From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2675 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Modeling infinitesimals with 2x2 matrices Date: Tue, 27 Apr 2004 22:13:13 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018821 5277 80.91.229.2 (29 Apr 2009 15:27:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:01 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Thu Apr 29 12:40:55 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Apr 2004 12:40:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BJDZc-0004A5-00 for categories-list@mta.ca; Thu, 29 Apr 2004 12:35:40 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 55 Original-Lines: 80 Xref: news.gmane.org gmane.science.mathematics.categories:2675 Archived-At: >But if you extend the domain to the algebra R(2) of 2x2 real matrices, >the columns indexed by singular matrices now lose some of their entries. >But not all, and so the solution space ceases to be rectangular. On reflection this is not so simple in the case when b in a/b is infinitesimal. First, noting that R(2) is noncommutative, the requirement should be phrased as two equations, a = bx and a = xb to prevent multiple solutions whose diagonal is not constant. But while this duplication then determines a unique real part (the diagonal), the two equations fail to pin down the infinitesimal part (the upper right entry). That's the sort of thing that happens with matrices of less than full rank. When there is no solution, certainly a/b should be considered undefined. But when there are multiple solutions, the question arises as to whether to punt completely (as with 0/0) or do something creative such as setting the undefined infinitesimal part to 0 (as with the ratio of two infinitesimals). One test is whether the "dominant" term is fixed, but this breaks down for 0/b. A better test would be to use the rank of b to decide how much of the quotient a/b to ignore---if b has rank 1 (a nonzero infinitesimal) then ignore the infinitesimal part of a/b. --------------- One virtue of Robinson's approach is its universality with respect to all first-order definable functions; this however is not sufficient to compensate for its more counterintuitive aspects. Now that I'm starting to see that the zero-divisor approach is less easily managed than I'd first thought, I'm not so sold on it either at this point (but maybe all its difficulties have been overcome somewhere...?). Meanwhile I remain convinced that Boole's finite difference approach to handling infinitesimals is superior (recall the trick here: setting h=0 instead of some positive quantity like 1 or Planck's constant introduces no artificial singularities with Boole's method). His 1860 *A Treatise on the Calculus of Finite Differences," substantially revised by J.F. Moulton for the 1872 edition after Boole's death, is 336 pages of inspired analysis. (You can get second hand copies for $10 from Amazon; my very second hand copy has "F.S. Curry, Trin. Coll., Feb. 1881" written on the inside cover.) The preface to the first edition starts out, "In the following exposition of the Calculus of Finite Differences, particular attention has been paid to the connexion of its methods with those of the Differential Calculus---a connexion which in some instances involves far more than a merely formal analogy. Indeed the work is in some measure designed as a sequel to my *Treatise on Differential Equations*. And it has been composed on the same plan." An updated version of this book incorporating the greatly matured perspective on linear algebra since then could be a worthwhile project for someone interested in improving on the existing explications of infinitesimals as real objects. While Boole's system beats the current crop hands down in principle (in my view anyway), in outlook it is showing its age. Category theory creatively applied might also help. I confess to having no idea how intuitionistic logic could be brought to bear effectively though. I can see that not cancelling certain double negations might preserve certain nuances that convey certain constructively motivated notions, but to my untrained eye these come across as nuances with a capital N when their contribution is assessed in the larger picture of alternative approaches to constructivizing infinitesimals. That makes me either a beer guzzler at a wine tasting or the owner of a screwdriver in a room full of hammer owners depending on one's outlook. :) YBMV (Your biases may vary.) Vaughan Pratt --------------------------