TECH

One of mathematics' greatest enigmas and AI

For decades, certain mathematical problems remained unsolved despite the efforts of experts worldwide. Now, artificial intelligence systems are venturing into this territory in ways that surprise even researchers accustomed to major breakthroughs. One company claims to have achieved dozens of new results in just a few weeks. Yet, while the speed is impressive, another question is gaining importance: how do we verify, understand, and transform these discoveries into mathematical knowledge?

In late August, OpenAI began internally training a model that has not yet been released to the public. Just 24 days later, the company made an extraordinary claim: the system had reportedly solved over 100 mathematical problems that had remained open for years.

These alleged results span various areas of mathematics. However, the case attracting the most attention involves a problem that has held a special place in the field for decades: Navier–Stokes.

OpenAI itself acknowledges that even its mathematicians did not anticipate such a rapid pace. According to the company, the model managed to produce results on a scale that raises a question beyond the mere ability to solve equations.

If an AI can generate potential solutions faster than experts can analyze them, the bottleneck shifts from simply finding an answer to verifying what has actually been discovered.

There is also a significant distinction between the claim regarding Navier–Stokes and the more than 100 other problems mentioned by the company. For Navier–Stokes, OpenAI published a manuscript and a formalization in Lean. For the other results, no complete list accompanied by public proofs has been presented so far.

This means that the figure of over 100 remains, for now, a claim made by the company itself rather than a set of results that has been independently examined by the mathematical community. The Navier–Stokes existence and regularity problem is one of the Clay Mathematics Institute's seven Millennium Prize Problems. Simply put, it seeks to determine whether the equations used to describe fluid motion in three dimensions can develop certain singularities in finite time.

OpenAI's model reportedly produced a solution indicating that this behavior can occur—and, more importantly, a formal proof in Lean.

This formalization is significant because Lean allows for the mechanical verification of a proof's logical steps. Even so, this does not mean the problem has been officially resolved.

The Clay Mathematics Institute acknowledged that the issue appeared to have been solved but noted that there is a specific process for recognizing a solution. The work must be published in an appropriate venue, remain available for scrutiny for at least two years, and gain general acceptance among mathematicians.

As this discussion unfolded, another issue arose: how should the community react to mathematical results produced by AI?

In September, 25 Fields Medalists and other mathematicians issued a statement criticizing how open problems were being used in the AI ​​race. The concern was not simply to deny the capabilities of these systems.

The point was different. In mathematics, solving a problem is only part of the process. It is also crucial to understand the idea behind the proof, explain why it works, acknowledge prior work, and enable other researchers to build upon that knowledge.

The company's response was announced on September 21: an advisory group comprising nine mathematicians began advising OpenAI on matters related to mathematics and artificial intelligence.

The group is affiliated with the Institute for Advanced Study in Princeton and includes prominent figures such as Timothy Gowers, Martin Hairer, Edward Witten, Ravi Vakil, and other distinguished researchers.

The group's role will be to help evaluate and communicate new results, recommend academic standards, and offer critical feedback to OpenAI itself when necessary. Its members will not be paid by the company and will be able to participate in determining the group's composition. But there is a clear boundary. The group lacks the power to dictate the pace of OpenAI’s internal research or to veto company decisions. Its role is to bridge the gap between AI labs and the mathematics community.

Mathematical enigmas and the controversy...These new achievements are part of a wave of AI results flooding the field of mathematics. Over the past year, AI has enabled major breakthroughs, leaving mathematicians grappling with rapid changes in their discipline. The new result, concerning one of mathematics' most important problems, represents "the spectacular culmination of the trajectory we’ve seen over the last 12 months," said OpenAI researcher Sébastien Bubeck during a press conference on September 8.

In a statement published on his website on September 7—alongside the solution regarding forced Euler flows—Buckmaster stated that his team's results mark a "Deep Blue-Kasparov moment," referencing the historic 1990s milestone when a supercomputer defeated the best human chess player. "The community needs to have a serious, unhurried discussion about the next steps," he said.

Mathematicians are certainly paying attention. Albritton learned of Buckmaster and Alpöge’s result around 1 a.m. while awake with his newborn baby. He stayed up until 6 a.m. discussing the matter with colleagues.

Despite the problem's mathematical significance, its solution will not have major practical implications, Eyink notes. The Navier-Stokes equations describe a fluid as a continuum, but real-world fluids are composed of individual molecules and atoms; thus, it is already known that there is a limit beyond which the equations cease to be valid. It is more a matter of prestige, says Eyink. "There is immense mathematical celebrity associated with these equations."

This also raises concerns regarding how credit for a discovery is distributed. After initial rumors began to circulate, Buckmaster says he held a series of discussions with OpenAI researchers about how to present the two results. In these discussions, he claims there was a request to exclude his co-author, Alpöge, who works for Anthropic, an OpenAI competitor. Buckmaster’s account also raises questions about whether AI agents had access to the progress made by Buckmaster and Alpöge. OpenAI denies that its AI agents had direct access but states: “while unlikely, we cannot rule out that anonymized data derived from their use of our products helped improve our models.”

The major implication of this breakthrough may be the issue it raises regarding the difficulty of assigning credit when AI is involved, says Eyink. "To me, this is the truly serious and ongoing problem and question that needs to be resolved."

This reveals the scale of the emerging problem.

The discussion is no longer just about whether an artificial intelligence can find a solution that no human had previously discovered. The question now becomes what happens when these machines generate potential discoveries faster than experts can verify, interpret, and incorporate them into existing knowledge.

And perhaps this is the true test of the new era of AI-driven mathematics: not just discovering more, but figuring out how to turn that speed into reliable science.

mundophone