From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!cs.utexas.edu!uunet!trwacs!erwin Tue Jan 28 12:15:04 EST 1992
Article 2961 of comp.ai.philosophy:
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>From: erwin@trwacs.UUCP (Harry Erwin)
Newsgroups: comp.ai.philosophy
Subject: Ideals and Reality
Keywords: plato ideals reality categories
Message-ID: <474@trwacs.UUCP>
Date: 21 Jan 92 18:55:53 GMT
Organization: TRW Systems Division, Fairfax VA
Lines: 25

I have a problem with the concept of platonic ideals as used in modern
philosophy and mathematics. Basically, I don't believe in them. When I
encounter constructivist approaches to mathematics, and when I encounter
non-standard models of arithmetic, I see examples where the standard
model is indistinguishable from other models as far as the arithmetic goes,
but where the superstructure differs markedly. 

Actually, modern mathematics has modernized Plato in the field of Category
Theory, which deals with universal objects, defined as objects, U, that
project onto any object of a given category, and where the projection
operator commutes with the functors of the category. Thus, there is a
universal object that represents integer arithmetic (a commutative ring)
and serves as a platonic ideal for that category. But... that basically
assumes Zorn's Lemma to be true, and if we assume that it is not true, we
end up without a universal object...

The alternative, though, is for much of mathematics to have foundations of
sand.

Cheers,

-- 
Harry Erwin
Internet: erwin@trwacs.fp.trw.com



