I.2.1. System of All Principles of Pure Understanding: Axioms and Anticipations

In the preceding chapter we have considered transcendental judgment with reference merely to the universal conditions under which it is alone justified in employing pure concepts of understanding for synthetic judgments. Our task now is to exhibit, in systematic connection, the judgments which understanding, under this critical provision, actually achieves a priori. There can be no question that in this enquiry our table of categories is the natural and the safe guide. For since it is through the relation of the categories to possible experience that all pure a priori knowledge of understanding has to be constituted, their relation to sensibility in general will exhibit completely and systematically all the transcendental principles of the use of the understanding.

Principles a priori are so named not merely because they contain in themselves the grounds of other judgments, but also because they are not themselves grounded in higher and more universal modes of knowledge. But this characteristic does not remove them beyond the sphere of proof. This proof cannot, indeed, be carried out in any objective fashion, since such principles [do not rest on objective considerations but] lie at the foundation of all knowledge of objects. This does not, however, prevent our attempting a proof, from the subjective sources of the possibility of knowledge of an object in general. Such proof is, indeed, indispensable, if the propositions are not to incur the suspicion of being merely surreptitious assertions.

Secondly, we shall limit ourselves merely to those principles which stand in relation to the categories. The principles of the Transcendental Aesthetic, according to which space and time are the conditions of the possibility of all things as appearances, and likewise the restriction of these principles, namely, that they cannot be applied to things in themselves, are matters which do not come within the range of our present enquiry. For similar reasons mathematical principles form no part of this system. They are derived solely from intuition, not from the pure concept of understanding. Nevertheless, since they too are synthetic a priori judgments, their possibility must receive recognition in this chapter. For though their correctness and apodeictic certainty do not indeed require to be established, their possibility, as cases of evident a priori knowledge, has to be rendered conceivable, and to be deduced.

We shall also have to treat of the principle of analytic judgments, in so far as it stands in contrast with that of synthetic judgments with which alone strictly we have to deal. For by thus contrasting them we free the theory of synthetic judgments from all misunderstanding, and have them in their own peculiar nature clearly before us.

The universal, though merely negative, condition of all our judgments in general, whatever be the content of our knowledge, and however it may relate to the object, is that they be not self-contradictory; for if self-contradictory, these judgments are in themselves, even without reference to the object, null and void. But even if our judgment contains no contradiction, it may connect concepts in a manner not borne out by the object, or else in a manner for which no ground is given, either a priori or a posteriori, sufficient to justify such judgment, and so may still, in spite of being free from all inner contradiction, be either false or groundless.

The proposition that no predicate contradictory of a thing can belong to it, is entitled the principle of contradiction, and is a universal, though merely negative, criterion of all truth. For this reason it belongs only to logic. It holds of knowledge, merely as knowledge in general, irrespective of content; and asserts that the contradiction completely cancels and invalidates it.

But it also allows of a positive employment, not merely, that is, to dispel falsehood and error (so far as they rest on contradiction), but also for the knowing of truth. For, if the judgment is analytic, whether negative or affirmative, its truth can always be adequately known in accordance with the principle of contradiction. The reverse of that which as concept is contained and is thought in the knowledge of the object, is always rightly denied. But since the opposite of the concept would contradict the object, the concept itself must necessarily be affirmed of it.

The principle of contradiction must therefore be recognised as being the universal and completely sufficient principle of all analytic knowledge; but beyond the sphere of analytic knowledge it has, as a sufficient criterion of truth, no authority and no field of application. The fact that no knowledge can be contrary to it without self-nullification, makes this principle a conditio sine qua non, but not a determining ground, of the truth of our [non-analytic] knowledge. Now in our critical enquiry it is only with the synthetic portion of our knowledge that we are concerned; and in regard to the truth of this kind of knowledge we can never look to the above principle for any positive information, though, of course, since it is inviolable, we must always be careful to conform to it.

Although this famous principle is thus without content and merely formal, it has sometimes been carelessly formulated in a manner which involves the quite unnecessary admixture of a synthetic element. The formula runs: It is impossible that something should at one and the same time both be and not be. Apart from the fact that the apodeictic certainty, expressed through the word ‘impossible’, is superfluously added—since it is evident of itself from the [very nature of the] proposition—the proposition is modified by the condition of time. It then, as it were, asserts: A thing which is something B, cannot at the same time be not-B, but may very well in succession be both B and not-B. For instance, a man who is young cannot at the same time be old, but may very well at one time be young and at another time not-young, that is, old. The principle of contradiction, however, as a merely logical principle, must not in any way limit its assertions to time-relations. The above formula is therefore completely contrary to the intention of the principle. The misunderstanding results from our first of all separating a predicate of a thing from the concept of that thing, and afterwards connecting this predicate with its opposite—a procedure which never occasions a contradiction with the subject but only with the predicate which has been synthetically connected with that subject, and even then only when both predicates are affirmed at one and the same time. If I say that a man who is unlearned is not learned, the condition, at one and the same time, must be added; for he who is at one time unlearned can very well at another be learned. But if I say, no unlearned man is learned, the proposition is analytic, since the property, unlearnedness, now goes to make up the concept of the subject, and the truth of the negative judgment then becomes evident as an immediate consequence of the principle of contradiction, without requiring the supplementary condition, at one and the same time. This, then, is the reason why I have altered its formulation, namely, in order that the nature of an analytic proposition be clearly expressed through it.

The explanation of the possibility of synthetic judgments is a problem with which general logic has nothing to do. It need not even so much as know the problem by name. But in transcendental logic it is the most important of all questions; and indeed, if in treating of the possibility of synthetic a priori judgments we also take account of the conditions and scope of their validity, it is the only question with which it is concerned. For upon completion of this enquiry, transcendental logic is in a position completely to fulfil its ultimate purpose, that of determining the scope and limits of pure understanding.

In the analytic judgment we keep to the given concept, and seek to extract something from it. If it is to be affirmative, I ascribe to it only what is already thought in it. If it is to be negative, I exclude from it only its opposite. But in synthetic judgments I have to advance beyond the given concept, viewing as in relation with the concept something altogether different from what was thought in it. This relation is consequently never a relation either of identity or of contradiction; and from the judgment, taken in and by itself, the truth or falsity of the relation can never be discovered.

Granted, then, that we must advance beyond a given concept in order to compare it synthetically with another, a third something is necessary, as that wherein alone the synthesis of two concepts can be achieved. What, now, is this third something that is to be the medium of all synthetic judgments? There is only one whole in which all our representations are contained, namely, inner sense and its a priori form, time. The synthesis of representations rests on imagination; and their synthetic unity, which is required for judgment, on the unity of apperception. In these, therefore, [in inner sense, imagination, and apperception], we must look for the possibility of synthetic judgments; and since all three contain the sources of a priori representations, they must also account for the possibility of pure synthetic judgments. For these reasons they are, indeed, indispensably necessary for any knowledge of objects, which rests entirely on the synthesis of representations.

If knowledge is to have objective reality, that is, to relate to an object, and is to acquire meaning and significance in respect to it, the object must be capable of being in some manner given. Otherwise the concepts are empty; through them we have indeed thought, but in this thinking we have really known nothing; we have merely played with representations. That an object be given (if this expression be taken, not as referring to some merely mediate process, but as signifying immediate presentation in intuition), means simply that the representation through which the object is thought relates to actual or possible experience. Even space and time, however free their concepts are from everything empirical, and however certain it is that they are represented in the mind completely a priori, would yet be without objective validity, senseless and meaningless, if their necessary application to the objects of experience were not established. Their representation is a mere schema which always stands in relation to the reproductive imagination that calls up and assembles the objects of experience. Apart from these objects of experience, they would be devoid of meaning. And so it is with concepts of every kind.

The possibility of experience is, then, what gives objective reality to all our a priori modes of knowledge. Experience, however, rests on the synthetic unity of appearances, that is, on a synthesis according to concepts of an object of appearances in general. Apart from such synthesis it would not be knowledge, but a rhapsody of perceptions that would not fit into any context according to rules of a completely interconnected (possible) consciousness, and so would not conform to the transcendental and necessary unity of apperception. Experience depends, therefore, upon a priori principles of its form, that is, upon universal rules of unity in the synthesis of appearances. Their objective reality, as necessary conditions of experience, and indeed of its very possibility, can always be shown in experience. Apart from this relation synthetic a priori principles are completely impossible. For they have then no third something, that is, no object, in which the synthetic unity can exhibit the objective reality of its concepts.

Although we know a priori in synthetic judgments a great deal regarding space in general and the figures which productive imagination describes in it, and can obtain such judgments without actually requiring any experience, yet even this knowledge would be nothing but a playing with a mere figment of the brain, were it not that space has to be regarded as a condition of the appearances which constitute the material for outer experience. Those pure synthetic judgments therefore relate, though only mediately, to possible experience, or rather to the possibility of experience; and upon that alone is founded the objective validity of their synthesis.

Accordingly, since experience, as empirical synthesis, is, in so far as such experience is possible, the one species of knowledge which is capable of imparting reality to any non-empirical synthesis, this latter [type of synthesis], as knowledge a priori, can possess truth, that is, agreement with the object, only in so far as it contains nothing save what is necessary to synthetic unity of experience in general.

The highest principle of all synthetic judgments is therefore this: every object stands under the necessary conditions of synthetic unity of the manifold of intuition in a possible experience.

Synthetic a priori judgments are thus possible when we relate the formal conditions of a priori intuition, the synthesis of imagination and the necessary unity of this synthesis in a transcendental apperception, to a possible empirical knowledge in general. We then assert that the conditions of the possibility of experience in general are likewise conditions of the possibility of the objects of experience, and that for this reason they have objective validity in a synthetic a priori judgment.

That there should be principles at all is entirely due to the pure understanding. Not only is it the faculty of rules in respect of that which happens, but is itself the source of principles according to which everything that can be presented to us as an object must conform to rules. For without such rules appearances would never yield knowledge of an object corresponding to them. Even natural laws, viewed as principles of the empirical employment of understanding, carry with them an expression of necessity, and so contain at least the suggestion of a determination from grounds which are valid a priori and antecedently to all experience. The laws of nature, indeed, one and all, without exception, stand under higher principles of understanding. They simply apply the latter to special cases [in the field] of appearance. These principles alone supply the concept which contains the condition, and as it were the exponents, of a rule in general. What experience gives is the instance which stands under the rule.

There can be no real danger of our regarding merely empirical principles as principles of pure understanding, or conversely. For the necessity according to concepts which distinguishes the principles of pure understanding, and the lack of which is evident in every empirical proposition, however general its application, suffices to make this confusion easily preventable. But there are pure a priori principles that we may not properly ascribe to the pure understanding, which is the faculty of concepts. For though they are mediated by the understanding, they are not derived from pure concepts but from pure intuitions. We find such principles in mathematics. The question, however, of their application to experience, that is, of their objective validity, nay, even the deduction of the possibility of such synthetic a priori knowledge, must always carry us back to the pure understanding.

While, therefore, I leave aside the principles of mathematics, I shall none the less include those [more fundamental] principles upon which the possibility and a priori objective validity of mathematics are grounded. These latter must be regarded as the foundation of all mathematical principles. They proceed from concepts to intuition, not from intuition to concepts.

In the application of pure concepts of understanding to possible experience, the employment of their synthesis is either mathematical or dynamical; for it is concerned partly with the mere intuition of an appearance in general, partly with its existence. The a priori conditions of intuition are absolutely necessary conditions of any possible experience; those of the existence of the objects of a possible empirical intuition are in themselves only accidental. The principles of mathematical employment will therefore be unconditionally necessary, that is, apodeictic. Those of dynamical employment will also indeed possess the character of a priori necessity, but only under the condition of empirical thought in some experience, therefore only mediately and indirectly. Notwithstanding their undoubted certainty throughout experience, they will not contain that immediate evidence which is peculiar to the former. But of this we shall be better able to judge at the conclusion of this system of principles.

The table of categories is quite naturally our guide in the construction of the table of principles. For the latter are simply rules for the objective employment of the former. All principles of pure understanding are therefore—

These titles I have intentionally chosen in order to give prominence to differences in the evidence and in the application of the principles. It will soon become clear that the principles involved in the a priori determination of appearances according to the categories of quantity and of quality (only the formal aspect of quantity and quality being considered) allow of intuitive certainty, alike as regards their evidential force and as regards their a priori application to appearances. They are thereby distinguished from those of the other two groups, which are capable only of a merely discursive certainty. This distinction holds even while we recognise that the certainty is in both cases complete. I shall therefore entitle the former principles mathematical, and the latter dynamical! But it should be noted that we are as little concerned in the one case with the principles of mathematics as in the other with the principles of general physical dynamics. We treat only of the principles of pure understanding in their relation to inner sense (all differences among the given representations being ignored). It is through these principles of pure understanding that the special principles of mathematics and of dynamics become possible. I have named them, therefore, on account rather of their application than of their content. I now proceed to discuss them in the order in which they are given in the above table.

Their principle is: All intuitions are extensive magnitudes.

Appearances, in their formal aspect, contain an intuition in space and time, which conditions them, one and all, a priori. They cannot be apprehended, that is, taken up into empirical consciousness, save through that synthesis of the manifold whereby the representations of a determinate space or time are generated, that is, through combination of the homogeneous manifold and consciousness of its synthetic unity. Consciousness of the synthetic unity of the manifold [and] homogeneous in intuition in general, in so far as the representation of an object first becomes possible by means of it, is, however, the concept of a magnitude (quantum). Thus even the perception of an object, as appearance, is only possible through the same synthetic unity of the manifold of the given sensible intuition as that whereby the unity of the combination of the manifold [and] homogeneous is thought in the concept of a magnitude. In other words, appearances are all without exception magnitudes, indeed extensive magnitudes, As intuitions in space or time, they must be represented through the same synthesis whereby space and time in general are determined.

I entitle a magnitude extensive when the representation of the parts makes possible, and therefore necessarily precedes, the representation of the whole. I cannot represent to myself a line, however small, without drawing it in thought, that is, generating from a point all its parts one after another. Only in this way can the intuition be obtained. Similarly with all times, however small. In these I think to myself only that successive advance from one moment to another, whereby through the parts of time and their addition a determinate time-magnitude is generated. As the [element of] pure intuition in all appearances is either space or time, every appearance is as intuition an extensive magnitude; only through successive synthesis of part to part in [the process of] its apprehension can it come to be known. All appearances are consequently intuited as aggregates, as complexes of previously given parts. This is not the case with magnitudes of every kind, but only with those magnitudes which are represented and apprehended by us in this extensive fashion.

The mathematics of space (geometry) is based upon this successive synthesis of the productive imagination in the generation of figures. This is the basis of the axioms which formulate the conditions of sensible a priori intuition under which alone the schema of a pure concept of outer appearance can arise—for instance, that between two points only one straight line is possible, or that two straight lines cannot enclose a space, etc. These are the axioms which, strictly, relate only to magnitudes (quanta) as such.

As regards magnitude (quantitas), that is, as regards the answer to be given to the question, ‘What is the magnitude of a thing?’ there are no axioms in the strict meaning of the term, although there are a number of propositions which are synthetic and immediately certain (indemonstrabilia). The propositions, that if equals be added to equals the wholes are equal, and if equals be taken from equals the remainders are equal, are analytic propositions; for I am immediately conscious of the identity of the production of the one magnitude with the production of the other. [Consequently, they are not] axioms, [for these] have to be a priori synthetic propositions. On the other hand, the evident propositions of numerical relation are indeed synthetic, but are not general like those of geometry, and cannot, therefore, be called axioms but only numerical formulas. The assertion that 7 + 5 is equal to 12 is not an analytic proposition. For neither in the representation of 7, nor in that of 5, nor in the representation of the combination of both, do I think the number 12. (That I must do so in the addition of the two numbers is not to the point, since in the analytic proposition the question is only whether I actually think the predicate in the representation of the subject.) But although the proposition is synthetic, it is also only singular. So far as we are here attending merely to the synthesis of the homogeneous (of units), that synthesis can take place only in one way, although the employment of these numbers is general. If I assert that through three lines, two of which taken together are greater than the third, a triangle can be described, I have expressed merely the function of productive imagination whereby the lines can be drawn greater or smaller, and so can be made to meet at any and every possible angle. The number 7, on the other hand, is possible only in one way. So also is the number 12, as thus generated through the synthesis of 7 with 5. Such propositions must not, therefore, be called axioms (that would involve recognition of an infinite number of axioms), but numerical formulas.

This transcendental principle of the mathematics of appearances greatly enlarges our a priori knowledge. For it alone can make pure mathematics, in its complete precision, applicable to objects of experience. Without this principle, such application would not be so self-evident; and there has indeed been much confusion of thought in regard to it. Appearances are not things in themselves. Empirical intuition is possible only by means of the pure intuition of space and of time. What geometry asserts of pure intuition is therefore undeniably valid of empirical intuition. The idle objections, that objects of the senses may not conform to such rules of construction in space as that of the infinite divisibility of lines or angles, must be given up. For if these objections hold good, we deny the objective validity of space, and consequently of all mathematics, and no longer know why and how far mathematics can be applicable to appearances. The synthesis of spaces and times, being a synthesis of the essential forms of all intuition, is what makes possible the apprehension of appearance, and consequently every outer experience and all knowledge of the objects of such experience. Whatever pure mathematics establishes in regard to the synthesis of the form of apprehension is also necessarily valid of the objects apprehended. All objections are only the chicanery of a falsely instructed reason, which, erroneously professing to isolate the objects of the senses from the formal condition of our sensibility, represents them, in spite of the fact that they are mere appearances, as objects in themselves, given to the understanding. Certainly, on that assumption, no synthetic knowledge of any kind could be obtained of them a priori, and nothing therefore could be known of them synthetically through pure concepts of space. Indeed, the science which determines these concepts, namely geometry, would not itself be possible.

In all appearances, the real that is an object of sensation has intensive magnitude, that is, a degree.

Perception is empirical consciousness, that is, a consciousness in which sensation is to be found. Appearances, as objects of perception, are not pure, merely formal, intuitions, like space and time. For in and by themselves these latter cannot be perceived. Appearances contain in addition to intuition the matter for some object in general (whereby something existing in space or time is represented); they contain, that is to say, the real of sensation as merely subjective representation, which gives us only the consciousness that the subject is affected, and which we relate to an object in general. Now from empirical consciousness to pure consciousness a graduated transition is possible, the real in the former completely vanishing and a merely formal a priori consciousness of the manifold in space and time remaining. Consequently there is also possible a synthesis in the process of generating the magnitude of a sensation from its beginning in pure intuition = 0, up to any required magnitude. Since, however, sensation is not in itself an objective representation, and since neither the intuition of space nor that of time is to be met with in it, its magnitude is not extensive but intensive. This magnitude is generated in the act of apprehension whereby the empirical consciousness of it can in a certain time increase from nothing = 0 to the given measure. Corresponding to this intensity of sensation, an intensive magnitude, that is, a degree of influence on the sense [i.e. on the special sense involved], must be ascribed to all objects of perception, in so far as the perception contains sensation.

All knowledge by means of which I am enabled to know and determine a priori what belongs to empirical knowledge may be entitled an anticipation; and this is undoubtedly the sense in which Epicurus employed the term πρόληψις. But as there is an element in the appearances (namely, sensation, the matter of perception) which can never be known a priori, and which therefore constitutes the distinctive difference between empirical and a priori knowledge, it follows that sensation is just that element which cannot be anticipated. On the other hand, we might very well entitle the pure determinations in space and time, in respect of shape as well as of magnitude, anticipations of appearances, since they represent a priori that which may always be given a posteriori in experience. If, however, there is in every sensation, as sensation in general (that is, without a particular sensation having to be given), something that can be known a priori, this will, in a quite especial sense, deserve to be named anticipation. For it does indeed seem surprising that we should forestall experience, precisely in that which concerns what is only to be obtained through it, namely, its matter. Yet, none the less, such is actually the case.

Apprehension by means merely of sensation occupies only an instant, if, that is, I do not take into account the succession of different sensations. As sensation is that element in the [field of] appearance the apprehension of which does not involve a successive synthesis proceeding from parts to the whole representation, it has no extensive magnitude. The absence of sensation at that instant would involve the representation of the instant as empty, therefore as = 0. Now what corresponds in empirical intuition to sensation is reality (realitas phaenomenon); what corresponds to its absence is negation = 0. Every sensation, however, is capable of diminution, so that it can decrease and gradually vanish. Between reality in the [field of] appearance and negation there is therefore a continuity of many possible intermediate sensations, the difference between any two of which is always smaller than the difference between the given sensation and zero or complete negation. In other words, the real in the [field of] appearance has always a magnitude. But since its apprehension by means of mere sensation takes place in an instant and not through successive synthesis of different sensations, and therefore does not proceed from the parts to the whole, the magnitude is to be met with only in the apprehension. The real has therefore magnitude, but not extensive magnitude.

A magnitude which is apprehended only as unity, and in which multiplicity can be represented only through approximation to negation = 0, I entitle an intensive magnitude. Every reality in the [field of] appearance has therefore intensive magnitude or degree. If this reality is viewed as cause, either of sensation or of some other reality in the [field of] appearance, such as change, the degree of the reality as cause is then entitled a moment, the moment of gravity. It is so named for the reason that degree signifies only that magnitude the apprehension of which is not successive, but instantaneous. This, however, I touch on only in passing; for with causality I am not at present dealing.

Every sensation, therefore, and likewise every reality in the [field of] appearance, however small it may be, has a degree, that is, an intensive magnitude which can always be diminished. Between reality and negation there is a continuity of possible realities and of possible smaller perceptions. Every colour, as for instance red, has a degree which, however small it may be, is never the smallest; and so with heat, the moment of gravity, etc.

The property of magnitudes by which no part of them is the smallest possible, that is, by which no part is simple, is called their continuity. Space and time are quanta continua, because no part of them can be given save as enclosed between limits (points or instants), and therefore only in such fashion that this part is itself again a space or a time. Space therefore consists solely of spaces, time solely of times. Points and instants are only limits, that is, mere positions which limit space and time. But positions always presuppose the intuitions which they limit or are intended to limit; and out of mere positions, viewed as constituents capable of being given prior to space or time, neither space nor time can be constructed. Such magnitudes may also be called flowings since the synthesis of productive imagination involved in their production is a progression in time, and the continuity of time is ordinarily designated by the term flowing or flowing away.

All appearances, then, are continuous magnitudes, alike in their intuition, as extensive, and in their mere perception (sensation, and with it reality) as intensive. If the synthesis of the manifold of appearance is interrupted, we have an aggregate of different appearances, and not appearance as a genuine quantum. Such an aggregate is not generated by continuing without break productive synthesis of a certain kind, but through repetition of an ever-ceasing synthesis. If I called thirteen thalers a quantum of money, I should be correct, provided my intention is to state the value of a mark of fine silver. For this is a continuous magnitude, in which no part is the smallest, and in which every part can constitute a piece of coin that always contains material for still smaller pieces. But if I understand by the phrase thirteen round thalers, so many coins, quite apart from the question of what their silver standard may be, I then use the phrase, quantum of thalers, inappropriately. It ought to be entitled an aggregate, that is, a number of pieces of money. But as unity must be presupposed in all number, appearance as unity is a quantum, and as a quantum is always a continuum.

Since all appearances, alike in their extensive and in their intensive aspect, are thus continuous magnitudes, it might seem to be an easy matter to prove with mathematical conclusiveness the proposition that all alteration (transition of a thing from one state to another), is continuous. But the causality of an alteration in general, presupposing, as it does, empirical principles, lies altogether outside the limits of a transcendental philosophy. For upon the question as to whether a cause capable of altering the state of a thing, that is, of determining it to the opposite of a certain given state, may be possible, the a priori understanding casts no light; and this not merely because it has no insight into its possibility (such insight is lacking to us in many other cases of a priori knowledge), but because alterableness is to be met with only in certain determinations of appearances, and because, in spite of the fact that the cause of these determinations lies in the unalterable, experience alone can teach what they are. Since in our present enquiry we have no data of which we can make use save only the pure fundamental concepts of all possible experience, in which there must be absolutely nothing that is empirical, we cannot, without destroying the unity of our system, anticipate general natural science, which is based on certain primary experiences.

At the same time, there is no lack of proofs of the great value of our principle in enabling us to anticipate perceptions, and even to some extent to make good their absence, by placing a check upon all false inferences which might be drawn from their absence.

If all reality in perception has a degree, between which and negation there exists an infinite gradation of ever smaller degrees, and if every sense must likewise possess some particular degree of receptivity of sensations, no perception, and consequently no experience, is possible that could prove, either immediately or mediately (no matter how far-ranging the reasoning may be), a complete absence of all reality in the [field of] appearance. In other words, the proof of an empty space or of an empty time can never be derived from experience. For, in the first place, the complete absence of reality from a sensible intuition can never be itself perceived; and, secondly, there is no appearance whatsoever and no difference in the degree of reality of any appearance from which it can be inferred. It is not even legitimate to postulate it in order to explain any difference. For even if the whole intuition of a certain determinate space or time is real through and through, that is, though no part of it is empty, none the less, since every reality has its degree, which can diminish to nothing (the void) through infinite gradations without in any way altering the extensive magnitude of the appearance, there must be infinite different degrees in which space and time may be filled. Intensive magnitude can in different appearances be smaller or greater, although the extensive magnitude of the intuition remains one and the same.

Let us give an example. Almost all natural philosophers, observing—partly by means of the moment of gravity or weight, partly by means of the moment of opposition to other matter in motion—a great difference in the quantity of various kinds of matter in bodies that have the same volume, unanimously conclude that this volume, which constitutes the extensive magnitude of the appearance, must in all material bodies be empty in varying degrees. Who would ever have dreamt of believing that these students of nature, most of whom are occupied with problems in mathematics and mechanics, would base such an inference solely on a metaphysical presupposition—the sort of assumption they so stoutly profess to avoid? They assume that the real in space (I may not here name it impenetrability or weight, since these are empirical concepts) is everywhere uniform and varies only in extensive magnitude, that is, in amount. Now to this presupposition, for which they could find no support in experience, and which is therefore purely metaphysical, I oppose a transcendental proof, which does not indeed explain the difference in the filling of spaces, but completely destroys the supposed necessity of the above presupposition, that the difference is only to be explained on the assumption of empty space. My proof has the merit at least of freeing the understanding, so that it is at liberty to think this difference in some other manner, should it be found that some other hypothesis is required for the explanation of the natural appearances. For we then recognise that although two equal spaces can be completely filled with different kinds of matter, so that there is no point in either where matter is not present, nevertheless every reality has, according to its quality, some specific degree (of resistance or weight) which can, without diminution of its extensive magnitude or amount, become smaller and smaller in infinitum, before it passes into the void and [so] vanishes [out of existence]. Thus a radiation which fills a space, as for instance heat, and similarly every other reality in the [field of] appearance, can diminish in its degree in infinitum, without leaving the smallest part of this space in the least empty. It may fill the space just as completely with these smaller degrees as another appearance does with greater degrees. I do not at all intend to assert that this is what actually occurs when material bodies differ in specific gravity, but only to establish from a principle of pure understanding that the nature of our perceptions allows of such a mode of explanation, that we are not justified in assuming the real in appearances to be uniform in degree, differing only in aggregation and extensive magnitude, and that we are especially in error when we claim that such interpretation can be based on an a priori principle of the understanding.

This anticipation of perception must always, however, appear somewhat strange to anyone trained in transcendental reflection, and to any student of nature who by such teaching has been trained to circumspection. The assertion that the understanding anticipates such a synthetic principle, ascribing a degree to all that is real in the appearances, and so asserting the possibility of an internal distinction in sensation itself (abstraction being made of its empirical quality), awakens doubts and difficulties. It is therefore a question not unworthy of solution, how the understanding can thus in a priori fashion pronounce synthetically upon appearances, and can indeed anticipate in that which in itself is merely empirical and concerns only sensation.

The quality of sensation, as for instance in colours, taste, etc., is always merely empirical, and cannot be represented a priori. But the real, which corresponds to sensations in general, as opposed to negation = 0, represents only that something the very concept of which includes being, and signifies nothing but the synthesis in an empirical consciousness in general. Empirical consciousness can in inner sense be raised from o to any higher degree, so that a certain extensive magnitude of intuition, as for instance of illuminated surface, may excite as great a sensation as the combined aggregate of many such surfaces less illuminated. [Since the extensive magnitude of the appearance thus varies independently], we can completely abstract from it, and still represent in the mere sensation in any one of its moments a synthesis that advances uniformly from o to the given empirical consciousness. Consequently, though all sensations as such are given only a posteriori, their property of possessing a degree can be known a priori. It is remarkable that of magnitudes in general we can know a priori only a single quality, namely, that of continuity, and that in all quality (the real in appearances) we can know a priori nothing save [in regard to] their intensive quantity, namely that they have degree. Everything else has to be left to experience.