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? "Dancing is necessary in a well ordered society." Thoinot Arbeau