The Logic of Hegel

by Georg Wilhelm Friedrich Hegel

The mathematical definition of magnitude as what may be increased or diminished, appears at first sight to be more plausible and perspicuous than the exposition of the notion in the present section. When closely examined, however, it involves, under cover of pre-suppositions and images, the same elements as appear in the notion of quantity reached by the method of logical development. In other words, when we say that the notion of magnitude lies in the possibility of being increased or diminished, we state that magnitude (or more correctly, quantity), as distinguished from quality, is a characteristic of such kind that the characterised thing is not in the least affected by any change in it. What then, it may be asked, is the fault which we have to find with this definition? It is that to increase and to diminish is the same thing as to characterise magnitude otherwise. If this aspect then were an adequate account of it, quantity would be described merely as whatever can be altered. But quality is no less than quantity open to alteration; and the distinction here given between quantity and quality is expressed by saying increase or diminution: the meaning being that, towards whatever side the determination of magnitude be altered, the thing still remains what it is.

One remark more. Throughout philosophy we do not seek merely for correct, still less for plausible definitions, whose correctness appeals directly to the popular imagination; we seek approved or verified definitions, the content of which is not assumed merely as given, but is seen and known to warrant itself, because warranted by the free self-evolution of thought. To apply this to the present case. However correct and self-evident the definition of quantity usual in Mathematics may be, it will still fail to satisfy the wish to see how far this particular thought is founded in universal thought, and in that way necessary. This difficulty, however, is not the only one. If quantity is not reached through the action of thought, but taken uncritically from our generalised image of it, we are liable to exaggerate the range of its validity, or even to raise it to the height of an absolute category. And that such a danger is real, we see when the title of exact science is restricted to those sciences the objects of which can be submitted to mathematical calculation. Here we have another trace of the bad metaphysics (mentioned in § 98, note) which replace the concrete idea by partial and inadequate categories of understanding. Our knowledge would be in a very awkward predicament if such objects as freedom, law, morality, or even God Himself, because they cannot be measured and calculated, or expressed in a mathematical formula, were to be reckoned beyond the reach of exact knowledge, and we had to put up with a vague generalised image of them, leaving their details or particulars to the pleasure of each individual, to make out of them what he will. The pernicious consequences, to which such a theory gives rise in practice, are at once evident. And this mere mathematical view, which identifies with the Idea one of its special stages, viz. quantity, is no other than the principle of Materialism. Witness the history of the scientific modes of thought, especially in France since the middle of last century. Matter, in the abstract, is just what, though of course there is form in it, has that form only as an indifferent and external attribute.

The present explanation would be utterly misconceived if it were supposed to disparage mathematics. By calling the quantitative characteristic merely external and indifferent, we provide no excuse for indolence and superficiality, nor do we assert that quantitative characteristics may be left to mind themselves, or at least require no very careful handling. Quantity, of course, is a stage of the Idea: and as such it must have its due, first as a logical category, and then in the world of objects, natural as well as spiritual. Still even so, there soon emerges the different importance attaching to the category of quantity according as its objects belong to the natural or to the spiritual world. For in Nature, where the form of the Idea is to be other than, and at the same time outside, itself, greater importance is for that very reason attached to quantity than in the spiritual world, the world of free inwardness. No doubt we regard even spiritual facts under a quantitative point of view; but it is at once apparent that in speaking of God as a Trinity, the number three has by no means the same prominence, as when we consider the three dimensions of space or the three sides of a triangle;—the fundamental feature of which last is just to be a surface bounded by three lines. Even inside the realm of Nature we find the same distinction of greater or less importance of quantitative features. In the inorganic world, Quantity plays, so to say, a more prominent part than in the organic. Even in organic nature when we distinguish mechanical functions from what are called chemical, and in the narrower sense, physical, there is the same difference. Mechanics is of all branches of science, confessedly, that in which the aid of mathematics can be least dispensed with,—where indeed we cannot take one step without them. On that account mechanics is regarded next to mathematics as the science par excellence; which leads us to repeat the remark about the coincidence of the materialist with the exclusively mathematical point of view. After all that has been said, we cannot but hold it, in the interest of exact and thorough knowledge, one of the most hurtful prejudices, to seek all distinction and determinateness of objects merely in quantitative considerations. Mind to be sure is more than Nature and the animal is more than the plant: but we know very little of these objects and the distinction between them, if a more and less is enough for us, and if we do not proceed to comprehend them in their peculiar, that is their qualitative character.

100.] Quantity, as we saw, has two sources: the exclusive unit, and the identification or equalisation of these units. When we look therefore at its immediate relation to self, or at the characteristic of self-sameness made explicit by attraction, quantity is Continuous magnitude; but when we look at the other characteristic, the One implied in it, it is Discrete magnitude. Still continuous quantity has also a certain discreteness, being but a continuity of the Many: and discrete quantity is no less continuous, its continuity being the One or Unit, that is, the self-same point of the many Ones.

(1) Continuous and Discrete magnitude, therefore, must not be supposed two species of magnitude, as if the characteristic of the one did not attach to the other. The only distinction between them is that the same whole (of quantity) is at one time explicitly put under the one, at another under the other of its characteristics. (2) The Antinomy of space, of time, or of matter, which discusses the question of their being divisible for ever, or of consisting of indivisible units, just means that we maintain quantity as at one time Discrete, at another Continuous. If we explicitly invest time, space, or matter with the attribute of Continuous quantity alone, they are divisible ad infinitum. When, on the contrary, they are invested with the attribute of Discrete quantity, they are potentially divided already, and consist of indivisible units. The one view is as inadequate as the other.

Quantity, as the proximate result of Being-for-self, involves the two sides in the process of the latter, attraction and repulsion, as constitutive elements of its own idea. It is consequently Continuous as well as Discrete. Each of these two elements involves the other also, and hence there is no such thing as a merely Continuous or a merely Discrete quantity. We may speak of the two as two particular and opposite species of magnitude; but that is merely the result of our abstracting reflection, which in viewing definite magnitudes waives now the one, now the other, of the elements contained in inseparable unity in the notion of quantity. Thus, it may be said, the space occupied by this room is a continuous magnitude, and the hundred men, assembled in it, form a discrete magnitude. And yet the space is continuous and discrete at the same time; hence we speak of points of space, or we divide space, a certain length, into so many feet, inches, &c., which can be done only on the hypothesis that space is also potentially discrete. Similarly, on the other hand, the discrete magnitude, made up of a hundred men, is also continuous: and the circumstance on which this continuity depends, is the common element, the species man, which pervades all the individuals and unites them with each other.

(b) Quantum (How Much).

101.] 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 whilst 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.

102.] 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.