all its development over the past few thousand years mathematics itself has continued to concentrate only on rather specific types of abstract systems—most often ones somehow derived from arithmetic or geometry. But the new kind of science that I describe in this book introduces what are in a sense much more general abstract systems, based on rules of essentially any type whatsoever.
One might have thought that such systems would be too diverse for meaningful general statements to be made about them. But the crucial idea that has allowed me to build a unified framework for the new kind of science that I describe in this book is that just as the rules for any system can be viewed as corresponding to a program, so also its behavior can be viewed as corresponding to a computation.
Traditional intuition might suggest that to do more sophisticated computations would always require more sophisticated underlying rules. But what launched the whole computer revolution is the remarkable fact that universal systems with fixed underlying rules can be built that can in effect perform any possible computation.
The threshold for such universality has however generally been assumed to be high, and to be reached only by elaborate and special systems like typical electronic computers. But one of the surprising discoveries in this book is that in fact there are systems whose rules are simple enough to describe in just one sentence that are nevertheless universal. And this immediately suggests that the phenomenon of universality is vastly more common and important—in both abstract systems and nature—than has ever been imagined before.
But on the basis of many discoveries I have been led to a still more sweeping conclusion, summarized in what I call the Principle of Computational Equivalence: that whenever one sees behavior that is not obviously simple—in essentially any system—it can be thought of as corresponding to a computation of equivalent sophistication. And this one very basic principle has a quite unprecedented array of implications for science and scientific thinking.
For a start, it immediately gives a fundamental explanation for why simple programs can show behavior that seems to us complex. For like other processes our own processes of perception and analysis can be