Finally there is a question on which I would invite the reader to reflect. The empirical origin of our sense of more and less, of quantity and of number I am willing to treat here as being irrelevant. But another question remains which can hardly be dismissed. How far does our arithmetic depend upon spatial schemata?1 l How far can we rid ourselves of the datum of space as perceived, and how far is this datum ultimately consistent and intelligible ? I raise no separate doubt as to time, since our developed perception of time itself appears largely to be spatial. How far, even to think of (I do not say to experience) the relation of object to subject, are we forced to make this a spatial relation to something which certainly is not in space? And the endless process of reflection on reflection, how far without a spatial scheme can any such process exist? And what in the end holds our ordinal series both apart and together? These questions to my mind are very relevant, but I can do no more than suggest them to the reader. Apart from any answer to them, I have however endeavoured to show that Prof. Royce s generation of number is, in the form in which he advocates it, not proof against criticism. I cannot however end without thanking him for the service which he has done in calling attention to issues, the importance of which, I am sure, he in no way exaggerates.
II. I have now to remark on some of the fundamental ideas used by Mr. Russell, and must endeavour to show that these ideas contain inconsistency. It is a task to which in one sense I am quite unequal. I am incompetent utterly to sit in judgement on Mr. Russell's great work (Principles of Mathematics). But, if the mathematical part is as good as the part which is philosophical, I am sure that he has produced a book of singular merit. To confine myself here to a one-sided criticism of ideas which I can only partially comprehend, is ungrateful to me, and I could not do it if I did not feel myself in a sense compelled to say something.
I understand Mr. Russell to hold that mathematical truth is true perfectly and in the end, since the principles as well as the inferences are wholly valid. The fundamental ideas, I under-
1 I of course do not mean visual schemata. Obviously that could not be true of every mind. But of how many minds it would be true is, again, another question.
