[1505] The limitation of Kant’s discussion to space, time, and causality is, of course, due to his acceptance of the current view that the concepts of infinity and continuity are derived from our intuitions of space and time. As we have already noted in discussing his intuitional theory of mathematical reasoning (above, pp. 40-1, 117 ff., 128 ff.), he fails to extend to mathematical concepts his own “transcendental” view of the categories, namely, as conditioning the possibility of intuitional experience. Such concepts as order, plurality, whole and part, continuity, infinity, are prior to time and space in the logical order of thought; and to be adequately treated must be considered in their widest application.
[1506] Cf. A 507 = B 535, and above, p. 431 ff.; below, pp. 501, 545-6.
[1507] Cf. Kant’s posthumously published Transition from the Metaphysical First Principles of Natural Science to Physics (Altpreussische Monatsschrift, 1882), pp. 279-80: “If we take in regard to space, not its definition, but only an a priori proposition, e.g. that space is a whole which must be thought only as part of a still greater whole, it is clear ... that it is an irrational magnitude, measurable indeed, but in its comparison with unity transcending all number.” “If space is something objectively existent, it is a magnitude which can exist only as part of another given magnitude.”
[1508] Cf. Schopenhauer, World as Will and Idea (Werke, Frauenstädt, ii. pp. 585-6; Eng. trans, ii. pp. 107-8). “I find and assert that the whole antinomy is a mere delusion, a sham fight. Only the assertions of the antitheses really rest upon the forms of our faculty of knowledge, i.e. if we express it objectively, on the necessary, a priori certain, most universal laws of nature. Their proofs alone are therefore drawn from objective grounds. On the other hand, the assertions and proofs of the theses have no other than a subjective ground, rest solely on the weakness of the reasoning individual; for his imagination becomes tired with an endless regression, and therefore he puts an end to it by arbitrary assumptions, which he tries to smooth over as well as he can; and his judgment, moreover, is in this case paralysed by early and deeply imprinted prejudices. On this account the proof of the thesis in all the four conflicts is throughout a mere sophism, while that of the antithesis is a necessary inference of the reason from the laws of the world as idea known to us a priori. It is, moreover, only with great pains and skill that Kant is able to sustain the thesis, and make it appear to attack its opponent, which is endowed with native power.... I shall show that the proofs which Kant adduces of the individual theses are sophisms, while those of the antitheses are quite fairly and correctly drawn from objective grounds.”
[1509] Cf. F. Erhardt’s Kritik der Kantischen Antinomienlehre (1888), a brief but excellent analysis of this section of the Critique.
[1510] § 1 n.
[1511] Cf. A 431-2 = B 460-1: “...the concept [of the infinite] is not the concept of a maximum; by it we think only its relation to any assignable unit, in respect to which it is greater than all number.”
[1512] Cf. Kant’s statement in the Observation to this antithesis, A 431-3 = B 459-61.
[1513] Kant regarded the point as a limit, i.e. as a boundary (Dissertation, § 14, 4; § 15, C: “The simple in space is not a part but a limit”; A 169-70 = B 211); whereas certain modern mathematicians take the point as one of the undefined elements. When the point is regarded in this latter manner, space may perhaps be satisfactorily defined as a set of points. In arguing for the antithesis, and in the passages just cited, Kant also assumes that, in the case of space, the properties of the class are determined by the properties of its elements. This questionable assumption is involved in his assertion that space can consist only of spaces.
[1514] A 438 = B 466.