FOM: fom: analysis in toposes

[Steve Awodey]

        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
>paraphrase them.

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.

Steve Awodey
Assistant Prof. of Philosophy
Carnegie Mellon University
http://www.andrew.cmu.edu/user/awodey/