This is very different from what happens in cases (a) and (b). For in these cases complex and seemingly random results are obtained even on the first of the previous two pages [150, 151]—when the original number has a very simple digit sequence. And the point is that these maps actually do intrinsically generate complexity and randomness; they do not just transcribe it when it is inserted in their initial conditions.

In the context of the approach I have developed in this book this distinction is easy to understand. But with the traditional mathematical approach, things can get quite confused. The main issue—already mentioned at the beginning of this chapter—is that in this approach the only attribute of numbers that is usually considered significant is their size. And this means that any issue based on discussing explicit digit sequences for numbers—and whether for example they are simple or complicated—tends to seem at best bizarre.

Indeed, thinking about numbers purely in terms of size, one might imagine that as soon as any two numbers are sufficiently close in size they would inevitably lead to results that are somehow also close. And in fact this is for example the basis for much of the formalism of calculus in traditional mathematics.

But the essence of the so-called chaos phenomenon is that there are some systems where arbitrarily small changes in the size of a number can end up having large effects on the results that are produced. And the shift map shown as case (d) on the previous two pages [150, 151] turns out to be a classic example of this.

The pictures at the top of the facing page show what happens if one uses as the initial conditions for this system two numbers whose sizes differ by just one part in a billion billion. And looking at the plots of sizes of numbers produced, one sees that for quite a while these two different initial conditions lead to results that are indistinguishably close. But at some point they diverge and soon become quite different.

And at least if one looks only at the sizes of numbers, this seems rather mysterious. But as soon as one looks at digit sequences, it immediately becomes much clearer. For as the pictures at the top of the facing page show, the fact that the numbers which are used as initial conditions differ only by a very small amount in size just means that their first several digits are the same. And for a while these digits are

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