Dear Organlearners,
Winfried Dressler <winfried.dressler@voith.de> writes:
> I have some questions regarding mathematics.
> In the theory of deterministic chaos, where nonlinear functions are
> applied iteratively while parameters are changed, I heard for the first
> time the word "bifurcation". Has this bifurcation in chaos theorie
> anything in common with your bifurcation at entropy saturation? Sometimes,
> iteration or changing the parameter is referred to as "increase in time".
> Do you see useful analogies between chaos theory and thermodynamics far
> from equilibrium?
Dear Winfried, this is one of those questions with a No amd Yes answer.
For the rest of you, dear readers, if mathematics destroy your passion for
learning, get of the bus now.
First the No. Mathematicans are a creative bunch of people. A
mathematician finds a pattern interesting when that pattern can lead to
other patterns. The mathematician then creates these other patterns, make
theorems out of them and ONLY then try to provide these theorems with
logical proofs in order to get them published. Thus, for the uninitiated,
mathematics seems to be mainly about logic. But for the initiated,
mathematics is primarily abstract creativity, leaving the material world
or the abstract world of others as soon as possible to play with your
newly found productive pattern. It is only when you want to get back into
the abstract world of other mathematicians that you have to employ logic,
the lingua franca of mathematicans.
At the turn of this century the great Frenceh mathematician Henry Poincare
tried to solve the three body problem of Newtonian mechanics. It means
that he tried to find a way to describe deterministically how the
positions of the three bodies change as time increases in terms of their
initial positions as well as the forces which they exert on each other. He
did not succeed - one of the famous negative results. (The Micheson-Morley
experiement was another one - and eventually led to Einstein's theory of
relativity.)
But during the course of his investigations, he noticed that some of his
approximate solutions had the property that a small change in the
approximation parameter produced a vast change in the how the solutions
determine the position and momentum of the bodies as time increases. Only
during the middle seventies did mathematicians like Lorenz began to
investigate this pattern. It became known as chaos theory. The famous
"butterfly" effect is one of the ways to illustrate chaos theory.
Quite a large number of different bifurcations (Hopf, Turing, Smale,
Arnold) have been studied. The ideas are
1 to know when a bifurcation can be expected,
2 to find a single parameter to predict the various outcomes.
The outcomes are not "classified" in any other way than by means of
this bifurcation parameter. In other words, the mathematician is
basically not interested in finding an application for his theorems.
I myself tried to categorise these bifurcations with respect to the
construction/destruction dichotomy, but soon gave it up because of
the arbritaryness of the whole project. There was no implicit
order/disorder in these bifurcations. What is the sense of forcing a
meaning into something if it does not invite such a meaning?
Now the Yes. Whether a mathematician takes a material example or an
abstract example and isolate/identify a dynamical pattern in it, even a
forking pattern, that pattern is not simply there. Its dynamics incolves a
history and a future. If these various patterns (which mathematicians
employ to stimulate their own mathematical thinking) are not related to
each other, then there is no reason to suspect a common pattern from which
they all have evolved before the mathematicains even have fixed their
attention on them.
Unfortunately, mathematicians seldom seek common relationships among the
patterns which they employ to produce their own patterns. They consider
that search to be the work of the experimentalists - phsyicists, chemists,
biologists, etc.
Thus, in the first place, two branches of mathematics can co-exist for
some time without much interaction between them. A nice example with
respect to mathematical chaos theory is mathematical catastrophy theory.
Whereas in chaos theory a pattern is investigated which can fork into two
or more different patterns at the bifurcation point, in catastrophy theory
a pattern is investigated which makes a discontinuous change at the
catastrophy point.
Chaos theory is like driving a car on an unknown road - at a certain stage
the road forks and you do not know why or which direction to take - thus
make a quick map yourself on the spur of the moment. Catastrophy theory is
like driving a car on a wet tar road - at a certain section the surface
suddenly becomes very slippery and you loose all control over the car -
thus make for a new direction as you regain control over the car.
In the second place, and especially with respect to mathematical chaos
theory and catastrophe theory, mathematicians seldom find the patterns
discovered by the experimentalists as also "interesting". For example,
practical examples of chaotic bifurcations and catastrophies always
involve much entropy production. Yet, if we take ten books on mathematical
chaos theory and ten books on mathematical catastrophy theory, the chances
are very good that not even one of them will mention the word entropy even
once! Do this little experiment yourself.
Here is an interesting thing about catastrophy theory. A certain Zeeman
first discovered the catastrophical behaviour in a simple device
consisting of two rubber bands, two pins and a carton disc. After
publishing his results, the famous and creative mathematician Rene Thom
answered him by showing that seven elementary catastrophies are possible
with the names "fold, cusp, swallowtail, butterfly, parabolic umbilic,
eliptic umbilic and hyperbolic unbilic".
You can imgine yourself how intrigued I was with Thom's work, trying to
open it up for a variety of phenomena at the edge of chaos due to entropy
production. However, nothing came of it. Eventually, in a moment of
frustration, I noted a comment made by a certain Saunders that catastrophy
theory develops into these seven cases because it is basically a
topological theory. This comment helped setting me on course, arriving at
the toposlogic (not topology) of mathematical category theory. From there
I proceeded to discover the seven patterns essential to mathematical and
chemical creativity.
> Mentioning relations, are you (or any other) familiar with Gottfried
> Guenters "proemialrelation", which seems to be a basic concept of second
> order cybernetics based on his polycontextural logic. It must be close to
> your concept of commutation.
No. It seems to be promising. I would be happy if you can send me more
information. Thanks.
Best wishes
--At de Lange Gold Fields Computer Centre for Education University of Pretoria Pretoria, South Africa email: amdelange@gold.up.ac.za
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