while, at the same time, we cannot, without involving ourselves in contradictions, overlook the general laws of nature, as it was in reference to them alone that this idea was employed. We cannot, I say, overlook the general laws of nature, and regard this conformity to aims observable in nature as contingent or hyperphysical in its origin; inasmuch as there is no ground which can justify us in the admission of a being with such properties distinct from and above nature. All that we are authorized to assert is that this idea may be employed as a principle, and that the properties of the being which is assumed to correspond to it may be regarded as systematically connected in analogy with the causal determination of phenomena.
For the same reasons we are justified in introducing into the idea of the supreme cause other anthropomorphic elements (for without these we could not predicate anything of it); we may regard it as allowable to cogitate this cause as a being with understanding, the feelings of pleasure and displeasure, and faculties of desire and will corresponding to these. At the same time, we may attribute to this being infinite perfection—a perfection which necessarily transcends that which our knowledge of the order and design in the world authorize us to predicate of it. For the regulative law of systematic unity requires us to study nature on the supposition that systematic and final unity in infinitum is everywhere discoverable, even in the highest diversity. For, although we may discover little of this cosmical perfection, it belongs to the legislative prerogative of reason to require us always to seek for and to expect it; while it must always be beneficial to institute all inquiries into nature in accordance with this principle. But it is evident that, by this idea of a supreme author of all, which I place as the foundation of all inquiries into nature, I do not mean to assert the existence of such a being, or that I have any knowledge of its existence; and, consequently, I do not really deduce anything from the existence of this being, but merely from its idea, that is to say, from the nature of things in this world, in accordance with this idea. A certain dim consciousness of the true use of this idea seems to have dictated to the philosophers of all times the moderate language used by them regarding the cause of the world. We find them employing the expressions wisdom and care of nature, and divine wisdom, as synonymous—nay, in purely speculative discussions, preferring the former, because it does not carry the appearance of greater pretensions than such as we are entitled to make, and at the same time directs reason to its proper field of action—nature and her phenomena.
Thus, pure reason, which at first seemed to promise us nothing less than the extension of our cognition beyond the limits of experience, is found, when thoroughly examined, to contain nothing but regulative principles, the virtue and function of which is to introduce into our cognition a higher degree of unity than the understanding could of itself. These principles, by placing the goal of all our struggles at so great a distance, realize for us the most thorough connection between the different parts of our cognition, and the highest degree of systematic unity. But, on the other hand, if misunderstood and employed as constitutive principles of transcendent cognition, they become the parents of illusions and contradictions, while pretending to introduce us to new regions of knowledge.
Thus all human cognition begins with intuitions, proceeds from thence to conceptions, and ends with ideas. Although it possesses, in relation to all three elements, à priori sources of cognition, which seemed to transcend the limits of all experience, a thoroughgoing criticism demonstrates that speculative reason can never, by the aid of these elements, pass the bounds of possible experience, and that the proper destination of this highest faculty of cognition is to employ all methods, and all the principles of these methods, for the purpose of penetrating into the innermost secrets of nature, by the aid of the principles of unity (among all kinds of which teleological unity is the highest), while it ought not to attempt to soar above the sphere of experience, beyond which there lies nought for us but the void inane.
The critical examination, in our Transcendental Analytic, of all the propositions which professed to extend cognition beyond the sphere of experience, completely demonstrated that they can only conduct us to a possible experience. If we were not distrustful even of the clearest abstract theorems, if we were not allured by specious and inviting prospects to escape from the constraining power of their evidence, we might spare ourselves the laborious examination of all the dialectical arguments which a transcendent reason adduces in support of its pretensions; for we should know with the most complete certainty that, however honest such professions might be, they are null and valueless, because they relate to a kind of knowledge to which no man can by any possibility attain. But, as there is no end to discussion, if we cannot discover the true cause of the illusions by which even the wisest are deceived, and as the analysis of all our transcendent cognition into its elements is of itself of no slight value as a psychological study, while it is a duty incumbent on every philosopher—it was found necessary to investigate the dialectical procedure of reason in its primary sources. And as the inferences of which this dialectic is the parent are not only deceitful, but naturally possess a profound interest for humanity, it was advisable at the same time, to give a full account of the momenta of this dialectical procedure, and to deposit it in the archives of human reason, as a warning to all future metaphysicians to avoid these causes of speculative error.
II. Transcendental Doctrine of Method If we regard the sum of the cognition of pure speculative reason as an edifice, the idea of which, at least, exists in the human mind, it may be said that we have in the Transcendental Doctrine of Elements examined the materials and determined to what edifice these belong, and what its height and stability. We have found, indeed, that, although we had purposed to build for ourselves a tower which should reach to Heaven, the supply of materials sufficed merely for a habitation, which was spacious enough for all terrestrial purposes, and high enough to enable us to survey the level plain of experience, but that the bold undertaking designed necessarily failed for want of materials—not to mention the confusion of tongues, which gave rise to endless disputes among the labourers on the plan of the edifice, and at last scattered them over all the world, each to erect a separate building for himself, according to his own plans and his own inclinations. Our present task relates not to the materials, but to the plan of an edifice; and, as we have had sufficient warning not to venture blindly upon a design which may be found to transcend our natural powers, while, at the same time, we cannot give up the intention of erecting a secure abode for the mind, we must proportion our design to the material which is presented to us, and which is, at the same time, sufficient for all our wants.
I understand, then, by the transcendental doctrine of method, the determination of the formal conditions of a complete system of pure reason. We shall accordingly have to treat of the discipline, the canon, the architectonic, and, finally, the history of pure reason.
This part of our Critique will accomplish, from the transcendental point of view, what has been usually attempted, but miserably executed, under the name of practical logic. It has been badly executed, I say, because general logic, not being limited to any particular kind of cognition (not even to the pure cognition of the understanding) nor to any particular objects, it cannot, without borrowing from other sciences, do more than present merely the titles or signs of possible methods and the technical expressions, which are employed in the systematic parts of all sciences; and thus the pupil is made acquainted with names, the meaning and application of which he is to learn only at some future time.
Chapter I. The Discipline of Pure Reason Negative judgements—those which are so not merely as regards their logical form, but in respect of their content—are not commonly held in especial respect. They are, on the contrary, regarded as jealous enemies of our insatiable desire for knowledge; and it almost requires an apology to induce us to tolerate, much less to prize and to respect them.
All propositions, indeed, may be logically expressed in a negative form; but, in relation to the content of our cognition, the peculiar province of negative judgements is solely to prevent error. For this reason, too, negative propositions, which are framed for the purpose of correcting false cognitions where error is absolutely impossible, are undoubtedly true, but inane and senseless; that is, they are in reality purposeless and, for this reason, often very ridiculous. Such is the proposition of the schoolman that Alexander could not have subdued any countries without an army.
But where the limits of our possible cognition are very much contracted, the attraction to new fields of knowledge great, the illusions to which the mind is subject of the most deceptive character, and the evil consequences of error of no inconsiderable magnitude—the negative element in knowledge, which is useful only to guard us against error, is of far more importance than much of that positive instruction which makes additions to the sum of our knowledge. The restraint which is employed to repress, and finally to extirpate the constant inclination to depart from certain rules, is termed discipline. It is distinguished from culture, which aims at the formation of a certain degree of skill, without attempting to repress or to destroy any other mental power, already existing. In the cultivation of a talent, which has given evidence of an impulse towards self-development, discipline takes a negative,[74] culture and doctrine a positive, part.
[74] I am well aware that, in the language of the schools, the term discipline is usually employed as synonymous with instruction. But there are so many cases in which it is necessary to distinguish the notion of the former, as a course of corrective training, from that of the latter, as the communication of knowledge, and the nature of things itself demands the appropriation of the most suitable expressions for this distinction, that it is my desire that the former terms should never be employed in any other than a negative signification.
That natural dispositions and talents (such as imagination and wit), which ask a free and unlimited development, require in many respects the corrective influence of discipline, every one will readily grant.
But it may well appear strange that reason, whose proper duty it is to prescribe rules of discipline to all the other powers of the mind, should itself require this corrective. It has, in fact, hitherto escaped this humiliation, only because, in presence of its magnificent pretensions and high position, no one could readily suspect it to be capable of substituting fancies for conceptions, and words for things.
Reason, when employed in the field of experience, does not stand in need of criticism, because its principles are subjected to the continual test of empirical observations. Nor is criticism requisite in the sphere of mathematics, where the conceptions of reason must always be presented in concreto in pure intuition, and baseless or arbitrary assertions are discovered without difficulty. But where reason is not held in a plain track by the influence of empirical or of pure intuition, that is, when it is employed in the transcendental sphere of pure conceptions, it stands in great need of discipline, to restrain its propensity to overstep the limits of possible experience and to keep it from wandering into error. In fact, the utility of the philosophy of pure reason is entirely of this negative character.
Particular errors may be corrected by particular animadversions, and the causes of these errors may be eradicated by criticism. But where we find, as in the case of pure reason, a complete system of illusions and fallacies, closely connected with each other and depending upon grand general principles, there seems to be required a peculiar and negative code of mental legislation, which, under the denomination of a discipline, and founded upon the nature of reason and the objects of its exercise, shall constitute a system of thorough examination and testing, which no fallacy will be able to withstand or escape from, under whatever disguise or concealment it may lurk.
But the reader must remark that, in this the second division of our transcendental Critique the discipline of pure reason is not directed to the content, but to the method of the cognition of pure reason. The former task has been completed in the doctrine of elements. But there is so much similarity in the mode of employing the faculty of reason, whatever be the object to which it is applied, while, at the same time, its employment in the transcendental sphere is so essentially different in kind from every other, that, without the warning negative influence of a discipline specially directed to that end, the errors are unavoidable which spring from the unskillful employment of the methods which are originated by reason but which are out of place in this sphere.
Section I. The Discipline of Pure Reason in the Sphere of Dogmatism The science of mathematics presents the most brilliant example of the extension of the sphere of pure reason without the aid of experience.
Examples are always contagious; and they exert an especial influence on the same faculty, which naturally flatters itself that it will have the same good fortune in other case as fell to its lot in one fortunate instance. Hence pure reason hopes to be able to extend its empire in the transcendental sphere with equal success and security, especially when it applies the same method which was attended with such brilliant results in the science of mathematics. It is, therefore, of the highest importance for us to know whether the method of arriving at demonstrative certainty, which is termed mathematical, be identical with that by which we endeavour to attain the same degree of certainty in philosophy, and which is termed in that science dogmatical.
Philosophical cognition is the cognition of reason by means of conceptions; mathematical cognition is cognition by means of the construction of conceptions. The construction of a conception is the presentation à priori of the intuition which corresponds to the conception. For this purpose a non-empirical intuition is requisite, which, as an intuition, is an individual object; while, as the construction of a conception (a general representation), it must be seen to be universally valid for all the possible intuitions which rank under that conception. Thus I construct a triangle, by the presentation of the object which corresponds to this conception, either by mere imagination, in pure intuition, or upon paper, in empirical intuition, in both cases completely à priori, without borrowing the type of that figure from any experience. The individual figure drawn upon paper is empirical; but it serves, notwithstanding, to indicate the conception, even in its universality, because in this empirical intuition we keep our eye merely on the act of the construction of the conception, and pay no attention to the various modes of determining it, for example, its size, the length of its sides, the size of its angles, these not in the least affecting the essential character of the conception.
Philosophical cognition, accordingly, regards the particular only in the general; mathematical the general in the particular, nay, in the individual. This is done, however, entirely à priori and by means of pure reason, so that, as this individual figure is determined under certain universal conditions of construction, the object of the conception, to which this individual figure corresponds as its schema, must be cogitated as universally determined.
The essential difference of these two modes of cognition consists, therefore, in this formal quality; it does not regard the difference of the matter or objects of both. Those thinkers who aim at distinguishing philosophy from mathematics by asserting that the former has to do with quality merely, and the latter with quantity, have mistaken the effect for the cause. The reason why mathematical cognition can relate only to quantity is to be found in its form alone. For it is the conception of quantities only that is capable of being constructed, that is, presented à priori in intuition; while qualities cannot be given in any other than an empirical intuition. Hence the cognition of qualities by reason is possible only through conceptions. No one can find an intuition which shall correspond to the conception of reality, except in experience; it cannot be presented to the mind à priori and antecedently to the empirical consciousness of a reality. We can form an intuition, by means of the mere conception of it, of a cone, without the aid of experience; but the colour of the cone we cannot know except from experience. I cannot present an intuition of a cause, except in an example which experience offers to me. Besides, philosophy, as well as mathematics, treats of quantities; as, for example, of totality, infinity, and so on. Mathematics, too, treats of the difference of lines and surfaces—as spaces of different quality, of the continuity of extension—as a quality thereof. But, although in such cases they have a common object, the mode in which reason considers that object is very different in philosophy from what it is in mathematics. The former confines itself to the general conceptions; the latter can do nothing with a mere conception, it hastens to intuition. In this intuition it regards the conception in concreto, not empirically, but in an à priori intuition, which it has constructed; and in which, all the results which follow from the general conditions of the construction of the conception are in all cases valid for the object of the constructed conception.
Suppose that the conception of a triangle is given to a philosopher and that he is required to discover, by the philosophical method, what relation the sum of its angles bears to a right angle. He has nothing before him but the conception of a figure enclosed within three right lines, and, consequently, with the same number of angles. He may analyse the conception of a right line, of an angle, or of the number three as long as he pleases, but he will not discover any properties not contained in these conceptions. But, if this question is proposed to a geometrician, he at once begins by constructing a triangle. 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; and he goes on to produce one side of his triangle, thus forming two adjacent angles which are together 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 immediately perceives that he has thus got an exterior adjacent angle which is equal to the interior. Proceeding in this way, through a chain of inferences, and always on the ground of intuition, he arrives at a clear and universally valid solution of the question.
But mathematics does not confine itself to the construction of quantities (quanta), as in the case of geometry; it occupies itself with pure quantity also (quantitas), as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity. In algebra, a certain method of notation by signs is adopted, and these indicate the different possible constructions of quantities, the extraction of roots, and so on. After having thus denoted the general conception of quantities, according to their different relations, the different operations by which quantity or number is increased or diminished are presented in intuition in accordance with general rules. Thus, when one quantity is to be divided by another, the signs which denote both are placed in the form peculiar to the operation of division; and thus algebra, by means of a symbolical construction of quantity, just as geometry, with its ostensive or geometrical construction (a construction of the objects themselves), arrives at results which discursive cognition cannot hope to reach by the aid of mere conceptions.
Now, what is the cause of this difference in the fortune of the philosopher and the mathematician, the former of whom follows the path of conceptions, while the latter pursues that of intuitions, which he represents, à priori, in correspondence with his conceptions? The cause is evident from what has been already demonstrated in the introduction to this Critique. We do not, in the present case, want to discover analytical propositions, which may be produced merely by analysing our conceptions—for in this the philosopher would have the advantage over his rival; we aim at the discovery of synthetical propositions—such synthetical propositions, moreover, as can be cognized à priori. I must not confine myself to that which I actually cogitate in my conception of a triangle, for this is nothing more than the mere definition; I must try to go beyond that, and to arrive at properties which are not contained in, although they belong to, the conception. Now, this is impossible, unless I determine the object present to my mind according to the conditions, either of empirical, or of pure, intuition. In the former case, I should have an empirical proposition (arrived at by actual measurement of the angles of the triangle), which would possess neither universality nor necessity; but that would be of no value. In the latter, I proceed by geometrical construction, by means of which I collect, in a pure intuition, just as I would in an empirical intuition, all the various properties which belong to the schema of a triangle in general, and consequently to its conception, and thus construct synthetical propositions which possess the attribute of universality.
It would be vain to philosophize upon the triangle, that is, to reflect on it discursively; I should get no further than the definition with which I had been obliged to set out. There are certainly transcendental synthetical propositions which are framed by means of pure conceptions, and which form the peculiar distinction of philosophy; but these do not relate to any particular thing, but to a thing in general, and enounce the conditions under which the perception of it may become a part of possible experience. But the science of mathematics has nothing to do with such questions, nor with the question of existence in any fashion; it is concerned merely with the properties of objects in themselves, only in so far as these are connected with the conception of the objects.
In the above example, we merely attempted to show the great difference which exists between the discursive employment of reason in the sphere of conceptions, and its intuitive exercise by means of the construction of conceptions. The question naturally arises: What is the cause which necessitates this twofold exercise of reason, and how are we to discover whether it is the philosophical or the mathematical method which reason is pursuing in an argument?
All our knowledge relates, finally, to possible intuitions, for it is these alone that present objects to the mind. An à priori or non-empirical conception contains either a pure intuition—and in this case it can be constructed; or it contains nothing but the synthesis of possible intuitions, which are not given à priori. In this latter case, it may help us to form synthetical à priori judgements, but only in the discursive method, by conceptions, not in the intuitive, by means of the construction of conceptions.
The only à priori intuition is that of the pure form of phenomena—space and time. A conception of space and time as quanta may be presented à priori in intuition, that is, constructed, either alone with their quality (figure), or as pure quantity (the mere synthesis of the homogeneous), by means of number. But the matter of phenomena, by which things are given in space and time, can be presented only in perception, à posteriori. The only conception which represents à priori this empirical content of phenomena is the conception of a thing in general; and the à priori synthetical cognition of this conception can give us nothing more than the rule for the synthesis of that which may be contained in the corresponding à posteriori perception; it is utterly inadequate to present an à priori intuition of the real object, which must necessarily be empirical.
Synthetical propositions, which relate to things in general, an à priori intuition of which is impossible, are transcendental. For this reason transcendental propositions cannot be framed by means of the construction of conceptions; they are à priori, and based entirely on conceptions themselves. They contain merely the rule, by which we are to seek in the world of perception or experience the synthetical unity of that which cannot be intuited à priori. But they are incompetent to present any of the conceptions which appear in them in an à priori intuition; these can be given only à posteriori, in experience, which, however, is itself possible only through these synthetical principles.
If we are to form a synthetical judgement regarding a conception, we must go beyond it, to the intuition in which it is given. If we keep to what is contained in the conception, the judgement is merely analytical—it is merely an explanation of what we have cogitated in the conception. But I can pass from the conception to the pure or empirical intuition which corresponds to it. I can proceed to examine my conception in concreto, and to cognize, either à priori or à posteriori, what I find in the object of the conception. The former—à priori cognition—is rational-mathematical cognition by means of the construction of the conception; the latter—à posteriori cognition—is purely empirical cognition, which does not possess the attributes of necessity and universality. Thus I may analyse the conception I have of gold; but I gain no new information from this analysis, I merely enumerate the different properties which I had connected with the notion indicated by the word. My knowledge has gained in logical clearness and arrangement, but no addition has been made to it. But if I take the matter which is indicated by this name, and submit it to the examination of my senses, I am enabled to form several synthetical—although still empirical—propositions. The mathematical conception of a triangle I should construct, that is, present à priori in intuition, and in this way attain to rational-synthetical cognition.
But when the transcendental conception of reality, or substance, or power is presented to my mind, I find that it does not relate to or indicate either an empirical or pure intuition, but that it indicates merely the synthesis of empirical intuitions, which cannot of course be given à priori. The synthesis in such a conception cannot proceed à priori—without the aid of experience—to the intuition which corresponds to the conception; and, for this reason, none of these conceptions can produce a determinative synthetical proposition, they can never present more than a principle of the synthesis[75] of possible empirical intuitions. A transcendental proposition is, therefore, a synthetical cognition of reason by means of pure conceptions and the discursive method, and it renders possible all synthetical unity in empirical cognition, though it cannot present us with any intuition à priori.
[75] In the case of the conception of cause, I do really go beyond the empirical conception of an event—but not to the intuition which presents this conception in concreto, but only to the time-conditions, which may be found in experience to correspond to the conception. My procedure is, therefore, strictly according to conceptions; I cannot in a case of this kind employ the construction of conceptions, because the conception is merely a rule for the synthesis of perceptions, which are not pure intuitions, and which, therefore, cannot be given à priori.
There is thus a twofold exercise of reason. Both modes have the properties of universality and an à priori origin in common, but are, in their procedure, of widely different character. The reason of this is that in the world of phenomena, in which alone objects are presented to our minds, there are two main elements—the form of intuition (space and time), which can be cognized and determined completely à priori, and the matter or content—that which is presented in space and time, and which, consequently, contains a something—an existence corresponding to our powers of sensation. As regards the latter, which can never be given in a determinate mode except by experience, there are no à priori notions which relate to it, except the undetermined conceptions of the synthesis of possible sensations, in so far as these belong (in a possible experience) to the unity of consciousness. As regards the former, we can determine our conceptions à priori in intuition, inasmuch as we are ourselves the creators of the objects of the conceptions in space and time—these objects being regarded simply as quanta. In the one case, reason proceeds according to conceptions and can do nothing more than subject phenomena to these—which can only be determined empirically, that is, à posteriori—in conformity, however, with those conceptions as the rules of all empirical synthesis. In the other case, reason proceeds by the construction of conceptions; and, as these conceptions relate to an à priori intuition, they may be given and determined in pure intuition à priori, and without the aid of empirical data. The examination and consideration of everything that exists in space or time—whether it is a quantum or not, in how far the particular something (which fills space or time) is a primary substratum, or a mere determination of some other existence, whether it relates to anything else—either as cause or effect, whether its existence is isolated or in reciprocal connection with and dependence upon others, the possibility of this existence, its reality and necessity or opposites—all these form part of the cognition of reason on the ground of conceptions, and this cognition is termed philosophical. But to determine à priori an intuition in space (its figure), to divide time into periods, or merely to cognize the quantity of an intuition in space and time, and to determine it by number—all this is an operation of reason by means of the construction of conceptions, and is called mathematical.
The success which attends the efforts of reason in the sphere of mathematics naturally fosters the expectation that the same good fortune will be its lot, if it applies the mathematical method in other regions of mental endeavour besides that of quantities. Its success is thus great, because it can support all its conceptions by à priori intuitions and, in this way, make itself a master, as it were, over nature; while pure philosophy, with its à priori discursive conceptions, bungles about in the world of nature, and cannot accredit or show any à priori evidence of the reality of these conceptions.
Masters in the science of mathematics are confident of the success of this method; indeed, it is a common persuasion that it is capable of being applied to any subject of human thought. They have hardly ever reflected or philosophized on their favourite science—a task of great difficulty; and the specific difference between the two modes of employing the faculty of reason has never entered their thoughts. Rules current in the field of common experience, and which common sense stamps everywhere with its approval, are regarded by them as axiomatic.
From what source the conceptions of space and time, with which (as the only primitive quanta) they have to deal, enter their minds, is a question which they do not trouble themselves to answer; and they think it just as unnecessary to examine into the origin of the pure conceptions of the understanding and the extent of their validity. All they have to do with them is to employ them. In all this they are perfectly right, if they do not overstep the limits of the sphere of nature. But they pass, unconsciously, from the world of sense to the insecure ground of pure transcendental conceptions (instabilis tellus, innabilis unda), where they can neither stand nor swim, and where the tracks of their footsteps are obliterated by time; while the march of mathematics is pursued on a broad and magnificent highway, which the latest posterity shall frequent without fear of danger or impediment.
As we have taken upon us the task of determining, clearly and certainly, the limits of pure reason in the sphere of transcendentalism, and as the efforts of reason in this direction are persisted in, even after the plainest and most expressive warnings, hope still beckoning us past the limits of experience into the splendours of the intellectual world—it becomes necessary to cut away the last anchor of this fallacious and fantastic hope. We shall, accordingly, show that the mathematical method is unattended in the sphere of philosophy by the least advantage—except, perhaps, that it more plainly exhibits its own inadequacy—that geometry and philosophy are two quite different things, although they go hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.
The evidence of mathematics rests upon definitions, axioms, and demonstrations. I shall be satisfied with showing that none of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians; and that the geometrician, if he employs his method in philosophy, will succeed only in building card-castles, while the employment of the philosophical method in mathematics can result in nothing but mere verbiage. The essential business of philosophy, indeed, is to mark out the limits of the science; and even the mathematician, unless his talent is naturally circumscribed and limited to this particular department of knowledge, cannot turn a deaf ear to the warnings of philosophy, or set himself above its direction.
I. Of Definitions. A definition is, as the term itself indicates, the