Definition of mathematics

When I use the term "mathematics" in this book what I mean is that field of human endeavor that has in practice traditionally been called mathematics. One could in principle imagine defining mathematics to encompass all studies of abstract systems, and indeed this was in essence the definition that I had in mind when I chose the name Mathematica. But in practice mathematics has defined itself to be vastly narrower, and to include, for example, nothing like the majority of the programs that I discuss in this book. Indeed, in many respects, what is called mathematics today can be seen as a direct extension of the particular notions of arithmetic and geometry that apparently arose in Babylonian times. Typical dictionary definitions reflect this by describing mathematics as the study of number and space, together with their abstractions and generalizations. And even logic—an abstract system that dates from antiquity—is not normally considered part of mainstream mathematics. Particularly over the past century the defining characteristic of research in mathematics has increasingly been the use of theorem and proof methodology. And while some generalization has occurred in the types of systems being studied, it has usually been much limited by the desire to maintain the validity of some set of theorems (see page 793). This emphasis on theorems has also led to a focus on equations that statically state facts rather than on rules that define actions, as in most of the systems in this book. But despite all these issues, many mathematicians implicitly tend to assume that somehow mathematics as it is practiced is universal, and that any possible abstract system will be covered by some area of mathematics or another. The results of this book, however, make it quite clear that this is not the case, and that in fact traditional mathematics has reached only a tiny fraction of all the kinds of abstract systems that can in principle be studied.