AI solves 80-year math problem, mathematicians push back with new declaration on AI guardrails

OpenAI AI Model Cracks the Erdős Unit Distance Conjecture

In mid-May, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had remained unsolved for 80 years. The unit distance problem, introduced by prolific mathematician Paul Erdős in 1946, asks a deceptively simple question: for any number of points on a flat surface, what is the greatest possible number of pairs that can be exactly one unit apart4

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? While mathematicians had believed the conjecture was true, the OpenAI AI model tried to disprove it instead—and succeeded4

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. Tim Gowers, who won the Fields Medal, wrote that "there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics". University of Toronto professor Daniel Litt called it "the first example of a result produced autonomously by an AI that I find exciting in itself".

The AI Math Breakthrough That Divided the Field

The achievement represents a significant step in the progression of AI in mathematics. Just three years ago, large language models struggled with basic arithmetic problems, and only last year did they start acing high school mathematics competitions. The rapid acceleration continued when Google DeepMind's AlphaProof achieved silver-level performance on the International Mathematical Olympiad in July 2024, solving four out of six questions3

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. A year later, both Google and OpenAI announced gold-level performance, making mathematicians "sit up" and realize the technology's advancing capabilities3

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. However, the OpenAI proof relied on perseverance rather than creative insight4

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. Melanie Matchett Wood of Harvard University noted that while it's "a beautiful piece of mathematics," she wasn't convinced this represents a breakthrough in Artificial Intelligence itself4

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. Thomas Bloom of the University of Manchester observed that it would have been "truly incredible" if the AI had managed to prove the conjecture, as that would require creative insight4

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Mathematicians Release Leiden Declaration Demanding Transparency

The OpenAI announcement triggered swift action from the mathematical community. On June 2, just days after the AI-generated proofs made headlines, a group of 16 mathematicians, computer scientists, and math historians published the Leiden Declaration on Artificial Intelligence and Mathematics2

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. The declaration aims to "frame the conversation about future directions," according to Dame Ursula Martin, one of the authors and a mathematician at Oxford5

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. As of June 5, the declaration had gathered 1,590 signatures4

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. The 11-page document lays out what mathematicians value about their research, how those values are threatened by AI, and how to address the situation2

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. Among the most important prescriptions: disclose the use of AI in research, ensure all papers receive peer review, and level the playing field between academia and for-profit companies through legal resources and public funding2

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Concerns About Research Integrity and the Unchecked Use of AI

The declaration addresses multiple threats to research integrity. Journal editors are already complaining about a flood of plausible-seeming AI-generated proofs that turn out to be incorrect in ways that are difficult for mathematicians to discern5

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. Bloom has seen people claim solutions to open problems using AI to generate "hundreds of pages of math that they can't understand or even read," raising the question: "Who's going to be able to check this?"4

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. The declaration also highlights how AI generates mathematical reasoning without showing what work inspired the ideas4

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. Wood raised concerns that the OpenAI paper did not appropriately reference "a history of closely related ideas in the literature"5

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. Rodrigo Ochigame, an anthropologist of AI at Leiden University, criticized the pattern: "The AI 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"5

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The Future of Human Mathematicians and Collaboration Between Humans and AI

The rapid progress has left many mathematicians questioning their future role. At a hastily organized meeting in San Francisco in April, there was "an air of excitement and curiosity," but also "an undeniable feeling of existential dread," according to Jacob Tsimerman at the University of Toronto3

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. Jeremy Avigad at Carnegie Mellon University wrote: "We are running out of places to hide. We have to face up to the fact that AI will soon be able to prove theorems better than we can"3

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. However, some see a complementary future. The current results point to a medium-term scenario where collaboration between humans and AI becomes standard: AI systems have broader knowledge of past work and more willingness to grind through tedious proof strategies, while humans can think more deeply about any one problem and ask more interesting questions. Terence Tao at UCLA suggests the field is moving from an era of "proof scarcity" to one of abundance, where mathematicians might race to be the first to understand a proof rather than find it3

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What Mathematicians Are Watching For

The declaration authors articulate that technology companies' involvement raises the risk that research questions are prioritized because of their amenability to AI methods rather than their deeper significance to understanding5

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. This disadvantages researchers who choose not to use the technology or lack access to it5

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. Jim Portegies at Eindhoven University of Technology notes that while almost every modern paper in math can be read for free on arXiv.org, tech companies often keep key details private—when Google DeepMind announced AlphaProof in 2024, it took more than a year before methods were published in a peer-reviewed journal2

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. The declaration is being endorsed by the International Mathematical Union and has a slot at the International Congress of Mathematicians in July in Philadelphia5

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. Most crucial, according to Ochigame, is the call for commercial AI companies to adhere to the declaration's principles and implement proper guardrails2

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