There is no question that AI systems, in addition to producing vast amount of junk known as AI slop, are also capable of solving major problems in mathematics and science. The problem is that the culture in which these private companies operate is very different for the culture of academic research. While academic researchers can be very competitive and argue over priority, there are some basic norms that they adhere to, such as acknowledging and giving appropriate credit to others in the field. This last week saw those two cultures collide.

Most people are familiar with Newton’s laws of motion that relates the forces acting on a system to its motion. The simplest case that is used in introducing the subject is where there is a constant force acting on single solid object. The situation becomes more complicated when there are multiple variable forces acting on many particles that are under various constraints, and thenthe equations take much more complicated forms.

One of the most important of such cases is where variable forces act on fluids, and the most general form of that situation is the Navier-Stokes equations that were developed by various people and achieved their final form around 1850. These equations are applicable to a vast array of physical phenomena, such as weather or shock waves or currents. They are partial differential equations which are notoriously difficult to solve and my own research did not deal with them at all. But the equations are of such importance that in 2000, one important aspect of the equations was selected as a Millennium Prize Problem, with a prize of $1 million to whoever solved it.

The Millennium Prize Problems were unveiled by the Clay Mathematics Institute (CMI) at a meeting in Paris in 2000 to celebrate “The Universality of Mathematical Thought”.

These problems encapsulate some of the most difficult challenges that mathematicians were grappling with at the turn of the millennium. CMI’s purpose in articulating these problems and attaching a $1M prize to each of them was threefold: to elevate in the consciousness of the general public the fact that in mathematics the frontier is open, close at hand, and abounds with important unsolved problems; to emphasize the abiding value of working towards a solution of the deepest, most difficult problems; and to recognize achievements in mathematics of historic magnitude.

Each of the Millennium Prize Problems concerns a classical question of compelling origin that had defied all attempts to resolve it over many years. These are not arbitrary puzzles akin to fiendish crosswords. Rather, they are fundamental challenges that mark the frontier of human knowledge and challenge us to develop new structures and methods. They provide foci for the continuing struggle, across generations and cultures, to deepen our human understanding of mathematics and the universe that it describes. The deep innovations that are required to make significant progress on these problems open new vistas of possibility that typically reach far beyond the domain of the problem.

The Millennium Problem concerning the motion of fluids has all these attributes. The problem asks about the existence and smoothness of Navier-Stokes solutions in 3-dimensional Euclidean space.

But a controversy has arisen between a pair of mathematicians who had been working on the problem and an AI company that claimed to have solved the problem.

In a nutshell, Tristan Buckmaster at New York University had been working with Levent Alpöge, a mathematician who works for Anthropic AI. They felt that they were close to solving the problem. But last week, this happened.

On Sunday [September 6], even though OpenAI had completed a proof of Navier-Stokes and Dr. Buckmaster had not, Sébastien Bubeck, the leader of the OpenAI effort, made what would seem to be enticing offers to Dr. Buckmaster: Dr. Buckmaster could publish his results first. He could also serve as the lead author of an OpenAI paper describing the proof or get ample computing resources from OpenAI to further his work. The company would also suggest that Dr. Buckmaster and his collaborator deserved to collect the $1 million prize.
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Dr. Buckmaster said since he and Dr. Alpöge had used both OpenAI and Anthropic tools, the best resolution would have been for the two companies to collaborate in this instance.

“The big sticking point was that they didn’t want Levent on the paper,” Dr. Buckmaster said as he sat in his spartan university office.

Buckmaster balked at omitting Lepoge. As he rightly said, this kind of work build on the efforts of many people, and he could not agree to abandoning Lepoge, while Open AI did not want to acknowledge someone from a rival company.

The way Open AI’s Bubeck reacted to Buckmaster is revealing, akin to a mobster saying “Nice research you got here. Too bad if something bad should happen to it.”

Dr. Buckmaster felt that OpenAI was trying to buy him off. He said no.

Frustrated, Dr. Buckmaster said he might take the dispute public. He said Dr. Bubeck of OpenAI replied, “Why would you ruin your career?”

Dr. Buckmaster said that when he asked how that would ruin the life of an academic researcher, Dr. Bubeck replied, “If you don’t want me to be nice, then I don’t have to be nice.”
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Peter Sarnak, a mathematician at the Institute for Advanced Study in Princeton, was among those whom Dr. Buckmaster phoned on Sunday. “This kind of bargaining I’ve never heard of,” Dr. Sarnak said. “I admire him for behaving the way mathematicians behave.”
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At 2:30 a.m., he finally left his office and went home. On Tuesday afternoon, OpenAI announced it had the Navier-Stokes proof.

Dr. Buckmaster said he was worried about what would happen next. “To go against a trillion-dollar company is super scary and frightening,” he said. “We are all scared of OpenAI.”

What has been lost in the fracas, Dr. Buckmaster says, is recognition and appreciation of what human mathematicians brought to the problem.

He points not to himself and Dr. Alpöge but to Diego Córdoba and Luis Martínez-Zoroa, two Spanish mathematicians who a couple of years ago came up with a new strategy for attacking problems related to Navier-Stokes, designing a series of kicks that could knock fluid into flowing at impossibly infinite speeds.

“The main intellectual credit has to go to Luis and Diego,” Dr. Buckmaster said.

There is no question that AI is making huge strides in solving problems in mathematics and theoretical physics. But there is more to this kind of research than solving specific problems, though those are the things that give a focus. When human beings address those major problems, as they chip away at them, they often create new fields of research. It is not clear if AI will do the same thing.

In a comment to his own article that I linked to above, reporter Kenneth Chang makes this important point.

Sebastien Bubeck admitted that even though they have a team of mathematicians, none of them are experts in partial differential equations. That is, none of them worked on Navier-Stokes until just a week ago.

Dr Buckmaster gave me a good high-level explanation of how these proofs work. Can anyone at OpenAI give a cogent account of how its agents constructed the proof? Its not just the result that matters.

That is very true.