[FOM] Infinitesimal calculus

[David Ross]

Monroe Eskew <meskew at math.uci.edu> wrote:
(Speaking about square-nilpotent infinitesimals)
> I worry that without showing that these new rules are sound in the
> appropriate way, this would seem like magic.  But showing that
> nonstandard analysis works is probably much more advanced than just
> doing standard analysis.

There are square-nilpotent infinitesimals in synthetic differential 
geometry/smooth infinitesimal analysis, and many proofs are very nice, even 
cleaner than in 'standard' nonstandard analysis; cf John Bell's little book 
on the subject.  In this universe all functions are smooth, so the theorems 
are not exactly the theorems of classical analysis.

There is an interesting ongoing project to teach Calculus with 
infinitesimals going on in Geneva; see the article by Richard O'Donovan in 
the recent LNL volume edited by Cutland, diNasso, and me.

Whether nonstandard analysis "works" is a foundational problem which need 
not appear in Calculus textbooks (as Keisler's text demonstrated); the fact 
that it is somewhat harder than showing that real numbers "work" is not very 
important.  The failure of modern infinitesimal calculus to catch on in the 
curriculum is fundamentally a matter of mathematical sociology and inertia, 
not practicality or correctness.

David Ross