The Principles of Mathematics (1903)

§ 150

But, concurrently with this purist's reform, an opposite advance has been effected. New branches of mathematics, which deal neither with number nor with quantity, have been invented; such are the Logical Calculus, Projective Geometry, and—in its essence—the Theory of Groups. Moreover it has appeared that measurement—if this means the correlation, with numbers, of entities which are not numbers or aggregates—is not a prerogative of quantities: some quantities cannot be measured, and some things which are not quantities (for example anharmonic ratios projectively defined) can be measured. Measurement, in fact, as we shall see, is applicable to all series of a certain kind—a kind which excludes some quantities and includes some things which are not quantities. The separation between number and quantity is thus complete: each is wholly independent of the other. Quantity, moreover, has lost the mathematical importance which it used to possess, owing to the fact that most theorems concerning it can be generalized so as to become theorems concerning order. It would therefore be natural to discuss order before quantity. As all propositions concerning order can, however, be established independently for particular instances of order, and as quatity will afford an illustration, requiring slightly less effort of abstraction, of the principles to be applied to series in general; as, further, the theory of distance, which forms a part of the theory of order, presupposes somewhat controversial opinions as to the nature of quantity, I shall follow the more traditional course, and consider quantity first. My aim will be to give, in the present chapter, a theory of quantity which does not depend upon number, and then to show the peculiar relation to number which is possessed by two special classes of quantities, upon which depends the measurement of quantities wherever this is possible. The whole of this Part, however—and it is important to realize this—is a concession to tradition; for quantity, we shall find, is not definable in terms of logical constants, and is not properly a notion belonging to pure mathematics at all. I shall discuss quantity because a thorough discussion is required for disproving this supposition; but if the supposition did not exist, I should avoid all mention of any such notion as quantity.(§ 150 ¶ 1)