The Fields Medalist Terence Tao is on the front lines of a debate about artificial intelligence and the future of his field.

© Ariana Drehsler for The New York Times

The education secretary is right to focus on what students need most: the ability to express themselves. As for maths, grab a calculator

At last the fourth R is born. Articulation – speaking – is to join reading, writing and arithmetic in the education canon. What has become the mighty struggle against the dictatorship of the screen can begin in earnest. The spoken word, human-to-human communication, must finally be recognised by the gods of reactionary schooling. The pundits call it oracy.

On Thursday, the education secretary, Lucy Powell, called the academics’ bluff. She wants a radical reform of the national curriculum in England to reduce “desktop” working in favour of verbal fluency and practical skills. This will take courage. For decades, education’s senior leaders have been ramming easily forgotten “subjects”, such as maths and French, into children’s brains so they can be judged and measured in league tables. These tables have grown ever more dominant as ministers seek to climb “global rankings”.

Simon Jenkins is a Guardian columnist

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© Photograph: Andrew Fox/Alamy

© Photograph: Andrew Fox/Alamy

© Photograph: Andrew Fox/Alamy

The solutions to today’s puzzles

Earlier today I set you these six puzzles. Here they are again with solutions.

The round table

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© Photograph: acceptfoto/Getty Images/iStockphoto

© Photograph: acceptfoto/Getty Images/iStockphoto

A puzzle for the ages

It’s made of wood, it’s in the shape of a circle, and it is the forum for many an important gathering.

It is also the setting for today’s puzzle.

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© Photograph: BBC/Studio Lambert/Euan Cherry/PA

© Photograph: BBC/Studio Lambert/Euan Cherry/PA

© Photograph: BBC/Studio Lambert/Euan Cherry/PA

Despite its tranquil reputation, knitting is anything but boring. One stitch slip-up can cause a defect that spreads across the entire fabric. But does it have to be that way? In a recent paper in Physical Review X, a team of researchers mathematically investigated knitting to invent a new, more resilient stitch. (Scientific American)

At the Los Alamos National Laboratory, Mary Tsingou Menzel’s computer programming helped set the stage for chaos theory and other fields.

Hundreds of new A.I.-generated findings moved the frontiers of higher math in a single day — dispelling any doubt that the field is forever changed.

© Ian C. Bates for The New York Times

OpenAI’s trove of new math results doesn’t fully solve any of these highly prestigious open questions, but it seems to make progress on some.

© Göttingen Library, Clay Mathematics Institute

The woman who helped put astronauts on the moon was a software engineer when the field was not prestigious or lucrative.

© Smithsonian National Air and Space Museum, via Agence France-Presse — Getty Images

The woman who helped put astronauts on the moon was a software engineer when the field was not prestigious or lucrative.

© Smithsonian National Air and Space Museum, via Agence France-Presse — Getty Images

After a previous solution angered mathematicians, the company characterized the new release as being more responsive to concerns about A.I. disrupting research.

© Jeff Chiu/Associated Press

In our field of applied mathematics, the long, human process of trial and error — not just the solutions themselves — is often what has led to progress.

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Emmy Noether, whom some consider the "Mother of Modern Mathematics," continues to shape modern research and the studies of mathematicians today; including research into the investigation of airplane crashes. Invited to Göttingen by Hilbert, Emmy Noether was denied a teaching post simply because she was a woman. For four years she lectured under Hilbert's name rather than her own, as she was not permitted to teach officially. When the faculty objected, Hilbert famously responded: "I do not see that the gender of the candidate is an argument against her admission. We are a university, not a bathing establishment."

Previously and previously.

Tristan Buckmaster was on the path toward an important proof when one of the A.I. giants used its staggering resources to get there first.

© Graham Dickie for The New York Times

A controversy around OpenAI’s latest math proof raises a fear that might become increasingly common: that A.I. tools could take valuable insights from one user and hand them to another.

© Kaoly Gutierrez for The New York Times

Claude Fable 5.1 solved a 370-year-old cipher in forty-four minutes.

This tracks with what I wrote about AIs doing mathematics: It’s good at things that involve lots of searching and testing.

EDITED TO ADD (9/14): I’m not sure if this result is correct. The blog comments have more discussion.

The company’s announcement is the most dramatic sign yet that artificial intelligence is fundamentally transforming the field of higher mathematics.

© Philippe Plailly/Science Source

The company’s announcement is the most dramatic sign yet that artificial intelligence is fundamentally transforming the field of higher mathematics.

© Philippe Plailly/Science Source

OpenAI announced today that it had solved the problem, one of seven “Millennium Problems” seen as among the most important in higher mathematics.

Jacob Tsimerman, a recent recipient of the Fields Medal, believes that higher mathematics can help curb the dangers of runaway artificial intelligence.

© Erin Blewett/Agence France-Presse — Getty Images

A Mesopotamian culture’s decorated ceramics show some early roots of math, researchers argued.

Joan Birman thought her major discoveries were behind her. Then came an email from a young neighbor — a girl who knew little but wanted to learn.

© Oliver Farshi for The New York Times

This essay was written with Kasra Rafi, and originally appeared in The Guardian.

Earlier this month, about 40 top mathematicians gathered at OpenAI’s offices to discuss the future of their profession. The meeting was off-the-record, but if recent articles by mathematicians are any guide, it was mostly pretty glum. People fear for their jobs, their careers and the work they love.

We think the contrary view is more likely, at least in the short-term. AI models are nowhere near as capable as experienced academic mathematicians.

This isn’t to say that AIs aren’t producing stunning mathematical results at the level of PhD researchers. In mid-May, OpenAI announced that its frontier AI model disproved the unit distance conjecture, a famous 80-year-old problem in discrete geometry. In July, Anthropic’s published two AI-derived results in academic cryptanalysis. Earlier this month, OpenAI published 10 new mathematical results from its latest AI model. And Anthropic published Claude’s attempt to prove the century-and-a-half-old Riemann hypothesis.

These results are both a vivid demonstration of the amazing capabilities of frontier AI in 2026 and an illustration of their limitations. In general, these AI-powered advances in mathematics fall into one of two categories. Some are counterexamples to mathematical statements that people had been trying to prove. Others are novel applications of known techniques to existing problems that human experts either did not know or did not think of using.

The counterexample to the Jacobian conjecture is the most notable example of the first kind. Once it had been found, checking it was quick and straightforward. The difficult part was finding it among a large number of possibilities. The AI seems to have combined some sort of intuition acquired through machine learning with extensive computational search, in order to find the right example.

An example of the second kind is the unit-distance conjecture. It was motivated by an elegant construction, and most mathematicians expected it to be essentially optimal—so they generally tried to prove rather than disprove it. The counterexample brings in ideas from elsewhere in mathematics: algebraic number theory. If an expert with that background deliberately set out to find a counterexample, they would probably have succeeded. But there was no reason for someone with precisely that expertise to focus on this problem. Because of its scope, AIs don’t have those same limitations.

These results are relatively low-hanging fruit for AI; none of them required developing an extensive new theory. This does not make the discoveries trivial, or the AI’s achievements less impressive. Choosing the right direction, and recognizing an unexpected connection between subjects, are themselves forms of creativity. They are the same sorts of capabilities that led to AIs playing the game of Go at the grandmaster level, or doing Nobel-prize level chemistry in the area of protein folding.

What we have not yet seen is an AI developing a substantial new conceptual framework in order to solve a mathematical problem. Much of mathematics proceeds by identifying the objects that are truly central to a question and then developing a theory that helps us understand them. Current AIs are very strong at searching and recombining existing ideas, but they are weak at building any deep and sustained new theory.

This speaks to a more general limitation of current AI systems. They are creative in the sense that they can recombine existing ideas in novel ways. But they are not creative in others: they have not yet developed conceptually new theories or structures. And while they have larger working memories than humans do, know more about more different things than any particular human does, and can process information faster than humans, can, true novelty is still largely beyond their reach.

Of course, that distinction may not survive for very long. Predictions are notoriously hard, especially about the future of AI. None of these mathematical capabilities were explicitly designed for, or planned. They’re all emergent properties of increasingly capable AI models. We are both confident that someday we will see AI models that are capable of the type of creativity required to do novel mathematics. Will that be in a few months, a few years or a few decades? Of course we don’t know, but our guess is sooner rather than later.

“The first rule of Mark Zuckerberg is never count out Mark Zuckerberg.”

© Photo Illustration by The New York Times; Image: Jason Henry for The New York Times