The concentration of advanced artificial intelligence in the hands of a small number of technology companies could allow private interests to control access to fundamental mathematical knowledge, leading researchers have warned. The concerns emerged at a meeting of Britain’s most senior mathematicians at the Royal Society in London last week, following OpenAI’s publication of solutions to 377 of the hardest mathematical problems, generated by its frontier AI models.
The meeting included three Fields medal winners, recipients of the highest distinction in mathematics, often described as the equivalent of the Nobel prize. Researchers said that while AI could accelerate mathematical discovery, the growing dependence on tools controlled by OpenAI, which makes ChatGPT, and Anthropic, the company behind Claude, could create new barriers to research if the firms chose to keep their findings or methods secret.
Martin Bridson, president of the Clay Mathematics Institute, which sets the Millennium Prize Problems, described the possibility of mathematical knowledge being restricted by commercial interests as a serious long-term concern. “There is a longer-term fear that knowledge could be hoarded behind paywalls,” he said, warning that researchers might eventually have to pay for access to AI chatbots to understand discoveries made using the technology.
Bridson, a professor of pure mathematics at the University of Oxford, said the more immediate concern was access to the tools themselves. “The more immediate fear is access to the tools of generating that knowledge,” he said. If advanced AI models become essential to mathematical research, decisions about who can use them could shape which researchers and institutions are able to contribute to the field.
The issue has already emerged in the work of Anthropic. Last month, Terence Tao, widely regarded as one of the world’s leading mathematicians, told The Observer that the company was paying mathematicians to formalise and write up solutions to problems solved by its frontier models, the most advanced AI systems not available to the public.
“That causes a different problem because they get to pick who writes up the paper,” Tao said.
The arrangement raises concerns about who controls the process of turning AI-generated discoveries into formal mathematical research. Even if solutions are eventually published, the companies developing the models may retain control over the tools needed to generate further results, leaving researchers dependent on decisions made outside the academic community.
Similar questions of access arise in other areas of scientific research. The Large Hadron Collider at Cern in Switzerland requires researchers to work through an established system in which access is determined by a research board advised by committees of scientists. The system provides a formal mechanism for deciding who can use the facility.
OpenAI and Anthropic have each committed $5m to a fund operated by the Simons Foundation, a leading supporter of mathematics research, to provide some researchers with access to their models. Bridson questioned the extent to which private companies should determine who receives access and the criteria used to make those decisions.
“Why should they get to say who has access to these tools, and on what basis do they choose who gets access?” he said.
Bridson acknowledged that some knowledge could legitimately be kept secret. Developing the ability to break cryptographic encryption, for example, could destroy online commerce, while intellectual property also required protection. But he distinguished such concerns from the control of fundamental mathematical discoveries.
“But the idea of fundamental knowledge being owned by people or by entities who might hoard it – that’s… deeply objectionable,” he said.
The debate also touches on a longstanding tension between secrecy and openness in mathematics. Scientific knowledge has not always been freely shared. The Pythagorean sect in ancient Greece operated as a secret society, while alchemists and Renaissance thinkers often guarded their methods. Bridson said the withholding of knowledge had historically enabled powerful groups to maintain their authority.
“There are moments in history where the priestly caste has maintained its power by having secret knowledge, and mathematics has played that role through the ages,” he said.
Early modern mathematicians also treated proofs as valuable intellectual property, using discoveries to secure employment and patronage rather than publishing them. In 1535, Niccolè Fontana, known as Tartaglia, became involved in a dispute with Antonio Marla Fior over cubic equations that continued for more than a decade. During one mathematical contest, Tartaglia solved all 30 of Fior’s problems, while Fior struggled to answer his opponent’s challenges.
Isaac Newton also worked on calculus in secret, claiming to have developed the method without publishing its details. When Gottfried Leibniz published his own version, Newton accused the German mathematician of plagiarism. British mathematicians subsequently refused to adopt Leibniz’s superior notation, a decision that set the field back for decades.
“[Openness] hasn’t been the norm throughout history, so it’s a very important principle to maintain,” Bridson said.
The Royal Society, which hosted the meeting, was founded partly to promote openness in science. Its vice-president, Professor Jon Keating, said AI could transform mathematical research, offering the prospect of major advances.
“The use of AI offers to usher in a golden era of mathematics,” Keating said. But he warned that the scientific community would need access to the underlying work to verify and understand the discoveries.
“However, if the mathematical community is to check these claims, and to benefit by understanding them, they need to be set out carefully and fully, not obscured or hidden from view,” he said.
The concerns were shared by researchers from outside Britain. Professor Ana Caraiani of Imperial College London, a member of the Association for Human Mathematics, said institutions in Romania, her native country, were not eligible to apply for access to the frontier AI models.
“Most countries in the world are in that position,” she said.
Martin Hairer, a Fields medal winner, said open-source AI models were unable to compete with the most advanced systems developed by private companies.
“This is a huge problem for the community,” he said. “It means that one of the main tools for mathematical research going forward is under the control of two US companies, and that’s obviously not the situation we want to be in.”
