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Path: utzoo!linus!philabs!pwa-b!mmintl!franka
From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.philosophy
Subject: Re: Metaphysics
Message-ID: <620@mmintl.UUCP>
Date: Mon, 26-Aug-85 21:02:52 EDT
Article-I.D.: mmintl.620
Posted: Mon Aug 26 21:02:52 1985
Date-Received: Thu, 29-Aug-85 23:42:53 EDT
References: <969@sphinx.UChicago.UUCP> <608@mmintl.UUCP> <761@cvl.UUCP>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 37
Summary: mathematics is not reducible to logic


In article <761@cvl.UUCP> westling@cvl.UUCP (Mark Westling) writes:
>
>From my limited knowledge of mathematical philosophy, it looks like the
>biggest group wasn't mentioned.
>
>The idea that numbers exist on their own was put forth last (I think) by
>Kant.  Kant believed that mathematical reasoning could not be derived from
>logic, but rather from intuitive, a priori notions of time and space:
>arithmetic arises from time, and geometry arises from space.  A major point
>was his claim that the figure is essential to all geometric proofs.  He also
>believed in the necessity of Euclid's axioms, which are naturally based on
>intuition.

Actually, this is the idea I didn't mention.  One can believe in mathematics
as a description of the real world; but this leaves little room for set
theory.

>The school of thought which was not mentioned in the original posting is the
>one founded by Frege and expanded by Russell.  Two major results provided
>the incentive for dismissing Kant's ideas.  Riemann and Lobachevsky showed
>that, in pure mathematics, non-Euclidean geometry also works, so there is no
>a priori need for Euclid's axioms.  What happens in the real world is
>irrelevant; we're talking strictly about mathematical truth.  Peano
>strengthened symbolic logic and set theory.  Frege took the next step and
>reduced mathematics to logic.  Russell continued along these lines, adding
>that the set theory used in his constructions was also reducible to logic.

Sorry, set theory is *not* reducible to logic.  The only ones who are willing
to treat it as such are the formalists -- who don't care about the existence
of the mathematical objects they deal with.  If one takes set theory as the
foundation, the original question remains: what kind of existence do sets
have?

>Recently, however, Quine and others have argued that the essential part is
>the set theory itself, not the logic.

Actually, I think you need both.