II.1. Doctrine of Method: The Discipline of Pure Reason in its Dogmatic Employment

If we look upon the sum of all knowledge of pure speculative reason as an edifice for which we have at least the idea within ourselves, it can be said that in the Transcendental Doctrine of Elements we have made an estimate of the materials, and have determined for what sort of edifice and for what height and strength of building they suffice. We have found, indeed, that although we had contemplated building a tower which should reach to the heavens, the supply of materials suffices only for a dwelling-house, just sufficiently commodious for our business on the level of experience, and just sufficiently high to allow of our overlooking it. The bold undertaking that we had designed is thus bound to fail through lack of material—not to mention the babel of tongues, which inevitably gives rise to disputes among the workers in regard to the plan to be followed, and which must end by scattering them over all the world, leaving each to erect a separate building for himself, according to his own design. At present, however, we are concerned not so much with the materials as with the plan; and inasmuch as we have been warned not to venture at random upon a blind project which may be altogether beyond our capacities, and yet cannot well abstain from building a secure home for ourselves, we must plan our building in conformity with the material which is given to us, and which is also at the same time appropriate to our needs.

I understand, therefore, by Transcendental Doctrine of Method the determination of the formal conditions of a complete system of pure reason. In this connection, we shall have to treat of a discipline, a canon, an architectonic, and finally a history of pure reason, and to provide (in its transcendental reference) what, in relation to the use of the understanding in general, the Schools have attempted, though very unsatisfactorily, under the title of a practical logic. For since universal logic is not confined to any particular kind of knowledge made possible by the understanding (for instance, not to its pure knowledge) and is also not confined to certain objects, it cannot, save by borrowing knowledge from other sciences, do more than present the titles of possible methods and the technical terms which are used for purposes of systematisation in all kinds of sciences; and this serves only to acquaint the novice in advance with names the meaning and use of which he will not learn till later.

Owing to the general desire for knowledge, negative judgments, that is, those which are such not merely as regards their form but also as regards their content, are not held in any very high esteem. They are regarded rather as the jealous enemies of our unceasing endeavour to extend our knowledge, and it almost requires an apology to win for them even tolerance, not to say favour and high repute.

As far as logical form is concerned, we can make negative any proposition we like; but in respect to the content of our knowledge in general, which is either extended or limited by a judgment, the task peculiar to negative judgments is that of rejecting error. Accordingly, negative propositions intended to reject false knowledge, where yet no error is possible, are indeed true but empty, that is, are not suited to their purpose, and just for this reason are often quite absurd, like the proposition of the Schoolman, that Alexander could not have conquered any countries without an army.

But where the limits of our possible knowledge are very narrow, where the temptation to judge is great, where the illusion that besets us is very deceptive and the harm that results from the error is considerable, there the negative instruction, which serves solely to guard us from errors, has even more importance than many a piece of positive information by which our knowledge is increased. The compulsion, by which the constant tendency to disobey certain rules is restrained and finally extirpated, we entitle discipline. It is distinguished from culture, which is intended solely to give a certain kind of skill, and not to cancel any habitual mode of action already present. Towards the development of a talent, which has already in itself an impulse to manifest itself, discipline will therefore contribute in a negative, culture and doctrine in a positive, fashion.

That temperament and our various talents (such as imagination and wit) which incline to allow themselves a free and unlimited activity are in many respects in need of a discipline, everyone will readily admit. But that reason, whose proper duty it is to prescribe a discipline for all other endeavours, should itself stand in need of such discipline may indeed seem strange; and it has, in fact, hitherto escaped this humiliation, only because, in view of its stately guise and established standing, nobody could lightly come to suspect it of idly substituting fancies for concepts, and words for things.

There is no need of a critique of reason in its empirical employment, because in this field its principles are always subject to the test of experience. Nor is it needed in mathematics, where the concepts of reason must be forthwith exhibited in concreto in pure intuition, so that everything unfounded and arbitrary in them is at once exposed. But where neither empirical nor pure intuition keeps reason to a visible track, when, that is to say, reason is being considered in its transcendental employment, in accordance with mere concepts, it stands so greatly in need of a discipline, to restrain its tendency towards extension beyond the narrow limits of possible experience and to guard it against extravagance and error, that the whole philosophy of pure reason has no other than this strictly negative utility. Particular errors can be got rid of by censure, and their causes by criticism. But where, as in the case of pure reason, we come upon a whole system of illusions and fallacies, intimately bound together and united under common principles, a quite special negative legislation seems to be required, erecting a system of precautions and self-examination under the title of a discipline, founded on the nature of reason and the objects of its pure employment—a system in face of which no pseudo-rational illusion will be able to stand, but will at once betray itself, no matter what claims it may advance for exceptional treatment.

But it is well to note that in this second main division of the transcendental Critique the discipline of pure reason is not directed to the content but only to the method of knowledge through pure reason. The former has already been considered in the Doctrine of Elements. But there is so much similarity in the mode of employing reason, whatever be the object to which it is applied, while yet, at the same time, its transcendental employment is so essentially different from every other, that without the admonitory negative teaching of a discipline, specially devised for the purpose, we cannot hope to avoid the errors which inevitably arise from pursuing in improper fashion methods which are indeed suitable to reason in other fields, only not in this transcendental sphere.

Mathematics presents the most splendid example of the successful extension of pure reason, without the help of experience. Examples are contagious, especially as they quite naturally flatter a faculty which has been successful in one field, [leading it] to expect the same good fortune in other fields. Thus pure reason hopes to be able to extend its domain as successfully and securely in its transcendental as in its mathematical employment, especially when it resorts to the same method as has been of such obvious utility in mathematics. It is therefore highly important for us to know whether the method of attaining apodeictic certainty which is called mathematical is identical with the method by which we endeavour to obtain the same certainty in philosophy, and which in that field would have to be called dogmatic.

Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts. To construct a concept means to exhibit a priori the intuition which corresponds to the concept. For the construction of a concept we therefore need a non-empirical intuition. The latter must, as intuition, be a single object, and yet none the less, as the construction of a concept (a universal representation), it must in its representation express universal validity for all possible intuitions which fall under the same concept. Thus I construct a triangle by representing the object which corresponds to this concept either by imagination alone, in pure intuition, or in accordance therewith also on paper, in empirical intuition—in both cases completely a priori, without having borrowed the pattern from any experience. The single figure which we draw is empirical, and yet it serves to express the concept, without impairing its universality. For in this empirical intuition we consider only the act whereby we construct the concept, and abstract from the many determinations (for instance, the magnitude of the sides and of the angles), which are quite indifferent, as not altering the concept ‘triangle’.

Thus philosophical knowledge considers the particular only in the universal, mathematical knowledge the universal in the particular, or even in the single instance, though still always a priori and by means of reason. Accordingly, just as this single object is determined by certain universal conditions of construction, so the object of the concept, to which the single object corresponds merely as its schema, must likewise be thought as universally determined.

The essential difference between these two kinds of knowledge through reason consists therefore in this formal difference, and does not depend on difference of their material or objects. Those who propose to distinguish philosophy from mathematics by saying that the former has as its object quality only and the latter quantity only, have mistaken the effect for the cause. The form of mathematical knowledge is the cause why it is limited exclusively to quantities. For it is the concept of quantities only that allows of being constructed, that is, exhibited a priori in intuition; whereas qualities cannot be presented in any intuition that is not empirical. Consequently reason can obtain a knowledge of qualities only through concepts. No one can obtain an intuition corresponding to the concept of reality otherwise than from experience; we can never come into possession of it a priori out of our own resources, and prior to the empirical consciousness of reality. The shape of a cone we can form for ourselves in intuition, unassisted by any experience, according to its concept alone, but the colour of this cone must be previously given in some experience or other. I cannot represent in intuition the concept of a cause in general except in an example supplied by experience; and similarly with other concepts. Philosophy, as well as mathematics, does indeed treat of quantities, for instance, of totality, infinity, etc. Mathematics also concerns itself with qualities, for instance, the difference between lines and surfaces, as spaces of different quality, and with the continuity of extension as one of its qualities. But although in such cases they have a common object, the mode in which reason handles that object is wholly different in philosophy and in mathematics. Philosophy confines itself to universal concepts; mathematics can achieve nothing by concepts alone but hastens at once to intuition, in which it considers the concept in concreto, though not empirically, but only in an intuition which it presents a priori, that is, which it has constructed, and in which whatever follows from the universal conditions of the construction must be universally valid of the object of the concept thus constructed.

Suppose a philosopher be given the concept of a triangle and he be left to find out, in his own way, what relation the sum of its angles bears to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and possessing three angles. However long he meditates on this concept, he will never produce anything new. He can analyse and clarify the concept of a straight line or of an angle or of the number three, but he can never arrive at any properties not already contained in these concepts. Now let the geometrician take up these questions. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of his triangle and obtains two adjacent angles, which together are equal to two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent angle which is equal to an internal angle—and so on. In this fashion, through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the problem.

But mathematics does not only construct magnitudes (quanta) as in geometry; it also constructs magnitude as such (quantitas), as in algebra In this it abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude. It then chooses a certain notation for all constructions of magnitude as such (numbers), that is, for addition, subtraction, extraction of roots, etc. Once it has distinguished in the universal concept of magnitudes the different relations in which the magnitudes may stand, it exhibits in intuition, in accordance with certain universal rules, all the various operations through which the quantities are produced and modified. When, for instance, one quantity is to be divided by another, their symbols are placed together, in accordance with the sign for division, and similarly in the other processes; and thus in algebra by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts.

Now what can be the reason of this radical difference in the fortunes of the philosopher and the mathematician, both of whom practise the art of reason, the one making his way by means of concepts, the other by means of intuitions which he exhibits a priori in accordance with concepts? The cause is evident from what has been said above, in our exposition of the fundamental transcendental doctrines. We are not here concerned with analytic propositions, which can be produced by mere analysis of concepts (in this the philosopher would certainly have the advantage over his rival), but with synthetic propositions, and indeed with just those synthetic propositions that can be known a priori. For I must not restrict my attention to what I am actually thinking in my concept of a triangle (this is nothing more than the mere definition); I must pass beyond it to properties which are not contained in this concept, but yet belong to it. Now this is impossible unless I determine my object in accordance with the conditions either of empirical or of pure intuition. The former would only give us an empirical proposition (based on the measurement of the angles), which would not have universality, still less necessity; and so would not at all serve our purpose. The second method of procedure is the mathematical one, and in this case is the method of geometrical construction, by means of which I combine in a pure intuition (just as I do in empirical intuition) the manifold which belongs to the schema of a triangle in general, and therefore to its concept. It is by this method that universal synthetic propositions must be constructed.

It would therefore be quite futile for me to philosophise upon the triangle, that is, to think about it discursively. I should not be able to advance a single step beyond the mere definition, which was what I had to begin with. There is indeed a transcendental synthesis [framed] from concepts alone, a synthesis with which the philosopher is alone competent to deal; but it relates only to a thing in general, as defining the conditions under which the perception of it can belong to possible experience. But in mathematical problems there is no question of this, nor indeed of existence at all, but only of the properties of the objects in themselves, [that is to say], solely in so far as these properties are connected with the concept of the objects.

In the above example we have endeavoured only to make clear the great difference which exists between the discursive employment of reason in accordance with concepts and its intuitive employment by means of the construction of concepts. This naturally leads on to the question, what can be the cause which necessitates such a twofold employment of reason, and how we are to recognise whether it is the first or the second method that is being employed.

All our knowledge relates, finally, to possible intuitions, for it is through them alone that an object is given. Now an a priori concept, that is, a concept which is not empirical, either already includes in itself a pure intuition (and if so, it can be constructed), or it includes nothing but the synthesis of possible intuitions which are not given a priori. In this latter case we can indeed make use of it in forming synthetic a priori judgments, but only discursively in accordance with concepts, never intuitively through the construction of the concept.

The only intuition that is given a priori is that of the mere form of appearances, space and time. A concept of space and time, as quanta, can be exhibited a priori in intuition, that is, constructed, either in respect of the quality (figure) of the quanta, or through number in their quantity only (the mere synthesis of the homogeneous manifold). But the matter of appearances, by which things are given us in space and time, can only be represented in perception, and therefore a posteriori. The only concept which represents a priori this empirical content of appearances is the concept of a thing in general, and the a priori synthetic knowledge of this thing in general can give us nothing more than the mere rule of the synthesis of that which perception may give a posteriori. It can never yield an a priori intuition of the real object, since this must necessarily be empirical.

Synthetic propositions in regard to things in general, the intuition of which does not admit of being given a priori, are transcendental. Transcendental propositions can never be given through construction of concepts, but only in accordance with concepts that are a priori. They contain nothing but the rule according to which we are to seek empirically for a certain synthetic unity of that which is incapable of intuitive representation a priori (that is, of perceptions). But these synthetic principles cannot exhibit a priori any one of their concepts in a specific instance; they can only do this a posteriori, by means of experience, which itself is possible only in conformity with these principles If we are to judge synthetically in regard to a concept, we must go beyond this concept and appeal to the intuition in which it is given. For should we confine ourselves to what is contained in the concept, the judgment would be merely analytic, serving only as an explanation of the thought, in terms of what is actually contained in it. But I can pass from the concept to the corresponding pure or empirical intuition, in order to consider it in that intuition in concreto, and so to know, either a priori or a posteriori, what are the properties of the object of the concept. The a priori method gives us our rational and mathematical knowledge through the construction of the concept, the a posteriori method our merely empirical (mechanical) knowledge, which is incapable of yielding necessary and apodeictic propositions. Thus I might analyse my empirical concept of gold without gaining anything more than merely an enumeration of everything that I actually think in using the word, thus improving the logical character of my knowledge but not in any way adding to it. But I take the material body, familiarly known by this name, and obtain perceptions by means of it; and these perceptions yield various propositions which are synthetic but empirical. When the concept is mathematical, as in the concept of a triangle, I am in a position to construct the concept, that is, to give it a priori in intuition, and in this way to obtain knowledge which is at once synthetic and rational. But if what is given me is the transcendental concept of a reality, substance, force, etc., it indicates neither an empirical nor a pure intuition, but only the synthesis of empirical intuitions, which, as being empirical, cannot be given a priori. And since the synthesis is thus unable to advance a priori, beyond the concept, to the corresponding intuition, the concept cannot yield any determining synthetic proposition, but only a principle of the synthesis of possible empirical intuitions. A transcendental proposition is therefore synthetic knowledge through reason, in accordance with mere concepts; and it is discursive, in that while it is what alone makes possible any synthetic unity of empirical knowledge, it yet gives us no intuition a priori.

There is thus a twofold employment of reason; and while the two modes of employment resemble each other in the universality and a priori origin of their knowledge, in outcome they are very different. The reason is that in the [field of] appearance, in terms of which all objects are given us, there are two elements, the form of intuition (space and time), which can be known and determined completely a priori, and the matter (the physical element) or content—the latter signifying something which is met with in space and time and which therefore contains an existent corresponding to sensation. In respect to this material element, which can never be given in any determinate fashion otherwise than empirically, we can have nothing except indeterminate concepts of the synthesis of possible sensations, in so far as they belong, in a possible experience, to the unity of apperception. As regards the formal element, we can determine our concepts in a priori intuition, inasmuch as we create for ourselves, in space and time, through a homogeneous synthesis, the objects themselves—these objects being viewed simply as quanta. The former method is called the employment of reason in accordance with concepts; in so employing it we can do nothing more than bring appearances under concepts, according to their actual content. The concepts cannot be made determinate in this manner, save only empirically, that is, a posteriori (although always in accordance with these concepts as rules of an empirical synthesis). The other method is the employment of reason through the construction of concepts; and since the concepts here relate to an a priori intuition, they are for this very reason themselves a priori and can be given in a quite determinate fashion in pure intuition, without the help of any empirical data. The consideration of everything which exists in space or time, in regard to the questions, whether and how far it is a quantum or not, whether we are to ascribe to it positive being or the absence of such, how far this something occupying space or time is a primary substratum or a mere determination [of substance], whether there be a relation of its existence to some other existence, as cause or effect, and finally in respect of its existence whether it is isolated or is in reciprocal relation to and dependence upon others—these questions, as also the question of the possibility of this existence, its actuality and necessity, or the opposites of these, one and all belong altogether to knowledge obtained by reason from concepts, such knowledge being termed philosophical. But the determination of an intuition a priori in space (figure), the division of time (duration), or even just the knowledge of the universal element in the synthesis of one and the same thing in time and space, and the magnitude of an intuition that is thereby generated (number),—all this is the work of reason through construction of concepts, and is called mathematical.

The great success which attends reason in its mathematical employment quite naturally gives rise to the expectation that it, or at any rate its method, will have the same success in other fields as in that of quantity. For this method has the advantage of being able to realise all its concepts in intuitions, which it can provide a priori, and by which it becomes, so to speak, master of nature; whereas pure philosophy is all at sea when it seeks through a priori discursive concepts to obtain insight in regard to the natural world, being unable to intuit a priori (and thereby to confirm) their reality. Nor does there seem to be, on the part of the experts in mathematics, any lack of self-confidence as to this procedure—or on the part of the vulgar of great expectations from their skill—should they apply themselves to carry out their project. For, since they have hardly ever attempted to philosophise in regard to their mathematics (a hard task!), the specific difference between the two employments of reason has never so much as occurred to them. Current, empirical rules, which they borrow from ordinary consciousness, they treat as being axiomatic. In the question as to the source of the concepts of space and time they are not in the least interested, although it is precisely with these concepts (as the only original quanta) that they are themselves occupied. Similarly, they think it unnecessary to investigate the origin of the pure concepts of understanding and in so doing to determine the extent of their validity; they care only to make use of them. In all this they are entirely in the right, provided only they do not overstep the proper limits, that is, the limits of the natural world. But, unconsciously, they pass from the field of sensibility to the precarious ground of pure and even transcendental concepts, a ground (instabilis tellus, innabilis unda) that permits them neither to stand nor to swim, and where their hasty tracks are soon obliterated. In mathematics, on the other hand, their passage gives rise to a broad highway, which the latest posterity may still tread with confidence.

We have made it our duty to determine, with exactitude and certainty, the limits of pure reason in its transcendental employment. But the pursuit of such transcendental knowledge has this peculiarity, that in spite of the plainest and most urgent warnings men still allow themselves to be deluded by false hopes, and therefore to postpone the total abandonment of all proposed attempts to advance beyond the bounds of experience into the enticing regions of the intellectual world. It therefore becomes necessary to cut away the last anchor of these fantastic hopes, that is, to show that the pursuit of the mathematical method cannot be of the least advantage in this kind of knowledge (unless it be in exhibiting more plainly the limitations of the method); and that mathematics and philosophy, although in natural science they do, indeed, go hand in hand, are none the less so completely different, that the procedure of the one can never be imitated by the other.

The exactness of mathematics rests upon definitions, axioms and demonstrations. I shall content myself with showing that none of these, in the sense in which they are understood by the mathematician, can be achieved or imitated by the philosopher. I shall show that in philosophy the geometrician can by his method build only so many houses of cards, just as in mathematics the employment of a philosophical method results only in mere talk. Indeed it is precisely in knowing its limits that philosophy consists; and even the mathematician, unless his talent is of such a specialised character that it naturally confines itself to its proper field, cannot afford to ignore the warnings of philosophy, or to behave as if he were superior to them.

I. Definitions.—To define, as the word itself indicates, really only means to present the complete, original concept of a thing within the limits of its concept. If this be our standard, an empirical concept cannot be defined at all, but only made explicit. For since we find in it only a few characteristics of a certain species of sensible object, it is never certain that we are not using the word, in denoting one and the same object, sometimes so as to stand for more, and sometimes so as to stand for fewer characteristics. Thus in the concept of gold one man may think, in addition to its weight, colour, malleability, also its property of resisting rust, while another will perhaps know nothing of this quality. We make use of certain characteristics only so long as they are adequate for the purpose of making distinctions; new observations remove some properties and add others; and thus the limits of the concept are never assured. And indeed 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 characteristics which we attach to it, is more properly to be regarded as merely a designation than as a concept of the thing; the so-called definition is nothing more than a determining of the word. In the second place, it is also true that no concept given a priori, such as substance, cause, right, equity, etc., can, strictly speaking, be defined. For I can never be certain that the clear representation of a given concept, which as given may still be confused, has been completely effected, unless I know that it is adequate to its object. But since the concept of it may, as given, include many obscure representations, which we overlook in our analysis, although we are constantly making use of them in our application of the concept, the completeness of the analysis of my concept is always in doubt, and a multiplicity of suitable examples suffices only to make the completeness probable, never to make it apodeictically certain. Instead of the term, definition, I prefer to use the term, exposition, as being a more guarded term, which the critic can accept as being up to a certain point valid, though still entertaining doubts as to the completeness of the analysis. Since, then, neither empirical concepts nor concepts given a priori allow of definition, the only remaining kind of concepts, upon which this mental operation can be tried, are arbitrarily invented concepts. A concept which I have invented I can always define; for since it is not given to me either by the nature of understanding or by experience, but is such as I have myself deliberately made it to be, I must know what I have intended to think in using it. I cannot, however, say that I have thereby defined a true object. For if the concept depends on empirical conditions, as e.g. the concept of a ship’s clock, this arbitrary concept of mine does not assure me of the existence or of the possibility of its object. I do not even know from it whether it has an object at all, and my explanation may better be described as a declaration of my project than as a definition of an object. There remain, therefore, no concepts which allow of definition, except only those which contain an arbitrary synthesis that admits of a priori construction. Consequently, mathematics is the only science that has definitions. For the object which it thinks it exhibits a priori in intuition, and this object certainly cannot contain either more or less than the concept, since it is through the definition that the concept of the object is given—and given originally, that is, without its being necessary to derive the definition from any other source. The German language has for the [Latin] terms exposition, explication, declaration, and definition only one word, Erklärung, and we need not, therefore, be so stringent in our requirements as altogether to refuse to philosophical explanations the honourable title, definition. We shall confine ourselves simply to remarking that while philosophical definitions are never more than expositions of given concepts, mathematical definitions are constructions of concepts, originally framed by the mind itself; and that while the former can be obtained only by analysis (the completeness of which is never apodeictically certain), the latter are produced synthetically. Whereas, therefore, mathematical definitions make their concepts, in philosophical definitions concepts are only explained. From this it follows:

(a) That in philosophy we must not imitate mathematics by beginning with definitions, unless it be by way simply of experiment. For since the definitions are analyses of given concepts, they presuppose the prior presence of the concepts, although in a confused state; and the incomplete exposition must precede the complete. Consequently, we can infer a good deal from a few characteristics, derived from an incomplete analysis, without having yet reached the complete exposition, that is, the definition. In short, the definition in all its precision and clarity ought, in philosophy, to come rather at the end than at the beginning of our enquiries. In mathematics, on the other hand, we have no concept whatsoever prior to the definition, through which the concept itself is first given. For this reason mathematical science must always begin, and it can always begin, with the definition.

(b) That mathematical definitions can never be in error. For since the concept is first given through the definition, it includes nothing except precisely what the definition intends should be understood by it. But although nothing incorrect can be introduced into its content, there may sometimes, though rarely, be a defect in the form in which it is clothed, namely as regards precision. Thus the common explanation of the circle that it is a curved line every point in which is equidistant from one and the same point (the centre), has the defect that the determination, curved, is introduced unnecessarily. For there must be a particular theorem, deduced from the definition and easily capable of proof, namely, that if all points in a line are equidistant from one and the same point, the line is curved (no part of it straight). Analytic definitions, on the other hand, may err in many ways, either through introducing characteristics which do not really belong to the concept, or by lacking that completeness which is the essential feature of a definition. The latter defect is due to the fact that we can never be quite certain of the completeness of the analysis. For these reasons the mathematical method of definition does not admit of imitation in philosophy.

2. Axioms.——These, in so far as they are immediately certain, are synthetic a priori principles. Now one concept cannot be combined with another synthetically and also at the same time immediately, since, to be able to pass beyond either concept, a third something is required to mediate our knowledge. Accordingly, since philosophy is simply what reason knows by means of concepts, no principle deserving the name of an axiom is to be found in it. Mathematics, on the other hand, can have axioms, since by means of the construction of concepts in the intuition of the object it can combine the predicates of the object both a priori and immediately, as, for instance, in the proposition that three points always lie in a plane. But a synthetic principle derived from concepts alone can never be immediately certain, for instance, the proposition that everything which happens has a cause. Here I must look round for a third something, namely, the condition of time-determination in an experience; I cannot obtain knowledge of such a principle directly and immediately from the concepts alone. Discursive principles are therefore quite different from intuitive principles, that is, from axioms; and always require a deduction. Axioms, on the other hand, require no such deduction, and for the same reason are evident—a claim which the philosophical principles can never advance, however great their certainty. Consequently, the synthetic propositions of pure, transcendental reason are, one and all, infinitely removed from being as evident—which is yet so often arrogantly claimed on their behalf—as the proposition that twice two make four. In the Analytic I have indeed introduced some axioms of intuition into the table of the principles of pure understanding; but the principle there applied is not itself an axiom, but serves only to specify the principle of the possibility of axioms in general, and is itself no more than a principle derived from concepts. For the possibility of mathematics must itself be demonstrated in transcendental philosophy. Philosophy has therefore no axioms, and may never prescribe its a priori principles in any such absolute manner, but must resign itself to establishing its authority in their regard by a thorough deduction.

3. Demonstrations.—An apodeictic proof can be called a demonstration, only in so far as it is intuitive. Experience teaches us what is, but does not teach us that it could not be other than what it is. Consequently, no empirical grounds of proof can ever amount to apodeictic proof. Even from a priori concepts, as employed in discursive knowledge, there can never arise intuitive certainty, that is, [demonstrative] evidence, however apodeictically certain the judgment may otherwise be. Mathematics alone, therefore, contains demonstrations, since it derives its knowledge not from concepts but from the construction of them, that is, from intuition, which can be given a priori in accordance with the concepts. Even the method of algebra with its equations, from which the correct answer, together with its proof, is deduced by reduction, is not indeed geometrical in nature, but is still constructive in a way characteristic of the science. The concepts attached to the symbols, especially concerning the relations of magnitudes, are presented in intuition; and this method, in addition to its heuristic advantages, secures all inferences against error by setting each one before our eyes. While philosophical knowledge must do without this advantage, inasmuch as it has always to consider the universal in abstracto (by means of concepts), mathematics can consider the universal in concreto (in the single intuition) and yet at the same time through pure a priori representation, whereby all errors are at once made evident. I should therefore prefer to call the first kind acroamatic (discursive) proofs, since they may be conducted by the agency of words alone (the object in thought), rather than demonstrations which, as the term itself indicates, proceed in and through the intuition of the object.

From all this it follows that it is not in keeping with the nature of philosophy, especially in the field of pure reason, to take pride in a dogmatic procedure, and to deck itself out with the title and insignia of mathematics, to whose ranks it does not belong, though it has every ground to hope for a sisterly union with it. Such pretensions are idle claims which can never be satisfied, and indeed must divert philosophy from its true purpose, namely, to expose the illusions of a reason that forgets its limits, and by sufficiently clarifying our concepts to recall it from its presumptuous speculative pursuits to modest but thorough self-knowledge. Reason must not, therefore, in its transcendental endeavours, hasten forward with sanguine expectations, as though the path which it has traversed led directly to the goal, and as though the accepted premisses could be so securely relied upon that there can be no need of constantly returning to them and of considering whether we may not perhaps, in the course of the inferences, discover defects which have been overlooked in the principles, and which render it necessary either to determine these principles more fully or to change them entirely.

I divide all apodeictic propositions, whether demonstrable or immediately certain, into dogmata and mathemata. A synthetic proposition directly derived from concepts is a dogma; a synthetic proposition, when directly obtained through the construction of concepts, is a mathema. Analytic judgments really teach us nothing more about the object than what the concept which we have of it already contains; they do not extend our knowledge beyond the concept of the object, but only clarify the concept. They cannot therefore rightly be called dogmas (a word which might perhaps be translated doctrines). Of the two kinds of synthetic a priori propositions only those belonging to philosophical knowledge can, according to the ordinary usage of words, be entitled dogmas; the propositions of arithmetic or geometry would hardly be so named. The customary use of words thus confirms our interpretation of the term, namely, that only judgments derived from concepts can be called dogmatic, not those based on the construction of concepts.

Now in the whole domain of pure reason, in its merely speculative employment, there is not to be found a single synthetic judgment directly derived from concepts. For, as we have shown, ideas cannot form the basis of any objectively valid synthetic judgment. Through concepts of understanding reason does, indeed, establish secure principles, not however directly from concepts alone, but always only indirectly through relation of these concepts to something altogether contingent, namely, possible experience. When such experience (that is, something as object of possible experiences) is presupposed, these principles are indeed apodeictically certain; but in themselves, directly, they can never be known a priori. Thus no one can acquire insight into the proposition that everything which happens has its cause, merely from the concepts involved. It is not, therefore, a dogma, although from another point of view, namely, from that of the sole field of its possible employment, that is, experience, it can be proved with complete apodeictic certainty. But though it needs proof, it should be entitled a principle, not a theorem, because it has the peculiar character that it makes possible the very experience which is its own ground of proof, and that in this experience it must always itself be presupposed.

Now if in the speculative employment of pure reason there are no dogmas, to serve as its special subject-matter, all dogmatic methods, whether borrowed from the mathematician or specially invented, are as such inappropriate. For they only serve to conceal defects and errors, and to mislead philosophy, whose true purpose is to present every step of reason in the clearest light. Nevertheless its method can always be systematic. For our reason is itself, subjectively, a system, though in its pure employment, by means of mere concepts, it is no more than a system whereby our investigations can be conducted in accordance with principles of unity, the material being provided by experience alone. We cannot here discuss the method peculiar to transcendental philosophy; we are at present concerned only with a critical estimate of what may be expected from our faculties—whether we are in a position to build at all; and to what height, with the material at our disposal (the pure a priori concepts), we may hope to carry the edifice.