Critique of Pure Reason

by Immanuel Kant

  I should therefore in vain philosophise, that is, reflect discursively on the triangle, without ever getting beyond the mere definition with which I ought to have begun. There is no doubt a transcendental synthesis, consisting of mere concepts, and in which the philosopher alone can hope to be successful. Such a synthesis, however, never relates to more than a thing in general, and to the conditions under which its perception could be a possible experience. In the mathematical problems, on the contrary, all this, together with the question of existence, does not concern us, but the properties of objects in themselves only (without any reference to their existence), and those properties again so far only as they are connected with their concept.

  We have tried by this example to show how great a difference there is between the discursive use of reason, according to concepts, and its intuitive use, through the construction of concepts. The question now arises what can be the cause that makes this twofold use of reason necessary, and how can we discover whether in any given argument the former only, or the latter use also, takes place?

  All our knowledge relates, in the end, to possible intuitions, for it is by them alone that an object can be given. A concept a priori (or a non-empirical concept) contains either a pure intuition, in which case it can be constructed, or it contains nothing but the synthesis of possible intuitions, which are not given a priori, and in that case, though we may use it for synthetical and a priori judgments, such judgments can only be discursive, according to concepts, and never intuitive, through the construction of the concept.

  There is no intuition a priori except space and time, the mere forms of phenomena. A concept of them, as quanta, can be represented a priori in intuition, that is, can be constructed either at the same time with their quality (figure), or as quantity only (the mere synthesis of the manifold-homogeneous), by means of number. The matter of phenomena, however, by which things are given us in space and time, can be represented in perception only, that is a posteriori. The one concept which a priori represents the empirical contents of phenomena is the concept of a thing in general, and the synthetical knowledge which we may have a priori of a thing in general, can give us nothing but the mere rule of synthesis, to be applied to what perception may present to us a posteriori, but never an a priori intuition of a real object, such an intuition being necessarily empirical.

  Synthetical propositions with regard to things in general, the intuition of which does not admit of being given a priori, are called transcendental. Transcendental propositions, therefore, can never be given through a construction of concepts, but only according to concepts a priori. They only contain the rule, according to which we must look empirically for a certain synthetical unity of what cannot be represented in intuition a priori (perceptions). They can never represent any one of their concepts a priori, but can do this only a posteriori, that is, by means of experience, which itself becomes possible according to those synthetical principles only.

  If we are to form a synthetical judgment of any concept, we must proceed beyond that concept to the intuition in which it is given. For if we kept within that which is given in the concept, the judgment could only be analytical and an explanation of the concept, in accordance with what we have conceived in it. I may, however, pass from the conception to the pure or empirical intuition which corresponds to it, in order thus to consider it in concreto, and thus to discover what belongs to the object of the concept, whether a priori or a posteriori. The former consists in rational or mathematical knowledge, arrived at by the construction of the concept, the latter in the purely empirical (mechanical) knowledge which can never supply us with necessary and apodictic propositions. Thus I might analyse my empirical concept of gold, without gaining anything beyond being able to enumerate everything that I can really think by this word. This might yield a logical improvement of my knowledge, but no increase or addition. If, however, I take the material which is known by the name of gold, I can make observations on it, and these will yield me different synthetical, but empirical propositions. Again, I might construct the mathematical concept of a triangle, that is, give it a priori in intuition, and gain in this manner a synthetical but rational knowledge of it. But when the transcendental concept of a reality, a substance, a power, etc., is given me, that concept denotes neither an empirical nor a pure intuition, but merely the synthesis of empirical intuitions, which, being empirical, cannot be given a priori. No determining synthetical proposition therefore can spring from it, because the synthesis cannot a priori pass beyond to the intuition that corresponds to it, but only a principle of the synthesis2 of possible empirical intuitions.

  A transcendental proposition, therefore, is synthetical knowledge acquired by reason, according to mere concepts; and it is discursive, because through it alone synthetical unity of empirical knowledge becomes possible, while it cannot give us any intuition a priori.

  We see, therefore, that reason is used in two ways which, though they share in common the generality of their knowledge and its production a priori, yet diverge considerably afterwards, because in each phenomenon (and no object can be given us, except as a phenomenon), there are two elements, the form of intuition (space and time), which can be known and determined entirely a priori, and the matter (the physical) or the contents, something which exists in space and time, and therefore contains an existence corresponding to sensation. As regards the latter, which can never be given in a definite form except empirically, we can have nothing a. priori except indefinite concepts of the synthesis of possible sensations, in so far as they belong to the unity of apperception (in a possible experience). As regards the former, we can determine a priori our concepts in intuition, by creating to ourselves in space and time, through a uniform synthesis, the objects themselves, considering them simply as quanta. The former is called the use of reason according to concepts; and here we can do nothing more than to bring phenomena under concepts, according to their real contents, which therefore can be determined empirically only, that is a posteriori (though in accordance with those concepts as rules of an empirical synthesis). The latter is the use of reason through the construction of concepts, which, as they refer to an intuition a priori, can for that reason be given a priori, and defined in pure intuition, without any empirical data. To consider everything which exists (everything in space or time) whether, and how far, it is a quantum or not; to consider that we must represent in it either existence, or absence of existence; to consider how far this something which fills space or time is a primary substratum, or merely determination of it; to consider again whether its existence is related to something else as cause or effect, or finally, whether it stands isolated or in reciprocal dependence on others, with reference to existence,—this and the possibility, reality, and necessity of its existence, or their opposites, all belong to that knowledge of reason, derived from concepts, which is called philosophical. But to determine a priori an intuition in space (figure), to divide time (duration), or merely to know the general character of the synthesis of one and the same thing in time and space, and the quantity of an intuition in general which arises from it (number), all this is the work of reason by means of the construction of concepts, and is called mathematical.

  The great success which attends reason in its mathematical use produces naturally the expectation that it, or rather its method, would have the same success outside the field of quantities also, by reducing all concepts to intuitions which may be given a priori, and by which the whole of nature might be conquered, while pure philosophy, with its discursive concepts a priori, does nothing but bungle in every part of nature, without being able to render the reality of those concepts intuitive a priori, and thereby legitimatised. Nor does there seem to be any lack of confidence on the part of those who are masters in the art of mathematics, or of high expectations on the part of the public at large, as to their ability of achieving success, if only they would try it. For as they have hardly ever philosophised on mathematics (which is indeed no easy task), th
ey never think of the specific difference between the two uses of reason which we have just explained. Current and empirical rules, borrowed from the ordinary operations of reason, are then accepted instead of axioms. From what quarter the concepts of space and time with which alone (as the original quanta) they have to deal, may have come to them, they do not care to enquire, nor do they see any use in investigating the origin of the pure concepts of the understanding, and with it the extent of their validity, being satisfied to use them as they are. In all this no blame would attach to them, if only they did not overstep their proper limits, namely, those of nature. But as it is, they lose themselves, without being aware of it, away from the field of sensibility on the uncertain ground of pure and ever transcendental concepts (instabilis tellus, innabilis unda) where they are neither able to stand nor to swim, taking only a few hasty steps, the vestiges of which are soon swept away, while their steps in mathematics become a highway, on which the latest posterity may march on with perfect confidence.

  We have chosen it as our duty to determine with accuracy and certainty the limits of pure reason in its transcendental use. These transcendental efforts, however, have this peculiar character that, in spite of the strongest and clearest warnings, they continue to inspire us with new hopes, before the attempt is entirely surrendered at arriving beyond the limits of experience at the charming fields of an intellectual world. It is necessary therefore to cut away the last anchor of that fantastic hope, and to show that the employment of the mathematical method cannot be of the slightest use for this kind of knowledge, unless it be in displaying its own deficiencies; and that the art of measuring and philosophy are two totally different things, though they are mutually useful to each other in natural science, and that the method of the one can never be imitated by the other.

  The exactness of mathematics depends on definitions, axioms, and demonstrations. I shall content myself with showing that none of these can be achieved or imitated by the philosopher in the sense in which they are understood by the mathematician. I hope to show at the same time that the art of measuring, or geometry, will by its method produce nothing in philosophy but card-houses, while the philosopher with his method produces in mathematics nothing but vain babble. It is the very essence of philosophy to teach the limits of knowledge, and even the mathematician, unless his talent is limited already by nature and restricted to its proper work, cannot decline the warnings of philosophy or altogether defy them.

  I. Of Definitions. To define, as the very name implies, means only to represent the complete concept of a thing within its limits and in its primary character.3 From this point of view, an empirical concept cannot be defined, but can be explained only. For, as we have in an empirical concept some predicates only belonging to a certain class of sensuous objects, we are never certain whether by the word which denotes one and the same object, we do not think at one time a greater, at another a smaller number of predicates. Thus one man may by the concept of gold think, in addition to weight, colour, malleability, the quality of its not rusting, while another may know nothing of the last. We use certain predicates so long only as they are required for distinction. New observations add and remove certain predicates, so that the concept never stands within safe limits. And of what use would it be to define an empirical concept, as for instance that of water, because, when we speak of water and its qualities, we do not care much what is thought by that word, but proceed at once to experiments? the word itself with its few predicates being a designation only and not a concept, so that a so-called definition would be no more than a determination of the word. Secondly, if we reasoned accurately, no a priori given concept can be defined, such as substance, cause, right, equity, etc. For I can never be sure that the clear representation of a given but still confused concept has been completely analysed, unless I know that such representation is adequate to the object. As its concept, however, such as it is given, may contain many obscure representations which we pass by in our analysis, although we use them always in the practical application of the concept, the completeness of the analysis of my concept must always remain doubtful, and can only be rendered probable by means of apt examples, although never apodictically certain. I should therefore prefer to use the term exposition rather than definition, as being more modest, and more likely to be admitted to a certain extent by a critic who reserves his doubts as to its completeness. As therefore it is impossible to define either empirically or a priori given concepts, there remain arbitrary concepts only on which such an experiment may be tried. In such a case I can always define my concept, because I ought certainly to know what I wish to think, the concept being made intentionally by myself, and not given to me either by the nature of the understanding or by experience. But I can never say that I have thus defined a real object. For if the concept depends on empirical conditions, as, for instance, a ship's chronometer, the object itself and its possibility are not given by this arbitrary concept; it does not even tell us whether there is an object corresponding to it, so that my explanation should be called a declaration (of my project) rather than a definition of an object. Thus there remain no concepts fit for definition except those which contain an arbitrary synthesis that can be constructed a priori. It follows, therefore, that mathematics only can possess definitions, because it is in mathematics alone that we represent a priori in intuition the object which we think, and that object cannot therefore contain either more or less than the concept, because the concept of the object was given by the definition in its primary character, that is, without deriving the definition from anything else. The German language has but the one word Erklärung (literally clearing up) for the terms exposition, explication, declaration, and definition; and we must not therefore be too strict in our demands, when denying to the different kinds of a philosophical clearing up the honourable name of definition. What we really insist on is this, that philosophical definitions are possible only as expositions of given concepts, mathematical definitions as constructions of concepts, originally framed by ourselves, the former therefore analytically (where completeness is never apodictically certain), the latter synthetically. Mathematical definitions make the concept, philosophical definitions explain it only. Hence it follows,

  a. That we must not try in philosophy to imitate mathematics by beginning with definitions, except it be by way of experiment. For as they are meant to be an analysis of given concepts, these concepts themselves, although as yet confused only, must come first, and the incomplete exposition must precede the complete one, so that we are able from some characteristics, known to us from an, as yet, incomplete analysis, to infer many things before we come to a complete exposition, that is, the definition of the concept. In philosophy, in fact, the definition in its complete clearness ought to conclude rather than begin our work;4 while in mathematics we really have no concept antecedent to the definition by which the concept itself is first given, so that in mathematics no other beginning is necessary or possible.

  b. Mathematical definitions can never be erroneous, because, as the concept is first given by the definition, it contains neither more nor less than what the definition wishes should be conceived by it. But although there can be nothing wrong in it, so far as its contents are concerned, mistakes may sometimes, though rarely, occur in the form or wording, particularly with regard to perfect precision. Thus the common definition of a circle, that it is a curved line, every point of which is equally distant from one and the same point (namely, the centre), is faulty, because the determination of curved is introduced unnecessarily. For there must be a particular theorem, derived from the definition, and easily proved, viz. that every line, all points of which are equidistant from one and the same point, must be curved (no part of it being straight). Analytical definitions, however, may be erroneous in many respects, either by introducing characteristics which do not really exist in the concept, or by lacking that completeness which is essential to a definition, because we can never be quite certain of the completeness of
our analysis. It is on these accounts that the method of mathematics cannot be imitated in the definitions of philosophy.

  II. Of Axioms. These, so far as they are immediately certain, are synthetical principles a priori. One concept cannot, however, be connected synthetically and yet immediately with another, because, if we wish to go beyond a given concept, a third connecting knowledge is required; and, as philosophy is the knowledge of reason based on concepts, no principle can be found in it deserving the name of an axiom. Mathematics, on the other hand, may well possess axioms, because here, by means of the construction of concepts in the intuition of their object, the predicates may always be connected a priori and immediately; for instance, that three points always lie in a plane. A synthetical principle, on the contrary, made up of concepts only, can never be immediately certain, as, for example, the proposition that everything which happens has its cause. Here I require something else, namely, the condition of the determination by time in a given experience, it being impossible for me to know such a principle, directly and immediately, from the concepts. Discursive principles are, therefore, something quite different from intuitive principles or axioms. The former always require, in addition, a deduction, not at all required for the latter, which, on that very account, are evident, while philosophical principles, whatever their certainty may be, can never pretend to be so. Hence it is very far from true to say that any synthetical proposition of pure and transcendental reason is so evident (as people sometimes emphatically maintain) as the statement that twice two are four. It is true that in the Analytic, when giving the table of the principles of the pure understanding, I mentioned also certain axioms of intuition; but the principle there mentioned was itself no axiom, but served only to indicate the principle of the possibility of axioms in general, being itself no more than a principle based on concepts. It was necessary in our transcendental philosophy to show the possibility even of mathematics. Philosophy, therefore, is without axioms, and can never put forward its principles a priori with absolute authority, but must first consent to justify its claims by a thorough deduction.