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

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 ass