From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3190 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: WHY ARE WE CONCERNED? I Date: Thu, 30 Mar 2006 09:10:53 -0800 Message-ID: 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 1241019146 7637 80.91.229.2 (29 Apr 2009 15:32:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:26 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Mar 30 18:58:14 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 18:58:14 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP64n-00071h-Ol for categories-list@mta.ca; Thu, 30 Mar 2006 18:57:13 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 136 Original-Lines: 46 Xref: news.gmane.org gmane.science.mathematics.categories:3190 Archived-At: In response to Peter Johnstone (and those who responded privately), my point about the Fundamental Theorem of Algebra was not that this particular proof (based on the limiting behaviors of small and large circles) was not known to anyone, but that it had not emerged, instead being effectively sat on by those in the know, even if not intentionally. At this risk of sounding like an Abu Ghraib interrogator, "who knew?" My claim is that no extant proof at all, that or any other, was considered fit for an elementary exposition more than a couple of decades ago. If that estimate is right, the 1982 Pontrjagin article cited by Nikita Danilov would be one of the earliest popular expositions based on the circles argument, assuming the section containing Fig. 6 is the relevant one (my Russian is even rustier than my algebra). I'd be very interested in seeing an earlier popular account that didn't claim that every proof necessarily either was long or depended on out-of-scope material. As a case in point, just now I checked a relatively recent Brittanica article on algebra (1987 ed.), which states flatly (p.260a) that "No elementary algebraic proof of [the FTAlg] exists, and the result is not proved here." (Not even "is known" but "exists"; an expository article should not assume that the reader knows the jargon meaning of this term as "exists in the literature".) The authors taking responsibility for this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter Hilton, and Paul Cohn. They go into detail to show that z^n = a has n roots, starting with the geometry of addition and multiplication in the Argand diagram, so it's not as if their exposition was at too elementary a level to talk in terms of mapping circles, or that "algebraic" ruled out simple geometric arguments. I submit their nonexistence claim as prima facie evidence for my claim that the very few who knew this argument weren't even letting the likes of Birkhoff, Hall, etc. in on it, let alone "the rest of us." The general message in the literature prior to the 1980's seemed to be, if Gauss couldn't find a simple proof in half a dozen tries, there isn't one. If you don't possess the necessary higher maths or the stamina for an intricate argument, we can't help you with that result, ask us about solvability of z^n = a. Good for Pontrjagin for promoting FTAlg to school children! Vaughan Pratt