Page:Mind (New Series) Volume 6.djvu/343

ON THE EELATIONS OF NUMBER AND QUANTITY. 327 intensive. But intensive quantity too, it will appear, must, if it be an intrinsic property of intensive quantities, be also a mere relation between them. The hypothesis that quantity is a category giving an intrinsic property will therefore have to be rejected. The hypothesis that quantity is a datum in sense will also be found to lead to contradictions. We shall be forced, therefore, to reject the view that quantity is an intrinsic property of quantities. We shall regard it, instead, as a category of comparison ; there is no common property, we shall say, among things that can be treated quantitatively, except what is involved in the extraneous property that there are other qualitatively similar things with which they can be quantitatively compared. This will turn quantity into measure, in the broadest sense, and with this, I think, our previous difficulties will cease. But at the same time, every connexion with number will cease quantity or measure, we shall say, is a wholly independent conception of comparison. But a discussion of the kind of comparison involved in measure will bring back our previous difficulties in a new form ; we shall find that the terms compared, though we no longer regard them as quantitative, are infected with contra- dictions similar to those which, in the first part of the paper, will have belonged to quantity itself. I shall conclude that quantity is only applicable to classes of actual and possible immediate data, and not to any fully understood material. In the ordinary procedure of mathematics, quantity ap- pears as a limit in the extension of number. Let us therefore examine, to begin with, how far this view of quantity is tenable. Number, throughout the following discussion, will be used only of discreta ; it will be taken as always the result, not of comparison as to the more or less, but of acts of synthesis (or analysis) of things whose qualitative or quantitative differences are disregarded. Pure number will denote the merely formal result of acts of synthesis, so far as any result can be known in total abstraction from the matter synthesised and from the specific qualities of the objects of the synthesis. Since a unit must be defined by some quality, pure number will thus have no reference to a unit, or rather its unit is the abstract object of any act of attention, of whatever kind this may be. Such an operation can only give rise to the natural numbers, the series of positive in- tegers. Fractions, negative numbers and imaginary numbers, I shall speak of as qualified numbers, for here we have always some more explicit reference to the specific qualities of the unit. Such numbers, unlike the positive integers,