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80 THE NATURE OF MATHEMATICS said : "In what follows we will assume that there are such things as fill up kinds of gaps in the system of rationals (or ratios)." Such a gap is shown by this. The rationals less than | and those greater than form two sets and | divides them. The rationals x such that x 2 is greater than 2 and those o;'s such that a? is less than 2 form two analogous sets, but there is only an analogue to the dividing number if we postulate a number ^/2. Thus by postulation we fill up these subtle gaps in the set of rationals and get a continuous set of real numbers. But we can avoid this postulation. If we define " ^2 " as the name of the class of rationals x such that x z is less than 2 and "()" as the name of the class of rationals x such that x is less than |. Proceeding thus, we arrive at a set of dosses, some of which correspond to rationals, as () to J, but the rest satisfy our need of a set without gaps. There is no reason why we should not say that these classes are the real numbers which include the irrationals. But we must notice that rationals are never real numbers ; is not (|), though analogous to it. We have much the same state of things as in the second chapter, where 2, + 2, and f were dis- tinguished and then deliberately confused because, with the mathematicians, we felt the importance of analogy in calculation. Here again v/e identify () with , and so on. Thus, integers, positive and negative " numbers," ratios, and real " numbers " are all different things : real numbers are classes, ratios and positive and negative numbers are relations. Integers, as we shall see, are classes. Very possibly there is a certain arbitrariness about this, but this is unimportant compared with the fact that in modern mathematics we have reduced the definitions of all " numbers " to logical terms. Whether they are classes or relations or propositions or other logical entities is comparatively unimportant. Integers can be defined as certain classes. Mathematicians like Weierstrass stopped before they got as far as this : they reduced the other numbers of analysis to logical developments out of the conception of integer, and thus freed analysis from any remaining trace of the sway of geometry. But it was obvious that integers had to be defined, if possible, in logical terms. It has long been recognised that two collections consist of the same number of objects if, and only if, these collections
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