Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/5/84; site cvl.UUCP
Path: utzoo!linus!philabs!cmcl2!seismo!umcp-cs!cvl!westling
From: westling@cvl.UUCP (Mark Westling)
Newsgroups: net.philosophy
Subject: Re: Metaphysics
Message-ID: <772@cvl.UUCP>
Date: Tue, 27-Aug-85 00:31:36 EDT
Article-I.D.: cvl.772
Posted: Tue Aug 27 00:31:36 1985
Date-Received: Thu, 29-Aug-85 07:25:33 EDT
References: <969@sphinx.UChicago.UUCP> <608@mmintl.UUCP> <761@cvl.UUCP>
Reply-To: westling@cvl.UUCP (Mark Westling)
Organization: Computer Vision Lab, U. of Maryland, College Park
Lines: 35
Summary: philosophy of mathematics

I am posting the following for a friend.  Send replies to John McLean
(arpanet: mclean@nrl-css, uucp: ...decvax!nrl-css!mclean).

------------------------------------------------------------------------------
Let's keep our epistemology and ontology straight.  Frank Adams was
giving the three traditional ontological views of numbers:  they don't
exist, they are mental constructions, and they are real. Frege and Quine both
fall into the third camp, but they disagree on mathematical epistemology.
For Frege numbers are logical objects; we know that arithmetic is true
because the laws of arithmetic are reducible to the laws of logic.  Of course,
his reduction was plagued by two problems: (1) his concept of logic was
larger than what is generally accepted as logic today (he included set
theory) and (2) his axiomatization of set theory was inconsistent.  Russell
solved the second problem with the theory of types, but not the first.  For
Quine, numbers exist on the grounds that Beth Christie echoed in a different
context:  our best theories are committed to them.  Physics needs arithmetic
and arithmetic says, among other things, that there is a number between 1 and
3.  We know about numbers just like we know about any other type of
object: we construct theories about them.  For people who like "isms",
Frege's view is called logicism and Quine's is called empiricism.  The
latter has a distinguished hitory dating at least as far back as Mill,
and is still a hot topic as can be seen by reading Hartry Field's latest
book (whose title is something like SCIENCE WITHOUT NUMBERS).

John McLean
...!decvax!nrl-css!mclean or
mclean@nrl-css
----------------------------------------------------------------------------

-- 
-- Mark Westling

ARPA: westling@cvl         CSNET: westling@cvl
UUCP: ...!{seismo,allegra}!umcp-cs!cvl!westling