From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4805 Path: news.gmane.org!not-for-mail From: Charles Wells Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology in definitions of limits Date: Wed, 21 Jan 2009 10:48:57 -0600 Message-ID: Reply-To: Charles Wells NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020184 14914 80.91.229.2 (29 Apr 2009 15:49:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:49:44 +0000 (UTC) To: Vaughan Pratt , catbb Original-X-From: rrosebru@mta.ca Wed Jan 21 19:58:13 2009 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Jan 2009 19:58:13 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LPmxE-0007aC-D9 for categories-list@mta.ca; Wed, 21 Jan 2009 19:57:52 -0400 Original-Sender: categories@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 63 Xref: news.gmane.org gmane.science.mathematics.categories:4805 Archived-At: Calculus teachers do something similar when they make an epsilon-delta proof into a game: The opponent picks an epsilon (the test object) and you have to come up with a delta. There is one big difference between epsilon-delta proofs and limits. To show that something is a limit you have to find, for each test object, the unique arrow specified by the definition of limit. Thus you are producing a function (indeed, a bijection). The delta for a given epsilon is not unique, and so there is no natural function giving a delta for each epsilon. I am pretty sure this makes epsilon-delta proofs harder for non-talented students than proving something is a limit. I know some calculus teachers talk about there being a function that takes epsilon to delta, but I suspect it is a mistake to bring that up. Charles Wells On Wed, Jan 21, 2009 at 1:34 AM, Vaughan Pratt wrote: > > Colin McLarty wrote: > >> I often call them "test objects" in talking with students (by analogy >> with "test particles" in General Relativity). I don't think I have ever >> done it in print. But I did use "T" as the typical name of such an >> object in my book. >> >> I am curious to know what others think. >> > > From a game-theoretic standpoint one can be either taking the test or > administering it. Both sides call it the test, showing that the name is > stable under perp (change of team). > > However that's not to say that "test" gives a helpful perspective in > either case. A right adjoint defined by its adjunction is simply a > specification of *all* homsets to it, and dually, in the case of left > adjoints, of all the homsets from it. What you're calling a "test" > object there is for me merely the variable being universally quantified > over in the definition of "all." > > Whether a student is going to find it helpful thinking of a universally > quantified variable as a "test object" is going to be less a question of > what the student thinks about that perspective than what the teacher > thinks about it and whether they can convey their point of view. The > mathematically talented student who immediately sees it is merely being > universally quantified over may be more puzzled than helped. > > But then how many of us are so lucky as to have a significant number of > mathematically talented students in our classes? > > Vaughan > > > > -- professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://www.gyregimble.blogspot.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm personal website: http://www.abstractmath.org/Personal/index.html