What Terence Tao thinks about mathematics
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Only Terence Tao on mathematics. See everyone on mathematics
34 dated positions, 2025 to 2026, in their own words. Our reading of what Terence Tao has said — not written or endorsed by them.
Our readingMost of these are about what AI changes in mathematics and what it does not: an experimental side arriving, formalisation flipping how papers are written and refereed, and a decade in which the bulk of the daily work goes over. What is still missing is named precisely: building cumulatively on partial progress.
Their wordsThere 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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Their wordsMost of the value that flows into the world from mathematics and the work of mathematicians will not be replaced by AI, but could be magnified.
↗A value-based approach to mathematicsterrytao.wordpress.com 1st of 3 in this piece
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Their wordsImproper usage of AI offloads thinking, understanding and learning, thus mortgaging future mathematical development.
↗A value-based approach to mathematicsterrytao.wordpress.com 2nd of 3 in this piece
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Their wordsThe value of math, like so many other forms of knowledge, also comes from the process, not just the product.
↗A value-based approach to mathematicsterrytao.wordpress.com 3rd of 3 in this piece
- 6 weeks earlier
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Their wordsI 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.
↗Mathematics in the age of AIarxiv.org All 5 from this paper
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Their wordsI 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.
↗Mathematics in the age of AIarxiv.org All 5 from this paper
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Their wordsConsequently, excessive optimization for one or two goals may cause the many previously aligned goals of mathematics to diverge from one another;
↗Mathematics in the age of AIarxiv.org All 5 from this paper
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Their wordsAn excessively AI-polished proof may sand away both the "artificial" friction (typos, awkward phrasing, disorganization) and the "natural" friction, leaving a text that is easy to read and hard to learn from.
↗Mathematics in the age of AIarxiv.org All 5 from this paper
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Their wordsBut passing an automatic filter is not a substitute for community acceptance; I do not believe that human referees can be removed from the publication process.
↗Mathematics in the age of AIarxiv.org All 5 from this paper
- 6 weeks earlier
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Their wordsWe 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.
↗From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors)arxiv.org All 4 from this paper
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Their wordsDespite these strides, we argue that current AI4Math systems still largely operate as solvers, excelling at isolated, well-defined proof generation rather than as researchers capable of expanding the boundaries of mathematical knowledge.
↗From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors)arxiv.org All 4 from this paper
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Their wordsThese developments inform our position that AI systems are now capable of meaningful contributions to formal mathematics, not merely informal problem-solving.
↗From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors)arxiv.org All 4 from this paper
- 4 months earlier
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Their wordsUm 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 5th of 24 in this recording
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Their wordsYeah, 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 7th of 24 in this recording
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Their wordsSo, 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 8th of 24 in this recording
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Their wordsUm 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 11th of 24 in this recording
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Their wordsUm 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 13th of 24 in this recording
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Their wordsThere 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 15th of 24 in this recording
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Their wordsI 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 17th of 24 in this recording
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Their wordsI 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 18th of 24 in this recording
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Their wordsso 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.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 19th of 24 in this recording
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Their wordsUm I think with within a decade, a lot of things that mathematicians currently do um we we spend a lot of the bulk of our time doing and a lot of stuff we put in our papers today can be done by AI. Um but we will find that that actually wasn't the most important part of what we do.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 22nd of 24 in this recording
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Their wordsI mean, I I I I do believe that that hybrid um human plus AIs will will dominate mathematics for a lot longer. It it's It will depend It will require some additional breakthroughs uh beyond what we already have.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 23rd of 24 in this recording
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Their wordsUm but now it's quite possible at the high school level or or whatever that that you could get involved in math project and actually make a real contribution because of all these AI tools and and and Lean and everything else.
↗Terence Tao – How the world’s top mathematician uses AIyoutube.com 24th of 24 in this recording
- 9 months earlier
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Their wordsIf 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 1st of 20 in this recording
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Their wordsI'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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 2nd of 20 in this recording
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Their wordsI 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 6th of 20 in this recording
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Their wordsVery 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 8th of 20 in this recording
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Their wordsAnd 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 9th of 20 in this recording
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Their wordsThere 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 10th of 20 in this recording
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Their wordsSo 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 13th of 20 in this recording
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Their wordsI 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 14th of 20 in this recording
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Their wordsCertainly 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.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 15th of 20 in this recording
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Their wordsIt's becoming much more a thing of the past. So I think you just have to be adaptable and flexible. I think people will have to get skills that are transferable, like learning one specific programing language or one specific subject of mathematics or something. That itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve when things go wrong.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 16th of 20 in this recording
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Their wordsso the mathematical community plural is incredibly super intelligent entity that no single human mathematician can come closer to replicating.
↗Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472youtube.com 18th of 20 in this recording