[FOM] Non-standard analysis
Alasdair Urquhart
urquhart at cs.toronto.edu
Mon Jul 28 11:21:45 EDT 2003
I posted the apparently paradoxical
assertion that there are results in
the theory of stochastic processes that
can be proved only by non-standard
methods. This seems to conflict with well-known
conservative extension results, so Steve
Simpson and Harvey Friedman rightly called
me on this.
I quote from the new monograph: "Model Theory
of Stochastic Processes" by Fajardo and Keisler,
(Lecture Notes in Logic), p. x.
"From the viewpoint of nonstandard analysis, our
aim is to understand why there is a collection of
results about stochastic processes which can only
be proved by means of nonstandard analysis.
Our explanation is that these applications require
adapted spaces with properties such as saturation
and homogeneity which are easy to get using nonstandard
methods but difficult or impossible without using them."
So, the explanation for the apparent paradox is this.
Standard stochastic process theory is interested in
the properties of distributions, but does not much
care about the space where these processes live.
Hence, we can get better results by using things
like saturated adapted spaces constructed by the
Loeb construction.
In other words, nonstandard analysis provides us not
just with short proofs of old theorems, but also with
new and interesting objects, that help in proving new
results. I stoutly adhere to my assertion that
nonstandard analysis is a major foundational advance.
Of course, Goedel and Turing are among my heroes, but so
is Abraham Robinson. I can call to my defence no less
a person that Kurt Goedel himself. Goedel wanted
Robinson to be his successor at the Institute for
Advanced Study, and in addition, allowed Robinson
to publish some very interesting (and highly positive) remarks of his on
non-standard analysis (see his Collected Works, Volume
II, pp. 307-311). These remarks (which Robinson
printed in the 2nd edition of his "Non-standard
Analysis") are the published version of comments that
Goedel made after a talk of Robinson at the IAS.
Some of the background to these remarks is explained
in a very funny and illuminating anecdote of Bill
Howard that is published in the interview by Amy
Shell-Gellasch in Volume 25, No. 1 of the
Mathematical Intelligencer (p. 41). I should add
that the whole interview with Bill Howard is a real
treat.
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