I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.
I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.