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

by Norman Kemp Smith

But in spite of its forcibleness this argument is nowhere presented in the Critique.

Similarly, in so far as particular spaces can be conceived only in and through space as a whole, and in so far as the former are limitations of the one antecedent space, the intuition which underlies all external perception must be a priori. This is in essentials a stronger and more cogent mode of formulating the second argument on space. But again, and very strangely, it is nowhere employed by Kant in this form.

The concluding sentence, ambiguously introduced by the words so werden auch, is tacked on to the preceding argument. Interpreted in the light of § 15 C of the Dissertation,[442] and of the corresponding fourth[443] argument[444] on time, it may be taken as offering further proof that space is an intuition. The concepts of line and triangle, however attentively contemplated, will never reveal the proposition that in every triangle two sides taken together are greater than the third. An a priori intuition will alone account for such apodictic knowledge. This concluding sentence thus really belongs to the transcendental exposition; and as such ought, like the third argument, to have been omitted in the second edition.

Kant’s proof rests on the assumption that there are only two kinds of representation, intuitions and concepts, and also in equal degree upon the further assumption that all concepts are of one and the same type.[445] Intuition is, for Kant, the apprehension of an individual. Conception is always the representation of a class or genus. Intuition is immediately related to the individual. Conception is reflective or discursive; it apprehends a plurality of objects indirectly through the representation of those marks which are common to them all.[446] Intuition and conception having been defined in this manner, the proof that space is single or individual, and that in it the whole precedes the parts, is proof conclusive that it is an intuition, not a conception. Owing, however, to the narrowness of the field assigned to conception, the realm occupied by intuition is proportionately wide, and the conclusion is not as definite and as important as might at first sight appear. By itself, it amounts merely to the statement, which no one need challenge, that space is not a generic class concept. Incidentally certain unique characteristics of space are, indeed, forcibly illustrated; but the implied conclusion that space on account of these characteristics must belong to receptivity, not to understanding, does not by any means follow. It has not, for instance, been proved that space and time are radically distinct from the categories, i.e. from the relational forms of understanding.

In 1770, while Kant still held to the metaphysical validity of the pure forms of thought, the many difficulties which result from the ascription of independent reality to space and time were, doubtless, a sufficient reason for regarding the latter as subjective and sensuous. But upon adoption of the Critical standpoint such argument is no longer valid. If all our forms of thought may be subjective, the existence of antinomies has no real bearing upon the question whether space and time do or do not have a different constitution and a different mental origin from the categories. The antinomies, that is to say, may perhaps suffice to prove that space and time are subjective; they certainly do not establish their sensuous character.

But though persistence of the older, un-Critical opposition between the intellectual and the sensuous was partly responsible for Kant’s readiness to regard as radical the very obvious differences between a category such as that of substance and attribute and the visual or tactual extendedness with which objects are endowed, it can hardly be viewed as the really decisive influence. That would rather seem to be traceable to Kant’s conviction that mathematical knowledge is unique both in fruitfulness and in certainty, and to his further belief that it owes this distinction to the content character of the a priori forms upon which it rests. For though the categories of the physical sciences are likewise a priori, they are exclusively relational,[447] and serve only to organise a material that is empirically given. To account for the superiority of mathematical knowledge Kant accordingly felt constrained to regard space and time as not merely forms in terms of which we interpret the matter of sense, but as also themselves intuited objects, and as therefore possessing a character altogether different from anything which can be ascribed to the pure understanding. The opposition between forms of sense and categories of the understanding, in the strict Kantian mode of envisaging that opposition, is thus inseparably bound up with Kant’s doctrine of space and time as being not only forms of intuition, but as also in their purity and independence themselves intuitions. Even the sensuous subject matter of pure mathematics—so Kant would seem to contend—is a priori in nature. If this latter view be questioned—and to the modern reader it is indeed a stone of stumbling—much of the teaching of the Aesthetic will have to be modified or at least restated.

Fifth (in second edition, Fourth) Argument.—This argument is quite differently stated in the two editions of the Critique, though the purpose of the argument is again in both cases to prove that space is an intuition, not a general concept. In the first edition this is proved by reference to the fact that space is given as an infinite magnitude. This characteristic of our space representation cannot be accounted for so long as it is regarded as a concept. A general conception of space which would abstract out those properties and relations which are common to all spaces, to a foot as well as to an ell, could not possibly determine anything in regard to magnitude. For since spaces differ in magnitude, any one magnitude cannot be a common quality. Space is, however, given us as determined in magnitude, namely, as being of infinite magnitude; and if a general conception of space relations cannot determine magnitude, still less can it determine infinite magnitude. Such infinity must be derived from limitlessness in the progression of intuition. Our conceptual representations of infinite magnitude must be derivative products, acquired from this intuitive source.

In the argument of the second edition the thesis is again established by reference to the infinity of space. But in all other respects the argument differs from that of the first edition. A general conception, which abstracts out common qualities from a plurality of particulars, contains an infinite number of possible different representations under it; but it cannot be thought as containing an infinite number of representations in it. Space must, however, be thought in this latter manner, for it contains an infinite number of coexisting parts.[448] Since, then, space cannot be a concept, it must be an intuition.

The definiteness of this conclusion is somewhat obscured by the further characterisation of the intuition of space as a priori, and by the statement that it is the original (ursprüngliche) representation which is of this intuitive nature. The first addition must here, again, just as in the fourth argument, be regarded as merely a recapitulation of what has already been established, not a conclusion from the present argument. The introduction of the word ‘original’ seems to be part of Kant’s reply to the objections which had already been made to his admission in the first edition that there is a conception as well as an intuition of space. It is the original given intuition of space which renders such reflective conception possible.

The chief difficulty of these proofs arises out of the assertion which they seem to involve that space is given as actually infinite. There are apparently, on this point, two views in Kant, which were retained up to the very last, and which are closely connected with his two representations of space, on the one hand as a formal intuition given in its purity and in its completeness, and on the other hand as the form of intuition, which exists only so far as it is constructed, and which is dependent for its content upon given matter.