simple programs to brains to our whole universe, the principle implies that there is a basic equivalence that makes the same fundamental phenomena occur, and allows the same basic scientific ideas and methods to be used. And it is this that is ultimately responsible for the great power of the new kind of science that I describe in this book.

## Relations to Other Areas

Mathematics. It is usually assumed that mathematics concerns itself with the study of arbitrarily general abstract systems. But this book shows that there are actually a vast range of abstract systems based on simple programs that traditional mathematics has never considered. And because these systems are in many ways simpler in construction than most traditional systems in mathematics it is possible with appropriate methods in effect to go further in investigating them.

Some of what one finds are then just unprecedentedly clear examples of phenomena already known in modern mathematics. But one also finds some dramatic new phenomena. Most immediately obvious is a very high level of complexity in the behavior of many systems whose underlying rules are much simpler than those of most systems in standard mathematics textbooks.

And one of the consequences of this complexity is that it leads to fundamental limitations on the idea of proof that has been central to traditional mathematics. Already in the 1930s Gödel's Theorem gave some indications of such limitations. But in the past they have always seemed irrelevant to most of mathematics as it is actually practiced.

Yet what the discoveries in this book show is that this is largely just a reflection of how small the scope is of what is now considered mathematics. And indeed the core of this book can be viewed as introducing a major generalization of mathematics—with new ideas and methods, and vast new areas to be explored.

The framework I develop in this book also shows that by viewing the process of doing mathematics in fundamentally computational terms it becomes possible to address important issues about the foundations even of existing mathematics.