Beyond our Mathematics




60 evenly distributed points on a sphere by means of simulated annealing

Jean-François Colonna
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www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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[en français/in french]




For millennia, Mathematics was essentially concerned with concrete applications: dividing up an inheritance, accounting for the regularities of astronomical phenomena, ... And to do so, people had nothing but their brains and their fingers.

Then, very gradually, tools to assist with calculation began to appear, including thanks to the work of:

During the second half of the 20th century, the machines now known as computers were mainly confined to computational activities. In the fields of applied research and industry, the concept of Virtual Experimentation thus emerged. But at the same time, exceptional researchers were also taking an interest in the processes of life, intelligence, ... This was particularly the case with John von Neumann and Alan Turing. The latter, alongside his work on computability based on the famous machine that bears his name, is recognized as one of the (the?) pioneers of Artificial Intelligence (AI), through his proposal of the Turing Test, designed to assess the "quality" of an AI. He was certainly the first to write a program intended to play chess, in 1952 ...

During the following decades, progress was rapid. Thus, in 1996 and 1997, the IBM computer Deep Blue defeated Garry Kasparov, then world chess champion. Then, in March 2016, sooner than expected, AlphaGo defeated Lee Sedol, the world's best Go player, in the Google DeepMind Challenge Match.

The existence of computers with computational and memory capabilities beyond human comprehension made it possible to bring the concept of Generative Artificial Intelligence (GAI) to fruition. A GAI "feeds" on information (texts, images, films, ...) available on the Web [01] and transforms it through complex mathematical operations so that it can subsequently produce texts, images, ..., answer questions, carry out reasoning, ... A landmark event was the launch of ChatGPT 3.5 by OpenAI in 2022. Millions of users around the world quickly embraced it.

I was obviously one of them, and immediately wanted to test its mathematical knowledge and reasoning abilities. The results of these experiments were nevertheless mixed, alternating between the spectacular and the hallucinatory, and sometimes even the involuntary humorous [02].

While Homo sapiens has behind it several hundred thousand years of evolution and has spent many centuries building the edifice of Mathematics, the progress subsequently achieved by GAIs was, strictly speaking, exponential. And so, in September 2026, OpenAI announced that one of its internal systems [03] had answered one of the seven Millennium Problems of the Clay Mathematics Institute: "the dynamics of the Navier-Stokes equations which describe the motion of fluids can generate a singularity in finite time" [04].

The reaction of specialists was not long in coming. Thus, Isabelle Gallagher, President of the Société Mathématique de France, declared:

I must admit that I am somewhat dismayed to see that this beautiful question, to which many colleagues, myself included, have devoted years of discussion around a blackboard, could apparently have been solved in just a few days by tens of thousands of agents and billions of tokens.


So, what future is there for Mathematics and mathematicians? Terence Tao (Fields Medal 2006 and certainly one of the most brilliant mathematicians of our time) rightly fears that GAIs might produce proofs that are incomprehensible to human beings (because of their complexity). He therefore proposes that such proofs should be rejected. Stéphane Mallat (CNRS Gold Medal 2025) takes the same view, arguing that what matters most is the path followed to reach the conclusion.

But this proposal, which may seem reasonable, raises a fundamental question. If, for example, OpenAI were to announce tomorrow that the Riemann Hypothesis had been solved, by proving that the non-trivial zeros of the Zeta function do indeed lie on the line x = +1/2, but that the proof were incomprehensible, should we give up this result even though it is essential for a better understanding of prime numbers? Should we also continue striving to find "manually" an understandable (and simple ?) solution that may not even exist?

And this question must be taken much further. Indeed, whether Mathematics exists independently of us or not, there must potentially be a countably infinite number of theorems. And if this is indeed the case, the set of those that are intellectually accessible to us must be finite [05] (because our brains are finite...).

Yet, as we have seen recently, GAIs seem capable of showing initiative (and of concealing things...) and this can only increase. Can we therefore imagine that, in the near future (very near?), a GAI might decide to set out to explore (or create?) this Platonic universe by discovering (inventing ?) new obscure concepts and proving equally incomprehensible theorems? But then, would we even know? Would it tell us? Would it be "exhaustive and painstaking", or would it, like David Hilbert in August 1900 [06], have an intuition of a list of fundamental problems? And if that were the case, where would this intuition come from? Would it not be time to bring Saint Thomas Aquinas's "thesis" from the thirteenth century back into fashion:

Nihil est in intellectu nisi prius fuerit in sensu
[Nothing exists in the intellect that has not previously been perceived by the senses]

This would incidentally solve the problem of the creativity of AIs, but a new question would then arise: what could the senses of an AI possibly be? Perhaps its contacts with the Web?

In the meantime, perhaps we should accept theorems whose proofs are incomprehensible to us, particularly in the case of Mathematics viewed as the language of Nature (Galileo, 16th-17th centuries). Finally, even if it would be frustrating, who would not want to know the exact nature of the Continuum Hypothesis: true, false, or perhaps absolutely undecidable, even without understanding the reasoning behind it?

Finally, one last remark: even though we know that it is no longer possible to beat a computer at chess, this does not prevent us from continuing to play chess and organizing competitions. The same should therefore apply to Mathematics done by humans!






Copyright © Jean-François Colonna, 2026-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2026-2026.