FOM: topos theory qua f.o.m.; a quote from Mac Lane

Stephen G Simpson simpson at
Fri Jan 16 11:50:42 EST 1998

Colin McLarty writes:
 > your goal of copying ZFC.

Copying ZFC is not a goal of mine.  My goal right now is to get you to
say something comprehensible about how good a job topos theory does in
providing a motivated, serious foundation for mathematics, including
enough real analysis to build bridges.  Note that this goal has
nothing to do with ZFC, unless *you* feel a need to copy ZFC, in order
to say something comprehensible.

As I said in my posting of yesterday:

 > My question is:
 >   What additional axioms are appropriate to add to topos theory to get
 >   enough real analysis to build bridges?  (By "enough real analysis to
 >   build bridges" I mean the standard undergraduate material on
 >   ordinary and partial differential equations, power series, Fourier
 >   analysis, etc., complete with applications, examples, etc.)

Note that ZFC is not mentioned in this question.

Do you accept the challenge of answering this question?  Or are you
going to duck this question?

If you accept the challenge and answer this question, my next question
will be: "What is the foundational motivation for the topos axioms and
the additional axioms which you have specified?"

Of course, the background for my questions is the fact that ZFC and
other well known f.o.m. schemes *do* suffice for real analysis (enough
to build bridges), and a great deal more, and *are* foundationally
well-motivated.  But this doesn't mean that I am asking you to copy
ZFC.  It's up to you whether you want to copy ZFC or not.

By the way, I got Mac Lane's book "Mathematics: Form and Function" out
of the library.  I found the following on page 406:

  Categories and functors are everywhere in topology and parts of
  algebra, but they do not as yet relate very well to most of

I would dispute the claim that categories and functors are everywhere
in topology.  But I can agree with Mac Lane that categories and
functors do not relate very well to most of analysis.  Do you agree
with this?  In view of this, do you still want to stand by your claim
that there is no problem about real analysis in a topos?

-- Steve

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