SECTION I. The Discipline of Pure Reason in Its Dogmatic Employment

THE DISCIPLINE OF PURE REASON IN ITS DOGMATIC EMPLOYMENT

In dealing with the distinction between mathematical and philosophical knowledge, Kant is here returning to one of the main points of his Introduction to the Critique. His most exhaustive treatment of it is, however, to be found in a treatise which he wrote as early as 1764, his Enquiry into the Clearness of the Principles of Natural Theology and Morals. The continued influence of the teaching of that early work is obvious throughout this section, and largely accounts for the form in which certain of its tenets are propounded.

“...one can say with Bishop Warburton that nothing has been more injurious to philosophy than mathematics, that is, than the imitation of its method in a sphere where it is impossible of application....”

So far from being identical in general nature, mathematics and philosophy are, Kant declares, fundamentally opposed in all essential features. For it is in their methods, and not merely in their subject-matter, that the essential difference between them is to be found. Philosophical knowledge can be acquired only through concepts, mathematical knowledge is gained through the construction of concepts. The one is discursive merely; the other is intuitive. Philosophy can consider the particular only in the general; mathematics studies the general in the particular. Philosophical concepts, such as those of substance and causality, are, indeed, capable of application in transcendental synthesis, but in this employment they yield only empirical knowledge of the sensuously given; and from empirical concepts the universal and necessary judgments required for the possibility of metaphysical science can never be obtained.

The exactness of mathematics depends on definitions, axioms, and demonstrations, none of which are obtainable in philosophy. To take each in order.

I. =Definitions.=--To define in the manner prescribed by mathematics is to represent the complete concept of a thing. This is never possible in regard to empirical concepts. We are more certain of their denotation than of their connotation; and though they may be explained, they cannot be defined. Since new observations add or remove predicates, an empirical concept is always liable to modification.

“What useful purpose could be served by defining an empirical concept, such, for instance, as that of water? When we speak of water and its properties, we do not stop short at what is thought in the word water, but proceed to experiments. The word, with the few marks which are attached to it, is more properly to be regarded as merely a designation than as a conception. The so-called definition is nothing more than a determining of the word.”

Exact definition is equally impossible in regard to a priori forms, such as time or causality. Since they are not framed by the mind, but are given to it, the completeness of our analysis of them can never be guaranteed. Though they are known, they are known only as problems.

“As Augustine has said, ‘I know well what time is, but if any one asks me, I cannot tell.’”

Mathematical definitions make concepts; philosophical definitions only explain them. Philosophy cannot, therefore, imitate mathematics by beginning with definitions. In philosophy the incomplete exposition must precede the complete; definitions are the final outcome of our enquiry, and not as in mathematics the only possible beginning of its proofs. Indeed, the mathematical concept may be said to be given by the very process in which it is constructively defined; and, as thus originating in the process of definition, it can never be erroneous. Philosophy, on the other hand, swarms with faulty definitions, which are none the less serviceable.

“In mathematics definition belongs ad esse, in philosophy ad melius esse. It is desirable to attain it, but often very difficult. Jurists are still without a definition of their concept of Right.”

II. =Axioms.=--This paragraph is extremely misleading as a statement of Kant’s view regarding the nature of geometrical axioms. In stating that they are self-evident, he does not really mean to assert what that phrase usually involves, namely, absolute a priori validity. For Kant the geometrical axioms are merely descriptions of certain de facto properties of the given intuition of space. They have the merely hypothetical validity of all propositions that refer to the contingently given. For even as a pure intuition, space belongs to the realm of the merely factual. This un-Critical opposition of the self-evidence of geometrical axioms to the synthetic character of such “philosophical” truths as the principle of causality is bound up with Kant’s unreasoned conviction that space in order to be space at all, must be Euclidean. Kant’s reference in this paragraph to the propositions of arithmetic is equally open to criticism. For though he is more consistent in recognising their synthetic character, he still speaks as if they could be described as self-evident, i.e. as immediately certain. The cause of this inconsistency is, of course, to be found in his intuitional theory of mathematical science. Mathematical propositions are obtained through intuition; those of philosophy call for an elaborate and difficult process of transcendental deduction. When modern mathematical theory rejects this intuitional view, it is really extending to mathematical concepts Kant’s own interpretation of the function of the categories. Concepts condition the possibility of intuitional experience, and find in this conditioning power the ground of their objective validity. Here, as in the Aesthetic, Kant fails adequately to distinguish between the problems of pure and applied mathematics.

=III. Demonstrations.=--Kant again introduces his very unsatisfactory doctrine of the construction of concepts: and he even goes so far as to maintain, in complete violation of his own doctrine of transcendental deduction, that where there is no intuition, there can be no demonstration. Apodictic propositions, he declares, are either dogmata or mathemata; and the former are beyond the competence of the human mind. But no sooner has he made these statements than he virtually withdraws them by adding that, though apodictic propositions cannot be established directly from concepts, they can be indirectly proved by reference to something purely contingent, namely, possible experience. Thus the principle of causality can be apodictically proved as a condition of possible experience. Though it may not be called a dogma, it can be entitled a principle! In explanation of this distinction, which betrays a lingering regard for the self-evident maxims of rationalistic teaching, Kant adds that the principle of causality, though a principle, has itself to be proved.

“...it has the peculiarity that it first makes possible its own ground of proof, namely, experience....”

This, as we have noted, is exactly what mathematical axioms must also be able to do, if they are to establish their objective validity.