Re: Infintesimals

From: Thomas Nordhaus (thnord2002_at_yahoo.de)
Date: 06/30/04 

Date: Wed, 30 Jun 2004 19:01:47 +0200

 israel@math.ubc.ca (Robert Israel) schrieb:

>>My question is: what is the opinion of other mathematicians as to the
>>existance of these types of numbers
>
>We get along quite well without them, but they are fun to play with,
>and sometimes useful.

I remember (hmm, about 20 years ago) a paper, where the existence of a
limit cycle was proven for a singular perturbation problem

Eps*u' = f(u) - v
v' = u

for small eps (f a cubic polynomial of u with 3 zeros).

We discussed that problem at length. Solved it the classical way by
means of singular perturbation theory (matching conditions, boundary
layers, "ugly" eps-delta-proofs).

The non-standard proof looked quite elegant. However we spent half of
the quarter with getting the definitions and concepts right. Still, at
the end, there was this "nagging" feeling: "Maybe I missed something
essential?"

So, maybe we should look a little more into this theory and make it
more easily accessible for "classicaly educated" mathematicians - or
are there serious technical difficulties in applying this theory that
I'm not aware of anymore?

Thomas