For ascertaining and knowing amounts, some contrivance is requisite. It is necessary to conceive some small amount, by the addition or subtraction of which, another becomes larger or smaller. This forms the instrument of ascertainment. Where one thing, taken separately, is of sufficient importance to form this instrument, it is taken. Thus, for ascertaining and knowing different amounts of men, one individual is of sufficient importance. Amounts of men are considered as increased or diminished by the addition or subtraction of individuals. A grain of gunpowder might also be taken; but it is not of sufficient importance; the quantity, taken as the instrument of measurement, must have an ascertainable influence upon the effect, for the sake of which, the ascertaining of the amount is of importance. In their simple state, men use principally the hand for their elementary ascertainments. A pinch, or as much as could be held between the finger and the thumb, was a small amount distinctly conceived, and formed the principle of measurement where small additions were important; a handful was not less distinctively conceived, and was the instrument, where only larger additions were of importance.
When one addition was made, or needed to be made, after another, and another after that, and so on, the next point of importance was to conceive exactly how often the addition was made. A few 91 additions are distinct to sense. Place one billiard-ball by another, the sight of the two is distinct. Place three or four, it is still distinct. Soon, however, it ceases to be so. Place a dozen, and you will not probably be able to distinguish them from eleven. You must count them, or divide them. If you divide them by the eye, into two parcels, you may see that one is six and another six; but to benefit by this, you must know the art of putting six and six together.
The next step, therefore, necessary in the process of ascertaining amounts, was, to mark these additions, one after another, in such a manner, as to make known to what extent they had gone. When men were familiar with the operation of assigning names as marks of their ideas, the course which would suggest itself to them is obvious; they would employ a name as the mark of each addition. They would say, one, for the first, two, for the second, three, for the third, and so on. These marks it was very useful to make connotative, that the other important ingredient of the process, the thing added, might be made known at the same time. Thus we say, one man, two men; one horse, two horses; and so of all other things, the enumeration of which we are performing.
Numbers, therefore, are not names of objects. They are names of a certain process; the process of addition; of putting one billiard-ball to another; not more mysterious than any other process, as walking, writing, reading, to which names are assigned. One, is the name of this once performed, or of the aggregation begun; two, the name of it once more performed; three, of it once more performed; and so on. The words, however, in these concrete forms, beside 92 their power in noting this process, connote something else, namely, the things, whatever they are, the enumeration of which is required.
In the case of these connotative, as of other connotative marks, it was of great use to have the means of dropping the connotation; and in this case, it would have been conducive to clearness of ideas, if the non-connotative terms had received a mark to distinguish them from the connotative. This advantage, however, the framers of numbers were not sufficiently philosophical to provide. The same names are used both as connotative, and non-connotative; that is, both as abstract, and concrete; and it is far from being obvious, on all occasions, in which of the two senses they are used. They are used in the connotative sense, when joined as adjectives with a substantive; as when we say two men, three women; but it is not so obvious that they are used in the abstract sense, when we say three and two make five; or when we say fifty is a great number, five is a small number. Yet it must, upon consideration, appear, that in these cases they are abstract terms merely; in place of which, the words oneness, twoness, threeness, might be substituted. Thus we might say, twoness and threeness are fiveness.[22] [23]
[22] The vague manner in which the author uses the phrase “to be a name of” (a vagueness common to almost all thinkers who have not precise terms expressing the two modes of signification which I call denotation and connotation, and employed for nothing else) has led him, in the present case, into a serious misuse of terms. Numbers are, in the strictest propriety, names of objects. Two is surely a name of the things which are two, the two balls, the two fingers, &c. The process of adding one to one which forms two is connoted, not denoted, by the name two. Numerals, in short, are concrete, not abstract names: they denote the actual collections of things, and connote the mental process of counting them. It is not twoness and threeness that are fiveness: the twoness of my two hands and the threeness of the feet of the table cannot be added together to form another abstraction. It is two balls added to three balls that make, in the concrete, five balls. Numerals are a class of concrete general names predicable of all things whatever, but connoting, in each case, the quantitative relation of the thing to some fixed standard, as previously explained by the author.—Ed.
[23] Here the process of numeration generally, together with the function of numbers carrying their separate names, are clearly set forth; after which we find the remark, that no distinction is made in the name of the number, when used as an abstract and when used as a concrete. Mr. James Mill thinks that it would have been conducive to clearness if such distinction had been marked by an inflexion of the name. “The names of numbers are used in the connotative (concrete) sense, when joined as adjectives with a substantive, as when we say, two men, three men: but it is not so obvious that they are used in the abstract sense, when we say three and two make five: or when we say fifty is a great number, five is a small number. Yet it must upon consideration appear, that in these cases they are abstract terms merely: in place of which, the words oneness, twoness, threeness, might be substituted. Thus we might say, twoness and threeness are fiveness.”
The last part of what is here affirmed cannot, in my judgment, be sustained. Connecting itself with one among the many arguments between Aristotle and Plato, it lays down a position from which both of them would have dissented. In the last book but one (Book M) of Aristotle’s “Metaphysica,” this argument will be found set forth at length; though with much obscurity, which is cleared up by the lucid commentary of Bonitz. Plato distinguished two classes of numbers—the mathematical, and the ideal. The first class were the Quanta of equal and homogeneous units (One, Two, Three, &c.), any or all of which might be added so as to coalesce into one total sum. The second class were, the ideal or abstract numbers, Two quatenus Two, &c., represented by Dyad, Triad, Tetrad, Pentad, Dekad, &c., the characteristic property of which was, that they could not be added together nor coalesce into one sum. These were uncombinable numbers, “ἀριθμοὶ ἀσύμβλητοι—numeri inconsociabiles.”—See Aristot. Metaph. M. 6. 1080. b. 12. Bonitz Comment. p. 540, 541, seq.
Plato regarded these uncombinable numbers as the highest representative specimens or coryphæi of the Platonic Ideas. In this character Aristotle reasoned against them, contending that they did nothing to remove the many objections against Plato’s ideal theory. With the question thus opened, I have no present concern: all that I wish to point out is the view which Plato originated and upon which Aristotle reasoned, viz.: That these ideal or abstract numbers could not be added together, or fused into one sum total. The abstract term Twoness means Two so far forth as two: so also Threeness and Fiveness. You cannot truly predicate anything of Twoness which would be inconsistent with this fundamental characteristic: you cannot add it to Threeness so as to make Fiveness, nor can you subdivide Fiveness into Twoness and Threeness, without suppressing the fundamental characteristic of each. Neither of them admit of increase or diminution. In like manner, a Triangle, or every particular Triangle, may have one of its sides taken away, or two more sides added to it: on each of which suppositions it ceases to be a triangle. But if we speak of a Triangle so far forth as Triangle, neither of these suppositions is admissible. We may say that its three angles are equal to two right angles, but we cannot subtract from it one of its sides, nor add to it one or two other sides. The subject of predication is so limited and specialised, that no predicate can be allowed which would efface its characteristic feature—Triangularity.
Bonitz remarks truly that the class of numbers set forth by Plato—the ideal or uncombinable numbers which could not be either added or subtracted—were divested of all the useful aptitudes and functions of numbers, and passed out of the category of Quantity into that of Quality. The Triad was one quality; the Pentad was another: there was no common measure into which both could be resolved (Bonitz, Comment. p. 540—553). Two, three, five, are quantifying names, designating each so many numerable units: and the units counted in each list may be added to, or subtracted from, the units counted in the others. But when we say, Twoness or the Dyad—Threeness or the Triad—Fiveness or the Pentad—we then recognise a peculiar quality, founded upon each separate variety of aggregation or quantification: so that these separate varieties are no longer resolvable into any common measure of constituent units. Each quality stands apart from the others, and has its own predicates. In the view of Plato and the Pythagoreans, the Dekad especially was invested with magnificent predicates.