[FOM] Infinitesimal calculus

[David Ross]

> Second order categoricity is fundamental to mathematician's acceptance
> of the fundamental structures of mathematics. [...] It is the best
> explanation as to why mathematicians are not going to overhaul the
> foundations of real analysis.

I think that rather it is an argument trotted out *after* mathematicians 
already decided not to overhaul these foundations.

Joe, I agree that one can give a better (if vague) idea of what a real 
number might be than what a hyperreal number might be at the Calculus I 
level.  However, if your students really worry about this, you are teaching 
very different students than I am.  Incidentally, diNasso, Benci, and Forti 
have written a couple of papers (notably "The eightfold path to nonstandard 
analysis") with descriptions of the hyperreals which might better fit your 
standards of an "intuitive description" than the ultraproduct construction.

I have on occasion taught out of Keisler's book to the same student cohorts 
as with conventional books, and found that they have no trouble at all with 
working with the hyperreals (or rather, no more trouble than with the 
regular reals).  Moreover, the 'nonstandard' students are much more 
comfortable with using the standard part map (which is the nonstandard 
expression of completeness of R) than the standard ones are with the LUB 
property.

Mathematicians are inertial.  When we switched to the Weierstrass approach 
in the late 19th century there was apparently great reluctance to do so, but 
at that time there was a critical problem to solve, the foundations being 
used had an obvious flaw, and the new approach solved the problem.  At this 
point there is no such problem to solve, so the question of whether 
infinitesimals are better or worse is irrelevant; the inertia wins.

David