Mathematical Preliminaries

An African explorer conversantwith the language of the Hottentot tribe asks a native, “How many children do you have?” The tribesman answers, “Many.” The determined explorer persists on. He shows his index finger, meaning “one?” Promptly comes the answer, “no.” He adds his middle finger, meaning “two”; the answer is “no”; “three?,” “no”; “four,” “no.” Now all the five fingers on the explorer's right hand are straight. Answer comes, “yes.” The puzzled explorer experiments with another tribesman. Over the next week, he discovers that they have only three kinds of numbers, one, two, and many.

It is an old story, but perhaps not without morals. The Hottentot tribesman does not have a way of naming the numbers more than two. How does he manage his cattle?

Our mathematical tradition has gone so far and so deep that it is indeed difficult to imagine living without it. In this small chapter, we will discuss a fragment of this tradition so that the rituals of learning the theory of computation can be conducted relatively easily. In the process, we will fix our notation.