From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5244 Path: news.gmane.org!not-for-mail From: Andre.Rodin@ens.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: pragmatic foundation Date: Thu, 12 Nov 2009 12:42:22 +0100 Message-ID: Reply-To: Andre.Rodin@ens.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1258047170 21079 80.91.229.12 (12 Nov 2009 17:32:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 17:32:50 +0000 (UTC) To: Colin McLarty , categories@mta.ca Original-X-From: categories@mta.ca Thu Nov 12 18:32:43 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N8dXA-000464-L7 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 18:32:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8d3t-0000sh-5E for categories-list@mta.ca; Thu, 12 Nov 2009 13:02:21 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5244 Archived-At: Selon Colin McLarty : > I myself am also confident that people will calm down and notice that > axiomatic categorical foundations such as ETCS and CCAF work perfectly > well, in formal terms, and relate much more directly to practice than > any earlier foundations. One hundred and fifty years of explicitly > foundational thought has made this progress possible. By now, that > can hardly qualify as "extraordinary"! > I do NOT believe that ETCS and CCAF "work perfectly well". Each of these = involve two foundational "layers", namely, the classical "bottom" and a categoric= al "superstructure". By the classical bottom I mean NOT an underlying Set th= eory but the "Elementary theory of categories" (ETC), i.e. a theory of categor= ies using the usual First-Order Logic (FOL) and relying on the standard Hilbert-Tarski-style axiomatic method. I agree with John Mayberry and som= e other people who argue that this aximatic method alone assumes a basic no= tion of set or collection. Unlike Mayberry I don't think that this fact implie= s that the project of categorical foundations, as a alternative to and replaceme= nt for set-theoretic foundations, is futile. Recall that the axiomatic method we= are talking about (which is, of cause, quite different from Euclid's method a= nd other earlier versions of axiomatic method) emerged together with Set the= ory. In order to make categorical foundations into a viable alternative of set-theoretic foundations we still need to provide Category theory with a= new axiomatic method rather than use the older axiomatic method as do ETCS an= d CCAF. Elements of this prospective axiomatic method are found in what I j= ust called the "categorical superstructure" of ETCS and CCAF but as far as th= ese theories are concerned the classical background (FOL+ETC) is indispensabl= e. This is why I say that ETCS and CCAF do NOT work perfectly weel as catego= rical foundations. Building of "purely categorical" foundations remains an open problem. It = is not a matter of a ideological purity but a matter of complete "rebuilding" (M= anin's word) of foundations: in my view, such a rebuilding is healthy and refres= hing in any circumstances (unless it clashes severely with practice). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]