CHAPTER IV ON THE NATURE OF THE CONCEPTS OF NUMBER, QUANTITY, MAGNITUDE, AND MEASURE Nature of the concept of Number.-Its independence of, but close association with, the concept of Order.—Interdependence of Numberconcepts.-Number-concepts as represented and as symbolized.-Transition from the representative to the symbolic image.-Distinction between Number and Quantity in applied Mathematics.-Stallo's criticism of the use of the term quantity in connexion with Algebraic symbols.-Use of the terms Number and Quantity in pure Algebra. LET us suppose an intelligent child to know the meaning of the word 'many', and of the word 'same', but not yet to have learnt to count: what might we suppose the child to be thinking about if he asked us the question: What is it called when I think of a same many? We should probably surmise that he was thinking of some number, that he expected that numbers must, like everything else, have names, and that he wished to know the name of the number he was thinking of. But if, on spreading out before him a few coins or other small objects and asking him to show us which 'same many' he meant, he pushed the things aside and said any same many ', we might indeed be astonished at this evidence of interest in abstract thought, but we should be obliged to conclude that the child had very definitely formed the abstract conception to which we give the general name 'number', and that he wished to know that name. The supposition no doubt implies a greater power of abstraction, as well as a higher degree of purposiveness in thought, than are at all likely to characterize any child in the earlier stage of learning language: I make use of it as illustrative, not as probable, as exemplifying what appears to me to be the essential trait in the conceiving of number, apart from the artifices involved in counting. Number-concepts, like all abstractions, originate in the innate capacity to discern likeness and difference and to concentrate attention on some one element of a complex presentation. But while the vast majority of familiar concepts arise in the discern ment of approximate likenesses and differences between individual things, the concepts of number spring from the discernment of exact likeness and difference between individual aggregates of things, merely as aggregates. The legs of a horse, of a cow, of a dog, of the chair upon which I am at present sitting; the limbs of my body; the fingers (excluding the thumb) of my hand; the sides of a square; the corners of a square in these things, taken individually and compared one with another, I discern various kinds and degrees of likeness and difference ; but taken in their several aggregates and compared merely as aggregates, I discern a certain exact likeness, or identity (the same many' of the child), which is the concept or abstraction to which I have been taught to give the name 'four ', and have learnt to associate with the mathematical symbol' 4'. The discernment of identity involves that of difference. It is impossible to have become aware of identity of aggregation without having pari passu become aware of that difference of aggregation which we indicate by the terms 'more' and 'less '.. The perception of non-identity may be merely that of 'more or less difference', that is, may be indefinite. But it is impossible to have become aware of the identity between the legs of a dog and those of a horse as aggregates, between either of these and the sides of a square, between any of these and the corners of a square, &c., without having become aware that the difference between this identity and that of the hind legs and tail of a dog and those of a horse, of either of these and the sides of a triangle, of any of these and the corners of a triangle, &c., is in every case the same difference, that is, is itself an identity, the identity called 'one', the name given to the individual as part of any aggregate.1 1 The interdependence of number-concepts is partly reflected on the symbolic side by the interdependence of number symbols or names in definition. No significant definition of the name or symbol of a number can be given without the aid of other number symbols or names whose meanings are already known; and in this as in every other case, definition ultimately abuts upon some mode of conveying meaning which is not definition, upon an appeal of some kind to the intuitional and nonconventional in personal intercommunication. In the opinion of recent investigators in the philosophy of mathematics, however, numbers (cardinal) can be defined each independently of all the others, and without assigning to them any order or relation (cf. Couturat, Les Principes des Mathématiques, 1905, p. 52)—a surprising feat if the conceiving of numbers |