Introduction to Mathematical Philosophy

by Bertrand Russell

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 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 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 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.