From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4810 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology in definitions of limits Date: Thu, 22 Jan 2009 10:07:13 -0200 Message-ID: Reply-To: "Eduardo J. Dubuc" 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 1241020187 14940 80.91.229.2 (29 Apr 2009 15:49:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:49:47 +0000 (UTC) To: Charles Wells , catbb Original-X-From: rrosebru@mta.ca Thu Jan 22 22:30:34 2009 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 22 Jan 2009 22:30:34 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LQBhr-0002Rx-80 for categories-list@mta.ca; Thu, 22 Jan 2009 22:23:39 -0400 Original-Sender: categories@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:4810 Archived-At: of course, by choice (and many times without choice), there are lots of functions \delta = f(\epsilon). It is a good question to see when there is a continous such "f". e.d. Charles Wells wrote: > 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 >> >> >> >> > >