AI is both the best and worst thing to ever happen to mathematics

The AI mathematics revolution has arrived

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Our best equations for describing the way fluids move were first written down two hundred years ago and ever since nobody has been able to answer a simple question about them – do they sometimes blow up? This explosive phrase has a precise mathematical meaning describing 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 – named after the pair who discovered them – 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 is not that fluids will start misbehaving but instead that the equations are not perfect. This is good news for your bathtub. Mathematics, however, may never be the same again.

The reason for this is not the result itself. The Navier-Stokes equations have already proven themselves many times over to be incredibly useful 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’s not to underplay the discovery, it is one of the biggest mathematical breakthroughs in decades. But the way it has happened, and the resulting fall out, is what will have the most dramatic effect on mathematics for years to come.

OpenAI began working on the problem after they got wind that a pair of mathematicians had made some progress, which hadn’t yet been published. Researchers at the tech company decided to set thousands of their AI agents on solving the problem. Just 88 hours later, they were done.

Over the last few months, AI has made breakthrough after breakthrough in mathematics with this most recent development being by far the most impressive. It wasn’t 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 longstanding mathematical conundrums will fall to AI is uncertain, but it will surely be many. OpenAI has already said it has made “substantial progress” on another of the Millenium Prize problems.

But AI proofs are not a straight replacement for human ones. Mathematicians want to know the why, AI’s not so much.

The great mathematician Paul Erdős 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 new understanding or a completely new way of thinking. Erdős would declare that a proof like this was “straight from The Book”. It’s fair to say, OpenAI’s effort wouldn’t make the cut.

This is because AIs tend to produce hard to follow and convoluted arguments. For the sake of only proving a theorem this doesn’t 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’s lacking.

So where does this leave mathematics? Clearly, the AI mathematical revolution has arrived. It’s a new era where we’ll have many more answers than explanations and unpicking it all will not be easy. It’s not just the Navier-Stokes equations AI has blown up, but mathematics itself.

These are unprecedented times and yet mathematics has gone through upheaval before. When René Descartes made the link between geometry and algebra in the 17^th 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 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 “scab of symbols”.

But 300 years later, it is indisputable that making this link between algebra and geometry was a good thing. Without it there’s no calculus or relativity, 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 on how to use it, who gets access and the role of the AI companies, but it’s a tool that has the potential to allow mathematicians to make much progress faster than before. AI may just end up being the best and worst thing to have ever happened to mathematics.