Terence Tao warns AI's opaque solutions to math problems could 'pollute' mathematical progress

Tao said that if the iterative process of solving an open mathematical problem is to generate new understanding, the person or system doing the work should not know the correct final ansatz in advance. He wrote that knowing the correct guess too early tends to suppress exploration of other routes, which may look like dead ends but often reveal why they fail. He also criticized AI companies for keeping the entire process, from initial attempts to final guesses, almost completely hidden from the public.

Tao cited the Navier-Stokes global regularity problem as an example. The problem, one of the most famous open questions in mathematics, had been regarded as a good case for testing AI-assisted research. But Tao said a new and increasingly realistic scenario is one in which the answer is mainly generated by AI and appears in a way that pollutes the problem, so that it no longer serves as a rich source of future mathematical progress.

Recent progress on the twin prime conjecture has brought this concern into focus. OpenAI said its newly released GPT-6 Astra, using a formal proof in Lean, achieved a new bound of 186 for the upper limit of gaps between consecutive primes, down from 246. Anthropic and Axiom have also announced narrower bounds of 188 and 212, respectively, according to the report. Tao said he was left speechless by the timing, and he noted that Julia Stadlmann, a mathematician at the University of Oxford, had published her analysis just before the question could be polluted. On Aug. 31, Stadlmann reduced the bound to 240 through new estimates for smooth moduli.

Tao said the answer itself is not the only thing that matters. Proving that solutions to the Navier-Stokes equations always remain regular, or constructing a finite-time blow-up example, would have theoretical significance. But because computational fluid dynamics is already mature, such an answer would not fundamentally change weather forecasting or climate research. The real value, he said, lies in the working process, which has historically produced foundational concepts and theorems in fluid mechanics, analysis and partial differential equations, including Leray-Hopf weak solutions, Gagliardo-Nirenberg-Ladyzhenskaya inequalities, Prodi-Serrin partial regularity theorems, Beale-Kato-Majda blow-up criteria and more.

In his own attempts on this problem, Tao said, he did not solve the Navier-Stokes blow-up question, but the effort unexpectedly connected fluid computation, Turing universality and symplectic topology. He said failures can generate valuable mathematics in that way. He now fears that AI-driven methods could kill this sort of wide-ranging development. For the Navier-Stokes problem, he noted, mathematicians increasingly believe the answer is likely negative, with finite-time singularities arising from special initial conditions. An explicit strategy has been outlined that involves designing an approximate self-similar ansatz, verifying residues numerically, proving stability in rescaled coordinates and checking the residual bound.

For humans, each of these steps is highly complex and intertwined. But AI, assisted by machine-learning simulation, rigorous interval arithmetic, formal proof systems and human guidance, may eventually handle them as a pipeline. Tao said the resulting construction could be so complex that only a formal proof could confirm it. If humans cannot understand the proof, however, much of the problem's value disappears. He also raised the possibility of an autonomous AI system, working behind corporate walls with massive computing power, that runs the entire trial-and-error loop and solves the problem without outside observers learning how. In that case, a famous open problem would be technically solved but mathematics itself would gain almost nothing.

Tao warned that if such opaque, AI-completed solutions become common, AI's overall effect on mathematics could switch from positive to net negative. He said mathematicians create new methods and concepts by struggling with problems, and later digest results to extract further insights. When a problem is solved too early by an inscrutable process, that productive cycle can be polluted. The deeper issue, he argued, is not whether AI should be used in mathematics, but that the process matters and transparency is needed.