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From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.philosophy
Subject: Re: Metaphysics
Message-ID: <623@mmintl.UUCP>
Date: Mon, 26-Aug-85 21:58:44 EDT
Article-I.D.: mmintl.623
Posted: Mon Aug 26 21:58:44 1985
Date-Received: Thu, 29-Aug-85 23:44:05 EDT
References: <969@sphinx.UChicago.UUCP> <608@mmintl.UUCP> <481@spar.UUCP>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 42
Summary: 


In article <481@spar.UUCP> ellis@spar.UUCP (Michael Ellis) writes:
>
>    Bertrand Russell's analysis of the nature of numbers is the most
>    appealing I've heard. He starts with the classical paradox..
>
>	I have red apples     => Each apple was red
>	I have ten fingers    => Each finger was ten???
>
>    From Russell's History of Western Philosophy (Simon and Schuster):
>
>    "The complete answer, as regards propositions in which `ten' occurs is
>    "that, when these propositions are fully analyzed, they are found to
>    "contain no constituent corresponding to the word `ten'. To explain
>    "this in the case of such a large number as ten would be complicated; let
>    "us therefore, substitute
>     		   `I have two hands'
>    "This means:
>
>	"There is a such that there is b such that a and b are not identical
>	"and whatever x may be, `x is a hand of mine' is true when, and only
>	"when, x is a or x is b
>
>    "Here the word `two' does not occur. It is true that two letters a and b
>    "occur, but we do not need to know that they are two.. Thus numbers are,
>    "in a precise sense, formal.
>
>    It is left as an exercise for the diligent reader to remove any
>    numerical reference from:
>
>       	       	  `I have ten fingers' 
>
>-michael

This works reasonably well for the natural numbers, and can be extended
without undo difficulty to the rational numbers and even the real numbers
(although the exact meaning in the latter case is not obvious).  It fails
when applied to, say, the complex numbers; and fails utterly when the
discussion turns to set theory.

There are those who only believe in mathematical objects for which a
description of this type can be given.  They are called finitists, and
they accept even less of modern mathematics than the intuitionists do.