Chapter 4: Systems Based on Numbers

Section 1: The Notion of Numbers

A note for mathematicians [about numbers]

Some mathematicians will at first find what I say in this chapter quite bizarre. It may help however to point out that the traditional view of numbers already shows signs of breaking down in many studies of dynamical systems done over the past few decades. Thus for example, instead of getting results in terms of continuous functions, Cantor sets very often appear. Indeed, the symbolic dynamics approach that is often used in dynamical systems theory is quite close to the digit sequence approach I use here—Markov partitions in dynamical systems theory are essentially just generalizations of digit expansions.

However, in the cases that are analyzed in dynamical systems theory, only shifts and other very simple operations are typically performed on digit sequences. And as a result, most of the phenomena that I discuss in this chapter have not been seen in work done in dynamical systems theory.

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