FOM: Explaining Simpson and McLarty to each other
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Fri Jan 16 18:30:55 EST 1998
Colin, I think Steve really wants to know the following:
If Set Theory did not exist, would it be necessary to invent it? That is,
can you come up with an alternative Analysis 101 curriculum, in which you prove
useful theorems like Fubini's theorem, Stokes's theorem (bridge to topology),
some useful formulation of Fourier Analysis (bridge to engineering), basic
properties of the Riemann zeta function (bridge to Number Theory), Chebyshev's
inequalities (bridge to probability and statistics), and so on, starting from
categories or toposes rather than set theory? If all you need are natural
number objects and right inverses those are not hard to motivate, the question
is can this really be done in a technically manageable fashion? (Harvey and
Steve would maintain, correctly, that you'd still have to invent set theory for
certain natural theorems whose logical strength corresponds to big ordinals).
Steve, is this correct? I think you're missing Colin's point--he doesn't need
to care about sets at all if he can really do analysis in a topos and specialize
it to get your bridges (especially if naively enough for undergrads!).-J Shipman
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