Recently there are signs that some branches of higher mathematics, among the most rarefied realms of human achievement, are vulnerable to a shake-up by artificial intelligence. Mathematicians, in turn, have been thinking about how to respond.
On Tuesday, a group of 16 mathematicians, in consultation with colleagues and math organizations worldwide, published the Leiden Declaration on Artificial Intelligence and Mathematics. It aims to “frame the conversation about future directions,” said Dame Ursula Martin, one of the authors, and a mathematician and computer scientist at Oxford.
This effort comes as A.I. models have been making headlines with successful results in research-level mathematics. In late May, OpenAI, the maker of ChatGPT, announced that one of its models had disproved a notable 80-year-old mathematics conjecture in the field of combinatorial geometry.
The conjecture is one of some 1,200 problems posed by the Hungarian mathematician Paul Erdos. While some of these “Erdos problems” are considered throwaway questions of narrow interest, others have proved influential and field shaping. Along with a research paper describing the proof, OpenAI released a companion paper by several independent mathematicians. Jacob Tsimerman of the University of Toronto, an expert in the adjacent subfield of number theory, commented: “This is a really impressive piece of work, and I would accept it for any journal without hesitation.”
Other figures in the field were less sanguine. Melanie Matchett Wood, a Harvard mathematician, was enthusiastic but raised concerns. For instance, she commented that the OpenAI paper did not appropriately reference “a history of closely related ideas in the literature.”
“It is a powerful tool, and I think it will be a great tool to accelerate mathematics research,” Dr. Matchett Wood said in an interview. But she noted that the community needs to figure out how to use A.I. “in a way that will maintain human understanding of the mathematics.”
Among the potential threats that the Leiden Declaration authors articulate are accuracy and reliability: Journal editors are already complaining about a flood of plausible seeming A.I.- generated papers and proofs that have turned out to be incorrect, and in ways that are difficult for mathematicians to discern.
Perhaps most pointedly, the authors raise the question of whether the many A.I. companies tackling mathematics — major players such as OpenAI, Google DeepMind and Anthropic, or start-ups such as Harmonic, Math, Inc. and Axiom Math — are keeping the field’s best interests in mind. “Technology companies’ involvement in research,” they write, “raises the risk that research questions are prioritized and incentivized because of their amenability to A.I. methods and models, rather than their deeper significance to understanding.” In turn, they point out, this disadvantages researchers who choose not to use the technology, and those who do not have access to it.
For Rodrigo Ochigame, a historian and anthropologist of computing and artificial intelligence at Leiden University in the Netherlands, and one of the statement’s authors, the latest OpenAI proof illustrates why this sort of collective reckoning in the discipline is necessary. “The story follows the same pattern as many other announcements by commercial A.I. developers,” Dr. Ochigame said. “The A.I. model is proprietary and unavailable to anyone outside the company. We get a flashy promotional video, while basic information needed to assess the scientific meaning of the result is kept secret. The company disclosed nothing about the methods, human-written prompts, training data, or computational resources consumed.”
The following conversation — conducted by videoconference and email with Dr. Ochigame, Dr. Martin and mathematician Michael Harris of Columbia University, author of the Substack newsletter “Silicon Reckoner” and another member of the declaration’s working group — was edited and condensed for clarity.
What is the Leiden Declaration?
MARTIN It’s a provocation, a stimulus for debate. There are more and more press stories about the mathematical achievements of A.I., and many mathematicians feel uneasy.
What OpenAI has done is throw a great deal of resources at finding a counterexample to this particular conjecture. That’s remarkable, and impressed the experts. We are not told about the model’s failures. If you put vast quantities of human effort into this problem, you likely would’ve solved it in the same way. But in math human effort is scarce, and just tends to be spent on different things.
To think of mathematics in terms of precise and neatly stated problems, like high school exams or the list of Erdos problems, is to misunderstand and diminish what makes mathematics so powerful and significant. Mathematics is not just about solving problems — it is also the cultivation of ideas, understanding, judgment, and human insight.
HARRIS The purpose, from my perspective, is to recover control of the narrative about the values and the goals of mathematics from the A.I. industry. Mathematicians are concerned that the values of the profession are being misrepresented, not intentionally but due to the media campaign on the part of the industry, which seems to want to promote the belief that they are in a position to transform mathematics — “the A.I. revolution in math,” as one headline put it not long ago.
If the people who make the decisions about funding base their decisions only on what’s being reported in most of the articles in the press, they could easily get the impression that A.I. is where the future of mathematics is.
We want to affirm certain values that have characterized the profession: openness, honesty, giving credit where credit is due, sharing, transparency about methodologies, and access for independent verification of results.
An aspect of mathematics that is cherished by mathematicians is that it is one of few successful examples of a gift economy — that is to say, its economy is somehow an island of idealism in our society. As director of graduate studies in the mathematics department at Columbia, I read the personal statements of all the applicants every year, several hundred, and they are still idealists.
The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics.
OCHIGAME Several A.I. companies are investing in dedicated teams focusing on mathematics, using problems as benchmarks and publications as training data. They are training their models to prove theorems not because they want to advance mathematical knowledge, but because they hope that such training will improve the models’ reasoning abilities more generally.
Those companies have repeatedly articulated this strategy in pitches to investors, so it is perhaps not a coincidence that OpenAI’s announcement about the unit distance conjecture came out the same day the news broke that the company is preparing to file for an I.P.O.
This situation has put mathematicians in a troubling ethical position. Without their consent, their published work is being used as strategic training data for the development of general-purpose A.I. The resulting models are being commercialized for many purposes, including military applications, that raise grave ethical concerns. Most mathematicians never imagined, much less consented, that their work would be used for such purposes.
MARTIN It’s important not to lose sight of the fact that what the A.I. companies are doing, what you can achieve with this technology, is absolutely extraordinary. I don’t think we’re challenging that. We’re challenging the framing, we’re challenging the behaviors around it.
OCHIGAME The group of authors included many people who are excited about the potential of new mathematical results, and people who have even contributed to the development of the technology. But the public discourse is so heavily tilted toward the very effective P.R. campaigns of the A.I. companies and the narratives they are pushing. We feel an obligation to be the voice for expressing the critical concern. But we certainly understand the enthusiasm.
HARRIS I would add that a lot of the enthusiasm and excitement is artificially generated by the corporations. The declaration warns against that: “Don’t believe the hype.” It is important for the mathematical community to have the last word on what, mathematically, is, and what is not, exciting.
Do you worry that the declaration might be seen as mathematicians embarking on a futile effort — circling the wagons in order to save an outdated profession that A.I. is threatening with obsolescence?
HARRIS Depending in part on how the story is reported, industry supporters are likely to frame it this way, but such a framing is hostile to mathematics as an intellectual achievement and not merely to mathematicians.
MARTIN It’s not either/or but both/and. Centuries of enduring work by mathematicians underpin every aspect of modern science, life and society. The authors of the declaration, alongside the world’s mathematical organizations, are committed to ensuring that mathematics continues to flourish through both intellectual rigor and practical application. We welcome A.I. companies as responsible partners in the spirit of the declaration.
OCHIGAME Mathematics is a rich form of cultural expression with an ancient history, and I am not worried that any technology will ever render it obsolete. Its most precious aspects, such as the collective quest to understand beautifully intricate ideas, and to explore the limits of the human imagination, cannot ever be automated. What I am worried about is that a handful of corporations are mobilizing their vast financial resources to impose an impoverished view of mathematics so forcefully — at a moment when scientific research is already under political attack — that they may well end up destroying the social institutions that allow mathematics to flourish. What could be futile about resisting that?
