“No AI will replace mathematicians, but I am concerned about education”

• Mathematics
• Research

AI recently made headlines for producing a proof related to the Navier-Stokes equations. This is one of the greatest unsolved problems in mathematics, which researchers have been working on for decades. Did that surprise you?
Alessio Figalli: The Navier-Stokes problem is one of the seven Millennium Prize Problems, a collection of exceptionally difficult mathematical challenges. So far, only one of them has been solved. At its heart lies the question of whether fluid flows always remain orderly and predictable, or whether seemingly harmless flows can develop extreme states. OpenAI's proof claims that, under certain conditions, such extreme states can indeed arise.

How did we get to this point?
For a long time, researchers tried to prove that such extreme states, known in mathematics as singularities, could never occur. In recent years, however, important progress has been made in understanding how they might arise. The AI proof builds on this body of work. That is why I was not entirely surprised by the result.

Will AI transform mathematics in the same way that the microscope transformed biology by revealing entirely new phenomena?
I don't think so. Unlike a microscope, AI does not make previously invisible phenomena observable for researchers to interpret. Its main contribution is to accelerate the production of results. That was true in the Navier-Stokes case as well: many researchers expected a result of this kind, but AI probably accelerated the process by years.

What can AI realistically achieve in mathematics?
AI primarily combines existing knowledge and builds on established work, methods and tools. When there are no recognised approaches, substantial literature or prior results to draw on, it reaches its limits.

More importantly, the purpose of mathematics is not simply to find answers as quickly as possible. Our goal is to uncover deep structures and understand relationships. That requires time, experience and creativity.

Even so, AI challenges the notion that only the human mind can solve exceptionally difficult mathematical problems.
Truly groundbreaking ideas often come from unconventional perspectives. People frequently learn more from failing to solve a problem than from obtaining an immediate answer. New tools, methods and insights emerge from those detours.

That is why, in mathematics, the answer itself is not the only thing that matters. The path towards it matters too. Major challenges such as the Millennium Prize Problems serve not only as goals but also as signposts, guiding entire fields towards new ways of solving difficult problems.

Looking ahead, will AI make mathematicians obsolete?
I don't think AI will replace mathematicians. AI is already very good at answering questions posed by humans, but as said, mathematics is not only about finding answers. A central part of research is identifying the right questions to ask, questions that open new directions and advance our understanding.

Developing that judgement requires learning how to do research, a process that involves struggle, failure, and time. Unfortunately, AI can create a temptation to bypass that struggle and failure, and this is where I see one of the greatest challenges, particularly for young researchers.

How is AI changing mathematical research?
AI could change research culture in significant ways. One major challenge is the enormous volume of AI-generated publications. It becomes increasingly difficult to identify which papers genuinely matter and advance knowledge.

The paradox is that this flood of publications could ultimately result in us learning less rather than more. The Navier-Stokes proof likewise illustrates both the remarkable capabilities and the current limitations of AI.

In what sense?
Human beings construct proofs through a clear sequence of logical steps. OpenAI's published proof may span 166 readable pages, yet it is not written in the way a human mathematician would normally construct and present a proof. Definitions suddenly appear without it being obvious why they matter, how they contribute to the proof, or what the genuinely new idea is.

AI cannot explain which insight made the decisive breakthrough possible or which parts of the proof are truly original. Nor can I invite an AI to give a seminar and ask it to explain the new ideas involved or clarify the logic behind its strategy. And when I encounter AI-generated proofs, I sometimes ask myself: who actually wants to read them?

You mentioned education. What risks do you see there in the age of AI?
Knowing when, how and whether AI should be used effectively requires considerable experience. In education, we guide students over many years towards tackling demanding problems. Along the way, they learn not only how to find solutions, but also how to fail, understand mistakes and develop new approaches.

Students who merely present solutions generated by AI do not gain those experiences. That is why universities and professors must now define which skills the next generation of mathematicians should acquire, rather than leaving that task to AI companies.

How do you see students using AI today?
Some students use AI constructively, for example by generating additional practice problems. This can strengthen their problem-solving abilities.

What concerns me is when students submit impeccably written assignments generated with AI but cannot explain how they arrived at their conclusions. As a lecturer, I therefore wonder whether we should increasingly rely on oral examinations in the future.

At the same time, many students feel pressured because they fear falling behind if they do not use AI while others do. Is that really the kind of pressure that promotes learning and the meaningful application of knowledge?

You also address this issue in the declaration "A Severe Misalignment of AI in Mathematics". One of its central messages is that we must not lose sight of the fundamental goals of our work, or of its value for people, amid all the changes brought about by AI.
The purpose of the declaration is to raise awareness and encourage discussion. Our concern is that, as AI becomes increasingly capable, we may begin optimising for outcomes that are easy to measure while overlooking what the activity itself was meant to achieve.

This is not only a question for mathematics. As we argue in the declaration, many intellectual and creative professions face the same challenge: how can we use AI to enhance what we do without losing sight of why we do it in the first place?

Fields medallists caution against AI misalignment in mathematics

In a declaration, 25 Fields Medal laureates, recipients of mathematics' highest honour, warn of problematic developments in the use of AI. As AI transforms mathematical research, they urge the community not to lose sight of the true purpose of mathematics.

They argue that using mathematical problem-solving as a benchmark for AI development risks harming the discipline. Furthermore, they argue that the goals of AI companies and those of the mathematical community are becoming increasingly misaligned.

external page A Severe Misalignment of AI in Mathematics

About the Author

Alessio Figalli is Professor of Mathematics at ETH Zurich and Director of the Institute for Mathematical Research (FIM). A specialist in partial differential equations and optimal transport theory, he has been honoured with the Fields Medal, the EMS Prize, and the Stampacchia Gold Medal. In recognition of his commitment to teaching, he has also received a Golden Owl award from VSETH.