From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3196 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: fundamental theorem of algebra Date: Thu, 30 Mar 2006 23:20:26 -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 1241019150 7663 80.91.229.2 (29 Apr 2009 15:32:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:30 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Fri Mar 31 19:24:20 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Mar 2006 19:24:20 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPStW-0004Yt-FB for categories-list@mta.ca; Fri, 31 Mar 2006 19:19:06 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 142 Original-Lines: 58 Xref: news.gmane.org gmane.science.mathematics.categories:3196 Archived-At: John Baez wrote: > I really doubt those authors were unaware of the topological proof > of the fundamental theorem of calculus in 1987. After all, it's Right, both my claim and its premises needed a fair bit of tuning (as with my recent question about the quasivariety "groups+free monoids" -- this is a good list to get corrective feedback from). (But a neat piece of historical research there, John.) The issue seems to be coming down to Mike Barr's question, which if I can paraphrase it without changing its intent, was, what is the proper status of an appeal to the very plausible in a proof? My suggestion in my last message to Peter Freyd was that the prover should point out the gap, its cause (lack of a simple proof), and its plausibility notwithstanding. This suggestion raises more questions than it answers. 1. Is a proof with a gap more acceptable for expository purposes when the bridgability of the gap is more plausible? (The case in point being an extreme example.) 2. How is plausibility to be judged? By consensus, or are there objective criteria? 3. It is certainly not necessary to prove A before B merely because B depends on A; indeed one common-sense practice when proving a two-lemma proof is to get the easier lemma out of the way first, even if it depends on the harder one. Is it kosher to truncate such a proof after the first lemma (or in this case the final result), call it an exposition, and point to the literature for the second lemma? Regarding 3, the authors of the Britannica article seemed not to think so, but perhaps this just reflects Garrett Birkhoff's attitude that "I don't consider this algebra, but this doesn't mean that algebraists can't use it" cited by Michael Artin when proving FTAlg in his 1991 book "Algebra". Who on this list considers the fundamental theorem of algebra "not algebra"? These questions are probably more appropriate for a philosophy of mathematics list than this one. What makes FTAlg such an interesting case study for those with something at stake in such questions is that the tensions here are so extreme. The final result (FTAlg) is not at all obvious, whereas the lemma it rests on, whether it be that |P(z)| attains its minimum, or that circles around a hole don't retract, or the intermediate value theorem, or the existence of a root for a real polynomial of odd degree, seems self-evident. Yet the one that is hard to see is easy to prove, while the one that is easy to see is hard to prove. If seeing is believing, what is proof? In the real world, when something is easy to see it is up to the opposition to demonstrate that it is nonetheless false. How did mathematics evolve to play by a different rule book? Vaughan Pratt