concerned with showing that even in traditional mathematical models, certain simple discrete changes could still occur. In this book I do not start from any assumption of continuity—and the types of behavior I study tend to be vastly more complex than those in catastrophe theory.

Chaos Theory. The field of chaos theory is based on the observation that certain mathematical systems behave in a way that depends arbitrarily sensitively on the details of their initial conditions. First noticed at the end of the 1800s, this came into prominence after computer simulations in the 1960s and 1970s. Its main significance is that it implies that if any detail of the initial conditions is uncertain, then it will eventually become impossible to predict the behavior of the system. But despite some claims to the contrary in popular accounts, this fact alone does not imply that the behavior will necessarily be complex. Indeed, all that it shows is that if there is complexity in the details of the initial conditions, then this complexity will eventually appear in the large-scale behavior of the system. But if the initial conditions are simple, then there is no reason for the behavior not to be correspondingly simple. What I show in this book, however, is that even when their initial conditions are very simple there are many systems that still produce highly complex behavior. And I argue that it is this phenomenon that is for example responsible for most of the obvious complexity we see in nature.

Complexity Theory. My discoveries in the early 1980s led me to the idea that complexity could be studied as a fundamental independent phenomenon. And gradually this became quite popular. But most of the scientific work that was done ended up being based only on my earliest discoveries, and being very much within the framework of one or another of the existing sciences—with the result that it managed to make very little progress on any general and fundamental issues. One feature of the new kind of science that I describe in this book is that it finally makes possible the development of a basic understanding of the general phenomenon of complexity, and its origins.

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