Space and Physics
PUBLISHED
People who think that math is boring don't know what they're talking about.
This week, in particular, a solution to one of the complex Millennium Prize Problems might have been put forward by the artificial intelligence company OpenAI. But the announcement was preceded by some heavy accusations from mathematician Tristan Buckmaster.
What are the Navier-Stokes equations, and why do they matter?
The solution in question looked at the Navier–Stokes equations. These equations describe the motion of fluids and are used in a bajillion technical applications.
They arise from Newton’s second law of motion, the famous F=ma, in which force is equal to the mass of an object times its acceleration, in combination with the assumption that stress in a fluid is a combination of its viscosity and pressure.
They are the generalization of what are called the Euler equations, which consider a fluid that is not viscous.
The equations are used in everything from the design of cars and airplanes and understanding how blood flows through our body, to how pollution spreads through the environment. But in pure mathematical thought, they represent a problem.
This is called the Navier–Stokes existence and smoothness problem.
The question is: are there always solutions for these equations in three dimensions, or do the equations break down, or ‘blow up’? This would be the presence of a nonsensical solution in which the velocity becomes infinite, something that does not happen in a physical system.
Because of this, the Navier-Stokes equations are part of something called the Millennium Prize.
What are the Millennium Prize Problems?
The Millennium Prize Problems are a set of seven complex mathematical problems selected by the Clay Mathematics Institute in 2000. If anyone solves any of these problems, then they will win a prize of $1 million.
To date, only one problem has been solved. In 2010 a Russian mathematician called Grigori Perelman solved the Poincaré conjecture, but he refused the money as the prize ignored Richard S. Hamilton, whose work was crucial for Perelman's solution.
The Clay Mathematics Institute has yet to assess OpenAI's claims to confirm if the problems have actually been solved. Whether or not the solution is correct, OpenAI has already said that it does not intend to claim the Prize.
Is this the moment that math changes forever?
So, back to the beef. On September 8, OpenAI, the company behind ChatGPT, announced that they had solved the Navier–Stokes equations and, indeed, everything did blow up.
In a press release, OpenAI stated that on September 1 that they heard rumors that two Millennium Prize Problems had been resolved, so they themselves set a new Large Language Model (LLM) to look at all six unsolved problems.
In a matter of days, the LLM appeared to have solved the problem by itself.
But then things got spicy. On September 7, mathematician Tristan Buckmaster from New York University published a long statement and several papers describing the work he and fellow mathematician Levent Alpӧge, who works for OpenAI’s competitor Anthropic, had done on the same the Navier–Stokes equations.
This showed that they had solved the problem for the Euler equation. This could be seen as a stepping stone to solving the full Navier-Stokes equations.
Buckmaster credits the idea behind their work to the mathematical explorations of Diego Córdoba and Luis Martínez-Zoroa. According to Buckmaster and Alpӧge, they used those ideas and combined them with several different LLMs, including Anthropic’s Claude and OpenAI’s Codex.
Buckmaster described the ability to solve such difficult problems in math in a matter of weeks with the use of LLMs as akin to a “Deep Blue-Kasparov moment”, a reference to when computers first beat Grand Masters at playing chess.
Serious accusations on how the proof came to be
In the statement, Buckmaster alleges that OpenAI was only able to develop a full proof for the Navier-Stokes problem because the company found out about the pair’s approach and copied it.
“Almost nobody else I know of was working on it,” Buckmaster wrote in the statement. “It is not the direction one arrives at in a few days by giving a model the problem statement. When I heard “forced,” it was a bright red flag.”
Buckmaster alleges that in private correspondence, someone at OpenAI told him that going public with his claims would ruin his career and that “If you don’t want me to be nice, then I don’t have to be nice.”
Buckmaster does not accuse OpenAI of plagiarism, though it would not be the first time that OpenAI has been accused of this.
In their announcement, OpenAI acknowledged the work of Buckmaster and Alpöge but stated that nobody was aware of the content.
“We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem,” says OpenAI. “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”
This is in direct contradiction to the timeline shared by Buckmaster. All we can do now is wait to see how this unfolds.
We told you: math is always full of drama.
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