Our Knowledge of the External World as a Field for Scientific Method in Philosophy

by Bertrand Russell

The way in which the problem is expounded in the above discussion is worthy of Galileo, but the solution suggested is not the right one. It is actually the case that the number of square (finite) numbers is the same as the number of (finite) numbers. The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. This is only an instance of the fact, now familiar to mathematicians, that the limit of a function as the variable approaches a given point may not be the same as its value when the variable actually reaches the given point. But although the infinite numbers which Galileo discusses are equal, Cantor has shown that what Simplicius could not conceive is true, namely, that there are an infinite number of different infinite numbers, and that the conception of greater and less can be perfectly well applied to them. The whole of Simplicius's difficulty comes, as is evident, from his belief that, if greater and less can be applied, a part of an infinite collection must have fewer terms than the whole; and when this is denied, all contradictions disappear. As regards greater and less lengths of lines, which is the problem from which the above discussion starts, that involves a meaning of greater and less which is not arithmetical. The number of points is the same in a long line and in a short one, being in fact the same as the number of points in all space. The greater and less of metrical geometry involves the new metrical conception of congruence, which cannot be developed out of arithmetical considerations alone. But this question has not the fundamental importance which belongs to the arithmetical theory of infinity.

(2) Non-inductiveness.—The second property by which infinite numbers are distinguished from finite numbers is the property of non-inductiveness. This will be best explained by defining the positive property of inductiveness which characterises the finite numbers, and which is named after the method of proof known as “mathematical induction.”

Let us first consider what is meant by calling a property “hereditary” in a given series. Take such a property as being named Jones. If a man is named Jones, so is his son; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his descendants in the direct male line are called Jones; this follows from the fact that the property is hereditary. Now, instead of the relation of father and son, consider the relation of a finite number to its immediate successor, that is, the relation which holds between 0 and 1, between 1 and 2, between 2 and 3, and so on. If a property of numbers is hereditary with respect to this relation, then if it belongs to (say) 100, it must belong also to all finite numbers greater than 100; for, being hereditary, it belongs to 101 because it belongs to 100, and it belongs to 102 because it belongs to 101, and so on—where the “and so on” will take us, sooner or later, to any finite number greater than 100. Thus, for example, the property of being greater than 99 is hereditary in the series of finite numbers; and generally, a property is hereditary in this series when, given any number that possesses the property, the next number must always also possess it.

It will be seen that a hereditary property, though it must belong to all the finite numbers greater than a given number possessing the property, need not belong to all the numbers less than this number. For example, the hereditary property of being greater than 99 belongs to 100 and all greater numbers, but not to any smaller number. Similarly, the hereditary property of being called Jones belongs to all the descendants (in the direct male line) of those who have this property, but not to all their ancestors, because we reach at last a first Jones, before whom the ancestors have no surname. It is obvious, however, that any hereditary property possessed by Adam must belong to all men; and similarly any hereditary property possessed by 0 must belong to all finite numbers. This is the principle of what is called “mathematical induction.” It frequently happens, when we wish to prove that all finite numbers have some property, that we have first to prove that 0 has the property, and then that the property is hereditary, i.e. that, if it belongs to a given number, then it belongs to the next number. Owing to the fact that such proofs are called “inductive,” I shall call the properties to which they are applicable “inductive” properties. Thus an inductive property of numbers is one which is hereditary and belongs to 0.

Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to 0 and are hereditary, they belong to 1; therefore, since they are hereditary, they belong to 2, and so on; by twenty-nine repetitions of such arguments we show that they belong to 29. We may define the “inductive” numbers as all those that possess all inductive properties; they will be the same as what are called the “natural” numbers, i.e. the ordinary finite whole numbers. To all such numbers, proofs by mathematical induction can be validly applied. They are those numbers, we may loosely say, which can be reached from 0 by successive additions of 1; in other words, they are all the numbers that can be reached by counting.

But beyond all these numbers, there are the infinite numbers, and infinite numbers do not have all inductive properties. Such numbers, therefore, may be called non-inductive. All those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers. The first of the infinite numbers has no immediate predecessor, because there is no greatest finite number; thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one, and the step-by-step method of proof fails. This is another reason for the supposed self-contradictions of infinite numbers. Many of the most familiar properties of numbers, which custom had led people to regard as logically necessary, are in fact only demonstrable by the step-by-step method, and fail to be true of infinite numbers. But so soon as we realise the necessity of proving such properties by mathematical induction, and the strictly limited scope of this method of proof, the supposed contradictions are seen to contradict, not logic, but only our prejudices and mental habits.

The property of being increased by the addition of 1—i.e. the property of non-reflexiveness—may serve to illustrate the limitations of mathematical induction. It is easy to prove that 0 is increased by the addition of 1, and that, if a given number is increased by the addition of 1, so is the next number, i.e. the number obtained by the addition of 1. It follows that each of the natural numbers is increased by the addition of 1. This follows generally from the general argument, and follows for each particular case by a sufficient number of applications of the argument. We first prove that 0 is not equal to 1; then, since the property of being increased by 1 is hereditary, it follows that 1 is not equal to 2; hence it follows that 2 is not equal to 3; if we wish to prove that 30,000 is not equal to 30,001, we can do so by repeating this reasoning 30,000 times. But we cannot prove in this way that all numbers are increased by the addition of 1; we can only prove that this holds of the numbers attainable by successive additions of 1 starting from 0. The reflexive numbers, which lie beyond all those attainable in this way, are as a matter of fact not increased by the addition of 1.

The two properties of reflexiveness and non-inductiveness, which we have considered as characteristics of infinite numbers, have not so far been proved to be always found together. It is known that all reflexive numbers are non-inductive, but it is not known that all non-inductive numbers are reflexive. Fallacious proofs of this proposition have been published by many writers, including myself, but up to the present no valid proof has been discovered. The infinite numbers actually known, however, are all reflexive as well as non-inductive; thus, in mathematical practice, if not in theory, the two properties are always associated. For our purposes, therefore, it will be convenient to ignore the bare possibility that there may be non-inductive non-reflexive numbers, since all known numbers are either inductive or reflexive.

When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers.

The logical definition of numbers, though it seems an essential support to the theory of infinite numbers, was in fact discovered independently and by a different man. The theory of infinite numbers—that is to say, the arithmetical as opposed to the logical part of the theory—was discovered by Georg Cantor, and published by him in 1882–3.[51] The definition of number was discovered about the same time by a man whose great genius has not received the recognition it deserves—I mean Gottlob Frege of Jena. His first work, Begriffsschrift, published in 1879, contained the very important theory of hereditary properties in a series to which I alluded in connection with inductiveness. His definition of number is contained in his second work, published in 1884, and entitled Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl.[52] It is with this book that the logical theory of arithmetic begins, and it will repay us to consider Frege's analysis in some detail.