Thanks, John. this is REALLY interesting. I will also use this opportunity to
respond to Lee's suggestion that I have retreated from the field of battle
beaten and bloodied.
Well, Lee is to some extent correct. I believe deeply in the usefulness of
occasionally tieing bright people up with dumb questions, but there is a limit,
and I try not to broach it. And, I have learned a lot. I have learned, for
one thing, that algebra is more fundamental to you folks than I ever took it to
be. Lee is correct in his suspicion that I dont quite "get" why a simple
logical account does not produce the mean valule theorem in three easy steps,
but I think it is time for me to go away and think about it by myself or with
the help of private tutors. One glaring weakness in my suppositioin was that
the mean is itself an algebraic concept.
One thing I did not find, which i expected to, was a philosophic split amongst
mathematicians, along some sort of dimentions such as inuitionist and formalist
(I am making these words up). You guys did close ranks to a remarkable
degree, and as a psychologist, where there are 20 ideologies for every ten
scientists, I found that quite remarkable.
there are still questions kicking aroiund in my haid In what way is algebra
NOT a simulation? But unless somebody is really keen to go on educating me,
perhaps we should save it for another day.
nick
----- Original Message -----
From: John F. Kennison
To: [EMAIL PROTECTED];[email protected]
Cc: Lee N. Rudolph; David Joyce
Sent: 7/25/2007 6:39:10 PM
Subject: RE: DIFFERENTIABILITY AND CONTINUITY
Nick,
The gem that you found is well put, and it shows Lee's style. I have said that
"algebra" is needed, but I find I would like to clarify that statement.
Continuity and differentiability are defined in terms of the real number
system, so any proof of any statement would have to go back to axioms we use
for the real numbers. The simplest axioms are essentially "algebraic" in that
they involve manipulating variables and operations, such as addition and
multiplication. One could try to imagine a purely geometric set of axioms, but
it is not clear (to me at least) what they would be like. My guess is that the
proofs would be much harder. One attempt at a more intuitive foundation for
calculus uses infinitesimals and some algebra --and some subtle logical
maneuvres.
You raise the issue of pedagogy. We use different pedagogical approaches to
meet different goals. We do want our students to leave calculus with an
intuitive understanding of how it works --and an intuitive appreciation of its
beauty. For this purpose, appeals to intuition and geometry are emphasized.
Since rigorous proofs usually fail to develop intuition, we do not spend all
our time doing proofs. But another goal is that students understand that the
results of calculus do have logical proofs and that intuition sometimes is
misleading, so we do spend some times on proofs. And we are careful to
distinguish intuitive plausibility arguments from logical proofs. It probably
seems to the students that most of our time is spent on proofs but to us it
seems that relatively little time is spent on rigor.
--John
----Original Message-----
From: Nicholas Thompson [mailto:[EMAIL PROTECTED]
Sent: Wed 7/25/2007 2:09 PM
To: [email protected]
Cc: John F. Kennison; Lee N. Rudolph; David Joyce
Subject: DIFFERENTIABILITY AND CONTINUITY
Deep down in the tangle of >>>>>'s I just found this gem. The record is
two confused for me to know who to thank so I will thank you ALL.
> What you have given is the "handwaving" version of the proof. The
> trouble is that human imagination can easily get us into trouble when
> dealing with infinities, which is necessarily involved in dealing with
> the concept of continuity. In the above example, you mention that
> continuity is important, but say nothing about differentiability. Are
> you aware that continuous curves that are nowhere differentiable
> exist? I fact most continuous curves are not differentiable. By most,
> I mean infinitely more continuous curves are not differentiable than
> those that are, a concept handled by "sets of measure zero".
OK. I AM BEING CALLED TO A MEAL AND YOU ALL KNOW WHAT HAPPENS WHEN ONE
DOESNT ANSWER THAT CALL. BAD KARMA
AM I WRONG THAT BOTH CONTINUITY AND DIFFERENTIABILTY OF AT LEAST THE
primary FUNCTION ARE A PREMISE OF THE MEAN VALUE THEOREM.
MORE TO THE POINT, ARE YOU ALL CONVERGING AROUND THE ASSERTION THAT THE
MEAN VALUE THEOREM CANNOT BE DONE WITH OUT ALGEBRA? AS OPPOSED THE THE
VIEW I WAS ENTERTAINING THAT THE MEAN VALUE THEORY IS A LOGICAL PROOF THAT
IS REPRESENTED ALGEBRAICALLY FOR PEDIGOGICAL PURPOSES.
SORRY TO TWIST EVERYBODY'S KNICKERS ABOUT THIS. BUT IRRITATING AS IT MAY
BE TO YOU ALL, THIS CONVERSATION HAS BEEN VERY HELPFUL TO ME.
NICK
nick
>
>
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