Fields Medalist Lafforgue: Without mathematicians' research and verification, AI-generated mathematical texts lose their meaning

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+1 Fields Medalist Laurent Lafforgue: Without mathematicians' research and verification, AI-generated mathematical texts lose their meaning

The Paper reporter Ji Jingjie 2026-10-30 07:37 Source: The Paper ∙ Science Pulse > Listen to the full text Font size

AI is triggering an earthquake in the mathematics community. Earlier this month, OpenAI released more than 700 mathematical preprints entirely generated by AI in one go. Fields Medalist Terence Tao publicly stated that the "romantic era" in which mathematicians pondered a problem for years and conquered it in a "heroic" posture may be coming to an end. But in the view of another Fields Medalist, the French mathematician Laurent Lafforgue, while AI is changing the way mathematics is written and produced on a large scale, it is also a good opportunity for mathematicians to collectively reflect on mathematical production. In the AI era, the status of mathematicians is also irreplaceable.

On October 28, in a lecture at Fudan University's "Pujiang Science Masters Forum," Lafforgue approached the relationship between mathematics and AI from another angle: large language models have achieved such astonishing success in text and images, and behind this there may also be hidden a structure that has not yet been identified—and this is precisely the field that mathematicians need to explore.

Fields Medalist and French mathematician Laurent Lafforgue (Laurent Lafforgue) delivers a lecture at the "Pujiang Science Masters Forum." Image from the organizers

Lafforgue mentioned that the success of AI applications has a common prerequisite: a massive amount of text written by real humans and a massive number of images captured by real humans. The reason large language models can predict the next token and complete a passage of text precisely suggests that within that text corpus written by countless people, there exists some kind of mathematical order. If the data were a completely irregular and discontinuous set, it would be impossible for the system to learn anything generalizable from it, let alone use finite computation to approximate a global structure. The problem lies in: "We know there is a structure, but we do not know what this structure looks like."

This is precisely what makes mathematicians uneasy. Engineers number the tokens, embed them as vectors, and repeatedly adjust parameters with massive computing power, and the model "works." But Lafforgue points out that these vectors do not come from any physical measurement, and are "even arbitrary to mathematicians." The success of AI is built precisely on a lack of understanding of the nature of this success: "For mathematicians, this is especially unsatisfactory."

To investigate what this "structure" is, Lafforgue chose to draw inspiration from Alexander Grothendieck. Grothendieck was one of the most revolutionary mathematicians of the 20th century and the founder of modern algebraic geometry. Traditional geometry cares about "what points there are" in a space and "what the coordinates of the points are," whereas Grothendieck instead asked whether points are an appropriate geometric concept at all.

He thus founded topos theory, redefining "space" in the language of "sheaves" and "coverings"—space is no longer a set composed of points, but can be described in a completely different way, in algebraic language.

Grothendieck's perspective

Lafforgue is a follower of Grothendieck. The work for which he won the Fields Medal in 2002—proving a class of Langlands correspondences over function fields of positive characteristic—was itself completed within the framework of algebraic geometry laid down by Grothendieck.

In his view, when facing the AI "black box," Grothendieck's approach is very useful: do not start by asking "what does this word mean" or "which point does this image correspond to," but instead regard all text or all images as a whole space, and ask whether they can be endowed with a continuous, classifiable geometric structure.

Mathematical research around large language models has already formed several quite different paths, such as statistical learning, dynamical systems, and combinatorics, but all of them take a given large language model as the object of study, treating it as a parameter space, a state trajectory, or a set of compositional rules of an architecture. But in the perspective of Grothendieck's theory, the problem is turned upside down: if human data itself possesses some unknown structure, then the large model is "isomorphic" to it, and success is a matter of course.

Although images and text can be defined in the form of data points and so on, Lafforgue reminds us that the strength of topos theory lies precisely in its abandonment of the definition of points—just as in life one has never truly "seen" an abstract mathematical point. The theory captures the overall patterns of covering, gluing, and continuous variation, and these patterns are more intuitive than points.

"For example, the concept of 'covering' means that you have some larger thing that is covered by smaller things—a city as large as Shanghai is covered by different districts; clothes are all woven together from smaller pieces of cloth to become a garment that covers our bodies," Lafforgue said.

What mathematicians need to do is turn this intuition of "piecing small things into big things" into a computable structure: given a space, first specify a family of local small pieces, then stipulate which small pieces can be pieced together and whether the way of piecing them together is unique. This set of rules of "covering and gluing" characterizes the overall shape of the space. And a "sheaf" is the data attached to this set of covering rules—it assigns an object to each local small piece (such as a passage of text, a function, a set of probabilities), and requires these local data to be compatible with each other at overlaps and to be able to glue together consistently.

Re-describing large language models in this language, Laforgue arrived at a distinctive picture: the model is not assigning a fixed meaning to each word, but rather, within a text space organized by covering rules, computing which local continuations can be consistently pieced together into a whole—this is precisely the counterpart of "sheaves" and "coverings" in large models.

Laforgue believes that if there really is a mathematical structure behind AI's text production, then topos theory offers a possible path to approximate it with a language that does not rely on "points." Of course, whether this idea will ultimately become a framework for understanding the laws of large models still requires mathematicians to continue exploring.

Beyond language

Laforgue mentioned that large language models operate entirely within the space of human language. What it processes is text that has already been written, and what it outputs is text that can be expressed as sequences of tokens. It can "pave roads" on the existing terrain of language with extreme efficiency, yet it cannot see those things that have not yet been named by language and have not yet entered the symbolic system.

A student in the audience asked Laforgue whether AI could generate a mathematical description of itself and thereby improve itself. Laforgue replied that such a description would certainly not be Grothendieck-style—mathematics is in a sense an interpretive science: we keep searching for certain things, and only after finding them and giving them a name do they enter graspable existence. If something has not been named, it exists outside the language system. The AI system "is entirely within our language space and domain, and it may not necessarily be able to see or understand those things that have not yet appeared in or been named by our language system."

He told reporters that mathematics before the 20th century was more literary in flavor, and that the modern mathematical writing paradigm—propositional, rigorously structured, and focused on problem-solving—was to a large extent driven by the American school and then adopted globally. Under the impact of AI, this highly standardized paradigm may loosen; when humans cannot match AI on the path of formal proof, a revival of reunderstanding mathematics as an activity connected with philosophy and literature, as in the Russian school, may occur, and entirely new styles may even be born.

Laforgue mentioned that he has participated in conferences and events in China many times, and that these conferences bring together people from completely different fields, which is very meaningful. In the history of science, mathematics's relationships with physics and computer science are all very important, and its relationships with philosophy and literature are much closer than one might imagine: Pascal was both a mathematician and one of France's greatest writers and philosophers, and was also the first person in history to design a calculator; and Grothendieck himself wrote extremely well and was often called a "poet."

Laforgue said that French professors are generally very satisfied with Chinese students: they study diligently, have solid knowledge, work seriously, and respect their teachers, but their shortcoming is that they do not like to ask questions. However, the emergence of AI has brought about a good phenomenon: many students first ask AI, and then gain the confidence to ask their professors questions.

"Professors are actually very happy to be asked difficult questions," he said. Responsible editor: Ji Jingjie; image editor: Zhu Weihui; The Paper news tip line: 021-962866; The Paper, unauthorized reproduction prohibited +1 Favorite I want to report View more