Das Kalenderblatt 100321

20/03/2010 - 13:50 von WM | Report spam
According to Russell, the structure of the infinite and the continuum
were completely revealed by Cantor and Dedekind, and the concept of an
infinitesimal had been found to be incoherent and was “banish[ed] from
mathematics” through the work of Weierstrass and others [1901, pp. 88,
90]. These themes were reiterated in Russell’s often reprinted
Mathematics and the Metaphysician
[1918] and further developed in both editions of Russell’s The
Principles of Mathematics [1903; 1937], the works which perhaps more
than any other helped to promulgate these ideas among historians and
philosophers of mathematics. In the two editions of the latter work,
however, the banishment of infinitesimals that Russell spoke of in
1901 was given an apparent theoretical urgency. No longer was it
simply that “nobody could discover what the infinitely little might
be,” [1901, p. 90] but rather, according to Russell, the kinds of
infinitesimals that had been of principal interest to mathematicians
were shown to be either “mathematical fictions” whose existence would
imply a contradiction [1903, p. 336; 1937, p. 336] or, outright “self-
contradictory,” as in the case of an infinitesimal line segment [1903,
p. 368; 1937, p. 368]. In support of these contentions Russell could
cite no less an authority than Georg Cantor, the founder of the theory
of infinite sets.
Having accepted along with Russell that infinitesimals had indeed been
shown to be incoherent, and that (with the possible exception of
constructivist alternatives) the nature of the infinite and the
continuum had been essentially laid bear by Cantor and Dedekind,
following the development of nonstandard analysis in 1961, a good
number of historians and philosophers of mathematics (as well as a
number of mathematicians and logicians) readily embraced the now
commonplace view that is typified by the following remarks:

In the nineteenth century infinitesimals were driven out of
mathematics once and for all, or so it seemed. [P. Davis and R. Hersh
1972, p. 78]

But ...

The German logician Abraham Robinson (1918–1974), who invented what is
known as non-standard analysis, thereby eventually conferred sense on
the notion of an infinitesimal greater than 0 but less than any finite
number. [Moore 1990; 2001, p. 69]

Indeed ...

nonstandard analysis ..., created by Abraham Robinson in the early
1960s, used techniques of mathematical logic and model theory to
introduce a rigorous theory of both [non-Cantorian] infinite and
infinitesimal numbers. This, in turn, required a reevaluation of the
long-standing opposition, historically, among mathematicians to
infinitesimals in particular. [Dauben 1992a, pp. 113–114]

[Philip Ehrlich: "The Rise of non-Archimedean Mathematics and the
Roots of a Misconception I: The Emergence of non-Archimedean Systems
of Magnitudes1,2", Arch. Hist. Exact Sci. 60 (2006) 1–121]

Und die Moral von der Geschicht? Noch vor ganz kurzer Zeit gab es in
der Mathematik eine fragwürdige Mehrheitsmeinung, die schließlich
überwunden wurde.

Das darf jetzt aber wirklich nimmer vorkommen! Wo bliebe den sonst das
Ansehen der einzigen Wissenschaft mit absoluter Sicherheit?

Gruß, WM

Lesen sie die antworten

#1 Vogel
21/03/2010 - 06:53 | Warnen spam
WM wrote in news:4ee71afe-af8a-4b7e-a90b-

Das darf jetzt aber wirklich nimmer vorkommen!

Hab ich vor kurzem mal im Fernsehen gehört:
"Jetzt darf es aber nicht mehr schneien, es hat schon genug geschneit"

Es ist wohl der grösste Selbstbetrug des Menschen zu glauben er sei ein
rationales Wesen.

Wo bliebe den sonst das
Ansehen der einzigen Wissenschaft mit absoluter Sicherheit?

Ansehen? Von was sprichst du?
Wieviel Dollar ist das wert?

Selber denken macht klug.

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