Das Kalenderblatt 090929

28/09/2009 - 19:19 von WM | Report spam
I have indicated that Zermelo-Fraenkel set theory does not have a
clear philosophical basis in either platonist or anti-platonist terms.
It therefore becomes reasonable to ask what the consequences would be
of rejecting ZFC as a foundation for mathematics.
Some philosophers may naturally be reluctant to pursue this
question because it could entail having to tell mathematicians that
they are practicing their subject incorrectly. The situation is not
quite as bad as that, since, after all, most mathematicians have
little interest in foundations and may have no particular commitment
to ZFC. (Maddy [Naturalism in Mathematics, 1997] paints a very
different picture, but her “mathematicians” really seem to be set
theorists.) {{Wie kann frau nur so weltfern sein und das noch mit
Realismus (1990) oder Naturalimus kennzeichnen? Bewusste Tàuschung, so
wie die Bezeichnung Realismus in der Philosophie der Mathematik?}}
However, it is now a settled fact that power sets of infinite sets
are not actually
needed for the vast bulk of mainstream mathematics. The philosophical
which admits the natural numbers but not its power set is called
predicativism; it was originally put forward by Bertrand Russell and
Henri Poincaré, and there is a long line of research stretching back
to Hermann Weyl which establishes in detail its ability to encompass
ordinary mathematics. [...] The basic idea is that we accept the
natural numbers and individual real numbers (or equivalently,
individual sets of natural numbers, which can still be pictured in
terms of physical instantiation) but do not assume the existence of a
well-defined set of all real numbers (which cannot be meaningfully
understood in terms of physical instantiation).

Gruß, WM

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#1 K. Schubser
14/11/2009 - 19:45 | Warnen spam

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