A Commentary to Kant's 'Critique of Pure Reason'

by Norman Kemp Smith

In the Observation on this thesis Kant shows consciousness of the defects of his argument. It does not apply to space, time, or change.

“We ought not to call space a compositum but a totum, because its parts are possible only in the whole, not the whole through the parts.”[1514]

As Kant further states, he is speaking only of the simples of the Leibnizian system. This thesis is “the dialectical principle of monadology.” Again in the Observation on the antithesis, in commenting on the mathematical proof of the infinite divisibility of matter, Kant even goes so far as to declare that the argument of the thesis is based on an illegitimate substitution of things in themselves, conceived by the pure understanding, for the appearances with which alone the antinomy is concerned.[1515]

“...it is quite futile to attempt to overthrow, by sophistical manipulation of purely discursive concepts, the manifest, demonstrated truth of mathematics.”

Antithesis.—No composite thing in the world consists of simple parts, and there nowhere exists in the world anything simple.

Proof.—Let us assume the opposite, namely, that a composite thing (as substance) consists of simple parts. As all external relation, and therefore all composition of substances, is only possible in space, space must consist of as many parts as there are parts of the composite that occupies it. Space, however, does not consist of simple parts, but of spaces. The simple must therefore occupy a space. Now as everything real which occupies a space contains in itself a manifold of constituents external to one another, and therefore is composite, and as a real composite is not composed of accidents (for without substance accidents could not be outside one another), but of substances, the simple would be a substantial composite, which is self-contradictory.

Criticism.—The Leibnizian standpoint is here completely deserted. Instead of proceeding to demonstrate the direct opposite of the thesis, Kant in this argument deals with the extended bodies of empirical intuition. The proof given ultimately reduces to an argument from the continuous nature of space to the continuous nature of the matter which occupies it. But as the thesis and the antithesis thus refer to different realities, the former to things in themselves conceived by pure understanding, and the latter to the sensuous, no antinomy has been shown to subsist. Antinomy presupposes that both the opposing assertions have the same reference. Kant, as already noted, argues in the Observation to this antithesis that all attempts “made by the monadists” to refute the mathematical proof of the infinite divisibility of matter are quite futile, and are due to their forgetting that in this discussion we are concerned only with appearances.

“The monadists have, indeed, been sufficiently acute to seek to avoid this difficulty by not treating space as a condition of the possibility of the objects of outer intuition (bodies), but by taking these and the dynamical relation of substances as the condition of the possibility of space. But we have a concept of bodies only as appearances, and as such they necessarily presuppose space as the condition of the possibility of all outer appearance.”[1516]

How Kant, after writing these words, should still have left standing the proof which he has given of the thesis may be partially explained as due to the continuing influence of his earlier view,[1517] according to which antinomy represents not a conflict between opposing views of the world of ordinary consciousness, but between the demands of pure thought and the forms of sensuous existence. That older view of antinomy here gains the upper hand, notwithstanding its lack of agreement with the general scheme of the Dialectic.

There is a further inconsistency in Kant’s procedure which may perhaps be taken as indicating the early origin of this portion of the Critique. He presents the mathematical proof of the continuity of matter as conclusive. Yet in the Metaphysical First Principles of Natural Science (1786) he most emphatically states that “the infinite divisibility of matter is very far from being proved through proof of the infinite divisibility of space.”[1518]