AI Can Do Math, but Is It Really Math?

“If nobody understands a mathematical proof, is it a proof at all?

 

OpenAI startled mathematicians last month by announcing that one of its models had disproved a longstanding conjecture, posed by the mathematician Paul Erdos (1913-96) in 1946, about the "unit distance problem." In a companion paper, prominent mathematicians verified and explained the argument.

 

The result was hailed as a breakthrough. But it also raises a question about the future of mathematics. If artificial intelligence continues to make serious mathematical progress, the field will need a new kind of human infrastructure: people and institutions devoted to choosing the right problems, checking machine-generated proofs, making those proofs intelligible, and judging when discoveries have consequences beyond mathematics.

 

The problem OpenAI tackled is relatively simple to picture. Put some number of dots on a sheet of paper. Count how many pairs of dots are exactly one inch apart. As you add more dots, how many such pairs can there be? Erdos conjectured that the maximum number of pairs, suggested by a gridlike system, could increase only slightly faster than the number of dots.

 

For decades, many mathematicians believed that Erdos's conjecture gave the right ceiling. OpenAI's model showed otherwise, finding arrangements that beat his supposed limit again and again, even as the number of dots grows arbitrarily large.

 

How OpenAI arrived at its result was as surprising as the result itself. The unit-distance problem belongs to a branch of math known as discrete geometry, which studies the arrangements of points, lines and shapes. But OpenAI's solution drew on methods from a seemingly unrelated field, algebraic number theory, which studies deep patterns in numbers and the structures built from them. Those tools helped it generate dot patterns that beat the best-known grid designs.

 

Cross-disciplinary progress isn't new in mathematics. Descartes helped revolutionize geometry in the 17th century by translating geometric problems into algebraic equations. What has changed, and what creates the opening for AI, is the scale of mathematics. A mathematician may spend a career in discrete geometry without developing any familiarity with the sophisticated tools of algebraic number theory. For AI the scope of mathematics seems limited only by the cost of computation.

 

The result is both exciting and unsettling. Mathematics is a system of symbols and accepted axioms created by human beings, which can be used as a language for describing the world. Math progresses when we pose the right problems and come up with comprehensible proofs. The qualities that led the 19th-century mathematician Carl Friedrich Gauss (1777-1855) to call mathematics the "queen of sciences" -- its rigor, logic and isolation from real-world phenomena -- are precisely what make it the realm of human inquiry that AI can best learn, imitate and expand on.

 

But would autonomous AI mathematics be mathematics at all? As the mathematician and philosopher Reuben Hersh (1927-2020) argued, mathematics isn't only a human activity but a social phenomenon. All sorts of mathematical truths aren't properly part of what we refer to as mathematics unless they are discovered and explained by human beings. Progress in mathematics is a matter of communication; a proof that is poorly explained is no better than a proof that is wrong.

 

This is why the human work was indispensable to making sense of OpenAI's discovery. Instead of merely reporting the model's output, OpenAI collaborated with leading mathematicians to verify the result and make it intelligible to experts in the field.

 

Without that communicability, the discovery would have a strange status. It might be true. But if humans had no way to verify it, what would its truth mean? On Hersh's view, it might not be a mathematical result at all.

 

Continued human involvement in mathematics is also necessary to safeguard the applied sciences and society at large. Gauss went further to say that number theory was the queen of mathematics. Number theory -- the study of patterns in numbers -- had long had an aura of abstract impracticability: math for math's sake. But in the 1970s it turned out to be the basis for modern cryptography. Much of contemporary computer privacy rests on mathematical ideas that once seemed entirely abstract.

 

That history should make us cautious about treating progress made by AI in pure mathematics as a mere intellectual curiosity. A new proof may begin as an internal event in an abstract field and then develop, perhaps with the aid of other AIs, into a tool to be used, for good or for ill, in practical fields. If the proof is unverified by humans and the pure math is shaky, the applied math built on it can be not only unsound but dangerous. A mistaken theorem can work its way into flawed financial, medical or engineering systems.

 

Mathematics will need to develop a research culture that can accommodate AI as a partner. This will involve journals that require verification, hiring and tenure arrangements that reward exposition and checking, and collaborative practices for the verification of proofs. Checking and explaining AI-generated mathematics must count as original intellectual labor. The stronger AI becomes, the more valuable this human expertise will be.

 

If there were ever a mathematician whose research practice foresaw these developments, it was Erdos, famous for his large number of collaborators and particularly skilled at coming up with the right problems to solve. An eccentric figure with an idiosyncratic language all his own, Erdos would say mathematicians had "left" when they died, and say that they had "died" when they stopped doing mathematics. AI can't and shouldn't cause mathematicians to die in Erdos's sense of the word.

 

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Mr. Kipnis is a statistician and an academic fellow at Columbia Law School.” [1]

 

1. AI Can Do Math, but Is It Really Math? Kipnis, Daniel.  Wall Street Journal, Eastern edition; New York, N.Y.. 15 June 2026: A17.