From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2038 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: Sketches and Platonic Ideas Date: Wed, 05 Dec 2001 04:36:21 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text X-Trace: ger.gmane.org 1241018361 2157 80.91.229.2 (29 Apr 2009 15:19:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:19:21 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Dec 5 16:16:53 2001 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Dec 2001 16:16:53 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 3.33 #2) id 16BiSe-0002VD-00 for categories-list@mta.ca; Wed, 05 Dec 2001 16:16:08 -0400 X-Originating-IP: [128.205.249.50] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 59 Xref: news.gmane.org gmane.science.mathematics.categories:2038 Archived-At: Certainly I did not mean to suggest that either John or Andree were supporting platonism as a philosophy of mathematics. In fact I had momentarily even forgotten that John had used the term. In my 1972 Perugia Notes I had made an attempt to characterize the relation between these sorts of mathematical considerations and philosophy by saying that while platonism is wrong on the relation between Thinking and Being, something analogous is correct WITHIN the realm of Thinking. The relevant dialectic there is between abstract general and concrete general. Not concrete particular ("concrete" here does not mean "real").There is another crucial dialectic making particulars (neither abstract nor concrete) give rise to an abstract general; since experiments do not mechanically give rise to theory, it is harder to give a purely mathematical outline of how that dialectic works, though it certainly does work. A mathematical model of it can be based on the hypothesis that a given set of particulars is somehow itself a category (or graph), i.e., that the appropriate ways of comparing the particulars are given but that their essence is not. Then their "natural structure" (analogous to cohomology operations) is an abstract general and the corresponding concrete general receives a Fourier-Gelfand-Dirac functor from the original particulars. That functor is usually not full because the real particulars are infinitely deep and the natural structure is computed with respect to some limited doctrine; the doctrine can be varied, or "screwed up or down" as James Clerk Maxwell put it, in order to see various phenomena. From: baez@math.ucr.edu >To: categories@mta.ca (categories) >Subject: categories: Sketches and Platonic Ideas >Date: Mon, 3 Dec 2001 19:42:40 -0800 (PST) > >Toby Bartels writes: > >> There could be multiple ideas that generate the same sketch; >> how do we decide which is the correct idea among equivalent ones? >> OTOH, if we take equivalence classes of ideas, then we're taking sketches. ... >> who has the right idea? > > I'm confused: in my understanding, a sketch basically amounts to ... >By the way, in response to Lawvere's comments: > > My use of the term "Platonic idea of X" for the free ... >versus concrete particulars. >Best, >John Baez