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From: westling@cvl.UUCP (Mark Westling)
Newsgroups: net.philosophy
Subject: Re: Metaphysics
Message-ID: <761@cvl.UUCP>
Date: Thu, 22-Aug-85 15:10:27 EDT
Article-I.D.: cvl.761
Posted: Thu Aug 22 15:10:27 1985
Date-Received: Sun, 25-Aug-85 01:31:02 EDT
References: <969@sphinx.UChicago.UUCP> <608@mmintl.UUCP>
Reply-To: westling@cvl.UUCP (Mark Westling)
Organization: Computer Vision Lab, U. of Maryland, College Park
Lines: 56
Summary: Are numbers real?

Keywords:

In article <608@mmintl.UUCP> franka@mmintl.UUCP (Frank Adams) writes:

>  Mathematical philosophers basically fall into three groups.  One is the
>  formalists, who insist they are just manipulating rules, and that any
>  applicability to anything else is strictly incidental.

>  The intuitionists believe that mathematical objects (numbers, for short),
>  are solely constructs of the mind.  ...

>  The third group, which is probably the largest, thinks numbers are real,
>  but doesn't really know what they are.  A good answer to that question
>  would be most appreciated.

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.

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

Formalists dislike this notion because they believe that logic is generally
less certain than mathematics; intuitionists dispute it because it allows
the existence of propositions which are unprovable (e.g.  with Goedel's
incompleteness theorem, or even quantification with infinite or just
unmanageably big classes:  how can you claim "all X are Y" if you can't test
every X empirically?).  Both provide alternatives to a logical or set
theoretical foundation of mathematics, but neither support a special,
"Platonic" existence of numbers.  (At least, that's how I understand it.)

It seems to me that there is more room for ontological investigation in the
philosophy of logic (especially quantification) than in the philosophy of
mathematics.  Any thoughts?


-- 
-- Mark Westling

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