Twenty-five Fields medallists published a letter this month arguing that AI companies are treating mathematics as a benchmark, and that a flood of machine-produced proofs will destroy the thing mathematics is actually for. We covered that letter here. Timothy Gowers — a Fields medallist himself — did not sign it, and instead wrote out his own position.

He agrees there is a crisis. He just thinks the signatories have named the wrong one.

Where he agrees, and where he stands

Gowers opens by ruling out two easy ways to dismiss him. He has contacts in the mathematics group at OpenAI, has been given early access to their models a few days before release, and has free access to their Pro models once out — but he has never been paid by OpenAI. And his group in Cambridge devoted to automatic theorem proving is not a reason to be pro-AI: their whole approach was to get computers to prove interesting theorems by first understanding how humans prove them, and LLMs turned out to do it without any of that insight. One of the main motivations for their work has disappeared.

He also agrees with the letter’s premise that something is breaking. His complaint is about the diagnosis.

Why he didn’t sign

  • He rejects the letter’s core claim that conceptual understanding is the goal and problem-solving is only a tool serving it. In an essay a quarter of a century ago he argued there is a spectrum of mathematical temperaments: some mathematicians are driven by the wish to solve problems and see understanding as the means, others are driven by the wish to understand and see problems as the means. Declaring one end of that spectrum the correct attitude tells a large fraction of mathematicians their temperament is wrong.
  • He sets up two futures instead. In the one he thinks is likelier, models better than virtually all mathematicians are released publicly, answers to long-standing questions arrive fast, and much of the output is obtained with zero human effort — prompts like “Thank you — please continue.” In the other, an international agreement blocks public release and mathematicians inside the labs hold off until a representative body decides which problems are worth solving.
  • On individual understanding, he thinks the loss is real but narrow: what atrophies is the part of the brain that spends months struggling with a hard problem. His analogy is satnav — nobody is forced to use it, but most people do. (He notes he tries not to, and was pleased with himself for navigating somewhere after looking up the route and then leaving his phone at home.) Even so, reading a paper actively — trying to prove each result before looking — is itself problem-solving, and a model that knows your background could feed you exactly the hint you need.
  • On collective understanding, he does not buy the claim that digestion becomes impossible. The output is not just true/false statements: the Navier–Stokes blow-up result builds on a great deal of earlier human work, even if the write-ups are often poor. Mathematics is specialized enough that 1,000 results in a year would be about 30 per subcommunity, each handled by a handful of obvious specialists. And too much material is not automatically worse than too little — his comparison is streaming services, where a surplus means you choose. If you need to know what exists, build a database or ask a model for a bird’s-eye view of an area.
  • The citation and credit problem is serious now but temporary, in his reading. He expects the credit system to collapse soon, because finding an amazing proof will be no more of an intellectual achievement than a citizen scientist spotting a new comet through a telescope. Careers still depend on credit until then, so it should be fixed — and he is puzzled it is not, since asking a model which ideas in a proof are already in the literature seems like a task models would be good at.
  • The risk he does take seriously is the one the letter underplays: not that mathematicians cannot absorb the results, but that the social structures which do the absorbing get destroyed and not replaced. His own honest version of this is motivational. His reason for becoming a mathematician was the dream of solving an unsolved problem; if that dream is gone, he is not sure he would have taken the path, and he does not know how many young people will. The downstream worry is funders and policymakers concluding that mathematicians are no longer needed.
  • Finally, he saw nothing to be gained from the letter’s implicit demand that AI companies slow down. The models are coming either way; criticism might only bring the flood forward a couple of months, and a controlled release by the labs may be less chaotic than random users getting the same capability later.

His conclusion is modest rather than resolved: recognize the changes that are coming, and work out the least unsatisfactory way of dealing with them.

The 308-comment thread on Hacker News mostly takes his side of the ledger and pushes on the part he left open — who pays for mathematicians once they are no longer the ones proving theorems.

What the thread adds

  • layer8 — the top branch of the thread (18 replies) is about money and the pipeline. The piece’s line that “we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts” is, they say, the main issue: “the Fields medallists’ letter failed to provide convincing arguments for why mathematicians should widely receive funding for merely understanding things, and how competition for postdoc and tenure positions would work under these circumstances.”
  • paimapi — answering that with history: imagine justifying Newton’s and Leibniz’s work on calculus in the 1700s, when maritime engineering ran on geometry and arithmetic, and the applications were a century or more away. “There’s no KPI to be derived from any academic field of study at the bleeding edge of theory.”
  • impendia — a working mathematician’s answer to what the new work looks like: every discipline insists on outwardly visible production, and their candidate for mathematics is the digestion genre, pointing at Terence Tao’s write-up of the recently proved Jacobian conjecture counterexample. “When computers can solve the underlying actual problem, this sort of work seems likely to rise in value.”
  • freehorse — dissent from the labour framing that dominates the thread. The deeper question, they argue, is what mathematics itself becomes: AI getting proofs “does not solve at all the question of how to produce new problems,” and which problems are worth attacking is a matter of taste built “through the exact friction that is more and more removed with AI.”
  • Chance-Device — the analogy to software engineering, where juniors are hired less and the ladder to seniority breaks; in the maximal case where AI does everything, what is left is hobbies, games and social status.
  • cmplxconjugate — the floor and the ceiling rise together: AI makes previously non-trivial tasks trivial, but you expand your scope and take on harder problems. The open question is whether that still holds once models clear the harder problems too, “or whether choosing and framing those problems remains the human part.”
  • waynecochran and kccqzy — on whether a 1,000-page machine proof nobody can read would change anything, given these results are assumed true anyway; kccqzy’s reply is that the article itself describes two mathematical cultures, “You are clearly on one end of the spectrum and do not seem to understand the other end.”
  • Almondsetat and augment_me — the mountain-climbing analogy for why humans should redo what a model already did, met immediately by the funder’s question: “Why should a government invest money into this? … historically in society it has been applied as a means to ends like social power, resource accumulation.”

The question the thread keeps asking

The essay’s weakest joint, by the thread’s own reckoning, is the value of a large population of human mathematicians once the proofs come from elsewhere. layer8 states it directly, augment_me lands on the same gap from the funder’s side, and paimapi answers it by analogy rather than by argument. Gowers says the explanation is urgent and does not supply it; nobody in the thread does either.

A note on reading comments as evidence: HN handles are pseudonymous and the site publishes no per-comment scores, so the ordering here is HN’s own ranking, not a vote. This is a slice of a 308-comment thread, not a consensus.