Some infinitesimal analysis questions

Some infinitesimal analysis questions

[Larry Stout]

I am teaching a course out of Bell's book "A Primer of Infinitesimal
Analysis" to senior mathmajors at my liberal arts college.  It is making
a nice capstone, since it lets them look at the material they started
college with (claculus) from a completely different viewpoint (that of
synthetic differential geometry).  It also lets me teach some of a topos
theoretic view on mathematics.  I am left with some questions in my own
mind about what one can and cannot do in a smoth world.  Specifically,
	1.  The usual inverse function theorem uses monotonicity to guarantee
the existence of an inverse function, a monotonicity obtained from the
mean value theorem.  It seems unlikely to me that the mean value theorem
holds in synthetic differential geometry, so how does one guarantee the
existence of an inverse for a function with strictly positive
derivative?
	2.  In a smooth world must the image of a closed interval be a closed
interval?   Can one characterize closed intervals without knowing what
their endpoints are purported to be?  (Since closed intervals are
microstable you can't actually know those endpoints uniquely).
	3.  How do you justify the leap from stationary points to maxima and
minima?

Have any of the other readers of this list tried teaching a course out
of this book? 
-- 
Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University
http://www.iwu.edu/~lstout

"Fiddling is a viol habit."  Anon?
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