what saves us is that proofs are text, and thus there are countably many proofs for any logic we might agree upon, where checking a proof is correct is decidable, and thus for any axioms we culturally agree on, a computer will be able to come up with all the proofs we would.
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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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A group of people with mutually accepted and practiced customs, or the customs themselves
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It seems to me that there is an ideal way to do mathematics (I know that this is a crass oversimplification), and the customs of mathematicians are not cultural but pragmatic approximations of this. Deforming them into an idiosyncratic culture perverts mathematics.
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(I think that a set of policies and behaviors only counts as a culture if it involves a choice, and is not imposed by the rules of the game one has to play.)
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There are a few choices for mathematicians: 1. Which logic to use for their metalogic 2. What axioms should be made explicit: Most papers don't declare "we now use modus ponens" 3. What level of detail to present their arguments: what is acceptable as a pragmatic approximation
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It seems to be a good approximation when we say that are no fundamental disagreements between mathematicians. There seem to be merely different areas and levels of expertise. In my view, philosophy is a culture (or several ones), because philosophers have tremendous disagreements
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You appear to be disagreeing with my definition of culture and not with the fundamental point my usage of the word culture and subsequent definition was intended to convey, which is not particularly interesting or insightful.
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I am in disagreement with what looks like postmodernism to me. Math is not a social construct. Much of philosophy is, but not the good parts.
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