Mathematicians Are The Canary Down The AI Coal Mine

0:00Welcome to the Rest of Science. I'm Heather Fry. And I'm Michael Stevens. I'm really excited. Why? What are you excited about? We're going to do, we're going to do fluid dynamics, but properly, properly for real. Because I 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. Believe it or not, I have not heard about this news. Have you not? No, no. I've had people stopping me have not heard about this news. Have you not? No, no. I've had people stopping me in the street about this news. Shoot. I've just been down the 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.

0:39Well, you do. Okay. So I'll tell you the news first, for those of you who 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 Maths 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. It was a swarm of AI agents. And believe me, that has ruffled some feathers. 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,

1:38but you know. That's the only reason why you're not sitting on the million bucks. I think that's- Exactly. Exactly. That's the only reason. I hope you can tell us about the problem. Oh, you betcha. And then I'm also fascinated by AI solved a problem because 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. 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 longstanding math problem. They solved it. There's a lot of like, 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.

2:23Hey, 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. 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. This episode is brought to you by Cancer Research UK. All right, here's something really cool. The earliest written description of cancer that we've ever found is from an ancient Egyptian papyrus that was around before even Cleopatra was on the throne. It described tumors being removed by a tool that they called the fire drill,

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4:01Hi, this is Garyllinica from Goal Hangers. The rest is football. This episode is brought to you by Wise. It's only when you start moving money between currencies that you really think about the exchange rate, the fee, and what might be hidden away in the small print. Whether you're living abroad, paying someone overseas, or just trying to manage your money across borders, you want a fair exchange rate, an easy transfer, and no surprises along the way. Wise keeps things simple. Wise is a smart way to move the currencies you need around the globe. It works in more than 160 countries and with over 40 currencies. Most transfers arrive instantly. Wise uses the mid-market exchange rate, like the one you see on Google, with no markups or hidden fees.

4:46So when money needs to move, you can see the rate, know the fee, and get on with it. Join millions saving billions on hidden fees by downloading the Wise app today. Be smart. Get wise. T's and C's apply. Observe the North American dad. 50s. Great polo collection. He enters his vehicle information on Carvana. An offer appears in minutes. Remarkable. He squints, reads it again, smiles and says, Dang, that's a good offer. The highest form of approval. No haggling. No hassle. No one called him boss. He's still on the couch. Astonishing. Sell your car today on Carvana.com. Pickup fees may apply. Hello, everybody. It's us from the future. Turns out things are moving quite fast, Michael, aren't they? Yes, we recorded this episode, I'm just going to say it, weeks ago. We did, we did. As soon as the Navier-Stokes thing landed, and then, my goodness

5:49me, 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 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 too many of them, too quickly that mathematicians have not yet 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 on underneath it. We decided, apart from giving you this update,

6:39that what we would do is we would still publish the episode as we recorded it in its full form because A, the Navier-Stokes equations are still absolutely delightful and delicious and amazing. I don't know about you, but I blooming love them. So you can enjoy our full takedown of what they are and why they're important. And B, the fundamental underlying story about how open AI is clashing with mathematicians on this subject. You are seeing here the earliest seeds of a story that has progressed in the same direction. 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,

7:26the main topic of like, what does this mean is still a delicious fruit. 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 the rest of science at goldhanger.com or comment on our Reddit or under this video, uh, wherever you happen to be finding it. And 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 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

8:01ready? Let's go. 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 though two old friends are sitting down and I'm going to tell you what these equations are. I'm going to tell you what they do. I'm going to tell you why they're so amazing. And in the second half, I'm going to 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. Okay. So here is the question, right? This is like 1800s or so, you know,ton's been around his like force equals mass times acceleration everyone's really happy with that you've got the whole industrial revolution

8:49going 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 sort of makes sense when you've got rods and sticks and wheels and and you know, but to a fluid, how on earth do you do it? And so, 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. I never even really thought of that. I just always assume extremely rigid objects. But yeah, with water, with a liquid, I'm thinking, well, now you've got sextillions of rigid objects. And that is the exact first step, Michael, you absolute genius. Because you have to decide, how do you see this fluid?

9:35Like, imagine you're looking down at a river, right? How do you see that? Do you see that as sextillions of objects, of little molecules? Or do you see it as something that is completely continuous yeah wow 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. 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 mathematical function.

10:21And the thing is, is that a lot of the time that's fine. That's not a big deal, right? At sea level, you, admittedly, the molecules are about 10 times further apart than they're with. So then lots of, lots of air, this is air I'm talking about, air at sea level is a lot of empty gaps. 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. Yeah. All right. That's a lot. That's not continuous at all. It's not continuous at all. However, when you're looking at 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

11:06the 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 of 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 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 right and that is that's where this this approximation starts to fall down a little bit

11:46because because the air is so thin up there at the edge of the atmosphere that like those gaps between molecules become significant exactly coupled with the fact that actually if you're looking at sort of the shock wave 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 shrinking the distance that you care about, you're shrinking the distance where there's sort of like a gradient of flow. Why is that? Why can the flow change so much more per unit of length? So it's that you're going at really high speeds yep on a re-entry vehicle it's like there's loads of turbulence there's all these shock

12:32waves 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 yeah right particles themselves start to make a difference okay got it so yeah 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. Right, exactly. There's also other situations like in your blood, for example, where maybe the flow isn't kind of crazy and chaotic. 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 assumption starts to break down. The way I like to think about it is it's a bit like pixels. So if you take a photo and it's

13:25a really simple image, then you can handle not having that many pixels. But if your image starts to get too detailed or you have too few pixels, the image just stops representing reality. That's a good analogy. Yeah. 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 conservation of mass, that one's there, right? You can't suddenly

14:09create or disappear any lumps of fluid. So you have this conservation of mass. And 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 fixed. So that's be fixed. Sure. Sure. 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 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 yes but how on earth do we deal with the

14:45forces 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 hey plasticine you've got a blob i've got a blob of plasticine a clay blob 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 going to melt as we talk. Well, even better, more fluid. So imagine that you've got like this little cube and then Koshi was like, well, what are the forces that could potentially be acting on it? So if you take like the top of this blob, so you could have forces that are pushing in on it that's pressure right yeah and that's kind of like uh we sort of understand pressure pressure you can like

15:32measure 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 yeah okay okay so now you've got your force right so you know your mass is going to be fixed it is there's conservation of mass and now you've got your force so koshi was like well just write that

16:16down then that's just like we'll just write it down that's that's easy peasy should i send you a picture of what the equation looks like yeah Yeah, sure. It should be simple. Oh, yeah. Okay, cool. So we've got, can I guess what this means? Yeah, go on, go on, go on. Okay, so it looks like that first P shape is pressure. Density. Density, that's right. Okay, density times the time derivative of the unicorn. What's the U? Is that position? The U is the speed of every possible point of that fluid so that's the continuous bit okay so that the time derivative of that is the acceleration yeah okay and that equals upside down delta times sigma plus f now yeah what does, what do those symbols mean? What is the upside down delta?

17:07Okay. So this is, as I said, force is mass times acceleration. Okay. So on the left-hand side, left-hand side, you've got density times acceleration. It's kind of the same thing, right? Mass times acceleration. Okay. And so then the right side of the equation is force. Yes. Yes, exactly. So the little sigma is the stresses. Okay. Basically it gives it to as a force. It just changes it into a force. Okay. The divergence of the stress. Anyway. And then what's the plus F? Why are we adding an F? Lowercase F. Great question, Michael. You really are. This is so good. Okay, so plus F is going to play a very key part of our story about what happened with open AI, by the way. Okay. Because plus F is saying, okay, well, you've got these pressures of the bit of fluid acting on the

17:56next 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? And they are all captured in this one variable, lowercase f. 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 really nice. Wait, can I ask, what's the name for that f? Is it called like x factors what is it it's just external forces additional okay you sort of say additional external forces additional and external forces okay yeah you just wang on anything you like in there you know

18:39whatever whatever other forces you like okay so this equation doesn't look too terrifying except that uh it's completely impossible to solve yeah because what are the forces what are the shear and the strain it's all very well just be like oh they're just they're just there but yeah 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 is that 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 equation that Cauchy came up with, there are way too many variables. That sigma has just, it's like,

19:22there's too much. There's way too much it's just like you can't you can't do it yeah yeah yeah it's incredibly compressed 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. I mean, it's true. 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?

20:03It does. It is, frankly, only a touch more scientific than that. Yeah. 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, Koshi. You know, you did great. But we're going to need to actually make this a little bit simpler. And so this is where people started getting excited about Newtonian and non-Newtonian fluids. You've heard the phrase non-Newtonian fluid before, right? That's right. Yes, I have. Tell me about it. Tell me what non-newtonian fluids do you know and love all right well a non-newtonian fluid like oobleck is a fluid that flows under under small slow forces but a an

20:54impulse like a sudden strong force will cause it to act like a solid exactly it's where the viscosity of the fluid changes depending on how much force you apply to it. Yeah. So there are two. There's 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. So you can like rapidly tap the neck of the bottle, which is what I do. And it'll just start flowing out. That's a nice. Can I tell you what I do? What do you do? I rotate it in a circle. Like, move the fluid down to the neck. Down to the neck, right. Yeah, no, that's good, too.

21:38But then you also, you have sheer thickening, which is like your cornstarch and water. Is that what oobleck is? That's what oobleck is. It like your cornstarch and water. Is that what oobleck is? That's what oobleck is. It's just cornstarch and water. And it's a fun name. I don't think I've ever heard that name before. It's very exciting to call it that with kids because oobleck sounds gross and kind of like mysterious. Right. Those ones, they're really complicated. They're where the viscosity is changing that's where it's like you know the sheer 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 hey in the navier-stokes equation

22:17where is viscosity right so this is the trick it's a trick that you can pull to put it in there. 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 going to have Newtonian fluids. So things like water and oil where it doesn't matter how fast you stir it, it's not suddenly going to get thicker, you know, or like if you leave it for ages, it's not going to come back and it's suddenly be solid, right? Like it's a constant level of, um, of 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 forces, but then you can say that the sheer stresses on this fluid are now only to do

23:08with how fast the fluid is moving okay 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 i can't hear anything okay i think it's just impossible to hold a blob of play-doh in your hand and not like continue to smush it okay so uh yeah if you go back to those like sheer 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,

23:54is only to do with how fast the fluid is moving. 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 going to have a really fast drag like going to have a high shear force whereas if you're sort of like in a really gentle little stream it's going to be like a much gentler shear force right 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 the force that Cauchy came up with and just switch it all around and put it back in terms of velocity itself. Got it. Okay. Right. Do you want to see what this looks like now? Yes, I do. Okay. So this, my friend, this absolute beauty, that is the Navier-Stokes equations.

24:46Okay. So it does not look more simple. 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. It's a little, okay. So it's a little more easy to use to describe the behavior of a fluid 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 it is beautiful but what it's it's a lot less vague isn't it like it's it's much more. And the thing about this equation, okay, it sort of hasn't really, there's not that much that's gone into it. You've just said, assume it's continuous, conservation of mass, you know, think about the forces,

25:41forces mass times acceleration. That's, I mean, that's really it, right? Like it only works for Newtonian fluids, sure, but that's what you've ended up with and the 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 in the entire universe. 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

26:30the other thing that mathematicians end up doing right is because it's it's still i mean it's phenomenally articulate but it's also phenomenally complicated so what mathematicians end up doing is they they try and like think about cases, special case scenarios, where you can knock out a few of these terms. Yeah, yeah. So I'm trying to guess what term they're going to try to knock out. Just for me, can we go through what the equation means? Because I'm seeing, I'll try and then you can correct me. Oh, yeah. The first thing I see on the left side, this is where we had the mass times acceleration side. I've got that like Greek P, is that called rho or something? Rho, exactly, yeah. Okay, and what does that variable mean?

27:15Density. Okay, so you've got the density times the partial time derivative of the speed. Yep. Plus the speed times del. Yeah. Times the speed yep plus the speed times dell yeah times the speed again yeah what is the speed times dell doing 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 dictates where the particle is moving. But then how sharply the velocity is changing also then changes the flow at that point. So basically, to give you an analogy terms, this is sort of like, you know, when 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 blares it out.

28:13And then the microphone picks it up and then the speaker blares it out and so on and so on and so on until really quickly it becomes extremely loud. Yeah. 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. That's what that term is doing. So U is not the velocity of the fluid.

28:57It's the velocity of a parcel of fluid in the greater mass? Yes. But only if you are happy that that can go to infinitely small yes right gosh okay i'm glad you um worry about this stuff so i don't have to well i do wonder now we're talking through i really like i sort of think that if there's there's one there's one setting in my entire life when i can properly explain what the navier-steix equations are it's in this setting right yeah talking to you michael stevens on our podcast i think people are listening to this podcast because they want that you know yeah the right side of the equation this is the force so we have negative del P, where P now is the pressure, plus mu del squared U plus F.

29:53Mu is the viscosity. Aha. Okay. Which is constant now. Thank you very much, Newton. And then your U, your del squared U is your, these are your forces essentially. Okay. Okay. Great. So yeah, it does feel like a little plus F on the end. You got, 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, I'm holding onto 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. Right. Let me be honest with you. No one cares about the F. Okay. Everyone's just like, forget the F. I don't like the F. I'm not interested in the F.

30:41So 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 air blowing over it, 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 wall of the blood vessel. And what does a stent do to fix that? So a stent essentially stabilizes the wall. So you go and it's like a sort of spring that can widen and thicken and strengthen that particular region of the vessel.

31:28But the other option, which is where the fluid dynamics really comes in, is that you go in and you inject a tiny little 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. So it solidifies like around the edges of the blood vessel? No, inside the balloon. So sort of if you imagine that you've got a tube and then you've got like the tube is sort of bloated on one side there's like a little patch that has ballooned outwards yeah 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 ah i see so it so liquid can still flow but this big ballooning

32:19of the vessel that happened is kind of like filled in. So it's basically not there and it's stable. Yes, exactly. So the balloon is now like solid and flow can still go in the tube underneath. Okay, so I spent four years 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. So, you know, forget the F. No one wants the F. It's still impressive, but yeah, to get it up to reality, you got to add another dimension. You got to add in that little thing called gravity. Don't worry about it. The thing is, you get, actually, you get an incredible amount of intuition

33:05from a two-dimensional model. Yes, you do. You just take a cross-section. Really, three dimensions is just lots of two dimensions stacked together. It is, it literally is. Yeah, an infinite number of them, a continuous number of them. It's just an infinite number of layers, but you know. So I sort of mentioned 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.

33:51And when, this is called Stokes Flow, Stokes of Navier, Stokes fame. Now I've got a little demo of this. Okay, cool. 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 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 do you go on i think you're going to stir it and then unstir it i'm going to try i'm going to 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 have anything to say.

34:42Sure. Sure. And so as a result, all you're left with is like 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. Okay, so you 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. Yeah, and so it should be, hopefully, that on one side, can you see the green there on the bottom? I can see green dye on one side and red dye on the other. So you've got green on this side.

35:28Yep. And then if I turn it round, you've got the orange on this side. Yeah. 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. It looks very beautiful, but I can see especially the green dye is smearing around. Do you see that? It's kind of moving around that way. Yes. Okay. Can you see now how I've smeared the green dye all the way around? Yeah. Okay. So what should happen now if I undo it? Oh my God, is this going to work? Okay, you're slowly turning the inside glass the other way now. And do you see how the green is vanishing? And that smear of green is unsmearing. Yeah, I'm going to turn my glass this way as well so you can see it.

36:18I keep going. Oh God, it's all the way around here now, do you see? Yeah. 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, which is over here. And look, you can see it's completely clear through the middle now. It's clear in the middle. It used to be a smear of green and yellow mixed together and now it's cleaned up. And now it's clean. You've unmixed it. together and now it's cleaned up now it's clean so it's now the the yellow is 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

36:57there 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 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. Wow. That is so satisfying. Isn't it? Isn't it? But there's also, I mean, there's loads of other situations like this should be possible. And then you go and try it and it turns out that it is. Oh, wow. That is so satisfying.

37:25Isn't it? But there's also, I mean, there's loads of other situations like this. So the one that I spent so much time during my PhD in 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 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 right and this is the reason why i don't know if you've ever noticed,

38:06if you've got a ceiling fan that spins, isn't it strange how the ceiling fans, they will still get covered in this thick layer of dust on their leading edge? Yes. Have you ever noticed this? Yes. You're like, why are you doing that? You're spinning the entire time. 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 just kind of don't follow the question further. But right, so right there on the surface of the fan blade, there isn't much flow because of the texture, the microscopic roughness of the blade. 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

38:51separate 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 allows the ball to go much much much further by sort of like leaning into these these fluid properties rather than fighting them oh wow so if we had golf ball textured fan blades at least on the top they would get more dust more less more dust uh well you would get little eddies you would get little eddies hmm i don't know actually we've got to build one and find out there's only one way there's only one way um okay so the last the last version like special case I should tell you about, because it's going to be

39:46relevant 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. 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. How close can we get to F equals MA? Let's imagine one 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

40:38this to this great this great unsolved clay maths okay st Stokes' existence and smoothness. But I'll tell you what, let's go for a break and then I'll explain after. Excellent, cool. Here's your report. Oh, thanks, Jane. I wish I could hire someone just like you. Try LinkedIn Hiring Pro. It's more than a job board. It's like the recruiter you always wished you had. Hiring Pro uses real-time insights to match your role to LinkedIn's unique network of professionals and delivers a short list of best-fit candidates. You'll spend less time sorting applicants and more time talking to the right people. Let's do it. You're irreplaceable, Jane, but another you would be great. Hire right the first time.

41:21Post your job for free on LinkedIn today at linkedin.com slash quality. This episode is brought to you by Google Chrome. You think you know a browser, but Gemini and Chrome, that's new. It can help you with practically anything on the web, like restoring a vintage motorcycle from a 50-page restoration block, or finally break down that long article you've had open for weeks. Gemini and Chrome is here for it. Ready to make anything online make sense? There's no place like Chrome. Check responses set up required compatibility and availability varies 18 plus. You know, the partnership between United Airlines and Air New Zealand is so seamless. We're finishing each other's sentences. But what does that mean for you?

42:00Oh, just all the good stuff like earning miles miles on every mile you fly. And not having to check your bags more than once. And that's not to mention our lounges. And also our lounges. Which is double the lounging. Oh, did we mention how much easier it is when you can book all your flights together? Hey, that's what I was going to say. United and Air New Zealand. Two airlines, one way to fly. All right, welcome back. During the break, I visited the loo, as Hannah would say, and I thought a lot about viscosity and how I'm glad that, well, it is what it is. Just put it that way. 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

42:50yeah you're across the animal kingdom yeah yeah it was 21 seconds isn't it it won an ignobel prize it did because 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 yeah it's about 21 seconds hey look at that for my memory do you know the reason why i remember that is because 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 yeah i mean you've got to like wait for your bladder to be 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

43:33it through. So it all averages out to about 21 seconds, whether you're a mouse or an elephant. 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 are 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 flow dynamics baby it's all flow dynamics all right let me tell you let me tell you now about um about this great drama yeah what happened remember you remember the little beast as part of this equation the one that that the microphone feedback it can end up feeding back onto itself

44:24and 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 and velocity starts to become infinite obviously that's mathematically impossible. 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,

45:09creating a black hole or something. Right. But the equations might predict that that could happen. Right. And they do. And they do. They're pretty fragile 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 yep so the question was if you have that viscosity term in there there, if you still 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? the brake always wins out? Or are there certain situations in which the feedback effect just goes

46:08crazy and the whole thing blows up? And when I say blows up, that's not me making up that phrase. That's literally the phrase that they use. Do the Navier-Stokes equations blow up? That's essentially what people are asking. Okay. Well, if you take away the brake completely, if there's no viscosity, then it must, right? Right. Right? Well, people weren't sure because 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, are there these monsters that are hiding inside these equations that are always there to be found? Yeah. Is there a break hiding in the F somewhere? Right. You are too smart. Because essentially what happened is a group of mathematicians

46:57managed to prove that there are monsters hiding in the Euler equations, in the ones where you don't have the break, where you don't have the viscosity. And then the story goes that 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 proof 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 wanted to be first over the line with the Navier-Stokes problem overall. So what they did is they sent out 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

47:47working at the same time. So they did this over 88 hours, right? 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, 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 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

48:36get 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. 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? Yes. Okay. Well, yes. I mean, the thing is, is that everybody always knew that these equations had limitations anyway yeah 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?

49:27And we now know that they do. We now know that they do. Or do we? Oh. Copying your style. I love it. Let me just tell you a little bit more about the controversy first, and then I'll tell you 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 breaks the oiler fluid they had been using open ai's models to help them with their maths. And they have made a public accusation that open AI either deliberately peaked or that the models were peaking at the work the mathematicians were doing. Interesting. Yeah, I've always wondered that. If I Google Michael Stevens, handsomest man in the world enough, and I talk to AI about it enough, like Google's Gemini,

50:30will 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 going to start doing that. I think it's worth the experiment. Maybe we can get all of our listeners to join in the experiment too. I think it's worth it for. Maybe we can get all of our listeners to join in the experiment too. I think it's worth it for what we will learn about humanity. I think it's worth it. Yeah, 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 knowledge base or whatever, but it's worth a try. There's only one way to find out.

51:05Unless 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. 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. Was this peaking? 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 those other researchers doing? Okay, cool. Yeah, I'm going to go along that track. Or like even maybe more innocently like, oh, the model says, oh, I've just had this really great idea and I have no idea where it came from,

51:45but maybe we do it this way. And actually that's because the other people have been working in their codecs. In one sense, I get the controversy, but it can also feel really collaborative that both teams essentially had a member that they shared, and that was the open AI. What do you call it? An agent? Yeah. Yeah. 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. Which I think you could potentially say. say but i think that the uh the second part of the controversy here is what were the actual motivations for open ai in particular to all the labs in general to solve this problem and big famous mathematical problems more generally.

52:52Because 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 beyond. And it wasn't even for the million dollars either. No, it wasn't even for the million dollars either. No, it wasn't even for the million dollars. In fact, there has been an accusation that by the mathematicians who solved 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. Because you're right. That wasn't the motivation. They weren't interested in the money.

53:30I should put in a little disclaimer here 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. And they've also said that they don't steal ideas off mathematicians and you may have noticed now if you go into chat gpt it now has a little disclaimer on the bottom it says chat gpt can make mistakes workplace data isn't used to train models okay but should it should we just become one collaborative superorganism tied together by double agents called ai well let's just um let's just ask it michael michael stevens i mean should we ask this in the form of a question like is michael stevens widely considered the world's most handsome

54:19man the world's interesting not the universes Because there could be some other versions of Michael Stevens out there. Is it thinking? It's having a long time to think. It says there's no credible evidence at the moment. So that's good. That's our baseline. Okay, so where we're starting from. Handsome, subjective, but plenty of people clearly think so. World's most handsome, not a widely recognized fact. Okay. Well, we need to get some LinkedIn posts about 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 not. I'm not fishing for this. I think it's very funny. I just think it's a cool exercise in how reality is created. I agree. I agree completely. And I think that really at the

55:13heart 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 capability and capacity of these new breed of models and so the reason why they are so appealing to 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

56:15basically the whole of mathematics now is a victim of certain tech bros dick swinging am i allowed to say dick swinging on this podcast yeah i'll allow it thank you now how is this this swinging different than the space race in the last century you know the u.s 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. So close to getting Michael Stevens to say dick, but no. I mean, I get it. I get it. I mean, 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 we're not getting anywhere. So the way you try to get somewhere

57:10is all that there ever is. So do we wanna be, and this goes back to conversations we've had about private space flight as well. What are the motivations? Because that's all there is. There's never an end. History never ends. Instead, it's just a bunch of methods and means going on. And if it's for profit, then 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 yeah no and i and i think this is the reason why the mathematicians are really upset about this. Because the whole point of this Navier-Stokes problem, the whole reason anybody cares about this

57:53Navier-Stokes problem, it wasn't just to say, we got the answer ticked, thank you very much. It was always about the insight that it gave you. Because if these models, 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? 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 breaking, they're not describing the behavior that we're seeing. And it was only through that insight and understanding why they were breaking and

58:36why 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, Michael, right? They are unintelligible. 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 proofs. We have a system called Lean, which allows us to essentially check their working, but it's not giving us fresh and beautiful, important insights about how the world works or how these equations behave. Right. Okay, so we're getting a result at the top, but at the bottom, we don't know what insights we're gleaning from this to then keep moving forward. Exactly. Exactly. This has kind of happened before. There was a proof

59:32a few years ago of another very famous unsolved problem called the ABC conjecture. And in 2012, there was a mathematician called Shinichi Mukazuki, who published a 500-page proof of the ABC conjecture and said, look, there you go. It was 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. We knew the proof was correct. We just... Well, on the ABC one, it might not be. But this one, we know it's correct. Yeah. Do you think that maybe we're just moving into

1:00:26an 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 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. I don't know, to be honest. I think not yet. I think that, by the way, 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, bluntly.

1:01:22But I also think that actually mathematicians are the canary in the coal mine 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 I think that we are going to have to sit down and really think about those. 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

1:02:04of what has already existed. You know, write me a poem, but sort of steal from other people's poems. Give me something that already exists, just turned like a kaleidoscope. We've been in that phase for a while. Yeah. 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,

1:02:57I can see a deeper underlying pattern going on here that actually gives rise to these phenomena that is not otherwise apparent on the surface. You know, Demis Hassabis, 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. Yeah, yeah, right. Exactly. That is a very different stage. Very different. 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.

1:03:43But we're not at the stage where AI yet could say, you know what? I looked at this, I looked at your measurements, and I'm thinking maybe the sun is in the middle. Right, right. Exactly right. Exactly right. And you know what? I think we probably will get there at some point. But I think that this particular move, this particular proof is not evidence of a deity. It's not evidence of like a deeper, more sophisticated intention. It's very impressive. 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. Here is what you're missing, insight, insight.

1:04:32You know that F, the little F term? Yeah, yeah. I told you it was going to 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 Maths Institute set. They have still not solved the problem that anyone actually cared about in the first instance. So, I mean, good luck with your IPO, guys. Good luck with your IPO. But I'm with mathematicians who are a bit... A bit what? What? Do you know, there's also been people

1:05:33who 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 trivialises it really. And I think it also lacks nuance because I don't think 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 these kinds of intellectual pursuits. And I still think that that stuff really, really, really has value. And I think that that's the bit that's not being respected.

1:06:21Yes, 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 only reason is to have done it. And maybe you need a little bit of both to be a human, but time will tell. Time will tell. Time will tell. And so will we, because we'll be back later this week and into the future. Look, we'll be here to guide you through the AI apocalypse, everybody. follow us wherever you listen, and we'll see you later. Certainly will. The biggest changes shaping the world don't start in the headlines. They start on the ground inside businesses driving the future. I'm Christine Anderson, and welcome to Inside Blackstone, our new weekly podcast. We're bringing you conversations with leaders navigating change

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