Posted by
Russell Standish on
URL: http://friam.383.s1.nabble.com/DISREGARD-math-and-the-mother-church-tp524257p524259.html
On Sun, Jul 22, 2007 at 06:12:58PM -0600, Nicholas Thompson wrote:
>
> So, here is my present understanding of the mathematician's argument for the mean value theorem. What I dont understand is why it takes three pages of algebra to get there!
I don't know where you get the 3 pages from. My analysis book does it
2 paragraphs of algebra, half a page at most. (That's including all
the necessary lemmas and definitions).
>
> Let us amagine that ab is a bit of a line. It could be straight, and the argument would still hold, but let us imagine that it is curved.... curved up, curved down, it does not matter. Let's imagine that is an inverted U, except that it doesnt have to be a straight up and down inverted U. In fact, it can be sitting so that somebody wobbled it so that it is, at the instant of being photographed, standing on one leg, about 30 degrees from the verticle. .
>
> What does matter is that the line be continuous ALL THE WAY FROM a to B. No gaps, not steps. Imagine that no matter how small the steps you are taking, you can walk along the points of the line from a to b and not get your feet wet, NOT AT ALL -- if of course your shoe size is small enough.
>
> Now draw a line that connects the bottom of the two legs of the inverted U. As we just said, that line will move off to the right, from a through b and beyond, at about a thirty degree angle from the horizontal. Thus the mean slope of the tilted inverted U is 30 degrees, right?
>
> Here is what that means, as I understand it. Every point on the tilted inverted U has a "slope", the slope of the line that is just tangent to the U at that point. Near point "a" that slope is VERY positive; near point "b", that slope is very negative. Now, imagine you set out to walk along the curve from "a" to "b". If you take tiny enough steps, you MUST step on the point where the slope is the same as the mean slope. That is what the mean value theorem says.
>
> But I just got there without any of the algebra usually devoted to that proof. So the question is, what is the VALUE of the algebra. If one can estab lish the truth of such an important MATHEMATICAL theorem in other than mathematical means, what is the value of the maths?
>
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".
To give an example, consider your interval joined by two line segments
so as to form a single kink in the middle at point c:
At all points on the interior, except for the c, the slope is either
s1 = (f(c)-f(a))/(c-a) or s2=(f(b)-f(c))/(b-c). At c, the slope is
undefined. But neither s1 nor s2 = (f(b)-f(a))/(b-1), so the mean
value theorem fails.
> I promise I am not MERELY trying to be a horses ass, here.
>
> Nick
>
>
Handwaving arguments are good for developing intuition. Great for
teaching during a lecture, and get the students to study the rigorous
proof later. Similarly, they're good for scientific seminars, but not
scientific papers.
Cheers
>
>
>
>
>
>
>
>
>
>
>
> .
>
>
>
>
> Nicholas S. Thompson
> Research Associate, Redfish Group, Santa Fe, NM (nick at redfish.com)
> Professor of Psychology and Ethology, Clark University (nthompson at clarku.edu)
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at
http://www.friam.org--
----------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder at hpcoders.com.au
Australia
http://www.hpcoders.com.au----------------------------------------------------------------------------