[FOM] constructivism and physics
Timothy Y. Chow
tchow at alum.mit.edu
Tue Feb 14 19:21:09 EST 2006
Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:
> It follows from this that to look for a theorem that
> you can only prove by nonstandard methods, but not
> by standard methods, is to go on a wild-goose chase.
> The strength of infinitesimal methods lies in their heuristic
> power, as Leibniz said several centuries ago.
If I have been following this thread correctly, the claim that there are
methods that can be justified by infinitesimals but not by rigorous
calculus is about "infinitesimals" in the classical nonrigorous sense, not
in the sense of the rigorous infinitesimals of nonstandard analysis.
I am also interested in examples, and I confess that I was unable to
understand the ones given by Antonino Drago. For example, I think there
wsa some example about all Taylor series converging. I'll grant that this
is a "result" that has not been recovered in modern rigorous analysis, but
is there really a *justification* of this claim via a nonrigorous argument
using infinitesimals? I had the impression that this was simply something
that was assumed without too much scrutiny, but I don't know my history
well enough. Can someone clarify?
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