3.345 chaos theory; science and humanities (93)
Willard McCarty (MCCARTY@VM.EPAS.UTORONTO.CA)
Fri, 11 Aug 89 20:55:08 EDT
Humanist Discussion Group, Vol. 3, No. 345. Friday, 11 Aug 1989.
Date: Fri, 11 Aug 89 19:29 EDT
From: Peter D. Junger <JUNGER@CWRU>
Subject: Chaos theory; science and the humanities
There appear to be two related subjects of general interest that
are currently being discussed on Humanist: The impact of 'Chaos' on
science, and perhaps the humanities, and the relations between the 'two
cultures' of Science and the Humanities.
I am now reading, and recommend highly to anyone who is
interested in the historical development and the implications of Chaos
theory, Ian Stewart's "Does God Play Dice? The Mathematics of Chaos"
(Basil Blackwell, Oxford and New York: 1989). Despite the title no
great mathematical knowledge is required of the reader, Stewart is one
of those rare mathematicians who can get the basic ideas across to
interested non-mathematicians.
Although I am only a third of the way through the book, I feel
that I can comment--with much more confidence than I would have had the
day before yesterday--on two points that have been touched on by others.
The type of 'indeterminancy' exhibited by chaotic systems is not
necessarily a product of complexity, in the sense of the system having many
variables (or dimensions or whatever). The behavior of the millions of
millions of molecules in a balloon appears random in part because of the
sheer number of molecules whose activities must be taken into account;
on the other hand a system containing only three bodies may exhibit
chaotic behavior--or, at least, the behavior of one of the three bodies
may be chaotic if that body is significantly smaller than the other two.
As an example of a very simple 'system' that can exhibit Chaos, Stewart
cites the equation f(x) = k(xx)-1, where k is a constant and xx stands
for x squared. Stewart includes the following BASIC program to explore the
behavior of successive iterations of this function.
10 INPUT k
20 x = 0.54321
30 FOR n = 1 TO 50
40 x = k*x*x-1
50 NEXT n
60 FOR n = 1 TO 100
70 x = k*x*x-1
80 PRINT x
90 NEXT n
100 STOP
Lines 10 through 50 are there to calculate, without printing,
enough iterations to allow things to get interesting. Stewart says:
"Chaos sets in around k = 1.5. After that, the bigger you make k, the
more chaotic things get.
"Or so it may seem. But it's not quite that easy."
It seems pretty clear from the fact that the output of this
program is chaotic that chaos can result from something that is very
simple.
The second point is that the new interest in chaos is not likely
to cause a scientific 'paradigm shift.' The problems that can best be
explored by chaos theory, i.e. by non-linear dynamics, are not those
that have been studied by traditonal scientists. Since Newton's day
scientists have, for the most part, restricted their interest to
problems that they could solve and--for the most part--that meant that
their interest was restricted to linear differential equations. Now
that computers have enabled us to study the behavior of non-linear
equations, new areas can be explored using new techniques. But for the
traditional problems of physics linear, deterministic equations will
still be the best tools. So there is not likely to be a paradigm shift
in physics attributable to the development of chaos theory; instead the
new paradigms will be used to study to new areas.
My guess is that many of these new areas of scientific study
will be in--or at least near--the humanities. One of the
characteristics of non-linear dynamic systems is that they are
unpredictable, even though their behavior can be described by simple
equations that are completely determined by the original conditions of
their variables. That suggests that mathematics will become another way
of describing the unpredictable behavior of systems that up to now have
only been describable by words. I would hope that such mathematical
descriptions would be considered as complementary, rather than oppossed,
to the more traditional descriptions of humanists.
Peter D. Junger--CWRU Law School--Cleveland, OH--JUNGER@CWRU.bitnet