Defining Complexity

Much of what I have done in this book has been concerned in one way or another with phenomena associated with complexity. But just as one does not need a formal definition of life in order to study biology, so also it has not turned out to be necessary so far in this book to have a formal definition of complexity. Nevertheless, following our discussion of randomness in the previous section, we are now in a position to consider how the notion of complexity might be formally defined.

In everyday language, when we say that something seems complex what we typically mean is that we have not managed to find any simple description of it—or at least of those features of it in which we happen to be interested. But the goal of perception and analysis is precisely to find such descriptions, so when we say that something seems complex, what we are effectively saying is that our powers of perception and analysis have failed on it.

As we discussed two sections ago, there are two ways in which perception and analysis can typically operate. First, they can just throw away details in which we are not interested. And second, they can remove redundancy that is associated with any regularities that they manage to recognize.

The definition of randomness that we discussed in the previous section was based on the failure of the second of these two functions. For what it said was that something should be considered random if our standard methods of perception and analysis could not find any short description from which the thing could faithfully be reproduced.

But in defining complexity we need to consider both functions of perception and analysis. For what we want to know is not whether a simple or short description can be found of every detail of something, but merely whether such a description can be found of those features in which we happen to be interested.

In everyday language, the terms "complexity" and "randomness" are sometimes used almost interchangeably. And for example any of the three pictures at the top of the next page could potentially be referred to as either "quite random" or "quite complex". But if one chooses to look

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