Here is how I see the reconciliation between points made by
@geoffreyhinton and
@davidbessis
Both are true:
1) yes, mathematics evolves by creating new ways of thinking. It expands by inventing new representational frameworks that change how we see.
2) yes, math is a closed system.
These both hold true because closure is exactly what makes generativity possible. Gödel, Turing, Hofstadter, Chaitin have all invoked this.
A system must be closed enough to recurse on itself, otherwise it can’t generate anything coherent.
If math wasn’t closed, there would be no recursion, no self-reference, no computation. The universality expressed by Gödel and Turing wouldn’t be possible.
So the closure in math does not define a static system that limits new ways of thinking, rather it is a constraint that enables the generative capacity new math depends on.
We already know that a system can be formally closed yet internally infinite, self-expanding, and creative.
So I agree, mathematics is never “solved” in the way Hinton seems to be suggesting. But not because math is open, because it is a closed self-referential system that can generate unbounded novelty.
Even a perfect proof-generating AI wouldn’t end mathematics, because math grows not only by solving within a system, but by redefining the system itself.
Math won’t end for the same reason stories won’t end; not because we run out of answers, but because we keep inventing new ways to ask questions.
This is structurally guaranteed.