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What Terence Tao thinks about mathematics

@terence-tao · 42 positions · 0 changes of mind

Mathematician at UCLA, Fields Medal 2006. Writes at terrytao.wordpress.com.

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16 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.

What AI still cannot do in mathematics is build cumulatively on partial progress; it is trial, error and brute force rather than an idea improving through use.

  1. Our readingAI will revolutionise mathematics by giving it an experimental side: large-scale data on which methods work, rather than the careful solving of individual problems.

    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

    Interesting

  3. Our readingWhat AI still cannot do in mathematics is build cumulatively on partial progress; it is trial, error and brute force rather than an idea improving through use.

    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

    Interesting

  5. Our readingWithin a decade AI will do the bulk of what mathematicians spend their time on, and we will discover that was never the important part of the job.

    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

    Interesting

  7. Our readingHybrid human-plus-AI teams will dominate mathematics for a lot longer, and a full replacement would need breakthroughs beyond anything we now have.

    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

    Interesting

  9. Our readingContributing to frontier mathematics no longer requires years of PhD training; with AI tools and Lean a high schooler can make a real contribution.

    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

    Interesting
  10. 9 months earlier

  11. Our readingWhat sets mathematicians apart is caring whether something holds in 100 percent of cases, when 99.99 percent is good enough for everyone else.

    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

    Interesting

  13. Our readingHe works as a fox, not a hedgehog: moving between fields, chasing analogies, and re-proving results he likes with the tools he favours.

    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

    Interesting

  15. Our readingLean and tools like GitHub will let experimental mathematics scale far beyond what one mathematician's spaghetti code allows today.

    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

    Interesting

  17. Our readingAI could compete with human mathematicians once it acquires a mathematical sense of smell: knowing which way of splitting a problem makes it easier rather than harder.

    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

    Interesting

  19. Our readingWhen formalising a proof costs no more than writing it, mathematics will flip: papers written in Lean first, and journals refereeing only for significance because correctness is certified.

    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

    formal proof

    Interesting

  21. Our readingHis prediction that research-level mathematics papers would be written in collaboration with AI by 2026 has already come true.

    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

    Interesting

  23. Our readingIn ten years there will be many results much closer to twin primes, perhaps not the whole thing; on the Riemann hypothesis he has no clue.

    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

    Interesting

  25. Our readingThe twin prime conjecture is certainly true, the random model gives overwhelming odds of it, and he just cannot prove it.

    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

    Interesting

  27. Our readingOn P versus NP, the evidence leans towards no, and the problem is unusual in how many approaches have been proven not to work.

    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

    Interesting

  29. Our readingThe stable one-profession career is becoming a thing of the past; what will still be needed alongside AI is reasoning with abstractions and problem-solving when things go wrong.

    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

    Interesting

  31. Our readingThe mathematical community as a whole is a superintelligent entity that no single mathematician comes close to replicating.

    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

    AI alignment

    Interesting