“A week after OpenAI made headlines with an A.I.-generated
proof, a new “declaration” by 16 experts raises concerns that the technology
threatens math as a discipline.
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 declaration is being endorsed by the International
Mathematical Union, and has a slot on the program at the International Congress
of Mathematicians, which will be held this July in Philadelphia. And it is now
open for signing by individuals and organizations such as national mathematical
societies.
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
Erdos problems, and got lucky with this one. 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?” [1]
1. As A.I. Makes Strides in Mathematics, Mathematicians Urge
Caution. Roberts, Siobhan. New York Times (Online) New York Times Company. Jun
2, 2026.
