Millennium Pastimes

Has math been solved by AI?

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Related: The Month AI Conquered Math, Human → Superhuman → Ultrahuman

The Alchemist (detail) by Joseph Wright of Derby, 1771

In 2000, mathematicians chose seven hard problems and created a challenge they thought would keep the field occupied for the next thousand years. Little did they know, it would end up killing mathematics by 2026.

In 2002, two years after the announcement, Grigori Perelman solved the Poincaré conjecture. Twenty-four years later, with the remaining six peoblems looking like they’d outlive the millennium, an unnamed OpenAI model solved the forced version of Navier-Stokes. One day later, OpenAI told the New York Times that they had made “substantial progress on another Millennium Prize problem.” One day later (again), rumors spread that Anthropic had set its sights on yet another. Welcome to the future, where 1,000 years are compressed into a week.

In this essay I explore what this means for mathematics and for the world. I’ll be quick, lest another problem is solved while you read.

The first place to look for hints about what will happen to mathematics is Terence Tao, possibly the best mathematician alive, deeply knowledgeable about AI, a regular user of frontier models, and neither reflexively pro- nor anti-AI, occupying instead a nuanced and unusually levelheaded middle ground (which can’t be said about his newfound detractors; more on this later). Pre-AI mathematics owes an enormous amount to Tao’s work; post-AI mathematics may end up owing just as much to his conscientiousness and willingness to keep up with what these systems can do.

So what does Tao say? In short: that struggling with a problem changes the mathematician, and when the struggle disappears, much of the value that a proof provides disappears with it.

As receivers of the Promethean gift that is math solutions à la carte, humanity has a duty to digest them. However, that’s only a fraction of the value that can theoretically be extracted. If you were to ask experts whether it’s better to learn 10 proofs after the fact or fully develop only 1, acquiring all the associated insights, they wouldn’t hesitate to choose the latter. But how could you deprive the world of nine proofs! Easy: because I understand that although it’s great that we can all be followers, it’s a self-condemnation if we all are. Someone has to blaze the trail. Likewise, it’s useful to learn Newton’s law of gravitation in textbooks—how convenient and helpful is not having to reinvent the wheel every time—but that prevents students from rediscovering it from first principles. It sounds preposterous to imply that the average high-schooler has any chance, but that’s the crux: it’s a genuine trade-off for professional mathematicians. They would grind for years—decades even—at problems they might solve one day, getting in return plenty of insight and often proof and status. But those problems AI can now solve in days.

Something beautiful is gained here, but something beautiful is lost with it.

So now you’re worshipping resistance? You implicitly associate achievement with hardship. Not quite: the idea is not to love struggle for the sake of it—the entire story of humanity has been one of dealing with struggle, no need for more—but rather to accept a painful growth insofar as it enables a better existence. To the extent that difficulty has value, it is because hard things force us to realize more of our potential, while also showing us where that potential lies. In this sense, difficulty is both vehicle and map: we become through labor.

Hegel says something to that effect: consciousness develops through labor, through encountering resistance and transforming both the object and oneself. Unfortunately, AI can give us the product of that process without humanity undergoing the process. (That’s exactly what’s happening in high schools around the world: students are getting greater performance but weaker competence; they sacrifice the learning to nail the homework.) We acquire the theorem while forfeiting whatever understanding would have accumulated during thirty or three hundred failed approaches (not to mention the thicker skin and tempered character that goes with it). As humanity’s stock of knowledge rises at light speed, we paradoxically become less knowledgeable.

The counterpart of not growing with one’s problems is to let one’s problems grow too much. This is referred to as proof abundance.

Imagine a machine producing 100,000 novel theorems a day. Truth becomes a commodity but truth without interpretation is useless; the receiver needs to make sense of the message for the message to be valuable. Oh, cool, you can generate marvels at the push of a button, but can you tell me which marvels matter? Which reveal new paths, which close old ones? Which are world-shattering and which are merely fun to know? Which are correct and which are—despite the all-knowing machine—ultimately wrong? Mathematics won’t so much resemble a cathedral anymore, in Thomas Bloom’s words, but Borges’s Library of Babel: it will become clear that having every answer available is closer to knowing nothing than to knowing everything. We will have accumulated so much hayslop that we’ll be deeply ignorant about where the mathematical needles hide.

And you will say: we will have the agents go through the hayslop maze and find the needles! This is the answer to every single objection, apparently. AI will digest the proofs, interpret them, rank them, connect them, explain them. Fine. Keep climbing the hierarchy; keep adding epistemic debt. But unless you intend to automate literally everything—*humanity has left the loop*—eventually the chain has to terminate in us. The world that wants to make use of all this knowledge is still a human world. Somewhere between the machine producing a theorem or something and a person deciding what to build, teach, fund, believe, or do because of it, somebody has to act as a bridge.

Whether the bridge is near the source or the sink makes the task easier or harder, but it ultimately doesn’t change the inescapable reality: the bridge can’t disappear.

Unfortunately, knowing that the needles are somewhere is itself a tremendous deterrent for needle-makers: would-be mathematicians will think twice before joining a dying discipline. I don’t blame them; I’d do the same. History rhymes: it happened when William Thurston became so good at foliations that graduate students were being advised not to go into the field; Thurston was “cleaning it out.” He “killed foliations” out of success and so no one saw a reason to follow his blazed trail. Until he left. But AI doesn’t leave. AI doesn’t tire or retire. AI is like Thurston but instead of foliations it’s everything. Will OpenAI devote 10,000 agents to scoop you on your favorite problem? Will they upend your dearest field? Sure they will.

This means, in practice, that we’re living through the disappearance of career-sized problems in both the sense that all the challenges that could have made someone’s career will be soon gone and in the sense that no one will risk trying to have career in a field that’s being bombarded from 30,000 feet high (that is, from the clouds). Bernard Williams wrote about “ground projects,” projects sufficiently central to one’s existence that they provide by themselves reasons to keep living a particular life. AI does not merely threaten to solve the problems we hoped to solve, which is a rather prosaic form of grief; it threatens to remove the problems around which they might have built a life. Don’t think for a moment this is exclusive to mathematics, because it’s all too easy to dehumanize what one doesn’t understand: you, wherever you are, whatever you do, are next. It just so happens that math is the first and also the most legible discipline where 1) AI can be cleanly rewarded during post-training to be really good and 2) working on a problem can provide enough purpose for a person to last for 40 years. But everything in life is like that: AI will eventually be aimed, like a laser, to cut through every little patch of purpose in what’s otherwise a sea of void.

That is quite the tragedy. And because I see it as such, I want to interleave a rant here.

I’ve seen more people than I can count respond to mathematicians’ grief with utter contempt (I’m not gonna name names). They will readily topple down their former idols and heroes—people whom I’ve seen them hold in high regard—with a disrespect I can only ascribe to envy: now you are a pleb, like me. All hail AI, the great equalizer. Here is what I think of them: they disgust me. They are often the SF-brain-wormed Twitter techno-bro type who finds it easier to summon empathy for a swarm of agents, the virtual connectome of a fruit fly playing video games, or the prawns dwelling on the ocean floor than for a group of human beings mourning the disappearance of the thing around which they built their lives. Why? Because nothing enrages this type more than someone standing in the path of “progress” (because, naturally, AI slop is progress, as is kids getting dumber). It barely matters whether the objections are measured, reasonable, or plainly correct; it barely matters whether the people grieving have contributed orders of magnitude more to civilization than those sneering at them. The moment they decide you are standing in the way of progress, they arrogate the permission to despise you—they convince themselves that despising you is their moral duty. I find these people disgraceful. There is something grotesque about extending your circle of concern to every conceivable substrate while somehow excluding the humans standing in front of you. I guess they can tolerate anything except their outgroup.

The more I read these takes, the more obvious their psychology becomes. People who won’t belong in history books are delighted to watch those whose faces will fill the walls of universities and the museum exhibitions brought down to their level. They call it “democratization” but it smells of resentment. When math people asked for a paced collaboration with the AI industry, they diagnosed a wounded ego, apparently incapable of recognizing the projection looking back at them from the dirty puddles.

That leads me to the opposite point: is one mathematician’s career more important than the benefit that ten proofs rather than one might bring humanity? Surely not. I’d gladly leave half the entire medical community without a job in exchange for the panacea. But 1) we’re not talking about a universal cure, just a set of pure math proofs relegated to one-upping the enemy AI company in a marketing war ahead of their IPO. Not quite the same sacrifice. And 2) we’re not talking about people’s careers either, which are contingent on having to “earn a living.” (Sure—let’s all be aristocrats. I’m in favor. Wasn’t that Asimov’s prediction? Maybe he missed the part where some get to be more aristocrat than others.) But no, we’re talking about the future of a field, or rather, if the oracles are right, of every field.

To be clear: I’d accept cutting down the number of doctors in the world in half in exchange for the panacea, but I wouldn't accept the end of medicine for it. It’s never a good bargain to offer a win now—as big as it might be—in exchange for all future wins.

That is Tao’s actual main point: the effects of having too many easy solutions pale beside the consequences of not having enough hard problems. Proof abundance is logistically messy and epistemically shallow but problem scarcity is an existential risk for mathematics itself. “AI could kill math through success” is not what you’d expect to hear after solving Navier-Stokes, but that’s the truth. Tao says that, to avoid Thurston’s fate, he chooses what problems to work on. To illustrate what can be a counterintuitive point, he uses an analogy: a city can suffer a shortage of drinking water despite having an ocean nearby. The history of agriculture gives us an even better one: fallow.

Fallow is one of agriculture’s oldest tricks. Prehistoric farmers already knew that land planted continuously kills fertility. To not turn a field into a wasteland you had to let the soil recover. An empty field would certainly look unproductive to the ignorant eye, but its emptiness was a feature, not a bug, of the system. But imagine a “gentleman farmer”; too wealthy to farm out of necessity, he sees the grasslands as this bucolic place people go to have fun. He arrives at the Iowan countryside with his large fleet of John Deere tractors and, clueless about the ancient practices, he sends his fleet to plow through his entire farm. The harvest is excellent; notably better than his neighbor’s. The next year he does the same. And the next. For a while he’s certain of his superior vision: those peasants over there are not maximizing the expected value. Then the soil thins and the yields fall and he learns the hard way why generations of farmers tolerated the ugly sight of unharvested fields.

Mathematics is like that guy’s farm: a bunch of uncaring gentlemen have arrived at the fields—number theory, topology, algebraic geometry, combinatorics, you name it—and have sent their fleets of machines to harvest them all. But they have not bothered to learn that some fields need to remain unharvested for the soil to recover. Twenty-five Fields medallists (the greatest recognition in math), Tao among them, just published a declaration—A Severe Misalignment of AI in Mathematics—saying exactly that: “the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.”

There’s another problem for which agriculture serves as the perfect parallel: industrialization. The poor farmers can’t compete with the tractor fleet and so they flee. Until recently, the only thing you needed to do mathematics at the highest level was pencil, paper, and an outlier brain. But what happens when having access to 100 megawatts of compute, the entire internet as pre-training data, and the narrow superintelligence of an agent swarm is a prerequisite to be competitive at the frontier? It happens that the math department at MIT or Harvard will have to become quant trading shops. Perelman was the first human to ever solve a Millennium problem. He’s also the last.

The means of intellectual production are means of production, period; they have managed to translate skill into capital.

Once again.

Chess found a way out of this trouble: if we can’t win, let’s separate competition between humans from competition against computers. To the extent that grandmasters were still relatively valuable as players, they could join mixed teams (until it was no longer the case; nowadays, a team of AI and human can at best tie an AI because the human is a net hindrance). Speedrunners do the same. You have the human gamer and then the TAS (tool-assisted speedruns). Does anyone care about TAS? Not really, except perhaps to watch what a literal “perfect play” looks like. But perfection is not the metric when you’re measuring the limits of what humans do or watching a sports competition.

So, can math define the same boundary? I don’t know: whereas a game is interesting only insofar as the process is interesting, discovery is interesting in itself. But maybe it’s not so far-fetched to try and create from scratch a category that sees math as a sport. Let machines dominate discovery. Let the field split into mathematics-as-research and mathematics-as-human-practice, just as people still run 100 metres despite cars existing. “No AI” math could, in principle, be a meaningful category; after all, learning math is also interesting in itself. That would be a hilarious reversal: the pinnacle of reason preserved as an amateur sport for humans whereas millennium problems become the idle pastimes of the machines.