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

by Norman Kemp Smith

What thus holds of time may likewise hold of events in time. If time is actually infinite, no proof can be derived from it in support of the assumption that the world has had a beginning in time.

The phrase “by means of a successive synthesis” gives a needlessly subjectivist colouring to Kant’s method of proof. The antinomy is professedly being stated from the realist standpoint, and ought not therefore to be complicated by any such reference. This objection applies, as we shall find, still more strongly to Kant’s proof of the second part of the thesis. The latter proof depends upon this subjectivist reference; the present proof does not.

Kant limits his problem to the past infinitude of time. The reason for this lies, of course, in the fact that he is concerned with the problem of creation. The limitation is, however, misleading.

Thesis b.—The world is limited in regard to space.

Proof.—Assume the opposite, namely, that the world is an infinite, given whole of coexisting parts. A magnitude not given within the determinate limits of an intuition can only be thought through the synthesis of its parts, and its totality through their completed synthesis. In order, therefore, that we may be able to think as a single whole the world which fills all space, the successive synthesis of the parts of an infinite world must be regarded as completed, i.e. an infinite must be regarded as having elapsed in the enumeration of all coexisting things. This, however, is impossible. An infinite aggregate of actual things cannot therefore be viewed as a given whole, nor as being given as coexistent. Consequently the world of spatial existences must be regarded as finite.

Criticism.—From the impossibility of traversing infinite space in thought by the successive addition of part to part, Kant here argues that “an infinite aggregate of actual things cannot be viewed as a given whole,” and consequently that the world cannot be infinitely extended in space. That is, from a subjective impossibility of apprehension he infers an objective impossibility of existence. But Kant has himself defined the infinite as involving this subjective impossibility; for in the proof of thesis a he has stated that the infinitude of a series consists in the very fact that it can never be completed through successive synthesis. Kant is therefore propounding against the existence of the infinite the very feature which by definition constitutes its infinitude. The implication would seem to be that the concept of the infinite is the concept of that which ex definitione cannot exist, and that there is therefore a contradiction in the very idea of the actual infinite.

Deferring for a moment the further objections to which such procedure lies open, we may observe that Kant, in arguing from a subjective to an objective impossibility, commits the fallacy of ignoratio elenchi. For when the conditions of objective existence are recognised in their distinction from those of mental apprehension, the supposed contradiction vanishes, and the argument ceases to have any cogency. The use of the words ‘given’ and ‘whole’ is misleading. If space is infinite, it is without bounds, and cannot therefore exist as a whole in any usual meaning of that term. For the same reason it must be incapable of being given as a whole. Its infinitude is a presupposition which analysis of actually given portions of it constrains us to postulate, and has to be conceived in terms of the definition employed in thesis a. The given must always be conceived as involving what is not itself given and what is not even capable of complete construction. In terms of this presupposition an actual infinite, not given and not capable of construction, can be represented with entire consistency.

But to return to the main assumption upon which Kant’s proof would seem to rest: it is all-important to observe that Kant does not, either in the Critique or in any other of his writings, assert that the concept of the actual infinite is inherently self-contradictory. This is a matter in regard to which many of Kant’s critics have misrepresented his teaching. Kant’s argument may, as we have just maintained, be found on examination to involve the above assertion; but this, if clearly established, so far from commending the argument to Kant, would have led him to reject it as invalid. The passage in the Dissertation[1510] of 1770, which contains his most definite utterance on this point, represents the view from which he never afterwards departed. It may be quoted in full.

“Those who reject the actual mathematical infinite do so in a very casual manner. For they so construct their definition of the infinite that they are able to extract a contradiction from it. The infinite is described by them as a quantity than which none greater is possible, and the mathematical infinite as a multiplicity—of an assignable unit—than which none greater is possible. Since they thus substitute maximum for infinitum, and a greatest multiplicity is impossible, they easily conclude against this infinite which they have themselves invented. Or, it may be, they entitle an infinite multiplicity an infinite number, and point out that such a phrase is meaningless, as is, indeed, perfectly evident. But again they have fought and overthrown only the figments of their own minds. If, however, they had conceived the mathematical infinite as a quantity which, when related to measure, as its unity, is a multiplicity greater than all number; and if furthermore, they had observed that measurability here denotes only the relation [of the infinite] to the standards of the human intellect, which is not permitted to attain to a definite conception of multiplicity save by the successive addition of unit to unit, nor to the sum-total (which is called number) save by completing this progress in a finite time; they would have perceived clearly that what does not conform to the established law of some subject need not on that account exceed all intellection. An intellect may exist, though not indeed a human intellect, which perceives a multiplicity distinctly in one intuition [uno obtutu] without the successive application of a measure.”

The concluding sentences of this Dissertation passage may be taken as Kant’s own better and abiding judgment in regard to the question before us. We must not argue from the impossibility of mentally traversing the infinite to the impossibility of its existence. Indeed the essentials of the above passage are restated in the ‘Observation’ on this thesis.[1511] Thus the concept of the actual infinite is not only, as a concept, perfectly self-consistent, it is also one which, in view of the nature of time and of space, we are constrained to accept as a correct representation of the actually given. The thesis of this first antinomy runs directly counter to admitted facts. That Kant is here arguing in respect to the world, and not merely in respect to space and time, does not essentially alter the situation. For if space and time are necessarily to be viewed as infinite, there can be no a priori proof—none, at least, of the kind here attempted—that the world-series may not be so likewise.