Chapter 4: Systems Based on Numbers

Section 7: Iterated Maps and the Chaos Phenomenon

Information content of initial conditions

Common sense suggests that it is a quite different thing to specify a simple initial condition containing, say, a single black cell on a white background, than to specify an initial condition containing an infinite sequence of randomly chosen cells. But in traditional mathematics no distinction is usually made between these kinds of specifications. And as a result, mathematicians may find it difficult to understand my distinction between randomness generated intrinsically by the evolution of a system and randomness from initial conditions (see page 299). The distinction may seem more obvious if one considers, for example, sequential substitution systems or cyclic tag systems. For such systems cannot meaningfully be given infinite random initial conditions, yet they can still perfectly well generate highly random behavior. (Their initial conditions correspond in a sense to integers rather than real numbers.)

From Stephen Wolfram: A New Kind of Science [citation]