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

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. Frege begins by noting the increased desire for logical strictness in mathematical demonstrations which distinguishes modern mathematicians from their predecessors, and points out that this must lead to a critical investigation of the definition of number. He proceeds to show the inadequacy of previous philosophical theories, especially of the “synthetic a priori” theory of Kant and the empirical theory of Mill. This brings him to the question: What kind of object is it that number can properly be ascribed to? He points out that physical things may be regarded as one or many: for example, if a tree has a thousand leaves, they may be taken altogether as constituting its foliage, which would count as one, not as a thousand; and one pair of boots is the same object as two boots. It follows that physical things are not the subjects of which number is properly predicated; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. This leads to a discussion of the very prevalent view that number is really something psychological and subjective, a view which Frege emphatically rejects. “Number,” he says, “is as little an object of psychology or an outcome of psychical processes as the North Sea…. The botanist wishes to state something which is just as much a fact when he gives the number of petals in a flower as when he gives its colour. The one depends as little as the other upon our caprice. There is therefore a certain similarity between number and colour; but this does not consist in the fact that both are sensibly perceptible in external things, but in the fact that both are objective” (p. 34). “I distinguish the objective,” he continues, “from the palpable, the spatial, the actual. The earth's axis, the centre of mass of the solar system, are objective, but I should not call them actual, like the earth itself” (p. 35). He concludes that number is neither spatial and physical, nor subjective, but non-sensible and objective. This conclusion is important, since it applies to all the subject-matter of mathematics and logic. Most philosophers have thought that the physical and the mental between them exhausted the world of being. Some have argued that the objects of mathematics were obviously not subjective, and therefore must be physical and empirical; others have argued that they were obviously not physical, and therefore must be subjective and mental. Both sides were right in what they denied, and wrong in what they asserted; Frege has the merit of accepting both denials, and finding a third assertion by recognising the world of logic, which is neither mental nor physical. The fact is, as Frege points out, that no number, not even 1, is applicable to physical things, but only to general terms or descriptions, such as “man,” “satellite of the earth,” “satellite of Venus.” The general term “man” is applicable to a certain number of objects: there are in the world so and so many men. The unity which philosophers rightly feel to be necessary for the assertion of a number is the unity of the general term, and it is the general term which is the proper subject of number. And this applies equally when there is one object or none which falls under the general term. “Satellite of the earth” is a term only applicable to one object, namely, the moon. But “one” is not a property of the moon itself, which may equally well be regarded as many molecules: it is a property of the general term “earth's satellite.” Similarly, 0 is a property of the general term “satellite of Venus,” because Venus has no satellite. Here at last we have an intelligible theory of the number 0. This was impossible if numbers applied to physical objects, because obviously no physical object could have the number 0. Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences. Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. Two general terms, such as “man” and “featherless biped,” which are applicable to the same collection of objects, will obviously have the same number of instances; thus the number depends upon the class, not upon the selection of this or that general term to describe it, provided several general terms can be found to describe the same class. But some general term is always necessary in order to describe a class. Even when the terms are enumerated, as “this and that and the other,” the collection is constituted by the general property of being either this, or that, or the other, and only so acquires the unity which enables us to speak of it as one collection. And in the case of an infinite class, enumeration is impossible, so that description by a general characteristic common and peculiar to the members of the class is the only possible description. Here, as we see, the theory of number to which Frege was led by purely logical considerations becomes of use in showing how infinite classes can be amenable to number in spite of being incapable of enumeration. Frege next asks the question: When do two collections have the same number of terms? In ordinary life, we decide this question by counting; but counting, as we saw, is impossible in the case of infinite collections, and is not logically fundamental with finite collections. We want, therefore, a different method of answering our question. An illustration may help to make the method clear. I do not know how many married men there are in England, but I do know that the number is the same as the number of married women. The reason I know this is that the relation of husband and wife relates one man to one woman and one woman to one man. A relation of this sort is called a one-one relation. The relation of father to son is called a one-many relation, because a man can have only one father but may have many sons; conversely, the relation of son to father is called a many-one relation. But the relation of husband to wife (in Christian countries) is called one-one, because a man cannot have more than one wife, or a woman more than one husband. Now, whenever there is a one-one relation between all the terms of one collection and all the terms of another severally, as in the case of English husbands and English wives, the number of terms in the one collection is the same as the number in the other; but when there is not such a relation, the number is different. This is the answer to the question: When do two collections have the same number of terms? We can now at last answer the question: What is meant by the number of terms in a given collection? When there is a one-one relation between all the terms of one collection and all the terms of another severally, we shall say that the two collections are “similar.” We have just seen that two similar collections have the same number of terms. This leads us to define the number of a given collection as the class of all collections that are similar to it; that is to say, we set up the following formal definition: “The number of terms in a given class” is defined as meaning “the class of all classes that are similar to the given class.” This definition, as Frege (expressing it in slightly different terms) showed, yields the usual arithmetical properties of numbers. It is applicable equally to finite and infinite numbers, and it does not require the admission of some new and mysterious set of metaphysical entities. It shows that it is not physical objects, but classes or the general terms by which they are defined, of which numbers can be asserted; and it applies to 0 and 1 without any of the difficulties which other theories find in dealing with these two special cases. The above definition is sure to produce, at first sight, a feeling of oddity, which is liable to cause a certain dissatisfaction. It defines the number 2, for instance, as the class of all couples, and the number 3 as the class of all triads. This does not seem to be what we have hitherto been meaning when we spoke of 2 and 3, though it would be difficult to say what we had been meaning. The answer to a feeling cannot be a logical argument, but nevertheless the answer in this case is not without importance. In the first place, it will be found that when an idea which has grown familiar as an unanalysed whole is first resolved accurately into its component parts—which is what we do when we define it—there is almost always a feeling of unfamiliarity produced by the analysis, which tends to cause a protest against the definition. In the second place, it may be admitted that