Re: terminology in definitions of limits

[Charles Wells <charles@abstractmath.org>]

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
>
>
>
>


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