Our best equations for describing the way fluids move were first written down 200 years ago. Ever since, we have been unable to answer a simple question about them: do they sometimes blow up? This explosive phrase has a precise mathematical meaning describing a situation where, apparently out of nowhere, part of the fluid starts to move infinitely fast. It would be like an infinite whirlpool forming in your bathtub in response to a flick of your toes.
Understanding the Navier-Stokes equations ā named after the pair who discovered them ā has been a long-running quest for mathematicians. So much so that the blow-up question sits on the list of prestigious Millennium Prize Problems, each of which comes with a $1 million reward for solving.
This week, OpenAI stunned the world of mathematics with its announcement that its AI agents have finally found the answer: the equations do sometimes blow up. The implication isnāt that fluids will start misbehaving, but instead that the equations arenāt perfect. This is good news for your bathtub. Mathematics, however, may never be the same again.
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The reason for this isnāt the result itself. The Navier-Stokes equations have already proven themselves to be incredibly useful many times over, in applications ranging from making better airplane wings to modelling the flow of fluids through artificial hearts. Mathematicians had a suspicion that blow-ups were possible; now they know for sure.
That isnāt to underplay the discovery ā it is one of the biggest mathematical breakthroughs in decades. But the way it has happened, and the resulting fallout, is what will have the most dramatic effect on mathematics for years to come.
OpenAI began working on the problem after it got wind that a pair of mathematicians had made some progress on it, which hadnāt yet been published. Researchers at the tech company decided to set thousands of AI agents on solving the problem. Just 88 hours later, they were done.
Over the past few months, AI has made breakthrough after breakthrough in mathematics, with this most recent development being by far the most impressive. It wasnāt that long ago that mathematics was a bit of an embarrassment for AI because it was so bad at it; now it is shockingly good. How many other long-standing mathematical conundrums will fall to AI is uncertain, but it will surely be many. OpenAI has already said it has made āsubstantial progressā on another of the Millennium Prize Problems.
But AI proofs arenāt a straight replacement for human ones. Mathematicians want to know the why; AIs not so much.
The great mathematician Paul ErdÅs believed that mathematical truth exists independently of humanity in a sort of celestial book, and it is the job of mathematicians to discover what is in it. Sometimes, a mathematician would discover a logical argument that was so insightful and so elegant that it would bring a fresh understanding or a completely new way of thinking. ErdÅs would declare that a proof like this was āstraight from The Bookā. Itās fair to say that OpenAIās effort wouldnāt make the cut.
That is because AIs tend to produce hard-to-follow, convoluted arguments. For the sake of proving a theorem, this doesnāt really matter, thanks to the magic of a process called formalisation, where proofs can be turned into code that can be rigorously checked by computers. But for the sake of gaining insight, it is lacking.
So, where does this leave mathematics? Clearly, the AI mathematical revolution has arrived. It is a new era where we will have many more answers than explanations, and unpicking it all wonāt be easy. It isnāt just the Navier-Stokes equations that AI has blown up, but mathematics itself.
These are unprecedented times, and yet mathematics has gone through upheaval before. When RenƩ Descartes made the link between geometry and algebra in the 17th century, showing that shapes could be written as equations, he completely transformed the toolkit available to mathematicians.
Geometry, however, had been the darling of mathematics since the ancient Greeks. It was trusted and considered pure. Mathematicians worried that the job would make them drones who simply manipulated symbols and came up with no new insights. Philosopher Thomas Hobbes described the approach as a āscab of symbolsā.
But 300 years later, it is indisputable that making this link between algebra and geometry was a good thing. Without it, there would be no calculus or relativity ā in fact, modern science would be completely unrecognisable. Perhaps AI will have a similar effect.
Mathematicians now have an imperfect truth machine that spits out mathematical answers. There are big challenges ahead regarding how to use it, who gets access and what role AI companies should play, but it is a tool that has the potential to allow mathematicians to make progress much faster than before. AI may end up being the best and worst thing to have ever happened to mathematics.
