FOM: "function" as a basic mathematical concept
Colin Mclarty
cxm7 at po.cwru.edu
Tue Jan 13 21:03:05 EST 1998
Reply to message from simpson at math.psu.edu of Tue, 13 Jan
Steve expressed scepticism about the possibility of doing
real analysis in a topos. I mentioned a large literature and cited
one book, Springer Lecture Notes in Mathematics no.753.
>Ah yes, I remember this conference proceedings volume from the 70's.
>Its title is "Applications of Sheaves". The editors express
>disappointment that analysis was not better represented at the
>conference; nevertheless, the volume contains many interesting
>articles about sheaves of real-valued functions on topological spaces.
>At a quick glance I don't see any articles about real analysis in a
>topos. In fact, the volume doesn't seem to contain many articles
>about topos theory at all.
>
>Categories of sheaves are toposes, but the notion of topos is much
>more general.
This is like saying Hermann Weyl's GROUP THEORY AND QUANTUM
MECHANICS contains no group theory, since it only concerns the
classical transformation groups and the abstract group concept is
much more general.
SLNM 753 concerns primarily categories of sheaves on posets.
But the method in nearly all the articles, quite explicitly, is to
give constructions and interpretations by uniform methods workable
in any topos with a natural number object--and see what they mean
in various specific sheaf categories.
A given sheaf in one of the articles will be called
the "sheaf of real numbers" or "the sheaf of real-valued functions
on the reals"; precisely because it is constructed within its sheaf
category the way the "real numbers" or "real-valued functions on the
reals" are constructed within any topos.
>I have my doubts as to whether general topos theory as
>a general theory of functions (if that's what it is) could provide a
>reasonably well-motivated foundation for real analysis, good enough
>for applications etc. I'm not saying it can't be done, I'm only
>saying that I have my doubts.
Following my last paragraph, the motivation for real analysis
in general topos theory is not only as good as the motivations in
particular cases in SLNM 753: It IS the motivation for all of them,
and for many newer specific applications.
The elementary topos definition of "Dedekind cut on the
rational numbers" or "Cauchy sequence of rational numbers" is exactly
the same as the ZF set theory definition.
Anyone who likes to, may "doubt" that the elementary topos
axioms can have any foundational role. But if you accept those
axioms themselves, real analysis proceeds from there by exactly
the same definitions as from the ZF axioms. Of course the results
differ from the ZF results in the ways I mentioned before. If you
require your topos to meet suitable further conditions you can
get exactly the same analysis as in ZF or ZFC or whatever you like.
As Feferman has noted, if you want to extend the topos axioms
to copy some particular fragment of extension of the ZF axioms, then
you necessarily have that fragment or extension of ZF in mind. I hope
Pratt's Star Trek parable has convinced people that this does not mean
topos theory secretly depends on set theory.
--Colin McLarty
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