[FOM] constructivism and physics

[Ben Crowell]

Todd Wilson wrote:
>I'm giving an informal talk on infinitesimal analysis at our math
>seminar in a couple of weeks, and I would love to include an example of
>a result obtained by infinitesimals that has not been confirmed by
>"rigorous calculus".  Can you give an example (or, better, a few)?

See http://en.wikipedia.org/wiki/Non-standard_analysis#Applications .
It gives an example of a new result that was first proved via
nonstandard analysis, but was then almost immediately proved again
with standard methods. If there are any results derived using nonstandard
analysis that haven't been confirmed through standard techniques,
it's probably just because nobody cared enough to do it. When you're
really just doing classical analysis, the infinitesimal language
can always be translated into the language of limits (as Cauchy did),
and the stuff about limits can always be translated into infinitesimals
(as Robinson did). The only reason to prefer one or the other is that the
idea you have in mind about classical analysis might be awkward to express
in one and not in the other.