<1>
It was sheer delight to read both Raman's original article and Fred's article.
I can see we are really into the core of the problems we've been discussing.
I don't think it would be fruitful for me to write something about both
articles. They are done too well to criticize.
I am only going to make some comments on a particular piece of writing by
Fred. My views are probably not standard.
<2>
' ... prerogative, and not too an unusual one. But the next claim is that
chaos is more evident at this level because of chance. I have already remarked
to the effect that chaos does not depend on chance. Deterministic equations
that produce chaos theoretically without the inclusion of a noise factor
are one of the hallmarks of dynamical systems exhibiting chaos. Definitions
of chaos are independent of the notion of chance and probability. So we
can only talk of the complexity of different levels in terms of the dimensionality
of the phenomena and the attractors, the number of the variables interacting
in a deterministic way, and the complexity of the patterns over time that
they exhibit. Which brings us to his final point, the absence of time in
point six; time being the penultimate concern of dynamical representation
of the function of systems.
<3>
It is here that I think I should say a few words. If there are any mathematicians
there who are more deeply into the dynamics of what chaos is, please do.
<4>
Long before this thing called deterministic chaos became something to study
for its own behavior, we (those of us who were forced to solve nonlinear
differential equations for whatever reason) studied what was called "Numerical
Analysis". In this course, we studied how to solve equations which
we could not solve in "closed form" numerically, that is, using
a digital computer. Since digital computers are discrete machines this naturally
meant that the continous problem (the differential equation) had to be represented
as a difference equation (discrete form). Various ways were attempted and
error estimates were made for all of them.
<5>
The basic idea is quite simple. Suppose we have y=f(x). If we know x exactly
we can compute y exactly. If however we know x only approximately (i.e.
with a small error term) what then happens to our computation of y? Naturally,
we cannot compute y exactly either; we get an error. So we can see that
this can be written as;
y + dy = f(x+dx)
where the dy, and dx are the "small variations" (i.e. errors)
in y and x. If we now subtract the original y=f(x) from this and rewrite,
we obtain
dy = f'(x)*dx
where f'(x) is the derivative of f(x). But dy was the error in y due to
the error in x. So we have e_y=f'(x)*e_x.
Now, if f(x) was linear then the error in y would be a constant times the
error in x. However, if f(x) is not linear then the error in the computation
of y, is function of both x and the error in x.
<6>
Strangely enough, this is the cause of chaos. The reason is that in solutions
of sets of nonlinear differential equations must be solved via the digital
computer. That means that we are always faced with a problem of iteration
in which we continually compute some number and calculate its next value.
But in this calculation we make errors which are variable because of the
nonlinearity. That means that any sets of nonlinear DEs which have periodic
solutions can have solutions in which the error term propagates ever so
much that we never get the trajectory to repeat exactly but always oscillate
about some periodic orbit.
<7>
What this means to me is a deeper question: Is nature discrete or continuous?
In other words is this a problem only of being able to solve some problems
numerically or are descriptions of some aspects of nature inherently discrete?
I have this feeling that chaos is a computation problem.
<8>
Aside from that, obviously it is quite useful for philosophical purposes,
especially as something that fits partway between determinism and random.
That is because we do not want human (or living matter) behavior to be either
random or deterministic.
We are upset if we are told that we are machines (because we are told that
machines are deterministic). And yet humans do not behave at random. We
are purposive in our behavior.
Obviously, deterministic chaos fits the bill almost perfectly; we are deterministic
and yet not predictable.
I guess this is what makes the chaos paradigm so attractive in the social
sciences.
Regards,
Mark
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
"640K ought to be enough for anybody."
- Bill Gates, 1981
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Mark Hubey
29 December 1997
[ H.M. Hubey is Associate Professor of Mathematics and Computer Science,
and works mainly in computer science, but with research interests in cognitive
science, economics, linguistics, psychology, history, and sociology. He
has written a book on linguistics, and is working on other books on economics,
and aspects of evolution, etc.
e-mail <hubey@amiga.montclair.edu>
http://csam.montclair.edu/Faculty/Hubey.html ]