[FOM] on bill tait's answers to GS's questions
a.hazen at philosophy.unimelb.edu.au
Thu Mar 23 03:10:28 EST 2006
Gabriel Stolzenberg wrote:
GS: Among classical mathematicians, constructive math is the study of
what can be proved without omniscience."
WT: I doubt that you are right about what classical mathematicians would
GS: Trust me. These were my people. And in a sense they still are.
Did you really never hear things like, "I want to do God's mathematics"?
(where I have inserted the initials identifying Stolzenberg's and
Tait's lines in the dialogue).
To cite one instance in print: Rudy Rucker in his book "Infinity and
the Mind" quotes Gaisi Takeuti as responding to a "What is set theory
about" question with "Set theory is about the thoughts of an infinite
mind." (I've lent my copy to a student, so that's from memory.)
On the general question of omniscience in mathematics... I
remember as a student (in the 1970s) being very puzzled by
expositions of intuitionism (I think the one in Beth's "Foundations
of Mathematics" was one) that justified the rejection of LEM in terms
of our LACK of omniscience: "What does it matter," I wanted to shout,
"that we don't know whether or not it is the case that P? That's not
what LEM says: it just says that it either IS or IS NOT the case that
P, whether we know it or not!" Dummett's account of the
philosophical stance appropriate to intuitionism is not above
criticism, but his was the first exposition I read that made it sound
like an understandable and non-silly one. (Stolzenberg has since
written a good, thoroughly un-Dummettian, account.)
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