From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4575 Path: news.gmane.org!not-for-mail From: "R Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Sun, 14 Sep 2008 11:24:18 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020036 13909 80.91.229.2 (29 Apr 2009 15:47:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:16 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Sep 14 13:58:59 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Sep 2008 13:58:59 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Keuqh-0006zb-Ns for categories-list@mta.ca; Sun, 14 Sep 2008 13:53:23 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 45 Original-Lines: 151 Xref: news.gmane.org gmane.science.mathematics.categories:4575 Archived-At: Dear All, The importance of Bourbaki should be stessed, as it was started when, so = we=20 are told, texts were very bad. There are many beautiful things in the boo= ks:=20 I developed part of an undergraduate course from the account of the=20 classification of closed subgroups of R^n. This relates to old questions = on=20 orbits of the planets, and also gives some nice exercises and even exam=20 questions of a calculation type. It is good to present students with a=20 classification theorem. The difficulties for Bourbaki seem to arise from the presentation (a) as = a=20 final and definitive view in toto, and (b) without enough context, as And= re=20 points out. On (a), there is the old childish joke: what happens if you put worms in = a=20 straight line from Marble Arch to Picadilly Circus? One of them would be=20 bound to wriggle and spoil it all! So some mathematical worms have not on= ly=20 wriggled but grown large and marched off in a different direction. On (b), there is the old debating society tag: text without context is merely pretext. See more questions in Tim and my article on `Mathematics in Context'. What is wrong is to present, or take, the whole account as totally=20 authoritative, and will last indefinitely. What Bourbaki also shows is the value for at least the writers of taking = a=20 viewpoint and following it through as far as it will go: if it seems in t= he=20 end to go too far, or to be inadequate, then that is valuable information= =20 for them and others. See my Dirac quote in `Out of Line'. Ronnie ----- Original Message -----=20 From: "Andre Joyal" To: Sent: Saturday, September 13, 2008 6:17 PM Subject: categories: Re: Bourbaki and Categories Dear Colin, Zoran, Robert, Eduardo and All, I find the present discussion on Bourbaki and category theory very=20 important. I recall asking the question to Samuel Eilenberg 25 years ago and more=20 recently to Pierre Cartier. If my recollection is right, Bourbaki had essentially two options: rewrit= e=20 the whole treaty using categories, or just introduce them in the book on homological algebra, The second option won, essentially because of the enormity of the task of= =20 rewriting everything. Other factors may have contributed on a smaller scale, like some unresolv= ed=20 foundational questions. In any cases, it was the beginning of end for Bourbaki. Bourbaki was a great humanistic and scientific enterprise. Advanced mathematics was made available to a large number of students, possibly over the head of their bad teachers. It defended the unity and rationality of science in an age of growing irrationalism (it was conceived in the mid thirties). I have personally learned a lot of mathematics by reading Bourbaki. Everything was proved, and the proofs were logically very clear. It was a like a continuation of Euclid Elements two thousand years later= ! But after a while, I stopped reading it. I had realised that something important was missing: the motivation. The historical notes were very sketchy and not integrated to the text. I remember my feeling of frustration in reading the books of functional=20 analysis, because the applications to partial differential equations were not=20 described. Everything was presented in a deductive order, from top to down. We all know that learning is very much an inductive process, from the particular to the general. This is true also of mathematical research= . Bourbaki is dead but I hope that the humanistic philosophy behind the=20 enterprise is not. Unfortunately, we presently live in an era of growing irrationalism. Science still needs to be defended against religion. Civilisation maybe at a turning point with the problem of climate change. Millions of people need and want to learn science and mathematics. Should we not try to give Bourbaki a second life? It will have to be different this time. Possibly with a new name. Obviously, internet is the medium of choice. What do you think? Andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de Colin McLarty Date: ven. 12/09/2008 14:46 =C0: categories@mta.ca Objet : categories: Re: Bourbaki and Categories From: zoran skoda Date: Friday, September 12, 2008 2:06 pm wrote, among other things > main points of departure. The remark that as a proponent of > "structures" Bourbaki > had to include categories is anyway a bit lacking an argument. > First of all, because > of the size problems one can not take big categories on equal > footing with, say groups, > and considering only small categories would be strange and lacking > most interesting examples. The claim is not that Bourbaki should have studied categories as structures. It is that Bourbaki was doomed to fail in trying to use their structure theory. Leo Corry shows in his book "Modern Algebra and the Rise of Mathematical Structures" (Birkh=E4user 1996) that they did fa= il. And they should have seen this coming, because their theory had been "superseded by that of category and functor, which includes it under a more general and convenient form" (Dieudonn=E9 "The Work of Nicholas Bourbaki" 1970). best, Colin