After Math

AI-generated proofs can now be formally verified, but understanding them requires human insight.

On September 8th, 2026, OpenAI announced it had generated a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. This announcement sparked debate regarding credit allocation and the respective contributions of humans and machines to the outcome. While one mathematician involved called it a "Deep Blue-Kasparov moment," the article argues that mathematics is not a game that can be "solved." The authors differentiate between a logical notion of proof, which AI can satisfy through formal verification like Lean, and an intelligible notion of proof, which requires understanding and communication to advance mathematical knowledge. OpenAI's submission is seen as potentially providing a formally valid answer but not necessarily a "fruitful solution" that offers the deep understanding mathematicians seek. The piece contends that even if AI produces proofs that are both logically correct and intelligible, it does not mean mathematics is "solved" in the same way chess or Go are, advocating for AI to be viewed as an assistant rather than a competitor.

The distinction between logical and intelligible proofs is critical as AI capabilities diverge. While AI can produce formally certified proofs that ensure logical validity, these may lack the intuitive understanding mathematicians require to build upon existing knowledge and develop new theories. This separation risks creating a situation where AI-generated proofs are correct but incomprehensible, hindering rather than advancing mathematical progress.

The narrative of AI "solving" mathematics is flawed because it oversimplifies the discipline to mere problem-solving. Mathematics encompasses concept development, theory building, community engagement, and the pursuit of beauty and depth, aims that go beyond simply finding answers. Viewing AI as an assistant, rather than a competitor, aligns with its technological nature and allows mathematicians to focus on these broader aspects of their field, ensuring that AI serves human purposes.