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Let an AI extend Mathlib forever
Math as code
Mathlib is a GitHub repository holding math written in code, where a computer can answer whether a proof is correct. It is very far from all of math, and writing those proofs by hand is labor-intensive. Picture a fork of it, kept free of human taste, handed to an AI with one instruction: extend it.
Letting it run
It could come up with its own conjectures, its own theories, and different definitions. Many would be useless, but the tree keeps growing without anyone checking in. Someone could press go, pour compute at it, look away for ten years, and come back to ask what it found. There would be something, and then the question becomes whether it is useful.
Self-play and axiom search
Go programs in the AlphaZero style work the same way, exploring a universe of their own against an automated reward. Natural-language math cannot do this, because a human still has to review each result. Terry Tao has described a related project that searches the possible axiom systems for algebra. Most collapse into nothing interesting, and every now and then an island appears that is rich in theorems, one that might later turn out to describe something like symmetry, the way the axioms for a group do.
How does a computer verify a proof?
Mathlib is written in the Lean proof assistant, which rests on a small core called the kernel. Every proof is broken down into steps that the kernel checks against the rules of its type theory. If the kernel accepts the final statement, the proof is correct, no matter how the proof was produced. That is why an untrusted source, even a machine, can generate proofs and still have them checked.
How did AlphaZero learn Go by itself?
AlphaGo Zero, published by DeepMind in 2017, was given only the rules of Go and played games against itself. It beat the earlier AlphaGo version that defeated Lee Sedol by 100 games to 0. The win or loss at the end of each game served as an automatic reward, with no human game records involved.
How did the group axioms emerge?
Arthur Cayley gave an abstract definition of a group in 1854, after Galois and others had worked with groups of permutations. By the 1880s the modern axioms of closure, associativity, identity, and inverses were in common use. Those four short rules turned out to describe the symmetries of shapes, the structure of number systems, and the classification of particles.
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