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

by Norman Kemp Smith

“Pure mathematics treats of space in geometry and of time in pure mechanics.”

But in the Dissertation the point is only touched upon in passing. The context permits of the interpretation that while geometry deals with space, mechanics deals with time in addition to space.

KANT’S VIEWS REGARDING THE NATURE OF ARITHMETICAL SCIENCE

In the Dissertation, and again in the chapter on Schematism in the Critique itself, still another view is suggested, namely, that the science of arithmetic is also concerned with the intuition of time. The passage just quoted from the Dissertation proceeds as follows:

“Pure mathematics treats of space in geometry and of time in pure mechanics. To these has to be added a certain concept which is in itself intellectual, but which demands for its concrete actualisation (actuatio) the auxiliary notions of time and space (in the successive addition and in the juxtaposition of a plurality). This is the concept of number which is dealt with in Arithmetic.”[510]

This view of arithmetic is to be found in both editions of the Critique. Arithmetic depends upon the synthetic activity of the understanding; the conceptual element is absolutely essential.

“Our counting (as is easily seen in the case of large numbers) is a synthesis according to concepts, because it is executed according to a common ground of unity, as, for instance, the decade (Dekadik).”[511] “The pure image ... of all objects of the senses in general is time. But the pure schema of quantity, in so far as it is a concept of the understanding, is number, a representation which combines the successive addition of one to one (homogeneous). Thus number is nothing but the unity of the synthesis of the manifold of a homogeneous intuition in general, whereby I generate time itself in the apprehension of the intuition.”[512]

This is also the teaching of the Methodology.[513] Now it may be observed that in none of these passages is arithmetic declared to be the science of time, or even to be based on the intuition of time. In 1783, however, in the Prolegomena, Kant expresses himself in much more ambiguous terms, for his words imply that there is a parallelism between geometry and arithmetic.

“Geometry is based upon the pure intuition of space. Arithmetic produces its concepts of number through successive addition of units in time, and pure mechanics especially can produce its concepts of motion only by means of the representation of time.”[514]

The passage is by no means explicit; the “especially” (vornehmlich) seems to indicate a feeling on Kant’s part that the description which he is giving of arithmetic is not really satisfactory. Unfortunately this casual statement, though never repeated by Kant in any of his other writings, was developed by Schulze in his Erläuterungen.