Ten Thousand Explorers

Yesterday, OpenAI announced that an internal AI system had produced a solution to the Navier–Stokes problem, one of the seven Millennium Prize Problems in mathematics.¹ The equations describe how fluids like water and air move. The open question was simple to state and brutally hard to answer: can a perfectly smooth flow, on its own, develop a point where the fluid moves infinitely fast – a kind of tear in the smoothness – within a finite stretch of time?²

»There’s a real difference between possessing a result and possessing a map of the search space around it«

It’s complicated.

And it doesn’t stop there. The announcement was immediately followed by a dispute over what scientists call »priority« – who gets credit for finding something first. NYU mathematician Tristan Buckmaster and Anthropic’s Levent Alpöge had spent almost a year on closely related fluid equations they believed could be extended into a full solution. Buckmaster says word of their progress reached OpenAI, which began its own effort soon after. OpenAI says it never accessed their specific work and that its result was independent, though it admits the effort began after hearing a rumour about theirs.³

For centuries, exploration had a physical character: there was the world, some of it known, most of it not, and there was a finite number of people brave and curious enough to explore it. Maps marked the edge of knowledge with blank space or monsters: »Here be dragons«. Science inherited the same structure for exploring its territories: having an idea where something might be was never the same as actually finding out.

That’s what makes the current episode strange. OpenAI’s system reportedly deployed around ten thousand agents against the problem at once. None of them were individually smarter than human mathematicians, but collectively they were able to explore many branches simultaneously. A mathematician has an intuition about a promising route; a large enough system sends explorers in every direction around it, all at once. The unknown itself hasn’t changed, but the cost of exploration around a promising direction has dropped substantially.

Sociologist Robert K. Merton described science’s official ethos: shared knowledge, common standards, disinterestedness, organised scepticism.⁴ But science also runs on a less idealistic economy: being first. Discovery produces recognition, and recognition is the currency careers are built on. That made sense when exploring was expensive. If you spent ten years working through a hard problem and someone else reached the answer two days after you published, there was still a real difference between the two journeys – the fact that they could follow your route didn’t erase the fact that you’d found it first.

But what if merely knowing the direction is enough? Buckmaster didn’t hand OpenAI a proof – by his own account, what reached them was the rumour of a direction, an inkling of where a proof might be found. There’s a real difference between possessing a result and possessing a map of the search space around it. An explorer can return from an unknown coast and say: there’s land beyond that horizon. A second expedition, sailing toward it, was never told what the coast looked like, but it doesn’t have to search the whole horizon either. It’s exploring the same unknown, but with a head start.

AI changes the economics of the mission: a system with thousands of agents doesn’t make the unknown less unknown, but it makes a region of it drastically cheaper to explore. One researcher proposes a conjecture, a lemma, a strange analogy; thousands of agents test variations and return with a map of the results. That’s a new division of labour, and it makes »discovery« less like a single event. Humans point at a promising region of the map, machines explore it with a speed and at a scale that no individual comes near. So what is the discovery, exactly? The first intuition? The first promising route? The first proof? The first machine-generated proof? Or the moment someone realised that a result in one territory could be carried into another?

As the number of computational explorers grows, the difference between discovering a territory and merely traversing it will get harder to see. The old maps had dragons because nobody knew what was beyond the edge. Our new maps may chart every dragon there is – without ever telling us whether we found it ourselves, or just followed someone else’s map to it.

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