[FOM] constructivism and physics
Timothy Y. Chow
tchow at alum.mit.edu
Thu Feb 16 18:52:00 EST 2006
Ben Crowell <fomcrowell06 at lightandmatter.com> wrote:
> 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.
Certainly that's your claim, and Alasdair Urquhart made the same claim.
It is uncontroversially true. But that is not what the discussion was
initially about. Antonino Drago wrote:
>Almost two centuries after, rigorous calculus confirmed almost all the
>results obtained by infinitesimals; but not all.
Clearly, "results obtained by infinitesimals" here does not refer to
nonstandard analysis, since nonstandard analysis did not precede rigorous
calculus by two centuries. Todd Wilson asked for examples, to which
Antonino Drago responded; I didn't understand the examples, but clearly
they were *not* examples (or even attempts at providing exmaples) of
results in nonstandard analysis.
> The problem with the nonrigorous treatment of infinitesimals wasn't that
> it was insufficiently powerful, it was just that it was logically
History is rife with examples of results that were derived from logically
inconsistent bases but that were nonetheless correct results in some
sense. I'm interested in whether there are examples of such coming from
the nonrigorous treatment of infinitesimals, that seem to have some
validity but that currently have no mathematically rigorous
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