Terence Tao

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I argue that it will become necessary to make these unwritten goals of mathematics much more explicit; but once we have thoroughly examined and codified them, our community will emerge stronger and more resilient than before.

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I believe that we are now entering a era of comparable turbulence in mathematics. This time, though, what is being stress-tested is not our foundational framework for mathematical truth, but rather the largely implicit framework of mathematical values and practices: what we consider a contribution to be, what we reward, what we regard as understood, and who - or what - we regard as having done the work.

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We argue that the next leap in AI4Math systems requires a decisive shift from predefined problem-solvers to research agents that can address frontier mathematical challenges with rigorous formal mathematical reasoning.

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Um And yeah, so I believe a lot in serendipity. Um And maybe there's a danger actually that you know, in the mo- modern society, it's not just AI, but we've become really good at optimizing everything.

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Um and uh I think we would very rapidly abandon any cryptography based on the primes because if there was a one pattern that we didn't know about there's probably more.

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I I I I do feel that, you know, fully autonomous one-shot approaches are not the right approach for these problems. I mean, I I think you you'll get a lot more mileage out of the interplay between between humans collaborating with these tools.

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I mean, some problems have been basically solved by pure brute force. The four color theorem is is a famous example. Um, we have still not found a conceptually elegant proof of this theorem.

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so so some people are concerned, you know, what if the Riemann hypothesis is proved with a completely incomprehensible proof? I I think once you have the artifact of a proof, we can do a lot of of of of of analysis on it.

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Um so yeah, the the the um the core of what I do, like actually solving um the most difficult part of of a math problem, that hasn't changed too much. I still use pen and paper for that.

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Uh yeah, so it's it's made papers sort of richer and broader, but not necessarily deeper.

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Um and then they can kind of suggest random things and it it but it it it um often I find that trying to chase them down and make them work and find they don't work, it wastes more time than it saves.

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So, they they they still make mistakes, but but um um I've I've tested these tools, you know, on on on on um like little tasks that I can do and and sometimes they pick up errors I make, sometimes I pick up errors that they make. It's it's about a tie right now.

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So, I think AI-type tools we really will will actually revolutionize the ex- the experimental side of math where where um you don't care so much about uh individual problems and and the process of solving them, but you you want to gather just large-scale data about about what things work, what things don't.

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Yeah, so they excel at breadth and humans excel at depth. Um like human experts at least. Yeah, so um I think they're very complementary. Um but our current uh way of doing math and science is focused on depth because that that's where the human uh expertise cuz humans can't do breadth.

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Yeah, you you you're on a new session it's forgotten what what it just did. Um, it it has you know, it has no new skills to to attach to to to build on on on related problems.

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There isn't this cumulative process which is, uh, sort of built up interactively. Um, it it it seems to be a lot more trial and error and just repetition, brute force, um, you know, which can see it scales and it can work amazingly well in in certain contexts, um, but yeah, this this idea this this sort of building up cumulatively from, um, from partial progress is kind of is what's still not quite there yet.

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um you know, these tools they either succeed or they fail um and they they've been really bad at creating sort of partial progress or identifying intermediate um um stages that you should you should focus on first.

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Uh but whenever we do a systematic study, um any given problem, an AI tool has a success rate of maybe 1 or 2%. Uh it's just that it's just that they can apply at scale and and you just pick the winners, it looks great.

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Um So, like with the Erdős problems, you know, like almost all of the 50 problems that were solved by AIs were ones for which basically there was no literature.

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Um we are seeing a lot fewer sort of pure AI solutions now where um they are just one shots the problem. Um so so there was there was a month where that happened and and that has stopped.

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you know, right now we're going through uh an in um an cognitive version of the Copernican revolution where we used to think that human intelligence is the center of the universe. And now we're actually seeing that there's there's very different types of intelligence um that that that are out there uh with very different strengths and weaknesses.

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So, you you can't look at any given scientific achievement purely in isolation and give it an objective grade without being aware of the context both in the the past and the future. And so it it it may never be something that you can just reinforcement learn the same way that that you can for much sort of more localized problems.

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So, we're now in a situation where suddenly people can generate thousands of theories for a given scientific problem. And now we have to to verify them, evaluate them and this is something which we we have to to change our structures of science to actually sort this out.

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So, I think AI has basically driven the cost of idea generation down to almost zero in a very similar way to how the internet drove the cost of communication down to almost zero, which is an amazing thing, but it you know, it it it doesn't make it doesn't create abundance by itself.

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Certainly more on the no than on the yes. The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem.

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So I think in 10 years we will have many more much closer results, we may not have the whole thing. So twin primes is somewhat close. The Riemann hypothesis I have no clue.

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I can tell you with complete certainty the twin prime conjecture is true. The random model gives overwhelming odds it is true, I just can't prove it.

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The dream is you just feed it all this data, and this is here is a new patent that we didn't see before, but it actually, even the current state of the art even struggles to discover old laws of physics from the data.

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Yeah, this decade I can see it making a conjecture between two things that people would thought was unrelated.

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There are certainly math results which could only have been accomplished because there was a human authentication and an AI involved, but it's hard to disentangle credit. I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, 30, 40%, so they can fill in gaps.

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Very rarely do you transform into a simpler problem. So if they can pick up a sense of smell, then they could maybe start competing with a human level of mathematicians.

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Yeah, these are great work that shows what's possible. The approach doesn't scale currently. Three days of Google's server time can solve one high school math format there. This is not a scalable prospect, especially with the exponential increase as the complexity increases.

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I think the platform that Lean and other software tools, so GitHub and things like that will allow experimental mathematics to scale up to a much greater degree than we can do now.

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And that's a phase shift, because suddenly it makes sense when you write a paper to write it in Lean first, or through a conversation with AI, which is generally on the fly with you, and it becomes natural for journals to accept.

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So right now I estimate that the time and effort taken to formalize it, proof is about 10 times the amount taken to write it out. So it's doable, but it's annoying.

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Yeah, that was a leading candidate for many decades. I think it's slowly pulling out of fashion. It's not matching experiment.

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But I have faith that we've been doing this for centuries and we've made progress before. There's no reason why we should stop.

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I'm much more comfortable with the fox paradigm. Yeah. So yeah, I like looking for analogies, narratives. I spend a lot of time… If there's a result, I see it in one field, and I like the result, it's a cool result, but I don't like the proof, it uses types of mathematics that I'm not super familiar with, I often try to re-prove it myself using the tools that I favor.

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If something holds 99.99% of the time, that's good enough for most things. But mathematicians are one of the few people who really care about whether really 100% of all situations are covered by it.