From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5339 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: A well kept secret? Date: Wed, 09 Dec 2009 07:40:43 +0000 Message-ID: Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1260370003 23209 80.91.229.12 (9 Dec 2009 14:46:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 9 Dec 2009 14:46:43 +0000 (UTC) To: "categories@mta.ca" Original-X-From: categories@mta.ca Wed Dec 09 15:46:36 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 1NINoJ-0000kL-8m for gsmc-categories@m.gmane.org; Wed, 09 Dec 2009 15:46:35 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NINLH-0005dX-EA for categories-list@mta.ca; Wed, 09 Dec 2009 10:16:35 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5339 Archived-At: In reply to Andr=E9 : What seems reasonable to do is analysis, namely what is behind the=20 success of category theory and how is this success related to the=20 progress of mathematics. Which implies asking questions of mathematics, some of which have been=20 aired in this discussion list. In this way, it should be possible to=20 avoid seeming partisan, but to ask serious questions, which should help=20 to steer directions, or suggest new ones. Of course lots of great maths=20 does not arise in this way, but by following one's nose, but that does=20 not mean that such analysis of direction is unhelpful. I know some argue that this excursion into what might be called the=20 theory of knowledge, or into methodology, seems unnecessary to some. In=20 reply I sometimes point to remarks of Einstein on my web site www.bangor.ac.uk/r.brown/einst.html or more mundanely retort that normal activities normally require some=20 meta discussion: if you want to go on a holiday, you do some planning,=20 you don't just rush to the station and buy some tickets. I develop this=20 theme in relation to the teaching of mathematics in an article What should be the output of mathematical education? on my popularisation and teaching page. I gave a talk to school children on `How mathematics gets into knots' in=20 the 1980s, and a teacher came up to me afterwards and said: `That is the=20 first time in my mathematical career that anyone has used the word=20 `analogy' in relation to mathematics.' Yet abstraction is about analogy,=20 and very powerful it is too. This was part of the motivation behind the=20 article 146. (with T. Porter) `Category Theory: an abstract setting for analogy=20 and comparison', In: What is Category Theory? Advanced Studies in=20 Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf There is also interest in the question of how category theory comes to=20 be successful, and more successful than, say, the theory of monoids.=20 This seems connected with the underlying geometric structure being a=20 directed graph, i.e. allowing a `geography of interaction'. A category=20 is also a partial algebraic structure, with domain of definition of the=20 operation defined by a geometric condition. Is this enough to explain=20 the success? It is worth noting that the article Atiyah, Michael, Mathematics in the 20th century, Bull. London Math.=20 Soc., {34}, {2002}, 1--15, suggests that important trends in the 20th century were: higher dimensions, commutative to non=20 commutative, local-to-global, and the unification of mathematics, but does not include the words `category' or `groupoid', let alone=20 `higher dimensional algebra'! This kind of analysis needs to be presented to other scientists, and to=20 the public, not only to mathematicians. There is a hunger for knowing=20 what mathematics is really up to, in common language as far as possible,=20 what new concepts, ideas, etc., and not just `we have solved Fermat's=20 last theorem'. If your analysis of what category theory should do suggests some gaps,=20 then that is an opportunity for work! Good luck Ronnie Brown Joyal wrote: Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directio= ns of research. It is probably the most important mathematical developpement of the 20th = century. But we cant say that publically. Andr=E9 Joyal =20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]