Path: utzoo!mnetor!uunet!lll-winken!lll-lcc!ames!hao!gatech!udel!princeton!mind!greg
From: greg@mind.UUCP (greg Nowak)
Newsgroups: sci.philosophy.tech
Subject: Re: Classifying the Axiom of Choice
Message-ID: <1938@mind.UUCP>
Date: 25 Feb 88 20:15:53 GMT
References: <7123@agate.BERKELEY.EDU> <8768@sunybcs.UUCP> <9734@shemp.CS.UCLA.EDU> <1937@mind.UUCP>
Reply-To: greg@mind.UUCP (greg Nowak)
Organization: Cabal of Fools
Lines: 31

In article <9734@shemp.CS.UCLA.EDU> hbe@math.ucla.edu (H. Enderton) writes:
>In article <8768@sunybcs.UUCP> rapaport@gort.UUCP (William J. Rapaport) writes:
>>I guess you [weemba@garnet.berkeley.edu have not read Kant.  All mathematical
>>propositions are synthetic apriori for him.  By that, I'd take the Axiom of
>>Choice as synthetic apriori, too.
>
>Kant's classification of mathematical truths as synthetic a priori seems
>in light of Godel to be more wrong than right.  The a priori statements
>form merely a recursively enumerable set, and hence cannot include even
>all the true statements of arithmetic.  I think that the analytic
>a posteriori is an unjustly neglected category in the philosophy of
>mathematics.  I am not sure, however, that I want to put the axiom of
>choice into it.

The "analytic a posteriori" category is neglected because it is EMPTY.
Anything which is analytic, i.e. true for semantic reasons, is a
priori (independent of experience) -- even if the terms refer to
objects we usually encounter with experience. Thus "all bachelors are
male" is an analytic a priori statement, because it is true by the
meanings of the words -- even though we DO have real-world experience
of males and bachelors, we NEED not to judge the truth of the
statement. "The sky is blue" is a synthetic a posteriori proposition,
because nothing in the definition of what a sky is requires it to be
blue. The suggestion about enumerating the set of propositions would
seem to be more applicable to the *analytic* propositions than the a
priori ones, thus strengthening the claim that there are synthetic a
priori propositions.

-- 


                              greg