09-25-2026, 04:46 PM
Jon Keating argues that artificial intelligence is forcing mathematics to confront a fundamental change in how the discipline defines research, education and mathematical achievement. The immediate catalyst is the claim that OpenAI has produced a solution to the Navier–Stokes problem, one of the seven Millennium Prize Problems. The Navier–Stokes equations describe the motion of fluids, typically written in simplified form as $\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\nabla p+\nu\nabla^2\mathbf{u}+\mathbf{f}$. If the claimed result survives mathematical scrutiny, it would represent a striking example of AI contributing to a problem that has resisted mathematicians for decades. Keating sees this as evidence that AI is moving from assisting mathematicians toward potentially outperforming them on certain difficult problems.
The article then raises the question of authorship and intellectual credit. Modern AI systems are trained on enormous bodies of existing mathematical knowledge, creating uncertainty about where the boundary lies between the contribution of an AI company and the generations of researchers whose work made the system possible. Keating specifically discusses questions surrounding OpenAI's reported inability to rule out connections between its Navier–Stokes work and research by Tristan Buckmaster and Levent Alpöge. More broadly, he asks whether mathematical progress should primarily be measured by obtaining solutions to famous problems or by creating new concepts, theories and questions. AI may become extremely effective at finding solutions, while its ability to formulate abstractions and construct genuinely new mathematical frameworks remains less certain.
A second major concern is mathematical communication. A proof is not valuable only because its conclusion is correct; mathematicians also care about understanding why the result works. Traditionally, researchers refine proofs, explanations and exposition so that other mathematicians can verify, learn from and build upon them. Keating worries that AI-generated mathematics could produce technically correct but opaque arguments that humans struggle to understand. This becomes particularly important for education: if AI generates an increasing number of solutions, universities must decide how students should learn mathematics when machines can already solve problems that once represented the frontier of human achievement.
Keating therefore calls for a calm and deliberate adaptation rather than either rejection or uncritical enthusiasm. Mathematicians are already divided between those rapidly adopting AI and those reluctant to use it, while younger researchers face uncertainty about future careers. He argues that mathematical societies, universities and researchers should help establish appropriate standards for verification, authorship, publication and education. His conclusion is ultimately that AI could radically transform mathematics without making human mathematics obsolete: just as photography changed painting without eliminating it, AI could become a tool through which mathematicians enter a new period of discovery.
Key takeaways
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The article then raises the question of authorship and intellectual credit. Modern AI systems are trained on enormous bodies of existing mathematical knowledge, creating uncertainty about where the boundary lies between the contribution of an AI company and the generations of researchers whose work made the system possible. Keating specifically discusses questions surrounding OpenAI's reported inability to rule out connections between its Navier–Stokes work and research by Tristan Buckmaster and Levent Alpöge. More broadly, he asks whether mathematical progress should primarily be measured by obtaining solutions to famous problems or by creating new concepts, theories and questions. AI may become extremely effective at finding solutions, while its ability to formulate abstractions and construct genuinely new mathematical frameworks remains less certain.
A second major concern is mathematical communication. A proof is not valuable only because its conclusion is correct; mathematicians also care about understanding why the result works. Traditionally, researchers refine proofs, explanations and exposition so that other mathematicians can verify, learn from and build upon them. Keating worries that AI-generated mathematics could produce technically correct but opaque arguments that humans struggle to understand. This becomes particularly important for education: if AI generates an increasing number of solutions, universities must decide how students should learn mathematics when machines can already solve problems that once represented the frontier of human achievement.
Keating therefore calls for a calm and deliberate adaptation rather than either rejection or uncritical enthusiasm. Mathematicians are already divided between those rapidly adopting AI and those reluctant to use it, while younger researchers face uncertainty about future careers. He argues that mathematical societies, universities and researchers should help establish appropriate standards for verification, authorship, publication and education. His conclusion is ultimately that AI could radically transform mathematics without making human mathematics obsolete: just as photography changed painting without eliminating it, AI could become a tool through which mathematicians enter a new period of discovery.
Key takeaways
- AI is moving toward frontier-level mathematical problem solving, potentially challenging the traditional role of mathematicians.
- The Navier–Stokes problem is a major test case: the equation concerns fluid dynamics and can be represented schematically as $\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\nabla p+\nu\nabla^2\mathbf{u}+\mathbf{f}$.
- Correctness is not the whole story: mathematicians need proofs that humans can verify, understand and reuse.
- Authorship and intellectual credit become complicated when AI systems rely on centuries of accumulated mathematical research.
- Problem solving and mathematical creativity may diverge: AI could become exceptionally powerful at solving existing problems while the ability to formulate concepts, abstractions and theories remains an open question.
- Mathematics education may need substantial redesign, particularly concerning what students should learn when AI can perform increasingly sophisticated calculations and proofs.
- Peer review and transparency become especially important for AI-generated mathematical results.
- The article argues for adaptation rather than simple acceptance or rejection: AI may fundamentally change mathematical practice while still leaving human mathematicians with important roles in conceptualisation, interpretation and communication.
- The central question shifts from “Can AI solve mathematics?” to “What should humans value and teach in mathematics when AI can solve much more of it?”
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