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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As for "mathematicians don't have fundamental disagreements" - Gödel's theorems came from a disagreement between him and hilbert. Forcing came from the dissatisfaction with the continuum hypothesis as an axiom which and is still contentious.
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The disagreement between Gödel and Hilbert was a mathematical and not a cultural one, and it was resolved by mathematics, not by culture.
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It is only resolved if you assume every axiom in the hierarchy but doing so is inconsistent. i.e. hilbert thought mathematics was consistent and wanted a proof. Gödel thought it was impossible to know, but proved (lots of people rly) that he couldn't prove that he couldn't know
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Yes, but the point is that in principle (and to a surprisingly large degree in practice) mathematicians can know and agree that this is the case.
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and because mathematicians agree that they can prove that they can't prove that they are consistent, the agreement is cultural and not mathematical. Statements such as "we don't condone pedophilia" also hold to a surprisingly large degree in practice and are cultural
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The difference is that societies have large degrees of freedom about whether they condone pedophilia, but mathematicians don't have many degrees of freedom about whether they accept a proof.
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