[2] [Greek ἑκὼν ἢ ἄκων (hekon e akon) = like it or not. Tr.]
What becomes of the exactness of mathematics according to Hume?
273. It is the necessity then, according to his theory, of 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 divisibility 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 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 versâ. 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.’ [1]
[1] Pp. 351-53. [Book I, part II., sec. IV.]
The universal propositions of geometry either untrue or unmeaning.
274. Such indefinite approach to exactness is all that Hume can allow to the mathematician. But it is undoubtedly another and an absolute sort of exactness that the mathematician himself supposes when he pronounces all right angles equal. Such perfect equality ‘beyond what we have instruments and art’ to ascertain, Hume boldly calls a ‘mere 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 making statements that are untrue or speaking of nonentities. If his ‘lines’ and ‘angles’ mean ideas that we can possibly have, his universal propositions are untrue; if they do not, according to Hume they can mean nothing. He says, for instance, that ‘two right lines cannot have a common segment;’ but of such ideas of right lines as we can possibly have this is only true ‘where the right lines incline upon each other with a sensible angle.’ [2] It is not true when they ‘approach at the rate of an inch in 20 leagues.’ According to the ‘original standard of a right line,’ which is ‘nothing but a certain general appearance, ’tis evident right lines may be made to concur with each other’. [3] Any other standard is a ‘useless and incomprehensible fiction.’ Strictly speaking, according to Hume, we have it not, but only a tendency to suppose that we have it arising from the progressive correction of our actual measurements. [4]
[1] P. 353. [Book I, part II., sec. IV.]
[2] Cf. Aristotle, Metaph. 998a, on a corresponding view ascribed to Protagoras.
[3] P. 356. [Book I, part II., sec. IV.]
[4] P. 354. [Book I, part II., sec. IV.]