Introduction to Mathematical Philosophy

WE have now seen how to define cardinal numbers, and also relation-numbers, of which what are commonly called ordinal numbers are a particular species. It will be found that each of these kinds of number may be infinite just as well as finite. But neither is capable, as it stands, of the more familiar extensions of the idea of number, namely, the extensions to negative, fractional, irrational, and complex numbers. In the present chapter we shall briefly supply logical definitions of these various extensions. One of the mistakes that have delayed the discovery of correct definitions in this region is the common idea that each extension of number included the previous sorts as special cases. It was thought that, in dealing with positive and negative integers, the positive integers might be identified with the original signless integers. Again it was thought that a fraction whose denominator is 1 may be identified with the natural number which is its numerator. And the irrational numbers, such as the square root of 2, were supposed to find their place among rational fractions, as being greater than some of them and less than the others, so that rational and irrational numbers could be taken together as one class, called "real numbers." And when the idea of number was further extended so as to include "complex" numbers, i.e. numbers involving the square root of -1, it was thought that real numbers could be regarded as those among complex numbers in which the imaginary part (i.e. the part [Pg 63] which was a multiple of the square root of -1) was zero. All these suppositions were erroneous, and must be discarded, as we shall find, if correct definitions are to be given. Let us begin with positive and negative integers. It is obvious on a moment's consideration that +1 and -1 must both be relations, and in fact must be each other's converses. The obvious and sufficient definition is that +1 is the relation of to , and -1 is the relation of to . Generally, if is any inductive number, will be the relation of to (for any ), and will be the relation of to . According to this definition, is a relation which is one-one so long as is a cardinal number (finite or infinite) and is an inductive cardinal number. But is under no circumstances capable of being identified with , which is not a relation, but a class of classes. Indeed, is every bit as distinct from as is. Fractions are more interesting than positive or negative integers. We need fractions for many purposes, but perhaps most obviously for purposes of measurement. My friend and collaborator Dr A. N. Whitehead has developed a theory of fractions specially adapted for their application to measurement, which is set forth in Principia Mathematica.[14] But if all that is needed is to define objects having the required purely mathematical properties, this purpose can be achieved by a simpler method, which we shall here adopt. We shall define the fraction as being that relation which holds between two inductive numbers , when . This definition enables us to prove that is a one-one relation, provided neither or is zero. And of course is the converse relation to . [14]Vol. III. * 300 ff., especially 303. From the above definition it is clear that the fraction is that relation between two integers and which consists in the fact that . This relation, like the relation , is by no means capable of being identified with the inductive cardinal number , because a relation and a class of classes are objects [Pg 64] of utterly different kinds.[15] It will be seen that is always the same relation, whatever inductive number may be; it is, in short, the relation of 0 to any other inductive cardinal. We may call this the zero of rational numbers; it is not, of course, identical with the cardinal number 0. Conversely, the relation is always the same, whatever inductive number may be. There is not any inductive cardinal to correspond to . We may call it "the infinity of rationals." It is an instance of the sort of infinite that is traditional in mathematics, and that is represented by "." This is a totally different sort from the true Cantorian infinite, which we shall consider in our next chapter. The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy. [15]Of course in practice we shall continue to speak of a fraction as (say) greater or less than 1, meaning greater or less than the ratio . So long as it is understood that the ratio and the cardinal number 1 are different, it is not necessary to be always pedantic in emphasising the difference. It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one. There is not any difficulty in defining greater and less among ratios (or fractions). Given two ratios and , we shall say that is less than if is less than . There is no difficulty in proving that the relation "less than," so defined, is serial, so that the ratios form a series in order of magnitude. In this series, zero is the smallest term and infinity is the largest. If we omit zero and infinity from our series, there is no longer any smallest or largest ratio; it is obvious that if is any ratio other than zero and infinity, is smaller and is larger, though neither is zero or infinity, so that is neither the smallest [Pg 65] nor the largest ratio, and therefore (when zero and infinity are omitted) there is no smallest or largest, since was chosen arbitrarily. In like manner we can prove that however nearly equal two fractions may be, there are always other fractions between them. For, let and be two fractions, of which is the greater. Then it is easy to see (or to prove) that will be greater than and less than . Thus the series of ratios is one in which no two terms are consecutive, but there are always other terms between any two. Since there are other terms between these others, and so on ad infinitum, it is obvious that there are an infinite number of ratios between any two, however nearly equal these two may be.[16] A series having the property that there are always other terms between any two, so that no two are consecutive, is called "compact." Thus the ratios in order of magnitude form a "compact" series. Such series have many important properties, and it is important to observe that ratios afford an instance of a compact series generated purely logically, without any appeal to space or time or any other empirical datum. [16]Strictly speaking, this statement, as well as those following to the end of the paragraph, involves what is called the "axiom of infinity," which will be discussed in a later chapter. Positive and negative ratios can be defined in a way analogous to that in which we defined positive and negative integers. Having first defined the sum of two ratios and as , we define as the relation of to , where is any ratio; and is of course the converse of . This is not the only possible way of defining positive and negative ratios, but it is a way which, for our purpose, has the merit of being an obvious adaptation of the way we adopted in the case of integers. We come now to a more interesting extension of the idea of number, i.e. the extension to what are called "real" numbers, which are the kind that embrace irrationals. In Chapter I. we had occasion to mention "incommensurables" and their discovery [Pg 66] by Pythagoras. It was through them, i.e. through geometry, that irrational numbers were first thought of. A square of which the side is one inch long will have a diagonal of which the length is the square root of 2 inches. But, as the ancients discovered, there is no fraction of which the square is 2. This proposition is proved in the tenth book of Euclid, which is one of those books that schoolboys supposed to be fortunately lost in the days when Euclid was still used as a text-book. The proof is extraordinarily simple. If possible, let be the square root of 2, so that , i.e. . Thus is an even number, and therefore must be an even number, because the square of an odd number is odd. Now if is even, must divide by 4, for if , then . Thus we shall have , where is half of . Hence , and therefore will also be the square root of 2. But then we can repeat the argument: if , will also be the square root of 2, and so on, through an unending series of numbers that are each half of its predecessor. But this is impossible; if we divide a number by 2, and then halve the half, and so on, we must reach an odd number after a finite number of steps. Or we may put the argument even more simply by assuming that the we start with is in its lowest terms; in that case, and cannot both be even; yet we have seen that, if , they must be. Thus there cannot be any fraction whose square is 2. Thus no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This seems like a challenge thrown out by nature to arithmetic. However the arithmetician may boast (as Pythagoras did) about the power of numbers, nature seems able to baffle him by exhibiting lengths which no numbers can estimate in terms of the unit. But the problem did not remain in this geometrical form. As soon as algebra was invented, the same problem arose as regards the solution of equations, though here it took on a wider form, since it also involved complex numbers. It is clear that fractions can be found which approach nearer [Pg 67] and nearer to having their square equal to 2. We can form an ascending series of fractions all of which have their squares less than 2, but differing from 2 in their later members by less than any assigned amount. That is to say, suppose I assign some small amount in advance, say one-billionth, it will be found that all the terms of our series after a certain one, say the tenth, have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have reached sooner or later a term in the series, say the twentieth, after which all terms would have had squares differing from 2 by less than this still smaller amount. If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-and-so many places, exactly fulfils the above conditions. We can equally well form a descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to later terms of the series, and differing, sooner or later, by less than any assigned amount. In this way we seem to be drawing a cordon round the square root of 2, and it may seem difficult to believe that it can permanently escape us. Nevertheless, it is not by this method that we shall actually reach the square root of 2. If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that, among those whose squares are not less than 2, all have their squares greater than 2. There is no maximum to the ratios whose square is less than 2, and