From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/329 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 11:13:50 -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 25212 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 11:14:01 1997 Original-Received: by mailserv.mta.ca; id AA17746; Wed, 5 Mar 1997 11:13:50 -0400 Original-Lines: 41 Xref: news.gmane.org gmane.science.mathematics.categories:329 Archived-At: Date: Wed, 5 Mar 1997 10:54:19 +0000 From: Steve Vickers > Constructive mathematics is a philosophy. Category theory is not. > The question doesn't even type-check. > > Of course they're different. Philosophy is the love of wisdom; type-checking is not. Of course category theory has its philosophy. To me it's "all things are connected" - you cannot fully describe anything purely in itself but only by the way it connects with others. Category theory makes the connections explicit (as morphisms) and then characterizes things by their universal properties. The philosophy plays a real role in categorical practice: for instance, in the idea that isomorphism between objects is more important than equality, which is not something that can be meaningful just in terms of the formal mathematics. The philosophy also yields a criterion for evaluating the theory: Is categorical structure adequate for describing the connections that we actually find? The strength of the categorical view of "connection structure" is amply confirmed by the power of the universal properties it can express (compare it with, say, graph theory); but if it does fail us anywhere, how might it advance beyond its present formalization? (There is already a plausible answer here: topology has a different way of describing the connections between a point and its neighbours, and the categorical and topological approaches combine to make topos theory.) I hesitate to try to reduce the philosophy of constructive mathematics to a single pithy phrase, not least because there are different schools of constructivism with apparently different philosopies. I shall therefore duck the question of comparing "the philosopies of constructive mathematics and category theory", but I don't believe it's a meaningless one. Steve Vickers.