Fields Medalist Laforgue: AI is getting better and better at generating, and more "human mathematics" is needed

Laurent Lafforgue (photo by Yuan Jing, Wen Hui Bao) This morning (October 31), the opening ceremony and main forum of the 2026 World Laureates Forum were held at the Lingang Center. Laurent Lafforgue, winner of the 2002 Fields Medal and a senior expert at Huawei France, delivered a keynote speech titled "The Mathematician Facing the Success of Large Language Models: Does a 'Text Space' Exist?" As a mathematician, Lafforgue, like many of his peers, has recently been confronting the "remarkable achievements" rapidly made by large language models. What surprises mathematicians is not only that large models can generate text and complete translations, but also that they can now solve certain problems, even problems that human mathematicians have never solved before. He recalled that when automatic translation systems emerged more than 20 years ago, many mathematicians did not believe in this technology. In the traditional understanding of mathematicians, behind an efficiently operating system there should be some theory that can explain what it is actually doing. To achieve automatic translation, one would first need to establish a complete explanatory theory of human language. But human language is too complex and too subtle, and it is hard to imagine a theory that encompasses everything. Therefore, automatic translation seemed almost impossible at the time. However, the emergence of large language models shattered this expectation. "At least we did not expect it to appear so soon," Lafforgue said. This raises a very practical question: as AI becomes able to do more and more of the work previously done by mathematicians, what will mathematicians' way of working become? The first to be affected will be mathematics doctoral students. In the past, doctoral students usually needed to solve a previously unsolved problem and write a thesis in order to graduate. But if AI can already complete part of mathematical proofs and even participate in proposing solutions, then how should doctoral students prove their research ability? This training model is bound to change. In Lafforgue's view, what is more worth pondering than "what AI can do" is what exactly is the difference between machine generation and human creation. The French mathematician Alexandre Grothendieck predicted in the 1980s that a machine capable of generating text might appear in the future. At the same time, the literary world also began imagining this possibility very early on. In the 1967 British television series The Prisoner, a computer capable of answering all kinds of questions appeared. He also mentioned Oulipo, the French literary group founded in the 1960s, namely the "Ouvroir de littérature potentielle." This literary group has long engaged in textual creation by setting various formal rules. In Lafforgue's view, some of what Oulipo does is similar to large language models: generating text through rules and computation. A large language model itself can also be seen as a "potential factory capable of generating text," ultimately generating text by computing the probability of the next text appearing. But today's large models still have an important difference from the machines that brought huge shocks during the Industrial Revolution: engineers in the 18th century generally knew what the machines they built were, and also knew the scientific principles behind them. Large language models are different. They can generate massive amounts of text, yet humans cannot fully understand how these texts are produced or why they are produced. At the underlying level, the model seems to consist only of simple mathematical rules, yet under the scale of enormous parameters and data it forms an extremely complex system. Laurent Lafforgue (photo by Yuan Jing, Wen Hui Bao) Lafforgue then mentioned the philosopher Simone Weil, sister of the mathematician André Weil, who long thought about mathematics from a philosophical perspective. She once proposed that "money, mechanization, and algebra are essentially the same." In Lafforgue's view, this view is still worth pondering when applied to today's large language models: currency converts things in reality into numerical values, while large language models convert language into tokens and vectors, and then generate new text through computation. In a sense, the world is increasingly being represented numerically. If machines become increasingly good at generating, what should humans do? Lafforgue believes the answer still lies in human creativity itself. He summarized Grothendieck's thinking on "mathematical creation" into three levels: posing questions that no one has posed before and creating new concepts; establishing new perspectives and understanding problems with new definitions and ways of expression; and connecting different viewpoints and concepts to form new understanding. In Lafforgue's view, Grothendieck himself was a representative of this creativity. He did not regard himself simply as a mathematician, but preferred to regard himself as a "writer." Just as writers create new texts and languages, mathematicians do not merely solve existing problems; they also create new concepts and "name" things that had not been defined before. This is also one of the differences between humans and AI: AI can process large amounts of existing knowledge and text, but humans can still pose new questions and create new concepts. "We can use AI, but we can do more than AI," Lafforgue said. At the end of the speech, he returned once again to the question of "text space": large language models can learn from massive amounts of text and generate new text, so is there some structure behind these texts that can be described mathematically? In Lafforgue's view, this may draw on Grothendieck's idea of the "topos" to further consider, from a mathematical perspective, the "text space" brought about by AI.