NUMBER, an abstract term for the integers i, 2, 3, . . . as well as for the various mathematical generalizations of the integers, such as fractions and irrational quantities. It is also used as a concrete term. The practice of specifying the multiplicity of classes of objects by means of marks (integers) is found at an early stage in all civilizations, and has led to various systems of numeration (see NUMERALS and ARITHMETIC). The ordinary decimal system is now in general use. While the particular system of numeration to be adopted is a matter of no theoretic impor tance, nevertheless, to insure the free development of the concept of number, some symbol for zero must be introduced, as well as symbols for addition, subtraction, multiplication and division. Over two thousand years ago the Greek philosopher Pythagoras (q.v.) foresaw the fundamental role of number and gave it a central place in his philosophy. The actual progress made however in the development of the concept of number is not to be found in the many philosophical speculations on the subject, but rather in the technical researches of mathematicians. In what follows our purpose will be to outline the essential ideas involved in this development.
The simplest and most fundamental type of number is that of the integers. This type may be analyzed in the following manner. Consider two classes of objects A and B, each of which has only a finite number of constituent elements. Suppose that the ele ments of A and of B can be put in "one-to-one correspondence." In other words suppose that the elements of A can be paired with the elements of B just as the fingers of one hand can be paired with those of the other. Suppose, furthermore, that the elements of A and of a third class C can be put in one-to-one correspondence in the same way. It is then evident that the ele ments of B and of C can also be put in one-to-one correspondence. Thus the various classes B, C, . . . in one-to-one correspondence with A are also in one-to-one correspondence with one another.
An integer may be defined as a mark associated with such a collection of classes A, B, C . . . ; e.g., the integer 5 is the mark associated with the class of fingers on a hand and with all other classes in one-to-one correspondence with this particular class. It is reasonable to believe that even before the dawn of primitive civilization the first step in the use of the integer must have been taken in some such way as the following. A herdsman might set one stone aside for each animal in the herd. By use of the pile of stones he would be able to determine at any time whether or not his herd was complete. Later a series of scratches might serve
as a simplified mark, and it is easy to see how the long continued use of such methods would lead to complete systems of numeration.
In basing the notion of the integer upon that of one-to-one correspondence, the classes were assumed to have only a finite number of constituent elements. Since the notion of a finite number of elements is thus involved at the outset, our analysis might seem to involve circular reasoning. This is not really the case since the notion of one-to-one correspondence of itself furnishes a method of distinguishing between finite and in finite classes. We may define an "infinite" class as one which can be put in one-to-one correspondence with a part of itself. For example, the class of integers 1, 2, 3, . . . will be infinite according to this definition since it may be put in one-to-one cor respondence with the class of even integers 2, 4, 6, . . . ; we need only make an integer n of the first class correspond to its double 2n of the second class : Likewise we may define a "finite" class as one which cannot be put in one-to-one correspondence with a part of itself.