Distinction between Hume’s doctrine and that of the hypothetical nature of mathematics.
275. Now it is obvious that what Hume accounts for by means of this tendency to feign, even if the tendency did not presuppose conditions incompatible with his theory, is not mathematical science as it exists. It has even less appearance of being so than (to anticipate) has that which is accounted for by those propensities to feign, which he substitutes for the ideas of cause and substance, of being natural science as it exists. In the latter case, when the idea of necessary connexion has been disposed of, an impression of reflection can with some plausibility be made to do duty instead; but there is no impression of reflection in Hume’s sense of the word, no ‘propensity,’ that can be the subject of mathematical reasoning. He speaks, indeed, of our supposing some imaginary standard—of our having ‘an obscure and implicit notion’—of perfect equality, but such language is only a way of saving appearances; for according to him, a ‘supposition’ or ‘notion’ which is neither impression nor idea, cannot be anything. A hasty reader, catching at the term ‘supposition,’ may find his statement plausible with all the plausibility of the modern doctrine, which accounts for the universality and exactness of mathematical truths as ‘hypothetical’—the doctrine that we suppose figures exactly corresponding to our definitions, though such do not really exist. With those who take this view, however, it is always understood that the definitions represent ideas, though not ideas to which real objects can be found exactly answering. Perhaps, if pressed about their distinction between idea and reality, they might find it hard consistently to maintain it, but it is by this practically that they keep their theory afloat. Hume can admit no such distinction. The real with him is the impression, and the idea the fainter impression. There can be no idea of a straight line, a curve, a circle, a right angle, a plane, other than the impression, other than the ‘appearance to the eye,’ and there are no appearances exactly answering to the mathematical definitions. If they do not exactly answer, they might as well for the purposes of mathematical demonstration not answer at all. The Geometrician, having found that the angles at the base of this isosceles triangle are equal to each other, at once takes the equality to be true of all isosceles triangles, as being exactly like the original one, and on the strength of this establishes many other propositions. But, according to Hume, no idea that we could have would be one of which the sides were precisely equal. The Fifth Proposition of Euclid then is not precisely true of the particular idea that we have before us when we follow the demonstration. Much less can it be true of the ideas, i.e. the several appearances of colour, indefinitely varying from this, which we have before us when we follow the other demonstrations in which the equality of the angles at the base of an isosceles is taken for granted.
The admission that no relations of quantity are data of sense removes difficulty as to general propositions about them.
276. Here, as elsewhere, what we have to lament is not that Hume ‘pushed his doctrine too far,’ so far as to exclude ideas of those exact proportions in space with which geometry purports to deal, but that he did not carry it far enough to see that it excluded all ideas of quantitative relations whatever. He thus pays the penalty for his equivocation between a feeling of colour and a disposition of coloured points. Even alongside of his admission that ‘relations of space and time’ are independent of the nature of the ideas so related, which amounts to the admission that of space and time there are no ideas at all in his sense of the word, he allows himself to treat ‘proportions between spaces’ as depending entirely on our ideas of the spaces—depending on ideas which in the context he by implication admits that we have not. [1] If, instead of thus equivocating, he had asked himself how sensations of colour and touch could be added or divided, how one could serve as a measure of the size of another, he might have seen that only in virtue of that in the ‘general appearance’ of objects which, in his own language, is ‘independent of the nature of the ideas themselves’—i.e. which does not belong to them as feelings, but is added by the comparing and combining thought—are the proportions of greater, less, and equal predicable of them at all; that what thought has thus added, viz. limitation by mutual externality, it can abstract; and that by such abstraction of the limit it obtains those several terminations, as Hume well calls them—the surface terminating bodies, the line terminating surfaces, the point terminating lines—from which it constructs the world of pure space: that thus the same action of thought in sense, which alone renders appearances measurable, gives an object matter which, because the pure construction of thought, we can measure exactly and with the certainty that the judgment based on a comparison of magnitudes in a single case is true of all possible cases, because in none of these can any other conditions be present than those which we have consciously put there.
[1] Part III. § 1, sub init.
Hume does virtually admit this in regard to numbers.
277. To have arrived at this conclusion Hume had only to extend to proportions in space the principle upon which the impossibility of sensualizing arithmetic compels him to deal with proportions in number. ‘We are possessed,’ he says, ‘of a precise standard by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard we determine their relations without any possibility of error. When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal’. [1] Now what are the unites here spoken of? If they were those single impressions which he elsewhere [2] seems to regard as alone properly unites, the point of the passage would be gone, for combinations of such unites could at any rate only yield those ‘general appearances’ of whose proportions we have been previously told there can be no precise standard. They can be no other than those unites which, not being impressions, he has to call ‘fictitious denominations’—unites which are nothing except in relation to each other and of which each, being in turn divisible, is itself a true number. We can easily retort upon Hume, then, when he argues that the supposition of infinite divisibility is incompatible with any comparison of quantities because with any unite of measurement, that, according to his own virtual admission, in the only case where such comparison is exact the ultimate unite of measurement is still itself divisible; which, indeed, is no more than saying that whatever measures quantity must itself be a quantity, and that therefore quantity is infinitely divisible. If Hume, instead of slurring over this characteristic of the science of number, had set himself to explain it, he would have found that the only possible explanation of it was one equally applicable to the science of space—that what is true of the unite, as the abstraction of distinctness, is true also of the abstraction of externality. As the unite, because constituted by relation to other unites, so soon as considered breaks into multiplicity, and only for that reason is a quantity by which other quantities can be measured; so is it also with the limit in whatever form abstracted, whether as point, line, or surface. If the fact that number can have no least part since each part is itself a number or nothing, so far from being incompatible with the finiteness of number, is the consequence of that finiteness, neither can the like attribute in spaces be incompatible with their being definite magnitudes, that can be compared with and measured by each other. The real difference, which is also the rationale of Hume’s different procedure in the two cases, is that the conception of space is more easily confused than that of number with the feelings to which it is applied, and which through such application become sensible spaces. Hence the liability to the supposition, which is at bottom Hume’s, that the last feeling in the process of diminution before such sensible space disappears (being the ‘minimum visibile’) is the least possible portion of space.
[1] P. 374. [Book I, part III., sec. I.]
[2] Above, par. 258.
With Hume idea of vacuum impossible, but logically not more so than that of space.