[FOM] Constructivism and physics

A.J. Franco de Oliveira francoli at kqnet.pt
Thu Feb 16 17:52:17 EST 2006

I dont agree. The predicate "standard" in, say, 
Nelson´s IST is not definable classically, nor 
are other concepts (infinitesimal, etc.) defined 
in terms of "standard". So, if one takes these 
new concepts seriously, say in modelisation of 
phenomena from physics, economy or whatever, or 
even pure mathematics, and gives nonstandard 
proofs of such results, there is no a priori 
reason to suppose that one must always find an 
equivalent classical proof. There may just not be 
one, simply because there is no way to translate 
the result in classical terms. Nelson, many years 
ago, mentioned some results of such an 
irreducible nature in brownian motion, and the 
french school around Reeb's descendants has found 
other such results pertaining to optics and to quantic mechanics.
At 16:01 10-02-2006, you wrote:

>Nonstandard analysis can be done in a formal system
>that is a conservative extension of standard set
>theory ZFC.  For example, Hrbacek set theory HST,
>which is a powerful version of nonstandard analysis,
>is equiconsistent with ZFC, and is conservative with
>respect to standard theorems of ZFC (see
>"Nonstandard Analysis, Axiomatically" by Kanovei
>and Reeken, p. 20).  Similar theorems can be
>proved for other theories such as Nelson's Internal
>Set Theory IST.
>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.
>Incidentally,  Dana Scott pointed out
>how expensive the book by Kanovei and Reeken is, as well
>as another book I mentioned earlier.
>However, Dover has recently
>re-issued three excellent books on nonstandard analysis in
>their admirable mathematics catalogue:
>"Infinitesimal Calculus" by Henle and Kleinberg -- an
>introductory calculus text using the hyperreals;
>"Nonstandard Analysis" by Alain Robert -- an
>introductory text using Nelson's IST, but including
>some more advanced material such as the Bernstein-
>Robinson theorem on invariant subspaces;
>"Applied Nonstandard Analysis" by Martin Davis --
>includes some advanced material on Hilbert space
>and real analysis.
>FOM mailing list
>FOM at cs.nyu.edu

A.J. Franco de Oliveira
Dep. Matematica
Univ. de Evora - CLAV
R. Romao Ramalho

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