[91]. Bradley, Appearance and Reality, chap. 3. Compare also chap. 15, “Thought and Reality.”
[92]. The reader who desires further knowledge of the researches in the theory of Numbers upon which Prof. Royce’s doctrine is based, may profitably consult Dedekind, Was sind und was sollen die Zahlen, and Couturat, L’Infini Mathématique.
[93]. Professor Royce’s own illustration of the map of England executed upon a portion of the surface of the country is really a typical instance of a self-contradictory purpose. He argues that such a map, to be theoretically perfect, must contain a reduced facsimile of itself as part of the country mapped, and this again another, and so on indefinitely. But the whole force of the reasoning depends on overlooking the distinction between the surface of England as it is before the map is made, and the surface of England as altered by the presence of the map. Prof. Royce assumes that you set out to represent in the map a state of things which can in fact have no existence until after the map is made. The previous existence of the map at a certain spot is falsely taken to be one of the conditions to which the map-maker is to conform in executing it. Every one of the supposed “maps within the map” will thus involve distortion and misrepresentation of the district it proposes to map. It is as if Hamlet had chosen “Hamlet” as the subject of the “play within the play.” The professor’s illustration thus does less than justice to his theory.
[94]. The fundamental defect in Professor Royce’s reasoning seems to me to lie in the tacit transition from the notion of an infinite series to that of an infinite completed sum. Thus he speaks of the series of prime numbers as a “whole” being present at once to the mind of God. But are the prime numbers, or any other infinite series, an actual sum at all? They are surely not proved to be so by the existence of general truths about any prime number.
[95]. See, e.g., Dedekind, op. cit., § 2: “It frequently happens that different things a, b, c ... are apprehended upon whatsoever occasion under a common point of view, mentally put together, and it is then said that they form a system; the things a, b, c ... are named the elements of the system”; and § 3 (definitions of whole and part).
[96]. Ante, Bk. II. chap. 2, § 5.
[97]. It is no answer to this view to urge that as soon as the intellect undertakes to reflect upon and describe Reality it unavoidably does so in relational terms. For it is our contention that the same intellect which uses these relational methods sees why they are inadequate, and to some extent at least how they are ultimately merged in a higher type of experience. Thus the systematic use of the intellect in Metaphysics itself leads to the conviction that the mere intellect is not the whole of Reality. Or, in still more paradoxical language, the highest truth for the mere intellect is the thought of Reality as an ordered system. But all such order is based in the end on the number-series with its category of whole and part, and cannot, therefore, be a perfectly adequate representation of a supra-relational Reality. Hence Truth, from its own nature, can never be quite the same thing as Reality.
[98]. Or does Dr. Stout merely mean that there may be a hat and a head, and also a relation of on and under (e.g., between the hat and the peg), and yet my hat not be on my head? If this is his meaning, I reply we have not really got the relation and its terms; if the hat is not on the head, hat and head are not terms in the relation at all. I do not see why, on his own principles, Dr. Stout should not add a fourth factor to his analysis, namely, qualifiedness, or the fact that the qualities are there, and so on indefinitely.
[99]. If you consider the lines a and α, as Dr. Stout prefers to do, I should have thought two views possible. (a) There are not two lines at all, but one, the “junction” at M being merely ideal. Then there remains nothing to connect and there is no relation of “immediate connection.” Or (b), the junction may be taken as real, and then you have a perfectly ordinary case of relation, the terms being the terminated lines a and α, and the relation being one of contact at M. On every ground (a) seems to me the right view, but it is incompatible with the reduction of continuity to “immediate connection.” Thus the source of the difficulty is that (1) immediate connection can only hold between the immediately successive terms of a discontinuous series, and yet (2) cannot hold between them precisely because they are discontinuous.