AI and the Scarcity of Meaningful Theorems

Terence Tao has expressed concern that AI is rapidly consuming the stock of open mathematical problems that mathematicians traditionally seek to solve, with the resulting theorem and proof carrying the name of the mathematician who formulated or solved the problem.

OpenAI's release on October 6, 2026, of more than 700 manuscripts covering some 370 mathematical research topics illustrates this development. The release includes claims concerning famous open problems, although the mathematical validity and significance of many results remain to be established.

Tao's concern is not simply that mathematicians will run out of problems. It is that solving problems by automated proof production may destroy opportunities for mathematical discovery and understanding.

But this raises a more fundamental question: What is a mathematical theorem, and what makes it worth proving?

Consider number theory, starting from Peano's axioms for the natural numbers.

There are infinitely many logical consequences of these axioms, and thus infinitely many possible theorems. But how many are worth stating, naming and remembering? A few dozen fundamental results? A few hundred? A few thousand?

The answer cannot be obtained from the axioms themselves. Logic distinguishes between statements that follow from axioms and those that do not. It does not distinguish important theorems from unimportant ones.

Consider the elementary statement that the sum of two even numbers is even. It conveys one clear message. The fundamental theorem of arithmetic, stating that every integer greater than one has a unique prime factorization, conveys another, considerably more powerful message.

Both are theorems. But their mathematical significance differs enormously.

A mathematical theorem of interest is therefore not merely a logical consequence of axioms. It is a statement conveying a specific, nontrivial and intelligible message, supported by a proof.

This suggests that the true scarcity in mathematics is not a scarcity of theorems or proofs, but a scarcity of meaningful mathematical messages.

AI can generate proofs on an industrial scale. But generating proofs is not necessarily the same as generating mathematical understanding.

The traditional mathematical publication system rewards the production of named theorems, with priority and authorship as central currencies. When AI can produce hundreds of proofs almost instantaneously, this system faces an obvious crisis.

Perhaps the real challenge for mathematics in the AI era is therefore not to find more open problems to solve, but to determine which mathematical statements actually deserve our attention.

The central question changes from "Can this be proved?" to "Why is this worth knowing?" 

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