3 ms·

It is fundamentally more complicated. Possible moves can be evaluated against one another in games. We have no idea how to make progress on many math conjecture

by qfiard 2y ago

It is fundamentally more complicated. Possible moves can be evaluated against one another in games. We have no idea how to make progress on many math conjectures, e.g. Goldbach or Riemann's. An AI would need to find connections with different fields in mathematics that no human has found before, and this is far beyond what chess or Go AIs are doing.

salamo 2y ago
Not a mathematician, but I can imagine a few different things which make proofs much more difficult if we simply tried to map chess algorithms to theorem proving. In chess, each board position is a node in a game tree and the legal moves represent edges to other nodes in the game tree. We could represent a proof as a path through a tree of legal transformations to some initial state as well. But the first problem is, the number of legal transformations is actually infinite. (Maybe I am wrong about this.) So it immediately becomes impossible to search the full tree of possibilities. Ok, so maybe a breadth-first approach won't work. Maybe we can use something like Monte Carlo tree search with move (i.e. math operation) ordering. But unlike chess/go, we can't just use rollouts because the "game" never ends. You can always keep tacking on more operations. Maybe with a constrained set of transformations and a really good move ordering function it would be possible. Maybe Lean is already doing this.
- grondilu 2y ago
  > But the first problem is, the number of legal transformations is actually infinite. I am fairly certain the number of legal transformations in mathematics is not infinite. There is a finite number of axioms, and all proven statements are derived from axioms through a finite number of steps.
  - hwayne 2y ago
    Technically speaking one of the foundational axioms of ZFC set theory is actually an axiom schema, or an infinite collection of axioms all grouped together. I have no idea how lean or isabelle treat them.
    - fspeech 2y ago
      Whether it needs to be a schema of an infinite number of axioms depends on how big the sets can be. In higher order logic the quantifiers can range over more types of objects.
    - 4ad 2y ago
      Lean (and Coq, and Agda, etc) do not use ZFC, they use variants of MLTT/CiC. Even Isabelle does not use ZFC.
      - robinzfc 2y ago
        Isabelle is generic and supports many object logics, listed in [1]. Isabelle/HOL is most popular, but Isabelle/ZF is also shipped in the distribution bundle for people who prefer set theory (like myself). [1] https://isabelle.in.tum.de/doc/logics.pdf https://isabelle.in.tum.de/doc/logics.pdf

by qfiard 2y ago

salamo 2y ago

Not a mathematician, but I can imagine a few different things which make proofs much more difficult if we simply tried to map chess algorithms to theorem proving. In chess, each board position is a node in a game tree and the legal moves represent edges to other nodes in the game tree. We could represent a proof as a path through a tree of legal transformations to some initial state as well. But the first problem is, the number of legal transformations is actually infinite. (Maybe I am wrong about this.) So it immediately becomes impossible to search the full tree of possibilities. Ok, so maybe a breadth-first approach won't work. Maybe we can use something like Monte Carlo tree search with move (i.e. math operation) ordering. But unlike chess/go, we can't just use rollouts because the "game" never ends. You can always keep tacking on more operations. Maybe with a constrained set of transformations and a really good move ordering function it would be possible. Maybe Lean is already doing this.

grondilu 2y ago
> But the first problem is, the number of legal transformations is actually infinite. I am fairly certain the number of legal transformations in mathematics is not infinite. There is a finite number of axioms, and all proven statements are derived from axioms through a finite number of steps.
- hwayne 2y ago
  Technically speaking one of the foundational axioms of ZFC set theory is actually an axiom schema, or an infinite collection of axioms all grouped together. I have no idea how lean or isabelle treat them.
  - fspeech 2y ago
    Whether it needs to be a schema of an infinite number of axioms depends on how big the sets can be. In higher order logic the quantifiers can range over more types of objects.
  - 4ad 2y ago
    Lean (and Coq, and Agda, etc) do not use ZFC, they use variants of MLTT/CiC. Even Isabelle does not use ZFC.
    - robinzfc 2y ago
      Isabelle is generic and supports many object logics, listed in [1]. Isabelle/HOL is most popular, but Isabelle/ZF is also shipped in the distribution bundle for people who prefer set theory (like myself). [1] https://isabelle.in.tum.de/doc/logics.pdf https://isabelle.in.tum.de/doc/logics.pdf