From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4615 Path: news.gmane.org!not-for-mail From: "Zinovy Diskin" Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Sat, 20 Sep 2008 13:17:19 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020058 14047 80.91.229.2 (29 Apr 2009 15:47:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:38 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Sep 21 10:50:37 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:50:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPFJ-0006mv-Oy for categories-list@mta.ca; Sun, 21 Sep 2008 10:45:05 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 85 Original-Lines: 75 Xref: news.gmane.org gmane.science.mathematics.categories:4615 Archived-At: Let me add my two cents :) On Tue, Sep 16, 2008 at 4:57 AM, Vaughan Pratt wrote: > If the subject Bourwiki is proposing to serve is mathematics, then > perhaps it is time that the American Mathematical Monthly, along with > the Putnam Mathematical Competition, the International Mathematics > Olympiad, and the Journal of the AMS, abandon their pretense of being > about mathematics and come up with a suitable name for their subject. We can think of two definitions of math. The first is based on the subject matter ("what"): math is the study of Bourbaki's structures. The other is based on "how:" math is the study of structures in a well-structured way. With this definition, a good computer programmer or, say, Henry Ford, who applied conveyor to assembling, are more mathematicians than some of the guys responsible for what Vaughan wrote about. If we imagine a two dimensional plane with the "what" axis being vertical and the "how" horizontal, then we get two mathematics and resp. two sorts of mathematicians: vertical and horizontal. (CT and CT-rists go, of course, along the harmonized diagonal :). Historically, Bourbaki put a bold point on the line of Euclid-Peano-Hilbert and indeed formed the vertical dimension of the modern mathematical space. However, while the very texts written by Bourbaki are mostly enjoyable, his epigones have created a special literary style, which is good for writing/producing but hardly for reading mathematical papers. Bourbaki should not be blamed for wide dissemination of this indigestible style but...Vladimir Arnold once said that "bourbakization" of modern mathematics should perhaps be called "oBourbachivanie" in Russian (which rhymes with the Russian "oDourachivanie", which refers to fooling with someone :). > Not only do categories, functors, natural transformations, adjunctions, > and monads go unused in these 20th century icons of mathematics, they go > unacknowledged. Clearly they have not gotten with the modern > mathematical program and fall somewhere between a throwback to a golden > age and a backwater of mathematics. When they die off like the > dinosaurs they are, real mathematics will be able to advance unfettered > into the 21st century and beyond. > Dinosaurs would not normally die off themselves. Some causes are needed, and here's one (somewhat speculative though). CT can change the very notion of what a formal definition is. In the modern style, the notion of ordered pair/tuple and its derivatives like formula and term are central. This quite simple syntax (as is often happens with simple syntax, think, for example, of a Java program) can hide complex structures so that a tuple-based formal definition is not actually formal and implicitly involves intuitive concepts. If CT will sometime indeed reshape the criteria of being a formal specification (of a Bourbaki's structure), then dinosaurs would be forced to acquire CT (or die off). Zinovy > Judging from the talks at BLAST in Denver last month (B = Boolean > algebras, L = lattices, A = (universal) algebra, S = set theory, T = > topology), at least the algebraic community is moving very slightly in > this direction. Things will hopefully improve yet further when > algebraic geometry gets over its snit with equational model theory. > > Meanwhile if you need a witness for seven degrees of separation, look no > further than AMM and CT. > > (I confess to being an unreconstructed graph theorist and algebraist > myself. I may have to preemptively volunteer myself for re-education > before it becomes involuntary.) > > Vaughan Pratt > > >