Chapter 4: Systems Based on Numbers

Section 7: Iterated Maps and the Chaos Phenomenon

Mathematical perspectives [on iterated map complexity]

Mathematicians may be confused by my discussion of complexity in iterated maps. The first point to make is that the issues I am studying are rather different from the ones that are traditionally studied in the mathematics of these systems. The next point is that I have specifically chosen not to make the idealizations about numbers and operations on numbers that are usually made in mathematics.

In particular, it is usually assumed that performing some standard mathematical operation, such as taking a square root, cannot have a significant effect on the system one is studying. But in trying to track down the origins of complex behavior, the effects of such operations can be significant. Indeed, as we saw on page 141, taking square roots can for example generate seemingly random digit sequences.

Many mathematicians may object that digit sequences are just too fragile an entity to be worth studying. They may argue that it is only robust and invariant concepts that are useful. But robustness with respect to mathematical operations is a different issue from robustness with respect to computational operations. Indeed, we will see later in this book that large classes of digit sequences can be considered equivalent with respect to computational operations, but these classes are quite different ones from those that are considered equivalent with respect to mathematical operations.

Image Source Notebooks:

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