Wolfram, Stephen. A New Kind of Science. Champaign, IL: Wolfram Media,

Inc., 2002. This is another blockbuster book by Stephen Wolfram, the creator of

Mathematica. The title is ambitious to say the least. The systematic study of

cellular automata -- the main research tool used by Wolfram for his book -- may well

become a new scientific discipline. But Wolfram's vision is much broader. He feels

he has identified some global principles, and he is deeply exploring the boundaries

and the interactions between the mental (mathematical) and physical worlds. This

makes him a major pioneer in Observer Physics (OP).

A key theme of Wolfram's work is what he calls the principle of "computational

equivalence" -- the ability of mathematics to mimic the physical world. If I

understand him correctly, he means by this idea that as a system grows in complexity,

there is a threshold beyond which all systems behave the same. (Does this mean that

God -- and the aliens -- are no smarter than we are? Is this anthropomorphic

jingoism? Or does it just mean that everyone is equally good at making a mess?)

Perhaps Wolfram is talking here about the limit where complexity becomes pure

randomness. Wolfram's idea of "Computational Irreducibility" refers, it seems, to

the notion that at some point any model of a system essentially reproduces the system

in another medium with the same complexity. Related to this is the mathematical

notion of "universality" -- that a certain program can be set up to emulate a whole

class of programs. A similar idea is the notion that you can emulate algebra with

geometry and vice versa.

Wolfram's starting point in the creation of his "New Kind of Science" was his

discovery that very simple programs can produce great complexity and even

randomness. This is not really a new discovery. Mathematicians have known this

since the discovery that a simple ratio such as pi is an irrational quantity that

generates a decimal with an infinite string of random digits. Linguists are familiar

with ways to generate randomness (or at least great complexity) from simple

grammars, and more recently the fractal and chaos people have made a great deal of

progress generating infinite complexity with simple mathematical structures such as

the Mandelbrot set. (In OP we discuss the example of the growth equation with the

Verhulst factor included so as to make the system nonlinear.) What may be special

about Wolfram's approach is the importance and generality he attaches to this

principle. Its flowering as a principle definitely is a product of the computer age