SORRY, i SENT THIS OFF BEFORE IT WAS DONE! THIS VERSION IS COMPLETE
Dear Friamers -- or Fry-Aimers, however it is that we are pronounced.
Ever since I first got to santa fe four years ago, the pot has been burbling
here concerning what can and cannot be done with mathematics that can or cannot
be done with computation. Some have taken the position that some complex
processes -- or aspects of complex processes --- can only be understood
through computational models while others -- or other aspects --- can only be
understoud through maths. I apologize to all for my starting of the isargument
in about three different places in the last week, but I have finally decided
that the FRIAM list, being the most comprehensive list, is the best place for
it.
What I THOUGHT I understood about this argument was that it was about inference
tickets. All deductive arguments give you inference tickets to travel from the
premises to the conclusions. How you get to the premises is your own business.
Mathematical arguments are deduductive. They tell you that if you can manage
to get from Boston to Albany, you can get a train to Chicago.
In order to get a better idea of what it meant to be mathematically
"on a train to Chicago", I decided to read a book for english majors on
calculus recommended to me by Mike Agar. I guess I thought this would be
helpful because if ever there were some powerful inference tickets lying about,
they would be in the calculus, no? And I thought that if I understood, how
mathematicians argue for the calculus, I would understand, perhaps, how they
argue.
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!
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?
I promise I am not MERELY trying to be a horses ass, here.
Nick
.
Nicholas S. Thompson
Research Associate, Redfish Group, Santa Fe, NM ([EMAIL PROTECTED])
Professor of Psychology and Ethology, Clark University ([EMAIL PROTECTED])============================================================
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