The Critique of Pure Reason, Immanuel Kant [english books to improve english TXT] 📗

them is to employ them. In all this they are perfectly right, if they do not overstep the limits of the sphere of nature. But they pass, unconsciously, from the world of sense to the insecure ground of pure transcendental conceptions (instabilis tellus, innabilis unda), where they can neither stand nor swim, and where the tracks of their footsteps are obliterated by time; while the march of mathematics is pursued on a broad and magnificent highway, which the latest posterity shall frequent without fear of danger or impediment.

As we have taken upon us the task of determining, clearly and certainly, the limits of pure reason in the sphere of transcendentalism, and as the efforts of reason in this direction are persisted in, even after the plainest and most expressive warnings, hope still beckoning us past the limits of experience into the splendours of the intellectual world—it becomes necessary to cut away the last anchor of this fallacious and fantastic hope. We shall, accordingly, show that the mathematical method is unattended in the sphere of philosophy by the least advantage—except, perhaps, that it more plainly exhibits its own inadequacy—that geometry and philosophy are two quite different things, although they go band in hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.

The evidence of mathematics rests upon definitions, axioms, and demonstrations. I shall be satisfied with showing that none of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians; and that the geometrician, if he employs his method in philosophy, will succeed only in building card-castles, while the employment of the philosophical method in mathematics can result in nothing but mere verbiage. The essential business of philosophy, indeed, is to mark out the limits of the science; and even the mathematician, unless his talent is naturally circumscribed and limited to this particular department of knowledge, cannot turn a deaf ear to the warnings of philosophy, or set himself above its direction.

I. Of Definitions. A definition is, as the term itself indicates, the representation, upon primary grounds, of the complete conception of a thing within its own limits.* Accordingly, an empirical conception cannot be defined, it can only be explained. For, as there are in such a conception only a certain number of marks or signs, which denote a certain class of sensuous objects, we can never be sure that we do not cogitate under the word which indicates the same object, at one time a greater, at another a smaller number of signs. Thus, one person may cogitate in his conception of gold, in addition to its properties of weight, colour, malleability, that of resisting rust, while another person may be ignorant of this quality. We employ certain signs only so long as we require them for the sake of distinction; new observations abstract some and add new ones, so that an empirical conception never remains within permanent limits. It is, in fact, useless to define a conception of this kind.

If, for example, we are speaking of water and its properties, we do not stop at what we actually think by the word water, but proceed to observation and experiment; and the word, with the few signs attached to it, is more properly a designation than a conception of the thing. A definition in this case would evidently be nothing more than a determination of the word. In the second place, no a priori conception, such as those of substance, cause, right, fitness, and so on, can be defined. For I can never be sure, that the clear representation of a given conception (which is given in a confused state) has been fully developed, until I know that the representation is adequate with its object. But, inasmuch as the conception, as it is presented to the mind, may contain a number of obscure representations, which we do not observe in our analysis, although we employ them in our application of the conception, I can never be sure that my analysis is complete, while examples may make this probable, although they can never demonstrate the fact. Instead of the word definition, I should rather employ the term exposition—

a more modest expression, which the critic may accept without surrendering his doubts as to the completeness of the analysis of any such conception. As, therefore, neither empirical nor a priori conceptions are capable of definition, we have to see whether the only other kind of conceptions—arbitrary conceptions—can be subjected to this mental operation. Such a conception can always be defined; for I must know thoroughly what I wished to cogitate in it, as it was I who created it, and it was not given to my mind either by the nature of my understanding or by experience. At the same time, I cannot say that, by such a definition, I have defined a real object. If the conception is based upon empirical conditions, if, for example, I have a conception of a clock for a ship, this arbitrary conception does not assure me of the existence or even of the possibility of the object.

My definition of such a conception would with more propriety be termed a declaration of a project than a definition of an object. There are no other conceptions which can bear definition, except those which contain an arbitrary synthesis, which can be constructed a priori.

Consequently, the science of mathematics alone possesses definitions. For the object here thought is presented a priori in intuition; and thus it can never contain more or less than the conception, because the conception of the object has been given by the definition—and primarily, that is, without deriving the definition from any other source. Philosophical definitions are, therefore, merely expositions of given conceptions, while mathematical definitions are constructions of conceptions originally formed by the mind itself; the former are produced by analysis, the completeness of which is never demonstratively certain, the latter by a synthesis. In a mathematical definition the conception is formed, in a philosophical definition it is only explained. From this it follows: [*Footnote: The definition must describe the conception completely that is, omit none of the marks or signs of which it composed; within its own limits, that is, it must be precise, and enumerate no more signs than belong to the conception; and on primary grounds, that is to say, the limitations of the bounds of the conception must not be deduced from other conceptions, as in this case a proof would be necessary, and the so-called definition would be incapable of taking its place at the bead of all the judgements we have to form regarding an object.]

(a) That we must not imitate, in philosophy, the mathematical usage of commencing with definitions—except by way of hypothesis or experiment. For, as all so-called philosophical definitions are merely analyses of given conceptions, these conceptions, although only in a confused form, must precede the analysis; and the incomplete exposition must precede the complete, so that we may be able to draw certain inferences from the characteristics which an incomplete analysis has enabled us to discover, before we attain to the complete exposition or definition of the conception. In one word, a full and clear definition ought, in philosophy, rather to form the conclusion than the commencement of our labours.* In mathematics, on the contrary, we cannot have a conception prior to the definition; it is the definition which gives us the conception, and it must for this reason form the commencement of every chain of mathematical reasoning.

[*Footnote: Philosophy abounds in faulty definitions, especially such as contain some of the elements requisite to form a complete definition. If a conception could not be employed in reasoning before it had been defined, it would fare ill with all philosophical thought. But, as incompletely defined conceptions may always be employed without detriment to truth, so far as our analysis of the elements contained in them proceeds, imperfect definitions, that is, propositions which are properly not definitions, but merely approximations thereto, may be used with great advantage. In mathematics, definition belongs ad esse, in philosophy ad melius esse.

It is a difficult task to construct a proper definition. Jurists are still without a complete definition of the idea of right.]

(b) Mathematical definitions cannot be erroneous. For the conception is given only in and through the definition, and thus it contains only what has been cogitated in the definition. But although a definition cannot be incorrect, as regards its content, an error may sometimes, although seldom, creep into the form. This error consists in a want of precision. Thus the common definition of a circle—that it is a curved line, every point in which is equally distant from another point called the centre—is faulty, from the fact that the determination indicated by the word curved is superfluous. For there ought to be a particular theorem, which may be easily proved from the definition, to the effect that every line, which has all its points at equal distances from another point, must be a curved line—that is, that not even the smallest part of it can be straight. Analytical definitions, on the other hand, may be erroneous in many respects, either by the introduction of signs which do not actually exist in the conception, or by wanting in that completeness which forms the essential of a definition. In the latter case, the definition is necessarily defective, because we can never be fully certain of the completeness of our analysis. For these reasons, the method of definition employed in mathematics cannot be imitated in philosophy.

2. Of Axioms. These, in so far as they are immediately certain, are a priori synthetical principles. Now, one conception cannot be connected synthetically and yet immediately with another; because, if we wish to proceed out of and beyond a conception, a third mediating cognition is necessary. And, as philosophy is a cognition of reason by the aid of conceptions alone, there is to be found in it no principle which deserves to be called an axiom. Mathematics, on the other hand, may possess axioms, because it can always connect the predicates of an object a priori, and without any mediating term, by means of the construction of conceptions in intuition. Such is the case with the proposition: Three points can always lie in a plane.

On the other hand, no synthetical principle which is based upon conceptions, can ever be immediately certain (for example, the proposition: Everything that happens has a cause), because I require a mediating term to connect the two conceptions of event and cause-namely, the condition of time-determination in an experience, and I cannot cognize any such principle immediately and from conceptions alone. Discursive principles are, accordingly, very different from intuitive principles or axioms. The former always require deduction, which in the case of the latter may be altogether dispensed with.

Axioms are, for this reason, always self-evident, while philosophical principles, whatever may be the degree of certainty they possess, cannot lay any claim to such a distinction. No synthetical proposition of pure transcendental reason can be so evident, as is often rashly enough declared, as the statement, twice two are four.

It is true that in the Analytic I introduced into the list of principles of the pure understanding, certain axioms of intuition; but the principle there discussed was not itself an axiom, but served merely to present the principle of the possibility of axioms in general, while it was really nothing more than a principle based upon conceptions. For it is one part of the duty of transcendental philosophy to establish the possibility of mathematics itself.

Philosophy possesses, then, no axioms, and has no right to impose its a priori principles upon thought, until it has established their authority and validity by a thoroughgoing deduction.

3. Of Demonstrations. Only an apodeictic proof, based upon intuition, can be