonderful statement around this that maybe these big companies are going to stop releasing the latest models immediately to the public for others to use and start using them internally to solve the problems that will actually make the money in the future. This is how they are going to recover their money. So there's a really interesting question about what are the problems which this could now solve and who's going to do that solving? Is it now going to be, are we going to share shift in how AI companies work and whether they're building the models for others to use or whether they're actually doing things themselves? Really interesting question. And the third thing which I think is most interesting is, well, what does this mean? And this is the question many mathematicians minds had a colleague from Caltech get in touch recently and

9:29we're saying my PhD students was worrying about AI for their thesis and I was telling them not to worry. Maybe on one, maybe AI is over the next two or three years, it's going to change the nature of mathematics so much that PhD students starting out now are only real trouble. And that's a really interesting discussion point. I don't think we're going to have answers to any of these but I'm keen to just start the discussion for at least those three topics as part of what we're discussing today. I think that really interesting topic. So I guess we'll start with one. Although I was only saying that one and three, but yeah, it takes it back to Terence Tals talk because he was reflecting on what is the goal of mathematical proof? Why do we do it? Does it matter if things are just in a machine variable language like lean that humans never understand? And his argument was that, well, yes, I mean, proving something is one step to problem solving but really it's about

10:32public adoption of the proof and recognition and how that result contributes to society or academic society or wider society or whatever. And this has always been the string of academia, together results in a good journal is not just about having a good result, it's about having an impactful result, one which you can demonstrate can contribute to another area or generate a genuinely new tool or link that helps advance other mathematics or physics or whatever. The first more reflection I've had is that the role and difficulty of being an academic particular mathematician is about how you storytelling your work, not just how you write the proof, it's about how you contextualize it, how you justify what you've done in the context of wider things. I think there's a really important skill and potentially one that's much less likely to be overrun by AI in the near time, anyway, compared to just being able to

11:37prove stuff. And yes, I've always thought that, well, yeah, if our time can be cleared up, we're getting AI to help actually do the proofs and work out the things, that might be a good thing. So this does go back to your third point of view, what does this mean for research and what does this mean for academics? Let me just come in because what you've made me think about is 30 years ago, whatever, no, it was not 30 years ago, it was 20 years ago. I still remember these discussions around the fact that mathematics was done very differently in the US than it was in Europe and the UK was somewhere in between. And this was something I was very interested in. And one of the things that I always found most visible about those differences was that because of the nature of funding in the US, the mathematics in the US was very, and it used the word fatty, as certain trends,

12:40there were things which are trending and which are really of the moment and everybody rushes to those things, progress is made, and then the next trend comes along and then the people rush to that. Whereas in Germany, I still remember when I was there, professors take their time, they work through their whole career, building out an area which is probably often considered a niche area where there's very little interest by the broader community at that point in time, but that's okay, they've got their niche and they're able to sort of make progress, slow and steady progress. And I love reflecting about the fact that neither of these were right or wrong, there were advantages and disadvantages to each. The trending nature meant that rapid progress was made in certain areas, whereas the slow and steady approach meant that the broader areas were things that were later trending emerged as they became important as people had made progress on them and so on.

13:40And I still remember this statement about relatively well known mathematician who was making extremely rapid progress in an area that was trending in the US and was rather surprised that people weren't happy about this or weren't happy in the way that he expected. And in fact, people were saying that he's killing the area because he could come out with proofs in that area so much faster than anyone else that he was single handedly, rapidly answering all the big questions which was leaving no work for anyone else to do and so note the errors can make a name for themselves in it and it was really interesting that actually he then recognised that to really contribute to the mathematics community, it wasn't just about what you proved yourself. It was also about how you built a community of people who understood and who thought about what you were doing. And so the real art, and this goes back 20 years, way before AI was involved,

14:42is about understanding and how to build community. And this is something which I've been aware of for 20 years and maybe I was very lucky to be exposed to this at a relatively young age for whom I supervise as others, but the maths was never just about proving it. It was always about community building and the proofs were part of that community building because they were these statements of things that were not known. This was about in mathematics what is so wonderful is you know the difference between what you know and what you don't know and where that limit of the knowledge is and that limit of the knowledge is defined by proofs even if there's something like the Navier-Stokes equation which for so long it has been expected that it is true. In fact so many things are built based on the fact that if it is true then this is what happens

15:47and therefore we're doing. So actually the breakthrough of proving the Navier-Stokes equation is not that important because most people have assumed it's true for many years anyway and they built on top of it. They're just not building on solid ground because that piece of the puzzle hasn't been totally verified yet and so mathematically it wasn't known. It's your reflection then so your point on this kind of this link between rapid progress, this community, this kind of fitting in the gaps is very similar story to the paramins proof of the primary conjecture currently beyond the millennium price of that insult and he refused the price essentially on the basis that he recognized that what he was doing was building upon everyone else's work. He was just the one to find that final link but he recognized that finding that final link is really a small part of the story and it is very much

16:48about as you say the community all those write-ups over the years that people have done the books that people have made all contribute to that one endpoint and in that way that endpoint is not so important and I think there's a really interesting parallel to draw with this Navier-Stokes thing in the paper that's been produced there are only 16 references and in an academic context with all the work that's gone into that it does raise some alarm bells of what are the aims here and I guess that does read us onto your second point which is what are these companies aiming for here are they aiming to advance the academic body of knowledge as a whole are they in it for money? Are they in it for power? Well unfortunately if they're not in it for the money then they've chosen

17:50the wrong business model. Open AI when it started out was a not-for-profit they flicked to be coming a full profit so that they can access the money to build the infrastructure which is enabling them to have results like this but that's borrowed money they have to get a return on investment for their investors really so if they're not in it for the money then their investors are going to be really unhappy with them that's exactly the transition that was made there if they were still a not-for-profit then we could be believing that there's all sorts of other things behind it but fundamentally right now if a for-profit company which is traded as they are being traded if they start making decisions which are not in the best interest of their investors not only are they likely to lose a lot of investments collapse and be in real trouble but further

18:51more they can be sued my understanding is particularly if you're registered in the US the managers the directors are accountable to serving their investors best interest so the only thing that they can do is try to maximize the money that they get back out of it and this is why it is dangerous this is the genuine danger in what there is let's say it isn't the Navi Estokes equation which I don't believe there is huge financial benefits to solving that it is very interesting that they've used that as a toy example to build real credibility around what the latest models can do but for them it's a toy example you know what about a health problem what about when you're actually now solving this to build a medicine which could solve a widespread problem that's something where maybe instead of releasing the model for others to use to solve that to do that it would be

19:55in their interest to become a pharmaceutical company now I'm really worried if that's how it starts to get used if they start saying well we have access to models which is so much more powerful so we can take those last steps because the same problems fundamentally these last steps building on the knowledge of others to take those next steps that's all that's happening in scientific research in a lot of these this is where they could catch a knowledge which has real value and where they could be trying to convert that value in ways that I think would not be in society's best interest I'm going to take malaria because malaria is still a really huge problem and there was real hope about this malaria vaccine recently a real cure to malaria this is life saving you put too high a price on it whose life gets saved and whose life is not worth saving these are the real problems if this is where it goes and the main driving force has to be a recovery of an investment

21:03and it's not just any old investment I mean these top companies have had a substantial proportion of the world's economy invested in them a cure for malaria that's nothing that's a drop in the ocean even if they are able to sell that at a huge price at a great cost to society the covering that is going to take something totally different there's real questions about how this gets integrated in what this does for society but we are seeing just how powerful these systems are and I do want to lay back in at some point and say the latest models and thinking about the latest models as being so powerful is I think underestimating the power that exists in much simpler models but in these multi-agent systems it is the multi-agent systems at the heart of what they're doing which are creating this power let's not be confused about that don't get me wrong

22:06of course the more powerful your agents the better their work in certain contexts but it is the nature of multi-agent systems which have made the difference here 10,000 people working on the Navier Stokes problem bringing everything together that doesn't surprise me if we could have got that sort of manpower working on it really seriously on top of the literature every one of those agents and expert this is exactly how the American system in mathematics was making progress for years because they were using funding to focus the effort of mathematicians on certain problems and make rapid progress now you're just able to do that with AI agents rather than just academics that that doesn't stop the fact that mathematics is not just that gold rush. Exactly yes and I think this may be a good night's to end on exactly what do we think this means

23:07to break academia well maybe not too much it's amazing to have such powerful tooling available but it's also amazing to have such funding available and this is something that academia is always struggled with you would never be able to assemble that manpower maybe it still isn't a genuine possibility that level of resource that's gone into this may never be possible to use it scale. Well this is what's not known and I think what's really interesting on this and this is where I do think as a mathematics community maybe it'll actually help us look in the mirror and say what do we actually value and I would like to finish by saying actually the mathematics community has always been much better than I think many portray it to be it has never simply valued those who squirrel away and proof it's and meanwhile did an amazing job squirrel and in a way proven from that's last

24:13there and here's the exception not the norm most mathematicians are highly collaborative it's all about community that's not to say that individuals who really focus on and just prove things weren't valued within the community they were but the community as a whole by and large has always been much more collaborative it has been much more about the mathematical community and the engagement of each other and people thinking about things in ways which advanced knowledge rather than the individual bits of knowledge being answered being solved that's never been the community I've known that's always been part of the community I know but in some sense it's always been my mind the unhealthy part of the community in the obsession the people who have solved a particular problem as opposed to those who are building the structures and the frameworks and the actual links and joining pieces and bringing people together and it has been as a community

25:19extremely good at still recognising many people with those skills and so the question really is going to be down to all of funding is the funding for mathematicians going to drastically change because they're not doing proofs in the same way probably not because although within the mathematics community that was something which was valued highly as a community its funding was really not determined by the fact that mathematicians did proofs it was determined by the value that community brought maybe the value the community could be increased because of what it could do using AI to help this value become more visible to others maybe it would be decreased because AI is sucking out funding from everywhere so we don't know but really the question is

26:19the mathematics community around the world is funded by academic institutions the success of the mathematics community is really mostly unfortunately due to its funding this is a funding issue whether PhDs are about proofing the future about they're taking another form that's what it's self-out the real question is is mathematics funding going to increase or decrease going forward if it decreases this is going to be difficult for the community because they're going to lose a lot if it increases what's that opportunity what's it going to become what's it going to emerge as and those are the questions these are the societal questions that we should be asking and this is where we come back to the point two which is where's that money going and what's it going to be useful and how does this relate to society as a whole those are the

27:20questions which are really hard and we don't have answers to but I do like the fact that it is sensible to refer back to Karen Howe's book none of this is surprising this is exactly part of what we would expect based on what down howe was articulating so well in her book The Empires of AI I think that's a great place to leave that thank you for this conversation there that's been good I'd enjoy it this is such a topic on moments so thank you nice time to be a mathematician god