Erdős and the problems he left behind

Paul Erdős, a Hungarian mathematician who died in 1996 at eighty-three, published almost fifteen hundred papers and left behind about a thousand unsolved problems. Solving an Erdős problem is a great honor.

Young Tao meets Erdős

In nineteen eighty-five, when Terence Tao was ten years old, Erdős visited Tao's home town of Adelaide, Australia. Erdős posed a problem for them to work on together. Tao never solved it.

Erdős problems catalogued online

In May of twenty twenty-three, mathematician Thomas Bloom began gathering known unsolved Erdős problems on a website, erdosproblems.com. Erdős described some as marshmallows and others as acorns, varying in difficulty.

Tao on AI and mathematics

Terence Tao, a Fields Medal winner and professor at UCLA, has been a prominent voice on AI in mathematics. He founded the Foundation for Science and AI Research in twenty twenty-five.

AI struggles with basic math

Early AI tools were conspicuously bad at basic math. When OpenAI's GPT-four Turbo was asked to count the number of L's in lollapalooza, it answered five. The tools have since improved dramatically.

AI wins at the International Mathematical Olympiad

In twenty twenty-four, AI tools began performing well at the International Mathematical Olympiad, scoring at the silver-medal level. In twenty twenty-five, they scored at the gold level.

AI starts solving Erdős problems

In early twenty twenty-six, mathematicians using AI tools started solving some of the more marshmallowy Erdős problems.In January, GPT-five point two Pro appeared to have solved Erdős Problem number seven twenty-eight largely on its own.

The unit-distance problem falls

On May twentieth, OpenAI published a proof of Erdős Problem number ninety, the unit-distance problem, a combinatorics problem involving arranging points on a plane. This was not a marshmallow.

The Navier-Stokes breakthrough

On September eighth, OpenAI announced it had solved the Navier-Stokes existence-and-smoothness problem, one of the Clay Mathematics Institute's seven Millennium Prize Problems, each with a million-dollar award.

Tao's crisis moment

Three days after the Navier-Stokes announcement, Tao gave a talk at Caltech wearing jeans and a black T-shirt. He declared, I have had the strangest week ever in my life.

“I have had the strangest week ever in my life”

The proof verification crisis

Solutions were being published with little explanation, relying on mathematicians to verify them. AI-generated submissions overwhelmed journals and databases. The field's volunteer refereeing system collapsed under the weight of low-quality proofs.

The loss of insight

Tao said, We have to slow down. It's insane, the pace. He explained that new problems emerge from the slow, deliberative process of working toward a solution, and this is what's being destroyed.

“We have to slow down. It's insane, the pace.”

Tao's city planning analogy

Tao compared the crisis to cities transitioning from horse and buggy to cars. You need parts for pedestrians and parts for cars, but you cannot design the whole city optimal for cars, because no one…

The Jacobian conjecture solved

On July nineteenth, mathematician Levent Alpöge at Anthropic posted that the Jacobian conjecture, an eighty-seven-year-old problem, was false. Claude Fable five had produced a counterexample. Within thirty-six hours, mathematicians understood why it was natural.

The Hadamard matrices problem

On August twelfth, Alpöge posted a sequence of Hadamard matrices, the next several in a series first described in eighteen sixty-seven. Mathematician Geordie Williamson felt as if an entire garden of mathematics had been killed.

The Fields Medal letter

An initial group of twenty-five Fields medallists, including Tao, responded with an open letter noting that solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight.

“solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight”

Mathematics as living tradition

The letter argued that mathematics is a living discipline and an oral tradition, not just a written one.

Urschel's AI collaboration

Mathematician John Urschel at MIT worked with GPT-five point six Sol and Claude Fable five on a Gaussian elimination problem.

The joy of struggle

Urschel said, There was no fun in this for me.The fun was had before I gave it to this computer program.

“There was no fun in this for me. The fun was had before I gave it to this computer program.”

Math's double life

Urschel suggested mathematics has been living a double life.To outsiders, math is important for applications like public-key cryptography and GPS.