Tuesday, September 18, 2012

The Simple Origins of Complexities

A New Kind of Science by Stephen Wolfram (Wolfram Media 2002)..

“Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world.  My purpose in this book is to initiate another such transformation, and to introduce a new kind of science that is based on the much more general types of rules that can be embodied in simple computer programs.” 

For example:
  1. All squares are white.
  2. Start with a black square.
  3. A cell is made black when it or either of its neighbors were black on the step before.
Or, for example:
  1. All squares are white.
  2. Start with a black square.
  3. A cell is black if either of its neighbors was black on the step before.
  4. A cell is white if both of its neighbors were white on the step before.
 Beginning with rules like these (and hundreds of others approximately no more complex), fantastic images, patterns, random and pseudo-random runs evolve. 

We know that truth is self-supporting.  No truth in biology can contradict a truth in chemistry, physics, psychology, or economics.  If they are, indeed, truths, statements support each other.  Here, too, the simplicity suggested by Stephen Wolfram has been explored by Benoit Mandelbrot.  Fractals of the form
zn+1 = zn2 + c
produce complex geometries based on reiteration of a basic pattern.  These fractals are related the Julia sets, which are also complicated yet generated from simple initial conditions expressed by terse functions.
Julia set (in white)
for the rational function associated to
Newton's method for ƒ:z→z3−1.
Coloring of Fatou set
according to attractor (the roots of ƒ)

Wolfram is associated with two major websites, Wolfram Science and Wolfram Mathematica.  Both offer you the opportunities to download samples, purchase home editions or professional tools.  Wolfram Mathematics was one of the supporters of the television series NUMB3RS.  Having written a couple of fractal programs for the IBM-PC and DEC VAX many years ago, I had some identification for Wolfram when I saw them go by in the credits for NUMB3RS.  My interest this time was sparked by an off-hand comment from Fred Bartlett, a consulting engineer whom I know from Objectivist discussion boards.

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