that I enter upon either undertaking. I am ignorant of mathematics, not willingly but through radical incapacity; and again (it is perhaps the same defect) I cannot follow any train of reasoning which is highly abstract. If under these circumstances what I am about to write proves worthless, no apology, it is clear, can help me. The reader in that case must judge of me as seems to him best.
I. I understand Prof. Royce to contend that number and truths about number can be constructed a priori, and that these truths are completely unconditional and self-consistent. The origin in time of our perception of number and quantity he, I understand, does not discuss, and we are concerned simply with what may be called an act of logical creation. I will ask first as to the nature of the process, and next as to the character of the result.
The process of creation appears to consist in reflection, a process more or less familiar to students of philosophy. We are to think of some object (no matter what), and then we are to think of our thought of this object, and so on indefinitely. In this way we gain (it is contended) an ordinal series where the process contains no unknown condition, and where the result is consistent. Now I agree that in the above way we produce some how a series which is ordinal, in the sense that each fresh product somehow contains and preserves what has gone before. I do not mean that, after reflecting in such a manner for a certain time, I know in fact where I am, and could say how many steps are included in my present result. To gain that knowledge I should say that a further operation is required. Still I admit (what is, I presume, the main point) that through the process of reflection an ordinal series is somehow generated. What I have to deny is first (a) that the generation consists in pure thought, and next (b) I have to deny that the product is consistent with itself.
(a) You have an object (O) before your self (S). You then go on to reflect that this is so; and in consequence you now have a new object (S—O) before you. A further reflection of the same kind gives an object (S—S|O), and thus you make an ordinal series which has in principle no end. Now what is the
