The Logic of Hegel

by Georg Wilhelm Friedrich Hegel

In arithmetic the several kinds of operation are usually presented as accidental modes of dealing with numbers. If necessity and meaning is to be found in these operations, it must be by a principle: and that must come from the characteristic elements 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, which together constitute number. But 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 ä 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.

(c) Degree.

103.] The limit (in a quantum) is identical with the whole of the quantum itself. As in itself multiple, the limit is Extensive magnitude; as in itself simple determinateness (qualitative simplicity), it is Intensive magnitude or Degree.

The distinction between Continuous and Discrete magnitude differs from that between Extensive and Intensive in the circumstance that the former apply to quantity in general, while the latter apply to the limit or determinateness of it as such. Intensive and Extensive magnitude are not, any more than the other, two species, of which the one involves a character not possessed by the other: what is Extensive magnitude is just as much Intensive, and vice versâ.