Fields Medalist Laforgue: Mathematicians should describe the "text space" behind large language models

In the keynote speech at the main forum of the 2026 World Laureates Forum, Laurent Lafforgue, winner of the 2002 Fields Medal and a senior expert at Huawei France, examined large language models from a mathematician's perspective and raised a question: behind the ability of machines to generate text, is there an as-yet-unidentified "text space" whose internal structure precisely requires mathematics to describe? Lafforgue admitted that most mathematicians initially did not believe automatic translation systems could be realized. The reason is simple: an efficiently operating system should usually be built on some kind of understanding and theory; and human language is too complex and subtle to be fully explained by an all-encompassing theory. The emergence of large language models broke this intuition—it not only generates text but can also handle mathematical problems, forcing mathematicians to confront a fact: people are using a tool whose principles they do not truly understand. He compared this situation to the transformation of science into industry in the 18th and 19th centuries: engineering systems at that time could be put into use, yet people did not necessarily understand why they worked. Today's large language models likewise "work" by relying on enormous computing power, but "we have never had a mature and complete theory of text." For mathematicians, this is especially unsatisfactory: tokens are numbered, embedded as vectors, and then parameters are adjusted through massive computation, yet these vectors do not come from any physical measurement and are even mathematically somewhat arbitrary. The success is real, and the "absence" of understanding is also real. Laurent Lafforgue, winner of the 2002 Fields Medal and a senior expert at Huawei France. Photo by Xu Cheng. The path of inquiry Lafforgue offered comes from Alexander Grothendieck. Traditional geometry cares about "what points are in a space" and "what the coordinates of the points are," whereas Grothendieck instead asked: are points the appropriate concept for describing space? Lafforgue tried to transfer this line of thought to text. In his view, large language models are not attaching a unique meaning label to each sentence, but rather observing, within a huge text space, which continuation paths work and how probable they are; so-called "understanding" perhaps does not correspond to some unique point, but to the overall structure of acceptable continuations. If this structure truly exists, the task of mathematics is to describe its geometric shape. Lafforgue did not regard AI as a replacement for mathematicians. He believes that mathematicians need to develop new ways of looking at mathematics and cannot focus only on solving problems; the relationship between the production of mathematical texts and human understanding of them deserves reexamination. He even conjectured that, influenced by AI, mathematical writing might return to a somewhat more traditional style, "and this is also my hope." In his speech, Lafforgue also drew an analogy to the French "Oulipo" (Ouvroir de littérature potentielle, workshop of potential literature). This literary group generates texts through constraint rules, echoing the way large language models compute the continuity of text under constraints. In Lafforgue's view, the real question is not whether machines can write mathematics, but whether humans are willing and able to understand the space that enables machines to succeed. The mathematician's place is precisely in inquiring into the structure of this space. [More reading] "The wish for a pay raise fell through": how do negative emotions crush us? Hu Hailan: after losing several days in a row, even a champion can become depressed.