From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5235 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: pragmatic foundation Date: Wed, 11 Nov 2009 11:38:17 -0500 Message-ID: Reply-To: Colin McLarty NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1257994150 29602 80.91.229.12 (12 Nov 2009 02:49:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 02:49:10 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Thu Nov 12 03:48:55 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 1N8Pjy-0002vb-Rc for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 03:48:55 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8PH9-0007b6-Qv for categories-list@mta.ca; Wed, 11 Nov 2009 22:19:07 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5235 Archived-At: 2009/11/6 Andre Joyal : writes > I invite everyone to read the interesting interview of Yuri Manin > published in the November issue of the Notices of the AMS: Manin is always entertaining but not very careful about what he says. Andr=E9 says: > The foundational framework of Bourbaki is very much in the tradition > of Zermelo-Fraenkel, Godel-Bernays and Russell. > I am aware that Bourbaki was more interested in the development of > mathematics than in its foundation. I agree. Naturally Bourbaki was in a better situation to make up a system that would work, since they had the others behind them. And still their system did not work in fact. Russell was more concerned with philosophic issues of logic, but his touchstone for logic was that it should work! (He was very clear about this by 1919, in his Principles Of Mathematical Philosophy.) He knew a lot less than Zermelo about what would work for two reasons: Russell got into it much earlier, and Russell studied math as a philosopher at Cambridge while Zermelo studied it as a mathematician with Hilbert in G=F6ttingen and in debates with Poincar=E9. All these people sought a foundation that would make sense in itself and would work. Naturally they had different emphases, partly shaped by the different resources they could draw on. Russell, Zermelo, and G=F6del all read each other (recalling that Russell was 59 years old, and two decades past his work on logic, when G=F6del published the incompleteness theorem, and everyone took years absorbing it). > In the interview, Manin also said that: > >>And so I don=92t foresee anything extraordinary >>in the next twenty years. Of course we do not expect to *foresee* extraordinary things. >> Probably, a rebuilding of what I call the =93pragmatic >> foundations of mathematics=94 will continue. That is a pretty safe bet. >>By this I mean simply a >>codification of efficient new intuitive tools, such >>as Feynman path integrals, higher categories, the >>=93brave new algebra=94 of homotopy theorists, as >>well as emerging new value systems and accepted >>forms of presenting results that exist in the minds >>and research papers of working mathematicians >>here and now, at each particular time. Yes, there will be progress on all of these things. 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"! best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]