The Natural Numbersby Richard Courant, Herbert Robbins, Ian Stewart
What Is Mathematics?: An Elementary Approach to Ideas and Methods, 1969
Number is the basis of modern mathematics. But what is number? What does it mean to say that half + half = 1, half x half = quarter, and (-1)(-1) = 1 ? We learn in school the mechanics of handling fractions and negative numbers, but for a real understanding of the number system we must go back to simpler elements. While the Greeks chose the geometrical concepts of point and line as the basis of their mathematics, it has become the modern guiding principle that all mathematical statements should be reducible ultimately to statements about the natural numbers, 1, 2, 3 ... . "God created the natural numbers; everything else is man's handiwork." In these words Leopold Kronecker (1823-1891) pointed out the safe ground on which the structure of mathematics can be built.
Created by the human mind to count the objects in various assemblages, numbers have no reference to the individual characteristics of the objects counted. The number six is an abstraction from all actual collections containing six things; it does not depend on any specific qualities of these things or on the symbols used. Only at a rather advanced stage of intellectual development does the abstract character of the idea of number become clear. To children, numbers always remain connected with tangible objects such as fingers or beads, and primitive languages display a concrete number sense by providing different sets of number words for different types of objects.
Fortunately, the mathematician such need not be concerned with the philosophical nature of the transition from collections of concrete objects to the abstract number concept. We shall therefore accept the natural numbers as given, together with the two fundamental operations, addition and multiplication, by which they may be combined.
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