DISREGARD: math and the mother church

[Nick Thompson]

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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[Russell Standish]

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

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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)
> ============================================================
> 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

--

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[Günther Greindl]

Hello 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

I disagree - why was that a handwaving proof? It was exactly the way
someone _understands_ what the proofs are about. Mathematical notation
is only meaningless symbolism unless it is interpreted. It is
interpreted by our intuitions (visualization, relation to other, more
basic concepts etc).

Mathematical notation is good for a number of things:

1) define your concepts exactly (again, somewhere it has to bottom out
intuitively like in the concept of set membership or the rules of inference)

2) use a convenient shorthand (=math notation) which let's us reason
more easily about the concepts than in natural language. Good math
notation captures some intuitive reasoning analogy in our brains about
the subject - no platonic reality about the structural relation in itself.

3) Mathematics is then used to reason about ever more complex subjects.
The notation has been developed in a way that inferential validity is
preserved when mindless symbol shunting is correctly followed. This
let's us "reason" about things where our intuition _fails_ to preserve
inferential validity.


So, actually, there is no _magic_ in math or in the notation: it is just
   a very clever way of performing reasoning.

But in essence, a three page proof in english (if diligently written)
differs not from a two paragraph proof in algebra (which is just more
condensed).

That is actually the reason (I think) why some people who are very
intelligent fail at math: not because they are to dumb, but because
somewhere in their education they had bad math teachers who failed to
teach the intuition/understanding on a certain essential and basic
formalism.

As maths will build on this formalism in more complex situations,
everybody who has failed to grasp the grounding "shorthand" will fail to
grasp anything else (or it will appear like magic anyway).

> 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.

I'm not sure - I think the focus on formalism and the deprecatory
attitude which one regards intuition nowadays is actually bad for
mathematics.

For a refreshingly different approach read for instance

Needham: Visual complex analysis

http://www.usfca.edu/vca/

which shows that you do not have to sacrifice rigor by being intuitive
(on the contrary!).

Cheers,
G?nther

--
G?nther Greindl
Department of Philosophy of Science
University of Vienna
guenther.greindl at univie.ac.at
http://www.univie.ac.at/Wissenschaftstheorie/

Blog: http://dao.complexitystudies.org/
Site: http://www.complexitystudies.org