Will it be a golden age or a dark age for mathematics (and science)?

[This is a guest post by Francesco Maggi. This blog post was initially written in a different file format and converted using AI. — T.] Large language models (LLMs) and the research systems built around them (and commonly known as “artificial intelligence”) have demonstrated remarkable abilities in generating mathematical proofs. While the quality and limits of these tools remain incompletely understood, their potential for accelerating discovery and enabling extraordinary advances in mathematics and, more broadly, across science and technology seems clear. Seizing on that potential will require the careful integration of LLMs into the various STEM ecosystems, with the priorities of empowering a new generation of scientists, developing new STEM expertise, and preserving an open, collaborative, and intellectually independent scientific enterprise: that is the “golden age” scenario in the title. In the “dark age” scenario, scientific and technological output would still increase spectacularly in the short term, but, at the same time, independent human scientific activity would progressively weaken, the production of new knowledge becoming increasingly concentrated in a small number of hands, who become increasingly incapable of understanding the knowledge produced. The real darkness here would be knowledge becoming alien to human understanding. Reality will likely combine these scenarios. In eight points, I will examine the choices shaping that outcome, moving from mathematics to science more broadly, and closing with actionable proposals. (1) The technological breakthrough is real! But… To state the obvious, LLMs have become powerful research tools. Recent LLM-generated proofs, like the construction of a finite-time blow-up solution to the Navier–Stokes equations with a forcing term, or the proposed proof of the quasi-Riemann Hypothesis, deserve the utmost attention from mathematicians. Assessing their significance will require continuous independent scrutiny, interpretation, and assimilation by the mathematical community. LLMs seem most at home when writing proofs in Lean. (Lean is a proof assistant that allows computers to check whether a formal argument establishes a stated claim under specified definitions and axioms.) For those hard theorems where we really want LLMs to help us, these Lean proofs are extremely long, like hundreds of thousands of lines of code long. A seemingly emergent trend is that the longer the Lean version of a proof, the messier and less faithful its machine-generated “natural language” translation seems to be. We don’t understand why LLMs seem so bad at doing this kind of translation, but, as of today, they seem to really struggle with that task. So, for the really important problems, LLMs can give us correct Lean proofs that we cannot directly read, and messy natural language translations that are so complex as to require intensive study by mathematicians before they can lead to human understanding. Although this specific problem translation problem may change in the future, it leads us directly to the fundamental question: Without human understanding, why do we care about studying mathematics, or even obtaining proofs, beyond benchmarking LLMs? Mathematics is a very peculiar human artifact. It seems to have deep biological roots in how humans cognize. We identify (people, animals, objects); therefore, we count, we have numbers. We seem to have innate concepts of space and time, and we quantify and measure them with numbers. Some basic forms of arithmetic and geometry seem to almost be innate to humans. Different civilizations have built different kinds of mathematics, and even during the most recent history of mathematics, the very notion of the domain of mathematical inquiry has expanded drastically, sometimes even accidentally. (We may even be in a situation where there is a mathematics of humans and a mathematics of LLMs. This is maybe true not only at a linguistic level, but also in a deeper sense, although it is just too early to say.) I therefore disagree with claims that mathematics is over because machines can prove theorems. Without humans who understand mathematics, there is, quite literally, no human mathematics—a fundamental part of human culture. If we agree that we should not abandon mathematics because poorly understood machines bruise our anthropocentric vision of it, then society will need more mathematicians, not fewer. Like… a lot more. (2) Healthy scientific skepticism and the limits of LLMs’ creativity. Given the extraordinary results obtained by LLMs in mathematics, there is now considerable excitement concerning future capabilities and, at times, I think a lack of cool-headedness that does not do justice to the scientific method. Extraordinary performance should be received as a phenomenon to study, not as evidence for speculative conclusions about intelligence, creativity, or even consciousness. The latter are among the hardest legitimate subjects of scientific inquiry. They should be approached with explicit definitions, testable hypotheses, and empirical evidence. The same goes for the sweeping predictions about inevitable social transformations that inevitably follow the amazement generated by observed capabilities. We seem to forget too easily that the social consequences of these technologies will not be solely determined by their capabilities, but, most importantly, by the choices we will make on how these capabilities are developed and integrated into existing institutions. There are no inevitable outcomes, but there is a sore need for scientific and political leadership. What do we know today about the mathematical abilities of LLMs? Theoretical and empirical insights both support the conclusion that LLMs are extraordinary at recombining existing intellectual structures and discovering unobserved connections among distant parts of the scientific literature. What is more, LLMs do this at a scale that is inaccessible to individual researchers. Whether their uncanny effectiveness at proving theorems can be entirely explained as hyper-sophisticated linguistic recombination, or whether it involves other forms of synthetic reasoning that are yet to be described and understood, is today an open scientific question. One point that seems particularly hard to decide, given that commercial developers are not (cannot be) transparent on the working of their models is the relation between LLMs’ performance on a problem and the richness of the related mathematical literature. For example, both the Navier-Stokes equations and the Riemann Hypothesis are the subject of a vast literature created by hundreds of the most brilliant mathematicians. There may be something here. After all, when OpenAI published their “Tuesday October 6, 2026, Drop”, they mentioned having tried 4000 problems, reporting success for some hundreds of them (as counted after “aggregation” and “selection”). The language used there is a bit vague, but it still seems to imply the existence of a potentially large group of problems on which the frontier LLM failed to find solutions: from a scientific viewpoint, it would be very interesting to know what these problems were. Maybe those problems are more interesting than NS and RH? Or is the ground around them less fertile for LLMs to come help? These are important questions for humans who want to advance mathematical progress: how do we put LLMs in the best position to help? (3) Mathematization. Mathematicians seem substantially agreed that some human mathematical functions are not reducible to theorem proving. A particularly important one is mathematization: The act of inventing the very mathematical language through which reality (including mathematical structures themselves) becomes quantitatively and rationally intelligible. Newton, Fourier, Riemann, Einstein, Shannon, Grothendieck: their greatest mathematical contributions could be described as the highest examples of mathematization, as they included the invention of mathematical languages and conceptual frameworks that transformed entire areas of knowledge. We do not know if LLMs can perform mathematization — my guess would be that they should be (or eventually become) capable of mathematization of mathematical structures, so, of a very specific part of reality, but it is much less clear if they can go beyond that, at least in ways that they can then translate efficiently into natural language — I am going back to human understanding here. In my opinion, the most exciting opportunity created by making proofs a more accessible commodity is that, in the near future, we may be able to do much more mathematization than ever. There are vast swaths of the natural and social sciences where our ability to mathematize has been largely dampened by the difficulty of obtaining proofs — indeed, without proofs, it can be difficult to establish a theory’s consequences and compare them with observed phenomena. But, as often observed in science, we may have just climbed the peak of the mountain only to find more land to explore. I think there should be considerable excitement around this idea. Indeed, whenever a system of phenomena, experiments, and data is successfully mathematized, we vastly increase the sophistication and depth of the accessible insights into that system. The most spectacular example of this assertion is probably the success of quantum mechanics, where the mathematical formalization of an original set of experimental data has famously produced many physically counterintuitive but testable predictions confirmed by experiments that would have otherwise been unthinkable — who would have proposed entanglement as a property of nature if not forced to do so by theoretical prediction? (4) Mathematical progress cannot be measured by checking off boxes on a problem list.