(b) Quantum (How Much)

Quantity, essentially invested with the exclusionist character which it involves, is Quantum (or How Much): i.e. limited quantity.

Quantum is, as it were, the determinate Being of quantity: whereas mere quantity corresponds to abstract Being, and the Degree, which is next to be considered, corresponds to Being-for-self. As for the details of the advance from mere quantity to quantum, it is founded on this: that while in mere quantity the distinction, as a distinction of continuity and discreteness, is at first only implicit, in a quantum the distinction is actually made, so that quantity in general now appears as distinguished or limited. But in this way the quantum breaks up at the same time into an indefinite multitude of quanta or definite magnitudes. Each of these definite magnitudes, as distinguished from the others, forms a unity, while on the other hand, viewed per se, it is a many. And, when that is done, the quantum is described as Number.

In Number the quantum reaches its development and perfect mode. Like the One, the medium in which it exists, Number involves two qualitative/factors or functions; Annumeration or Sum, which depends on the factor discreteness, and Unity, which depends on continuity.

In arithmetic the several kinds of operation are usually presented as accidental modes of dealing with numbers. If necessary and meaning is to be found in these operations, it must be by a principle: and that must come from the characteristic element in the notion of number itself. (This principle must here be briefly exhibited.) These characteristic elements are Annumeration on the one hand, and Unity on the other, of which number is the unity. But this latter Unity, when applied to empirical numbers, is only the equality of these numbers: hence the principle of arithmetical operations must be to put numbers in the ratio of Unity and Sum (or amount), and to elicit the equality of these two modes.

The Ones or the numbers themselves are indifferent towards each other, and hence the unity into which they are translated by the arithmetical operation takes the aspect of an external colligation. All reckoning is therefore making up the tale: and the difference between the species of it lies only in the qualitative constitution of the numbers of which we make up the tale. The principle for this constitution is given by the way we fix Unity and Annumeration.

Numeration comes first: what we may call, making number; a colligation of as many units as we please. But to get a species of calculation, it is necessary that what we count up should be numbers already, and no longer a mere unit.

First, and as they naturally come to hand, Numbers are quite vaguely numbers in general, and so, on the whole, unequal. The colligation, or telling the tale of these, is Addition.

The second point of view under which we regard numbers is as equal, so that they make one unity, and of such there is an annumeration or sum before us. To tell the tale of these is Multiplication. It makes no matter in the process, how the functions of Sum and Unity are distributed between the two numbers, or factors of the product; either may be Sum and either may be Unity.

The third and final point of view is the equality of Sum (amount) and Unity. To number together numbers when so characterised is Involution; and in the first instance raising them to the square power. To raise the number to a higher power means in point of form to go on multiplying a number with itself an indefinite amount of times. Since this third type of calculation exhibits the complete equality of the sole existing distinction in number, viz. the distinction between Sum or amount and Unity, there can be no more than these three modes of calculation. Corresponding to the integration we have the dissolution of numbers according to the same features. Hence besides the three species mentioned, which may to that extent be called positive, there are three negative species of arithmetical operation.

Number, in general, is the quantum in its complete specialisation. Hence we may employ it not only to determine what we call discrete, but what are called continuous magnitudes as well. For that reason even geometry must call in the aid of number, when it is required to specify definite figurations of space and their ratios.