This 24-Year-Old Raised $64M to Build an AI Smarter Than the World's Best Mathematicians

27:20Carina Hong:It's sort of the pre-conjecturing step. That's the fascinating part of, I think, mathematicians. Like their reasoning is a very like specific and specialized process. And these examples guide their reasoning. um it's it's very difficult i think if you do math without being able to do any constructions or examples some of the best mathematicians um they are great at generating examples that go one way or the other so either examples that tell them hey that's a direction worth pursuing or constantly having examples to eliminate all other possible routes you can you can sort of search your space that way.

28:05Carina Hong:And Axiom's vision is formal math is great and mathematical discovery is great. And the sort of informal approach of like, you know, like literature search, explaining concepts, quick, you know, like quick query in the knowledge, that's also great. It's not a one size fits all like AI for math shoot. We have a very comprehensive vision that We need all these. So we need to build all of them out. And the pattern boost led by François Chartrand and Alberto Aferino that's on other works such as End to End, these are the two toolkits we have on the mathematical discovery side. So pattern boost is a generative method.

28:47Carina Hong:It generates interesting examples that play a pivotal role. And those examples are novel. They are out of the box examples. they sort of help mathematicians derive interesting intuitions just by looking at them and the into in part is the translative methods into in is like int to int the idea is if you have a problem and you have a solution right then you have a problem solution pair and you learn all on all these problem-solution pair. And so next time you have a problem and we want a solution, the model just predicts that solution. It's that sort of like translation approach. And there are interesting tricks related to this translation as to which direction.

29:37Carina Hong:So back then in like natural language processing, people find, you know, one direction of translations usually easier than the other. So, you know, a model that has seen a lot of English likely find French to English being easier than English to French, right? We have currently auto formalization being harder than auto informalization. So lean to English is a lot easier than English to lean because model has not seen much lean. In these sort of problem solution translation problems, if you have, for example, taking the derivative, the first derivative or the differentiation of a function, if you think about it in high school, like in K-12 education, we are asked to take the derivative.

30:21Carina Hong:It's actually relatively easy. Do the integral is hard. Like the integration part is difficult, right? Because like, that's why you have these integration bees, you know, having the math widths, like trying to crack those hard integration problem. And they're sort of inverse of the other, right? Like you take the integration and then you take the derivative and then you have the same thing. So one direction of the translation is easier than the other one. And we take advantage of that in some of the works where int to int is applied to. So, sorry, you asked about pattern boost. That's okay. No, that's good, yeah.

30:54Carina Hong:Yeah. Specifically, we were asking about that. So speaking of pattern boost, there was the 30-year-old conjecture in graph theory. And then I think in int to int, you solved a 130-year-old problem, right? About the Liapunov function, if you want to go into either of those. Yeah, yeah, 100%. So Liapunov functions are important mathematical objects. They show up in the three-body problem. That's kind of like a very famous, you know, case. And these sort of dynamical systems and Liapunov functions have been studied. And there are many kind of like semi-global Liapunov functions, like many different things.

31:35Carina Hong:Poincaré, Newton, Lagrange have all studied on them. so like we have found you know these mathematical objects to be extremely hard to find like humans generally relate on like you know pure luck to to find them or deep intuition like but no one notes those intuitions down it's like you know like i don't know if their their notebook that has like run out of like space like the famous for mass joke it's very hard to figure out how to find them. There's no general algorithm. And, you know, Leopoldo functions, they represent the global stability of a dynamical system. There are cases like, you know, polynomial system, non-polynomial systems, and both are very, very hard.

32:21Carina Hong:By training on a dozen million synthetic example, like Francois and Alberto, they taught the model to reliably guess these functions, something that humans cannot do. And it's, you know, it's a dozen million, which is actually not a lot. So a smaller model, not a lot of compute, can actually crack something this sort of fundamental that bothers humans for many decades and centuries. And they went for even harder cases like the non-polynomial systems that were previously just considered completely out of reach and not tractable. So I think it's very interesting where the Liepunov functions case is solved, but it's also very interesting that the end-to-end, it's a code base, it's a toolkit that you can apply to many problems that will benefit from the translative method.

33:12Carina Hong:So if you have another problem that people are excited about recently, we're thinking about something along the line of number theory, keep it awake, but the end-to-end code base likely is going to be equally useful there as well for all kinds of problems that will benefit from the translative approach. That is so cool. It is. So you describe a loop where your AI generates new conjectures, tries to prove them, learns from successes and failures, and gets better, which is super amazing on its own. But how do you prevent it from getting millions of trivial, useless theorems as well in a thousand different ways?

33:57Carina Hong:That's a key question. That's a very, very good question. so first of all is there are different programming languages in the past that we have seen provide interesting math results that are in a way disappointing i actually had this conversation with um stephen wolfram and um the the wolfram program language is like really amazing but there sometimes are like you know results that you look at him you're like why why does it you know he he wrote it in the blog himself that there are moments like that. For Lean, because I think it has seen a lot of math, it's a pretty sort of robust language. It has seen a lot of undergraduate algebra and analysis.

34:42Carina Hong:And by a lot, I mean, compared to some other programming languages and the sort of abstraction level that it functions is, I don't think it's necessarily the optimal abstraction level, but it's passable. It's pretty good. It is actually able to do some independent exploration and to generate new theorems and that of reasonable mathematical merit. Of course, when it comes to taste, there are many ways to do it, right? One is to, you know, have LMS judges to see, hey, like, you know, how would you rank the interestingness of this mathematical result. There are other ways to more programmatically program them.

35:24Carina Hong:Like we can look at the dependency graph of that proof. So, okay, well, this proof is the child of two previously unconnected branches, algebra and combinatorics, you know, like just like the 2022. We should check this one out. Yeah. Yeah. Then that would be novel, right? If you can have actually a correct proof that that bridges two previously unconnected views. That's novel. And the same sort of way I think about it is machine-assisted mathematics or science actually promotes a diffusion of ideas between the differences. It doesn't kind of get bounded by the human boundary of like, okay, well, this is a probability conference.

36:09Carina Hong:This is a geometry conference. I remember I was invited to a conference called the Doktor Seminar in Germany. and we our conference was i think it was called like pattern avoidance you know it's a combinatorics like serratical computer science one and there's one next door on like continual learning my math research at the time was on continual learning so i was very excited i would like sneak into their talks all the time but that's sort of like diffusion of ideas right like it doesn't happen unless you literally put like you know two adjacent like workshops next to each other in the same castle in the rural area in Germany.

36:48And they're drinking coffee together and having weird conversations.

36:51Carina Hong:Yeah, exactly. So I think that's something I'm really excited about, about AI for math and AI for science is for the diffusion of ideas across fields. One could also, I think, you know, look at how many new theorems or notes in the knowledge base that got extended because of introduction of one definition or one lemma that I think Yosha Bengio has a really good paper on that. It's about thinking about AI mathematician from a compression lens, like first order logic lens. And you can say that something is good if it's useful, that's reasonable, isn't useful in the mathematical playground. Yeah, I think that Those are like some, and you can have also sort of ideas where you have a conjecture and that because that is plugged with the prover, you can have the prover provide reward for the conjecture.

37:48Carina Hong:So something that can be proved, you know, and suppose, okay, suppose something is there's some rough proxy of that thing being interesting and that thing can be proved, then it's likely to be a good thing, right? And if you have something that just cannot be proved, when it shouldn't be that hard, could it be wrong? There are interesting disentanglement mechanisms one can do that we consider very interesting. That's awesome. Something you said sparked something. So you were talking about how the diffusion of ideas, AI can move beyond our human boundaries, which I think is really accurate. it.

38:31Do you think that we as humans have sort of one of our skill sets, right, is that as a collective, we have all specialized into all these different subfields and subdomains and organized that well so that we could work within it. But those things also sometimes hold us back. And do you think that one of the better things about AI is that it can then take all the information and go and actually find new patterns that we humans are sort of like stuck on?

38:56Carina Hong:Yeah, I think there are many exciting opportunities when AI comes into the game. So first of all, AI can be deployed, AI mathematician can be deployed to work with applied scientists. That's very exciting. If you have the best brilliant math human mind, they likely are in Princeton IAS and do advanced study and staring at the deer thinking about their pure math problems. They're not collaborating with mechanical engineers or people who really need their hard partial differential equation to be solved to make industrial real societal progress. So that's number one. Number two, they don't even necessarily collaborate within mathematics if there are different fields, as we talked about.

39:44Carina Hong:Now, the third thing that I find interesting is like when software engineering projects are like sometimes like hundreds and thousands of people working the same one, you don't have math being that sort of collaborated and refactored that's suited for collaboration. We don't have... Polymath projects actually are the first instances of mathematicians or even math enthusiasts from all over the world collaborate on pushing the bound, squeezing the bound of the prime gap to smaller. And that is really amazing. That's in the informal math side, right? They are reasoning in the natural language for that.

40:21Carina Hong:And then you have the lean, big formalization projects led by Terence Tao, Alex Kontorovic, and others like the Prime Number Theory and Project. They are able to divide the task to very fine-grained levels. So people who don't have the big contacts, who don't have necessarily the mathematical ability to see the 3 ,000 feet view of the problem can contribute on those tasks. that they have the sort of polymath but lean version. That I find very exciting. What we want to do at Axiom is through product. That's why we really think that we should be a product company after we have a really good model and technology.

41:06Carina Hong:And it's to bring these into people's hands for them to be using and collaborating and to work on generating these large-scale collaborative lean effort like at a much faster like speed and hopefully you know people can find it more and more accessible so that will sort of lower the entrance bar because now I think a lot of very good mathematicians use lean but not not all of the mathematicians they're necessarily there isn't like necessarily a you know correlation between mathematical ability and lean proficiency but one could argue that if you know, someone is just like learning how to do proof, then it might be challenging, right?

41:46Carina Hong:So how do we sort of serve those people for them to be generating these very exciting work products? And that will also in turn, like data flywheel, makes the system a lot more intelligent. Okay. So when your AI finds something new that no human has ever verified, how do you go forward and know that this is correct and not just something that looks correct. Like, who checks the checker, of course? Yeah, exactly. So suppose the statement is correctly formalized and the model finds a proof and then it compiles. Yay. Like, by the Curry Howard, like, you know, correspondence, like, here you go, you have a proof.

42:35Carina Hong:That's really amazing. What has happened in the past, and we have seen some other companies model release is that the statement are not correctly formalized. Not all of them, say like two out of 49 in one instance were not correctly formalized. So I think that part is, you know, you need checking. But interestingly, right, the proof is like, sorry, like the statement is like this long. It's like two, three lines. Okay, your proof is like this long. And this thing does not have to be checked. Like, isn't that like amazing already? Like, I think that's - Oh, it's one. Yeah. Yeah. And that's because it can be verified through the system.

43:09Carina Hong:Yeah, exactly. And purge can go on and on and on. Exactly. And suppose we have a really good construction engine baked into it, and we ask the construction engine, hey, I don't know if this statement is right. Find me a counter example. And they found one, and then, okay, well, there's a problem. That's why what I mean by construction stress test like lemmas. It's very exciting. And any logical argument, if you think about this, any logical argument can be refuted by counter example. Like, yeah, I don't know. Mom, all Chinese restaurants have chopsticks. Here's one that doesn't. I don't know. There will be one, yeah, inevitably.

43:55Carina Hong:And that's really like if you apply that to these sort of more general domains beyond math and like, you know, coding, like edge cases, code verification. Suppose you have like legacy code at a company that serves critical business function. You want to, you know, take them out. You want to like replace them with state of the art. And I want to make sure that all my core business function use cases are still covered. You really need to verify that code, especially given we have so many AI agents, coding agents now generating code. And the model that can do formal verification while en masse can formally verify code.

44:34Carina Hong:That is a huge opportunity, similar to other cryptographic protocol verifications, smart contract verification, et cetera. So you're able to do those verifications from multiple angles over time? Yeah, and we intend to diligently sort of pursue them one at a time. But they're all built on the core technology, starting with AI for formal math. Gotcha. So you raised$64 million to build this so far. You're solving 130-year-old math problems. That's intellectually very cool. but what's the first practical thing that you expect Axiom to do that makes someone you know want to write a check like you mentioned productizing this at some point like or maybe you're there now and what problems do you expect to solve you know this year or 2026 that could potentially directly immediately be used in business or industry research lab etc yeah 100 % so we want the model on the technology and we wanted to have a few important capabilities that are practically grounded.

45:39Carina Hong:One is we really want the model to do induction very well. That will have very important implications in, say, circuit property testing. These are... So in a way, the model is going to be smart and right, like verified superintelligence sort of have these two goals. If the model is smart and right, it can help, say, quant trading. if the model is right and not that smart yet, it can already solve a lot of problems. If you think about other big tech companies like AWS, they have a whole neurosymbolic reasoning team doing very similar things. And a lot of customers actually chose them because of this feature, especially heavily compliant, required or regulated enterprises.

46:32Carina Hong:We need to prove a financial audit, et cetera. they actually have the sort of considerations of formal verification in mind. Now, LM together with formal verification is going to make formal verification a lot less painful than 1980. That's the bet that Axiom is taking. And so, I mean, by I think like, you know, next year, we should have an extremely strong math system. And this system is going to have really good transfer learning capability on other important benchmarks that have commercial relevance. And then partnering with very selective folks, we are going to try to see if these technology gang can, how much we need to build to bridge from what we have to what they want.

47:27Carina Hong:And that middle layer is actually not the blocker, right? Like the core engine is the blocker because that is something that is differentiated and very difficult for these customers to. I mean, if they can't have it in the Frontier Lab models and they are definitely not going to build it themselves, then Axiom is like a very natural choice. And especially in this grand market opening of us being one of the leaders in the startup space for AI for math. Yeah, for sure. Well, let's say Axiom succeeds with its main goal, which is to build this mathematical superintelligence right now. Five years from now, how does, let's say, a software engineer, a financial analyst use what you've built?

48:14Can you kind of paint us a picture of what their workday would look like?

48:19Carina Hong:Yeah. So you open up your laptops like, you know, like an interface. And we want to build the default reasoning platform for these people who are engaging with very complex quantitative reasoning tasks in their daily workflow. It's some sort of, you know, IDE. And you can put in like the problem you're solving, a spec. Sorry, like you put in the problem, you put in like a spec, which is some sort of, you know, your preliminary intuitions of how to go about it. And then the model immediately starts suggesting the intermediate steps for you to proof-reject. That could be like the first sort of co-pilot thing.

48:54Carina Hong:It will still require the very smart person's supervision, but it will be more and more automatic as time goes and as we collect more high-quality data. I think also the model is going to have the computer program, right? It's a LIMP program underlying that mass proof running for the tracker. and then you can see that checkmark knowing that output is sound and hallucination-free. The model can help you explore. If you don't have that spec intuition, you can have the examples, the discovery part that we just mentioned, pattern-boost generative methods to help you generate interesting examples or translative methods that help you find the function by learning previously the problem-function pairs.

49:40Carina Hong:and you know it can handle all sorts of like translative like tasks it can be numeric to numeric it can be symbolic to numeric, numeric to symbolic and symbolic to symbolic right like it's really symbolic are like things like functions, numeric are things like values so give me a function, give me a degree output is a degree that's symbolic to numeric give me a like you know give me a, this is another example of like numeric to symbolic I mean give me some sort of a I guess a give me a give me a matrix I guess I give me a matrix and then I'll give you like the set of set of equations like that that will be like numeric to symbolic and these are very important objects in that quantitative researchers like daily workflows and you can you know have that part of Canva to explore it requires a lot of things going together I mean the we have tried different sort of, you know, explorations.

50:39Carina Hong:And because the core technology has really not been in place for some of the obvious choices of big companies, big labs, we couldn't get it to work. I mean, we couldn't have GPT generate a Fourier, like, transform with the symbolic, you know, like, notations baked in, like, reliably or correctly. That's quite interesting. So have you been building out like your own data sets of mathematical equations and proofs and solutions to put together? You've been building these out as a team, I assume, and gathering them elsewhere as well from places you trust? It seems like your step one need is a lot of math.

51:27is the way.

51:28Carina Hong:I'm so glad you asked this question because one thing that we do very differently from, say, our competitors is we take a very bold synthetic data generation approach. While there is a co-start problem, we are totally solving it. We are converting the informal mass data into formal mass data by the system. And we are generating more examples, synthetic examples, based on the existing formal data that we have. Oh, wow. Instead of sort of paying all the lean developers from all over the world to gather the proof. We do some of that as to solve the cold start problem, similar to when you develop Chats and Go at the beginning.

52:11Carina Hong:You would have Chats and Go players, experts to give you certain configurations, the starting point. We do a very tiny part of that. And we most do that with statements. So we are taking a very different, I think, in both data bet. We're feeling pretty good about it. Yeah, that's great. That's awesome. Well, we think AI for science is really cool. What do you see as being the biggest bottleneck in the way of building this super intelligence for both you at Axiom Math and in general for the entire industry? Is it the data or is it the compute? What do you think? Yeah, so I think AI for science is like a big, you know, word.

52:58Carina Hong:It has multiple parts. The digital part, the algorithmic part, that axiom, AI for math, we are totally dedicated to solving. I think a lot of the AI for science companies like, you know, periodic labs, et cetera, they are actually building more like real world part. So we do not have, I mean, like Laila said, we, for example, we do not have robotics. At this company, there's no robotics labs. We cannot have robots do the biology experiments for us. But we can solve some of the hard theoretical problems that they will encounter. So it's not so much an AI for science company. I wouldn't put us in that category just because we don't have the real world testing part.

53:44Carina Hong:But a lot of the similar philosophies and intellectual journey of how we got here, it's similar to AI for Science, which is that do not take for granted that fundamental science transform into real world application. That timeline is what it was before. It's no longer what it was before. It's no longer 300 years. Like from, say, Hardy in a mathematician's apology be like, wow, the stuff I work on will have no use whatsoever. And he's, I mean, like you couldn't see it, but we all got surprised when that was the sort of the backbone of cryptography, right? And that number three, the big part in that, a lot of the fundamental science can be, in a very short amount of time, transformed into real world applications.

54:32Carina Hong:And I think investment in that space generally are very bullish, like AI for science, AI for math. The one thing I find interesting is people who are bullish about the AI for science part. And I think it really needs a coherent thesis to also be bullish about AI for math. You kind of can't, you'd be like, okay, hey, I am bullish about the root word testing part. I'm not bullish about this radical breakthrough. So then, you know, making try and error like engineering approach without any sort of theoretical guidelines, guidance only in the in the real world. And that's incredibly expensive. I think that's that's like one part that's missing.

55:16So I think I say, don't you need one for the other? I would think you need. I would say programming, too. You would think. it's interesting to me that software development through through ai has moved as far as it has without a product like yours at its base but do you mean like a symbolic like math kind of prover at the base not like in a symbolic logic way i'm meaning like like you you would think that that math bedrock again for lack of a better word i'm gonna use bedrock again uh yeah would have been such a necessary step for tools like quad code for uh for codex to have that built under there and i know there's definitely a strong math core under there but what you're doing is is leaps beyond that i feel like yeah yeah i do say that you're

56:10Carina Hong:Right, in that it requires the same sort of hierarchical planning. And that's the high-level planning. Also, I guess, like low-level, like detailed execution. You want to bridge them together to build a very strong coding technology. That I agree. And the similar ability of having many lemmas forming a huge complex mass proof is a little bit also similar to blocks of code forming like a big comprehensive query. And we have seen coding models being great at making local changes and suggestions. The ability to understand multiple files, make changes to them, that backbone logic, we have not seen that.

56:58Is that something that you think what you're building could solve, especially with long context stuff?

57:02Carina Hong:I think so. That's amazing. Karina, this has been absolutely fascinating. I thank you so much for being here. Thank you so much. This is a fascinating conversation. It's been a lot of fun. If people want to learn more about Axiom Math, keep up with what you all are doing. What's the best way to find you? I think our blog recently updated with the discovery part. We are going to introduce some key members of our discovery team through multiple videos over the recent three weeks. And we are organizers of many big AI for Math conferences and sponsors of many conference workshops as well. You can find us at NeurIPS.

57:43Carina Hong:We are a sponsor of AI for Math workshop at NeurIPS. And then at the joint math meetings in January, we're also going to give some invited talks. After JNN, we have ARHUS, which is an AI for Math discovery workshop that we are co-organizing. And that is the sort of next thing after Oberwolvog, which we also co-organize. So we are having a lot of community gatherings, and please do find us at one of them. That's so amazing. We'll make sure we have links, right, Grant? Yeah. Awesome. Great. Thank you so much. Excellent. Well, I'd like to thank everyone who took the time out to watch or listen today, wherever you're viewing this from.

58:22Mathematical superintelligence might sound abstract, but the applications from chip design to aircraft safety to furthering science are so concrete. And the good news is if you're watching this, you're already thinking about the future and where AI is heading. And hopefully they'll give you a leg up. We hope you enjoyed today's show. If you don't already, please like and subscribe so we can continue to bring amazing conversations like this to you every week. And join more than half a million others who read the Neuron AI newsletter every morning. Just visit the neuron.ai today to sign up. And on that note, until next time, farewell, humans.

59:06Thank you.