Re: Linear Thinking

[FVoehl@aol.com]

In a message dated 7/28/98 12:39:59 AM, ilx@execpc.com wrote:

<<I believe the term linear thinking has been driven by Senge. He has been
trying to explain the way people think and=20
how that differs from the way systems work.  He points out that Systems
tend to be a combination of enforcing actions and counter actions. >>

Although Senge is credited with popularizing the term linear thinking, it's
roots go back to the turn of this century.  In the 1960s, it was advanced
forward in the writings and teachings of Edward Lorenz, a Deming -style or
type of physicist who, in the 1960s, wrote about this subject under the
concept of deterministic non-periodic flow.  Lorenz taught that the solvable
systems are the ones shown in the textbooks because they behave nicely.
Confronted with nonlinear systems, scientists would have to substitute linear
approximations or, in the words of James Glieck, find some other backdoor
approach for textbooks showed only the rare nonlinear techniques that would
yield to such techniques.  

In other words, they did not display sensitive dependence upon initial
conditions.  Nonlinear systems with real chaos were rarely taught and even
more rarely learned.  When people stumbled upon such things, all of their
training argued for dismissing them as aberrations.  Only a handful remembered
that the solvable, orderly, linear systems were the aberrations.  For example,
the mathematician Stanislaw Ulam remarked *to call the study of chaos
nonlinear science was like calling zoologogy the study of nonelephant
animals.*

A readable essay on linearity, nonlinearity, and the historical use of
computers in understanding the difference is *Experimental Mathematics: the
Role of Computation in Nonlinear Science* by David Campbell, James
Crutchfield, J.D. Farmer, and Erica Jen (Communications of the Association for
Computing Machinery 28, 1985, p.p. 374-384). 

Ulam also describes the origin of another important thread in the
understanding of nonlinearity, the Fermi-Pasta-Ulam theorem.  Looking for
problems that could be computed on the new MANIAC computer at Los Alamos,
these scientists tried a  dynamical system that was simply vibrating string,
which was a simple model having a physically correct nonlinear term, in Ulam's
words.  *The results were extremely different from what even Fermi, with his
great knowledge of wave motions, had expected.  To our surprise, the string
played a game of musical chairs.*  Fermi considered the results unimportant at
the time, and they were never widely published until a few mathematicians and
physicists followed them up and the theorem became part of the Los Alamos lore
surrounding nonlinearity.

When and if we get into the topic of Complexity, Chaos, and Deming, these
issues will help us to see nonlinear systems in a new way.  Perhaps.

Frank Voehl (FVoehl@aol.com)
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