FOM: infinitesimals, infeasible numbers, f.o.g.
Rick Sommer
sommer at Csli.Stanford.EDU
Wed Nov 12 00:30:01 EST 1997
This note is response to comments on nonstandard analysis made in
postings by Machover (Nov 6,11), Barwise (Nov 7), Franzen (Nov 8),
Davis (?), and Friedman (Nov 11), as well as the discussion of
feasible numbers that has been ongoing (e.g., Sazonov), and various
mentions of foundations of geometry and geometrical foundations for
mathematics (see p.s.s. below).
In e-mail of Tue, 11 Nov 1997 05:35:35 +0100, Harvey Friedman
indicates that he tends to side with the following view of
non-standard analysis:
"There is no fundamental notion of infinitesimal and/or infinitesimal
reasoning. It is simply a confused way of thinking which people can get by
with at an elementary level. It was replaced by something more coherent and
powerful...."
But, intuitions about infinitesimals and infinite natural numbers had
extremely important roles in the development of analysis, and those
intuitions are put to frequent use by applied mathematicians,
physicists and engineers. A careful look might reveal that the
"coherence and power" of epsilon-delta arguments is actually inferior
to the power the mathematician has to reason about infinitesimals
all-the-while being consistent with observed reality. This can't be
overlooked.
Friedman's justification for his view is based on the fact that "the
reals are unique" and "all nonstandard models of arithmetic are
pathological. There is nothing unique about any of them." Barwise
raises the same issue (Fri, 7 Nov 1997 08:21:16); this is a standard
criticism. So, in some respects, nonstandard models yield an
unnecessarily complicated picture of mathematical reality.
My way of confronting this is to first recognize the great value of
the mathematical arguments (formal and informal) that use
infinitesimal and infinite numbers (now justified by Robinson's
landmark work), and then set out to understand how we get our
intuitions about these objects and how these objects fit in with other
mathematical objects that seem more grounded in reality.
Trying to pick the correct model of nonstandard arithmetic doesn't
seem to give us the sought after understanding (and Harvey gives good
reasons for that). Casting the world of infinitesimal and infinite
numbers in ultraproducts does not help with this problem either (so
when connecting infinitesimals to reality, I think we should ignore
Robinson's development).
To me the best explanation for these objects is through identifying
infinite integers with something akin to the "infeasible numbers" that
have been discussed in recent postings. Roughly, the idea is to
finitize the empirically relevant part of mathematical reality
(including infinitesimal and infinite numbers). Since the empirically
relevant parts of the multitude of nonstandard models are all
isomorphic, the problem of many distinct models is no longer an issue.
Together with Pat Suppes, I am working on problems of this type (see
references below).
So clearly I'm in agreement with Machover in questioning Franzen's
claim: "non-standard analysis as it now exists is not a foundational
subject." and I agree with Goedel in the quote cited by Machover from
the preface in the second (1977) edition of Robinson's book:
" ... there are good reasons to believe that NSA, in some version or
other, will be the analysis of the future."
--Rick Sommer
P.S. The following references are a start at addressing these
questions; the plan is to develop much further the ideas contained
therein.
Sommer and Suppes, Dispensing with the Continuum, Journal of
Mathematical Psychology, Vol. 41, No. 1, March 1997.
Sommer and Suppes, Finite Models of Elementary Recursive Nonstandard
Analysis. Notas de la Sociedad Matematica de Chile, Vol. 15, No. 1,
Jan. 1996.
These can be found in postscript and dvi form at
ftp://gauss.stanford.edu/pub/papers/sommer/
(get there from http://math.stanford.edu.)
P.S.S. Another related foundational issue, that Suppes and I are
working on, is connected to the idea that the intuitive concept of
infinitesimal is a geometric one, and so a theory of infinitesimals
could provide a geometric (in place of arithmetic) foundation for
analysis.
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