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

by Norman Kemp Smith

Russell,[1519] in discussing the thesis and antithesis on their merits, from the point of view of certain present-day mathematical theories, makes the following criticism of Kant’s procedure.

“Here, again, the argument applies to things in space and time, and to all collections, whether existent or not.... And with this extension[1520] the proof of the proposition must, I think, be admitted; only that terms or concepts should be substituted for substances, and that, instead of the argument that relations between substances are accidental (zufällig), we should content ourselves with saying that relations imply terms and complexity implies relations.”

Russell further argues that Kant’s assumption in the antithesis, that “space does not consist of simple parts, but of spaces,” cannot be granted. It

“...involves a covert use of the axiom of finitude, i.e. the axiom that, if a space does consist of points, it must consist of some finite number of points. When once this is denied, we may admit that no finite number of divisions of a space will lead to points, while yet holding every space to be composed of points. A finite space is a whole consisting of simple parts, but not of any finite number of simple parts. Exactly the same thing is true of the stretch between 1 and 2. Thus the antinomy is not specially spatial, and any answer which is applicable in Arithmetic is applicable here also. The thesis, which is an essential postulate of Logic, should be accepted, while the antithesis should be rejected.”

But, as above observed,[1521] those mathematicians who adopt this view so alter the meaning of the term point that it would perhaps be equally true to say that the thesis, as thus interpreted by Russell, coincides with what Kant believes himself to be asserting in the antithesis.

THIRD ANTINOMY

Thesis.—Causality according to the laws of nature is not the only causality from which the appearances of the world can be deduced. There is also required for their explanation another, that of freedom.

Proof.—Let us assume the opposite. In that case everything that happens presupposes a previous state upon which it follows according to a rule. That previous state is itself caused in similar fashion, and so on in infinitum. But if everything thus happens according to the mere laws of nature, there can never be a first beginning, and therefore no completeness of the series on the side of the derivative causes. But the law of nature is that nothing happens without a cause sufficiently determined a priori. If, therefore, all causality is possible only according to the laws of nature, the principle contradicts itself when taken in unlimited universality. Such causality cannot therefore be the sole causality possible. We must admit an absolute spontaneity, whereby a series of appearances, that proceed according to laws of nature, begins by itself.

Criticism.—The vital point of this argument lies in the assertion that the principle of causality calls for a sufficient cause for each event, and that such sufficiency is not to be found in natural causes which are themselves derivative or conditioned. As the antecedent series of causes for an event can never be traced back to a first cause, it can never be completed, and can never, therefore, be sufficient to account for the event under consideration. Either, therefore, the principle of causality contradicts itself, or some form of free self-originative causality must be postulated. This argument cannot be accepted as valid. Each natural cause is sufficient to account for its effect. That is to say, the causation is sufficient at each stage. That the series of antecedent causes cannot be completed is due to its actual infinitude, not to any insufficiency in the causality which it embodies.[1522] To prove his point, Kant would have to show that the conception of the actual infinite is inherently self-contradictory; and that, as we have already noted, he does not mean to assert. His argument here lies open to the same criticism as we have already passed upon his argument in proof of the thesis of the first antinomy.