Episode 17 - What happens now that AI is good at math?

the models they are able to help field medalists in their day-to-day work. So really the the jump is is just simply astounding. And maybe if I can build a a little bit more on that. Um Something which is important to understand is that everybody has been surprised by this progress, including us. So to tell you a a story, um a year and a half ago uh I was at a workshop at a conference with other fellow mathematicians and there was a debate that I participated in on whether LLMs scaling LLMs

will help us resolve major open problems. So this was a debate, you know, a year and a half ago. And and the room was very divided. In fact, I did a poll at the beginning and I think it was like 80% said no, impossible that this would happen. So then the debate unfolded and you know by the end of the debate it was more like fifty fifty, so you know, pr pretty good progress during that that hour.

This obviously was just so wrong in hindsight. Like just mere eight months later, the model were starting to be able to do research level mathematics.

So that was amazing news and that demonstrated that, well, at least for the competition level mathematics, uh the models are capable, very highly capable, um, only um on par with the the top human high school contestants. Um but uh well competition problems are canned problems. Uh they have relatively short solutions because they are meant to be solved in it within a few hours, and they're not novel because somebody came up with it, as there's a solution. So it's not research level math.

Um so then I got curious, and a lot of people got curious, can ChatGPT do uh research level mathematics? And there was a lot of debate online. And then I thought to myself, I should try it on my own problem problems. Maybe I'll try it for myself and make up my own minds as opposed to, you know, listening to what other people say, because I'm a mathematician myself.

So I took a uh uh a a classical open problem in uh in optimization theory, which is a a branch of applied mathematics that I work on work in. And um the question specifically is there's a uh a famous algorithm called the Nestrov accelerated gradient method. And does this have this convergent behavior or is it possible that the um for you know in in certain bad cases can there be a certain divergent behavior?

This question was genuinely open in the sense that people know that in most cases the algorithm behaves well. It's convergent. But people really did not know: like, is there a bad instance? Does it, in the worst case, could it diverge? The answer turned out to be yes. And the way I d discovered it is um I remember it distinctly. So so um my bedtime for my son is eight PM and then I try not to stay awake after midnight. So I had four hours of usually

uh evening hours to myself if I wanna focus on something. So I decide, okay, I'm gonna spend a few days working on this. So over the the course of three days, so that's 12 hours total, I interacted with Chat GPT on this question. It wasn't as simple as me just putting in the prompt and getting a solution. And I played the role of the verifier. I told whenever the model made a mistake, I corrected it. I also tried to point the conversation into areas that I felt, approaches that I felt were not.

And after a while, the proof uh uh there was a proof and I checked it. I also asked ChatGPT to double check it, and it it was correct. And that's how this uh forty two year old open problem got resolved. And once I got this uh uh this solution Um, I thought to myself, what what would be the most fun thing for for me, fun way for me to um publicize this?'Cause I could just write a paper and that would be but that would be less fun. So I decided, let me go to Twitter and and talk about this and um

What does that mean from like a scale of a problem? You know, we like understand addition, subtraction, multiplication. Could you give me an example of understanding like? Where we went from, from like, you know, first Chat GPT, which could kind of sort of use it, then it could do a could do math, it could use a tool, but then the model sort of implicits the understanding.

Imagine a scenario where we like the three of us went camping together and then I paid for this, Spr paid for this. Um And and then Andrew, you pay for whatever. And then we might want to clear the ledger. We want to split things evenly at the end. Can ChatGPT do the calculations for us? And it's this is moderately complicated if you have like 17 items that we've purchased. In twenty three, twenty-four, and also in twenty early twenty five, I remember, uh the models couldn't do this.

Another example would be I'm in, let's say in Korea, uh Seb's in Paris, Andrew, you're in California and you want to set up a Zoom meeting. Like what would be a good hour to do so? Um again, in early 25, the models uh couldn't do this. But then just suddenly things just um changed. Uh and I wasn't in open AI at the time, so I'm not at all I don't I'm not quite privy to what exactly y you did, but

um suddenly the models started solving IMO problems and then uh furthermore it started solving research problems. And the uh the way I sort of calibrate this right now is that um unless you are a professional mathematician trying to discover new mathematics. If you are somebody who's like, let's say a physicist or a chemist who who uses relatively complicated mathematics like differential equations, differential geometry, things like this. But um but you're not inventing new math.

then Chat GPT can do all of the math that you would need. So any basically user of high-level mathematics it from STEM can now use

uh ChatGPT to basically have their math taken care of. And you would want to you want to exercise some some degree of caution, you know, to check the check whether things are right, you know, run simulations just to double check. The models can make mistakes. But now um Any math problem that you would want to solve, most people for 99% of the population, the models can do it.

It's not just about scaling the models. So when you ask what happened, or you know, when you you're asking what happened middle of last year when suddenly the model were able to solve mass problems, well a lot of things happen. We do a lot of research and and All of this has to progress at the same time. So I can't really point to a single element.

And I fell from my chair. I was so impressed. What was I impressed by? That the model I could give it the coordinates of points in the plane and it would give me a line that goes through those points. Like when I say that, you know, now it's it's almost hard to understand. What are you talking about? Obviously a model can do that.

So I think we we have kind of forgotten how quickly things have happened. And now, yeah, you know, Ernest was saying that it it's basically at the point where unless you're trying to invent new mathematics, it's kind of at the right level already. I would say we're already seeing glimmers that even to invent new mathematics it's it's it's getting there.

Although you can put a pin on that because we will talk about, you know, in research level it's not that simple anymore to evaluate. But before research level it's very easy to evaluate. So th mathematics was just the perfect benchmark. to see the model making progress during the last four years.

Now we'd say we have kind of saturated that aspect. And you can ask, okay, now, now, okay, fine. The models do mathematics. We have understood. What about the next steps? And for the next step, I would say that. Having our models be good at mathematics is going to be good for many, many other things, and let me explain why. A key feature of mathematics is that to resolve a problem you have to think for a long time. Be it days, weeks, sometimes years.

So this long thinking, not only do I have to think for a long time, but you also have to think consistently for a long time. If at some point in your chain of reasoning there is a mistake, this will kill the entire argument. It doesn't matter if everything after that is correct. If there is one single failure point,

Everything the entire argument is destroyed. So this property makes it that this is what you want out of reasoning models. That if they make mistakes, they will be able to correct themselves. So we are hoping that this properties that they acquire through mathematics will generalize to other domain. Which by the way is exactly the same thing with human beings. Why do we train human beings in mathematics? I mean it's a very fun topic. I love it. We did it professionally.

Uh maybe we still do some of it a little bit. Uh but wh why do we train humans in mathematics? Exactly for the same reason. It gives you this kind of very logical thinking.

Explain what is the progress, in particular because this progress is so fast. So, you know, for example, I I I maybe we can talk a little bit about the AirDoche problems, uh, you know, and and and and some of the controversies that happen uh around that. So Um there was a first example. So there was first, you know, Ernest example and then there were a few other problems that were solved.

Not all the questions that he asked were interesting. Let me just say that uh right away. But still it was very productive and you know they The research committee wrote a lot of papers with him. There is even this concept of an AirDosh number, which is you know, how far away are you in a chain of collaborators from having authored uh a paper with with AirDosh? Uh my Airdosh number is is two. I have coursed

a paper with someone who crossed out with Feudor. So, you know, yeah, I'm I'm pretty happy about that.

So I think there is like a thousand problem or something like that on that website. And Thomas himself has done the work of trying to find, you know, he's an expert in combinatorics. So you can kind of say okay, this is open, this is you know, uh resolved, this has some complicated status, you know, for every every uh problem. Of course it doesn't

Necessarily knows the answer to all of them. So if there is a paper which is marked open, it is not necessarily true that nobody knows how to solve it. But it is also a very interactive website where people can go on it and, you know, add comments to every uh problem and explain whether there is a solution, etcetera. So it's a very dynamic, uh great website.

So of course, once we started to have uh GPT be able to solve research mass problems, this sounded like a treasure trove of problem to try our models on. And um we tried a a couple and to our great surprise the model came back with answers to some of them that were marked as open.

So we got really excited uh about this. The the first one, you know, that I tweeted about uh I don't remember when it was, maybe it was in October or something like that, uh last year. It was a a deep literature search. So let me explain what that means. It means that what GPT did is that it did a vast literature search, trying to scan, you know, thousands of papers.

And it found in some unrelated field the answer to the question. Now it's really important to understand that it's not like in that, you know, unrelated field the person said, okay, I'm solving an AirDosh problem. It was written in a completely different language. It was different mathematics. You have to do work to connect the two pieces, and GPT did that. So that was kind of amazing. And and this was very ad hoc. Like you know, we just tried by hand basically in the chat GPT interface.

Once we saw that, um uh Mark Selke, who is, you know, uh in in in our team also decided to have a a a more systematic approach of trying all of the problems, and he tried that and the the model came back with solutions to ten. Airdosh problem. And this was you, you have to remember at that point, there was still, I think, a very dynamic discussion about whether, you know, those models could go beyond the state of the art and discover, invent new mathematics.

So I I I got very excited about uh this this result and and and I tweeted about it and and you know it's it's kind of an infamous tweet because people misunderstood it. as kind of saying it really found the solution to ten open problems that are very hard and the solution is completely new and did not exist in the literature. But that's not what happened. It was connected of course to the previous case where it is a deep literature.

So there was some you know food with uh with Google about uh you know with endemists about whether you know this this is uh the right way to talk about such results. But now the the punchline is kind of amazing, which is A few months later, so again I said ten uh solutions to uh open problems and these were solutions in the literature, and then the question is can you find solutions that are not in the literature? By now we have more than ten.

actual solutions that are completely new that are publishable in top journal in combinatorics completely obtained by You know, some by Chat GPT and some by our internal uh models. So just within again, this this really speaks to the acceleration. In the span of just a few months, we went to it's kind of a ridiculous statement to say that there would be ten solutions to AirDosh problems to it's actually happening for real and it's accelerating.

Or are there those brilliant, you know, sparks of insight? Everybody of course points to Einstein's, you know, relativity. I'm I'm not even sure that really counts to be honest. So I think the jury is still out. on whether this process of just recombination and a little bit of thinking, whether you can kind of increase, you know, human knowledge with with no limit, or do you really need the sparks of genius that would be somehow only human?

Well, one reason is because it's very easy to benchmark. It's very easy to see that progress. But we have full expectation that this is gonna happen in all sciences. It's not gonna be limited to mathematics.

One point, of course, is that it's compressing those timelines greatly. In earnest story, you know, of solving this problem in 12 hours. I mean, I don't know. Without ChatGPT, how long would it have taken you?

This is kind of needed if we want to go beyond the current level. The current level of interaction, you know, the professor student interaction where the student comes back after a week, it's gonna be very hard with that mode of interaction to do real break.

to solve actually longstanding, you know, research problems or to make problem progress in in you know very difficult fields in biology where you need to interact, you know, with the wet lab and do all kinds of experiments. So once you want to go towards a real breakthrough, We will need to work over longer timelines. And this is where the automated researcher uh comes in. Maybe let me say it in a in a slightly different way. One one concept that I'm a big fan of is this concept of AGI time.

So you can have AGI seconds, minutes, hours, days, and so on. So that really means you have an AI and for like it can mimic human thinking, but for how long? So as Ernest was saying, you know, two years ago, maybe models were mimicking, you know, a high school student who thinks for a few minutes on a problem. Now we can mimic a researcher who can think for hours, maybe a few days.

We really want to go towards and and this progress has has been going on for now, you know, very consistently for four years, where we went literally from seconds to minutes to hours to days, and now we are roughly at days slash one week. We want to go to weeks if not months. This is open research. You know, I I don't think anyone on the planet knows exactly how to do

But this goes back to we are doing a lot of research, a lot of innovation, and I think once everything will be put together, we're just seeing this arc of progress where we keep making progress in AGI time. But this is this is the direction of the automated research.

Because a lot of math papers are longer than 50 pages. And also the the the thought, the human thought that went into to produce, let's say, a 10 or 30 page paper is usually, well, much orders of magnitude longer than the final output. Um so there's a limitation with the the limited context wind window. But um for users uh but people who've used codex will will know that you can actually have very long work session sessions with codex. So you just keep, you know.

giving instructions as to what kind of code you want to write and then the code itself that that you're working on, the repository of your code, uh, which in the math sense, uh the analogy would be that would be analogous to like math notes that you write down. That can be very, very, very long.

Um but Codex has a is is pretty good at dealing with that. It it it at once in a while it compactifies its conta its its conversations, um and it has its way and b of becoming this um really um uh amazing agent that can do really complex jobs over uh huge repositories of code over a long, a really long uh context of conversation. Um and this uh I believe is going to happen with mathematics research as well. So we will be able to have uh LLMs be able to

solve problems that are longer than just, you know, that require more than 50 pages of thinking. And that's what humans do. That's what human mathematic mathematicians do. We peep when people think for a day on a certain problem and then we kind of summarize our ideas and then put it into notes. the next day or the next week we come back to it. And then over several months we've thought for so so long, but um uh it's

sort of summarize, it's sort of organized in a way that becomes manageable and in the end, the final output becomes a 30 page paper uh uh that summarizing the thoughts over you know many, many about months or even years. So yeah, I think that's gonna happen.

my own benchmark here and just generate the data for that. Yeah. And five minutes later I had it. And that was magical to me because I'm in the middle of working on the tool. that would have involved me all of a sudden, okay, I gotta spend a few hours, go do a generator, go produce this sort of stuff. Absolutely. And it runs in the background. I can't imagine what it's like for you guys doing grown up problems.

Uh the the flip side is all of course like that scientists in other disciplines they can also use more advanced mathematics now, thanks to challenge.

I there is no reason I anybody who would look at the situation would say, okay, a year from now you will have systems that can think for weeks, two years from now systems that can think for you know uh years. Not only that, but already today we're finding that our models

They are able to really surpass humans in the sense that they can find mistakes in papers. You know, we had system, we had agents uh that internally that have been able to come up with to to find papers and say, hey, actually this is wrong. Here is the correct answer. Not only that, but people st tend to think that um AI is only good at answering questions.

Actually no, it's also pretty good at asking questions. Of course you need to be, you know, you you again, you need some research innovation there, which we had. And now our models are very good at asking questions.

So good in fact that humans are looking at those questions and saying, Hey, maybe I should write a paper based on this question. So so this is, you know, really, really already happening now. So I think w what I'm trying to say is that In a year, in two years, yes, models could do basic more or less everything that human researchers So now what? What is the role of human? Well, why is it that we're doing science? What's the point? You know the point is not to s I mean

I at least n it it shouldn't be to just solve problem for for the fun of solving problems. We're solving problems because we're trying to understand something. The understanding piece is key. We're not solving problems to write papers. To show to say that we can write, you know, ten times more papers than than our neighbour. That's not that's not the point. They you know you can do ch competitive chess if that's if that's your your kind of deal. We're trying to really understand deeper things.

Why are we trying to understand deeper things? Because we want to have better control over our environment. We want to be able to cure diseases. We want to be able to build things, you know, better, faster, more robust, more solid, all of those things. So I think we there is a chance that we're looking at a very, very bright future using those tools as long as the human stays in control and guides what are the problems that matter.

Problems that you know the AI doesn't care about curing disease. I mean, you know, they will not suffer from the same disease as we do. But we do care. So we have to control them and to guide them towards those problems.

That's not the case, that computation opened up a whole new branch of that. And he that's what's gonna continue. That we're just gonna see that the mathematician that's in high school today is gonna have a very exciting future thirty years from now because of what's happening here.

hu living humans right now that will care about this paper, but you like the result. So you put you put it put it out and then the five other people appreciate it. So they they read it. But then you know 20 years later it's gonna well it's gonna be in the archive somewhere and nobody will read. But now that we have AI, the AI will have Reddit. And if there is a useful connection, as uh Sebastian mentioned, it will surface it and then people people, you know, hundred years down the line will

discover it and use it for whatever they want to use. So there's uh I I would now have much more confidence that my results that are just like that I just put out there will be used if there is a use in the in in the future. And also I'm now able to access the mathematics in a much broader way. There are fields that I've not studied, but if a result comes up, then I would still have to study that field to be able to use that particular result in my research.

But there is no way I could have found that result w without the assistance of AI, but now it's accessible. The model tells me, hey, you can use this to solve your problem. And then well, okay, uh, I'll go and try to try to use that. So, math is going to be m a much more interconnected enterprise. And also, verifying correctness of mathematics is actually.

quite non-trivial because imagine there's a proof written by some you know some somebody uh that's it's the it's three hundred pages long and it it claims to solve uh a really important problem. And this person is a very reputable person. So like there's there and and the paper at at surface looks, you know, uh plausible.

How do you know? Well, I mean re this these are this is a process that takes years to to verify and it's it's also not enough that one person reads it. Mul many people need to read it and then uh uh try to extend it and then look into the details. It d this is a process that take e takes years. And sometimes

um uh like fatally incorrect proofs are are published. So that's also a pro very slow process where the field initially accepts a result but later on discovers that it's unsalvageable. So then it it needs to get filtered out. This is going to be so much more accelerated with with AI. So right now, our ChatGPT and our AI models are not perfect at verifying mathematics, but it's very good. And also it has much more patience than than humans.

Um have much more certainty as to which results are correct, which results are incorrect, and we'll have a much faster feedback on this. A paper published, uh put out uh a week ago, we could get a verification on that, and then we could trust and build on that.

As opposed to waiting for five years to really ascertain the correct So it's overall it's uh math is going to be much more fun, it's gonna be much more interconnected, um we'll uh it'll it'll be we'll be able to trust the results more, we'll be able to move faster, and the mathematicians will solve harder and more interesting problems.

which would be that we kind of hand the keys to the castle, to to the AIs and that humans just start to trust the system a lot more and that they don't do the hard work that we, you know, kind of did to own our skills and to own our skills to to be able to verify and to sit patiently, you know, for hours, you know, many days in a row or many weeks in a row to try to understand deeply a result.

And instead just kind of ask ChatGPT to explain it to us in in in simpler terms. So basically I'm worried about potentially having a shallower understanding of things because we rely too much on the tool. So I think it's really important for the audience, for everyone listening to us, to understand that expertise is even more valuable than it ever was.

The reason why we are able to, you know, squeeze out those results from ChatGPT is because of all of those years of training and our deep understanding of the subject. If it wasn't for that, we would not be able to push the state of the art. And we're seeing it. It's not like we're seeing, you know, thousands of people like non-mathematicians suddenly being able to to prove new results. In fact, if anything, we have seen recent examples in in social media where

non-mathematician have tried to use those tools to prove theorem and and come up with, you know, many tens of pages of of of proof, and then it turns out to be just wrong. So so this is a danger that we have to grapple with.

Obviously, open AI cannot do everything, you know, just to to to say it out loud. And this is where the existing institution have a very big role to play. So academia needs to both understand the the the rate of progress and and you know how fast this is going, but also to kind of reclaim their role in in that process.

you know, completely top out, you know, a benchmark doesn't mean that you go, okay, we're done. We we we we solved grade school math. Congratulations, everybody. AI is done. It's like, no, there's a next level and the next level and you're gonna need people.

flagging specific potential issues. So kind of bringing to the front, okay, hey, maybe this this part I'm not totally sure about it. So that will accelerate, that that will help you know the human to have less to verify.

I would uh habitually go to Wikipedia and then it's just very dense. And I'm like, okay, well, after like thirty seconds I I I go, okay, let me ask Chat GPT and then I ask it and then I also ask follow-up questions and and when I do so it it it gives me so much uh so much more helpful information that is tailored to the the parts of my knowledge that is missing because I'm asking the questions tailored towards that.

And you could you could ex imagine explaining to ChatGPT your mathematical background, the the the the things that you've the books that you've read, the material that you've learned. And then ask it to come up with a question that would be open and also would be understandable with your level of expertise.

Sebastian mentioned this. I think uh you know people I don't think people yet appreciate that uh these LLMs are able to come up with good questions, but I think they can. So having this companion that you can talk up talk. uh mat uh talk about math with uh and uh uh had talk talk about questions. You could ask the model to help you solve it. And uh once you have a solution then you could m keep talking and uh come up with the next question, you know, variations of this.

Uh it it's becomes a much more Um even though you're still in your room alone, it it feels much less of a solitary process. And that that's what really makes uh mathematics uh fun. Because math, I think, it really is a social endeavor.

And start asking these questions. Next thing you know, you're starting to do more and more complex mathematics and realize how much it affects you. Yeah. Uh, gentlemen, this is great. Sebastian, Ernest, thank you very much.