The Online Library of Liberty

A project of Liberty Fund, Inc.

• Dedication
• Preface 1
• Translator’s Preface
• Translator’s Preface to Second Edition
• Introduction [p. 1]
• I: The Idea of Transcendental Philosophy
• II: Division of Transcendental Philosophy
• I: The Elements of Transcendentalism
• Part I: Transcendental Æsthetic [p. 19]
• Section I: Of Space
• Section II: Of Time
• Part II: Transcendental Logic [p. 50]
• Introduction the Idea of a Transcendental Logic
• I: Of Logic In General
• II: Of Transcendental Logic
• III: Of the Division of General Logic Into Analytic and Dialectic
• IV: Of the Division of Transcendental Logic Into Transcendental Analytic and Dialectic
• Division I: Transcendental Analytic In Two Books, With Their Chapters and Sections
• Book I: Analytic of Concepts
• Chapter I: Method of Discovering All Pure Concepts of the Understanding
• Section I: Of the Logical Use of the Understanding In General
• Section II: Of the Logical Function of the Understanding In Judgments [p. 70]
• Section III: Of the Pure Concepts of the Understanding, Or of the Categories
• Chapter II: Of the Deduction of the Pure Concepts of the Understanding [p. 84]
• Section I: Of the Principles of a Transcendental Deduction In General
• Section II: Of the a Priori Grounds For the Possibility of Experience
• I: Of the Synthesis of Apprehension In Intuition
• II: Of the Synthesis of Reproduction In Imagination
• III: Of the Synthesis of Recognition In Concepts [p. 103]
• IV: Preliminary Explanation of the Possibility of the Categories As Knowledge a Priori
• Section III: Of the Relation of the Understanding to Objects In General, and the Possibility of Knowing Them a Priori [p. 115]
• Book II: Analytic of Principles
• Introduction of the Transcendental Faculty of Judgment In General
• Chapter I: Of the Schematism of the Pure Concepts of the Understanding [p. 137]
• Chapter II: System of All Principles of the Pure Understanding [p. 148]
• Section I: Of the Highest Principle of All Analytical Judgments
• Section II: Of the Highest Principle of All Synthetical Judgments [p. 154]
• Section III: Systematical Representation of All Synthetical Principles of the Understanding
• I: [of the Axioms of Intuition 1 Principle of the Pure Understanding
• II: [ Anticipations of Perception
• III: [ the Analogies of Experience
• IV: The Postulates of Empirical Thought In General
• Chapter III: On the Ground of Distinction of All Subjects Into Phenomena and Noumena
• I.: Identity and Difference
• II.: Agreement and Opposition
• III.: The Internal and the External
• IV.: Matter and Form
• Note On the Amphiboly of Reflective Concepts
• Division II: Transcendental Dialectic In Two Books, With Their Chapters and Sections [p. 293]
• Introduction: 1.
• Book I: Of the Concepts of Pure Reason [p. 310]
• First Section of Ideas In General
• Second Section of Transcendental Ideas [p. 321]
• Third Section System of Transcendental Ideas [p. 333]
• Book II: Of the Dialectical Conclusions of Pure Reason
• Chapter I: Of the Paralogisms of Pure Reason [p. 341]
• Chapter II: The Antinomy of Pure Reason
• Section I: System of Cosmological Ideas
• Section II: Antithetic of Pure Reason
• Section III: Of the Interest of Reason In These Conflicts [p. 462]
• Section IV: Of the Transcendental Problems of Pure Reason, and the Absolute Necessity of Their Solution
• Section V: Sceptical Representation of the Cosmological Questions In the Four Transcendental Ideas [p. 485]
• Section VI: Transcendental Idealism As the Key to the Solution of Cosmological Dialectic
• Section VII: Critical Decision of the Cosmological Conflict of Reason With Itself
• Section VIII: The Regulative Principle of Pure Reason With Regard to the Cosmological Ideas [p. 508]
• Section IX: Of the Empirical Use of the Regulative Principle of Reason With Regard to All Cosmological Ideas
• Chapter III: The Ideal of Pure Reason
• Section I: Of the Ideal In General
• Section II: Of the Transcendental Ideal ( Prototypon Transcendentale )
• Section III: Of the Arguments of Speculative Reason In Proof of the Existence of a Supreme Being
• Section IV: Of the Impossibility of an Ontological Proof of the Existence of God [p. 592]
• Section V: Of the Impossibility of a Cosmological Proof of the Existence of God [p. 603]
• Section VI: Of the Impossibility of the Physico-theological Proof
• Section VII: Criticism of All Theology Based On Speculative Principles of Reason [p. 631]
• Chapter I: The Discipline of Pure Reason
• Section I: The Discipline of Pure Reason In Its Dogmatical Use
• Section II: The Discipline of Pure Reason In Its Polemical Use
• Section III: The Discipline of Pure Reason With Regard to Hypotheses
• Section IV: The Discipline of Pure Reason With Regard to Its Proofs
• Chapter II: The Canon of Pure Reason [p. 795]
• First Section: of the Ultimate Aim of the Pure Use of Our Reason
• Section II: Of the Ideal of the Summum Bonum As Determining the Ultimate Aim of Pure Reason
• Section III: Of Trowing, Knowing, and Believing [p. 820]
• Chapter III: The Architectonic of Pure Reason [p. 832]
• Chapter IV: The History of Pure Reason [p. 852]
• Supplement I: Motto to Second Edition
• Supplement II: Preface to the Second Edition. 1787. [p. Vii]
• Supplement III: Supplement Iv
• Supplement V
• Supplement Vi
• Supplement V
• Supplement Vi
• Supplement Vii
• Supplement Viii
• Supplement Ix
• Supplement X
• Supplement Xi
• Supplement Xii
• Supplement Xiii
• Supplement Xiv
• Supplement Xv
• Supplement Xvi a
• Supplement Xvi B
• Supplement Xvii
• Supplement Xviii
• Supplement Xix
• Supplement Xx
• Supplement Xxi
• Supplement Xxii
• Supplement Xxiii
• Supplement Xxiv
• Supplement Xxv
• Supplement Xxvi
• Supplement Xxvii
• Supplement Xxviii

Section I

The Discipline of Pure Reason in its Dogmatical Use

The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. Such examples are always contagious, particularly when the faculty is the same, which naturally flatters itself that it will meet with the same success in other cases which it has had in one. Thus pure reason hopes to be able to extend its domain as successfully and as thoroughly [p. 713] in its transcendental as in its mathematical employment; particularly if it there follows the same method which has proved of such decided advantage elsewhere. It is, therefore, of great consequence for us to know whether the method of arriving at apodictic certainty, which in the former science was called mathematical, be identical with that which is to lead us to the same certainty in philosophy, and would have to be called dogmatic.

Philosophical knowledge is that which reason gains from concepts, mathematical, that which it gains from the construction of concepts. By constructing a concept I mean representing a priori the intuition corresponding to it. For the construction of a concept, therefore, a non-empirical intuition is required which, as an intuition, is a single object, but which, nevertheless, as the construction of a concept (of a general representation) must express in the representation something that is generally valid for all possible intuitions which fall under the same concept. Thus I construct a triangle by representing the object corresponding to that concept either by mere imagination, in the pure intuition, or, afterwards on paper also in the empirical intuition, and in both cases entirely a priori without having borrowed the original from any experience. The particular figure drawn on the [p. 714] paper is empirical, but serves nevertheless to express the concept without any detriment to its generality, because, in that empirical intuition, we consider always the act of the construction of the concept only, to which many determinations, as, for instance, the magnitude of the sides and the angles, are quite indifferent, these differences, which do not change the concept of a triangle, being entirely ignored.

Philosophical knowledge, therefore, considers the particular in the general only, mathematical, the general in the particular, nay, even in the individual, all this, however, a priori, and by means of reason; so that, as an individual figure is determined by certain general conditions of construction, the object of the concept, of which this individual figure forms only the schema, must be thought of as universally determined.

The essential difference between these two modes of the knowledge of reason consists, therefore, in the form, and does not depend on any difference in their matter or objects. Those who thought they could distinguish philosophy from mathematics by saying that the former was concerned with quality only, the latter with quantity only, mistook effect for cause. It is owing to the form of mathematical knowledge that it can refer to quanta only, because it is only the concept of quantities that admits of construction, that is, of a priori [p. 715] representation in intuition, while qualities cannot be represented in any but empirical intuition. Hence reason can gain a knowledge of qualities by concepts only. No one can take an intuition corresponding to the concept of reality from anywhere except from experience; we can never lay hold of it a priori by ourselves, and before we have had an empirical consciousness of it. We can form to ourselves an intuition of a cone, from its concept alone, and without any empirical assistance, but the colour of this cone must be given before, in some experience or other. I cannot represent in intuition the concept of a cause in general in any way except by an example supplied by experience, etc. Besides, philosophy treats of quantities quite as much as mathematics; for instance, of totality, infinity, etc., and mathematics treats also of the difference between lines and planes, as spaces of different quality, it treats further of the continuity of extension as one of its qualities. But, though in such cases both have a common object, the manner in which reason treats it is totally different in philosophy and mathematics. The former is concerned with general concepts only, the other can do nothing with the pure concept, but proceeds at once to intuition, in which it looks upon the concept in concreto; yet not in an [p. 716] empirical intuition, but in an intuition which it represents a priori, that is, which it has constructed and in which, whatever follows from the general conditions of the construction, must be valid in general of the object of the constructed concept also.

Let us give to a philosopher the concept of a triangle, and let him find out, in his own way, what relation the sum of its angles bears to a right angle. Nothing is given him but the concept of a figure, enclosed within three straight lines, and with it the concept of as many angles. Now he may ponder on that concept as long as he likes, he will never discover anything new in it. He may analyse the concept of a straight line or of an angle, or of the number three, and render them more clear, but he will never arrive at other qualities which are not contained in those concepts. But now let the geometrician treat the same question. He will begin at once with constructing a triangle. As he knows that two right angles are equal to the sum of all the contiguous angles which proceed from one point in a straight line, he produces one side of his triangle, thus forming two adjacent angles which together are equal to two right angles. He then divides the exterior of these angles by drawing a line parallel with the opposite side of the triangle, and sees that an exterior adjacent angle has been formed, which is equal to an interior, etc. In this way he arrives, through a chain of conclusions, though always guided by intuition, at a thoroughly [p. 717] convincing and general solution of the question.

In mathematics, however, we construct not only quantities (quanta) as in geometry, but also mere quantity (quantitas) as in algebra, where the quality of the object, which has to be thought according to this quantitative concept, is entirely ignored. We then adopt a certain notation for all constructions of quantities (numbers), such as addition, subtraction, extraction of roots, etc., and, after having denoted also the general concept of quantities according to their different relations, we represent in intuition according to general rules, every operation which is produced and modified by quantity. Thus when one quantity is to be divided by another, we place the signs of both together according to the form denoting division, etc., and we thus arrive, by means of a symbolical construction in algebra, quite as well as by an ostensive or geometrical construction of the objects themselves in geometry, at results which our discursive knowledge could never have reached by the aid of mere conceptions.

What may be the cause of this difference between two persons, the philosopher and the mathematician, both practising the art of reason, the former following his path according to concepts, the latter according to intuitions, which he represents a priori according to concepts? If we remember what has been said [p. 718] before in the Elements of Transcendentalism, the cause is clear. We are here concerned not with analytical propositions, which can be produced by a mere analysis of concepts (here the philosopher would no doubt have an advantage over the mathematician), but with synthetical propositions, and synthetical propositions that can be known a priori. We are not intended here to consider what we are really thinking in our concept of the triangle (this would be a mere definition), but we are meant to go beyond that concept, in order to arrive at properties which are not contained in the concept, but nevertheless belong to it. This is impossible, except by our determining our object according to the conditions either of empirical, or of pure intuition. The former would give us an empirical proposition only, through the actual measuring of the three angles. Such a proposition would be without the character of either generality or necessity, and does not, therefore, concern us here at all. The second procedure consists in the mathematical and here the geometrical construction, by means of which I add in a pure intuition, just as I may do in the empirical intuition, everything that belongs to the schema of a triangle in general and, therefore, to its concept, and thus arrive at general synthetical propositions.

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, [p. 719] 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 [p. 720] 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 [p. 721] 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 [p. 722] 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 synthesis1 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 [p. 723] 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 [p. 724] 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 [p. 725] 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), they 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 even transcendental concepts (instabilis tellus, innabilis unda) where they are neither able to stand nor to [p. 726] 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 [p. 727] 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.1 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 [p. 728] 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 [p. 729] 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 [p. 730] 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 [p. 731] in its complete clearness ought to conclude rather than begin our work;1 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, [p. 732] 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, [p. 733] 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 twicetwo 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. [p. 734]

III. Of Demonstrations. An apodictic proof only, so far as it is intuitive, can be called demonstration. Experience may teach us what is, but never that it cannot be otherwise. Empirical arguments, therefore, cannot produce an apodictic proof. From concepts a priori, however (in discursive knowledge), it is impossible that intuitive certainty, that is, evidence, should ever arise, however apodictically certain the judgment may otherwise seem to be. Demonstrations we get in mathematics only, because here our knowledge is derived not from concepts, but from their construction, that is, from intuition, which can be given a priori, in accordance with the concepts. Even the proceeding of algebra, with its equations, from which by reduction both the correct result and its proof are produced, is a construction by characters, though not geometrical, in which, by means of signs, the concepts, particularly those of the relation of quantities, are represented in intuition, and (without any regard to the heuristic method) all conclusions are secured against errors by submitting each of them to intuitive evidence. Philosophical knowledge cannot claim this advantage, for here we must always consider the general in the abstract (by concepts), while in mathematics we may consider the general in the concrete, in each single intuition, and yet through pure representation a priori, where every mistake becomes at once manifest. I should prefer, [p. 735] therefore, to call the former acroamatic, or audible (discursive) proofs, because they can be carried out by words only (the object in thought), rather than demonstrations, which, as the very term implies, depend on the intuition of the object.

It follows from all this that it is not in accordance with the very nature of philosophy to boast of its dogmatical character, particularly in the field of pure reason, and to deck itself with the titles and ribands of mathematics, an order to which it can never belong, though it may well hope for co-operation with that science. All those attempts are vain pretensions which can never be successful, nay, which can only prove an obstacle in the discovery of the illusions of reason, when ignoring its own limits, and which must mar our success in calling back, by means of a sufficient explanation of our concepts, the conceit of speculation to the more modest and thorough work of self-knowledge. Reason ought not, therefore, in its transcendental endeavours, to look forward with such confidence, as if the path which it has traversed must lead straight to its goal, nor depend with such assurance on its premisses as to consider it unnecessary to look back from time to time, to find out whether, in the progress of its conclusions, errors may come to light, which were overlooked in the principles, and which render it necessary [p. 736] either to determine those principles more accurately or to change them altogether.

I divide all apodictic propositions, whether demonstrable or immediately certain, into Dogmata and Mathemata. A directly synthetical proposition, based on concepts, is a Dogma; a proposition of the same kind, arrived at by the construction of concepts, is a Mathema. Analytical judgments teach us really no more of an object than what the concept which we have of it contains in itself. They cannot enlarge our knowledge beyond the concept, but only clear it. They cannot, therefore, be properly called dogmas (a word which might perhaps best be translated by precepts, Lehrsprüche). According to our ordinary mode of speech, we could apply that name to that class only of the two above-mentioned classes of synthetical propositions a priori which refers to philosophical knowledge, and no one would feel inclined to give the name of Dogma to the propositions of arithmetic or geometry. In this way the usage of language confirms our explanation that those judgments only which are based on conceptions, and not those which are arrived at by the construction of concepts, can be called dogmatic.

Now in the whole domain of pure reason, in its purely speculative use, there does not exist a single directly synthetical judgment based on concepts. We have shown that reason, by means of ideas, is incapable of any synthetical judgments which could claim objective validity, while by means of the concepts of our understanding it establishes no doubt some perfectly certain principles, [p. 737] but not directly from concepts, but indirectly only, by referring such concepts to something purely contingent, namely, possible experience. When such experience (anything as an object of possible experience) is presupposed, these principles are, no doubt, apodictically certain, but in themselves (directly) they cannot even be known a priori. Thus the proposition that everything which happens has its cause, can never be thoroughly understood by means of the concepts alone which are contained in it; hence it is no dogma in itself, although, from another point of view, that is, in the only field of its possible use, namely, in experience, it may be proved apodictically. It should be called, therefore, a principle, and not a precept or a dogma (though it is necessary that it should itself be proved), because it has this peculiarity that it first renders its own proof, namely, experience, possible, and has always to be presupposed for the sake of experience.

If, therefore, there are no dogmata whatever in the speculative use of pure reason, with regard to their contents also, all dogmatical methods, whether borrowed from mathematics or invented on purpose, are alike inappropriate. They only serve to hide mistakes and errors, and thus deceive philosophy, whose true object is to shed the clearest light on every step which reason takes. The method may, however, well be systematical; for our reason (subjectively) is itself a system, though in its [p. 738] pure use, by means of mere concepts, a system intended for investigation only, according to principles of unity, to which experience alone can supply the material. We cannot, however, dwell here on the method of transcendental philosophy, because all we have to do at present is to take stock in order to find out whether we are able to build at all, and how high the edifice may be which we can erect with the materials at our command (the pure concepts a priori).

[1 ]In the concept of cause I really pass beyond the empirical concept of an event, but not to the intuition which represents the concept of cause in concreto, but to the conditions of time in general, which in experience might be found in accordance with the concept of cause. I therefore proceed here, according to concepts only, but cannot proceed by means of the construction of concepts, because the concept is only a rule for the synthesis of perceptions, which are not pure intuitions, and therefore cannot be given a priori.

[1 ]Completeness means clearness and sufficiency of predicates; limits mean precision, no more predicates being given than belong to the complete concept; in its primary character means that the determination of these limits is not derived from anything else, and therefore in need of any proof, because this would render the so-called definition incapable of standing at the head of all the judgments regarding its object.

[1 ]Philosophy swarms with faulty definitions, particularly such as contain some true elements of a definition, but not all. If, therefore, it were impossible to use a concept until it had been completely defined, philosophy would fare very ill. As, however, we may use a definition with perfect safety, so far at least as the elements of the analysis will carry us, imperfect definitions also, that is, propositions which are not yet properly definitions, but are yet true, and, therefore, approximations to a definition, may be used with great advantage. In mathematics definitions belong ad esse, in philosophy ad melius esse. It is desirable, but it is extremely difficult to construct a proper definition. Jurists are without a definition of right to the present day.