Solving Math’s Greatest Problems Was an Art Form. Then Came AI

From Suno composing uncanny elevator music to Tilly Norwood delivering customer-service-inflected one-liners, artificial intelligence has rattled creative industries. You can add math to the list after OpenAI said that, using thousands of agents, it had solved a decades-old puzzle known as the Navier-Stokes existence and smoothness problem. “The artists and the musicians have already gone through this,” says Juspreet Singh Sandhu, a mathematician at Colorado State University. Despite what timed high school exams might have you believe, mathematicians aren’t focused on getting the right answer as fast as possible. In fact, contrary to math’s image as a practical subject, the development of new mathematical ideas often resembles artistic exploration, akin to inventing a game or puzzle. Mathematicians typically follow a thoughtful, deliberate process to develop their ideas. In contrast, OpenAI approached the proof with brute force, which shortcut the process in a way that threatens to undermine human understanding. “A mathematician, like a painter or poet, is a maker of patterns,” wrote the English mathematician G. H. Hardy in his 1940 essay, A Mathematician’s Apology. As Hardy describes it, a painter uses shapes and colors, and a poet uses words, while a mathematician constructs patterns out of ideas. A pacifist, Hardy wrote the essay during World War II to argue that people should pursue mathematics for its own sake, separate from applications, particularly wartime ones. His essay promotes “useless” mathematics. He cites famed mathematician Carl Friedrich Gauss’ oft-quoted statement about number theory, the subfield of math which involves the study of integers, as the epitome of beautiful, useless math. While Hardy’s comparison of math to art holds up, he would turn out to be wrong on some of the specifics: After being useless for centuries, number theory would prove valuable for encryption protocols widely used today to secure your emails and bank accounts. At this point, though, useless is exactly what OpenAI’s proof is. The puzzle it solves is simply one that’s been interesting to mathematicians for decades. Its name comes from the Navier-Stokes equations, which 19th-century scientists developed to describe the flow of viscous fluids. Engineers use the equations to model airflow for airplane design. But mathematicians became enamored with the equations themselves, not their applications. “Mathematicians’ main interest in the equations was certainly not engineering,” says Jared Speck, a mathematician at Vanderbilt University. They pursued answers to the problem, he says, because of the “mathematical richness, the puzzle aspect of it.” The puzzle’s solution will not help anybody design a more aerodynamic airplane wing. Put another way: Many cakes are cylindrical, but studying the equations that describe a cylinder won’t necessarily help you bake a better cake. “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. The equations describe fluid flow, approximately, in the world we live in. But mathematicians imagined how the equations would apply in a fringe, almost sci-fi context, just because it intrigued them. They formulated a puzzle: They wanted to know whether the equations implied that in unrealistic conditions, a fluid could explode for no physical reason. Mathematicians expect approximate equations like Navier-Stokes to imply such nonsensical situations, which they find particularly interesting because sometimes they can lead to brand-new mathematical ideas. For years the community had been developing “a deep and beautiful theory” around the equations, says Speck. They were on the verge of cracking the problem before OpenAI’s proof found that yes, the Navier-Stokes equations did imply a sci-fi fluid explosion. While people do perform results-oriented math, such as to design new cryptography protocols or optimization algorithms for robotics, the mathematics community is by and large “interested in more than just a yes or no answer to a problem,” says Speck. There’s also the problem of understanding the final answer itself. When humans write proofs, they organize their arguments based on previous research, which they cite. They spend months, if not years, writing and communicating the steps of their proofs to try to convince colleagues of their insights. Consider it akin to workshopping a poem, but with logic instead of emotion. That didn’t happen with OpenAI’s Navier-Stokes 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. Mathematicians have criticized the company'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. “We don't even know how autonomous it was, or how much human expertise is necessary to scaffold the process,” says Sandhu. Because of these and other factors, “basically nobody in my community really understands what's going on,” says Speck. “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.” Sandhu has signed an online declaration titled “A Severe Misalignment of AI in Mathematics,” which says mass-producing proofs “could destroy fertile ground instead of breathing life into new ideas.” The declaration’s initial signatories were 25 winners of the Fields Medal, often called the Nobel Prize in mathematics. The declaration is not against the use of AI, says Sandhu, who, like many mathematicians, already uses AI in his work. To Sandhu, the declaration states that “mathematical culture values understanding,” and that the speed through which LLMs tear through proofs could mean that “we will have solved without understanding.” In response to the declaration, OpenAI this week formed an advisory group of mathematicians to guide the company’s use of AI. “We believe those criticisms [in the declaration] highlight the need for thoughtful engagement between AI companies and the math community,” an OpenAI spokesperson writes in an email, adding that the company wants “mathematicians to have a meaningful voice” in how AI is used. However they’re used, the speed of LLMs may hinder human creativity. The models, while still error-prone, can churn through long calculations that human mathematicians find unpleasant. But mathematicians may miss valuable insights by outsourcing all these calculations, says Lorenzo Gavassino, a theoretical physicist at the University of Cambridge in the UK. “When a calculation turns out to be harder than expected, to the point of it being discouraging, that is when the greatest progress is possible,” he says. The struggle inspires mathematicians to invent new concepts. Gavassino gives the invention of the imaginary number, i, which is the square root of –1, as an example. Mathematicians invented i in the 16th century after generations of trying to solve cubic equations, expressions that contain an x3 term and no higher exponents. This new concept gave birth to an entirely new subfield of math, known as complex analysis. Centuries later, physicists would use imaginary numbers in the equations for describing quantum mechanics, and they were able to use mathematicians’ theorems about i to build their understanding of electrons, atoms, and, now, quantum computers. Like many mathematical concepts, the imaginary number was initially mostly useless. Often, a mathematical concept’s practical success unfolds over generations. First, mathematicians noodle extensively over the logic behind the concept. After the field develops a solid understanding of the ideas, people find applications in contexts their inventors never intended. Mathematicians have a saying, Gavassino tells me: “Good mathematicians prove theorems; great mathematicians propose conjectures; the greatest of all provide definitions.” He explains what it means using an analogy that compares the mathematical process to a game. Good mathematicians win a game; great mathematicians propose ways to win a game; the greatest of all invent the game in the first place. In the case of Navier-Stokes and other areas where AI systems have come up with proofs, the LLM demonstrated how to win a game, but Gavassino thinks the technology is still far from being able to invent new ones. Gavassino isn’t afraid that LLMs will do math better than he can. He thinks of LLMs as just another tool—one with the reliability, currently, of a “drunk guy”—and that it’s possible to use them responsibly and creatively. But he distrusts how leaders may exploit inflated ideas of their performance for political purposes. “I am afraid, instead, that certain people with not great intentions may start to say, ‘You know what? There is no need to finance research. We can let AI do the discovery in place of the researchers,’” says Gavassino. Notably, OpenAI also spent millions—equivalent to the annual pay of dozens of academic mathematicians—on computational power to generate the proof. A similar sum could fund dozens of graduate students, all with the potential to unlock new creative solutions and puzzles. The mathematical community has long relied on a teacher-apprentice structure to discover new ideas sustainably over generations. “When I was starting out, I was given problems that people senior to me probably could have solved more easily themselves, or at least done more quickly,” says Speck. “But they were investing in me. They were giving me an opportunity to develop.” Now, AI seems to be capable of picking math’s lowest-hanging fruit, and thereby threatening to scoop younger researchers and deprive students of training opportunities. That in turn could turn off the spigot of human creativity that’s built the entire field—and led to far-flung advances some mathematicians may never have even considered. “Trying to answer big questions in general forces us to develop new ideas, techniques, strategies, and languages that then become useful elsewhere,” says Gavassino. That creative process gave rise to LLMs themselves. The study of differential equations and complex analysis has enabled the invention of the lasers for etching the computer chips where the models reside. The software itself consists of neural networks, which crunch numbers using the tools of linear algebra, a mathematical field at least as old as the 17th century. AI companies have fattened their models with generations of human insight. Now that the models are fed, and trillion-dollar valuations must be sought, the companies may very well trample the community that created their product in the first place.