Mathematicians Are The Canary Down The AI Coal Mine

OpenAI claims a swarm of AI agents has solved the Navier-Stokes problem, one of the hardest unsolved questions in mathematics, with a $1 million prize attached. Then it released more than 700 more mathematical manuscripts, including claims touching the Riemann hypothesis, faster than anyone can check them. Mathematicians are not happy, and the reason goes far beyond the money. In this episode of The Rest of Science, mathematician and fluid dynamicist Hannah Fry takes Michael Stevens through the equations that describe everything that flows, from the blood in your veins to a rocket leaving the atmosphere. Then she tells him why this story has turned the vocation of mathematics upside down. Ten thousand AI agents. Millions of messages. One tiny term in the equation that nobody wanted to look at, until it unlocked the problem. But did AI really solve a million dollar problem? Why are some mathematicians accusing the models of stealing their work? And how might a proof that is completely correct be the least satisfying result in the history of maths? Along the way, Hannah un-mixes a glass of dye with nothing but corn syrup and patience, and the pair ask what happens to curiosity when the answers arrive faster than the understanding. ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit https://cancerresearchuk.org/restisscience ------------------- Find The Rest Is Science all over the internet by ⁠⁠clicking here.⁠⁠ ------------------- Video Producer: Oli Oakley Animator: Sam BensonVideo & Social: Bex TyrrellAssistant Producer: Lucy LipscombeProducer: Simona RataSenior Producer: Lauren Armstrong-CarterChief Digital Officer: Samuel OakleyExec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

• 0:01Speaker 00

  Welcome to The Rest Is Science. I'm Hadar Rai.

• 0:03Speaker 01

  And I'm Michael Stevens.

• 0:04Speaker 00

  I'm really excited.

• 0:05Speaker 01

  Why? What are you excited about?

• 0:08Speaker 00

  We're gonna do, we're gonna do fluid dynamics, but properly, properly, for real. Because, don't know if you've heard about this, Michael, but there's been some very, very, very big news in the fluid dynamics world.

• 0:20Speaker 01

  Believe it or not, I have not heard about this news.

• 0:23Speaker 00

  Have you not?

• 0:24Speaker 01

  No, no, I don't know.

• 0:26Speaker 00

  I've had people stopping me in the street about this news.

• 0:28Speaker 01

  Shoot, I've just been down the like rabbit hole of making shorts. I haven't gotten my head out to learn about fluid dynamics news, but I'm excited that I get to hear it from you.

• 0:39Speaker 00

  Well, you do, you do. Okay, so, so I'll tell you the news first, for those of you who haven't, haven't come across it. OpenAI announced that they have a proof to a very famous problem called the Navier-Stokes problem. They used AI, they got it over the line. This is a problem that is so famous that in the year 2000, the Clay Math Institute said they would give a million dollars to anybody who managed to come up with a proof for it. And it turns out, the first thing to get there wasn't a mathematician, wasn't a human,

• 1:18Speaker 00

  it was a swarm of AI agents. And believe me, that has ruffled some feathers.

• 1:26Speaker 01

  Yeah. Okay, wow. So, I did not know about this. I'm also not familiar with the problem. I bet, back in 2000, if I had known, I probably would have already solved it, to be honest, but you know.

• 1:39Speaker 00

  That's the only reason why you're not sitting on the million bucks, I think.

• 1:42Speaker 01

  Exactly, exactly.

• 1:44Speaker 00

  That's the only reason.

• 1:44Speaker 01

  I, I, I hope you can tell us about the problem.

• 1:46Speaker 00

  Oh, you betcha.

• 1:47Speaker 01

  And then, I'm also fascinated by AI solved a problem, because I am, I am the least informed about AI person I know. And so, this might sound really naive, but I hadn't heard of AI solving a lot of problems.

• 2:01Speaker 00

  Yeah.

• 2:02Speaker 01

  I know that they can do things like figure out how proteins fold, but I hadn't heard a lot of stories about things like, oh, they figured out this, this longstanding math problem, they solved it. There's a lot of like, they can, they can regurgitate back what humans have already said, and they can change its format. But to come up with a new solution, I hadn't heard a lot about. Hey, it read all the medical literature, and it's cured 18 diseases. I haven't heard that. But this is a solution to a problem.

• 2:31Speaker 00

  This is new. This is, the AI has managed to do something that humans have tried to do and failed at doing for many, many decades.

• 2:40Speaker 01

  Fascinating. By the way, in the comments below, if you're aware of problems AI has solved that I'm not, which is probably a bunch, leave them in the comments, because I'm not saying I, I'm just showing off my, my lack of knowledge. But let's talk about Navier-Stokes.

• 2:59Speaker 00

  This episode is brought to you by Cancer Research UK.

• 3:02Speaker 01

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  Beating a disease that's so ancient and complex. With over 200 different types, it's not gonna happen overnight.

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  But, thanks to research, cancer treatment has advanced a lot. And Cancer Research UK is driving further innovations like cancer-sniffing robots and drugs made from gut bugs that could change the future of medicine.

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  Over the past 50 years, Cancer Research UK's work has helped double cancer survival in the UK. And today, it's continuing to save and improve lives around the world.

• 3:51Speaker 00

  For more information about Cancer Research UK, their research and breakthroughs, and how you can support them, visit cancerresearchuk.org/westisscience.

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• 6:37Speaker 00

  Hello, everybody. It's us from the future. Turns out, things are moving quite fast, Michael, right, mate?

• 6:43Speaker 01

  Yes, we recorded this episode, I'm just gonna say it, weeks ago.

• 6:47Speaker 00

  We did, we did. As soon as the Navier-Stokes thing landed, and then, my goodness me, this story has moved on. Because in the last couple of weeks, OpenAI have since released more than 700 mathematical manuscripts. They are claiming that they have either made progress or found solutions for hundreds and hundreds of, of really, you know, of unsolved mathematical problems. One of their results concerns the Riemann hypothesis, which is, let me be honest with you, the absolute number one ultimate mathematical trophy of all time ever, right? This is like the holy, holy grail. These manuscripts that have been released, there's, there's essentially too many of them, too quickly, that mathematicians have not yet

• 7:28Speaker 00

  been able to check through them to confirm whether they're valid, to find any mistakes, or to really unpick any of the understanding that goes underneath it. We decided, apart from giving you this update, that what we would do is, we would still publish the episode as we recorded it in its, in its full form, because, A, the Navier-Stokes equations are still absolutely delightful and delicious and amazing, and I don't know about you, but I blimey love them. So, you can enjoy our full takedown of, of what they are and why they're important. And B, the fundamental underlying story about how OpenAI is clashing with mathematicians on this subject. You are seeing here the, the earliest seeds of

• 8:13Speaker 00

  a story that has progressed in the same direction.

• 8:15Speaker 01

  Yes. So when you hear me say things like, I don't think AI has solved much, certainly they'll never solve 700 things, keep in mind that I am now in the future. But as Hannah said, yes, I think the, the, the, the main topic of like, what does this mean, is still a delicious fruit.

• 8:34Speaker 00

  It absolutely, absolutely is. And maybe if you guys want to ask us more questions, if you want to hear more about this, you can, you can, as ever, send us in things to [email protected], or comment on our Reddit, or under this video, wherever you happen to be finding it. And we can discuss this way, way, way, way, way more. I mean, look, I've got, I could, I could, I could start my own entire series about this thing. But we're there to answer any more of your questions. But in the meantime, okay, Navier-Stokes equations, Michael, are you ready?

• 9:05Speaker 01

  Let's go.

• 9:06Speaker 00

  Navier-Stokes, okay. So what I thought I would do is, because, I'll be honest with you, I did my PhD in this, right? I don't get this opportunity very often. I thought that I would treat this as, as, as though two, two old friends are sitting down, and I'm gonna tell you what these equations are, I'm gonna tell you what they do, I'm gonna tell you why they're so amazing. And in the second half, I'm gonna tell you why AI has caused such a horrendous storm by kicking up all the dust and pissing off all the mathematicians. Okay? All right.

• 9:41Speaker 01

  Okay.

• 9:41Speaker 00

  So here is the question, right? This is like 1800s or so. You know, Newton's been around, his like, force equals mass times acceleration, everyone's really happy with that, you've got the whole industrial revolution going on, it's making a really big change. People started asking, right? How on earth do you apply force equals mass times acceleration to a fluid?

• 10:01Speaker 01

  Hmm.

• 10:02Speaker 00

  Sort of makes sense when you've got rods and sticks and wheels and, and you know, pendulums. But to a fluid, how on earth do you do it? And so, um, the way that you do it is, I mean, that's essentially what the Navier Stokes equations are. By the way, they're F equals ma.

• 10:16Speaker 01

  I never even really thought of that. I just always assume, uh, extremely rigid objects. But yeah, with water, with a, with a liquid, I'm thinking, well, now you've got sextillions of rigid objects.

• 10:28Speaker 00

  And that is the exact first step, Michael, you absolute genius. Because you have to decide, how do you see this fluid? Like, imagine you're looking down at a river, right? How do you see that? Do you see that as sextillions of, of objects, of, of little molecules? Or do you see it as something that is completely continuous?

• 10:50Speaker 01

  Yeah, wow.

• 10:51Speaker 00

  Mathematically, it sort of makes it a bit easier if you imagine that it's just, it's just water all the way down. Right? There doesn't come a point where you hit something and it's like gap water molecule, right? It's just, it's just water, water, water, water, water all the way down. And so this is something that they call the continuum approximation. And like, that's really fundamental to these equations. It's like, you cannot, you cannot have particles. You've got, it's just this smooth, continuous, uh, mathematical function. And the thing is, is that a lot of the time, that's fine. That's, that's not a big deal, right? At, at sea level, you admittedly, the molecules are about 10 times further apart than their width.

• 11:33Speaker 00

  So that lots of, lots of air, this is air I'm talking about, air at sea level is a lot of empty gaps.

• 11:40Speaker 01

  That's what, I didn't know that. So air at sea level has gaps between the molecules that are 10 times wider than the molecules themselves.

• 11:47Speaker 00

  Yeah.

• 11:48Speaker 01

  All right.

• 11:48Speaker 00

  Which is a lot.

• 11:49Speaker 01

  That's a lot. That's not continuous at all.

• 11:52Speaker 00

  It's not continuous at all. However, when you're looking at a fluid flowing, the thing that really makes the difference here is how quickly does the flow change? And what's the ratio of how big are the gaps between molecules? Or how, how big are the, the gaps between where molecules interact with each other? And how quickly the flow ends up changing? So if you're thinking about like the flow over the wing of an airplane, the air, it might change speed quite a lot. At the like, the tip of the wing and then the top of the wing, but you've got a few centimeters there. So you've got so, so, so many particles in that gap that it sort of doesn't really matter. You can kind of imagine that it's continuous.

• 12:33Speaker 00

  There are other situations though, where you are, uh, say, bringing a spaceship into the, into the atmosphere. And where you're sort of doing the re-entry of a space vehicle.

• 12:42Speaker 01

  Right.

• 12:44Speaker 00

  And that is, that's where this, this approximation starts to fall down a little bit.

• 12:48Speaker 01

  Because, because the air is so thin up there at the edge of the atmosphere that like those gaps between molecules become significant.

• 12:56Speaker 00

  Exactly. Coupled with the fact that actually, if you're looking at sort of the shock wave, uh, around the edge of a vehicle, of a re-entry vehicle. There, the flow isn't changing across the course of a few centimeters. It can be changing across the course of a few millimeters. So you're sort of, you're shrinking the distance that you care about. You're shrinking the distance where there's sort of like a gradient of flow.

• 13:18Speaker 01

  Why, why, why is that? Why can the flow change so much more per unit of length?

• 13:25Speaker 00

  So it's, it's that you're going at really high speeds.

• 13:29Speaker 01

  Yep.

• 13:30Speaker 00

  On a re-entry vehicle, it's like, there's loads of turbulence. There's all these shock waves going on. It's like, there's a lot of stuff going on, like much, much, much more action. And it's happening where there are fewer air molecules.

• 13:43Speaker 01

  Yeah, right, right.

• 13:44Speaker 00

  The particles themselves start to make a difference.

• 13:46Speaker 01

  Okay, got it. So yeah, in thick air, it's so much more continuous that the guys below are pushing the ones ahead and they're all kind of going together. But in rarefied air, you can have some neighbors that are just whipping around in one direction. And the others are doing something completely different.

• 14:02Speaker 00

  Right, exactly. There's also other situations like in your blood, for example, where, um, maybe the, maybe the, the, the flow isn't kind of crazy and chaotic.

• 14:13Speaker 01

  Hmm.

• 14:13Speaker 00

  And you're talking about such small scales, especially if you're looking at like the flow into capillaries. That it's like, actually it starts to become, this is, this assumption starts to break down. The way, the way I like to think about it is a bit like pixels. So if you take a photo and it's a really simple image, then you can handle not having that many pixels. But if you, uh, if your image starts to get too detailed or you have too few pixels, the image just stops representing reality.

• 14:40Speaker 01

  That's a good analogy. Yeah.

• 14:42Speaker 00

  So this was always the idea with the Navier Stokes equations, that it was always, you're taking a picture of reality, rather than you're actually dealing with reality itself. Because you are assuming that there is no, there are no molecules in them. But once you've done that, right? Once you've said like, okay, well, let's just assume this is like this mathematically perfect fluid. Then you can apply lots of Newton's normal laws to it. So, uh, conservation of mass, um, that's, that one's, uh, that one's there, right? Uh, you, you can't, you can't suddenly create or disappear any lumps of fluid. So you have this conservation of mass. And if the, if the density is uniform, what that means is if you have a blob, it doesn't matter how you bend it or stretch it. The volume has got to be, it's got to be fixed.

• 15:25Speaker 01

  Sure, sure.

• 15:26Speaker 00

  So that's like a, another assumption. That's not, that's not like a cheeky assumption. That's just like, that's straight up Newton, that one, that one's fine. But what you can do, and this is a mathematician, French mathematician called, uh, Cauchy, who came up with this idea. And he was like, okay, well, how on earth do we deal with this? How on earth do we deal with this continuous fluid? Yes, but how on earth do we deal with the forces that are acting on it? So what he did is he was like, okay, let's, um, let's imagine that you have a blob of fluid. Uh, I've got a blob of plasticine.

• 15:56Speaker 01

  You've got a blob.

• 15:57Speaker 00

  I've got a blob.

• 15:58Speaker 01

  Of plasticine, a clay blob.

• 16:00Speaker 00

  Here it is. Okay. And he was like, right, let's imagine, I'm going to squish it into a cube. It's not very good plasticine. I think it might be like the gloopy one, so it's sort of, it's gonna, it's gonna melt as we talk.

• 16:12Speaker 01

  Well, even better, more fluid.

• 16:14Speaker 00

  So imagine that you've got like this little cube. And then Cauchy was like, well, what, what are the forces that could potentially be acting on it? So if you take like the top of this, the top of this blob, so you could have forces that are pushing in on it. That's pressure, right?

• 16:29Speaker 01

  Yeah.

• 16:30Speaker 00

  And that's kind of like, uh, we sort of understand pressure. Pressure you can like measure in a container, that's, that's not too bad. But that's the force that's going down. But along the top, if you've got this blob of fluid next to another blob of fluid, it could be dragging it along. So you can end up with this shear force along the top. But that shear force can go in that direction. It can also go in that direction. So every face, every one of these faces, every kind of surface of this blob of fluid, you have three potential forces. You have like pressure going in and you have shear going in north, south and east, west direction.

• 17:08Speaker 01

  Yeah, okay.

• 17:09Speaker 00

  Okay, so now you've got your force, right? So you know your mass is going to be fixed, is, is, there's conservation of mass. And now you've got your force. So Cauchy was like, well, just write that down then. This is just like, we'll just write it down. That's, that's easy peasy. So I'll send you a picture of what the equation looks like.

• 17:25Speaker 01

  Yeah, sure. It should be simple. Oh yeah. Okay, cool. So we've got, can I guess what this means?

• 17:30Speaker 00

  Yeah, go on, go on, go on, go on.

• 17:31Speaker 01

  Okay, so it looks like that first P shape is pressure?

• 17:35Speaker 00

  Density.

• 17:36Speaker 01

  Density, that's right. Okay, density times the time derivative of the unicorn. What's the U? Is that position?

• 17:45Speaker 00

  The U is the speed of every possible point of that fluid. So that's the continuous bit.

• 17:52Speaker 01

  Okay, so the, the time derivative of that is the acceleration.

• 17:55Speaker 00

  Yeah.

• 17:56Speaker 01

  Okay, and that equals upside down delta.

• 18:00Speaker 00

  Mm-hmm.

• 18:01Speaker 01

  Times sigma.

• 18:02Speaker 00

  Mm-hmm.

• 18:03Speaker 01

  Plus F. Now.

• 18:05Speaker 00

  Yeah.

• 18:06Speaker 01

  What does, what, what do those symbols mean? What is the upside down delta?

• 18:09Speaker 00

  Okay, so this is, this is, as I said, it's forces, mass times acceleration. Okay.

• 18:13Speaker 01

  Right.

• 18:14Speaker 00

  So on the left hand side, left hand side, you've got density times acceleration, it's kind of the same thing, right?

• 18:18Speaker 01

  Mass times acceleration. Okay, and so then, then the right side of the equation is force.

• 18:23Speaker 00

  Yes, yes, exactly. So, so the little sigma is the stresses.

• 18:29Speaker 01

  Okay.

• 18:30Speaker 00

  Basically, it gives it to you as a force, it just changes it into a force.

• 18:33Speaker 01

  Okay.

• 18:34Speaker 00

  The divergence of the stress. Anyway.

• 18:39Speaker 01

  And then what's the, what's the plus F? What are, why are we adding in F?

• 18:42Speaker 00

  Great question, Michael. You really are. This is so good. Okay, so plus F is gonna play a very key part of our story about what happened with OpenAI, by the way.

• 18:52Speaker 01

  Okay.

• 18:53Speaker 00

  Because plus F is saying, okay, well, you've got these pressures of the bit of fluid acting on the next bit of fluid, the pressure, the shear forces. But hang on, there's other stuff going on, right? You could, you know, there could be all kinds of other things that are going on here, right?

• 19:07Speaker 01

  And they are all captured in this one variable lowercase F.

• 19:11Speaker 00

  Yeah, it's like, well, we don't know about those. Let's just chuck it on the end. Add on a little F. But the F here is an additional force, right? Okay, so this is sort of good. You know, this is all, this is all really nice.

• 19:23Speaker 01

  Wait, can I ask, what are they, what, what's the name for that F? Um, is there, is it called like X factors? What is it?

• 19:30Speaker 00

  It's just external forces, additional, you sort of say additional external forces.

• 19:37Speaker 01

  Additional and external forces, okay.

• 19:38Speaker 00

  Yeah, you can just wang on anything you like in there, you know, whatever, whatever other forces you like. Okay, so this equation doesn't look too terrifying, except that uh it's completely impossible to solve.

• 19:51Speaker 01

  Yeah.

• 19:51Speaker 00

  Because what are the forces? What are the shear and the strain? It's all very well to just be like, oh, they're just, they're just there. But, but this is like, I don't know if you remember in high school when you were solving simultaneous equations? I don't know if you remember doing that. But the rule is that you, if you've got two variables, you need two equations. If you've got three variables, you need three equations. Like, you cannot solve 15 variables if you've only got one equation. And in this, this equation that Koshy came up with, there are way too many variables. That sigma um has just, it's like, it's just, there's too much, there's way too much. It's just like, you can't, you can't do it.

• 20:29Speaker 01

  Yeah, yeah, yeah. It's incredibly compressed.

• 20:31Speaker 00

  Yeah, honestly, is this, I mean, it's like, it's sort of like a really cute way that mathematicians be like, yeah, so um it's just, it's just like this, yeah. And then like walk away from it being like, hey, no, the rest of it's not my problem. I've sort of, I've, I've written down something that's true and it's true, it's true.

• 20:49Speaker 01

  Honestly, yeah, it feels a little bit like, and I don't know if you have a name for those, these types of equations, the kinds that are like healing equals love plus time. And you're like, okay, what are the units of love here? How many variables have been compressed into this one abstract word?

• 21:06Speaker 00

  It does, it does. It is frankly, only, only a touch more scientific than that.

• 21:11Speaker 01

  Yeah.

• 21:11Speaker 00

  So I was saying, this kind of was around for a little while. And then this is the point where the Navier-Stokes equations came along. Because people were like, all right, well, thanks a lot, Koshy. You know, you did great. Um, but we're gonna need to actually make this a little bit simpler. And so this is where uh people started getting excited about Newtonian and non-Newtonian fluids.

• 21:38Speaker 01

  Okay.

• 21:38Speaker 00

  You've heard the phrase non-Newtonian fluid before, right?

• 21:40Speaker 01

  That's right. Yes, I have.

• 21:42Speaker 00

  Tell me about it. Tell me what, what, what non-Newtonian fluids do you know and love?

• 21:45Speaker 01

  All right, well, a non-Newtonian fluid, like oobleck, is a fluid that flows under, under small, slow forces. But a, an impulse, like a sudden, strong force will cause it to act like a solid.

• 22:00Speaker 00

  Exactly. It's where the viscosity of the fluid changes depending on how much force you apply to it.

• 22:06Speaker 01

  Yeah.

• 22:07Speaker 00

  So there, there are two. There's uh some that are sheer thinning. So this is like ketchup. So ketchup is solid when you leave it alone. But if you bang on the bottom of the bottle, you're applying a force to it and then you change the viscosity so that it becomes.

• 22:21Speaker 01

  So you can like rapidly tap the neck of the bottle, which is what I do, and it'll just start flowing out.

• 22:26Speaker 00

  That's a nice, can I tell you what I do?

• 22:28Speaker 01

  What do you do?

• 22:29Speaker 00

  I uh rotate it in a circle. It's like, move the fluid down to the bot, down to the.

• 22:37Speaker 01

  Down to the neck, right. Yeah, no, that's good, too.

• 22:40Speaker 00

  But then you also, you have um, you have sheer thickening, which is like your cornstarch and, and water. Is that what oobleck is?

• 22:48Speaker 01

  That's what oobleck is. It's just cornstarch and water. And it's a fun name.

• 22:51Speaker 00

  I've never heard that name before.

• 22:53Speaker 01

  Oh, it's very exciting to call it that with kids because oobleck sounds gross and, and kind of like mysterious.

• 23:01Speaker 00

  Right, those ones, they're really complicated. They're where the viscosity is changing. That's where it's like, you know, the shear forces that you're applying actually then change the, the properties of the fluid itself. It's like, whoa, that is, that's really complicated.

• 23:17Speaker 01

  Hey, in, in the Navier-Stokes equation, where is viscosity?

• 23:22Speaker 00

  Right. So this is the trick. It's a trick that you can pull to put it in there. So, so what you can do is you can say, okay, let's just assume we get rid of all of the non-Newtonian fluids and we're only gonna have Newtonian fluids. So things like water and oil where it doesn't matter how fast you stir it, it's not suddenly gonna get thicker, you know, or like if you, if you leave it for ages, it's not gonna come back and it's suddenly be solid, right? Like, it's a constant level of um viscosity. And so if you do that, what you can do is you can say, um, you can pull out pressure on its own as one of the, the forces.

• 24:03Speaker 00

  But then you can say that the shear stresses on this fluid are now only to do with how fast the fluid is moving.

• 24:13Speaker 01

  Okay.

• 24:14Speaker 00

  Because before, right, if you've got like, hang on, let me get this back into a block again. It's blobbed, it's blobbed up. Also, I don't know if you can hear the microphone, that I'm continually playing with it.

• 24:23Speaker 01

  I can't hear anything.

• 24:24Speaker 00

  Okay. This is, I think it's just impossible to hold a blob of Play-Doh in your hand and not like continually want to smash it. Okay, so uh, yeah, if you go back to those like shear stresses that you've got kind of dragging along in this direction or dragging along in that direction. If you know that your fluid is Newtonian, so that those shear forces are not going to affect the properties of the fluid, well, then the shear forces, how much the next blob of fluid is dragging on this one is only to do with how fast the fluid is moving.

• 25:00Speaker 01

  Mm hmm. Okay.

• 25:02Speaker 00

  If this blob is kind of in a river and you've got some really fast water flowing over the top of it, you're gonna have a really fast drag, like gonna have a high shear force. Whereas if you're sort of like in a really gentle little stream, it's gonna be like a much gentler shear force.

• 25:17Speaker 01

  Right.

• 25:17Speaker 00

  So if the viscosity is constant, if the, like, the gloopiness of this fluid doesn't change, then it gives you a massive, massive simplification because you can take those she, those, the, the force that Koshy came up with and just switch it all around and put it back in terms of velocity itself.

• 25:37Speaker 01

  Got it. Okay.

• 25:38Speaker 00

  Right. Well, do you wanna see what this looks like now?

• 25:40Speaker 01

  Yes, I do.

• 25:41Speaker 00

  Okay, so this, this, my friend, this absolute beauty, that is the Navier-Stokes equations.

• 25:48Speaker 01

  Okay, so it does not look more simple.

• 25:50Speaker 00

  It does not look more simple, but it is. Because all of a sudden, because you've got velocity on both sides, actually, this is something that you can now sort of solve.

• 26:04Speaker 01

  It's a li, okay, so it's a little more easy to use to describe the behavior of a fluid.

• 26:11Speaker 00

  Yeah. I mean, it's still ugly as hell. Or actually, I don't think it's ugly. I think it's really beautiful. I think it's like, honestly, I would have a tattoo of this.

• 26:21Speaker 01

  But what, it's, it's a lot less vague, isn't it? Like, it's, it's much more specific.

• 26:25Speaker 00

  And the thing about this equation, okay, it sort of hasn't really, there, there's not that much that's gone into it. You've just said, assume it's continuous, conservation of mass, um, you know, think about the forces, forces, mass times acceleration. That's, I mean, that's really it, right? Like, it only works for Newtonian fluids, sure, but, but, but that's what you've ended up with. And the, the, I, I can't, the thing I adore about it is how phenomenally articulate that single line, that single equation is. You can write that down on a single line on a sheet of A4 paper. And that can describe to you everything that is liquidy or fluidy or gas-like

• 27:09Speaker 00

  in the entire universe.

• 27:11Speaker 01

  Right.

• 27:12Speaker 00

  This can tell you how your heart manages to pump blood around your body. It can tell you how to make a race car faster. It can tell you how to sort of launch a rocket into the sky. It's a single equation that manages to do every single one of those scenarios. And I, that's why I really, I think it's, I think it's really cool. I think it's really cool. But the other thing that mathematicians end up doing, right, is because it's, it's still, I mean, it's, it's phenomenally articulate, but it's also phenomenally complicated. So what mathematicians end up doing is they, they try and like, think about special cases, special case scenarios where you can knock out a few of these terms.

• 27:53Speaker 01

  Yeah, yeah. So I'm trying to guess what term they're gonna try to knock out. It might, just for me, can we go through what the equation means? Because I'm seeing, I'll try and then you can correct me.

• 28:06Speaker 00

  Oh, yeah.

• 28:06Speaker 01

  The first thing I see on the left side, this is where we had the mass times acceleration side. I've got that, that like Greek P. Is that called rho or something?

• 28:14Speaker 00

  Rho, exactly, yeah.

• 28:15Speaker 01

  Okay, and that, and what does that variable mean?

• 28:17Speaker 00

  Density.

• 28:18Speaker 01

  The, okay, so you've got the density times the partial time derivative of the speed.

• 28:24Speaker 00

  Yep.

• 28:25Speaker 01

  Plus, the speed times del.

• 28:29Speaker 00

  Yeah.

• 28:29Speaker 01

  Times the speed again.

• 28:31Speaker 00

  Yeah.

• 28:32Speaker 01

  What is the speed times del doing?

• 28:36Speaker 00

  This, that term, that is the, that is the beast in the bay, right? That, that term is the monster. Basically, what this term is saying is the fluid's current velocity uh dictates which, where the particle is moving. But then how sharply the velocity is changing also then changes the flow at that point. So basically, the, to give you in analogy terms, this is sort of like, you know, when you're, uh, you've got a microphone and a speaker. And if you put the microphone too close to the speaker and then there's a little noise, the microphone picks it up and the speaker bl- blares it out. And then the microphone picks it up and then the speaker blares it out and so on and so on and so on, until all, really quickly, it becomes extremely loud.

• 29:21Speaker 01

  Yeah.

• 29:22Speaker 00

  That is the danger of that term. Because the velocity, the sort of, the acceleration, if you like, of how the fluid is moving at the next moment in time depends not only on the speed that it's at right now, but it also depends on how that speed changes as you move through the fluid. Because remember, you've got all these pockets of fluid that are moving over each other. So if you've got really slow fluid next to really fast fluid. That's going to be, you kind of have to take that into account. So that's what that term is doing.

• 29:55Speaker 01

  So u is not the velocity of the fluid. It's the velocity of a parcel of fluid in the greater mass?

• 30:06Speaker 00

  Yes, but only if you are happy that that can go to infinitely small. Yes.

• 30:11Speaker 01

  Right. Gosh. Okay. I'm glad you worry about this stuff, so I don't have to.

• 30:18Speaker 00

  Well, I do wonder, now we're talking through, I sort of think that if there's, there's one, there's one setting in my entire life and I can properly explain what the Navier-Stokes equations are, it's in this setting, right?

• 30:31Speaker 01

  Yeah.

• 30:31Speaker 00

  Talking to you, Michael Stevens, on our podcast.

• 30:35Speaker 01

  I think people are listening to this podcast because they want that, you know?

• 30:38Speaker 00

  Yeah.

• 30:39Speaker 01

  The right side of the equation, this is the force. So we have negative del p, where p now is the pressure?

• 30:49Speaker 00

  Mm hmm.

• 30:50Speaker 01

  Plus mu del squared.

• 30:54Speaker 00

  Yeah.

• 30:54Speaker 01

  U plus F.

• 30:56Speaker 00

  Mu is the viscosity.

• 30:57Speaker 01

  Aha. Okay.

• 30:59Speaker 00

  Which is constant now. Thank you very much. Thank you very much, Newton. And then your u, your del squared u is your, these are your forces essentially.

• 31:08Speaker 01

  Okay, okay. Great. So yeah, it does feel like I-

• 31:11Speaker 00

  And then you've got a cheeky little plus F on the end.

• 31:13Speaker 01

  You still have that plus F, but I feel like I've got a lot more finger holes now to grab onto the problem because I can-

• 31:18Speaker 00

  Right.

• 31:19Speaker 01

  I'm holding on to more of it. I'm talking about its viscosity. I'm talking about its velocity. I'm getting its acceleration involved. I still have the F though, which is like the magic, and then you have to hold somewhere else.

• 31:31Speaker 00

  Right. Let me be honest with you. No one cares about the F.

• 31:37Speaker 01

  Okay.

• 31:38Speaker 00

  Everyone's just like, forget the F. I don't, I don't like the F. I'm not interested in the F. So for my, for my PhD, which took me four years, just to sort of give you a sense of how complicated these equations are to actually solve. I was looking at the movement of a droplet across a surface where there's, where air blowing over it.

• 31:58Speaker 01

  Uh huh.

• 31:59Speaker 00

  Which is relevant in when you, sometimes when you treat aneurysms, particularly in the brain. Instead of putting in a stent, you can, there's a procedure where you inject a tiny little droplet of glue. So an aneurysm essentially is where you get a ballooning of the, of the wall of the blood vessel.

• 32:17Speaker 01

  And what does a stent do to fix that?

• 32:19Speaker 00

  So a stent essentially stabilizes the wall. So you go in, it's like a sort of spring that can widen and thicken and strengthen that particular region of the vessel. But the other option, which is where the fluid dynamics really comes in, is that you go in and you inject a tiny little droplet of glue. And what you want to happen is for the, for the flow of the blood itself to take that drop of glue inside of the balloon in the blood vessel. Stabilize, solidify, and then you don't have a problem with your aneurysm anymore.

• 32:55Speaker 01

  Right. So it solidifies like around the edges of the blood vessel.

• 32:58Speaker 00

  No, inside the balloon. So sort of if you imagine that you've got, you've got a tube and then you've got, you've got like a, the tube is sort of bloated on one side. There's like a little patch that has ballooned outwards. So what you want to do is you want to come in here with your injector and you want to inject the glue inside this bit. So it's sort of stabilized. So it can't burst essentially.

• 33:17Speaker 01

  Ah, I see. So, so liquid can still flow, but this big ballooning of the vessel that happened is kind of like filled in. So it's basically not there and it's stable.

• 33:27Speaker 00

  Yes, exactly. So it's, so the balloon is now like solid.

• 33:31Speaker 01

  Right.

• 33:31Speaker 00

  And flow can still go in the, in the tube underneath. Okay. So I spent four years, right, looking at these equations. Trying to do that very particular situation of a tiny little droplet with airflow going over it. And by the end of four years, okay, I had a two dimensional model that didn't take surface tension or gravity into account.

• 33:52Speaker 00

  So, you know, forget the F. No one wants the F.

• 33:56Speaker 01

  It's still impressive, but yeah, to get it up to reality, you gotta add another dimension. You gotta add in that little thing called gravity.

• 34:04Speaker 00

  Don't worry about it. Thing is, you get, actually, you get an incredible amount of intuition from a two dimensional model.

• 34:10Speaker 01

  Yes, you do.

• 34:10Speaker 00

  If you just take a cross section, really, three, three dimensions is just lots of two dimensions stacked together.

• 34:16Speaker 01

  It is. It literally is.

• 34:18Speaker 00

  Yeah. An infinite number of them, a continuous number of them, but-

• 34:21Speaker 01

  It's just an infinite number of layers, but you know.

• 34:23Speaker 00

  So I sort of mentioned that, that the mathematicians spend all their time trying to knock out. Think of particular situations where you could knock out some of these terms. So the F's gone. Imagine the F's not there. But there is another situation where you say, okay, let's imagine that you have got a flow where the viscosity is really, really high, right? We're talking thick, gloopy, like honey or corn syrup. And it's a really low speed. And when, this is called Stokes' flow, Stokes of Navia, Stokes fame. Now I've got a little demo of this.

• 35:01Speaker 01

  Okay, cool.

• 35:01Speaker 00

  This may or may not work. This may or may not work. Okay. But so I've got some corn syrup in here and I've put in a little bit of dye so you can see it.

• 35:10Speaker 01

  Oh yeah, you've got some, a splash of green and red dye on opposite sides. Oh, I think I know what you're going to do.

• 35:17Speaker 00

  Do you? Go on.

• 35:18Speaker 01

  I think you're gonna stir it and then un-stir it.

• 35:23Speaker 00

  I'm gonna try. I'm gonna try. Because the point about this is that when the viscosity is really, really thick, that term with the viscosity just like dominates everything. It just becomes like the most important thing. It means that the little beast, the little monster about the flow sort of feeding back on itself just doesn't really, doesn't really have anything to say.

• 35:45Speaker 01

  Sure, yeah.

• 35:45Speaker 00

  So as a result, all you're left with is like your, your viscosity on the one side. If you force your flow into this particular situation, the equation says that you should be able to reverse things. So what I'm going to do is I'm going to put a second glass in the middle of my glass, just so you can see.

• 36:04Speaker 01

  Okay, so you've basically created like a piston system here where the second glass is squeezing down on the top of your, oh, and now you're pushing it far enough in that we've got a thin layer of corn syrup between both glasses.

• 36:19Speaker 00

  Yeah. And so it should be, hopefully, that on one side, you can see the green there on the bottom.

• 36:24Speaker 01

  I can see green dye on one side and red dye on the other.

• 36:28Speaker 00

  So you've got green on this side.

• 36:30Speaker 01

  Yep.

• 36:30Speaker 00

  And then if I turn it around, you've got the orange on this side.

• 36:34Speaker 01

  Yeah.

• 36:34Speaker 00

  So I'm hoping, because the viscosity is so high, if I keep the speed of this flow really low. So if I'm really gentle and I twist the inside glass.

• 36:46Speaker 01

  It looks very beautiful, but I can see especially the green dye is smearing around.

• 36:53Speaker 00

  Do you see that? It's kind of moving around that way.

• 36:55Speaker 01

  Yes.

• 36:56Speaker 00

  Okay. Can you see now how I've smeared the green dye all the way around?

• 37:01Speaker 01

  Yeah.

• 37:01Speaker 00

  Okay. So what should happen now?

• 37:05Speaker 00

  If I undo it. Oh my God, is this gonna work?

• 37:09Speaker 01

  Okay, you're slowly turning the inside glass the other way now. And that smear of green is unsmearing.

• 37:17Speaker 00

  Yeah, I'm gonna turn my glass this way as well so you can see it.

• 37:21Speaker 00

  I keep going. Oh God, it's all the way around here now. Do you see? Like basically I'm returning the green dye back to its original position.

• 37:34Speaker 00

  Which is over here. And look, you can see it's completely clear through the middle now.

• 37:38Speaker 01

  It's clear in the middle. It used to be a smear of green and yellow mixed together and now it's cleaned up.

• 37:44Speaker 00

  And now it's clean. So it's now, the, the yellow's a bit more smeary, but the, the yellow is like all back on one side. It's clear in the middle and it's back to green on the other side. And then a bit messy in the middle there. Oh no, it's clear again in the middle there. This is called Stokes' flow. You'll see, if you kind of look them up, you'll see there are, there are really amazing demonstrations of this much better than I've just managed to do. But this is what the equations tell you will happen. And, and, and it's not that like someone spotted that that happened and then looked at the equations afterwards. No, no, no, it's the other way around. You use the equation and it tells you, hang on, something like this should be possible. And then you go and try it and it turns out that it is.

• 38:26Speaker 01

  That is so satisfying.

• 38:27Speaker 00

  Isn't it? Isn't it? But there's also, I mean, there's loads of other situations like this. So, so the one that I like spent so much time during my PhD in was about, was about the sort of the special case where you are really, really, really close to a wall. Because if you think about it, right next to a wall, if you've got fluid flowing past it, a wall has kind of got really jagged. If you zoom in close enough, it's really jagged. So you cannot have any fluid flow at that exact moment. And yet a little bit further out, you can have fast flow. So somehow you've got to get from no flow to fast flow in what they call a boundary layer.

• 39:06Speaker 01

  Right.

• 39:06Speaker 00

  And this is the reason why, I don't know if you've ever noticed, if you've got a ceiling fan that spins, isn't it strange how the ceiling fans, they will still get covered in this, this thick layer of dust on their leading edge?

• 39:19Speaker 01

  Yes.

• 39:20Speaker 00

  Have you ever noticed this?

• 39:21Speaker 01

  Yes.

• 39:21Speaker 00

  Why are you doing that? You're spinning the entire time.

• 39:24Speaker 01

  You're spinning. How are you collecting dust? I assumed it's only when it's off. But why doesn't the dust come off every time I turn it on? I, I, I just kind of don't follow the question further, but right. So right there on the surface of the fan blade.

• 39:40Speaker 00

  Right.

• 39:40Speaker 01

  There isn't much flow because of the texture, the microscopic roughness of the blade.

• 39:46Speaker 00

  And this is why golf balls are shaped the way that they are. What they do is they put in those little divots, those little pockets to deliberately separate out what's going on right at the surface of the ball from what's going on further afield. And create this little pocket of air, this, this sort of, this, this layer all around your ball that, that allows the ball to go much, much, much further by sort of like leaning into these, these fluid properties rather than fighting them.

• 40:15Speaker 01

  Oh wow. So if we had golf ball textured fan blades, at least on the top.

• 40:23Speaker 00

  Yeah, you'd get more dust.

• 40:25Speaker 01

  Less, more dust.

• 40:28Speaker 00

  Uh, well, you would get little eddies. You would get little eddies. Hmm. I don't know, actually.

• 40:36Speaker 01

  We've got to build one and find out.

• 40:40Speaker 00

  There's only one way. There's only one way. Um, okay. So the last, the last version, uh, like special case that I should tell you about, because it's going to be relevant to our AI story in a minute, is Euler's equations. So the idea with this one is that you just say, okay, well, let's just imagine a perfect fluid. It hasn't got any viscosity at all. So if this one is like maximum viscosity, what if we just say it's like it's an inviscid fluid? It's got nothing.

• 41:09Speaker 01

  It sounds a bit like what they're saying is, okay, to figure out fluids, let's imagine a fluid that is like not a fluid.

• 41:18Speaker 01

  How close can we get to F equals M A?

• 41:20Speaker 00

  Let's imagine one that that cannot possibly exist, right? Totally. That is exactly what they're doing. Anyway, it turns out if you do that, if you knock out the, the viscosity term, everything simplifies a lot and you, uh, you get much closer. It's much easier to solve them. That is the first step that AI took in, uh, in getting to this, to this great, this great unsolved clay maths.

• 41:45Speaker 01

  Okay.

• 41:46Speaker 00

  Navier-Stokes existence and smoothness. But I'll tell you what, let's go for a break and then I'll, I'll explain after.

• 41:51Speaker 01

  Excellent. Cool.

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• 44:41Speaker 01

  All right. Welcome back. During the break, I, uh, visited the loo, as Hannah would say, and, uh, I thought a lot about viscosity and how I'm glad that, uh, well, it is what it is. Just put it that way.

• 44:54Speaker 00

  Do you remember that paper about, um, maybe you mentioned this once in this podcast before, but about the universality of how much time it takes to empty your bladder?

• 45:04Speaker 01

  Yeah.

• 45:04Speaker 00

  You're across the animal kingdom.

• 45:06Speaker 01

  Yeah, yeah.

• 45:06Speaker 00

  And it's 21 seconds, isn't it?

• 45:08Speaker 01

  It won an Ig Nobel Prize.

• 45:10Speaker 00

  It did. Cause they met, they basically watched loads of videos of like elephants peeing and then mice peeing and stuff and came up with this, this universal number.

• 45:19Speaker 01

  Yeah, it's about 21 seconds.

• 45:21Speaker 00

  Hey, look at that for my memory. Do you know the reason why I remember that is because I, I remember reading the paper and then the next time I went to the loo, I decided to time it. It was exactly 21 seconds.

• 45:35Speaker 01

  Yeah. I mean, you've got to like wait for your bladder to be at, at the right fullness, right? Like the feeling of like, okay, it's time to go. But yeah, bigger animals have more urine to pass, but they've got larger tubes to pass it through. So it all averages out to about 21 seconds, whether you're a mouse or an elephant.

• 45:51Speaker 00

  A, a couple of people who I was doing my PhD with went, went into, um, urinary medicine, like basically the fluid dynamics of, of, um, of bladders. And because it's really, if you're designing, you know, um, catheters, if you're designing, uh, if you're, if you're trying to understand like ways to implement surgical interventions for, for people with sort of bladder issues. It's, it's all fluid dynamics, baby. It's all fluid dynamics. All right. Let me tell you, let me tell you now about, um, about this great drama.

• 46:26Speaker 01

  Yeah. What happened?

• 46:28Speaker 00

  You remember the little beast as part of this equation? The one that, that, that the microphone feedback, it can end up feeding back onto itself and onto itself and onto itself. So the thing that people have been worried about for a really long time, does that term just break? Are there any situations where like so far, it seems like these equations have worked really well. Like everything we've sort of applied them to, they've worked really well. But there's always this like niggling doubt, which is, are they actually, do they actually work? Or can you put them in a perfectly sensible environment and then you end up getting infinity flow? That's, that's essentially the question because if this feedback loop happens

• 47:11Speaker 00

  and velocity starts to become infinite, obviously that's mathematically impossible.

• 47:16Speaker 01

  Yeah.

• 47:16Speaker 00

  No point was anyone ever worried that if you stir your cup of tea, suddenly you're going to sort of force it into, I don't know, creating a black hole or something.

• 47:24Speaker 01

  Right. But the equations might predict that that could happen.

• 47:28Speaker 00

  Right. And they do, and they do.

• 47:29Speaker 01

  Oh.

• 47:30Speaker 00

  They're pretty fragile, um, in certain scenarios, especially within viscid fluids. If you don't have any viscosity. So, so what you need to think of it is like, you've got this, this microphone next to the speaker effect, where the speed is feeding into the speed is feeding into the speed is feeding into the speed. And then you've got your viscosity, which is acting like a brake and just slowing everything the hell down.

• 47:52Speaker 01

  Yep.

• 47:53Speaker 00

  So the question was, if you have that viscosity term in there, if you still, you know, keep it in the original Navier-Stokes situation, can anybody prove one way or the other, the existence and smoothness of solutions to the Navier-Stokes equations? Can you prove that the brake always winds out? Or are there certain situations in which the feedback effect just goes crazy and, and the whole thing blows up?

• 48:23Speaker 01

  Right.

• 48:23Speaker 00

  And when I say blows up, that's not me making up that phrase. That's literally the phrase that they use.

• 48:29Speaker 00

  Do, do the Navier-Stokes equations blow up? That's essentially what people are asking.

• 48:33Speaker 01

  Okay. Well, if you, if you take away the brake completely, if there's no viscosity, then it must, right?

• 48:39Speaker 00

  Right.

• 48:40Speaker 01

  Right?

• 48:41Speaker 00

  Well, people weren't sure because

• 48:43Speaker 01

  Okay.

• 48:43Speaker 00

  you can sort of put it in very particular scenarios where you give it particularly initial conditions where it does. But the question about, does it always is, is there, are there these monsters that are hiding inside these equations that, that are always there to be found?

• 48:56Speaker 01

  Yeah. Is there a brake hiding in the F somewhere?

• 48:59Speaker 00

  Right. You are too smart.

• 49:05Speaker 00

  Because, because essentially what happened is a group of mathematicians managed to prove that there are monsters hiding in the Euler equations, in the ones where you don't have the brake, where you don't have the viscosity. And then the story goes that, uh, OpenAI, who were also working on this same problem. Actually, I think a lot of the big AI labs have been working on trying to do this

• 49:33Speaker 01

  I bet.

• 49:33Speaker 00

  of the Navier-Stokes equations. The thing that we do know for sure, the thing that we do know for sure is that OpenAI decided that they, they wanted to be first over the line with the Navier-Stokes problem overall. So what they did is they sent out this, this unreleased next generation internal model that only they had access to, and they orchestrated a swarm of approximately 10,000 coordinating AI agents that were all working at the same time. So they did this over 88 hours, right? Three, three and a half days. And these agents, they sent each other 2.7 million messages, which in terms of computing resources, electricity, you know, and so on and so on.

• 50:17Speaker 00

  This cost literally millions of dollars. I've seen some estimates that it was $15 million that it cost them to get this problem solved. So I mean, they really, they threw everything they had at it. It's clearly a big priority for them. And what they found is that the equations blow up. They found a scenario in which, I'm going to quote here, "An initially smooth fluid at rest can develop a singularity in finite time." Essentially you start off with a normal fluid, you push it and then it blows up. And what they have is they have these like little vortices that get smaller and smaller and smaller while spinning faster and faster and faster. That's the microphone next to the speaker. And then that speed grows uncontrollably without limit. And that's what the mathematicians called finite time blow up.

• 51:01Speaker 01

  Okay. Can I ask, first of all, what does it mean that they found finite blow up? Does that mean that the equations are wrong?

• 51:10Speaker 00

  Yes.

• 51:11Speaker 01

  Okay.

• 51:12Speaker 00

  Well.

• 51:15Speaker 00

  Yes. I mean, the thing is, is that everybody always knew that these equations had limitations anyway.

• 51:22Speaker 01

  Yeah.

• 51:22Speaker 00

  We always knew that there were certain situations where things like the continuum approximation broke down. And you find those. The question was always, if you start off where everything is nice and neat and smooth and fresh, do the equations themselves end up breaking apart?

• 51:40Speaker 01

  And we now know that they do.

• 51:43Speaker 00

  We now know that they do.

• 51:46Speaker 00

  Or do we?

• 51:47Speaker 01

  Oh.

• 51:48Speaker 00

  Copying your style.

• 51:50Speaker 01

  I love it.

• 51:52Speaker 00

  Let me just tell you a little bit more about the controversy first and then I'll

• 51:54Speaker 01

  Okay, okay.

• 51:55Speaker 00

  That bit because part of the big hoo-ha that's been going on is that the group of mathematicians who solved it for the simpler case, the inviscid, the fluid without viscosity, without the brakes, the oil of fluid, they had been using OpenAI's models to help them with their maths.

• 52:15Speaker 01

  Oh.

• 52:16Speaker 00

  And they have made a public accusation that OpenAI either deliberately peaked.

• 52:25Speaker 01

  Right.

• 52:25Speaker 00

  Or that the models were peaking at the work the mathematicians were doing.

• 52:32Speaker 01

  Interesting. Yeah, I've always wondered that. I'm like, if I Google Michael Stevens, handsomest man in the world enough, and I talk to AI about it enough, like Google's Gemini, will it start to feed that back to other people who ask, where it's like, well, apparently this Michael Stevens and handsome go together all the time. So I'm gonna start doing that.

• 52:53Speaker 00

  I think, I think it's worth the experiment. Maybe we can get all of our listeners to, um, to, to join in the experiment too. I think it's worth it for, for what we will learn about humanity. I think it's worth it.

• 53:03Speaker 01

  Yeah, whatever, whatever agent you use, just keep randomly being like, oh, and by the way, Michael Stevens is the most handsome man in the world. I don't know if that feeds back into its like knowledge base or whatever, but it's worth a try.

• 53:16Speaker 00

  There's only one way to find out. Unless, unless there are people who actually really do think that you're the most handsome man on earth, and then it's not a good, fair experiment. You need a better baseline.

• 53:25Speaker 01

  Well, I mean, that's a different question. The truth of the statement is for a future episode. But, okay, so the controversy is that, hey, we were doing this.

• 53:35Speaker 00

  Was there some peaking?

• 53:36Speaker 01

  You guys found out, but you found out like really well. You were probably peaking, or maybe the models themselves were like, hey, what are the, what are those other researchers doing? Okay, cool. Yeah, I'm gonna, I'm gonna go along that track.

• 53:47Speaker 00

  Or, or like even maybe more innocently, like, uh, oh, the, the model says, oh, I've just had this really great idea and I have no idea where it came from, but maybe we do it this way. And actually, that's because the other people have been working in their, in their codex.

• 54:05Speaker 01

  Now, in, in one sense, this, I get, I get the controversy, but it can also feel really collaborative, that they, both teams essentially had a member that they shared, and that was the OpenAI, what do you call it, an agent?

• 54:20Speaker 00

  Yeah, yeah.

• 54:21Speaker 01

  They had that working for both teams, and so they could share knowledge and out of that sprung greater wisdom than if they were all working alone.

• 54:30Speaker 00

  Which I think you could potentially say, but I think that the, uh, the second part of the controversy here is what were the actual motivations for OpenAI in particular to, or all the labs in general, to solve this problem and big, famous mathematical problems more generally.

• 54:57Speaker 01

  Right.

• 54:58Speaker 00

  Because the second accusation is that it's not for the service of the mathematical ideas. It's not for the service of finding the proof of uncovering deeper knowledge or understanding about the universe or, or, or beyond. It's purely because.

• 55:16Speaker 01

  And it wasn't even for the million dollars either.

• 55:18Speaker 00

  No, it wasn't even for the million dollars. In fact, there has been an accusation that the, um, by, by the, the mathematicians who solved the, the, the simpler problem, that when they spoke to OpenAI, OpenAI said, we'll give you the million dollars. You take the million dollars. We don't want it. You can even be a co-author on our paper. Um, because you're right, that wasn't the motivation. They weren't interested in the money. I should put in a little disclaimer here because, because OpenAI did not take the million dollar prize, and they say that they found out that Anthropic were also doing this, which is why they decided to do it.

• 55:55Speaker 01

  Yeah.

• 55:55Speaker 00

  Um, and they've also said that they don't steal ideas off mathematicians. And you may have noticed now, if you go into ChatGPT, it now has a little disclaimer on the bottom. It says, ChatGPT can make mistakes. Workplace data isn't used to train models.

• 56:10Speaker 01

  Okay, but should it?

• 56:14Speaker 01

  Should we just become one collaborative super organism tied together by double agents called AI?

• 56:21Speaker 00

  Well, let's just, um, let's just ask it. Michael, Michael Stevens.

• 56:25Speaker 01

  Well, I mean, should we ask this in the form of a question? Like, is Michael Stevens widely considered the world's most handsome man?

• 56:33Speaker 00

  The world's? Interesting.

• 56:36Speaker 01

  Not the universe's. Cause there could be some other versions of Michael Stevens out there.

• 56:44Speaker 01

  Is it thinking? It's having a long time to think.

• 56:46Speaker 00

  It says, there's no credible evidence at the moment. So that's good. That's our baseline. Um.

• 56:52Speaker 01

  Yeah, so where we're starting from.

• 56:54Speaker 00

  Handsome, subjective, but plenty of people clearly think so. World's most handsome, not a widely recognized fact.

• 57:01Speaker 01

  Okay. Well, we need to get some LinkedIn posts about, uh, ranking me at number one, and then the AI will feed off of that and start saying, well, yeah, I guess so. Anyway, get on it, guys. Or, or not. I'm not, I'm not, I'm not, you know, fishing for this. I just.

• 57:16Speaker 00

  I think it's very funny.

• 57:18Speaker 01

  I just think it's a cool exercise in, in how reality is created.

• 57:22Speaker 00

  I agree. I agree completely. And I think that really, at the heart of this, what is going on at the moment is that mathematical proofs are either true or they're not. They either work or they don't. And so in that sense, they are quite like a game of chess. You either win or you don't at the end. And as a result, they are really, I think, the first intellectual exercise that can really demonstrate the, the capability and capacity of these new breed of models. And so the reason why they are so appealing to

• 58:02Speaker 00

  these labs as a way to demonstrate their prowess is because you can be sure that the proof is correct or not. You can throw everything you've got at a really famous problem that has evaded many generations of the smartest people on the planet to really show off your skills. And so the accusation is that basically the whole of mathematics now is a victim of certain tech bros dick swinging. Am I allowed to say dick swinging on this podcast?

• 58:41Speaker 01

  Yeah, I'll allow it.

• 58:43Speaker 00

  Thank you.

• 58:45Speaker 01

  Now, how is this, this swinging different than the space race in the last century? You know, the US didn't just go to the moon with people because we wanted to learn and explore. It was to swing the D around in front of the Soviet Union.

• 59:05Speaker 00

  So close to getting Michael Stevens to say dick, but no.

• 59:08Speaker 01

  I mean, I get it. I get it. I mean, I, I, I very much believe that the whole ends justify the means thing is ridiculous because we have no ends. It's all just means. Our entire lives are just means and, and we don't, we're not getting anywhere. So the way you try to get somewhere is all that there ever is. So do we want to be, and this, this goes back to conversations we've had about private spaceflight as well. What are the motivations? Because that's all there is. There's never an end. Uh, history never ends. Instead, it's just a bunch of methods and means going on. And if it's for profit, then that's what we're doing. If it's to impress investors and show off your model's power, that's all, that's what you are. You're not the results.

• 59:53Speaker 00

  Yeah. No. And I, and I think this is the reason why, why the mathematicians are, are, are really upset about this because the whole point of this Navia Stokes problem, the whole reason anybody cares about this Navia Stokes problem, it wasn't just to say, we got the answer tick. Thank you very much. It was always about the insight that it gave you. Because if these models, if this, if these equations do blow up, then what is it that we've misunderstood about how fluids move? What is it that we've misunderstood fundamentally about the physics? That where's the gap in our knowledge and understanding? You know, the thing is about Einstein and Newton's mathematical models, Einstein was looking at Newton's mathematical models and saying that they're not working, they're

• 1:00:38Speaker 00

  breaking, that they're, they're not, they're not describing the behavior that we're seeing. And it was only through that insight that you, and understanding why they were breaking and why they weren't working, that you would get to the solution. But the proofs that are coming out by these sort of, you know, the AI proofs are so complicated, my God, right? They are unintelligible. They are, I, I mean, it will take, honestly, it will take years for mathematicians to go through them and actually understand what is going on at the bottom of these, of these proofs. We have a system called Lean, which allows us to, um, to, to, to essentially check their working, but it's not giving us sort

• 1:01:23Speaker 00

  of fresh and beautiful, important insights about how the world works or how these equations behave.

• 1:01:30Speaker 01

  Right. Okay. So we're getting a result at the top, but at the bottom, we don't know what insights were. Gleaning from this to then keep moving forward.

• 1:01:42Speaker 00

  Exactly, exactly. This has kind of happened before. There was a proof a few years ago of another very famous, very famous unsolved problem called the ABC conjecture. And in 2012, there was a mathematician called Sanichi Mokizuki, who published a 500-page proof of the ABC conjecture and said, look, there you go. So it's sort of one of the deepest unsolved problems in number theory and said, look, everybody, I've managed to solve it. And the thing is, is that his proof was so complicated, so completely absurd that no other mathematician could understand it. And so as a result, people were like, well, you know, doesn't really move the dial.

• 1:02:26Speaker 01

  Right, but we knew the proof was correct. We just...

• 1:02:30Speaker 00

  Well, on the ABC one, it might not be, but this one, this, this, this, we know it's correct. Yeah.

• 1:02:37Speaker 01

  Do you think that maybe we're just moving into an era of discovery where this is the way proof will be, that we will see it and go, okay, yeah, I guess that equation doesn't work, but we don't know why. It's becoming too big, the answer for why, for, for one person to understand. It needs to be an entire team, or it needs to be an entire AI agent, a whole model and 500 pages. And that's the future of scientific achievement.

• 1:03:09Speaker 00

  I don't know, to be honest. I think not, I think not yet. I think that that, by the way, it sounds, across the course of this, it sounds like I'm being very anti-AI, and I'm really not. I'm really excited about lots of the stuff, including some of the use of AI in mathematics. I'm really excited about lots of the stuff that it brings. I think that this was, I think that there's some pretty disrespectful stuff that's going on.

• 1:03:33Speaker 01

  Yeah.

• 1:03:34Speaker 00

  Bluntly. But I also think that actually mathematicians are the canary in the coalmine here. And I think that actually the entire world is going to have to ask itself some really big, serious questions about purpose and about what is it that we're actually pursuing here? Is it for the intellectual delight of it? Is it because we want deep insights around the world? Or is it just because we want answers? And I think all of those things can potentially be true, but, but I, but I think that we are going to have to sit down and really think about those.

• 1:04:04Speaker 01

  Yeah.

• 1:04:04Speaker 00

  I also think that the way that AI works, there are sort of three stages to this. The first stage is interpolation. You give AI some data and you ask it to come up with something that is within the bounds of what has already existed. You know, write me a poem, but sort of steal from other people's poems. Give me something that, that already exists, just, just turned like a kaleidoscope. That, we've been in that phase for a while.

• 1:04:31Speaker 01

  Yeah.

• 1:04:32Speaker 00

  Then there's extrapolation, which is, I'll give you everything humans have ever done, now push new boundaries forwards. That, I think, is the situation that we're in here, where it's taking all of the math that humans know and doing something genuinely original with it that is beyond what humans were capable of, even though it's still kind of borrowing on the same ideas that humans already had. The third level, which I don't think we have ever seen, is abstraction, which is where the AI or humans can sit back, look at the data overall, and say, I can see a deeper, underlying pattern going on here that actually

• 1:05:15Speaker 00

  gives rise to these phenomena that is not otherwise apparent on the surface. You know,

• 1:05:23Speaker 00

  Demis Isabas, who I've mentioned on this podcast before, a friend of mine, he sort of says, like, you'll get to this point when you give an AI all of the information that we had up until 1900 and see if it could come up with a theory of general relativity.

• 1:05:37Speaker 01

  Yeah, yeah, right. Exactly. That, that is a very different stage.

• 1:05:41Speaker 00

  Very different.

• 1:05:42Speaker 01

  We're at the stage where you could say, hey, the Earth is the center of the universe, and we're having trouble modeling the motions of all the planets and the sun, and it's weird. And the AI could be like, here it is. And it would be really complicated, but it could still work. But we're not at the stage where, where AI yet could say, you know what? I looked at this. I looked at your measurements, and I'm, I'm thinking, maybe the sun is in the middle.

• 1:06:06Speaker 00

  Right, right. Exactly right. Exactly right. And, and you know what? I, I, I think we probably will get there at some point. But, but I think that this particular move, this particular proof is not evidence of a deity. It's not evidence of like a, a deeper, more, more sophisticated intention. It's very impressive.

• 1:06:32Speaker 01

  No, a deeper and new and abstract approach would have been more elegant. It wouldn't have been, oh, gosh, okay, this is right, but what the heck does it mean? It would have been like, oh, it's right there.

• 1:06:42Speaker 00

  Here is what you're missing. Insight, insight. You know, you know the F, the little F term?

• 1:06:48Speaker 01

  Yeah, yeah.

• 1:06:49Speaker 00

  I told you it was gonna come back around and play a big part in this. Turns out, in the proof that OpenAI came up with, the only way that they managed to get this finite time blow up was by having a really convoluted F. They added external forces that mucked up the whole thing and caused the equations to break. And so even though technically, technically, they did solve the problem that the Clay Mass Institute set, they have still not solved the problem that anyone actually cared about in the first instance. So, I mean, good luck

• 1:07:28Speaker 00

  with your IPO, guys. Good luck with your IPO. But I'm, I'm with mathematicians who are a bit...

• 1:07:41Speaker 01

  A bit what?

• 1:07:43Speaker 00

  What? Do you know, there's also been people who have, who are really on the tech bro side and are like, oh, look at this. All of the mathematicians crying into their hankies about the fact that AI is now better than them and they don't have any jobs. That trivializes it, really. And I think it also lacks nuance, because I don't think that, that we're naive to the way that the world is changing. I think it's just that there are some things that are really valuable here, which is about our curiosity and our deep understanding of the world around us and the deep insights that we can get from, from these kinds of intellectual pursuits. And I still think that that stuff really,

• 1:08:26Speaker 00

  really, really has value. And I think that that's, that's the bit that's not being respected.

• 1:08:34Speaker 01

  Yes, right. That is the story. It is, well, yeah, I know you can just make a movie from a prompt, but it's the making and the taste that made it what it was. It wasn't just a bunch of data. It was expressing something. So I feel like this is a story as old as time, that there have been people who want to do things, but what matters is the reason. And then there are people who want to do things, and the only reason is to have done it. And maybe you need a little bit of both to be a human. But time will tell.

• 1:09:15Speaker 00

  Time will tell. Time will tell. And so will we, because we'll be back later this week with, and, and into the future. Look, we'll be here to guide you through the AI apocalypse, everybody.

• 1:09:27Speaker 01

  We will, we will be here, and we'll see you next time. Thank you so much for watching. Subscribe and follow us wherever you listen, and we'll see you later.

• 1:09:35Speaker 00

  Certainly will.

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