II. ON THE RELATIONS OF NUMBER AND QUANTITY. 1 BY B. EUSSELL. I WISH, in this paper, to discuss one of the most fundamental questions of mathematical philosophy. On the view we take of this relation must depend our interpretation of the Infinitesimal Calculus and all its consequences in a word, of all higher mathematics. The very idea of the continuum an idea which, in philosophy as in mathematics, has become gradu- ally more and more prominent, and has, of late especially, ousted the atomic views which were shared by Hume and Kant must stand or fall, I think, with the relative justifica- tion of quantity in mathematics as against number. It will not be necessary, however, to deal with mathematical con- siderations here ; it will be sufficient to consider number and quantity in their purely logical aspects. I shall use quantity, always, as equivalent to continuous quantity, and I shall en- deavour, in the course of the paper, to make clear the mean- ing of the word continuous. My argument will be as follows : First, I shall discuss number, and show that its extensions beyond the positive integers result from a gradual absorption of the properties of the unit, and give a gradually diminishing information as to the whole. Then I shall discuss the application of number to continua, and shall endeavour to show that number per se gives no information as to quantity, but only compari- son with an already quantitative unit. It will appear, there- fore, that quantity must be sought in an analysis of the unit. Assuming quantity to be an intrinsic property of quantities, I shall discuss two hypotheses. The first regards quantity as an irreducible category, the second regards it as an im- mediate sense-datum. On the first hypothesis, we shall see that extensive quantities are rendered contradictory by their divisibility, and must be taken as really indivisible, and so 1 Read before the Aristotelian Society.
