Re: terminology in definitions of limits

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

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
>>
>>
>>
>>
>
>