Notation

In the main text, I have almost entirely avoided any kind of formal symbolic notation—usually relying instead on diagrammatic pictures. In these notes, however, it will often be convenient to use such notation to give precise and compact representations of objects and operations. In the past, essentially the only large-scale notation available for theoretical science has been traditional mathematical notation. But on its own this would do me little good—for I need to represent not only traditional mathematics, but also more general rules and programs, as well as procedures and algorithms. But one of the reasons I created the Mathematica language was precisely to provide a much more general notation. So in these notes I use this language throughout as my notation. And this has many important advantages—and indeed it is hard to imagine that I would ever have been able to write these notes without it. One point is that it is completely uniform and standardized: there can never be any hidden assumptions or ambiguity about what a particular piece of notation means, since ultimately it is defined by the actual Mathematica software system and its documentation (see below). In cases where there is traditional mathematical notation for something, the corresponding Mathematica notation is normally almost identical—though occasionally a few details are changed to avoid ambiguity. The concept that everything is a symbolic expression allows Mathematica notation, however, to represent essentially any kind of abstract object. And when it comes to procedures and algorithms, the primitives in the Mathematica language are chosen to make typical steps easy to represent—with the result that a single line of Mathematica can often capture what would otherwise require many paragraphs of English text (and large amounts of pseudocode, or lower-level computer language code). Another very important practical feature of Mathematica notation is that by now a large number of people are familiar with it—certainly more than are for example familiar with sophisticated traditional notation in, say, mathematical logic. And the final and very critical advantage of Mathematica notation is that one can not only read it, but also actually execute it on a computer, and interact with it. And this makes it both vastly easier to apply and build on, and also easier to analyze and understand.