ON THE RELATIONS OF NUMBER AND QUANTITY. 331 said, is an intrinsic property of our arbitrary unit. If so we ought to be able to say something, as regards the quantity of our unit, which shall not consist merely in comparison with another quantity. But this, so far as I can see, we are unable to do. As soon as we begin to make judgments about quantity, they are always judgments of comparison the vaguest possible judgments of quantity are : " This is greater (or smaller) than that ". We seem thus to be landed in an antinomy. Though every judgment of quantity is necessarily a comparison, the essence of quantity must not lie in comparison, but in the terms compared. In other words, quantities have an intrinsic nature, but this consists wholly in difference from something else. Our difficulty may be illustrated by the ordinary definition of quantity, as that which can be increased or diminished. Since increase and diminution are themselves quantitative, this definition involves a vicious circle. But it suggests, as Hegel points out, an important characteristic of quantity. Quantity may be altered without producing any other change in the thing whose quantity is altered. This will give us some guidance as to the sort of things which can be regarded as quantitative. It explains why quantity appears as a purely extrinsic determination. For it places the whole essence of a quantity in being different from other quanti- ties : by a quantitative change, a thing is changed in nothing except quantity, and quantity cannot, therefore, be defined by reference to the nature of the quantitative thing, since this nature, apart from quantity, is unaffected by a quantita- tive change. In this extrinsic reference, quantity is on a par with space. The whole essence of one part of space is to be external to another part, just as the whole essence of one quantity is to differ from some other quantity. This ex- plains why space is quantity par excellence, and why all other quantities have to be reduced to spatial equivalents before they can be quantitatively treated. But just as the extrinsic reference of space leads to an antinomy which shows it to be mere relativity, so the extrinsic reference of quantity leads to an antinomy which forces us, it would seem, to abandon the position that quantity is an intrinsic property of quantitative things. But if we abandon this position, what are we to say of the terms of quantitative comparison ? We can no longer say, apparently, that quantitative comparison compares quantities, or that measurement is measurement of quanti- ties. Two ways of escape suggest themselves, by which we can endeavour to rescue the intrinsic nature of quantity.
