FOM: Responses to Hersh's answers to my questions JSHIPMAN at
Tue Jan 20 16:53:18 EST 1998

Reuben Hersh sent answers to my questions which were quite revealing.  I don't
think my questions 5) and 7) really do fit his definition, because in 5) there
is not a complete consensus on the version of the Axiom of Choice needed to get
Banach-Tarski, and the statement 7) is NOT established by any sort of
consensually accepted proof.  But in both cases Hersh can save himself by the
standard move "this isn't a proof of A, it's a proof of B -> A where B is the
axiom (AC or MC) about which there is no consensus".  Since set theorists
habitually say for example "Measurable Cardinal implies V not = L" rather than
"V not = L because there really is a measurable cardinal" in order to attain a
Hershian consensus, and it's still considered good practice to note when AC is
needed, my question 1) seems the biggest threat to his thesis.  -- Joe Shipman

1) Is the most interesting case, I'll take it up in a later posting.
2) I agree with you
3) I agree because to make this precise requires us to refer to human

capacities, not abstractions only
4) I provisionally agree, but you could probably make this precise with
algorithmic information theory and then it would be mathematics
5) You won't get a consensus from intuitionists or constructivists on this one!
6) But you will here because this statement is provable in a very weak system!
7) This is technical set theory, constructible in the sense of Godel's "L" --
you get much less of a consensus here then in 5) because ZFC isn't enough
8) But here again you get a consensus. -- JS

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