NOTES AND CORRESPONDENCE. 143 A la fin, M. Russell n'a pas 1'air tres satisfait de ce que je dis de la probability. Je n'en suis pas tres satisfait non plus et je serais heureux si M. Eussell avait quelque chose de plus satisfaisant a proposer. M. Poincare's reply to my review in MIND, July, 1905, calls for a few words of explanation. On the subject of mathematical induction I await his forthcoming article in the Revue de M&taphysique et de Morale, But I should like to clear up a misunderstanding as to the sense in which, as I hold, mathe- matical induction does not proceed from the particular to the general. (Mathematical induction, by the way, does not define integers, but finite integers.) The principle may be stated as follows: "A number n is said to obey mathematical induction if it possesses every property which (1) belongs to 0, and (2) belongs to m + 1 whenever it belongs to m". Here the principle itself is doubly general, since (a) it makes a. statement about all properties, (b) it makes a statement about all num- bers. The statement about all numbers occurs in (2) above. And when we have taken a particular property, and thus ceased to concern our- selves with the general principle of induction, we still have a general statement about all numbers. Let us take an instance : Suppose we wish to prove that if n obeys mathematical induction, then n is not equal to n + 1. We prove (a) that is not equal to 1, (6) that if m is not equal to m + 1, then m + 1 is not equal to m + 2. Here (b) is a statement about all numbers. It is only from (a) and (b) together that we reach the desired conclusion. The generality of (b) is not the kind of generality that M. Poincare supposes me to mean when he suggests that I wish to adduce the principle of mathematical induction itself as a necessary premiss in all its applications. As regards geometry, I do not think it is necessary to my point to decide what is meant by perception. My point is that relations of order, as opposed to metrical relations, are in some sense given in experience, and that this appears to show that spatial relations are to some extent empirically determined. I regret that my remark about " Abracadabra " appeared to be a mere epigram. I meant to suggest that what it is convenient to suppose must have some meaning, and I did not suppose that I was " profiting by an ambiguity," which I should be most unwilling to do consciously. B. RUSSELL. MR. MACCOLL'S VIEWS ON LOGICAL EXISTENCE. Mr. MacColl in the last number of MIND replied to my note in the previous number. He has put the matter so clearly that no doubt whatever can be entertained as to the position he occupies, but I still think that there are certain important considerations that prevent the general adoption of his view. Mr. Russell has well expounded from one standpoint the doctrine commonly held. The following criticisms may throw some further light upon the subject. In my view it is not permissible to consider the Universe of Discourse as made up of two universes. The Universe of Discourse in Symbolic Logic means all the things that we are talking about. Now such a universe may be divided into two compartments, but each of these does not form a universe by itself. This is not a matter of mere words, but is one of principle. Within our Universe of Discourse there is not a universe of unrealities : all the members of the Universe of Dis-
