When Mathematics faculty member Mike Dabkowski saw the news that OpenAI had solved the Navier-Stokes equation he felt a pit in his stomach. His UM-Dearborn colleague, Yunus Zeytuncu, experienced sadness, fear and uncertainty all at once. They weren’t alone. The announcement that a problem mathematicians had pondered for more than 100 years was solved by AI in 88 hours has created an existential crisis across the field.
The Navier-Stokes problem— deemed one of seven "Millennium Prize problems” by the Clay Mathematics Institute because it is so complex — is a series of equations pertaining to the movement of liquids and gases. These equations are integral to weather forecasting and can also explain other phenomena related to fluid dynamics, such as ocean currents and blood flow.
Dabkowski, an associate professor of mathematics whose dissertation explored a problem that is Navier-Stokes adjacent, says to think of a water balloon: If you squeeze the balloon, the water doesn’t compress, the balloon just deforms a little bit. It’s that incompressibility that makes the equations difficult for mathematicians to analyze. Work to date had suggested there could be a point — even if completely theoretical — at which the equations become unstable. If that point were reached, it would mean the laws of physics had broken down and the water in the balloon could spontaneously explode.
This conundrum may not vex engineers, who are largely concerned with the ways liquids and gases function in reality. In the conceptual world of math, however, “the equations were very, very challenging to understand,” Dabkowski says.
But, he continues, attempting to solve very challenging equations — even if the answer is never reached in one’s lifetime — is the joy of mathematics. For many mathematicians, solving a math puzzle is akin to writing a poem or painting a work of art. The inspiration comes not from the final product but the creative process that leads there. Now, says Zeytuncu, who is a professor of mathematics and interim vice provost-elect for research, he and his peers across the world are asking: What does it actually mean to solve a problem?
“Mathematicians will spend three hours, three days, maybe three months to solve a problem on their own,” he explains. “They know the solution is on the next page but they don't care about that. It is such a drug that when you can solve it on your own. The satisfaction you will get from it is so high that you will keep doing this.”
And satisfaction is not the only allure. “The amount of mathematics you will learn in the process by trying different things is such powerful training that sooner or later, you will solve a problem where there's no solution on the next page,” he says.
OpenAI’s model, on the other hand, raced straight to the answer with no understanding of how it arrived there. This is causing profound soul searching among the humans who delight in working their way to the end.
“Now the question is, Does it mean a lot if you saw something, but no one yet understands it?” Zeytuncu says. “It reminds me of this very philosophical question, If a tree falls in a forest where there's no one, will anyone hear it?”
Another reason the mathematics community was shocked by OpenAI’s announcement was because two of their own — Tristan Buckmaster and Levent Alpöge — were poised to solve the problem themselves, building on the work of scores of others. The community is particularly close-knit and author credit is given generously (and alphabetically, rather than by degree of contribution), so seeing no acknowledgement of their peers was deeply troubling to Dabkowski and Zeytuncu.
“We've thrown hundreds and hundreds and hundreds of thousands of hours at this problem. It’s a monumental human effort of generations of mathematicians at this point who've had cracks at this problem,” Dabkowski says, holding up a dog-eared blue paperback titled “Navier-Stokes Equations.” “So that can be a little bit disheartening to see the fact that it's very thin on credit. It's saying a company did this. It's really a community that built this apparatus.”
Plus, mathematicians are now missing out on the jubilation of seeing people they truly care about achieve this milestone. “If Tristan Buckmaster would have announced that he and some other mathematicians together solved the Navier-Stokes equation, I would be super excited,” Zeytuncu says. “I would have said, ‘Hey, good job, Tristan. Good job, Levent.’ I would message Levent and congratulate him and celebrate him because I would feel like it is part of our mathematics family.”
A month after the Navier-Stokes announcement, OpenAI threw another curveball when it released solutions to an additional 700 long-standing problems. None were as thorny as Navier-Stokes, but Zeytuncu says the pace at which the company is solving problems, without a careful vetting process, creates a slippery slope that will soon have implications far beyond mathematics.
“What happens if six months from now OpenAI says they’ve developed a drug and it cures cancer?” he asks. “How are we going to verify that? Who's going to test it? And how are we going to manage the financial and medical fallout from that?”
Still, Dabkowski remains hopeful that, ultimately, the situation will settle and some positive outcomes will emerge. “Growth occurs in crisis, right?” he says. “Mathematicians thrive by talking to each other, by thinking about a problem, by working diligently together. It's our sense of community which is our real strength.”
Zeytuncu, too, sees room for optimism. “The gain is that now the biggest conversation in the mathematics community is, ’Oh, wait a minute. It is really important to understand your solution and, furthermore, to explain it to other mathematicians and, at some point, to explain those new ideas to graduate students, to undergraduate students.' So digestion is really important,” he says.
This is not the first time technology has upended the math community, Dabkowski points out. “Now that computers are better than humans at chess, we didn't throw our chessboards away,” he says. “We play chess because it's fun to play. It’s a challenge. So as technology evolves and gets better than us at certain things, it's still useful for us to do that thing in the first place.”
Still, Dabkowski and Zeytuncu agree that this is a time of turbulence for their field. And to weather it, they will need to do what anyone must: keep on going. “We're here to teach, we're here to develop and create, we're artists in that form,” Dabkowski says. “So, sure, this is a little bit unsettling, but at the same time, it's not going away. As mathematicians, in order to preserve our beautiful legacy of academic creation, we will find a way. I'm confident that we'll find a way forward.”
Story by Kristin Palm
