purely in terms of size, one should make no distinction between numbers that are sufficiently close in size. And this implies that in choosing initial conditions for a system like the shift map, one should therefore make no distinction between the exact number 1/2 and numbers that are sufficiently close in size to 1/2.
But it turns out that if one picks a number at random subject only to the constraint that its size be in a certain range, then it is overwhelmingly likely that the number one gets will have a digit sequence that is essentially random. And if one then uses this number as the initial condition for a shift map, the results will also be correspondingly random—just like those on the previous page.
In the past this fact has sometimes been taken to indicate that the shift map somehow fundamentally produces randomness. But as I have discussed above, the only randomness that can actually come out of such a system is randomness that was explicitly put in through the details of its initial conditions. And this means that any claim that the system produces randomness must really be a claim about the details of what initial conditions are typically given for it.
I suppose in principle it could be that nature would effectively follow the same idealization as in traditional mathematics, and would end up picking numbers purely according to their size. And if this were so, then it would mean that the initial conditions for systems like the shift map would naturally have digit sequences that are almost always random.
But this line of reasoning can ultimately never be too useful. For what it says is that the randomness we see somehow comes from randomness that is already present—but it does not explain where that randomness comes from. And indeed—as I will discuss in Chapter 7—if one looks only at systems like the shift map then it is not clear any new randomness can ever actually be generated.
But a crucial discovery in this book is that systems like (a) and (b) on pages 150 and 151 can show behavior that seems in many respects random even when their initial conditions show no sign of randomness and are in fact extremely simple.
Yet the fact that systems like (a) and (b) can intrinsically generate randomness even from simple initial conditions does not mean that they