Posted by
Nick Thompson on
URL: http://friam.383.s1.nabble.com/DISREGARD-math-and-the-mother-church-tp524257.html
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
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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)
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