What changed
[This is a guest post by Tapio Schneider . This blog post was initially written in a different file format and converted using AI. — T.] [This will be cross-posted on the CliMA blog .] The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a debate about the role of AI in mathematics. The proof was produced with a system of some 10,000 AI agents that explored many approaches in parallel; it was then formalized and verified in Lean. The formal verification supports its correctness, but mathematicians are still working to digest it. A proof settles the truth value of a statement, but Terry Tao and others have argued that this is only part of the point of proofs; the other part is to advance our conceptual understanding of “ basic structures of shapes, numbers, and natural phenomena .” As Yehuda Rav put it a quarter-century ago, “ theorems are the headlines, proofs are the inside story .” A proof that is correct but incomprehensible, or undigested by the mathematical community, gives us the headline without the story. The issue is not whether the prover is a human or a machine, but whether the reasoning advances our collective understanding of methods and structures, which can spur new thinking and further advances. As Timothy Gowers has noted, the inadequate AI-generated write-ups of proofs are likely a temporary annoyance. I want to argue that the same point about understanding holds in the natural sciences and engineering, where it also has intrinsic value and, in addition, acquires instrumental value when predictions must be trusted before they can be verified empirically. One role of science is what ancient philosophers called episteme , roughly explanatory understanding; here, understanding is the goal itself. Another role is techne , roughly the craft of predicting and making: forecasting how natural or engineered systems will behave under circumstances not yet observed. Whether the two roles can be separated depends on how easily predictions can be checked. When they can, techne can stand on its own. A black-box AI weather prediction model can predict tomorrow’s weather, and we can trust it because its forecasts can be checked every day. Techne can then also serve episteme as an instrument. For example, AlphaFold predicts 3D protein structures without providing explanat…
