Chaos theory versus complexity theory

[Chris Burford <cburford@xxxxxxxxxx>]

Lisa:
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Please distinguish for me between chaos theory and complexity theory?
I haven't seen this before, as Gleick does not distinguish.

Of course, on a rapidly moving cutting edge, Gleick is not up to
date, pub way back in 1987.

Chris B:
--------

[You will understand why I have hesitated to reply to your question.
If my answer below has flashes of creativity, it is not because
I am confident by any means but because I feel safe accepting your
request to push my understanding to the chaotic limits of my intelligence.]

This distinction between chaos theory and complexity theory is not made
clearly in most of the discussions. For
some reasons I do not understand, I suspect that there is something
inherent that makes them difficult to distinguish. But

1) chaos theory has its roots in kosher maths that goes back over
100 years. The development of computers made it possible to do
repeated calculations establishing that its phenomena were not just
random errors, noise etc, which should be dismissed from our pure clean
and rather mechanical science, but were *under certain conditions*
indisputably part of the scientific scene.

By contrast someone like Mandelbrot who has played around with
the animation of a science fiction film in creating their almost real
landscapes, is dismissive of the Santa Fe upstarts, a handful of people
who have got together to market yet another new word. (Caution: what I
have just said is not even a paraphrase of Mandelbrot's words and should
not be quoted - I am explaining my interpretation of why he is
dismissive)

2. Chaos theory takes a simple equation of a non-linear nature, that is
with an expression that is raised to a power or reduced to a power (that
has postive or negative feedback to use language I understand better).
If this is repeated many times, the resulting pattern *can* at times
oscillate between two results, four results, many results, or to appear
to have no pattern at all.

(This last state is called chaotic - NB two
usages of "chaos" in this new science a) such a state b) an equation which
*mostly* produces *regular* patterns, but under certain conditions may
produce such a chaotic non-pattern).

So chaos maths starts with a simple equation and shows it can produce
complex results, entirely of itself, without any outside influences.
(This is maths of a deterministic system producing non-deterministic
results. For the mathematicians a deterministic system is one in which
all the inputs are specified eg a simple equation. This NEVER (I shall
shout) happens in real life, but what is interesting about this model
of maths is that even in their purely artificial models of non-linear
equations, such deterministic systems may produce non-deterministic
consequences)

Chaos maths starts then with a simple expression and shows it can produce
complex results. Complexity theory, starts with a complex situation and
shows it can produce simple results! I believe it has also been developed
by people who call it "Anti-chaos" theory. Again it is maths. Again it
is an artificial closed, deterministic system. But the essence is it
starts with a very large number of the same things with a couple of
simple rules for their interaction, and shows that they produce patterns,
rather than random noise. One of the most important contributors to this
field is Stewart Kaufman who made simple computer models of "life forms" and
showed that, behold, they went forth and multiplied. Further that they
appeared to evolve. And further that they appeared to evolve into certain
patterns that were more common than other patterns. He has a major book
coming out this year.

3.  In order to get a purchase on the difference between chaos theory and
complexity theory, I find it helpful to understand what is the rule and
what is the exception. They are named after their exceptions!
Chaos theory says that simple non-linear expressions when iterated on
themselves repeatedly are usually regular in the pattern but can
occasionally go chaotic. But in many ways chaos theory is most useful
in *possibly* explaining why patterns in life remain as broadly stable
as they do. Eg why capitalism does NOT collapse, but only looks
vulnerable to a phase shift at certain times.

Complexity theory says that 10,000 widgets are usually ten thousand
widgets, but if they are able to interact with each other according to
a simple rule, they may exceptionally create a complex pattern at the
macro-level. Mostly dust is dust, strewn stochastically (by patterns
of chance across the floor). But occasionally, occasionally... Mandelbrot
has referred to the creative mathematical theory of dirt. There is more
to dirt than meets the eye. It may be a bridge between chaos theory
and complexity theory.

[I suspect that life, and everything, is largely shaped by an interaction
between non-linear (whether chaos or complex) and stochastic processes.]

4. So there is unity and struggle between chaos theory and complexity
theory. Where is the unity? They are both mathematical deterministic
systems highly abstract from the real world, in which as Engels says,
everything is connected with everything else. They are both about
feedback demonstrated by innumerable computerised repetitions over time.
Stuart Kaufman's work provides one bridge. It is clearly complexity
theory and he is persuasive that it tends to show up attractors in the
form of a limited number of patterns which emerge. Now the word "attractor"
is a word at the heart of chaos theory, so it is interesting that a
complexologist is using it. (How's that for a neologism?)

Another connection is the population equation in chaos theory. You
remember well from Gleick the model of the lake with a fish species in it.
A simple equation can make a fair approximation to the situation where
its reproduction year on year may have difficulty maximising the energy
potential of the lake in a smooth way, so that the fish numbers are always
at their very maximum. Instead it may oscillate between two or more levels.

Now this is a model from chaos theory. The number of fish is just taken
as a number for the purpose of the simple non-linear equation. But we
are talking about a supposedly homogeneous population, exactly the sort
of thing that complexity theory could model too. I do not know if it has
been done, but intuitively it seems to me that there is overlap in the
approach.

5. Let us now in the traditional marxist manner, proceed from appallingly
abstract discussions to try to apply things to reality in a way that
has a glimmer of familiarity. Let us take our own dear example of an
economy. Complexity theory would start off saying there are 6 billion
people in the world, half of whom can trade their labour power as a
commodity for commodities in return. ie 3 billion widgets who can interact.
Will they interact absolutely smoothly and evenly or will patterns
creating inequalities build up? Inequalities might indeed build up.

Chaos theory would say 3 billion is merely one number, a constant for
the purposes of our simple deterministic equation -

(and maybe this is
the only point where these highly artificial mathematical models that
ignore all other conditions might approximate to reality - if we
apply them to the entire globe)

 - and would ask what is the positive or the
negative feedback in the equation we want to iterate. One application
is to say that while increasing technology means endless pressure on
use values to increase, there is a ceiling on exchange value, rather like
the limits on the size of the lake in which the fish breed. The total
amount of exchange value on the planet is limited to the population selling
their labour power. Hence the population equation of chaos theory might
model to some extent the fluctuations in the capitalist business cycle.

I am totally confident that P8475423 (Steve, can't you do anything about
your inhuman name? ) will have read this post. He has referred to Goodwin's
interesting predator-prey model for the economy. I have got Goowin's book
but am not confident with the maths, and get destracted by this marxism
list. I am hoping Steve can say whether this predator-prey model is the
same as the population equation in chaos theory.

I am hoping he will explain the term "limit-cycle", which he has suggested
the capitalist economy looks like to him, Marx, and intuitively to me too.
But I am not quite sure of this terminology. It sounds like the fish and
the lake. What is a limit cycle?

I am hoping that Steve, Lisa and others, will comment on the apparent
similarities and the apparent differences between chaos theory and
complexity theory, and on how and why they may interact.

All of course from an impeccably marxist perspective - well, Steve, I
know you try :).


Chris B, London.




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