I cannot therefore agree with Mr. James Mill in his opinion that, “when we say three and two make five, we use these numbers in the abstract sense.” We clearly do not mean that three, so far forth as three, and two, so far forth as two, make five. But this would be what we should mean, if we used these names of numbers in the abstract sense. What we do mean is, that the units constituting three may be added to those constituting two, so as to make five: and that this is equally true, whether the units are men, horses, stones, or any other objects. Two, three, five, &c., are general or universal terms, capable of being joined with units of indefinite variety: but they do not become abstract terms, until we limit them by quâtenus, καθόσον, ᾗ, so far forth as, &c., or by a suffix such as ness. Such abstracts would have been of little use as to the ordinary functions of numbers; and accordingly they have never got footing in familiar speech, though they are occasionally employed in metaphysical discussions.—G.
93 It is necessary to observe, that the process, marked by the names called numbers, though used for the 94 purpose of ascertaining synchronous order, is in the mind successive; one addition follows another. 95 Numbers, therefore, in reality, name successions; and are easily applied to mark certain particulars of the 96 successive order, when the marking of those particulars is of importance.
It is of importance, when successions take place all of one kind; and when consequences of importance depend upon the less or greater length of the train. It is then of importance, to mark the degrees of that length, which is correctly done by the enumeration of the links.
To take a simple and familiar instance, that of the human steps. They are successions all of one kind. Consequences of importance may, and often do result from a knowledge of the length of any particular series of steps. The ascertainment of an aggregate, in this order, is made in the same way, as that which we have traced in the synchronous order. An element of aggregation is taken; by its successive aggregations, the amount of the aggregate is correctly conceived; and, by a proper mark for each successive aggregation, it is also correctly denoted. The continued successions of day and night are all of one kind; and it is of the greatest importance for us to know accurately the length of a series of those successions; of the series between such and such events; between the sowing of the seed in the ground, for example, and the maturity of the crop. This is done, accurately, by putting a several mark upon each 97 several succession, one for the first, two for the one after that, three for the one after that, and so on.
If there be no mystery in one sensation after another, or one idea after another; and, if having them in that order and associating the idea of the antecedent with the sensation of the consequent be to know that they are in that order; then there is no mystery in Numbers, for they are only marks to shew that one is after another.
That there is no mystery in the ideas of priority and posteriority, which are relative terms, has been shewn under the [preceding] head of discourse.
The word Number itself, which is only a name of the names, one, two, &c., nothing being a number but some one of those names, has also been explained, when the class of words which are distinguished as Names of Names was under consideration.
In using the terms, one, two, three, four, and so on, the object is to ascertain with precision, the amount of the aggregate in question. In some cases, however, it is of importance to ascertain the order of aggregation, as well as the amount; and that, whether a synchronous, or a successive, aggregate be the object in view. This purpose is answered by a set of names, called the ordinal numbers, which, applied to the units of aggregation in the order in which they are taken, mark precisely the order of each. Thus, when we say, first, second, third, fourth, and so on; each of these concrete, or connotative names, notes a certain position, if in the synchronous order; a certain link, if 98 in the successive; and connotes the precise object which holds that position, or forms that link.
As there is no difficulty whatsoever in tracing the ideas, which, on each occasion, receive those marks, there is no need of multiplying words in their illustration.