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 <pratt@cs.stanford.edu>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