Page:The World and the Individual, First Series (1899).djvu/321

two objects, say two curves, or two variable quantities, or two collections of objects, — one of them a collection of symbols, the other a collection of objects to be symbolized, — a relation of correspondence can be established, or assumed, between these two objects, or collections, in the most manifold and, in one sense, in the most arbitrary fashion. Necessary to the relations of correspondence is only this, that you shall be able to view the two corresponding objects together, in a one-to-one relation, or in some other definite way, and, with some single purpose in mind, shall then be able in some one perhaps very limited aspect to affirm of one of them the same that you, at the same time and in the same limited sense, affirm of the other. In consequence, with reference to this one affirmation, you could in some specified wise substitute one of them for the other, whole for whole, part for part, element for element. Thus, if you have before you a collection of counters, and a collection of other objects, you can make these collections correspond, if you are able to arrange both sets of objects in a definite order, and then to say, that the first of your counters agrees with the first of your other objects precisely, and perhaps solely, in being the first of its series; while the second counter agrees with the second of the objects precisely in being the second of the series, and so on. The result will then be that by counting the counters, you can afterwards, perhaps more conveniently, enumerate the objects to be counted. Ordinary counting depends, in fact, upon making the members of a number series, one, two, three, four, etc., arbitrarily correspond to the distinguishable objects of the collection that you number. The result is,