268. If thought were no more than Hume takes feeling to be, this objection would be valid. But if by thought we understand the self-conscious principle which, present to all feelings, forms out of them a world of mutually related objects, permanent with its own permanence, we shall also understand that the relations by which thought qualifies its object are not qualities of itself—that, in thinking of its object as made up of parts, it does not become itself a quantum. We shall also be on the way to understand how thought, detaching that relation of simple distinctness by which it has qualified its objects, finds before it a multitude of units of which each, as combining in itself distinctions from all the other units, is at the same time itself a multitude; in other words, finds a quantum of which each part, being the same in kind with the whole and all other parts, is also a quantum; i.e. which is infinitely divisible. When once it is understood, in short, that quantity is simply the most elementary of the relations by which thought constitutes the real world, as detached from this world and presented by thought to itself as a separate object, then infinite divisibility becomes a matter of course. It is real just in so far as quantity, of which it is a necessary attribute, is real. If quantity, though not feeling, is yet real, that its parts should not be feelings can be nothing against their reality. This once admitted, the objections to infinite divisibility disappear; but so likewise does that mysterious dignity supposed to attach to it, or to its correlative, the infinitely addible, as implying an infinite capacity in the mind. From Hume’s point of view, the mind being ‘a bundle of impressions’—though how impressions, being successive, should form a bundle is not explained—its capacity must mean the number of its impressions, and, all divisibility being into impressions, it follows that infinite divisibility means an infinite capacity in the mind. This notion however arises, as we have shown, from a confusion between a felt division of an impossible ‘compound feeling,’ and that conceived divisibility of an object which constitutes but a single attribute of the object and represents a single relation of the mind towards it. There may be a sense in which all conception implies infinity in the conceiving mind, but so far from this doing so in any special way, it arises, as we have seen, from the presentation of objects under that very condition of endless, unremoved, distinction which constitutes the true limitation of our thought.
What are the ultimate elements of extension? If not extended, what are they?
269. When, as with Hume, it is only in its application to space and time that the question of infinite divisibility is treated, its true nature is more easily disguised, for the reason already indicated, that space and time are not necessarily considered as quanta. When Hume, indeed, speaks of space as a ‘composition of parts’ or ‘made up of points,’ he is of course treating it as a quantum; but we shall find that in seeking to avoid the necessary consequence of its being a quantum—the consequence, namely, that it is infinitely divisible—he can take advantage of the possibility of treating it as the simple, unquantified, relation of externality. We have already spoken of the dexterity with which, having shown that all divisibility, because into impressions, is into simple parts, he turns this into an argument in favour of the composition of space by impressions. ‘Our idea of space is compounded of parts which are indivisible.’ Let us take one of these parts, then, and ask what sort of idea it is: ‘let us form a judgment of its nature and qualities.’ ‘’Tis plain it is not an idea of extension: for the idea of extension consists of parts; and this idea, according to the supposition, is perfectly simple and indivisible. Is it therefore nothing? That is impossible,’ for it would imply that a real idea was composed of nonentities. The way out of the difficulty is to ‘endow the simple parts with colour and solidity.’ In words already quoted, ‘that compound impression, which represents extension, consists of several lesser impressions, that are indivisible to the eye or feeling, and may be called impressions of atoms or corpuscles endowed with colour and solidity.’ (Part II. § 3, near the end.)
Colours or coloured points? What is the difference?
270. It is very plain that in this passage Hume is riding two horses at once. He is trying so to combine the notion of the constitution of space by impressions with that of its constitution by points, as to disguise the real meaning of each. In what lies the difference between the feelings of colour, of which we have shown that they cannot without contradiction be supposed to ‘make up extension,’ and ‘coloured points or corpuscles’? Unless the points, as points, mean something, the substitution of coloured points for colours means nothing. But according to Hume the point is nothing except as an impression of sight or touch. If then we refuse his words the benefit of an interpretation which his doctrine excludes, we find that there remains simply the impossible supposition that space consists of feelings. This result cannot be avoided, unless in speaking of space as composed of points, we understand by the point that which is definitely other than an impression. Thus the question which Hume puts—If extension is made up of parts, and these, being indivisible, are unextended, what are they?—really remains untouched by his ostensible answer. Such a question indeed to a philosophy like Locke’s, which, ignoring the constitution of reality by relations, supposed real things to be first found and then relations to be superinduced by the mind—much more to one like Hume’s, which left no mind to superinduce them—was necessarily unanswerable.
True way of dealing with the question.
271. In truth, extension is the relation of mutual externality. The constituents of this relation have not, as such, any nature but what is given by the relation. If in Hume’s language we ‘separate each from the others and, considering it apart, form a judgment of its nature and qualities,’ by the very way we put the problem we render it insoluble or, more properly, destroy it; for, thus separated, they have no nature. It is this that we express by the proposition which would otherwise be tautological, that extension is a relation between extended points. The ‘points’ are the simplest expression for those coefficients to the relation of mutual externality, which, as determined by that relation and no otherwise, have themselves the attribute of being extended and that only. If it is asked whether the points, being extended, are therefore divisible, the answer must be twofold. Separately they are not divisible, for separately they are nothing. Whether, as determined by mutual relation, they are divisible or no, depends on whether they are treated as forming a quantum or no. If they are not so treated, we cannot with propriety pronounce them to be either further divisible or not so, for the question of divisibility has no application to them. But being perfectly homogeneous with each other and with that which together they constitute, they are susceptible of being so treated, and are so treated when, with Hume in the passage before us, we speak of them as the parts of which extended matter consists. Thus considered as parts of a quantum and therefore themselves quanta, the infinite divisibility which belongs to all quantity belongs also to them.
‘If the point were divisible, it would be no termination of a line.’
Answer to this.
272. In this lies the answer to the most really cogent argument which Hume offers against infinite divisibility ‘A surface terminates a solid; a line terminates a surface; a point terminates a line: but I assert that if the ideas of a point, line, or surface were not indivisible, ’tis impossible we should ever conceive these terminations. For let these ideas be supposed infinitely divisible, and then let the fancy endeavour to fix itself on the idea of the last surface, line, or point, it immediately finds this idea to break into parts; and upon its seizing the last of these parts it loses its hold by a new division, and so on ad infinitum, without any possibility of its arriving at a concluding idea’. [1] If ‘point,’ ‘line,’ or ‘surface’ were really names for ‘ideas’ either in Hume’s sense, as feelings grown fainter, or in Locke’s, as definite imprints made by outward things, this passage would be perplexing. In truth they represent objects determined by certain conceived relations, and the relation under which the object is considered may vary without a corresponding variation in the name. When a ‘point’ is considered simply as the ‘termination of a line,’ it is not considered as a quantum. It represents the abstraction of the relation of externality, as existing between two lines. It is these lines, not the point, that in this case are the constituents of the relation, and thus it is they alone that are for the time considered as extended, therefore as quanta, therefore as divisible. So when the line in turn is considered as the ‘termination of a surface.’ It then represents the relation of externality as between surfaces, and for the time it is the surfaces, not the line, that are considered to have extension and its consequences. The same applies to the view of a surface as the termination of a solid. Just as the line, though not a quantum when considered simply as a relation between surfaces, becomes so when considered in relation to another line, so the point, though it ‘has no magnitude’ when considered as the termination of a line, yet acquires parts, or becomes divisible, so soon as it is considered in relation to other points as a constituent of extended matter; and it is thus that Hume considers it, ἑκὼν ἢ ἄκων [2], when he talks of extension as ‘made up of coloured points.’
[1] P. 345. [Book I, part II., sec. IV.]