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

by Norman Kemp Smith

“Since geometry has space and arithmetic has counting as its object (and counting can only take place by means of time), it is evident in what manner geometry and arithmetic, that is to say pure mathematics, is possible.”[515]

Largely, as it would seem,[516] through Schulze, whose Erläuterungen did much to spread Kant’s teaching, this view came to be the current understanding of Kant’s position. The nature of arithmetic, as thus popularly interpreted, is expounded by Schopenhauer in the following terms:

“In time every moment is conditioned by the preceding. The ground of existence, as law of the sequence, is thus simple, because time has only one dimension, and no manifoldness of relations can be possible in it. Every moment is conditioned by the preceding; only through the latter can we attain to the former; only because the latter was, and has elapsed, does the former now exist. All counting rests upon this nexus of the parts of time; its words merely serve to mark the single steps of the succession. This is true of the whole of arithmetic, which throughout teaches nothing but the methodical abbreviations of counting. Every number presupposes the preceding numbers as grounds of its existence; I can only reach them through all the preceding, and only by means of this insight into the ground of its existence do I know that, where ten are, there are also eight, six, four.”[517]

Schulze was at once challenged to show that this was really Kant’s teaching, and the passage which he cited was Kant’s definition of the schema of number, above quoted.[518] It is therefore advisable that we should briefly discuss the many difficulties which this passage involves. What does Kant mean by asserting that in the apprehension of number we generate time? Does he merely mean that time is required for the process of counting? Counting is a process through which numerical relations are discovered; and it undoubtedly occupies time. But so do all processes of apprehension, in the study of geometry no less than of arithmetic. That this is not Kant’s meaning, and that it is not even what Schulze, notwithstanding his seemingly explicit mode of statement, intends to assert, is clearly shown by a letter written by Kant to Schulze in November 1788. Schulze, it appears, had spoken of this very matter.

“Time, as you justly remark, has no influence upon the properties of numbers (as pure determinations of quantity), such as it may have upon the nature of those changes (of quantity) which are possible only in connection with a specific property of inner sense and its form (time). The science of number, notwithstanding the succession which every construction of quantity demands, is a pure intellectual synthesis which we represent to ourselves in thought. But so far as quanta are to be numerically determined, they must be given to us in such a way that we can apprehend their intuition in successive order, and such that their apprehension can be subject to time....”[519]

No more definite statement could be desired of the fact that though in arithmetical science as in other fields of study our processes of apprehension are subject to time, the quantitative relations determined by the science are independent of time and are intellectually apprehended.

But if the above psychological interpretation of Kant’s teaching is untenable, how is his position to be defined? We must bear in mind the doctrine which Kant had already developed in his pre-Critical period, that mathematical differs from philosophical knowledge in that its concepts can have concrete individual form.[520] In the Critique this difference is expressed in the statement that the mathematical sciences alone are able to construct their concepts. And as they are pure mathematical sciences, this construction is supposed to take place by means of the a priori manifold of space and of time. Now though Kant had a fairly definite notion of what he meant by the construction of geometrical figures in space, his various utterances seem to show that in regard to the nature of arithmetical and algebraic construction he had never really attempted to arrive at any precision of view. To judge by the passage already quoted[521] from the Dissertation, Kant regarded space as no less necessary than time to the construction or intuition of number. ”[The intellectual concept of number] demands for its concrete actualisation the auxiliary notions of time and space (in the successive addition and in the juxtaposition of a plurality)” A similar view appears in the Critique in A 140 = B 179 and in B 15. In conformity, however, with the general requirements of his doctrine of Schematism, Kant defines the schema of number in exclusive reference to time; and, as we have noted, it is to this definition that Schulze appeals in support of his view of arithmetic as the science of counting and therefore of time. It at least shows that Kant perceived some form of connection to exist between arithmetic and time. But in this matter Kant’s position was probably simply a corollary from his general view of the nature of mathematical science, and in particular of his view of geometry, the “exemplar”[522] of all the others. Mathematical science, as such, is based on intuition;[523] therefore arithmetic, which is one of its departments, must be so likewise. No attempt, however, is made to define the nature of the intuitions in which it has its source. Sympathetically interpreted, his statements may be taken as suggesting that arithmetic is the study of series which find concrete expression in the order of sequent times. The following estimate, given by Cassirer,[524] does ample justice both to the true and to the false elements in Kant’s doctrine.

”[Even discounting Kant’s insistence upon the conceptual character of arithmetical science, and] allowing that he derives arithmetical concepts and propositions from the pure intuition of time, this teaching, to whatever objections it may lie open, has certainly not the merely psychological meaning which the majority of its critics have ascribed to it. If it contained only the trivial thought, that the empirical act of counting requires time, it would be completely refuted by the familiar objection which B. Beneke has formulated: ‘The fact that time elapses in the process of counting can prove nothing; for what is there over which time does not flow?’ It is easily seen that Kant is only concerned with the ‘transcendental’ determination of the concept of time, according to which it appears as the type of an ordered sequence. William [Rowan] Hamilton, who adopts Kant’s doctrine, has defined algebra as ‘science of pure time or order in progression.’ That the whole content of arithmetical concepts can really be obtained from the fundamental concept of order in unbroken development, is completely confirmed by Russell’s exposition. As against the Kantian theory it must, of course, be emphasised, that it is not the concrete form of time intuition which constitutes the ground of the concept of number, but that on the contrary the pure logical concepts of sequence and of order are already implicitly contained and embodied in that concrete form.”

Much of the unsatisfactoriness of Kant’s argument is traceable to his mode of conceiving the “construction”[525] of mathematical concepts. All concepts, he seems to hold, even those of geometry and arithmetic, are abstract class concepts—the concept of triangle representing the properties common to all triangles, and the concept of seven the properties common to all groups that are seven. Mathematical concepts differ, however, from other concepts in that they are capable of a priori construction, that is, of having their objects represented in pure intuition. Now this is an extremely unfortunate mode of statement. It implies that mathematical concepts have a dual mode of existence, first as abstracted, and secondly as constructed. Such a position is not tenable. The concept of seven, in its primary form, is not abstracted from a variety of particular groups of seven; it is already involved in the apprehension of each of them as being seven. Nor is it a concept that is itself constructed. It may perhaps be described as being the representation of something constructed; but that something is not itself. It represents the process or method generative of the complex for which it stands. Thus Kant’s distinction between the intuitive nature of mathematical knowledge and the merely discursive character of conceptual knowledge is at once inspired by the very important distinction between the product of construction and the product of abstraction, and yet at the same time is also obscured by the quite inadequate manner in which that latter distinction has been formulated. Kant has again adhered to the older logic even in the very act of revising its conclusions; and in so doing he has sacrificed the Critical doctrines of the Analytic to the pre-Critical teaching of the Dissertation and Aesthetic. Mathematical concepts are of the same general type as the categories; their primary function is not to clarify intuitions, but to make them possible. They are derivable from intuition only in so far as they have contributed to its constitution. If intuition contains factors additional to the concepts through which it is interpreted, these factors must remain outside the realm of mathematical science, until such time as conceptual analysis has proved itself capable of further extension.