FOM: Toposy-turvey

[Solomon Feferman]

I didn't think I would get drawn again into a discussion of category
theory vs. set theory as a foundation of mathematics, but Aczel's FOM
drugs are at work.  First, I appreciate Peter's effort to clarify what the
conflict is about.  But I think he has left out something essential, and
that was what I emphasized in an earlier related posting of 19 Nov 23:46.
Namely, the notion of topos is a relatively sophisticated mathematical
notion which assumes understanding of the notion of category and that in
turn assumes understanding of notions of collection and function.  I do
not mean these in the sense identified in the cumulative hierarchy or any
supposed standard model for that, but rather in the informal everyday
mathematical sense.  Thus there is both a logical and psychological
priority for the latter notions to the former.  'Logical' because what a
topos is requires a definition in order to work with it and prove theorems
about it, and this definition ultimately returns to the notions of
collection (class, set, or whatever word you prefer) and function
(or operation).  'Psychological' because you can't understand what a topos
is unless you have some understanding of those notions.  Just writing down
the "axioms" for a topos does not provide that understanding.  If that is
granted, and I can't continue further in this discussion if it is not,
then what work in f.o.m. has to accomplish is a clarification of the
notions of collection (or class) and function (or operation).  I am not
assuming that axiomatic set theory has accomplished that, though it is
certainly one effort to do so and, because so much is known about that
effort, has to be given serious consideration for that reason.  This is
not in conflict with my negative views about the supposed standard model
for set theory.	That so many mathematical notions, including possibly the
real numbers, can be reinterpreted in topos-theoretic terms does not give
foundational priority to the notion of a topos or of a category more
generally.  

Sol Feferman