It is real just in so divisibility becomes a matter of course. What are the ultiments of mate ele extension? If not ex 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 tended, 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 Colours or coloured points? What is the difference? True way of dealing with the question. 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.) 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. 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, from 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. 272. In this lies the answers 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." 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 1 P. 315. 1 What becomes of ness of mathe matics ac Hume? "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, ExÒväкwv, when he talks of extension as 'made up of coloured points.' 273. It is the necessity then, according to his theory, of the exact making space an impression that throughout underlies Hume's argument against its infinite divisibility; and, as we have seen, the same theory which excludes its infinite divisicording to bility logically extinguishes it as a quantity, divisible and measurable, altogether. He of course does not recognize this consequence. He is obliged indeed to admit that in regard to the proportions of greater, equal and less,' and the relations of different parts of space to each other, no judgments of universality or exactness are possible. We may judge of them, however, he holds, with various approximations to exactness, whereas upon the supposition of infinite divisibility, as he ingeniously makes out, we could not judge of them at all. He asks the mathematicians, what they mean when they say that one line or surface is equal to, or greater or less than, another.' If they maintain the composition of extension by indivisible points,' their answer, he supposes, will be that lines or surfaces are equal when the numbers of points in each are equal.' This answer he reckons 'just,' but the standard of equality given is entirely useless. For as the points which enter into the composition of any line or surface, whether perceived by the sight or touch, are so 6 minute and so confounded with each other that 'tis utterly impossible for the mind to compute their number, such a computation will never afford us a standard by which we may judge of proportions.' The opposite sect of mathematicians, however, are in worse case, having no standard of equality whatever to assign. For since, according to their hypothesis, the least as well as greatest figures contain an infinite number of parts, and since infinite numbers, properly speaking, can neither be equal nor unequal with respect to each other, the equality or inequality of any portion of space can never depend on any proportion in the number of their parts.' His own doctrine is that the only useful notion of equality or inequality is derived from the whole united appearance, and the comparison of, particular objects.' The judgments thus derived are in many cases certain and infallible. 'When the measure of a yard and that of a foot are presented, the mind can no more question that the first is longer than the second than it can doubt of those principles which are most clear and self-evident.' Such judgments, however, though sometimes infallible, are not always so.' Upon a 'review and reflection' we often pronounce those objects equal which at first we esteemed unequal,' and vice versa. Often also we discover our error by a juxtaposition of the objects; or, where that is impracticable, by the use of some common and invariable measure which, being successively applied to each, informs us of their different proportions. And even this correction is susceptible of a new correction, and of different degrees of exactness, according to the nature of the instrument by which we measure the bodies, and the care which we employ in the comparison.' (Pp. 351-53.) either un 274. Such indefinite approach to exactness is all that The uniHume can allow to the mathematician. But it is undoubtedly versal propositions another and an absolute sort of exactness that the mathema- of geotician himself supposes when he pronounces all right angles metry equal. Such perfect equality beyond what we have instru- true or unments and art' to ascertain, Hume boldly calls a mere meaning. fiction of the mind, useless as well as incomprehensible."1 Thus when the mathematician talks of certain angles as always equal, of certain lines as never meeting, he is either 1 P. 353. |