In this chapter we provide a practical introduction to basic diagrammatic reasoning, namely how to perform computations and solve problems using diagrams. We also demonstrate why diagrams are far better in many ways than traditional mathematical notation. The development and study of diagrammatic languages is a very active area of research, and intuitively obvious aspects of diagrammatic reasoning have actually taken many years to get right. Luckily, the hard work needed to formalise the diagrams in this book has already been done! So, all that remains to do is reap the benefits of a nice, graphical language.

Along the way, we will encounter Dirac notaton. Readers who have previously studied quantum mechanics or quantum information theory may have already seen Dirac notation used in the context of linear maps. Here, we'll explain how it arises as a one-dimensional fragment of the two-dimensional graphical language. Thus, readers not familiar with Dirac notation will learn it as a special case of the graphical notation we use throughout the book.

We also introduce the notion of a process theory, which provides a means of interpreting diagrams by fixing a particular collection of (physical, computational, mathematical, edible, etc.) systems and the processes that these systems might undergo (being heated up, sorted, multiplied by two, cooked, etc.).

As we pointed out in Chapter 1, taking process theories as our starting point represents a substantial departure from standard practice in many disciplines. Rather than forcing ourselves to totally understand single systems before even thinking about how those systems compose and interact, we will seek to understand systems primarily in terms of their interactions with others. Rather than trying to understand Dave the dodo by dissecting him (at which point, he'll look pretty much like any other fat bird), we will turn him loose in the world and see what he does.

This turns out to be very close in spirit to the aims of category theory, which we will meet briefly in the advanced material at the end of this chapter.