What is a continuum? Russell knocks Hegel's logic (1903)

Bertrand Russell, in his Principles of Mathematics (1903), comments on G.W. Hegel's Logic:

271. The notion of continuity has been treated by philosophers, as a rule, as though it were incapable of analysis. They have said many things about it, including the Hegelian dictum that everything discrete is also continuous and vice versâ. This remark, as being an exemplification of Hegel’s usual habit of combining opposites, has been tamely repeated by all his followers. But as to what they meant by continuity and discreteness, they preserved a discreet and continuous silence; only one thing was evident, that whatever they did mean could not be relevant to mathematics, or to the philosophy of space and time. -- Chapter XXXV, Section 271.

Though Russell gives a page number from William Wallace's translation of Hegel's logic, I could not pin down the reference exactly. Still, the following excerpt from Wallace's version of "Being Part One of the Encyclopaedia of The Philosophical Sciences (1830)" [known as The Science of Logic] gives you an idea of what Russell meant.

From Wallace's translation:

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, etc., 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 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.

§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. 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.

I would mildly disagree with Russell to the extent that continuity and discreteness do seem to imply each other and do seem to be two sides of the same coin, rather like Change and the Absolute. If one, for example, divides a line segment into say 2ⁿ finite segments, then we assume that "at" infinity the finite segments have reached zero length, and yet each location (point) is discrete. But how can a discrete point of 0 width abut another discrete point of 0 width? The points sort of have a peculiar state of discreteness and non-discreteness that are, one might say, superposed.

Russell took a dim view of Hegel's logic in general. And, from the above passage, one can see why. Yet, we should at least grant that Hegel's Idealism meant that the mind of the observer was a candle of the flame of Spirit (or Mind), so that Russell's traditional objectivism would not do for Hegel. (Even so, Russell eventually came to the view that the cosmos must be made up of something weird, such as a mind-matter composite.)