From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9649 Path: news.gmane.org!.POSTED!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Point-free affine real line? Date: Thu, 7 Jun 2018 09:52:38 -0700 Message-ID: Reply-To: Vaughan Pratt NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1528637218 10486 195.159.176.226 (10 Jun 2018 13:26:58 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 10 Jun 2018 13:26:58 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun Jun 10 15:26:54 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1fS0Mv-0002Zs-C1 for gsmc-categories@m.gmane.org; Sun, 10 Jun 2018 15:26:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54508) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fS0No-0000GO-Ez; Sun, 10 Jun 2018 10:27:48 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fS0NC-0004vY-4y for categories-list@mlist.mta.ca; Sun, 10 Jun 2018 10:27:10 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9649 Archived-At: > Not quite: the affine rational line doesn't have a definable total order, > since it has order-reversing automorphisms, so any definition using > Dedekind sections is problematic. Morphism-wise, since the affine transformations are just the composition of a linear transformation with a translation, and translation of the rational line preserves order, affinity can't be the problem here. Structure-wise, one can equip the rational line with either its linear combinations or its linear order, or both. Using both eliminates the order-reversing linear transformations. "Affine" only makes sense in the context of having the linear combinations, as "affine" limits the linear combinations to those whose coefficients sum to one. If it is ok for the linear combinations and the linear order to coexist, it must be even more ok for the affine combinations and the linear order to coexist. So whether one considers the morphisms or the structure they preserve, affinity (affineness?) must be a red herring here: any problem for the rational line as an affine space is surely also a problem for it as a vector space. Vaughan Pratt On Tue, Jun 5, 2018 at 3:55 AM, Peter Johnstone wrote: > On Sun, 3 Jun 2018, Vaughan Pratt wrote: > > > The question about the affine real line represents a challenge to this >> >>> geometric approach, and I'd like to form a better idea of whether it is >>> simply a difficult problem, or a fundamental limitation to my approach. >>> >> >> An affine space over any given field differs *k* from a vector space over >> *k* only in its algebraic structure, not its topological structure. >> Whereas the algebraic operations of a vector space over *k* consist of all >> finitary linear combinations with coefficients drawn from *k*, those of an >> affine space consist of the subset of those combinations whose >> coefficients >> sum to unity, the barycentric combinations. Since the former includes the >> constant 0 as a linear combination while the latter does not, a >> consequence >> is that 0 is a fixpoint of linear transformations but not of affine >> transformations, whence the latter can include the translations. >> >> This is equally true whether *k* is the rationals or the reals. So >> whatever method you use to obtain the real line from the rational line >> should also produce the affine real line from the affine rational line. >> >> Not quite: the affine rational line doesn't have a definable total order, > since it has order-reversing automorphisms, so any definition using > Dedekind sections is problematic. However, it does have a ternary > `betweenness' relation, and it should be possible to rewrite the > geometric theory of Dedekind sections of Q, as presented on p. 1015 > of `Sketches of an Elephant', in terms of this relation (but note that > sections will have to be unordered rather than ordered pairs of > subobjects of Q). > > Peter Johnstone > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]