Introduction to Mathematical Philosophy

If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses—that it is, and that it is not, a member of itself—each implies its contradictory. This is a contradiction. There is no difficulty in manufacturing similar contradictions ad lib. The solution of such contradictions by the theory of types is set forth fully in Principia Mathematica,[27] and also, more briefly, in articles by the present author in the American Journal [Pg 136] of Mathematics,[28] and in the Revue de Metaphysique et de Morale.[29] For the present an outline of the solution must suffice. [27]Vol. I., Introduction, chap. II., * 12 and * 20; vol. II., Prefatory Statement [28]"Mathematical Logic as based on the Theory of Types," vol. XXX., 1908, pp. 222-262. [29]"Les paradoxes de la logique," 1906, pp. 627-650. The fallacy consists in the formation of what we may call "impure" classes, i.e. classes which are not pure as to "type." As we shall see in a later chapter, classes are logical fictions, and a statement which appears to be about a class will only be significant if it is capable of translation into a form in which no mention is made of the class. This places a limitation upon the ways in which what are nominally, though not really, names for classes can occur significantly: a sentence or set of symbols in which such pseudo-names occur in wrong ways is not false, but strictly devoid of meaning. The supposition that a class is, or that it is not, a member of itself is meaningless in just this way. And more generally, to suppose that one class of individuals is a member, or is not a member, of another class of individuals will be to suppose nonsense; and to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything. Thus if there are individuals in the world, and classes of individuals, we cannot form a new class, consisting of both individuals and classes and having members. In this way the attempt to escape from the need for the axiom of infinity breaks down. I do not pretend to have explained the doctrine of types, or done more than indicate, in rough outline, why there is need of such a doctrine. I have aimed only at saying just so much as was required in order to show that we cannot prove the existence of infinite numbers and classes by such conjurer's methods as we have been examining. There remain, however, certain other possible methods which must be considered. Various arguments professing to prove the existence of infinite classes are given in the Principles of Mathematics, § 339 (p. 357). [Pg 137] In so far as these arguments assume that, if is an inductive cardinal, is not equal to , they have been already dealt with. There is an argument, suggested by a passage in Plato's Parmenides, to the effect that, if there is such a number as 1, then 1 has being; but 1 is not identical with being, and therefore 1 and being are two, and therefore there is such a number as 2, and 2 together with 1 and being gives a class of three terms, and so on. This argument is fallacious, partly because "being" is not a term having any definite meaning, and still more because, if a definite meaning were invented for it, it would be found that numbers do not have being—they are, in fact, what are called "logical fictions," as we shall see when we come to consider the definition of classes. The argument that the number of numbers from 0 to (both inclusive) is depends upon the assumption that up to and including no number is equal to its successor, which, as we have seen, will not be always true if the axiom of infinity is false. It must be understood that the equation , which might be true for a finite if exceeded the total number of individuals in the world, is quite different from the same equation as applied to a reflexive number. As applied to a reflexive number, it means that, given a class of terms, this class is "similar" to that obtained by adding another term. But as applied to a number which is too great for the actual world, it merely means that there is no class of individuals, and no class of individuals; it does not mean that, if we mount the hierarchy of types sufficiently far to secure the existence of a class of terms, we shall then find this class "similar" to one of terms, for if is inductive this will not be the case, quite independently of the truth or falsehood of the axiom of infinity. There is an argument employed by both Bolzano[30] and Dedekind[31] to prove the existence of reflexive classes. The argument, in brief, is this: An object is not identical with the idea of the [Pg 138] object, but there is (at least in the realm of being) an idea of any object. The relation of an object to the idea of it is one-one, and ideas are only some among objects. Hence the relation "idea of" constitutes a reflexion of the whole class of objects into a part of itself, namely, into that part which consists of ideas. Accordingly, the class of objects and the class of ideas are both infinite. This argument is interesting, not only on its own account, but because the mistakes in it (or what I judge to be mistakes) are of a kind which it is instructive to note. The main error consists in assuming that there is an idea of every object. It is, of course, exceedingly difficult to decide what is meant by an "idea"; but let us assume that we know. We are then to suppose that, starting (say) with Socrates, there is the idea of Socrates, and so on ad inf. Now it is plain that this is not the case in the sense that all these ideas have actual empirical existence in people's minds. Beyond the third or fourth stage they become mythical. If the argument is to be upheld, the "ideas" intended must be Platonic ideas laid up in heaven, for certainly they are not on earth. But then it at once becomes doubtful whether there are such ideas. If we are to know that there are, it must be on the basis of some logical theory, proving that it is necessary to a thing that there should be an idea of it. We certainly cannot obtain this result empirically, or apply it, as Dedekind does, to "meine Gedankenwelt"—the world of my thoughts. [30]Bolzano, Paradoxien des Unendlichen, 13. [31]Dedekind, Was sind und was sollen die Zahlen? No. 66. If we were concerned to examine fully the relation of idea and object, we should have to enter upon a number of psychological and logical inquiries, which are not relevant to our main purpose. But a few further points should be noted. If "idea" is to be understood logically, it may be identical with the object, or it may stand for a description (in the sense to be explained in a subsequent chapter). In the former case the argument fails, because it was essential to the proof of reflexiveness that object and idea should be distinct. In the second case the argument also fails, because the relation of object and description is not [Pg 139] one-one: there are innumerable correct descriptions of any given object. Socrates (e.g.) may be described as "the master of Plato," or as "the philosopher who drank the hemlock," or as "the husband of Xantippe." If—to take up the remaining hypothesis—"idea" is to be interpreted psychologically, it must be maintained that there is not any one definite psychological entity which could be called the idea of the object: there are innumerable beliefs and attitudes, each of which could be called an idea of the object in the sense in which we might say "my idea of Socrates is quite different from yours," but there is not any central entity (except Socrates himself) to bind together various "ideas of Socrates," and thus there is not any such one-one relation of idea and object as the argument supposes. Nor, of course, as we have already noted, is it true psychologically that there are ideas (in however extended a sense) of more than a tiny proportion of the things in the world. For all these reasons, the above argument in favour of the logical existence of reflexive classes must be rejected. It might be thought that, whatever may be said of logical arguments, the empirical arguments derivable from space and time, the diversity of colours, etc., are quite sufficient to prove the actual existence of an infinite number of particulars. I do not believe this. We have no reason except prejudice for believing in the infinite extent of space and time, at any rate in the sense in which space and time are physical facts, not mathematical fictions. We naturally regard space and time as continuous, or, at least, as compact; but this again is mainly prejudice. The theory of "quanta" in physics, whether true or false, illustrates the fact that physics can never afford proof of continuity, though it might quite possibly afford disproof. The senses are not sufficiently exact to distinguish between continuous motion and rapid discrete succession, as anyone may discover in a cinema. A world in which all motion consisted of a series of small finite jerks would be empirically indistinguishable from one in which motion was continuous. It would take up too much space to [Pg 140] defend these theses adequately; for the present I am merely suggesting them for the reader's consideration. If they are valid, it follows that there is no empirical reason for believing the number of particulars in the world to be infinite, and that there never can be; also that there is at present no empirical reason to believe the number to be finite, though it is theoretically conceivable that some day there might be evidence pointing, though not conclusively, in that direction. From the fact that the infinite is not self-contradictory, but is also not demonstrable logically, we must conclude that nothing can be known a priori as to whether the number of things in the world is finite or infinite. The conclusion is, therefore, to adopt a Leibnizian phraseology, that some of the possible worlds are finite, some infinite, and we have no means of knowing to which of these two kinds our actual world belongs. The axiom of infinity will be true in some possible worlds and false in others; whether it is true or false in this world, we cannot tell. Throughout this chapter the synonyms "individual" and "particular" have been used without explanation. It would be impossible to explain them adequately without a longer disquisition on the theory of types than would be appropriate to the present work, but a few words before we leave this topic may do something to diminish the obscurity which would otherwise envelop the meaning of these words. In an ordinary statement we can distinguish a verb, expressing an attribute or relation, from the substantives which express the subject of the attribute or the terms of the relation. "Cæsar lived" ascribes an attribute to Cæsar; "Brutus killed Cæsar" expresses a relation between Brutus and Cæsar. Using the word "subject" in a generalised sense, we may call both Brutus and Cæsar subjects of this proposition: the fact that Brutus is grammatically subject and Cæsar object is logically irrelevant, since the same occurrence may be expressed in the words "Cæsar was killed by Brutus," where Cæsar is the grammatical subject. [Pg 141] Thus in the simpler sort of