On October 6th, OpenAI released more than 700 mathematics papers written by an AI model it hasn’t made public. Each one tackles a problem that professional mathematicians had left unsolved, some of them for decades. Some run past a hundred pages.
Before I became a software engineer, I studied mathematics and physics and did research in mathematical physics, so I started reading. That night I wrote to a friend: Je sais même pas quoi en penser, ou par où commencer. I don’t even know what to think, or where to start.
The machine chose where to start
Some of the papers have been checked by a computer, line by line. The rest haven’t, and OpenAI says some could contain mistakes. I can’t check a hundred-page proof in a week, so I’m assuming most of them hold up.
What struck me first was how they’re built. A proof in a university course is direct: you start from the problem, and each line follows from the one before, along a route someone found long ago. An unsolved problem can’t be handled that way, or someone would have done it already. You have to step back and build new ideas first, ones the problem never mentions, and you can’t tell in advance which of them will matter. You don’t know what you don’t know. The hardest part is deciding where to start.
The papers do exactly that. Each one builds its own tools before it reaches the main argument. If a person had written them, you’d call that person a genius and an expert in the field, and nobody would doubt they had a deep understanding of the subject.
The usual objection is that a model can’t understand anything, because it only predicts the next word. That’s how it’s trained, but it doesn’t explain much here. To predict the next line of a correct proof that nobody has ever written, the model has to get that line right, and an idea chosen on page three has to pay off on page eighty. There was nothing like it to copy. If that still counts as prediction, then people are prediction machines too.
For a while now, I’ve argued that what sets people apart from every other system, alive or not, is that we can choose where to aim our attention. In the first piece I published on Substack, I compared the conscious mind to a prompt and the subconscious to the model behind it. The prompt holds no knowledge. It points, and all the capability sits on the other side.
These papers fit that picture closely. The machine didn’t decide to work on any of these problems. People at OpenAI chose about four thousand of them and pointed the model at each one for a few hours. The model supplied the thinking, and finding a way into a problem turned out to be part of it. On problems like these, that thinking now matches what human experts can do, and on some of them it goes further. Choosing what to aim at is the one ability still on our side.
A proof used to be my best evidence that someone understood a subject. If a machine can write one, what does understanding mean?
Proofs were always a stand-in for understanding
That question has already reached Hugo Duminil-Copin, who won the Fields Medal, the top prize in mathematics, in 2022. His area of expertise is percolation, the study of how things change state, the way water turns to steam. A couple of days ago I watched an interview with him, in French, about the book he wrote on what mathematics is for.
He described when AI reached his own field. In April, a colleague typed a problem his team had worked on for five or six years into an AI model. It came back with a one-page proof, short and elegant, built on a paper the team itself had written the year before. Over the summer, he says, another AI model solved the biggest open problem in his field. He described watching it fall overnight as a strange, mixed feeling.
His explanation for the machine’s edge is simple. A mathematician knows a few hundred papers well. A model has read tens of thousands, so it can spot a connection between two distant ideas that no single person would have seen.
What worries him is something else. One recent AI proof, the interviewer said, comes with a paper of about 150 pages that experts reportedly can’t follow past page 30. A computer can confirm the proof is correct while no human understands it. Duminil-Copin imagines AI settling one of the most famous problems in mathematics with a 150,000-page proof, “and then what do we do with that?” (my translation).
His answer is that solving problems was always a stand-in. Prizes go to whoever cracks the famous problem, but what mathematicians valued underneath was understanding a field and passing that understanding on. Now that machines can crack the problems, he says, the stand-in is worth very little.
He also talked about failure. For years he tried to solve the biggest problem in his field and couldn’t. Stuck, he turned to a related problem about magnets, and there he met one of his closest friends and collaborators. Much of the work he’s known for came from that detour. Each failed attempt was a guess about where to start, and he warns against pointing AI at every important problem at once, because the search is what keeps a field alive.
In mathematics, a machine finding the connection people missed is mostly good news. At my job, I depend on some of those connections never being found.
Online security is a bet that some problems stay unsolved
I’m a software engineer at Stripe, and almost everything Stripe does relies on a kind of cryptography called public-key cryptography. When a business connects to Stripe, it keeps the connection private and proves the business is really talking to Stripe. It also protects employee logins and bank connections, and it confirms that the software Stripe ships hasn’t been tampered with.
It works like a padlock that anyone can snap shut and only the keyholder can open. Underneath is a mathematics problem that’s easy in one direction and hard in the other. Multiplying two huge prime numbers takes a computer a fraction of a second. Working backward from the result to the two numbers would take longer than anyone could wait, as far as anyone knows. Nobody has proved it has to be hard. It’s an unsolved problem, and most of the security on the internet is a bet that it stays unsolved.
That bet can fail in two ways.
The first is quantum computers, a new kind of machine that could open these padlocks quickly. For years, a machine powerful enough looked decades away. In March, Google estimated that one about twenty times smaller than previously thought could break some of today’s most common padlocks in minutes. A week later, the cryptographer Filippo Valsorda moved his deadline for switching to quantum-safe padlocks from 2035 to 2029. “Remember that you are betting with your users’ lives,” he wrote.
The second is AI doing mathematics. Vitalik Buterin, who co-founded Ethereum, pointed out on X that today’s padlocks already had to grow much larger over the decades, because mathematicians kept finding shortcuts through the problems underneath. His worry is that more shortcuts exist that nobody has been clever enough to find, and that AI soon will be. “If AI will bring us 50 years of math in 2 years,” he wrote, those fifty years could include shortcuts through the new quantum-safe padlocks too. In his view they’d survive, but they’d need to get much bigger to stay safe.
AI can push on both threats. It already helps detect errors inside quantum computers, and the kind of model that solves open problems might also find faster quantum attacks, though that part is my speculation. On ordinary computers, the risk is the one Duminil-Copin described in his field: a machine that has read everything finding the connection people missed.
I don’t speak for Stripe here; this is how it looks from where I sit. Stripe was already working on its move to quantum-safe cryptography before this week. There’s a good reason not to wait: someone could record encrypted payment traffic today and unlock it later, once the right machine exists. Until now, shortcuts against these padlocks arrived at the pace of human mathematicians, a handful over several decades. This week we watched a machine work through hundreds of unsolved problems at a few hours each, and the work feels more urgent to me now.
I still don’t know where to start
When I told my friend I didn’t know where to start, I meant I didn’t know how to think about a machine that writes proofs. A week later, I have two answers, and they pull in different directions. As a former researcher, I’d want to do what Duminil-Copin says he might: look for a proof of my own, knowing a machine already found one, just to see where the search leads. At work, I have to assume the machines will find the shortcuts people missed, and want our systems ready before they do.
I’ve argued that choosing where to aim is what separates us from machines. This week a machine found its own way into hundreds of problems nobody had solved. I’m less sure of that than I was on last week.