constructive mathematics is not time-less like classical math. a function that has not been computed yet does not have a defined value. pi is a function, not a number. the axiom of choice does not hold. true infinities and continua cannot be constructed
I don't like the idea that "we" "culturally agree" on axioms. If by "we" you include non-mathematicians, you are probably wrong in most cases, and if you characterize mathematics as a culture instead of a formal exercise, you miss its point.
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no, we doesn't include non-mathematicians. Mathematics is a culture. This is what gödel's thms actually show: there are provably infinitely many possible independent axioms forming a hierarchy of logics by relative consistency, and at the top assuming all of them is inconsistent
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How do you define "culture"?
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restatement: for any axioms a specific mathematician comes up with, a computer will be able to come up with all the correct proofs he could if they both lived forever, regardless of the mathematician's internal theorem proving algorithm
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