Page:The Conception of God (1897).djvu/345

the individual quantity A, B is free to be either one of the two square roots of A. The n different roots of an algebraic equation of the nth degree illustrate a still more complex instance of this sort of ambiguity. In general, let us suppose a system of various ideas, a, b, c, d, e, etc. Let us suppose certain relationships, r, r' r'', etc., so that I know in advance that a stands to b in relation r, b to c in relation r', c to d in relation r'' , etc. Then each one of these relations may be, as to its logical definition, perfectly exact, yet each one of them may be such that if the first member is determined, two, three, or an indefinite number of possibilities may be permissible in the determination of the remaining number. In other words, the relation may be such that the relation r permits the equation which would express it to have two, three, or an indefinite number of roots. In such cases, the system of ideas would be such that when I undertook to give it individual embodiment, having first chosen the individual embodiment A of one of the ideas, I should still be able, without inconsistency, to choose from a considerable number of possibilities in defining B, and from still a new list of possibilities in defining C, and so on. And all this ambiguity, or rather multiplicity, this freedom, would not mean that my relationships were necessarily inexactly conceived. The conception of each, in its kind, might be rigidly exact, just as there is no ambiguity of a logically objectionable character in the definition of a root of an algebraic equation, although such a definition leaves it necessarily ambiguous which one of several roots we shall in a given