Re: John Gabriel's Theorem Revisited.

From: Jason (logamath_at_yahoo.com)
Date: 02/18/05 

Date: 17 Feb 2005 20:48:31 -0800

Hello Yan,

It's simple: w/n can never be zero, no matter how large you make n.
Thus
the denominator in gabriel's quotient may never be zero. Why? Because
infinity is not a number. In the classic definition, no matter how
small
you make w, it will eventually reach zero. Why? Because zero is a
number.
The difference between zero and infinity is that we can perform finite
calculations/arithmetic using zero but not using infinity. This is the
major difference.

The next difference is that gabriel's version leads to the integral
concept whereas the classic version does not. How? Well, when we
integrate, we are summing an infinite number of areas or partitions to
compute the total area/integral. Partitions are part of gabriel's
version but the classic version shows only one partition, i.e w.

Now for the most important difference:
Gabriel's theorem forges the link between the mvt and ftoc to explain
everything in terms of the average tangent/derivate gradient/value.
Gabriel's version allows one to pull out the inner limit because both
are taken to infinity. The classic version cannot be used because the
limits are calculated differently. This appears to work in gabriel's
proof of the average tangent theorem. However, what seems to be suspect
in his proof is the use of what he calls positional derivatives (*). If
indeed these positional derivatives exist, then it appears to agree
with his requirement that the whole interval be differentiable. This
makes sense because this is exactly what happens when we integrate:

  f(x+w) - f(x) 1 n-1 ws 1 x+w
  ------------- = Lim - Sigma f'(x+ --) = - INTEGRAL
f'(x)dx
         w n->Infinity n s=0 n w x

The middle portion in the above is gabriel's ATT (Average of Infinite
limit
of the sum of derivatives). It follows that all the interval must be
differentiable except possibly at x+w.

                                    1 n-1 ws
Average Derivative = Lim - Sigma f'(x+ --)
                        n->Infinity n s=0 n

In the classic format we have:

                     x+w
  f(x+w) - f(x) = INTEGRAL f'(x)dx
                      x

This is how Gabriel explained it to me. Where my wheels come off are
his positional derivatives are used. They appear to be elusive
quantities
since they cannot be calculated but he claims they exist. His proofs
use
these positional derivatives.

> No offense sir, but I think I deserve the same respect to my post as
I
> did to yours, reading through it and asking specific questions around

> each point in context. I can guess that you did not read my post very

> carefully if you did not bother to answer any of the questions I have

> posed. Since in your past posts you have used as an argument against
> many mathematicians' intellects for failing to answer direct
questions,
> I think you should return the favor.

So I think I have returned the favour.

> I am afraid there are at least two problems at stake, one more
important
> than the other:
> 1) We are already depending a lot on real analysis with definition of
> continuity and limits anyway. I have always been curious exactly
> which part of real analysis you have problems with, can you point
> them out again?
> 2) In some sense, we "are" doing real analysis. Though this is just
> a matter of word choice.

No. We are not depending at all on R.A (real analysis) when we use
gabriel's theorem. This forum is not about the problems I have with R.A
and there are too many to enumerate here. R.A does not satisfactorily
provide
proof for the mvt - this is the main problem. And Rudin's 'proof' does
it no justice at all.

> Rudin is very rigid and stable. May I ask your objections against it?

Read the previous paragraph.

> So you think it is bad to have n in the denominator if n goes to 0,
but
> not bad to have w/n in the denominator if n goes to infinity?

See the first paragraph in this response.

> Can you be specific - i.e. mathematically, step by step pointing out
> this difference? This logic sounds hand-waivy, and I am not competent

> enough to understand it in its current form if it is currect. Also,
> exactly how WHAT is different to WHAT I have stated. I cannot follow
> this rebuttal. Replies to problems I have posed in the previous post
> would also be appreciated.

I think you are quite able to understand it. Probably more competent
than any one else in this forum so far. If you stay with it, you may
even be able to find the answers to my questions before I do. :-)

(*) I use the word suspect with caution because he demonstrates quite
convincingly how to arrive at the arc length formula using only his
average sum and average tangent theorem.

Jason