The rise of artificial intelligence has pushed mathematics to a crisis point. In our latest video, we speak with Terence Tao, one of the world’s top mathematicians and author of Six Maths Essentials, to diagnose the state of the field right now and explore if it has a future even in the face of increasing automation.

Why should anyone learn to do maths with a pen and a piece of paper right now? What is the power of abstract thinking and can we afford to lose it? And what is on the research wish list of one of the most prolific and decorated mathematicians of our time? Join us as we take a grand tour of modern mathematics, following one of its best guides through the rapidly changing science. We’ll explore the fundamental character of mathematics research, as well as the principles and questions that guide the field – and how AI can be in discord with them.

Why prime numbers might not be random after all

The Riemann hypothesis may hold the answer to the distribution of the primes, the only problem is, no-one has yet been able to prove it

OpenAI, the maker of ChatGPT, claims its AI models have disproven the exponential time hypothesis – a fundamental assumption that underpins computer science.

“This came as a complete surprise, a shock,” says at the University of Bergen in Norway. “Nobody I know expected it.”

Proposed nearly 30 years ago, the exponential time hypothesis (ETH) tries to put a limit on how quickly computers can solve certain logical puzzles.

The simplest way would be to try every possible combination of true or false for each part of the puzzle. ETH proposes that, although it may be possible to do this quicker with shortcuts, the amount of work needed has to grow exponentially as the puzzle gets bigger.

OpenAI, however, says its models have devised an algorithm that can beat the previously assumed speed limit for solving logical puzzles, apparently proving ETH false. “Assuming that it is confirmed, this is an enormous breakthrough,” says at the University of Oxford.

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Many computer science findings assume ETH is true. If it isn’t, the conclusions they establish about computational limits become useless. “It would require a complete revision of several areas of algorithms and complexity,” says Fomin, who says a textbook he wrote on the subject would become obsolete if the finding holds.

ETH is linked to a long-standing maths problem known as P versus NP, which concerns whether a problem whose answer is easy to check is also easy to solve. If ETH were true, it would have implied that the answer was no. But disproving ETH wouldn’t mean the answer is yes. “This doesn’t have a direct impact on P versus NP, so we should not view it as progress towards P versus NP,” says Goldberg.

The result was announced alongside other breakthroughs in maths and physics on 26 October.

In mathematics, OpenAI says its AI has proved a 1991 conjecture that waves cannot remain intensely focused for long. If confirmed, this would also answer enduring questions about how thin a shape can be while also containing line segments that point in every direction, as well as how waves spread.

In theoretical physics, OpenAI claims to have proven that quantum field theories – including quantum chromodynamics, which describes the strong force holding quarks together – make sense mathematically, rather than just producing predictions that match experiments. The findings also touch on the existence of pure Yang–Mills theory, part of a million-dollar mathematical problem, but don’t resolve the entire challenge. That work hasn’t been released publicly while an external group led by Nobel laureate David Gross examines it.

Hannah Fry: How OpenAI has caused an existential crisis in mathematics

OpenAI has been throwing everything at solving long-standing maths problems. This should have been a glorious moment for mathematicians – but instead we’re having a massive collective wobble, says New Ů��С��Ƶ’s new columnist, Professor Hannah Fry

OpenAI says it verified the ETH result using Lean, a piece of software that checks mathematical proofs. at TU Wien in Austria has looked at what OpenAI has released and says he expects it to withstand scrutiny, although he says researchers still need to check that the formally proven statement means what its authors say it means.

To do so, OpenAI has commissioned an independent group, including Russell Impagliazzo and Ramamohan Paturi, both at the University of California, San Diego, who helped propose ETH, to examine the findings. In , the group says it could “change an important part of theoretical computer science”.

The speed at which AI is solving maths problems and the way AI firms are releasing their findings have concerned some academics. at Royal Holloway, University of London, describes the phenomenon as “mathematics done by press”, worrying that proprietary models can suddenly overturn months of researchers’ work.

at the Eindhoven University of Technology in the Netherlands describes OpenAI’s work on ETH as “ingenious”, “spectacular” and “awe-inspiring – but also humbling.” “It really makes you think about the future of mathematics, algorithms and academia,” he says.

at the University of Sydney says proving something and understanding why it is true aren’t the same thing. “The web collapsed, and we looked behind,” he says of the network of findings relying on ETH. “We have no idea why yet.”

OpenAI says it is experimenting with an approach in which independent academics organise efforts to verify and explain the results. But Fomin welcomes the disruption. “This result destroys a lot, but it also opens many new opportunities and new questions that nobody expected,” he says. “One shouldn’t complain about having too many interesting mathematical breakthroughs to try to digest,” says Wahlström.

A few years ago, when I first started talking about mathematics in public, a professor of algebraic number theory pulled me to one side and gave me some sharp words of warning: “The world fears mathematicians, Hannah. Let’s keep it that way.”

He was being, to put it mildly, a bit of a snob. Maths had an intimidating mystique to maintain, and I was apparently jeopardising it by titting around on the internet making videos and podcasts, rather than focusing on a perfectly respectable research career.

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I have thought a lot about that comment since. Partly because I have spent the intervening years doing everything I can to completely ignore him (and getting a New Ů��С��Ƶ column feels like a rather lovely reward for doing so).

But buried beneath his snobbery, I have come around to the idea that he was right about something. And I have never felt that more than in the past few weeks, since a swarm of AI agents have been busy solving our most famous long-standing problems.

I’m talking, of course, about OpenAI’s announcements over the past few weeks that it has found proofs for hundreds of mathematics problems that had previously resisted human attempts to solve them.

The furore began, most notably, with a proof (by way of finding a counterexample) that the Navier-Stokes equations can break apart. This particular gem was such a big deal to the mathematical community that at the turn of the millennium, the Clay Mathematics Institute promised they’d give $1 million to whoever managed to solve it.

Those Navier-Stokes equations are what I did my PhD on. They are basically Isaac Newton’s second law of motion: F=ma, but applied to fluids. They are, in my not entirely unbiased opinion, staggeringly beautiful, able to articulate everything from the drip of a water droplet on a leaf and the churn of an entire atmosphere to the gases swirling around in a galaxy in the sky.

The thing is, these equations have a tricky little bit that pops out of the acceleration part, which contains a feedback loop within it. The faster fluid flows, the faster the gradient can get between two adjacent streams, and so the faster fluid flows. It’s a bit like the fluid version of putting a microphone next to a speaker. The feedback loop can get louder and louder until everything ends up breaking. Or, in the case of the Navier-Stokes equations, the speed of the flow heads off to infinity.

To be clear: no one was ever worried about fluid dynamics tearing open a black hole when you stir your tea or anything. It was just that the weird behaviour of this little term left the possibility that the equations themselves would break.

On the flip side, there’s another term in the equation – the viscosity – that acts as a brake and calms everything down, stopping the flow from getting completely out of control. So, the question was: does the brake always kick in?

If you have been following the story, then by now you will know what happened. Various groups of mathematicians had been using AI to circle in on this problem, finding special cases where the feedback loop goes crazy, without quite getting to the full Navier-Stokes.

But supposedly, once OpenAI heard that others were getting close, they threw everything at it: 10,000 AI agents, sending each other 3 million messages over 88 hours, costing millions of dollars in tokens, to find a single example where everything blows up. The Navier-Stokes equations do indeed fall. And an AI proved it.

This should have been a glorious moment of scientific exposition, but instead, mathematicians have spent the last few weeks having a massive collective wobble.

Part of the reason is that nobody yet understands the answer. The OpenAI solution is almost hilariously ugly. It is a proof that’s 166 pages long, accompanied by hundreds of thousands of lines of formal code – almost impenetrable symbols – slapped down on the table and left for the mathematicians to pick apart, while the AI labs move on to the next thing. They came for the glory without giving much back.

So, we are left with a proof that almost no one wants to read, a $1 million prize that no one will want to , and no one who can stand at a blackboard and explain why this particular vortex makes everything break, when nothing else we tried ever did.

And now, to top it off, the new tranche of proofs that landed last week claim to have polished off hundreds of other important, unsolved mathematical problems, including a big result in the biggest trophy in all of mathematics: the Riemann Hypothesis. All tipped onto the code-sharing website GitHub for someone else to check.

So, mathematicians are having their own existential crisis about their role and purpose, like Lee Sedol did after AlphaGo, Gary Kasparov before him with Deep Blue and probably every literature graduate, musician and artist since ChatGPT was launched.

Please don’t get me wrong. I want to use this column, at least in part, to talk about the incredible, world-changing things that are happening with AI in science. I know that there is some inevitability to machines outpacing the humans, and I know that at its best, AI has a great deal to offer.

But at its worst, AI is an engine of extraction. Not just text, or images or equations. The real sting in the tail is that OpenAI set its sights on mathematics because the prestige, the respect and the glory was just another thing that could be mined.

It didn’t need to be that way.

They so easily could have been open and collaborative. But the moment when one of humanity’s greatest intellectual traditions became vulnerable, they chose conquest over stewardship.

And so, I am left thinking about that professor again. He was right that there is a special mystique to mathematics, but wrong about who would puncture it. And I’m sorry to say that, while the world might fear mathematicians, the machines certainly don’t.