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

by Norman Kemp Smith

“The proposition—the angles of a triangle are together equal to two right angles—Kant regards as synthetic. It is indeed deduced from the axiom of parallels (with the aid of auxiliary lines), and to that extent is understood in accordance with the principle of contradiction.... The angles in the triangle constitute a special case of the angles in the parallel lines which are intersected by other lines. The principle of contradiction thus serves as vehicle in the deduction, because once the identity of A and A´ is recognised, the predicate b, which belongs to A, must also be ascribed to A´. But the proposition is not for that reason itself analytic in the Kantian sense. In the analytic proposition the predicate is derived from the analysis of the subject concept. But that does not happen in this case. The synthetic proposition can never be derived in and by itself from the principle of contradiction; ... but only with the aid of that principle from other propositions. Besides, in this deduction intuition must always be resorted to; and that makes an essential difference. Without it the identity of A and A´ cannot become known.”

Pure mathematics.[268]—“Pure,” as thus currently used, is opposed only to applied, not to empirical. Kant here arbitrarily reads the latter opposition into it. Under this guise he begs the point in dispute.

7 + 5 = 12.[269]—Though 7 + 5 = 12 expresses an identity or equality, it is an equality of the objects or magnitudes, 7 + 5 and 12, not of the concepts through which we think them.[270] Analysis of the concepts can never reveal this equality. Only by constructing the concepts in intuition can it be recognised by the mind. This example has been already cited in the first edition.[271] It is further elaborated in the Prolegomena, § 2 c, and is here transcribed. Kant’s mode of stating his position is somewhat uncertain. He alternates between “the representation of 7 and 5,” “the representation of the combination of 7 and 5,”[272] and “the concepts 7 and 5.”[273] His view would seem to be that there are three concepts involved. For the concept of 7 we must substitute the intuition of 7 points, for the concept of 5 the intuition of 5 points, and for the concept of their sum the intuitive operation of addition.

Call in the assistance of intuition, for instance our five fingers.[274]—This statement, repeated from the Prolegomena,[275] does not represent Kant’s real position. The views which he has expressed upon the nature of arithmetical science are of the most contradictory character,[276] but to one point he definitely commits himself, namely, that, like geometrical science, it rests, not (as here asserted) upon empirical, but upon pure intuition.[277] Except indirectly, by the reference to larger numbers, Kant here ignores his own important distinction between image and schema.[278] The above statement would also make arithmetic dependent upon space.

Segner: Anfangsgründe der Arithmetik,[279] translated from the Latin, second edition, Halle, 1773.

Natural science (physica) contains synthetic a priori judgments.[280]—There is here a complication to which Vaihinger[281] has been the first to draw attention. In the Prolegomena[282] Kant emphasises the distinction between physics and pure or universal science of nature.[283] The latter treats only the a priori form of nature (i.e. its necessary conformity to law), and is therefore a propaedeutic to physics which involves further empirical factors. For two reasons, however, this universal natural science falls short of its ideal. First, it contains empirical elements, such as the concepts of motion, impenetrability, inertia, etc. Secondly, it refers only to the objects of external sense, and not, as we should expect in a universal science, to natural existences without exception, i.e. to the objects of psychology as well as of physics.[284] But among its principles there are, Kant adds, a few which are purely a priori and possess the universality required: e.g. such propositions as that substance is permanent, and that every event has a cause. Now these are the examples which ought to have been cited in the passage before us. Those actually given fall entirely outside the scope of the Critique. They are treated only in the Metaphysische Anfangsgründe. They belong to the relatively, not to the absolutely, pure science of nature. The source of the confusion Vaihinger again traces to Kant’s failure to hold fast to the important distinction between immanent and transcendent metaphysics.[285] His so-called pure or universal natural science (nature, as above noted, signifying for Kant “all that is”) is really immanent metaphysics, and the propositions in regard to substance and causality ought therefore to be classed as metaphysical. This, indeed, is how they are viewed in the earlier sections of the Prolegomena. The distinction later drawn in § 15 is ignored. Pure natural science is identified with mathematical physics, and the propositions which in § 15 are spoken of as belonging to pure universal natural science are now regarded as metaphysical. “Genuinely metaphysical judgments are one and all synthetic.... For instance, the proposition—everything which in things is substance is permanent—is a synthetic, and properly metaphysical judgment.”[286] In § 5 the principle of causality is also cited as an example of a synthetic a priori judgment in metaphysics. But Kant still omits to draw a distinction between immanent and transcendent metaphysics; and as a consequence his classification of synthetic a priori judgments remains thoroughly confused. They are taken as belonging to three spheres, mathematics, physics (in the relative sense), and metaphysics. The implication is that this threefold distinction corresponds to the threefold division of the Doctrine of Elements into Aesthetic, Analytic, and Dialectic. Yet, as a matter of fact, the propositions of mathematical physics, in so far as they are examples of applied mathematics, are dealt with in the Aesthetic, and in so far as they involve concepts of motion and the like fall entirely outside the scope of the Critique, while the Analytic deals with those metaphysical judgments (such as the principle of causality) which are of immanent employment.[287]

As the new paragraphs in the Introduction to the second edition are transferred without essential modification from the Prolegomena, they are open to the same criticism. To harmonise B 17 with the real teaching of the Critique, it must be entirely recast. Instead of “natural science” (physica) we must read “pure universal natural science [= immanent metaphysics],” and for the examples given we must substitute those principles of substance and causality which are dealt with in the Analytic. The next paragraph deals with metaphysics in its transcendent form, and accordingly states the problem peculiar to the Dialectic.

Metaphysics.[288]—This paragraph deals explicitly only with transcendent judgments, but as the terms used are ambiguous, it is possible that those of immanent metaphysics are also referred to. The paragraph is not taken from the Prolegomena. The corresponding passage[289] in the Prolegomena deals only with the judgments of immanent metaphysics.

The real problem of pure reason is contained in the question: How are synthetic a priori judgments possible?[290]—Cf. above, pp. 26 ff., 33 ff., 43 ff.

David Hume.[291]—Cf. above, pp. 61 ff.