From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5243 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: pragmatic foundation Date: Thu, 12 Nov 2009 00:25:45 -0800 Message-ID: <4AFBC689.6000009@cs.stanford.edu> References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1258047100 20759 80.91.229.12 (12 Nov 2009 17:31:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 17:31:40 +0000 (UTC) To: categories list Original-X-From: categories@mta.ca Thu Nov 12 18:31:33 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 1N8dW8-0003SB-95 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 18:31:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8d0q-0000Tv-5D for categories-list@mta.ca; Thu, 12 Nov 2009 12:59:12 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5243 Archived-At: Colin McLarty wrote: > [responding to Manin thoughts] 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. Thanks, Colin. There I was nicely calmed down and then you got me all worked up again. :) I prefer the Euclidean plane over sets as a suitable starting point for understanding mathematics. What advantage is there to making geometry rest on set theory as opposed to vice versa? What is wrong with starting from a geodesic space as a place where it is always determined, given two points, what is the next one, subject to some simple equational principles? This is a common basis for the second postulate of Book I of Euclid's *Elements*, Newton's first law of motion, Einstein's theory of general relativity that a falling body is merely following a geodesic in a space curved by a nearby mass, and the notion of Hamiltonian flow of a vector field for an energy function defined on the cotangent space of a manifold as an expression of the principle of least action. In this framework a *set* is simply a geodesic space where the next point after x and y is x. (So if I ask what is the next element in the sequence 3,4,... the answer is 3, not 5.) More on this at http://boole.stanford.edu/pub/consgeom.pdf . A geodesic space or geode, aka kei, is related to a quandle (see http://en.wikipedia.org/wiki/Quandle ), the difference being that for abelian groups, quandles are merely sets whereas flat geodes (those satisfying Euclid's 5th postulate) form a symmetric monoidal closed category fully and reflectively extending Set (properly of course). Moreover its subdirect irreducibles are those of Ab except for those of even order as per the last slide. Quandles are for knot theory, not geometry. The difference between sets and geodesic spaces in foundations is like the difference between scales and Fur Elise for piano students. Both are good ways to get started but the second is more interesting. (Apologies again to Eduardo for my impenetrable writing, in this case I can only counsel patience since these ideas seem to come with a certain viscosity that inhibits any royal road of the kind Eduardo would like.) Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]