Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.2.WHAT IS A NUMBER?

As this is probably difficult to answer, let us ask a different question: “Can you give an example of a number?” Probably, the answer will be something like 5, or five. But then, what about V or ||||| or 3 + 2 or cinque?

Clearly, the symbol 5 is not a number—it is just a symbol. It is a common mistake to take a symbolic representation for the “real thing.” But this mistake is very understandable because our everyday language does not distinguish between them and calls everything a number. But as long as we talk about the “meaning of numbers,” we have to be precise: A symbol, like 5, serves to designate a number, but it is not the number itself. Indeed, the number five can be represented by quite different symbols—for example, by the Roman symbol V or the Chinese 五. The number five can even exist without any written symbol at all—it was probably used by Homo sapiens long before the invention of writing and expressed by showing the fingers of one hand.

In the same way, the spoken word five (a combination of sounds) and the written word five (a combination of letters) are just representations of the number five. The number itself is an abstract idea, and it can be expressed in many different ways and by other words. For example, the word for five in French is cinque, in German it is fünf, and in Japanese it is go. In any case, all these different representations—symbol, word, sound, or even a dot pattern like —should evoke the same idea of the number five. In linguistics, a word designating a number, like five, or twenty-four (no matter whether it is spoken or written), is called a numeral or a number word.

So far, we have not really explained what a number is; rather, we have said what it is not: It is not a symbol or a number word, which are just names. We are going to distinguish between the abstract idea number and the words or symbols used to designate numbers. The abstract idea is unique and invariable; symbols and words are a mere matter of convention and hence quite arbitrary. Moreover, there is a difference between the idea of a number and its different (although related) applications. The number described by the symbol 5 could be used, for example, to describe the fifth place in a sequence (as an ordinal number) or the number of objects in a collection (as a cardinal number) or the length of a flagpole in yards (as a measuring number).

In this chapter, we want to describe the “thing behind the symbol,” the genesis, true meaning, and scope of the abstract idea number, which belongs to the greatest inventions of humankind.

In order to approach this concept, we shall first concentrate on the most basic aspect of numbers: their ultimate and original raison d’être. A first reason for the existence of numbers is that they can be used for counting.