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 PRAGMATISM AND PSEUDO-PRAGMATISM. 387 how or of what it is, in his opinion, an illustration. I cannot see, that is, how the ' transfinites ' can be used as an illustration of 'useless truth,' for the simple reason that they appear themselves to exemplify the usefulness of number. They are, that is, not numbers at all, properly speaking, but a use of number, proceeding from an application of number to space, and the paradoxes they involve would seem to be ultimately reducible to the old difficulty of representing the continuous by the discrete. Since Zeno's days this has been recognised as a thorny problem ; but how can it possibly affect the validity of number ? I cannot therefore understand how the use of ' transfinites ' in numbering space intervals can possibly refute any theory as to the nature of number, and can only infer that Prof. Taylor has either mistaken the nature of the ' transfinites,' or (more probably) that of the issue between us. If however Prof. Taylor dissents from this conclusion I must beg him to make his point a little more explicit. To make his ' illustration ' good, let him show us (1) that the ' empirical ' and the ' rationalist ' theories of number are the only two conceivable and that one of them is right (for if they are both wrong, they will not illustrate his contention) ; (2) that they are strictly incom- patible, and (3) that there is an issue between them to which the ' transfinites ' are relevant. He should next show us (4) how he conceives the ' transfinites ' to arise, and (5) to bear on either theory, and (6) explain what are the ' marked logical advantages ' thereby accruing to one of them. Lastly (7) he should try to show how these ' advantages ' constitute one theory of number a ' proved truth ' and the other a ' demonstrable contradiction '. As regards this last point his language was indeed far from confident, though it would hardly lead one to infer that many competent mathematicians (very reasonably) consider the ' transfinites ' as equally compatible with either theory. As regards the other points he was wholly silent. To have argued them all would no doubt have been a long and arduous task, but an illustration which cannot be rendered intelligible without technicalities which had to be avoided was surely a bad one. And it was at least necessary to show that it was relevant, which, so far, it does not appear to be. I pass therefore to Prof. Taylor's third illustration, that from metaphysics. This I may claim to have answered by anticipation. If it were true that neither Berkeleian nor any other idealism made any practical difference to the ' every day realism ' on which we act, it would add nothing to our knowledge and be held to collapse into a meaningless subtlety, a distinction without a differ- ence. Needless to say, however, this was not Berkeley's opinion : he ascribed to his theory the utmost moral value as disposing of materialism. Nor is it mine ; Prof. Taylor can find a discussion of the practical difference a real belief in a real idealism should make in Humanism, pages 197-198.