THE TRANSCENDENTAL MAIN QUESTION—FIRST PART.
How is pure Mathematics possible?
Here is a great and established branch of knowledge, already of remarkable compass, and promising unbounded extension in the future, carrying with it a thorough apodictic certainty, i.e., absolute necessity, and thus resting on no empirical grounds, but being a pure product of the Reason, besides thoroughly synthetic. “How is it possible for the human Reason to bring about such a branch of knowledge entirely à priori?” Does not this capacity, as it does not and cannot stand on experience, presuppose some ground of knowledge à priori, lying deep-hidden, but which might reveal itself through these its effects, if their first beginnings were only diligently searched for?
But we find that all mathematical knowledge has this speciality, that it must present its conception previously in intuition, and indeed à priori, that is, in an intuition that is not empirical but pure, without which means it cannot make a single step; its judgments therefore are always intuitive, whereas philosophy must be satisfied with discursive judgments out of mere conceptions; for though it can explain its apodictic doctrines by intuition, these can never be derived from such a source. This observation respecting the nature of mathematics, itself furnishes us with a guide as to the first and foremost condition of its possibility, namely, that some pure intuition must be at its foundation, wherein it can present all its conceptions in concreto and à priori at the same time, or as it is termed, construct them. If we can find out this pure intuition together with its possibility, it will be readily explicable how synthetic propositions à priori are possible in pure mathematics, and therefore, also, how this science is itself possible. For just as empirical intuition enables us, without difficulty, to extend synthetically in experience the conception we form of an object of intuition, by new predicates, themselves afforded us by intuition, so will the pure intuition, only with this difference: that in the last case the synthetic judgment à priori is certain and apodictic, while in the first case it is no more than à posteriori and empirically certain, because the latter only contains what is met with in chance empirical intuition, but the former what is necessarily met with in the pure intuition, inasmuch as being intuition à priori, it is indissolubly bound up with the conception before all experience or perception of individual things.
But the difficulty seems rather to increase than to diminish by this step. For the question is now: How is it possible to intuite anything à priori? Intuition is a presentation, as it would immediately depend on the presence of the object. It seems therefore impossible to intuite originally à priori, because the intuition must then take place without either a previous or present object to which it could refer, and hence could not be intuition. Conceptions are indeed of a nature that some of them, namely, those containing only the thought of an object in general, may be very well formed à priori, without our being in immediate relation to the object (e.g., the conceptions of quantity, of cause, &c.), but even these require a certain use in concreto, i.e., an application to some intuition, if they are to acquire sense and meaning, whereby an object of them is to be given us. But how can intuition of an object precede the object itself?
Were our intuition of such a nature as to present things as they are in themselves, no intuition à priori would take place at all, but it would always be empirical. For what is contained in the object in itself, I can only know when it is given and present to me. It is surely then inconceivable how the intuition of a present thing should enable me to know it as it is in itself, seeing that its properties cannot pass over into my presentative faculty. But granting the possibility of this, the said intuition would not take place à priori, that is, before the object was presented to me, for without it no ground of connection between my presentation and the object could be imagined; in which case it must rest on inspiration (Eingebung). Hence there is only one way possible, by which my intuition can precede the reality of the object and take place as knowledge à priori, and that is, if it contain nothing else but that form of sensibility which precedes in my subject all real impressions, by which I am affected by objects. For, that objects of sense can only be intuited in accordance with this form of sensibility, is a fact I can know à priori. From this it follows, that propositions merely concerning the form of sensible intuition, will be valid and possible for all objects of sense; and conversely, that intuitions possible à priori, can never concern other things than objects of our sense.
Hence, it is only by means of the form of sensuous intuition that we can intuite things à priori, but in this way we intuite the objects only as they appear to our senses, not as they may be in themselves; an assumption absolutely necessary if synthetic propositions à priori are to be admitted as possible, or in the event of their being actually met with, if their possibility is to be conceived and defined beforehand.
Now, such intuitions are space and time, and these lie at the basis of all the cognitions and judgments of pure mathematics, exhibiting themselves at once as apodictic and necessary. For mathematics must present all its conceptions primarily in intuition, and pure mathematics in pure intuition, i.e., it must construct them. For without this it is impossible to make a single step, so long, that is to say, as a pure intuition is wanting, in which alone the matter of synthetic judgments à priori can be given; because it cannot proceed analytically, that is, by the dissection of conceptions, but is obliged to proceed synthetically. The pure intuition of space constitutes the basis of geometry—even arithmetic brings about its numerical conceptions by the successive addition of units in time; but above all, pure mechanics can evolve its conception of motion solely with the aid of the presentation of time. Both presentations, however, are mere intuitions; for when all that is empirical, namely, that belongs to feeling, is left out of the empirical intuitions of bodies and their changes (motion), space and time still remain over, and are therefore pure intuitions, lying à priori at the foundation of the former. For this reason, they can never be left out, but being pure intuitions à priori, prove that they are the bare forms of our sensibility, which must precede all empirical intuition, i.e., the perception of real objects, and in accordance with which objects can be known à priori, though only as they appear to us.
The problem of the present section is therefore solved. Pure mathematics is only possible as synthetic knowledge à priori, in so far as it refers simply to objects of sense, whose empirical intuition has for its foundation a pure intuition à priori (that of time and space), which intuition is able to serve as a foundation, because it is nothing more than the pure form of sensibility itself, that precedes the real appearance of objects, in that it makes them in the first place possible. Yet this faculty of intuiting à priori does not concern the matter of the phenomenon, i.e., that which is feeling (Empfindung) in the latter, for this constitutes the empirical element therein; but only its form, space and time. Should anybody cast the least doubt on the fact that neither of them are conditions of things in themselves, but only dependent on their relation to sensibility. I should be glad to be informed how he deems it possible to know à priori, and therefore before all acquaintance with the things, that is, before they are given us, how their intuition must be constructed, as is here the case with space and time. Yet this is quite conceivable, as soon as they both count for nothing more than formal determinations of our sensibility, and the objects merely as phenomena, for in that case the form of the phenomenon, that is, the pure intuition, can be conceived as coming from ourselves, in other words, as à priori.
To contribute something to the explanation and confirmation of the above, we have only to consider the ordinary and necessary procedure of geometricians. All the proofs of complete likeness between two given figures, turn at last upon the fact of their covering each other; in other words, of the possibility of substituting one, in every point, for the other, which is obviously nothing else but a synthetic proposition resting on immediate intuition. Now this intuition must be given pure and à priori, for otherwise the proposition in question could not count as apodictically certain, but would possess only empirical certainty. We could only say in that case, it has been always so observed, or it is valid so far as our perception has hitherto extended. That complete space, itself no boundary of a further space, has three dimensions, and that no space can have more than this number, is founded on the proposition that not more than three lines can bisect each other at right angles in a single point. But this proposition cannot be presented from conceptions, but rests immediately on intuition, and indeed on pure à priori intuition, because it is apodictically certain that we can require a line to be drawn out to infinity (in indefinitum), or that a series of changes (e.g., spaces passed through by motion) shall be continued to infinity, and this presupposes a presentation of space and time, merely dependent on intuition, namely, so far as in itself, it is bounded by nothing, for from conceptions it could never be concluded. Pure intuitions à priori, then, really lie at the foundation of mathematics, and these make its synthetic and apodictically valid propositions possible, and hence our transcendental deduction of conceptions in space and time explains at the same time the possibility of pure mathematics, which without such a deduction, and without our assuming that “all which can be given to oursenses (the outer in space, the inner in time) is only intuited by us, as it appears to us, and not as it is in itself,” might indeed be conceded, but could in nowise be understood
Those who are unable to free themselves from the notion, that space and time are real qualities (Beschaffenheiten) appertaining to the things in themselves, may exercise their wits on the following paradoxes, and when they have in vain attempted their solution, may suppose, being freed from their prejudices at least for a few moments, that perhaps the degradation of space and time to the position of mere forms of our sensible intuition, may have some foundation.
When two things are exactly alike [equal] in all points that can be cognised in each by itself (i.e., in all respecting quantity or quality), it must follow, that one can in all cases and relations be put in the place of the other, without this substitution occasioning the least cognisable difference. This indeed applies to plane figures in geometry; but there are many spherical figures, which in spite of this complete internal agreement exhibit in their external relations an agreement falling short of admitting one to be put in the place of the other.
For instance, two spherical triangles on opposite hemispheres, having an arc of the equator as a common base, are perfectly equal both in respect of their sides and their angles, so that in neither of them, if separately and at the same time completely described, would anything be found which was not equally present in the other; and yet notwithstanding this, one cannot be put in the place of the other, i.e., on the opposite hemisphere, and herein consists the internal difference of both triangles, that no understanding can indicate as internal, but which reveals itself only by means of the external relation in space. I will now adduce some more ordinary cases taken from common life.
What can more resemble my hand or my ear, and be in all points more like, than its image in the looking-glass? And yet I cannot put such a hand as I see in the glass in the place of its original; for when the latter is a right hand, the one in the glass is a left hand, and the image of the right ear is a left one, which can never take the place of the former. Now, here there are no internal differences that could be imagined by any understanding. And yet the differences are internal, so far as the senses teach us, for the left hand cannot, despite all equality and similarity, be enclosed within the same bounds as the right (they are not congruent); the glove of one hand cannot be used for the other. What then is the solution? These objects are not presentations of things as they are in themselves, and as the pure understanding would cognise them, but they are sensuous intuitions, i.e., phenomena, the possibility of which rests on the relations of certain unknown things in themselves to something else, namely, to our sensibility. Now, space is the form of the outward intuition of these, and the inward determination of every space is only possible through the determination of outward relations to the whole space, of which each [separate] space is a part (i.e., by its relation to the outward sense); in other words, the part is only possible through the whole, which though it could never be the case with things in themselves, namely, with objects of the mere understanding, can very well be so with mere phenomena. Hence we can render the difference of similar and equal, though incongruent things (e.g., spirals winding opposite ways ) intelligible by no single conception, but only by the relation of the right and left hands, which refers immediately to intuition.
Pure mathematics, and especially pure geometry, can only possess objective reality under the condition that they merely refer to objects of sense, in view of which, however, the axiom holds good that our sensuous presentation is in nowise a presentation of things in themselves, but only of the manner wherein they appear to us. Hence it follows that the propositions of geometry are not the mere determinations of a creation of our poetic fancy, which therefore cannot be referred with confidence to real objects, but that they are necessarily valid of space, and consequently of everything that may be found in space; because space is nothing more than the form of all external phenomena, under which alone objects of sense can be given us. Sensibility, the form of which lies at the foundation of geometry, is that whereon the possibility of external phenomena rests; so these can never contain anything but what geometry prescribes for them. It would be quite different if the senses had to present the objects as they are in themselves. For in that case it would by no means follow from the presentation of space (which the geometrician posits with all its properties as an à priori basis), that all this, together with what is deduced therefrom, is exactly so constituted in Nature. The space of the geometrician would be regarded as a mere fiction, and no objective validity ascribed to it, because we do not see why things must necessarily conform to the image that we make of them spontaneously and beforehand. But when this image, or rather this formal intuition, is the essential property of our sensibility by means of which alone objects are presented to us; and yet this sensibility presents not things in themselves, but only their appearances, it is quite easy to conceive, and at the same time incontrovertibly proved, that all the external objects of our sense-world must necessarily conform with the most complete accuracy to the propositions of geometry. For sensibility, by its form of external intuition (space) with which the geometrician is occupied, makes those objects themselves (though as mere appearances) primarily possible. It will always remain a remarkable phenomenon in the history of philosophy that there has been a time when even mathematicians who were also philosophers began to doubt, not indeed of the correctness of their propositions in so far as they concerned space, but of the objective validity and application of this conception, with all its geometrical determinations, to Nature. They were concerned lest a line in Nature might consist of physical points, and the true space in the object, accordingly of simple parts, whereas the space the geometrician has in his mind can never consist of such. They did not recognise that this space in thought makes the physical space, i.e., the extension of matter, itself possible; that the latter is no quality of things in themselves, but only a form of our sensible faculty of presentation; that all objects in space are mere phenomena, i.e., are not things in themselves, but presentations of our sensuous intuition; and hence that space, as the geometrician thinks it, is exactly the form of sensuous intuition we find à priori in ourselves, containing the ground of possibility of all external phenomena (as regards their form); and that these must necessarily and in the most exact manner agree with the propositions of the geometrician, which he draws from no fictious conception, but from the subjective foundation of all external phenomena, namely, the sensibility itself. In such and no other manner can the geometrician be ensured as to the indubitable objective reality of his propositions against all the cavils of an arid metaphysics, however strange it may seem to him, owing to his not having reverted to the sources of his conceptions.
All that is given us as object, must be given us in intuition. But all our intuition takes place by means of the senses alone; the understanding intuites nothing, but only reflects. Inasmuch then as the senses, according to what is above observed, never enable us to cognise, not even in one single point, the things in themselves, but only their phenomena, while these are mere presentations of sensibility, “all bodies, together with the space in which they are found, must be held to be nothing but mere presentations, existing nowhere but in our thoughts.” Now is this not the plainest idealism?
Idealism consists in the assertion that there exist none but thinking entities; the other things we think we perceive in intuition, being only presentations of the thinking entity, to which no object outside the latter can be found to correspond. I say, on the contrary, things are given as objects discoverable by our senses, external to us, but of what they may be in themselves we know nothing; we know only their phenomena, i.e., the presentations they produce in us as they affect our senses. I therefore certainly admit that there are bodies outside us, that is, things, which although they are wholly unknown to us, as to what they may be in themselves, we cognise through presentations, obtained by means of their influence on our sensibility. To these we give the designation of body, a word signifying merely the phenomenon of that to us unknown, but not the less real, object. Can this be termed idealism? It is indeed rather the contrary thereof.
That without calling in question the existence of external things, it may be said of a number of their predicates that they do not belong to the things in themselves, but only to their phenomena, and have no self-existence outside our presentation, is what had been generally accepted and admitted long before Locke’s time, but more than ever since then. To these belong heat, colour, taste, &c. No one can adduce the least ground for saying that it is inadmissible on my part, when for important reasons I count in addition the remaining qualities of bodies called primarias, such as extension, place, and more especially space, together with what is dependent thereon (impenetrability or materiality, figure, &c.) amongst the number of these phenomena. And just as little as the man who will not admit colours to be properties of the object in itself, but only to pertain as modifications to the sense of sight, is on that account called an idealist, so little can my conception be termed idealistic because I find in addition that all properties which make up the intuition of a body belong merely to its appearance. For the existence of a thing, which appears, is not thereby abolished as with real idealism, but it is only shown that we cannot cognise it, as it is in itself, through the senses.
I should like to know how my assertions must be fashioned, if they are not to contain an idealism. I should doubtless have to say, that the presentation of space is not alone completely in accordance with the relation of our sensibility to objects, for that I have already said, but that it is exactly similar to the object itself; an assertion to which no sense can be attached, just as little as that the feeling of red has a similarity with the cinnabar producing this feeling in me.
Hence we may readily set aside an easily foreseen but pointless objection: namely, that through the ideality of space and time, the whole sense-world would be changed to sheer illusion. All philosophical insight into the nature of sensuous cognition was ruined from the first by making sensibility to consist simply in a confused mode of presentation, by which we cognise the things as they are, without having the capacity to bring everything in this, our cognition, to clear consciousness. On the other hand, it has been proved by us that sensibility does not consist in this logical distinction of clearness and obscurity, but in the genetic distinction of the origin of knowledge itself, since sensuous cognition does not present the things as they are, but only the manner in which they affect our senses; and that therefore through them mere phenomena, and not the things themselves, are given to the understanding for reflection. After this necessary correction, a consideration presents itself, arising from an inexcusable and almost purposeless misapplication, as though my doctrine changed all the objects of sense into mere illusion.
When an appearance is given us we are quite free as to what we thence infer with regard to the matter. The former, namely, the appearance, rests on the senses, but the judgment on the understanding; and the only question is, whether or not there is truth in the determination of the object. But the distinction between truth and dream is not decided by the construction of the presentations, which are referred to objects, for they are alike in both, but by the connection of the same according to the rules determining the coherence of presentations in the conception of an object, and by whether they can stand together in an experience or not. Hence the fault does not lie with the phenomena, if our cognition takes the illusion for truth, i.e., if an intuition, whereby an object is given, is held to be the conception of the object or its existence, which the understanding alone can cogitate. The senses present to us the course of the planets as first forwards and then backwards, and in this there is neither falsehood nor truth, because so long as it is considered as an appearance only, no judgment is yet formed as to the objective character of their motion. But inasmuch as when the understanding does not take great care lest this subjective mode of presentation be held for objective, a false judgment may easily arise; it is said, they seem to go back; the illusion, however, is not to be laid to the account of the senses, but of the understanding, whose province alone it is to form an objective judgment on the phenomenon.
In this manner, even if we did not reflect on the origin of our presentations, and let our intuitions of sense contain what they may, if it be but connected according to the coherence of all knowledge in an experience, [we shall find that] deceptive illusion or truth will arise according as we are negligent or careful; for it concerns solely the use of sensuous presentations in the understanding, and not their origin. In the same way, if I hold all presentations of sense together with their form, namely, space and time, to be nothing but phenomena, and the latter to be a mere form of sensibility not present in the objects external to it, and I make use of these presentations only in reference to a possible experience, there is not therein the least temptation to error, neither is there an illusion implied in my regarding them as mere appearances; for in spite of this they can rightly cohere according to the rules of truth in an experience. In such wise all the propositions of geometry respecting space are valid just as much of all the objects of sense, and therefore in respect of all possible experience, whether I regard space as a mere form of sensibility or as something inhering in the things themselves. But in the first case alone can I conceive how it is possible to know à priori the above propositions concerning objects of external intuition. Otherwise everything remains in respect to all merely possible experience just as though I had never undertaken this departure from the popular judgment.
But, let me only venture with my conceptions of space and time beyond all possible experience, which is unavoidable if I give them out as qualities appertaining to the things in themselves (for what should prevent me from assuming them as valid of these same things, even though my senses were differently constructed, and whether they were suited to them or not?) then a serious error may arise, resting on an illusion giving out as universally valid what is a mere condition of the intuition of things pertaining to my subject (certain for all the objects of sense, and thereby for all possible experience), because I refer them to things in themselves and fail to limit them to the conditions of experience.
So far, then, from my doctrine of the ideality of space and time reducing the whole sense-world to mere illusion, it is rather the only means of ensuring the application of some of the most important cognitions, namely, those propounded à priori by mathematics, to real objects, and of guarding them from being held as illusion. For without this observation it would be quite impossible to ascertain whether the intuitions of space and time we borrow from no experience, but which nevertheless lie à priori in our faculty of presentation, were not mere self-made cobwebs of the brain, to which no object, or at least no adequate object, corresponded, and geometry itself therefore a mere illusion; instead of which, its incontestable validity in respect of all objects of the sense-world, owing to these being simply phenomena, has been able to be demonstrated by us.
Secondly, so far from my principles, because they reduce the presentations of the senses to phenomena, turning the truth of experience into illusion, they are rather the only means of guarding against the transcendental illusion, whereby metaphysics has always been deceived and misled into childish endeavours to grasp at soap-bubbles, by taking phenomena, which are mere presentations, for things in themselves; whence have resulted the remarkable assumptions of the antinomy of the Reason, of which I shall make mention farther on, and which are abolished by the single observation that appearance, as long as it is used simply in experience, produces truth, but as soon as it passes beyond the bounds of the latter and becomes transcendent, nothing but pure illusion.
Inasmuch, then, as I leave their reality to the things we intuite to ourselves through the senses, and only limit our sensuous intuition of those things in that they in no particular, not even in the pure intuitions of space and time, represent more than the appearance of the above things, and never their constitution as they are in themselves; this is no thorough-going illusion of my own invention [applied to] Nature. My protestation against all supposition of an idealism is so decisive and clear, that it might seem superfluous were it not for incompetent judges, who like to have an old name for every departure from their distorted although common opinion, and who never judge of the spirit of philosophical terminology, but cling simply to the letter, being ready to put their own delusion in the place of well-defined perceptions, and so to distort and deform them. For the fact of my having myself given my theory the name of transcendental idealism, can justify no one in confounding it with the idealism of Descartes (though this was only a problem, on account of whose insolubility every one was free, in the opinion of Descartes, to deny the existence of the bodily world, because it could never be satisfactorily solved), or with the mystical and visionary idealism of Berkeley, against which and other similar cobwebs of the brain our Critique rather contains the best specific. For what is by me termed idealism, does not touch the existence of things (the doubt of the same being what properly constitutes idealism in the opposite sense), for to doubt them has never entered my head, but simply concerns the sensuous presentation of things, to which space and time chiefly belong; and of these and of all phenomena I have only shown that they are neither things (but only modes of presentation), nor determinations belonging to things in themselves. But the word transcendental, which with me never implies a reference to our knowledge of things, but only to our faculty of knowledge (Erkenntnissvermogen) should guard against this misconception. Rather, however, than occasion its further continuance, I prefer to withdraw the expression, and let it be known as critical (idealism). If it be indeed an objectionable idealism, to change into mere presentations real things (not phenomena), what name shall be applied to that which conversely turns mere presentations into things? I think we may term it the dreaming idealism, in contradistinction to the foregoing, that may be termed the visionary, but both of which ought to have been obviated by my elsewhere so-called transcendental, but better, critical, idealism.