From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4350 Path: news.gmane.org!not-for-mail From: "Al Vilcius" Newsgroups: gmane.science.mathematics.categories Subject: exploiting similarities and analogies Date: Mon, 31 Mar 2008 15:51:39 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019888 12870 80.91.229.2 (29 Apr 2009 15:44:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:48 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Mar 31 21:52:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 31 Mar 2008 21:52:29 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JgUVu-0001e1-A7 for categories-list@mta.ca; Mon, 31 Mar 2008 21:38:10 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 114 Original-Lines: 55 Xref: news.gmane.org gmane.science.mathematics.categories:4350 Archived-At: Dear Categorists, Has anyone explored, either formally or informally, the connection betwee= n the Melzak Bypass Principle (MBP) and adjoints? The MBP (aka "the conjugacy principle" which embraces and generalizes Jacobi inversion) Ref MR696771 =20 http://www.ams.org/mathscinet/pdf/696771.pdf (and no, it does not appear in either Wikipedia or PlanetMath, yet) is somewhat heuristic in character, suggesting : Transform the problem (T), Solve(S), Transform back(T^1), as a "bypass" given by (T^1)ST, which looks like conjugation. Melzak himself refers to adjoints (quite tangentially) as "being bypasses= , though dressed up and served forth exotically" p.106 ibid. (I do recall that adjoints were generally seen as pretty exotic in the early 1970's when I was a graduate student at UBC, to my great chagrin). The MBP is acclaimed in MM vol.57 No.3 May 1984 as "a device for exploring analogies", or as "a dazzling attempt to comprehend complexity". Perhaps "bypass" could also be seen in the words of W.W. Tait (1996) "the propositions about the abstract objects translate into propositions about the things from which they are abstracted and, in particular, the truth o= f the former is founded on the truth of the latter". http://home.uchicago.edu/~wwtx/frege.cantor.dedekind.pdf . For me, "bypass" is a kenning for quasi-inverse or ad joint pairs. Motivation for seeking such a connection is not really to revive a 25 or 30 year old idea (as brilliant as Melzak's insights were, of course), but rather "to facilitate invention and discovery" (in Melzak's own words), and to find additional (as well as interdisciplinary) sources of instance= s of adjoints, possibly as a way to make adjoints more immediately relevant in any introductory discussion of categories, since, of course, adjoints are undoubtedly one of the most successful concepts within category theory. Further inspiration could be found in the Brown/Porter "Analogy" paper http://www.bangor.ac.uk/~mas010/eureka-meth1.pdf and others, along with a passion for invention and discovery through the continued pursuit of Unity and Identity of Opposites (UIO) - obviously referring to Bill Lawvere. Furthermore, I would also like to see this connection developed for practical reasons, applied to various situations, in particular to the structure of the www as an anthropomorphic creation that could benefi= t from further categorical perspective, given by the learned categorists I respect the most. I look forward to your thoughts and comments. ..... Al Al Vilcius Campbellville, ON, Canada