¹ A recent distinguished example is Strawson, P. F. (The Bounds of Sense, p. 176).Google Scholar

² For a general survey see G. Martin, Kant's Metaphysics and Theory of Science, Chap. II. But this is far from complete.

³ Principles of Mathematics, p. 459.

⁴ It could be pointed out that Russell's argument is unsatisfactory, in that when the inference is from two earlier states of the Universe to a later and still future one, an extra hypothesis, that of Induction, has to be employed.

⁵ Kemp Smith, A pp. 427–9/B pp. 454–7 and A pp. 432–3/B pp. 460–61.

⁶ Chap. VI, p. 159f.

⁷ Ibid., p. 161.

⁸ Whitrow, G. J., The Natural Philosophy of Time, pp. 31–32Google Scholar, has some pertinent remarks about this, and also about Russell's other criticisms, cf. pp. 147–9.

⁹ It has been doubted whether Kant used this term in the way I suggest. His usage is indeed rather confusing, but he frequently describes the transcendental realist as one who interprets outer appearances, whose reality he takes for granted (A p. 369. Compare A p. 491/B p. 519, where the transcendental realist becomes an empirical idealist). Some confusion is caused by the fact that in the discussion of the Antinomies Kant calls the supporters of the theses transcendental dogmatists and their opponents empiricists, and that at A p. 543/B. 571 the transcendental realist is apparently against nature and freedom, and on the side of the empiricists. But this is because both sides of the antinomy are the views of transcendental realists, who become involved in contradictions and cannot finally maintain their positions.

¹⁰ For a recent discussion see Furley, D. J., ‘Aristotle and the Atomists on Infinity’, pp. 92–96, in Naturphilosophie bei Aristoteles und Theophrast, ed. Düring, I. (Heidelberg, 1969).Google Scholar

¹¹ In context this seems quite clearly an argument for the infinity of matter, notwithstanding the fact that Leibniz elsewhere denied that there could be an actual infinite in mathematics. In contrast to Leibniz are the views of Newton and Locke, who argue for a finite world in infinite space, and Locke's arguments in many ways anticipate those of Kant. He does not, however, go about the matter straightforwardly: in Book II, Chap. XVII, on Infinity, he argues at length that space is infinite, but rejects any kind of real infinity; in Chap. XIII, § 21, he has the passing remark ‘if body be not supposed infinite, which I think no one will affirm’, but we are left to conjecture why he believed that no one would affirm the infinity of matter. Presumably it is because matter is real, in the way that space is not.

¹² See, for instance, North, J. D., The Measure of the Universe (Oxford 1965), Note XIX, p. 422.Google Scholar

¹³ Cf. Locke II, ch. 17, §7, where he distinguishes between ‘the idea of the infinity of space and the idea of space infinite: the first is nothing but a supposed endless progression of the mind over what repeated ideas of space it pleases; but to have actually in the mind the idea of space infinite, is to suppose the mind already passed over, and actually to have a view of all those repeated ideas of space which an endless repetition can never totally represent to it; which carries in it a plain contradiction’.

¹⁴ The Measure of the Universe, p. 393.

¹⁵ Of course there are exceptions, like the prime numbers smaller than ten, the parents of Abraham Lincoln, and the points of the compass. But this is irrelevant to the point I am making.

¹⁶ Remarks on the Foundations of Mathematics, IV, 6.

¹⁷ van Fritz, Kurt, ‘Das ἄπειρον bei Aristoteles’ in Naturphilosophie bei Aristoteles und Theophrast, ed. Düring, , pp. 65–84, discusses Cantor in relation to Aristotle.Google Scholar

¹⁸ Some Main Problems of Philosophy, pp. 195–6 (Allen-Unwin 1953).

¹⁹ An early draft of this paper was seen by John Faris and Martha Kneale, for whose comments I am grateful, and it was also read to senior seminars hi Manchester and Liverpool.