OpenAI Researchers on the Future of Mathematical Reasoning

0:00Often as a practicing not sufficient, you have an idea and then you kind of think it might work then you try for a few hours a few weeks and at some point you give up. Whereas for GPT, like okay, I keep in touch with you to do this, let's just do this. And so that's why we're serving this around us on some like reachable results. This is the best part about this problem, which is really nobody had any idea. Because the model just guessing in some insane way. It doesn't seem like there's a limit so far, but it doesn't have that context yet. It'd be nice for the world to find mathematics on a lot faster. The ceiling for a difficulty of a math problem is pretty high. Even if AI continues getting like exponentially better at math, plausible will never solve something like P versus N. What's the ideal way that this is being taken up by the math community? Probably most of this point are like, okay, AI is obviously doing some non-trivial stuff. Um, so AI isn't just getting better at math benchmarks. It's beginning to make progress on mathematical problems that have resisted humans for decades. In this episode, A16Z InfraPartner, Lisha Lee,

1:00sits down with open AI mathematicians, Metab Swani, and Mark Selkie, to understand what's actually changing. They walk through recent results in sphere packing, coding theory and group theory, and explain why these advances can't be reduced to brute force. The models try different approaches, abandoned dead ends, connect ideas across fields, and in some cases, produce reasoning that reads surprisingly like the notes of a human mathematician. They also tackle the bigger question, what happens to mathematics when proving a result becomes less of a bottleneck? From mathematical taste and human judgment, to understanding an explosion of new results, Lisha, Metab, and Mark explore how AI could change not just what problems get solved, but what it means to practice mathematics. Well, thank you guys for coming. This is really exciting because I think math has been moving so fast, with AI, there's love to get to both practicing mathematicians and who are good open AI to chat on some of these results.

2:00So we have with Mark Selkie and Metab Swani. We're connected actually because UFA was actually your advisor. So both of you guys have worked much more deeply in math since I have quit. Many, many, like over a decade ago. So this is very exciting to kind of hear. You download your thoughts on how open AI has been sort of approaching this, and also just like where you think math is going with the incredibly rapid advance of how AI has been helping. We can start off with some very basic questions. What do you do to accept that you can of course share? And how did you come from being a practicing mathematician to working at OpenAI? Yeah, I mean, I guess we both broadly got excited last year when the models started to really take off in math. Yeah. So I joined a little bit. Before I met Tab, I saw the IMO Gold Medal last summer basically. I thought this is amazing. I want to see what the heck they did. Let me go see. And then, yeah, I guess in the fall, Mark gave me a GPT-5 account.

3:02I started playing with the models and very quickly became convinced that, yeah, it was extremely exciting to play with that. And you two were collaborating before. Yeah, we've known each other for a while. With one paper we actually wrote jointly. Yeah. So GPT-5 was your conversion? What was the magic that sort of, what question do you throw at it? What process? Yeah, so I think actually, yeah, so I think how this started was, at least for me, the starting moment was something like, there's a collection of problems that were called, so air polish is a very kind of thought. But additionally, it posed a bunch of problems. And so they've now all been collected on this site. And so I specifically worked in Cominatronics and a lot of these questions are among the most important. So it's always fun to flick through the slide. But one thing that often happened to me that was extremely frustrating was I would look at a question, see that it's marked as open and then not actually know if it's correct, not actually know if it was still on salt, because the literature is often quite hard to search. And one instance I just plugged it into GPT-5 and five minutes later,

4:03it found a reference. And this was a case where a few of my friends actually started thinking about the problem on the site. I was talking with them. And I mean, we'd spent a few hours, it wasn't clear if the problem was within reach. And it was very nice. Okay, to be told the access isn't reached here, how you do it. And yeah, GPT-5 told me this and then I told Mark about this and this. Yeah, this is sort of, yeah, this for me was quite a surprising moment. Yeah. And then we looked more into it and we found 10 more cases sort of like this. At the time, I feel like being better at maybe making connections between is your saying like the search for whether there's been a result or a related thing earlier is just kind of humanly hard, but maybe better for machine. But I imagine as the progress has happened in the last year, what has been impressive has kind of reached beyond that. And maybe through talking about more abstractly or if it's more natural to talk about it through one of the problems that has been recently announced through, you know, Astra, and kind of enlighten me as to like how the recent progress has been a lot more than just searching through more areas, making these connections between the field and perhaps

5:07just actually deeper, more mathematical reasoning that's similar to working at the tissue. Yeah, I mean, I think this like search point of being familiar with everything is still definitely like a relative strength that maybe informed was like the types of problems that AI is solving now. I think there are some other relative strengths and weaknesses. Another relative strength that's pretty noticeable is just like it's very good at executing on some like idea once it has it. Whenever you have an idea, there's usually some amount of getting everything lined up as epsilon like smaller than delta, this kind of thing. You have to get everything correct. And like for a human, it's easy to get lost in these kinds of details. And the AI is just kind of always nail these kinds of arguments I find. Yeah, I feel like you guys will know more detail on this, but like for the unit distance problem, it was just like the approach, there was definitely contributions from OpenAI, but like the approach perhaps was suggested even originally by Erdog and then it's just that

6:12the actual reasoning was a very, very like momentous feat. And so for human, you're like, well, I only have a limited amount of time. And if after so many steps, it is still not clear. I mean, maybe the year like Andrew Wiles and you actually spent 10 years alone and do something, but like it's not clear that the risk reward is not good enough. Whereas for GPT, okay, I'll like a human told me to do this, let's just do this. And so that's why we're serving the Serenostons of like reachable results. Does that track and did you feel like with the astro results, is that sort of like where the strengths have been primarily or there's an extreme ingredient or magic here? I feel like I do a unit distance example is about it's quite telling in the sense of maybe the exact construction. You can make it look very similar to what people had tried before, but I think, I mean, often as a practicing mathematician, you have an idea and then you kind of think it might work then you try for a few hours or a few days or a few weeks and at some point you give up and then

7:13a not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work. So somehow getting an idea to work can even be a large portion of the battle. And I think especially in the case of the unit distance contractor, there's just a lot of extraordinarily finicky details and very often when you're doing that, you're kind of gambling against the problem. You're like, maybe I should try this approach, but it seems really unlikely and just not worth my time. And the model, I think in several of this cases, both by combining what you and sort of having good taste kind of made the correct path. And you can kind of see this in the summarized chain of thought release. You can sort of look at it. It's reasoning like a mathematician and because it knows a few very correct bits, it makes the right decisions and eventually able to prune the search tree. It's not really trying everything. It tries a lot of different things. It's extremely dog-in. I mean, it can't try every idea. It has to try a limited set of ideas and it's able to kind of use its knowledge plus good mathematical judgment and find the right path to go along. So I mean, that for me was like, because this was a problem which a lot of people have thought about.

8:16The fact that the idea is not so foreign probably indicates that a lot of people have tried it. Or at least a few very serious mathematicians have tried it. And I think that's when made it really interesting to see. I think something else that I feel when I see these proofs is like, if I have an idea and I'm trying to execute, it might be that I have some wrong plan for like how to get things to work. And as a human, if you have some wrong path, you go down for a while, it can be hard to like rewire your brain to start over and try a different path. The initial idea is kind of linked in your brain with these other things that ended up not working. It's sort of like your context window is like a little polluted and you can't just make another clone of yourself from last week and say, don't do this, try something else, build your intuition in another direction. But it's very easy to do this with an AI. So I think this is another reason that it's like getting the details right once you have some good general direction is much less of a barrier all of a sudden. And when you say it's much easier to do with AI, it's like it's not actually being directed with human interference too. It's as you're saying, the reasoning traces, it's like

9:18making these choices. Maybe it backtracks, but then it's able to not be distracted by maybe the context in which it's thinking about the problem, like via these machinery. And it's like, do you see it kind of go back as well or is it just like good choices? Like is it a lucky sample or is it actually reasoning like a mathematician or okay, it doesn't do well in this path, but it goes back, but then it doesn't let that pollute. No, I mean, it definitely makes mistakes and then it goes back and thinks about it. I think it's somehow very calculating, very correct. I mean, yeah, as human offmages, you're always perfect at making these decisions. The first time something doesn't work, you automatically kind of downgrade how likely this approach is to work and you keep doing this a few times. The model somehow is much better able to like, it seems for several of the solutions you see, it somehow seems it seems much better able to update like how likely the path is to work like versus rejecting a path versus a human doing it. But I think even if it weren't, the fact that you could just start another model session over, it's like, you know, it's always going to be the case that got it. So in some sense, it is like still leveraging the fact that you

10:19could like run kind of parallel, you know, agents on the problem. But if it were kind of backtracking, then does make it seem much more like a human, you know, mathematician and perhaps it is kind of doing some of that stuff too, because like, obviously, like, we have to like make mistakes in in order to like give and gain intuition for like why that solution space is like not, you know, not in the set of paths that it could be in. I mean, I think this kind of thing happens with humans too, or like, if you get stuck on some approach, you might tell another human, you're kind of general idea, and then they'll come back and like figure out how to get it to work. And you know, it's just yeah, it takes more time to do this, like with humans. I wonder, I mean, you know, maybe this gets to extend that you can actually talk about sort of like, obviously, don't talk about the training recipes or whatever, but like it's it's interesting that if you're just studying, for instance, from math papers, it's like a very poor training set, like a priori for math, because I mean, maybe math textbooks are even a pure example of this. It's like really bad at actually reconstructing the motivation for why things were, you know, like, don't, I mean, maybe some people

11:22like it, but don't learn real analysis from Ruda. It's like, it's very clean, already in crisp. And I think that that's bad because it doesn't show the struggle that made us formulate definitions in a certain way. Like, why do we even need to have real numbers be to find this like super abstract way, et cetera. And so, you know, I think papers also, I mean, unless you're most people don't write papers with the the context of I need to educate somebody to do, to be a mathematician. And so like, the actual maybe curriculum of like learning math is not inherent in like a lot of our artifacts as mathematicians. So maybe another way to ask this question is if the reason Tracers are actually producing things that's like, okay, this is actually more close to mathematical thought, like, how does that arise? I mean, I guess OpenAI has been like the pioneer of reasoning models and, you know, teaching AI to reason in this way. So, you know, we're doing a lot of work at kind of, kind of,

12:24kind of all possible directions on, you know, teaching models to reason better and for longer and and all kinds of different domains. I mean, I think I think we're training general purpose reasoning models and if you know, kind of one, a lot of these behaviors that we're describing mathematically like backtracking or kind of starting again, I mean, these are these are not really specific to math. I mean, we're seeing them specifically in mathematics and these examples, but kind of their general purpose tools are reasoning. And I think if you work hard at reasoning, you should see these patterns eventually. So, it's just an immersion because it's, I mean, I do think that's why the OpenAI approach was so, I mean, it's like, it doesn't rely on, you know, doing auto formalization in order to like guide the reasoning. I think that's like obviously more like us. But it's just like so not obvious that if you're just like training on, say, a corpus of like math proofs, maybe auto formalizing that you get the sort of like projection of like how to think well, like put another way like with maybe

13:30maybe we think about it with code like code is such a good corpus to train on because it's one of the few data sets that have such large contacts. You just like, I mean, maybe you see this kind of with books, but they're less structurally interconnected. There's just like less structure there, I think, it's safe to say I'm like on average in a book compared to like a piece of code. And so like with math papers, I feel like maybe what we're still bad at with coding models is stuff that that data set doesn't contain, which is like kind of the semantics, like the syntax is there, but there's a little bit of like the higher level semantics of what produce like, why do I have to write it this way is not. I'm kind of getting too much of the philosophical, but it is just like really interesting how it's still emergence that it's doing good mathematics. And we'll probably get into this more detail if you guys, you know, wanted to talk in more detail about some of the problems, which is just like, it's not just doing like the expected like we'll push the brute force thing like you clearly are

14:31impressed with some of the reasoning traces. And it's just not obvious that's gleaned from, you know, what we would imagine would be the easy training set here. Yeah, absolutely. I mean, I think this kind of thing is one reason we decided it was important to release like these summarized chains of thought for these kinds of results, because if you if you've never seen these and you just see all these proofs coming out, you're kind of you're not sure what it means like like is the model just guessing in some same way like is it thinking in some totally foreign like what's going on, but but actually it's it's reasoning kind of shockingly like an expert human would. Yeah, yeah. Yeah, it's it's very much like reading a colleague's like notes. I mean, it's a little more to organize in some way the kind of like as soon as you work close enough with the cloud, sometimes you'll just see them like spill out their thoughts and an email to you. And it kind of it feels like reading a lot of those chain together. So it's yeah, it's very it's quite surprised of first two times. Were you two sort of very involved in choosing the problems to to release in

15:37this like last 10 problem set that Astra is applied to? Which was your favorite. Yeah, we have definitely involved. Do you want to start? Yeah, I mean, I'm sorry. Yeah, I guess yeah, so I guess my personal favorite among these problems is the following. It's it's extremely simple question, which is just like it's just about how efficiently can you put a my circles are not very good and they're not all the same size. But we're assuming. Yeah, so the question is just like how dense can you place a bunch of so you have a bunch of spheres, you have a bunch of spheres of radius one and d dimensions. So the question is how densely can they pack? And so yeah, so so in two dimensions, it's kind of like the so d equals one. This is not an interesting question. Kind of it's just the real line. Yeah, you've been cut it up in a sphere and dimension one is just a unit segment. So okay, you can cover everything.

16:38So indeed, equals two, it's kind of the picture that you know that everybody loves. It's just like it's just about your spheres, which sort of form like hexagonal lattice. Hopefully, I've drawn it well enough, but I can draw the hexagon. Kind of betraying my name. It's a sprung. Is that like obvious? Is it like a very elegant proof that it's not so yeah, it's not so obvious that this should work. I was only proven in the 60s, I think. There's a short argument, but it's not it's not so easy. Yeah, um, we're just into it. Like we're just kind of like the I mean, scenery of the argument. I mean, it definitely looks like it should work. That's why I'm I think it's something. Yeah, I think this is the best part about this problem, which is really nobody has any idea. So yeah, so I mean, honestly, the best situation I have for this is that like bees do this. And if there was a more efficient way, then probably bees would pack how he comes them other way. It's evolution is efficient. Yeah, I think beyond that, like I don't have a great argument. I mean, and I think how little we know is demonstrated by the fact, so

17:40okay, D equals three. The the answer is just like, um, it's what it's how you pack like oranges in a grocery store. And this was only this was proved by hails sometime in the 2000s. And like, and we don't have a short proof of this. Like I think the shortest proof is like a few hundred pages. What area does it like a draw from? So it's a lot of linear programming arguments and it's very delicate like geometry. It's it's quite ugly. Actually, it's like, this is like a famously ugly argument. Oh, no. And then the two most famous results are D equals eight and 24. Eight and 24 must be somewhat weird subspace. Yes. Yeah, exactly. So this was done in like gluing. Yes, this was done in 27. So slightly prettier, those. Yeah. So yeah. So so the reason it works out in these two very special dimensions is that so this is called a lattice packing. So it's like kind of very regular. And it turns out in these two dimensions,

18:41there are two very special lattices. They're called the E8 and leech lattice. And they're very nice and they're like unusually dense. Like kind of they're just very, very pretty structures coming from other areas of math. And it turns out that they're the optimal structures. But they're still like regular. Yeah, they're very regular. But I mean, beyond this, so we don't know any more exact dimensions. We know these five dimensions and we kind of don't know anything else. And I mean, like to give an indication of how little we know. So there are two very surprising things about this. So you can define like Delta D to be like the densest spear packing deer, data mentions. So there's kind of an easy lower bound of like two to the minusty. Basically, yeah, this is not so hard to show. Basically any packing where you can't put it in another spear has to have this density. So okay, it's not ridiculously small. And we know that it has to decay exponentially. So it has to decay, like it grows like one minus C for some, at least for

19:46some quality. So in large dimensions, you can only kind of cover like a vanishingly small portion. But we know like basically nothing else. And so for us, just because like the high-dimensional spear like, yeah, where they occupy is just like the volume of behaviors where yeah, so basically, I mean, basically they don't want to touch next to each other. I don't know if I don't think there's a particularly short way to see that it's like exponentially small, but it's known to be exponentially small. And for a long, long time, the best bound was something like this funny number like two to the minus point five nine nine D. And this was proved by two mathematicians in the seventies. Kapitanski. That's a weird number. Where's that? It's not like combinatorial. It's the answer to some extremely ugly optimization. There's like a nice underlying strategy. Yeah, I'll say one last thing about this. Yeah,

20:47yeah, these were these two Russian mathematicians in the seventies. I think it's very hard to find their paper. Like one page, it's like two pages long. Yeah, they don't write very many details because paper was fine. But yeah, and so two native dudes, there's like a square lattice, like the dumb other. So yeah, it's actually not so easy to, so the argument for this is as follows. Basically, imagine that you have a set of spheres and I tell you can construct a set of spheres so that like you can't put down another sphere because if you could put down an actor's sphere, you just keep putting it down. So you have a set of spheres so that there's no other sphere which you can put down. That's here. It's like almost like a just yeah, just take any such packing. Okay, yeah, yeah. And I claim that this has to cover at least two to the minus D fraction. The reason is that like if you blew up each of these spheres by a factor of two, then like they have to cover every point in space. And the reason is otherwise you could put down, if there was any empty space, you could put down a sphere there. Yeah, so I guess if you take the usual lattice,

21:49there are actually like more places you can put things kind of diagonally. Yeah. Okay, yeah, so that's actually not it's like a worse bound. Yeah, you can just keep plopping things in. Yeah, this is like more okay. Yeah, this is related to this like really funny factory like put a sphere on every point on you take a cube and hide dimensions, you put a sphere on every point. It's like finishing. It's vanishing. It's so small that you can put another sphere in the middle. Yeah, yeah, it fits. Yeah, it's very weird. And so okay, so the great part is that so the model shows the following. So I'll write two things. So this is a astra, I guess, probably the right way to refer to this. So okay, I'm going to write something called the LP bound. I'll explain it to the second. And it shows that it's smaller than this very nice number doing a state equals. Yeah, it's equal. Actually, the two pi plus little o1 to the D. And if you you can work out what this number is, it's like,

22:57it's like roughly something like two to the minus point six. Yeah, it's surprising. But this one, you know, you're like, oh, maybe there's some like nicer kind of like structure there that that fell out. And it's most like this is like roughly something like two to the minus point six zero one dot dot dot D. That's the numerics. I thought it was six or four. Great. Yeah, let's just mine. Yeah. So okay, so there are a couple of things. So first, what is this LP of D? So via Zosu's work actually builds on some earlier work. It turns out that there's a way to attack spear packing via what's called a linear programming ground. So LP just stands for linear programming. So Conan Elke's gave an approach for spear packing based on linear programming. So it's like a it's like a linear optimization problem over a convex set, but it's all kind of infinite to be sure. Oh, oh,

23:57and basically what this reduces down to is you try to understand the follow. So what you try to show is you basically you construct a function f. So this is in data mentions and it's mapping to up and it has the following properties. So first, um, aquavax. So this is a function in data mentions. So it's always less than zero if like the size of aquavax is bigger than one. And you second have that the Fourier transform of x. This is always non-negative. So this is just, this is a linear program because the Fourier transform is a linear operator and uh, nitrace. So you're taking just some arbitrary f that says yeah, it's property. Yeah, so you can take any f that satisfies these properties. And what they prove is that delta D is bounded by the ratio of the Fourier transform of zero to its, uh, to the, um, the, the Fourier transform of zero,

24:57um, f of zero to the, to over the Fourier transform of zero times the volume of the ball of radius one half and data mentions. And so, okay, this, this proof is not so shorted for experienced math. Petitions like it's a half a paragraph to prove it, but a little tricky. And the point is, um, so it turns out, so this is a relaxation problem. There's no guarantee that taking the optimal will give you a good bound on delta D. Um, but so what V is osc it did. And this was sort of the key. I mean, a large part of the reasons you want to feel as metal. And, um, in 2020 was that, or in 2022, was that, um, she constructed a function in eight in 24 dimensions such that this upper bound matches exactly these two very special lines. And these are kind of miracles of nature that both you can construct this function and that it gives you the optimal bound. Um, but you can just, this is a very, very natural problem. I mean, it's a function

25:57with a very two very simple properties and you just want to understand how this behaves in for large dimensions D. And that was a big mystery. Um, we, there was a numeric paper by cone and several others, which conjecture that, um, just based on doing numeric that this was the answer, but they had no idea why this would be the answer. And what the model shows is that actually, um, the linear programming bound in large dimensions has this extremely nice asymptotic behavior. And the proof kind of explains where this is called. And because you understand this LP bound perfectly, this actually just gives a better bound on deltady. It turns out that this old bound can be kind of reinterpreted in the framework. And what the model does is it shows you the best possible bound you can get that to framework. So the model sort of made the connection. And yes, what is the sort of like, what do you know, what do I mean? So I think, so the model gives a function, which so first it constructs a function of which gives you this bound. And

26:59then it shows that there's no function F, which does any better. So it's in a quality, um, which is quite strong. So it like sort of it, we now understand this, this problem in high dimensions very well. And that's pretty remarkable. Um, and the model is just kind of told like, analyze this linear program and high dimensions, you know, go have fun. Got it. And we give an indication of like how it was known. I think this conjecture was based basically only by doing new networks, um, extremely clever new networks, but new, and so yeah, you have to kind of figure out why this is the right thing to aim for. And it does. And that was pretty remarkable. Yeah, I mean, I actually thought about this problem for about six months at some point when I was a graduate student. And yeah, just I remember making like, absolutely zero progress on it. So it was very nice to be like explained why it was. Yeah, why it was true. Um, so that was a positive thing. I think also in general, it was it was one of these solutions, which I knew several people had tried the problem. It's pretty remarkable because like the model solution, especially for this being

28:01like the LP can't do better than this was like quite short. It's a few pages of like complex analysis, but it's kind of exactly the right approach. Like once you see it, it's kind of, it's like unbelievable. Like why why haven't somebody done this before? It was like, there are many types of good mathematics. But I think one of them is just like, you see it and you're like, oh man, why didn't I think of this? And it was really fun. I mean, I sort of knew why I didn't think of it, but it was quite nice to see it. And it was fun to see. That's why I like this. All the luck. Yeah, so this is the first of the 10 problems that Astro saw. But the second is actually closely related. So this was fear packing. The second one is a spherical and binary codes. So what's like your draw code? You drew a packing. It's going to be the same otherwise I'm going to have his picture. Like a diagonal packing. Yeah, a spherical code is literally just a sphere packing, but on another sphere. Yeah, so I mean a spherical code is basically just a sphere packing on the surface of another sphere.

29:13So yeah, it looks like this. Okay, so same pictures before except you're kind of on a curved surface. Okay, okay. So why is it called a code? Well, you can I guess the reason is because of binary codes, which is again the same sort of thing, but now it's on a cube. Okay, yeah, let me draw a picture of a cube and some like simplest possible code on it. So when you're like sending, so this is really like about error correcting codes. So what are error correcting codes? So it's like I send you some string of bits, right? Like and maybe I'm like worried that some of the bits I send you get corrupted, right? So

30:18so maybe like just because of some errors in my system, like this one gets gets changed. And we want some communication protocols so that like you can decode this like small amount of error and like recover what I was trying to tell you. Uh, and you know, like normal English language kind of has the sort of property, right? If I make a few typos, you know, you're going to be able to understand what I'm saying. But if I, if we have some like really brittle communication scheme, it's not going to work. So, um, uh, codes are kind of the way you solve this and mathematically, it just means like, you know, what's a binary string like this is a fixed length? It's like a point on some hyper cube. And we want a dictionary of allowable code words that are like separated from each other. So like in this case, if I don't want any to to be adjacent, I would kind of take these four vertices kind of the like even ones if you sum up the digits, right? And like, okay, I guess, uh, okay, in this case, I guess if I if I have an error, you can't tell which one it's

31:21from. But at least you can tell it's like not, uh, at least you can tell there wasn't error. Oh, I see, I see. Yeah, because it's like kind of sparse, um, in, uh, um, yeah, it's like, it's not she would adjacent. So that like, is this like a one of the hamming distance? Yeah, yeah, yeah, yeah, yeah, right, right. You want, yeah. So you want like a large hamming distance between any distinct, okay. And you're in your dictionary. And yeah, I guess if you take two opposite corners, then if I have like a single bit error, I can always like recover which point it was coming from. One that it's definitely closest to. Yeah. So there's, there's kind of the, you know, same question in both of these cases, like in a very high dimensional setting, what kind of rate can you get? And like for binary codes, it's really like, you know, an extremely practical question. It's, it's sort of like, if I send you, um, like an end bit string, and there's like, you know, one percent error rate, like how much longer does my message have to become to tolerate that amount of errors? Right. It's like some fundamental fundamental information theoretic limit of like, like,

32:21you know, like communication. And, um, you know, but, but you can see like, like certainly this, uh, the spherical case, uh, is like, it looks very much like sphere packing. For example, if you like, if you make all these little spheres really small, uh, then like the curvature of the big sphere is kind of not going to matter so much. And it looks like just packing spheres in full space. Um, and in fact, yeah, like these, these problems turned out to be very related. Uh, so, so for, for these, uh, problems, there was like, there were similar bounds coming from the KL authors and, and like, there's like something for the sphere and something for the cube, but it's all kind of the same stuff. And, um, our models found, uh, better bounds for, for these cases as well. And, uh, like, it, you, I mean, the techniques look pretty different actually if you, if you like, write them out. So, uh, so this, uh, this full space analysis of this linear programming was

33:22using like, just complex analysis. Um, but if you, uh, like the, the method for, for these cases, we're using representation theory. Like, like both the sphere and the cube have a lot of symmetry. And basically the, um, the idea of the proof was to really leverage this symmetry. Uh, like, there's some amount of this in the, the previous like existing method and, and really the improvement is to like lean into the representation theory, like really hard and kind of make the, the algebraic symmetry like, and turn a more sophisticated way. Um, and then it like, turns out that from the representation theory formulas, if you kind of take this like, like, small sphere limit in the spherical code case, you, you recover like part of this result and you recover this, this value. Um, so, uh, like this result doesn't have special case. You kind of only one direction of the bound from, from looking at it from the code's point of view, but like there's like a very close connection. Okay. Yeah. This you guys let this like run in parallel. So it's like

34:24kind of discovering or because you're not sort of like feeding it. Uh, so, so yeah. So actually, uh, this was the one case where there was some interactivity. Oh, interesting. So for, for all of, so except for this pair, it was just, you know, we had some problems. We fed them in and we, um, you know, the model, the model came back with some solutions. Um, what happened here was actually pretty interesting. So, so we first asked it to, uh, improve the bounds for the codes. Uh, and it came back with an improvement that like used some amount of representation theory. And then, uh, we kind of asked it, hey, can you like push this further? Like, you know, what happens? And then it came back with some like much more sophisticated representation theory. And I like, it turned out that, uh, you got this, uh, conjectured value for full space, sphere packing, uh, like out of that method by pushing it as far as it can go. So then we, we kind of asked to directly analyze this guy and try to complete the picture. Okay. Yeah.

35:26So, so the relationship like isn't a coincidence. Yeah. Yeah. It's like, interesting when you're saying the first prompt, which is, you know, maybe so basic or just like, can you push this further? It does require some judgment from mathematicians and but like, eventually you would imagine by scaling the models, you don't need to do that. Or there's another view that the harness actually does matter. And this is kind of part of the harness apparatus. Do you guys have any views on that with your working with Astra? Especially generations of models and how, how much do you have to kind of input or how much the harness matters versus not? I mean, I, I guess there have been some like funny quirks like this that just come from like exactly what you asked the model to do basically. Like, like in this case, what the model was asked to do originally for codes was to improve the bounds by like some exponential factor. So it really like shows up in this like leading constant up here. Um, and, you know, it, it improved the bounds and it didn't try to push things too much further. Like sometimes you see a do, but sometimes it

36:27just doesn't bother. But yeah, you know, you just ask it again and it goes further. So it wasn't like a capability issue. It just, yeah, I didn't feel like it at the time. Do you call that judgment? Or like, what is the, because like there's a, yeah, what do you call that? Models tend to be pretty task-oriented. But if you, you tell it to do a task, it's pretty, it's pretty happy. So yeah, the task-orientedness, it's like, but do we expect that level to kind of ascend up to, it's not that they will be less good at being task-oriented. It's like they'll ascend to a level of like, okay, no, let's, let's go in this direction. You'll have the judgment too. Because you guys have the judgment too. Like, okay, this is pretty promising. Looks like you're using a lot of representation theory. It doesn't seem like there's a limit so far. Um, but it doesn't have that context yet. But like, I guess what I'm trying to say is like, this one is hard to, maybe, harder to extrapolate, but from like previous generations when you had to give it more, maybe prompting, more of that harness work, but eventually probably had to give it less. So probably gives you some confidence that there's this like really fast ascension. And do you see,

37:32yeah, like, what are you talking about? So now I'm solving a harder math problem. Like, you have to solve many smaller, like somewhat less hard math problems. And the fact that the math problems are getting harder is kind of an indication that you're, the model is able to take on more and more work and like a single, continuous unit. Um, and I think that's, that's the thing that looks very promising somehow. Like any of these solutions, it's not like one idea, then you're kind of home free. You need several, you need several pieces to kind of interact and talk to each other. The model is like, the model doesn't come up with all the ideas that want, right? It doesn't pull everything out in an instance of kind of the fact that it needs to sort of see how this piece interacts with another piece that's kind of like solving a problem itself or piecing together many problems in itself. It could just be that, okay, when you're telling it, okay, push this even further, that was of the same order of like magnitude as like all the smaller things it's solving as well and each way. And so you don't think that this kind of like a privilege direction is just sort of like, hey, let's give it like one more help. Or you actually think that

38:32there's, I guess we'll try to get at a bigger question is like, is there a good sense of like, you know, taste because like when people talk about, for instance, how well the models are getting at like doing research, for instance, that's a little, you know, well, a little bit of our side. And sort of like, there's surprising things about how that improves and then there's like the, oh, you know, maybe right now it's a level of still like a junior researcher. It's like not really asking like the right problems. And so I'm just trying to get like a maybe a sense of like where you're seeing that progress through the model advancements each generation. And what is taste even? Yeah, I think I had to be pretty utilitarian by view of taste and like, if you're able to solve problems faster by making better judgments, like I think that's like the best like general proxy. I have for taste and some of the fact that solving harder problems means it has kind of by definition means that has better taste. I think there are these no, yeah, I think occasionally,

39:36occasionally because they are task oriented and you do occasionally get these symptoms of like, oh, it clearly has made a breakthrough. It kind of understands it's made a breakthrough and then it doesn't kind of push all the way to the limit because that's not what you asked. But that seems yeah, that seems rather minor compared to like the set, the state of progress we've seen so far. Okay, yeah, I think that's a pretty clear. I think it's like maybe you're liable to get confused if you're trying to like do a concrete, long horizon task and show taste kind of at the same time. But like, you know, if you have like one model that's responsible for taste and one model that's responsible for going out and like, you know, working for a long time at solving a hard problem kind of as the like, you know, underling of the supervising AI, I feel like that's kind of going to be fine currently. Oh, interesting. Because that is like saying that they these two things are somewhat

40:39separate, if not separate at least they shouldn't kind of pollute each other's context, which is a little bit, I mean, it could be potentially like a strong stronger statement. Then, I guess, you know, it's just kind of interesting because it might just be like to your point, it's, you know, let's take the each hillitarian answer solving harder and harder problems. It's doing a lot more than just like, you know, enforcing something that's making choices. It's like pruning, you know, vastly large space of possible pads into something that's like really, you know, it's both tractable, but then ends up being like it's a diminishing, least small path within that space. But like having like why would we would be like a separate model, a separate generation or something that's a different version of the model that would contribute to taste, or maybe that's totally like it's too abstract, doesn't make any sense, you know, we should just let the actual. This might like a related question be like, you know, what is the thing that gets us to

41:40a better version of intelligence, the harness and the model, is it just the model? And it's like, we see this in, you know, at least in a apply day eye or, you know, the startups where it's like, it's a continual battle of like you need the harness, but then the harness that adapts very poorly to new model because sometimes like a very, very minimal harness is still the best way to expose to the raw power of the model. But then now we also have these like training regimes where we require the harness to be, you know, trained with many part of this is to keep things more proprietary and harder to harder for other people to use it. But I think partially it's, it's maybe actually that helps have more control on like the reasoning traces you care about. It's a long, rambling way of saying it's like, yeah, I don't actually, like this is so interesting to see how the models have gotten better at math. And maybe something that's like very abstract and hard to describe like taste is a way to tease out like what is actually necessary here. I think my only like non-truly-althought here is that like when you're working, I mean, just when you're doing any tasks, occasionally you get pigeonholed and you like work really

42:41hard and just having a friend look over your shoulder and be like, what are you doing? And then just like just having that one bit of like step back for 10 seconds, like this is often very useful. Yeah, yeah. I mean, I see no reason why humans would be so different than models somehow. Yeah, yeah. Having, or models would be so different than humans, having, having a few humans working together is often more powerful than just having one. Yeah, it's like a most kind of collaborative thing you actually kind of, yeah, artificially created it, but it's very similar in dynamic. But I think a lot of taste is also like having a sense of what problems you or like some method you have in mind are going to be good at solving. I mean, certainly there's some amount of like absolute aesthetic point, right? But there's also just like, you know, having a nose for what what you might want to pursue because you'll be able to make progress. And you know, I think for that like there's, you know, you would expect that I was a side product of being good at completing

43:42past you would you would get there sort of right? Let me know if we still want to do like a section on soft at groups because I think, you know, up to you guys, it's definitely a super interesting. So maybe the first question is what is a group of some minor selves? So a group is a set of elements with some multiplication operation. And basically, this is how mathematicians think about symmetry. So you're like, basically like if G and HR elements of your group, then GH has some is some other well-defined element of your group. And you have like a sociativity.

44:47And you have an inverse. So for every G, there's some inverse. And there's some like specific element in the group that is kind of the identity. Okay. So you know, it's it's some like abstraction of like, composing operations. So these could be like numbers. They could be like multiplying matrices. They could be like like, like, rotating something, which is, you know, a special case of multiplying matrices. And our group is so thick. What if? Well, there's some, you know, precise definition. But, you know, roughly, it means it. All right. So I should say like groups that can be finite or infinite. So like, you know, if you have like a square, like all the rotations of it form a group with like four elements,

45:48if you have like a circle, then the rotations form a group with like, uncountably any many elements. And so, so if it groups are either finite or countable, you should think of them as being countably infinite. So there's like the same number of elements as like integers. And if it's so thick, if in some sense, it can be approximated by finite groups. So we didn't know if there was a non-sofic group. So the result that Astro proved is simply that there exists a non-sofic group. Yeah. And without like, I mean, we can, you know, before going to, to that prove it is like, you know, it's like, I feel like a lot of the programs from math is like, okay, we are such finite creatures. Let's see how well our finite approximations are, you know, do. And in this case, especially for the countable case, it's like, maybe you'll be relating it to like the Aldous Leon's thing. It's just like, it helps kind of anchor the picture of like, it seems like such a, I mean,

46:51it's a nice result. If it were true, but it's not. And it seems almost like reasonable. And so, yeah, I actually didn't, didn't go, I would love to hear the explanation of like how it found a counter example. Yeah, I mean, I would say that like, you know, the hope that there was no non-sofic group. So every group has this kind of approximation. Like, maybe this is sort of like people hoping that there's a miracle because it turns out that groups like this have a lot of nice properties. Because you can run certain proofs for finite groups and then, you know, kind of approximate them in whatever way the definition of being so thick lets you approximate them and I get the result. So, so like, there's this notion of being a surdjunctive group. So there's there's some fact that any group which is sofic is also surdjunctive.

47:52Surdjunctive is some property of like dynamical systems on the group. And I guess the original question was whether every group is surdjunctive. This is some question of goth shalk from the 70s. And this, this fact that follows this like pattern of prove it for finite groups and then do this approximation is what motivated the question about if there's a non-sofic group. Yeah, maybe I'll say a little bit about this all just slions conjecture. Yeah, sure. Yeah. So I guess I had like heard of this a little bit beforehand because there's a related stronger conjecture in probability that was made popular by all this in lions. This conjecture roughly what it says is like any like infinite graph with some nice property called you know modularity. In modular random graph can be approximated by large finite graphs.

49:14So maybe the way to like explain what these kinds of things are trying to say without getting into technical weeds is to say what they mean about the integers. Like how would I draw the integers as a graph? So this is called the keely graph. You're just going to connect nearest neighbors. Okay, so there's some kind of canonical way in which this is like the graph that represents the integers. Okay. And there's some sense in which you can approximate this by finite graphs. Why? Well, if you look at integers mod n then you kind of get the same picture. But like you have like a big circle instead of infinite line. And the point is if you like look at any point here and any point here like in nearby things look

50:14the same. You have to go like very far away to kind of see this global geometric structure that you have a circle and not a line. And in fact the integers and integers mod n are both groups just by like adding numbers or adding numbers mod n. So these integers mod n are like sofic approximations for the it-full integers. So like this approximation is kind of why the integers are a sofic group. So the the soficity the the statement that every group is sofic is sort of a generalization of the fact that you can do this approximation with groups. And this all those lines conjecture is kind of a broader conjecture that like any network you can do this and it like you don't require as much algebraic structure roughly. So it's kind of a broader conjecture. So so this conjecture was disproved earlier like two years ago. And it was kind of a really torrid of force work. Like it was like 250 pages building on another 200 pages it uses like

51:20like quantum complexity theory. So it really builds this like you know very complicated bridge and like you know I think not many people could understand this. Right. So since this is a stronger conjecture the disproof like is weaker than disproving this statement that all groups are sofic. But it turns out that the direct proof that all that there's a nonsofic group was like much shorter and easier than this than this really amazing disproof of all those lines conjecture. It's like like 15 pages maybe and it doesn't have any of this like very complicated connection with quantum complexity. It just kind of stays in group theory land. I mean it uses some important like you know existing results by other mathematicians like Kuhn and Kuhn and Tom. But it's like it's like a very reasonable normal kind of proof. Yeah. And it's kind of spelling maybe out the obvious but like the connection between the sofic group statement is just you take the kelly graph and that's the one that is like what they

52:22use or for the oldest land. Yeah. And so that's why it's like a you know a subset of yeah. Basically what happens is yeah. So for a group you can take exactly a kelly graph. So you take some like elements that like generate the group and you kind of connect elements that are adjacent. So in this case like this is a kelly graph of the insurgers. So when you do that from a group you get like a deterministic graph. You just get like a single graph. You fix some set of generators. So this conjecture is stronger basically because it allows a broader set of graphs that aren't deterministic. It allows them to be random but have some extra you know modularity property that constrains exactly how it can be random. But yeah. But basically that's that's the difference. Here you kind of have to give a deterministic network instead of a random one. Yeah. Anything kind of interesting surprising about the result. I mean you mentioned some things

53:23which is like it's it's stayed within group theory. The techniques. I mean I think maybe it's like a nice example of this general pattern that theorems produce by AI have generally been like the proofs are pretty short generally. They're like. Like with a counter example so far. Yeah. But yeah. This one it's like okay. It's sort of a counter example but like there's some you know there's some like stuff you have to do to analyze things. The difficult part here is that like the property of being a sofa group is not so easy to get your hands on. So you have to find like a concrete way of saying like like producing a producing way of saying this group cannot be so big. And like yeah. And the proof is actually it's very short. It's almost it's a combinatorics argument. It's like a very very delicate combinatorics argument. The model like somehow you need to both have the right statement and know what piece of literature and then I execute it correctly. That's very nice. Like the difficulty in this problem is that like it's just a really really hard. It's

54:28like very hard to get your hands on like being approximate by any possible finite group. Yeah. So it's like what is happening at that countable infinity that's like resisting this approximation. Like do you guys kind of do give a sense of like when do you do post mortem we're like okay aster explain to me like what is the oh yeah we did the yeah. Yeah. Well it's a good explanation. You got out of it. I think there's like some concrete like combinatorial instruction. Basically it's like it's hard to explain but there's like some concrete combinatorics of structure which if you read the previous papers you realize that that's what they couldn't to rule out and aster found a way to kind of say okay no no if you add this one extra algebraic fact this this like weird conspiracy can't happen. It's like very clearly trying to rule out a conspiracy it and the sort of previous author said implicitly written about. Yeah. And those were the actual suspects. Yeah. It turned out so they were sort of on the right track and then this did the last mile of well whatever however you quantify that. But I think it's like like a year ago I would have been

55:32very surprised to learn that like all of these AI proofs are like very short and elegant. Yeah. Like they're you know you you're kind of like afraid that they're going to like generate all these thousand pages. Yeah. But it's been kind of the opposite. Yeah. Like only humans can generate like 200 page proofs right now. Yeah. Well and also I was like asking like if you do that post mortem it ends up usually engendering more mathematics. So when you do that with humans like that's what you know breeds new mathematics. So maybe maybe if you kind of alter the prompt a little bit and be like how would you you know and generalize this or something like I yeah I don't know if that's been a technique for you guys to like have it explore and exploit what it has already developed. Well there has been some there has been follow up on this actually by by Kunim Tom who this was always built on so they like. It's not coming news coming. Yeah. Yeah which is kind of what we're hoping. Yeah. You know we don't want to be you know writing lots of follow-up papers ourselves but if there's some interesting follow-up that you

56:35know it's like we're very very happy that there's some there's some follow-up building out these ideas more and giving like more examples of non-sofic groups in this case. Yeah well actually maybe it's a great segue into like how you know what's the ideal way that this is being taken out by the math community because I feel like there's a spectrum of answers from working mathematicians sense of like some you know I probably most of this point are like okay AI is obviously doing some non-trivial stuff it would be disadvantaged not to admit that in my workflow. I've definitely heard some stories where people are kind of you know would find it hard to either take AI as a co-author or like how do you even do kind of attribution this way but I don't know like what maybe to paint the more optimistic picture so you're saying you want the mathematicians to be building on these results it definitely generates a lot more results to be verified so you know puts pressure on the community and the profession like how do you how do you kind of expect the evolution of about taking in collaboration with mathematicians. I mean given that the fact

57:41that the models can produce sophisticated mathematics means that they can help you understand like sophisticated mathematics. I mean like I don't know KShare like I enjoy looking at the archive and I want to understand and prove and like I could read the introduction but in practice it's just much faster take the PDF put it into my favorite model and then like get an output of like what is the rough proof strategy and somehow this like along I mean of course models are going to help us produce exponentially more mathematics but they also make it much easier to absorb it. Yeah and right now okay it's still a bit of a challenge back and forth but I think it's for me at least a much much faster understanding it's much much faster to understand a piece of mathematics with a model than without it. So it's helping solve the problem it creates anyways. Yeah I feel like that at least and it's you know I don't I don't view it as like creating much room. Again like I don't have such you know high stakes and like okay I'm gonna get I'm not gonna get tenure etc so like I agree like making it more accessible like if I'm not spending so

58:44much time absorbing an area I can like put it into chat GPT and then expect to I mean you guys having a more powerful model hopefully really sing. It's for other people to enjoy as well but like it's I think like the positive version of that is actually more people can participate in mathematics. There's like people might be coming with other intuitions and they could actually maybe generate good mathematics. Is that sort of like closer to the vision of what you're hoping this is you know pushing towards or like what what things do you think path of to just to be wary of to kind of adapt fast enough to take advantage of AI. Yeah I mean I think certainly there will be a lot of changes right like I guess in math like there are there are a lot of things that are kind of important for for like a given result right you need someone to come up with it but you also need people to understand and absorb it and like you know internalize it enough to do more with it and like figure out where it fits into like humanity is understanding right and like a couple of years ago

59:52like the proving the result was like so hard that kind of the other stuff was just kind of coming along for the ride right you know like if you if you manage to like prove this thing yourself you're automatically going to understand it quite well you're kind of responsible for like maintaining it in some sense and like you know explaining it to other people and yeah now this kind of what was the main bottleneck before is kind of much less of a bottleneck and you know these other kind of constraints come into play so yeah the like optimal structuring for you know organizing knowledge could could look rather different yeah how does that look I mean did does this make the field a lot more kind of empirical what people do sort of the hard like the first thing that was scarce which is like all the reasoning and then more I mean not that it's like a bad thing to make it empirical but it's almost like it functions is a very

1:00:52different discipline like a lot of the fun stuff is understanding you know and so go into standing communicating maybe assembling having still the human taste is that sort of remain verified and and that's how you know current mathematicians need to adapt and end reward you know contributions or or is this too much of a caricature it's like something else I think certainly understanding how to put as we get more and more mathematics put in like a proper frame where can sort of how sort of like be able to explain it to other humans so that they can also appreciate I mean so implicitly we valued this but it was usually because you were the person proving the results that gave everybody else the understanding but I think increasingly would be a function of like you're sort of helping you're the human who can sort of give this understanding to other people and sort of help them with it I think that more that communal understanding will I think become is much more implicit in how we view math and gene but I think it'll be an increasingly more explicit and doubtable part of the subject I mean a nice thing about math is that

1:01:57the ceiling for a difficulty of a math problem is pretty high so even if you know kind of even if I get you know continues getting like exponentially better at math like it might you know plausible will never solve something like p versus np and it could be that like the the field kind of becomes more you know attached to like like these big mysteries and less to like smaller mysteries that are more like routine now yeah yeah I think that's a positive there's another I mean also like I don't know there are things I spent like months or years of my life wondering about not getting to know and yeah I'm excited about this like renaissance of results and understanding and I feel like I mean this is such an infinite you know field like okay no pun intended but like it's just like it's it's just there's so much that you can

1:03:00actually create here so I mean especially me for somebody like me who's not gonna have the time to actually like practice mathematics now there's like a lot more that you can actually do the activity you know so yeah I think the the like the ability of someone who's not working on math is like their literal job all the time so like understand what's going on and like you know learn about some of the mysteries they might have wondered about we'll go up quite a lot um also you know if you're if you're like if you're working on something that requires some math no suddenly you don't need to like find a world expert on topic to to be able to you know use it in your own work you can sorry mathematicians it's getting me no it's true I mean I think there was just like a dearth of actual like people who could could do that and so I think this is helpful maybe it's helpful for the theoretical physics like we'll see um but a lot of other applied areas as well it'd be nice for the world to apply mathematics on a lot faster yes I mean I'm I'm over that well thank you guys for joining this is a lot of fun and I'm you know just

1:04:03so excited for how much the models are advancing so maybe we'll have you guys back as soon yeah thanks so much for having us yeah thanks for having us thanks for listening to this episode of the a 16z podcast if you like this episode be sure to like comment subscribe leave us a rating or review and share it with your friends and family or more episodes go to youtube apple podcast and spotify follow us on x a 16z and subscribe to our sub stack at a 16z dot sub stack dot com thanks again for listening and I'll see you in the next episode as a reminder the content here is for informational purposes only should not be taken as legal business tax or investment advice or be used to evaluate any investment or security and is not directed at any investors or potential investors in any a 16z fund please note that a 16z and its affiliates may also maintain investments in the companies discussed in this podcast for more details including a link to our investments please see a 16z dot com forward slash

1:05:05disciples