[FOM] Infinitesimal calculus
ross at math.hawaii.edu
Mon May 25 19:07:06 EDT 2009
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.
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