Page:Mind (New Series) Volume 6.djvu/355

ON THE RELATIONS OF NUMBER AND QUANTITY. 339 Finally, the reduction of measure to number must, on the view here taken, appear as extraneous and merely convenient. Measure, on our theory, is an independent kind of compari- son, yielding judgments of more or less. The reduction of extensive to intensive quantities, at an earlier stage of this paper, still holds good ; the two do not differ logically, but only in the manner in which they are given. At most they differ in this, that extensive quantities are relations, while intensive quantities are adjectives. Relations and adjectives alike, however, are indivisible, and the division, on which the application of number is based, is logically a delusion, and due only to our hypostatising space and time. Even so, the application of number to spaces and times rests on equality, which is here a quantitative relation, having no accurate counterpart in number. 1 Quantity or measure, therefore, as the mere more or less, should be put among conceptions of relation, and wholly separated from number. The connexion of quantity with number so we must con- clude is due partly to motives of convenience, but mainly to a confusion between two fundamentally distinct ways of regarding space and time. Thus measure removes the contradictions, in the con- ception of quantity, which appeared in the first part of this paper. To this extent, it is a real advance upon quantity regarded as intrinsic. But it must not be supposed that all contradictions are removed from the conception of measure. The chief difference is, that these contradictions now appear in the properties of terms compared. For although quantity, in our present sense, is not a common property of quantities, some common properties are involved. When we investigate these we shall see, I think, that they render measure de- pendent on the inadequacy of thought to sense, and thus still logically self-contradictory. Our result so far is as follows : First, we saw that two 1 It is important to observe that the conception of equality proper does not occur in the treatment of number. Equality does not occur, to begin with, in applied number, for here we have no guarantee of the equality of the units the fact that the number of two collections is the same is no ground for affirming their equality. Equality therefore, if it occurs in number at all, must occur in pure number. In pure number, however, what we get is not equality, but identity. Thus the numerical measurement of extensive quantities consists of two steps : in the first, which alone is quantitative, we assure ourselves that we have a collection of equal quantities ; in the second, we count these equal quantities as we might count anything else. The single extensive quantity which results from our counting is thus numerically measured in terms of any one of our original equal quantities.