From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/328 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Wed, 5 Mar 1997 11:13:00 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016894 25209 80.91.229.2 (29 Apr 2009 14:54:54 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:54:54 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Mar 5 11:13:55 1997 Original-Received: by mailserv.mta.ca; id AA20713; Wed, 5 Mar 1997 11:13:02 -0400 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:328 Archived-At: Date: Wed, 05 Mar 1997 00:56 -0500 (EST) From: Fred E J Linton <0004142427@mcimail.com> Any philosophy category theory may have would have at its core, I think, the notion that mathematical objects are known *not* in isolation but in the context of their comrades. The group of rational integers, accompanied *only* by its identity map, and the Thom space of the tangent bundle of some exotic manifold, accompanied once again *only* by its identity map, are, as categories, indistinguishable. Plucked out of their original contexts, there is no longer any social setting where one can find any difference between them that really *makes* a difference. According to some other views of mathematics, the group of rational integers, that particular Thom space, the real number {pi}, and my current left shoe, all have unique mathematical personalities that let them be "obviously" distinguished one from another, without any reference even to what I would call their "natural ambient environments". >>From my perspective, admittedly that of a categorist, these views result from a simple failure to recognize that what passes for the "intrinsic structure" of a mathematical object is in fact nothing more (nor less) than a clear understanding of its relations with its mates, of roughly similar character, in some category (that "went without saying") they all jointly inhabit -- even the phrase "roughly similar character" is justifiable *only* by virtue of the fact that they *do* all inhabit some same category. I hope I'm actually making myself clear, and not just preaching to the converted. -- Fred