This is a good question. There is no question that Goodwin's
approach appeals to me personally because mechanics plays such
a crucial role in the proposed theory of form. But I think there's
more to it than that. To me an explanation based on fundamental
principles and devoid of historical contingency is much more
satisfying. Of course, a historian (or geologist :) might disagree.
There are many examples I might give here. The shortest is an
example used by Goodwin all the time. "Why does the earth
go around the sun in an elliptical orbit." The historical
explanation would be that last year it went around the sun in
an elliptical orbit. And the year before and ....
There's nothing wrong with this in so far as it goes. But it
is much more satisfying to understand the regularity in terms
of something more fundamental, i.e. Newton's laws.
This example is good only because it is so short. A better,
but lengthier example (also from Goodwin) is spiral phyllotaxis.
I wrote something on this about a year ago and so I'm going
to just copy it here with slight modifications. It is written
in a narrative form because of original context which might
seem strange here, but I'm taking the easy way out (an obvious
example of historical contingency :).
Spiral Phyllotaxy
====== ==========
Suppose you are a botanist investigating possible
geometrical growth laws in plants. In particular, you
are studying a group of plants displaying a growth
pattern which has come to be referred to as spiral
phyllotaxy. But you don't know this obviously, since
you are the lucky scientist who is going to make this
discovery :). As you look down the stem of a plant
from the top you note that successive leaves form a
spiral pattern as you move up the stem with a constant
angle of divergence. Careful measurements reveal this
angle to be very nearly 137.5 degrees. As you study
more and more plants with this spiral pattern you
find this same constant divergence angle again and
again.
Well, this is not particularly surprising. Its not
really surprising that the divergence angle should
be a constant. This constant must be some number,
why not 137.5? As to why the same angle in so many
plants one imagines two possible explanations:
(1) historical contingency (frozen accident) or
(2) natural selection (this particular angle confers
some advantage and was thus selected for during evolution).
OK, fine. Several weeks later you are reading your
favorite "joy of math" book during one of your many
"time-outs" imposed by the Emperor, err, I mean the
Department Chair. You are fascinated to learn about
the Golden Rectangle and the mystical and magical
Golden Ratio. The ratio that Kepler referred to as
the "Divine Proportion" and a "precious jewel", one
of the two "great treasures" of geometry, the other
being the theorem of Pythagoras.
Now the thought occurs to you: What angle will
divide a circle into the divine proportion?
IOW, consider a circle of circumference A and
some angle that divides the circumference into two
parts B and C (A = B + C) in such a way that the
ratio C/B = B/A = R, the Golden Ratio. This is
a fairly simple problem and after a few moments
you discover, to your great horror :), that the
required angle is 137.5 degrees.
And so you have discovered that the divergence
angle during the growth of the plants you have
been studying divides a circle into the Divine
Proportion. So you have to add a 3rd possibility
to the list of explanations above: (3) magic :)
Just kidding. The actual explanation (more correctly,
the explanation I find more appealing :) is that
there are fundamental laws at play, the operation of which
give rise to a restricted set of possible divergence
angles. Learning how this works (beautiful mechanics :)
is very satisfying.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"All kinds of private metaphysics and theology have
grown like weeds in the garden of thermodynamics"
-- E. H. Hiebert