From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2667 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Modeling infinitesimals with 2x2 matrices Date: Sat, 24 Apr 2004 23:58:39 -0700 Message-ID: <200404250658.i3P6wdYo009866@coraki.Stanford.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018817 5229 80.91.229.2 (29 Apr 2009 15:26:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:57 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Apr 25 18:09:43 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 25 Apr 2004 18:09:43 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BHqnn-00060G-00 for categories-list@mta.ca; Sun, 25 Apr 2004 18:04:39 -0300 X-Mailer: exmh version 2.6.3 04/04/2003 with nmh-1.0.4 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 46 Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:2667 Archived-At: I'm told that Bell's "microlinear calculus" in his 1998 book on infinitesimals is equivalent to the matrix approach I suggested, so that was not new after all, other than perhaps its formulation in terms of 2x2 matrices. On the other hand it is apparently mixed in with Bell's strongly intuitionistic outlook, whereas it would seem intuitively that something so simple as a model of d^2 = 0 should transcend whether one is working intuitionistically or classically. A more classical version of Bell's account might be of interest (perhaps to relatively few people on the categories mailing list though, which seems to have a strongly intuitionistic slant). Meanwhile I received the April issue of Mathematics Magazine just now, and it has an article on pp. 118-129 on "Geometry of Generalized Complex Numbers" by Anthony and Joseph Harkin. The microlinear calculus, under the names "Study product" and "parabolic complex numbers," apparently dates back to Study's 1903 book Geometrie der Dynamen. The Harkins associate i^2 = -1,0,1 with respectively Ordinary (i.e. complex) product, Study product, and Clifford product (though Clifford algebras include ordinary product as well, the quaternions being a Clifford algebra). The article makes no mention of infinitesimals, and it would be interesting to try to find the appropriate infinitesimal interpretations of the geometric properties of the parabolic complex plane. One approach I very much like to infinitesimals that I haven't seen in the nonstandard analysis/infinitesimal literature (but would certainly appreciate pointers) is one that does all the work with what one might call finitesimals. A finitesimal h is just a positive real that you plan one day to reduce to zero, and thus organize everything around it to that end. Polynomials in R[x] of degree d form a (d+1)-dimensional vector space. The usual basis for this space is the d+1 monomials x^i for i in 0..d. However if one fixes h > 0 and takes the basis to be 1, x, x(x-h), x(x-h)(x-2h),... then Boole's difference calculus works essentially identically to the infinitesimal calculus for polynomials represented in the monomial basis. Since h is a free variable throughout the development, one can do all the work first and then drive h to 0 uniformly everywhere at the end. Expressions such as x^i (Knuth writes an underbar under the i and calls it "x to the falling i") mention h only implicitly and hence don't change (as symbolic expressions) as h changes, though their numerical values at any given x change. The Stirling numbers of the first and second kind, organized as matrices, constitute linear transformations from the bases for h=1 to h=0 and back again, respectively. I've looked from time to time at how one might extend this to exponentials and logarithms, but have never been satisfied with the results. It would be nice to know how to deal exactly with exp(it) for nonzero h. If this were possible it might give an even nicer constructive treatment of infinitesimals than the others, and one that didn't care at all whether one was classically or intuitionistically inclined. Vaughan Pratt