[FOM] constructivism and physics
Ben Crowell
fomcrowell06 at lightandmatter.com
Wed Feb 15 11:50:26 EST 2006
Timothy Y. Chow wrote:
>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.
No, I think the claim is that even rigorous nonstandard analysis is incapable
of proving results that can't be proved using limits. At least, that's my
claim. Of course there are results *about* the hyperreal number system that
can't be proved without using the hyperreals, but that's different. The
problem with the nonrigorous treatment of infinitesimals wasn't that it was
insufficiently powerful, it was just that it was logically inconsistent.
>For example, I think there
>wsa some example about all Taylor series converging.
The whole idea of limits and convergence loses some of its interest in
nonstandard analysis, since limits are never unique. That is, if you
write down the Cauchy/Weierstrass definition of the limit, and apply
it to the hyperreals, then a sequence like 1, 1/2, 1/3, 1/4, ...
has infinitely many limits, all of them infinitesimally close to zero.
The existence and uniqueness of limits is essentially the definining
characteristic of the real number system, as compared with the rationals
or the hyperreals. Historically, I believe there was a proposal to put
calculus on a rigorous foundation by assuming that every function had
a Taylor series -- it seems like nonsense with hindsight, but I guess
that's a symptom of the confused state things were in.
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