[From Bill Powers (960313.1945 MST)]
Bruce Gregory (960314.1730 GMT) --
There is a test for the presence of chaos, just as there is for the
presence of control. The test is called sensitivity to initial
conditions. If two systems differing in initial conditions by only
a small delta begin to diverge exponentially, chaos is said to
apply.
This is what I meant by the "weak form" of chaos -- the divergence of
outcomes resulting from infinitesimal differences in initial conditions.
What I think of as the "strong" (er) form involves strange attractors,
nonlinear oscillators, and so on (what a friend of mine studied, in the
1950s, as "almost-periodic functions"). But I assure you, I don't know
what I'm talking about.
The amount of time it takes for such a divergence to appear can be
very long. Chaos in the solar system can be a matter of hundreds
of millions, if not billions, of years. By this test, it would
appear that there may be many non-chaotic systems in nature.
Yes, there may be. But relative to the behavior of organisms, there
would be essentially none if it were not for the active negative-
feedback control that forces outcomes to match predetermined reference
conditions. Ordinary behavior is an excellent example of the Butterfly
Effect, the hypersensitivity to initial conditions, if the feedback
effects are ignored. Every behavior begins where the previous one ended;
there is never any resetting to initial conditions. The main thing that
creates high sensitivity to initial conditions is time integration. The
more time integrations that intervene between a starting point and some
later state of affairs, the more difference any small variation in
initial conditions makes. An organism moving on a horizontal surface or
swimming in water is a living example of multiple time integrations that
have no basic constraints to assure that any particular end-state will
in general be reached, although of course there can be specific
conditions that create constraints. Even a simple common behaviour like
getting out of your easy chair, finding the car keys, walking to the
garage, opening the door, getting in the car, starting it, backing out
into the street, and driving to work involves a continuing series of
time integrations with nothing but feedback effects to prevent the car
from going off the road in the next five seconds. Without continuous
feedback control, each integration would be a bifurcation, with
subsequent movements diverging farther and farther from any predictable
form.
I think that engineers have been less impressed by chaos theory than
mathematicians and mathematical physicists have been. The engineers
always knew that time integrations are not to be trusted for long,
something that a mathematician working with symbolic equations would
have little reason to suspect. Digital computer models, likewise, give a
false sense of the relationship between computed and observed realities.
When you can run a computer model of the solar system backward and
forward for 10,000 years and predict positions within a few miles, you
get a false sense of the regularity of nature. Very few processes in
nature will fit an equation calculated to 100 binary bits of precision,
with random errors being on the order of one bit in 10^16. A model that
works forever in a digital computer might fail in a few seconds when
used to predict real observations.
If I had a signoff file, it would say
ALL ELSE IS NEVER EQUAL
or perhaps
YOU CAN NEVER STEP IN THE SAME RIVER TWICE
···
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Peter Cariani, 960313, 1100 --
Colloquially, we talk and act as if the discrete numerical
approximations that we use when we go to "compute" these variables
are the same as the continuous variables themselves, but they are
not. The quantity pi is only exactly defined in terms of
circumferences and diameters of circles, not in terms of ratios
between integers. We do not have an effective procedure (one that
yields an exact result in a finite number of steps) for computing
the value of pi, although we do have such procedures for
approximating that value. There is a qualitative difference here
between the two situations, and it is a ultimately the result of
the incommensurability of notational systems based on circles and
those based on integers.
That's good stuff, but I'm thinking of a still different aspect of the
problem. Even the variables we use, and the functions that relate them,
are only descriptions; they are maps, not territories.
When you apply a force to a mass, you get an acceleration. We represent
this relationship as f = ma. But what is it that makes the mass
accelerate when the force is applied? It is not the algorithm, f = ma.
The symbols f, m, and a stand for measurements -- i.e., perceptions. The
perceptions are not what create the behavior. In fact, even f, m, and a
are themselves perceptual representations of some external sea of
variables and interactions which can't possibly be as simple as the
symbols that we use to speak about them. What f=ma does is to _impose
order_ on our experiences. It simplifies our interactions with the
external world, making them comprehensible. But our symbolic
representations of the world no more make it work than the words "200-
horsepower engine" make a car run.
Algorithms are "as if" constructs. They are only ways of perceiving.
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I have not heard from anyone who wishes to join in the rat research that
Bruce Abbott (with me kibitzing) is now conducting.
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Best to all,
Bill P.