There are seven questions in pure mathematics deemed important enough by the Clay Mathematics Institute, a nonprofit foundation devoted to “furthering the beauty, power and universality of mathematical thought,” that each of their solutions carries a prize of $1 million. To date only one of these “Millennium Prize Problems,” as the foundation calls them, has been settled: the Poincaré conjecture. A problem in topology—the study of the properties of shapes that survive stretching and bending—it had been “open” for almost a century before it was proved in the early 2000s by the Russian mathematician Grigory Perelman. But on September 8 OpenAI announced that it had settled a second Millennium Problem. While Perelman had turned down the prize money (as well as mathematics’ most prestigious award, the Fields Medal), rejecting both the notion of celebrity for mathematicians and the conceit that credit should be given to him alone, OpenAI issued a press release. The company’s claim, still awaiting the community’s judgment, concerns the solutions of the Navier-Stokes equations.

Equations are compact symbolic expressions that say that two ways of doing something mathematically are equivalent. They are most interesting when there is something unknown involved, in which case a “solution” is the thing you “plug in” to give you the same thing “on both sides.” In this way, an equation gives you an indirect description of a mathematical object. A familiar one is x² + 4 = 5x, which says, “I am thinking of a number that, when I multiply it by itself and add four, is the same as if I just multiplied it by 5.” Only two numbers fit that description: 1 and 4.

A differential equation is an equation about change. In a simple example, it might be about some quantity changing with respect to time. Think of an infant on a playground swing. You launch her from a height and let gravity take over. She goes back and forth; at the highest points she stops for an instant and then travels back. As she moves back and forth her position relative to the center changes over time. She starts from “0,” then goes maybe a few feet behind, then a few feet ahead, and the corresponding numbers swing from positive to negative. Her acceleration also changes, and the farther out she swings the more strongly she is pulled back toward the center. The relationship between the position and acceleration can be encoded as a differential equation (including a mathematical caveat for air resistance). But here a solution is not a number. It is a record of the swing’s position at every moment, what mathematicians call a function. Drawn as a graph, it looks like a run of Formula-1 S-curves tapering toward a finish line.

When the unknown quantity depends on changes in more than one other factor—say both where you look and when (space and time)—then we have a partial differential equation. The traditional wave equation is one famous example with many applications. It even applies to a guitar string, when we consider the up-down displacement of any point on the string from rest. Here the wave equation encodes a relationship between that displacement and changes in the position on the string and the time since plucking it. At the beginning and end, where the string is pinned down, there is no movement. The closer you get to where you plucked it, the more it goes back and forth. Here the solutions include the famous sinusoidal waves that we associate with harmonics. Analogous equations can be written for a drumhead or the surface of a pond.

The Navier-Stokes equations, named for the French engineer Claude-Louis Navier and the Cambridge mathematician and physicist George Stokes, are also partial differential equations. They are foundational to a mathematical description of liquids and gases, and as such are essential to the study of fluid dynamics. A solution has two components: one function that gives the velocity of the fluid at every point in space and time (the “velocity field”) and another that gives the pressure at every point in space and time.

Like any good math problem, these have some conditions. The fluid is incompressible, meaning that whatever flows into a region must flow out. There is a viscosity parameter that gauges the “stickiness” of the fluid (molasses has high viscosity, WD-40 has low viscosity). Of course the fluid need not just sit there (it could, but that would be simple to describe), so we allow for an external force to generate the action. It can be strong and even violent in its effect, but it must be mathematically (and quintessentially) “smooth” throughout. Even the geometry of the container is idealized: we imagine that the fluid flows throughout three-dimensional space. No corners, no rough surfaces, free and unfettered, driven only by perfectly smooth Platonic paddlers.

The mathematician Leonhard Euler is generally credited as the first (in 1757) to formulate equations that model how fluid flows. The Euler equations are simply Navier-Stokes equations without viscosity. Euler later applied them to derive a model for blood flowing through arteries. In the early 1800s Navier tried to build on the Euler equations to create a more detailed description that accounted for viscosity. By the 1840s Stokes had cleaned up this attempt to give us the equations we use today.

Navier-Stokes is so fundamental—and provides such a good model of fluid dynamics—that we live its success every day. It underlies the mathematical modeling necessary for tasks ranging from weather prediction to airplane design. It is applied mathematics at its best: terse yet precise, as simple as possible, but not simpler, as Einstein would say. Its simplicity belies the difficulty of understanding the full range of what it encodes—that is, the full range of its possible solutions.

What makes the solutions especially tricky to figure out is a subtlety in their description: there is one “term,” like a phrase in a stanza or a chord in a measure, accounting for the fact that any flowing fluid at once moves with the current and makes the current. Think of a river. There are fast places and slow places, straight stretches and eddies. A leaf on the river—or leaves scattered about the river—materialize the velocity field. But we also have to remember that there is water moving through all this, effectively a continuum of infinitesimal packets of water both making up the current and riding the current. How fast and in what direction the water is moving therefore interacts with how the current varies from place to place. This gives the equations what mathematicians call a nonlinearity. It is also the source of turbulence, both mathematically and—for those trying to solve the equations—analytically.

For generations nobody could say whether the equations, starting from a perfectly smooth flow, could produce a point at which, in a finite time, the speed of the fluid becomes infinite. Such a “singularity” or “blow-up” would be a system spinning wildly out of control—and since real water cannot move infinitely fast, it would be the place where the equations stop describing the world. Resolving this lacuna is the Millenium Problem. It doesn’t keep us from applying Navier-Stokes in real-world settings, but it has been a nagging gap in the landscape of our mathematical knowledge. It now seems that we know the answer: despite the relatively innocuous-sounding assumptions, blow-ups can occur. The coincidence of simplicity of description and complexity of behavior is amazing, and surprising.

The story of the problem’s solution has by now been widely reported. In mid-August the NYU mathematician Tristan Buckmaster and his collaborator Levent Alpöge, an employee of Anthropic, found a blow-up for a version of the Euler equations. Buckmaster and Alpöge built on the work of two Spanish mathematicians, Diego Córdoba and Luis Martínez-Zoroa, who (building in turn on Martínez-Zoroa’s dissertation work) had by 2023 found a strategy to create a singularity out of an “infinite cascade” of nonsingular solutions, each one amplifying the effects of the previous ones. Buckmaster and Alpöge also used a range of large language models in their work. Córdoba and Martínez-Zoroa used none.

OpenAI has a group of researchers studying the use of the company’s models for math research. While Alpöge and Buckmaster were working to finish a write-up of their solution, OpenAI heard rumors that significant progress had been made toward solving two of the open Millennium Problems. Not knowing which two, but now aware that the solutions were close, the company presumably saw an opportunity for their own research and business efforts. They put their latest, unreleased model to work on all six problems, along with some easier but still important warm-ups. Among those warm-ups were the Euler equations. After around fifty hours of compute time by nearly one hundred AI agents, they too came up with a solution that produced a singularity, which gave them confidence to go all-in on attacking Navier-Stokes.

To this task, by their estimation, they dedicated “on the order of” ten thousand agents. The agents were given the new Euler proof and centuries’ worth of published mathematical work. They very well may also have benefited from the residue of several years of mathematicians’ interactions with OpenAI’s wide range of tools. This swarm of agents was set up in such a way that they could efficiently share their findings among themselves—and share they did. They “talked” to one another for eighty-eight hours, generating 2.7 million messages—130 billion tokens of output, the equivalent of about several hundred thousand books of scratch work.

OpenAI’s human mathematicians oversaw the process, periodically using a second model to consolidate the most promising threads and feed them back. OpenAI has not said how much computing the effort consumed, but outside estimates put it at several hundred thousand GPU-hours, which translate to roughly the amount of electricity it takes to power a town for the four or so days that the model ran. Eventually the problem fell. The machine’s work still needs to be verified by humans, but initial assessments suggest that the problem has been settled.

The controversy over these results began as OpenAI strategized about how to make them public. The night before the company’s announcement, Buckmaster made a post on Mastodon sharing his and Alpöge’s achievements, including the Euler result. He included links to preprint versions of the work as well as supporting materials, the latter of which included a “Lean-4” version of their work that enables a computer program to verify their proof.

Also attached was a document giving his version of the timeline and process, crediting the early work of Córdoba and Martínez-Zoroa (whom Buckmaster recommended for a Fields Medal) and claiming that he and Alpöge had also made significant progress toward resolving the Millennium Problem. He apologized for the somewhat unpolished nature of their write-ups (“AI slop” is how he described the write-up of the Euler result), which he explained he had hurried out the door because he felt “pressured by outside factors.” The next day OpenAI issued their press release.

By Buckmaster’s account, he had reached out to OpenAI several days earlier, after word got back to him that rumors about his and Alpöge’s work were circulating there. He then, he said, had a tense back and forth with the company regarding how they had achieved their result and how they were going to announce it. Buckmaster had concerns that OpenAI had used work of his, which he had shared with the company by using its tools over the preceding months, never thinking that OpenAI was or would be working on Navier-Stokes. He alleged that in a phone call two days earlier an OpenAI scientist had offered him authorship on the company’s paper provided that Alpöge—an Anthropic employee—be left off. OpenAI denies trying to withhold credit from Alpöge (although it has said that it did want to bring only Buckmaster in as a coauthor). OpenAI also now claims that nothing Buckmaster typed could have reached the model. The company’s September 8 announcement credits both men, recognizes their priority on Euler, and says that it will not claim the prize. Whatever happened on that call, the episode cut against the oldest norms of scholarly credit and collaboration and fed the cynicism many of us already feel about the AI industry’s public professions of altruism.

Priority controversies are hardly new in academia, or in mathematics, for that matter, but there are greater stakes here. It is meaningful that the AI giants have chosen mathematics as the arena for a cage match. Mathematics is a touchstone of intelligence, even of genius, and there is no dispute about when a problem is solved. A problem that withstood generations of officially very smart people is, in this sense, the cleanest possible advertisement for a technology whose product is thinking. The machine that solved Navier-Stokes was not built to solve Navier-Stokes. That is the whole point of telling us that it did. What we are witnessing is a spectacle of computational power staged to acquire market share, and with it control of a landscape that extends well beyond mathematics.

OpenAI showed that the Navier-Stokes equations can “experience a singularity.” But did we? The model solved one of the world’s most important math problems. But it’s just a math problem. These are, for sure, complicated puzzles, some generated by thinking about the world around us and many that come completely from a world inside us, vistas on some elaborate ziggurat of the intellect that we continue to build. Solving them brings joy and understanding to a relatively small number of people around the world.

Unfortunately, the most important problems that we face are not math problems. Some, like climate change and even AI’s “alignment problem,” have mathematical components, but the most important problems are people problems, and no matter how much we might try to make them into math problems, they aren’t. Mathematicians have been solving long-standing math problems for centuries. If this were a sign of superintelligence, I’d like to think that at least one of these problem solvers could have brought more lasting happiness to the human race.

Did OpenAI’s model do math? Kind of. Mathematicians have been using machines to help prove theorems for over fifty years. Whatever the machine did here, however, it did not do it in the way humans would. One way to say what it did was that it “simulated” the act of doing mathematics—albeit to great effect. Any given piece of that scratch work might look like a dashed-off expression on a blackboard or an email between collaborators. And yet the process as a whole—a group of people serving up and organizing ideas for a machine—is, well, very un-mathlike.

For some problem-solvers, settling a question is perhaps the only point of pure mathematics. The focus on the goal makes professional mathematics seem like big-game hunting and the profession like a competition. That is at least part of what Buckmaster might have meant when he called OpenAI’s solution a “Deep Blue/Kasparov moment.” But there is also something about the human journey in mathematical research. For most of us the process is itself at least as important, even a source of pleasure in its own right: looking at delicate symbolic hieroglyphics, puzzling alone and with others over a complicated expression or idea, the simple satisfaction of thinking hard about something, turning it over purposely and unconsciously, marinating in a problem. Living with a problem. We talk to ourselves and we talk to others, and when everything finally falls into place we take our time to give a careful, bullet-proof explanation. A mathematics paper is a human activity, written, not generated. Asked why he had not used the new tools, Córdoba spoke for the collaboration that made the whole thing possible: “I don’t use AI. I have Luis.”

I am reminded of Walter Benjamin’s well-known  essay “The Work of Art in the Age of Mechanical Reproduction,” first drafted in 1935. It is a deep reflection on what the technologies of photography and especially filmmaking had done to the practice—the art—of image-making. In a deft rhetorical display he argues that, once mechanical reproduction untethers a work of art from its singular existence, society reaches a dialectical fork. On the one hand, photography and film can democratize image-making for all kinds of positive effects: good things can happen when everyone is an author, or even more, feels as though they can be an author. These technologies also “bring out those aspects of the original that are unattainable to the naked eye.” But with reproducibility “quantity has been transmuted into quality,” setting the stage for a battle between concentration and distraction. The result is to promote a public that, according to Benjamin, “is an examiner, but an absent-minded one.”

There is a political danger here as well. Ease of access, for Benjamin, is a form of sleight of hand—it gives the illusion of voice without transferring power or changing the structure of property ownership. This is a Marxist argument, which Benjamin wrote in the shadow of Nazism and the fascist takeover of Italy. At our historical remove much of it reads as prescient.

We are now confronted with machines that are able to produce things that we’ve historically viewed as works of thought. But do they reproduce thinking? The mapping of Benjamin’s argument onto such a question is not one-to-one, but the resonances are hard to miss. His central observation is that the technique of reproduction “detaches the reproduced object from the domain of tradition.” In its most quoted form this is the famous loss of “aura,” perhaps the essay’s best-known concept. It has nostalgic overtones but also points to new opportunity. With freedom comes possibility, art as a vehicle for political action.

We again stand at a historical juncture that threatens the “aura” of a long-standing human practice. One reading is that we are at a gateway to a period of rapid self-improvement and accelerated research, and soon to be released from traditional and increasingly irrelevant educational structures. Even proofs can now be certified by machines. Mathematicians—including Buckmaster and Alpöge—regularly advance their own research with AI tools.

But something has also been lost, and that loss cannot be taken lightly. Connected to the Greek word for “breath,” “aura” links to the body itself. Thought is human. “Aura” also speaks to the amazement that we should feel in the presence of creative work. Its loss suggests a marginalization of human activity and a disregard or even disrespect for the creative act. A proof only loosely attached to the people who spend their lives thinking becomes detached from the history of thought, and therefore from the foundation of meaning that it created. When a new technology untethers thoughts from their thinkers, we go from thinking for ourselves to thinking that we are thinking for ourselves. And because the aura has not disappeared but migrated, from the thinker to the machine, we are more inclined to believe what we are told, and ultimately to start valuing its “thoughts” more than our own.

Our thoughts are our own because they come from our cognitive processes, our biology, our history. No two thoughts are ever exactly the same. Now that so many of us pass our thoughts and ideas through a handful of common AI midwives, the many small singularities that were each of us collapse into a handful of larger ones. The wisdom of crowds becomes the wisdom of a few. Receiving the machine’s words in a room alone or wrapped up in headphones, we may experience them as our own ideas, but they aren’t. And the machines producing all those seemingly individual voices are owned by a very small number of powerful companies.

In the essay’s epilogue, Benjamin makes a direct connection to fascism. He reminds us of the Futurists and their celebration of the speed, power, and beauty of the machine. He quotes Marinetti, by then closely identified with Italian Fascism, extolling even modern warfare as an aesthetic triumph of technology. When speed and power become objects of admiration in themselves, the human activity they were supposed to serve becomes the slow thing, the obstacle. For Benjamin, writing in the interwar years, amid both memories of the technology-enabled carnage of World War I and the specter of fascism, the endpoint of that logic was war. Its’s not difficult to bring at least the first part of that analogy to OpenAI’s achievement: eighty-eight hours, ten thousand agents, 130 billion tokens, and one historic theorem. The announcement’s implicit message—intentional or not—is that humans are the past and LLMs are the future.

For Benjamin, this was a battle of aesthetics that ultimately played out on a real battlefield. Perhaps ours is also a battle of aesthetics that is now playing out in the environment, economy, and university. Whether we go on thinking for ourselves and with others—carefully and, yes, even slowly— when faster options are a click away is a choice. Sometimes the faster way is the better one. But sometimes you need Luis.