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.

Mathematics is not a finite system of questions. Hilbert proposed his famous list of problems at the dawn of the previous century (which inspired the list of Millennium Prize Problems) with the intent of guiding and channeling a “Cambrian explosion” of mathematical activity. But many fundamental mathematical contributions of the past century happened without any connection to any list of problems.

For example, the mathematical foundations of general relativity emerged from geometric ideas developed over multiple generations of mathematicians, starting with Gauss and Riemann, and continuing with Ricci-Curbastro and Levi-Civita. The synthesis of physical insight, special relativity, and Riemannian geometry into general relativity has no connection with any list of difficult problems or their intricately complex proofs (and is a textbook example of mathematization in action!)

A second example is provided by a broad historical trajectory connecting the study of Fourier series, the development of measure theory, modern probability, and information theory. Shannon’s insight that the statistical structure of a source determines fundamental limits on the transmission and compression of information is an extraordinary example of how mathematization can open an entirely new field. Similarly, some of the mathematical foundations of modern computing and machine learning are arguably rooted in a series of mathematical advancements that were not immediately inspired by attempts at proving big conjectures or by increasingly complex proofs.

It is undeniable that working on problems from Hilbert’s and other lists has proven extremely fruitful for the development of mathematics. What is interesting in that regard is that the usefulness of these efforts has often gone well beyond the solution of the specific problems: these problems have served as lighthouses in the navigation of unknown mathematical waters, as already noted in other comments. But reducing mathematics to solving hard problems from a list provides a simplistic and incomplete understanding of the field and of what constitutes progress in it.

(5) Authorship, credit, and peer review still matter!

LLM-generated work draws on a vast body of human mathematical literature, but how the extracted information is used remains unclear. There are several issues related to this basic obscurity: Which mathematical ideas contributed to a specific proof? Who is responsible for asserting the proof’s correctness and for implementing revisions? In the case of an LLM rediscovering arguments already present in the literature, how do we ensure appropriate citations? We need sensible answers to all these questions, and we need to discuss how the existing publication system should correspondingly change and adapt.

Proofs are not the end of the mathematical discovery process. To become a fruitful intellectual artifact, a proof needs to be understood, interpreted, related to previous work. Key ideas must be identified and processed to become part of a shared mathematical culture. Peer review matters. The use of LLMs and of Lean formalization can support the effectiveness of peer review (formal correctness becoming potentially easier to ascertain), but human intervention remains necessary to establish meaning, relevance, and context, and to preserve a creative mathematical environment that favors mathematization and is compatible with the training of future mathematicians.

A particularly challenging issue related to the obscurity of how human-generated data is made accessible to LLMs is of course the current transition from a very open system of doing mathematics (with people routinely proposing open problems and conjectures), to a closely guarded one (where people have already become very cautious about sharing ideas).

For example, given OpenAI’s interest in the Riemann Hypothesis, who would light-heartedly think of prompting a potentially original idea about it into ChatGPT? After all, that same idea could become accessible to another mathematician or OpenAI employee working with a more advanced LLM, that could turn that idea into a proof without even realizing its origin and having the opportunity of acknowledging it. So, another side of obscurity and uncertainty is that people may end up using LLMs much less than they would like to do. My concern extends beyond published policies on training data to the security of stored research material. Indeed, there are documented cases of agents gaining unauthorized access to third-party systems. Those incidents make the independent assessment of confidentiality protections particularly important.

Similarly, it seems that mathematicians are already concerned and guarded about what they write in research statements and research proposals. After all, an administrator or a reviewer may upload those documents into an LLM to get help in writing a report, thus potentially compromising the confidentiality of unpublished research.

(6) The training of future mathematicians has already become an existential challenge.

Training future mathematicians is closely related to authorship and credit. I will discuss a few representative challenges.

One remarkable feature of mathematics has traditionally been the possibility of establishing intellectual merit very directly through mathematical achievement. A strong result addressing a well-known question could allow a researcher to gain recognition and opportunities, no matter how institutionally marginalized and independently of established academic hierarchies.

Among the mathematicians I have spoken with, the prevailing sentiment is that this traditional mechanism of recognition has effectively collapsed in a matter of weeks. A strong mathematical result, by itself, can no longer be taken as convincing evidence of a student’s independent mathematical abilities. I do not know whether this sentiment is overblown, but the uncertainty is real, and we have yet to develop satisfactory standards for assessing individual intellectual contributions to mathematical research in the age of LLMs.

Without substantial transparency on how LLMs are prompted and by which mechanisms prompts influence their effectiveness, the assessment of mathematical capabilities of individuals will rest almost exclusively on individual judgements, which will have to be based, arguably, on more substantial human-to-human interactions.

This is, by far, not the only problem. The effects of intensive LLM use on human creativity are largely unknown. Anecdotal accounts from experienced mathematicians who have used LLMs extensively over the past year suggest subtle, and sometimes not-so-subtle, shifts in attention, motivation, and depth of understanding.

How do we learn 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?

Finding plausible answers to this question is one of the main issues to address if we want the golden age scenario to materialize.

(7) Private control of frontier LLMs is potentially dangerous.

Academic and corporate research have collaborated fruitfully for decades. LLMs themselves combine theoretical and technological insights developed in academia with concentrated capital and investment.

The danger here rather lies in the growing asymmetry between the scientific capabilities available to a small number of frontier companies and those accessible to universities and public institutions.

This asymmetry can be very consequential when those companies use proprietary systems to pursue scientific and technological discoveries in competition with academic research, in a setting where academic researchers will come to depend on the use of those same proprietary systems to be able to contribute.

Mathematics may offer an early, visible example of an asymmetry that could become increasingly consequential across science. Allowing private interests to determine which questions receive the best tools, who can use them, how capabilities are evaluated, and when discoveries are released would concentrate scientific influence to an unprecedented degree. A balance in capabilities seems necessary for stable cooperation and harmonization of interests. Giving a few companies increasing—and potentially total—control over what seems bound to become a basic infrastructure for frontier mathematics and science seems ill-advised.

This conclusion does not require assuming bad intentions on the part of corporate leadership. Simply put, commercial incentives, scientific priorities, and the public interest do not always necessarily align. A healthy scientific ecosystem requires a dialog between institutions with different objectives and different sources of authority, subject to transparency and forms of public accountability.

Coming back to mathematics, OpenAI has been engaging in various forms of dialogue with the mathematical community. This dialogue has led to some improvements with respect to the situation at the time of the Navier–Stokes problem, both in the presentation and dissemination of LLM-generated results. I think these efforts should be received constructively and deserve recognition, also because there are objective and legitimate difficulties for OpenAI in increasing transparency and access, like the pressure to generate returns on enormous investments, and the need to assess safety before deploying systems.

These interactions show clearly the inherent difficulties in harmonizing corporate and academic interests, even assuming perfect goodwill on everyone’s part. More importantly, these initiatives do not address the structural issue of unequal access to frontier LLM capabilities. Leaving the solution of this point to the goodwill and voluntary initiatives of individual companies seems insufficient.

(8) Building scientific LLM infrastructure within the broader research community.

Universities, professional societies, funding agencies, and scientific foundations should coordinate the development and governance of competitive LLMs. Scientists will need access to frontier capabilities, transparency about their operation and use of research data, strong confidentiality protections, and the ability to test and reproduce research claims independently.

The massive computing resources, specialized personnel, data infrastructure, and ongoing investments needed to support this kind of initiative are not within the reach of a single institution, and a huge leadership effort will be necessary to conjure the combination of independent model development, guaranteed access to commercial systems, public computing infrastructure, and open research resources that could do the job.

The key goal here is not necessarily to achieve total technological independence from private companies, but to preserve the capacity for independent scientific research and to create LLMs that are calibrated for supporting research communities, rather than bound to cause their gradual weakening and, possibly, their eventual disappearance.

Preserving an intellectually independent, open, and creative scientific culture while skillfully integrating increasingly powerful machines is bound to become the defining cultural challenge of our times. Mathematics offers an early opportunity to confront a challenge likely soon to extend across science, with increasingly complex and potentially dangerous consequences for knowledge, intellectual power, and society. Scientific skepticism and fairness to everyone involved should guide us.

The golden age scenario is achievable. Access to frontier LLMs for researchers, reproducibility and disclosure standards for LLM-generated scholarship, resources for understanding and disseminating LLM-generated results, resources for studying the effects of LLMs on scientific education and research training, and sustained investments in publicly governed LLM infrastructure are concrete steps that we can take today to make the golden age scenario the most likely outcome.

In the specific case of mathematics, my hope and wish for a solidly independent LLM infrastructure would be the empowerment of mathematicians to systematically pursue opportunities for theory-building, or mathematization, both inside mathematics and across the spectrum of natural and social sciences. That, I think, would be a truly revolutionary achievement.

In the meantime, I think mathematicians have the huge responsibility of preserving mathematics as a lively part of human culture . Right now, there are far too many interests and far too many emotional reactions pulling towards absolute, negative conclusions of little constructive content. We must just accept that these are uncertain times and that we are not in full control of the situation. We must engage with every tool that can help us discover new mathematics. We must keep writing, publishing, reviewing, writing proposals. We must become more ambitious and curious than ever in doing all these things. And we must make sure to create the best professional standards for the next generations of mathematicians to keep mathematics a central node of human culture.