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Defining the Notion of Randomness

Many times in this book I have said that the behavior of some system or another seems random. But so far I have given no precise definition of what I mean by randomness. And what we will discover in this section is that to come up with an appropriate definition one has no choice but to consider issues of perception and analysis.

One might have thought that from traditional mathematics and statistics there would long ago have emerged some standard definition of randomness. But despite occasional claims for particular definitions, the concept of randomness has in fact remained quite obscure. And indeed I believe that it is only with the discoveries in this book that one is finally now in a position to develop a real understanding of what randomness is.

At the level of everyday language, when we say that something seems random what we usually mean is that there are no significant regularities in it that we can discern—at least with whatever methods of perception and analysis we use.

We would not usually say, therefore, that either of the first two pictures at the top of the facing page seem random, since we can readily recognize highly regular repetitive and nested patterns in them. But the third picture we would probably say does seem random, since at least at the level of ordinary visual perception we cannot recognize any significant regularities in it.

So given this everyday notion of randomness, how can we build on it to develop more precise definitions? The first step is to clarify what it means not to be able to recognize regularities in something. Following the discussion in the previous section, we know that whenever we find regularities, it implies that redundancy is present, and this in turn means that a shorter description can be given. So when we say that we cannot recognize any regularities, this is equivalent to saying that we cannot find a shorter description.

The three pictures on the facing page can always be described by explicitly giving a list of the colors of each of the 6561 cells that they contain. But by using the regularities that we can see in the first two

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