FOM: fom: analysis in toposes
awodey at cmu.edu
Sat Jan 17 17:28:54 EST 1998
I agree completely with Bill Tait that n-th order arithmetic is a
perfectly natural logical foundation for analysis - both conceptually and
historically. And I presume that the intimate connection that he alludes
to between that higher-order logic and the notion of a topos is clear
enough to all (ref.s available on request).
The further questions Steve is asking, regarding the use of
excluded middle and choice in developing analysis in a topos, are not
essentially different in topos theory than in the context of higher-order
logic. In particular, there's nothing new to be said.
Regarding the "motivation" for topos theory: certainly the
higher-order logical aspect just mentioned is a part of the story. Topos
theory develops this intuition in the "structural" style of general
category theory; and this is also where the "general theory of functions"
interpretation that Steve alludes to comes in - say, as developed in Mac
Lane's "form & function" book. Yet another motivation is the "variable
sets" interpretation developed by Lawvere; and following Grothendieck's
original lead, there's the geometric interpretation.
Of course, these kind of remarks are terribly vague - almost
useless. But there's plenty of good literature around for those who are
really interested in such questions, and not just trying to pick a fight.
>Date: Fri, 16 Jan 1998 23:26:44 -0600 (CST)
>From: Dave Marker <marker at math.uic.edu>
>Subject: FOM: "list 2"
>... I think you
>should reread some of our posts before you attempt to
Here, here! Another case in point:
>Date: Fri, 16 Jan 1998 19:11:15 -0500 (EST)
>From: Stephen G Simpson <simpson at math.psu.edu>
>Subject: FOM: Explaining Simpson and McLarty to each other
>... McLarty says that right
>inverses don't exist in categories of sheaves, except in trivial
>cases. On the other hand, Awodey says they do exist in categories of
>sheaves over complete Boolean algebras. ... I say these guys
>need to get together and compare notes.
There's no disagreement between McLarty's statements and mine: he was
talking about sheaves on topological spaces and I mentioned sheaves on
(complete) boolean algebras. This was quite clear from our respective
posts, to anyone who cared to understand what was being said.
The talk-radio style, inflammatory moderation on this list
(presumably intentional?) may elicit more postings, but like talk radio
itself I suspect that it makes most thoughtful readers tune out.
Assistant Prof. of Philosophy
Carnegie Mellon University
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