From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3181 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: WHY ARE WE CONCERNED? I Date: Wed, 29 Mar 2006 12:10:40 -0800 Message-ID: References: <4429A1E1.8080907@math.upenn.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019140 7595 80.91.229.2 (29 Apr 2009 15:32:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:20 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Mar 29 23:49:21 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Mar 2006 23:49:21 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FOo9I-0000oP-O4 for categories-list@mta.ca; Wed, 29 Mar 2006 23:48:40 -0400 X-Accept-Language: en-us, en In-Reply-To: <4429A1E1.8080907@math.upenn.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 127 Original-Lines: 39 Xref: news.gmane.org gmane.science.mathematics.categories:3181 Archived-At: jim stasheff wrote: > now for a mathematical subject a math proof is sometimes but not always > necessary Absolutely. I would add publication date as a factor here. As an example, a few decades ago an elementary exposition of the Fundamental Theorem of Algebra would not be expected to include an elementary proof since the extant proofs were either lengthy arguments or nonelementary appeals to the minimum modulus principle, properties of holomorphic functions such as Liouville's theorem, or other results the reader would be unlikely to be on top of. The dominant belief was that the only short proofs were nonelementary ones. But for an audience aware only that z^i for any nonnegative integer i maps circles at the origin to i-fold circles of radius r^i at the origin, an entirely elementary notion, an expositor today would be morally obligated to include a full proof since there is hardly anything left to explain. The polynomial a_d z^d + ... + a_0 maps little circles to the neighborhood of a_0 and big circles to a loop tending to a very big d-fold circle of radius a_d r^d, whence the smoothly growing image, under the polynomial, of a smoothly growing circle is obliged to cross the origin at some stage. Still a topological argument, but now an entirely elementary one. Except, that is, for the theorem that a loop wound d times around the hole in the punctured plane cannot be continuously retracted to a point, which was tacitly smuggled in there. But that statement is less intimidating than anything based on holomorphic functions. This slick proof seems only to have emerged in the past couple of decades. It is an interesting commentary on mathematics that it took this long for people to come up with an argument "for the rest of us." Maybe some people "knew" it all along, but in that case they were keeping pretty quiet about it. Vaughan Pratt