A decades-old mathematics puzzle has been cracked, according to OpenAI, which says it used thousands of agents to solve the Navier-Stokes existence and smoothness problem. The result is a 166-page proof, yet mathematicians are not celebrating. Experts argue that the solution, created by agents, fails to include the deliberate reasoning they regard as essential to the discipline.
Scientists in the 19th century built the Navier-Stokes equations to explain how thick liquids move. These formulas offer an approximate picture of fluid flow in our actual world. Yet mathematicians turned their attention toward the equations themselves, setting aside any practical use they might have.
What the Proof Actually Does
Experts questioned what the formulas might produce under impossible circumstances. A problem was shaped around this doubt: Whether the equations themselves suggested that, in such conditions, a liquid could suddenly burst apart without any physical cause.
The equations known as Navier-Stokes are approximate, and mathematicians believe they should not permit physically impossible scenarios. Yet those very contradictions are what makes them especially intriguing, since they sometimes give rise to entirely new mathematical concepts. According to Jared Speck, a mathematician at Vanderbilt University, the field had been building “a deep and beautiful theory” around these equations for years. The community was close to solving the question when OpenAI’s proof showed that yes, the Navier-Stokes equations do indeed imply a sci-fi fluid explosion.
How Mathematicians Work
Mathematicians don’t always rush to get the right answer as quickly as possible. The creation of new mathematical ideas can actually look a lot like artistic exploration, with mathematicians taking a careful, considered approach to developing their thoughts. This process often involves spending months or years writing out and sharing the steps of their proofs in an attempt to persuade other colleagues of their conclusions.
That didn’t happen with OpenAI’s proof. LLMs don’t reliably cite their work, and they don’t explain themselves clearly to humans. The 166-page proof remains under peer review, and experts have not yet had time to confirm its validity.
A mathematician at Colorado State University says “We don’t even know how autonomous it was, or how much human expertise is necessary to scaffold the process,”.
The Declared Misalignment
Sandhu has signed an online declaration titled “A Severe Misalignment of AI in Mathematics.” The declaration says mass-producing proofs “could destroy fertile ground instead of breathing life into new ideas.” Its initial signatories were 25 winners of the Fields Medal, often called the Nobel Prize in mathematics.
Sandhu is not opposed to the use of AI, he explains; he and many other mathematicians already employ it in their work. His objection, rather, is to how AI solves problems: it does so in a manner that undercuts human comprehension, even as it finds solutions.
“We don’t even know how autonomous it was, or how much human expertise is necessary to scaffold the process.”
Why the Proof Is Unreadable
The proof’s lack of transparency is the central concern. Mathematicians have criticized OpenAI’s lack of transparency in how it arrived at the solution, and Tristan Buckmaster, a mathematician at New York University, has suggested that OpenAI may have used his and others’ work without proper attribution.
“When problems resist solution, they take on a bit of lore,” says Speck. The draw of Navier-Stokes is similar to why people play Sudoku or chess, both of which have no utility other than being fun and intellectually stimulating.
What Comes Next
While the proof remains pending peer review and has not been validated by experts, the larger issue concerns whether AI can make a genuine contribution to mathematics.
Speck says it plainly: “Basically nobody in my community really understands what’s going on. In fact, the proof was done in a very unfamiliar order. First, the result was given to us by the computer, and now people are like, ‘Let’s try to understand what’s going on.’ We’re at the very beginning of that.”
Mathematicians care about how discoveries come into view. The path from question to answer carries meaning for them. OpenAI’s method breaks that chain, turning the steps around so the final piece arrives before anyone can say what it means.
Here is how the stages of the problem unfolded:
- Mathematicians developed “a deep and beautiful theory” around the Navier-Stokes equations for decades.
- The community was on the verge of cracking the problem.
- OpenAI’s proof found that the equations did imply a sci-fi fluid explosion.
- Experts are now trying to understand what the proof means.
The accomplishment is genuine. Yet the path to it is not. OpenAI has constructed a device capable of resolving issues that have long stumped mathematicians. Still, the machine does not grasp those issues in the manner humans do.
The difference between the two approaches matters greatly. A demonstration that nobody can understand fails to advance mathematics. It stands instead as a tribute to processing capability that reveals nothing about the underlying issue at hand.
OpenAI says it has resolved the Navier-Stokes existence and smoothness problem. Whether that counts as a service rendered to mathematics remains open to doubt.
Source material: “Solving Math’s Greatest Problems Was an Art Form. Then Came AI,” WIRED.
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