Computing

From elementary school to high school we learn to compute with one form of data: numbers. Our first use of numbers is to count real things, say, three apples, five friends, twelve bagels. Later we use numbers without any appeal to concrete objects, but we have learned that numbers represent information in the real world.

Computing with software is algebra for all kinds of data, not just numbers. Nowadays, computer programs process representations of music, molecules, law cases, electrical diagrams, architectures of houses, and poems. Fortunately, we have learned to represent information with other forms of data than just numbers. Otherwise, computing and programming would become extremely tedious tasks.

Above all, we shouldn't forget that computing means manipulating data through proper basic operations. Some operations create new values. Others extract values from values. Yet others modify values. Finally, there are also basic operations for determining to which class a piece of data belongs. Built-in operations and functions are of course just another class of data. Definition is value creation; application is a form of value extraction.

When we define a function, we combine basic data operations. There are two fundamental mechanisms for combining functions: function composition and conditional expressions. The former means that the result of one function becomes the argument of another one. The latter represents a choice among several possibilities. When we eventually apply a function, we trigger a computation.

In this book we have studied the laws of basic operations and the laws of operation combination. Using these laws we can understand, in principle, how any function processes its input data and how it produces its results and effects. Because the computer is extremely fast and good at using these laws, it can perform such evaluations for more data and for larger programs than we can do with paper and pencil.