Subject: [HM] Monsters domesticated and harnessed
From: Moshe' Machover (moshe.machover@kcl.ac.uk)
Date: Fri Aug 25 2000 - 11:57:22 EDT
Robert Tragesser wrote:
> I have a question about the other direction. When did the monsters
> become gods? (As a fractal theorist once put it.) Was Norbert
> Wiener's "discovery" that everywhere continuous, nowhere differentiable
> functions were at the core of the theory of Brownian motions the first?
> More generally, G-C Rota once characterized mathematical
> development thus (in paraphrase): mathematics at any moment is like
> the circus. There is the Big Top. And then there is the Side Show.
> Part of the Side Show is the Freak Show. The thing about the Freak
> Show is that we do not know whether these are the monsters they appear
> to be, or the beginning of an altogether new and splendid form of
> life.
> What other "freaks" of the late 19th/20th century have turned
> out to be real performers?
My favourite case is that of nonstandard models (of arithmetic, analysis
and other formalized standard mathematical theories).
Skolem proved in 1934 the existence on nonstandard models of arithmetic
(although there is a hint about them in his 1921/22 paper, in which he
proves, inter alia, the existence of nonstandard models of what we now
call first-order ZF).
For a long time there was an embarrassed silence. From about 1949/50,
there was a growing interest in such nonstandard models *as monsters*.
In 1960 A Robinson invented nonstandard analysis, in which nonstandard
models (with some special saturation properties) are the heroes.
Skolem, T [1922] "Einige Bemerkungen zur axiomatichen Begruendung der
Mengenlehre". (Engl. trans. in Van Heijenoort's collection, From Frege
to Goedel.)
--------- [1934] "Ueber die Nichct-charakterisierbarkeit der Zahlenrehe
mittels endlich oder abzaehlbar unendlich vieler Aussagen mit
ausschliesslich Zahlenvariablen". Fund. Math 23, 150--161.
Robinson, A [1961] "Non-Standard Analysis". Indagationes Math. 23, 432--440.
Greetings from sunny(!) London,
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