Terence Tao’s blog on Sunday, Oct. 18, carried a guest essay asking a blunt question: “Will it be a golden age or a dark age for mathematics (and science)?” The author is Francesco Maggi, a mathematician at the University of Texas at Austin, and his answer is that the outcome hasn’t been decided. “Reality will likely combine these scenarios,” he wrote, before laying out eight points on the choices he thinks will settle it. (Tao added a note that the post was first written in a different file format and converted using AI.)
Maggi defines both futures carefully. The golden age means integrating large language models into science while empowering a new generation of scientists and “preserving an open, collaborative, and intellectually independent scientific enterprise.” In the dark age, output would still jump in the short term, but independent human research would weaken and the production of knowledge would concentrate in a small number of hands, “who become increasingly incapable of understanding the knowledge produced.” In his words, “The real darkness here would be knowledge becoming alien to human understanding.”
Proofs nobody can read
He doesn’t play down the technology. Maggi points to recent LLM-generated results, including the construction of a finite-time blow-up solution to the Navier–Stokes equations with a forcing term and a proposed proof of the quasi-Riemann Hypothesis, and says they “deserve the utmost attention from mathematicians.” Assessing them, he adds, will take continuous independent scrutiny by the community.
The catch is in how those proofs arrive. LLMs seem most at home writing proofs in Lean, a proof assistant that lets a computer check whether a formal argument holds, and for hard theorems those proofs can run to hundreds of thousands of lines of code. The longer the Lean proof, Maggi writes, the messier and less faithful the machine-generated plain-language translation tends to be. So for the most important problems, mathematicians can end up with “correct Lean proofs that we cannot directly read,” plus translations that need intensive study before anyone understands them.
That leads to the question at the center of the essay: “Without human understanding, why do we care about studying mathematics, or even obtaining proofs, beyond benchmarking LLMs?” Maggi’s reply is that mathematics is a human artifact with deep biological roots in how people think, and that without humans who understand it, “there is, quite literally, no human mathematics.” He rejects claims that the field is finished because machines can prove theorems. If anything, he argues, society will need more mathematicians, not fewer. “Like… a lot more.”
Skepticism, and the problems AI didn’t solve
Maggi also asks for cooler heads. Extraordinary performance, he writes, “should be received as a phenomenon to study, not as evidence for speculative conclusions about intelligence, creativity, or even consciousness.” What researchers do know is that LLMs are very good at recombining existing ideas and finding connections between distant parts of the literature, at a scale no individual can match. Whether their theorem-proving goes beyond that kind of recombination is, in his view, “today an open scientific question.”
He raises a specific puzzle. Both the Navier–Stokes equations and the Riemann Hypothesis have huge bodies of literature behind them, and he wonders whether LLM success tracks how much has already been written on a topic. When OpenAI published its “Tuesday October 6, 2026, Drop,” Maggi notes, it said it had tried 4,000 problems and reported success on some hundreds. That implies a large group of problems the model couldn’t crack, and he would like to know what they were.
More than a list of problems
Much of the essay is an argument about what counts as progress. Maggi singles out “mathematization,” which he defines as “the act of inventing the very mathematical language through which reality (including mathematical structures themselves) becomes quantitatively and rationally intelligible.” He names Newton, Fourier, Riemann, Einstein, Shannon and Grothendieck as its greatest practitioners. Nobody knows whether LLMs can do it. His guess is that they may manage it within mathematics itself, though it’s much less clear whether they can go beyond that, at least in a form they could translate into natural language.
Here he sees the biggest opening. Large parts of the natural and social sciences have been held back because proofs were hard to get, and without proofs it’s difficult to work out a theory’s consequences and test them. If proofs become cheaper, Maggi writes, “we may be able to do much more mathematization than ever.” His model case is quantum mechanics, where formalizing experimental data produced testable predictions nobody would otherwise have made. “Who would have proposed entanglement as a property of nature if not forced to do so by theoretical prediction?” he asks.
For the same reason, he warns against judging mathematics by problems checked off a list. Hilbert’s famous problems, which inspired the Millennium Prize Problems, were very productive. But the mathematics behind general relativity grew from geometric ideas passed from Gauss and Riemann to Ricci-Curbastro and Levi-Civita, and Shannon’s information theory came out of a line of work running from Fourier series through measure theory and probability. Neither came from a problem list. Reducing the field to hard problems, he writes, “provides a simplistic and incomplete understanding of the field and of what constitutes progress in it.”
Credit, secrecy and students
LLM-generated work draws on a vast human literature, and Maggi lists questions nobody has settled: which ideas fed a given proof, who is responsible for its correctness, and how citations get made when a model rediscovers an argument that’s already published. Lean can make formal correctness easier to check, he says, but people are still needed to establish meaning, relevance and context.
He also describes a field that is getting more guarded. Mathematicians have traditionally shared open problems and conjectures freely, and now, he writes, many have become cautious about sharing ideas. Who would type an original idea about the Riemann Hypothesis into ChatGPT, he asks, given OpenAI’s interest in the problem? Some are careful about what goes into research statements and proposals as well, in case a reviewer or administrator uploads them to an LLM for help writing a report.
Training may be the hardest problem. A strong result on a well-known question used to be a way for an unknown researcher to earn recognition. Among the mathematicians Maggi has spoken with, the prevailing sentiment is that “this traditional mechanism of recognition has effectively collapsed in a matter of weeks,” because a strong result no longer proves a student’s independent ability. He says he doesn’t know whether that sentiment is overblown. He adds that experienced mathematicians who have used LLMs heavily over the past year describe shifts in attention, motivation and depth of understanding, and he frames the open question as how to use LLMs “in ways that amplify our intellectual abilities without increasingly delegating the very cognitive activities through which mathematical expertise is learned and acquired?”
Who owns the tools
Maggi’s sharpest worry is the gap between what a few frontier AI companies can do and what universities and public institutions can do, especially when those companies pursue discoveries in competition with academics who rely on the same proprietary systems. He says this requires no assumption of bad intent, only that “commercial incentives, scientific priorities, and the public interest do not always necessarily align.” He credits OpenAI’s dialogue with mathematicians, which he says has improved how LLM-generated results are presented since the Navier–Stokes episode, but he calls voluntary company initiatives insufficient.
His proposal is for universities, professional societies, funding agencies and scientific foundations to coordinate the development and governance of competitive LLMs for research, with access to frontier models, disclosure and reproducibility standards for LLM-generated work, and strong confidentiality protections. The goal, he writes, is not total independence from private companies but preserving “the capacity for independent scientific research.”
In the meantime, he asks mathematicians to keep working at full tilt. “We must keep writing, publishing, reviewing, writing proposals,” Maggi wrote. “We must become more ambitious and curious than ever in doing all these things.”