No More Learning

Our transcendental doctrine of the faculty of judgment will contain two chapters. The first wild treat of the sensuous condition under which alone pure conceptions of the under standing can be employed, --that of the schematism of the pure understanding. The second will treat of those synthetical judgments which are derived priori from pure conceptions of the understanding under those conditions, and which lie
priori at the foundation of all other cognitions, that
to say, will treat of the principles of the pure understanding,
For, as doctrine, that as an endeavour to
understanding.
enlarge the sphere of the understanding in regard to pure priori cognitions, philosophy worse than useless, since from all the attempts hitherto made, little or no ground has been gained. But, as critique, order to guard against the mistakes of the
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? OT THE SCHEMATISM OF THE CATEGORIES.
107
TRANSCENDENTAL DOCTRINE OF THE FACULTY OF JUDGMENT,
On, Analytic of Principles.
CHAPTER I.
Of the Schematism of the Pure Conceptions of the Understanding.
In all subsumptions of an object under a conception, the representation of the object must be homogeneous with the conception ; in other words, the conception must contain that which is represented in the object to be subsumed under it. For this is the meaning of the expression, An object is con tained under a conception. Thus the empirical conception of a plate is homogeneous with the pure geometrical conception of a circle, inasmuch as the roundness which is cogitated in the former is intuited in the latter.
But pure conceptions of the understanding, when compared with empirical intuitions, or even with sensuous intuitions in
general, are quite heterogeneous, and never can be discovered in any intuition. How then is the tubsumption of the latter under the former, and consequently the application of the cate gories to phenomena, possible 1 -- For it is impossible to say, for example, Causality can be intuited through the senses, and is contained in the phsenomenon. --This natural and important question forms the real cause of the necessity of a transcen dental doctrine of the faculty of judgment, with the purpose, to wit, of shewing how pure conceptions of the understand ing can be applied to phenomena. In all other sciences, where the conceptions by which the object is thought in the general are not so different and heterogeneous from those which represent the object in eoncreto--as it is given, it is quite unnecessary to institute any special inquiries concerning the application of the former to the latter.
Now it is quite clear, that there must be some third thing, which on the one side is homogeneous with the category, and with the phsenomenon on the other, and so makes the applica tion of the fo rmer to the latter possible. This mediating repre
? ? ? ? 108 &KAXYTI0 Or PBIHOTPLBS.
eentation must be pure (without any empirical content), and yet must on the one side be intellectual, on the other sensuous. Such a representation is the transcendental schema.
The conception of the understanding contains pure syn thetical unity of the manifold in general. Time, as the formal condition of the manifold of the internal sense, oonsequently of the conjunction of all representations, contains a priori a manifold in the pure intuition. Now a transcendental deter mination of time is so far homogeneous with the category, which constitutes the unity thereof, that it is universal, and rests upon a rule a priori. On the other hand, it is so far ho mogeneous with the phenomenon, inasmuch as time is con tained in every empirical representation of the manifold. Thus an application of the category to phenomena becomes possible. by means of the transcendental determination of time, which, as the schema of the conceptions of the understanding, mediates the subsumption of the latter under the former.
After what has been proved in our deduction of the catego ries, no one, it is to be hoped, can hesitate ns to the proper de cision of the question, whether the employment of these pure conceptions of the understanding ought to be merely empirical or also transcendental ; in other words, whether the categories, as conditions of a possible experience, relate h priori solely to phenomena, or whether, as conditions of the possibility of things in general, their application can be extended to objects as things in themselves. For we have there seen that con ceptions are quite impossible, and utterly without signification, unless either to them, or at least to the elements of which they consist, an object be given ; and that, consequently, they cannot possibly apply to objects as things in themselves without re gard to the question whether and how these may be given to us ; and further, that the only manner in which objects can be given to us, is by means of the modification of our sensibility ; and finally, that pure h priori conceptions, in addition to tli? function of the understanding in the category, must contain a priori formal conditions of sensibility (of the internal sense, namely), which again contain the general condition under which alone the category can be applied to any oliject. This formal and pure condition of sensibility, to which the conception of tho understanding is restricted in its employment, we shall name the schema of the conception of th-' understandinij, and thf
? ? ? ? Or THJS SCHEMATISM Or THE CATEGORIES. 109
procedure of the understanding with these schemata, we shall oull the Schematism of the pure understanding.
The Schema is, in itself, always a mere product of the ima
gination. * But as the synthesis of imagination has for its aim no single intuition, but merely unity in the determination of sen
sibility, the schema is clearly distinguishable from the image. Thus, if I place five points one after another, this is an image of the number five. On the other hand, if I only think a number in general, which may be either five or a hun dred, this thought is rather the representation of a method of representing in an image a sum g. thousand) in con formity with conception, than the image itself, an image which should find some little difficulty in reviewing, and comparing with the conception. Now this representation of general procedure of the imagination to present its image to conception, call the schema of this conception.
In truth, not images of objects, but schemata, which lie at the foundation of our pure sensuous conceptions. No image could ever be adequate to our conception of triangle general. For the geueralness of the conception never could attain to, as this includes under itself all triangles, whether
right-angled, acute-angled, &c. , whilst the image would always be limited to single part of this sphere. The schema of the triangle can exist nowhere else than in thought, and indi cates rule of the synthesis of the imagination in regard to pure figures in space. Still less an object of experience, or an image of the object, ever adequate to the empirical concep tion. On the contrary, the conception always relates imme diately to the schema of the imagination, as rule for the de termination of our intuition, in conformity with certain ge neral conception. The conception of dog indicates rule, according to which my imagination can delineate the figure of
four-footed animal in general, without being limited to any particular individual form which experience presents to me, or indeed to any possible image that can represent to myself in eoncreto. This schematism of our understanding in regard to phenomena and their mere form, an art, hidden in the depths of the human soul, whose true modes of action we shall only with difficulty discover and unveil. Thus much only enii
Sec note at 34. -- IV.
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we say : --The image is a product of the empirical faculty of the productive imagination, --the schema of sensuous conceptions (of figures in space, for example) is a product, and, as it were, a monogram of the pure imagination a priori, whereby and according to which images first become possible, which, however, can be connected with the conception only mediately by means of the schema which they indicate, and are in themselves never fully adequate to it. On the other hand, the schema of a pure conception of the understanding is something that cannot be reduced into any image, --it is nothing else than the pure synthesis expressed by the category, con formably to a rule of unity according to conceptions. It is a transcendental product of the imagination, a product which concerns the determination of the internal sense,
? according to conditions of its form (time) in respect to all representa
tions, in so far as these representations must be conjoined H priori in one conception, conformably to the unity of apper
ception.
Without entering upon a dry and tedious analysis of the
essential requisites of transcendental schemata of the pure conceptions of the understanding, we shall rather proceed at once to give an explanation of them according to the order of the categories, and in connection therewith.
For the external sense the pure image of all quantities (quantorum) is space ; the pure image of aU objects of sense in general, is time. But the pure schema of quantity (quantila- tis) as a conception of the understanding, is number, a re presentation which comprehends the successive addition of one to one (homogeneous quantities). Thus, number is no thing else than the unity of the synthesis of the manifold in a homogeneous intuition, by means of my generating time* it self in my apprehension of the intuition.
Reality, in the pure conception of the understanding, is that
which corresponds to a sensation in general ; that, conse
quently, the conception of which indicates a being (in time). Negation is that the conception of which represents a not-
The opposition of these two consists there fore in the difference of one and the same time, as a time filled or a time empty. Now as time is only the form of intuition,
>> I generate time because I generate succession, namely, in the sue- ecMive addition of one to one. -- Tr.
being (in time).
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consequently of objects as plienomena, that which in objects corresponds to sensation is the transcendental matter of all objects as things in themselves {Sachheit, reality). Now every sensation has a degree or quantity by which it can fill time, that is to say, the internal sense in respect of the representation of an object, more or less, until it vanishes into nothing (=0= negatio). Thus there is a relation and connection between reality and negation, or rather a transition from the former to the latter, which makes every reality representable to us as a quantum ; and the schema of a reality as the quantity of something in so far as it fills time, is exactly this continuous and uniform generation of the reality in time, as we descend in time from the sensation which has a certain degree, down to the vanishing thereof, or gradually ascend from negation to the quantity thereof.
The schema of substance is the permanence of the real in time ; that the representation of as substratum of the empirical determination of time substratum which there fore remains, whilst all else changes. (Time passes not, but in
passes the existence of the changeable. To time, therefore,
which itself unchangeable and permanent, corresponds that which in the phenomenon unchangeable in existence, that
substance, and only by that the succession and co existence of phsenomena can be determined in regard to time. ) The schema of cause and of the causality of thing the
real which, when posited, always followed by something else. consists, therefore, in the succession of the manifold, in so
far as that succession subjected to rule.
The schema of community (reciprocity of action and re
action), or the reciprocal causality of substances in respect of their accidents, the co-existence of the determinations of the one with those of the other, according to general rule.
The schema of possibility the accordance of the synthesis
of different representations with the conditions of time in ge
neral (as, for example, oppopites cannot exist together at the same time in the same thing, but only after each other), and therefore the determination of the representation of thing
at any time.
The schema of reality* existence in determined time.
WWklichkeit. Id the table of categoric! called Bxiitenc* (Dueyn). -- Tr.
OF TlIE SCHEMATISM Of THE CATESORIEB.
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? 112 Of ASAltflO tiltUCIPLES.
The schema of necessity is the existence of an object in All time.
It is clear, from all this, that the schema of the category of
quantity contains and represents the generation (synthesis) of time itself, in the successive apprehension of an object ; the schema of quality the synthesis of sensation with the repre sentation of time, or the filling up of time ; the schema of relation the relation of perceptions to each other in all time (that according to rule of the determination of time) and finally, the schema of modality and its categories, time itself, as the correlative of the determination of an object --whe ther does belong to time, and how. The schemata, there fore, are nothing but priori determinations of time according to rules, and these, in regard to all possible objects, following the arrangement of the categories, relate to the series in time,
the content in time, the order in time, and finally, to the com
plex or totality in time.
Hence apparent that the schematism of the under
standing, by means of the transcendental synthesis of the ima gination, amounts to nothing else than the unity of the mani fold of intuition in the internal sense, and thus indirectly to the
dimity of apperception, as function corresponding to the in ternal sense receptivity). Thus, the schemata of the pure conceptions of the understanding are the true and only condi tions whereby our understanding receives an application to objects, and consequently significance. Finally, therefore, the categories are only capable of empirical use, inasmuch as they serve merely to subject phenomena to the universal rules of synthesis, means of an u priori necessary unity (on account of the necessary union of all consciousness in one original ap
? and so to render them susceptible of complete connection in one experience. But within this whole of pos sible experience lie all our cognitions, and in the universal re lation to this experience consists transcendental truth, which antecedes all empirical truth, and renders the latter possible.
however, evident at first sight, that although the schemata of sensibility are the sole agents in realizing the categories, they do, nevertheless, also restrict them, that they limit the categories by conditions which lie beyond the
sphere of . understanding --namely, in sensibility. Hence the bchcma properly only the phenomenon, or the sensuous
perception)
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? SYSTEM OV ALL FRINOIPLES. 113
conception of an object in harmony with the category. (Au- merus est quantitas phenomenon,* --sensatio realitas phseno- menon ; constant et perdurabile rerum substantia phenomenon --mternitas, neeetsitas, phsenomena, &c. ) Now, if we re move a restrictive condition, we thereby amplify, it appears,
the formerly limited conception. In this way, the categories in their pure signification, free from all conditions of sensibi lity, ought to be valid of things as they are, and not, as the schemata represent them, merely as they appear, and consequently the categories must have a significance far more extended, and wholly independent of all schemata. In truth, there does always remain to the pure conceptions of the under standing, after abstracting every sensuous condition, a value and significance, which however, merely logical. But in this case, no object given them, and therefore they have no meaning sufficient to afford us conception of an object. The notion of substance, for example, we leave out the sensuous determination of permanence, would mean nothing more than
something which can be cogitated as subject, without the possibility of becoming predicate to anything else. Of this representation can make nothing, inasmuch as does not indicate to me what determinations the thing possesses which must thus be valid as premier subject. Consequently, the categories, without schemata, are merely functions of the un
derstanding for the production of conceptions, but do not represent any object. This significance they derive from sensibility, which at the same time realizes the understanding and restricts it.
TRANSCENDENTAL DOCTRINE OF JUDGMENT, OB ANALYTIC OF PRINCIPLES.
CHAPTER II.
System of all Princifles of the Pure Understanding.
In the foregoing chapter we have merely considered the ge neral conditions under which alone the transcendental faculty of judgment justified in using the pure conceptions of the
onderstanding for synthetical judgments. Our duty at pre- Phenomenon heie an adjective. --Tram,
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sent is to exhibit in systematic connection those judgments which the understanding really produces d priori. For this purpose, our table of the categories will certainly afford us the
natural and safe guidance. For it is precisely the categories whose application to possible experience must constitute all pure & priori cognition of the understanding ; and the rela tion of which to sensibility will, on that very account, pre sent us with a complete and systematic catalogue of all the transcendental principles of the use of the understanding.
Principles <k priori are so called, not merely because they contain in themselves the grounds of other judgments, but also because they themselves are not grounded in higher and more general cognitions. This peculiarity, however, does not raise them altogether above the need of a proof. For although there could be found no higher cognition, and therefore no objective proof, and although such a principle rather serves as the foundation for all cognition of the object, this by no means hinders us from drawing a proof from the subjective sources of the possibility of the cognition of an object. Sucb a proof is necessary moreover, because without it the prin ciple might be liable to the imputation of being a mere gratu itous assertion.
In the second place, we shall limit our investigations to those principles which relate to the categories. For as to the principles of transcendental esthetic, according to which space and time are the conditions of the possibility of things as phenomena, as also the restriction of these principles, namely, that they cannot be applied to objects as things in themselves ; -- these, of course, do not fall within the scope of our present enquiry. In like manner, the principles of ma thematical science form no part of this system, because they are all drawn from intuition, and not from the pure concep tion of the understanding. The possibility of these principles, however, will necessarily be considered here, inasmuch as they are synthetical judgments h priori, not indeed for the purpose of proving their accuracy and apodeictic certainty, which is unnecessary, but merely to render conceivable and deduce the possibility of such evident << priori cognitions.
But we shall have also to speak of the principle of analy tical judgments, in opposition to synthetical judgments, which is the proper subject of our enquiries, because this very oppo
? ? ? ? SYSTEM O1 PRINCIPLES. 113
<<ition will free the theory of the latter from nll ambiguity, ami place it clearly before our eyes in its true nature.
Ststem of the Principled or the Pure Under standing.
SECTION FIRST.
Of the Supreme Principle of all Analytical Judgments.
Whatever may be the content of our cognition, and in whatever manner our cognition may be related to its object, the universal, although only negative condition of all our judgments is that they do not contradict themselves ; other wise these judgments are in themselves (even without respect to the object) nothing. But although there may exist no contradiction in our judgment, it may nevertheless connect conceptions in such a manner, that they do not correspond to the object, or without any grounds either a priori or a pos teriori for arriving at such a judgment, and thus, without being self-contradictory, a judgment may nevertheless be either false or groundless. "
Now, the proposition, No subject can have a predicate that contradicts it," is called the principle of contradiction, and is an universal but purely negative criterion of all truth. But it belongs to logic alone, because it is valid of cognitions, merely as cognitions, and without respect to their content, and declares that the contradiction entirely nullifies them. We can also, however, make a positive use of this princi ple, that not merely to banish falsehood and error (in bo far as rests upon contradiction), but also for the cog nition of truth. For the judgment analytical, be affirmative or negative, its truth must always be recognizable
means of the principle of contradiction. For the contrary of that which lies and cogitated as conception in the cogni tion of the object will be always properly negatived, but the conception itself must always be affirmed of the object, inas much as the contrary thereof would be in contradiction to the object.
We must therefore hold the principle contradiction to be the universal and fully sufficient principle all analytical tocnitiox. But as sufficient criterion of truth, has no further
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utility or authority. For the fact that no cognition can be at variance with this principle without nullifying itself, consti tutes this principle the sine qua non, but not the determining ground of the truth of our cognition. As our business at present is properly with the synthetical part of our knowledge only, we shall always be on our guard not to transgress this inviolable principle ; but at the same time not to expect from it any direct assistance in the establishment of the truth of any synthetical proposition.
There exists, however, a formula of this celebrated principle --a principle merely formal and entirely without content -- which contains a synthesis that has been inadvertently and quite unnecessarily mixed up with it. It is this : -- " It is impossible for a thing to be and not to be at the same time. " Not to mention the superfluousness of the addition of the word impossible to indicate the apodeictic certainty, which ought to be self-evident from the proposition itself, the propo sition is affected by the condition of time, and as it were says :
? " A thing-- A, which is something=J? , cannot at the same time be non-B. " But both, B as well as non-B, may quite well exist in succession. For example, a man who is young cannot at the same time be old ; but the same man can very well be at one time young, and at another not young, that old. Now the principle of contradiction as merely logical propo sition must not any means limit its application merely to relations of time, and consequently formula like the pre ceding quite foreign to its true purpose. The misunder standing arises in this way. We first of all separate predicate of thing from the conception of the tiling, and afterwards connect with this predicate its opposite, and hence do not establish any contradiction with the subject, but
only wit'. i its predicate, which 1ms been conjoined with the
? ubject synthetically, --
contradiction, moreover, which ob tains only when the first and second predicate are affirmed in the same time. If say " man who ignorant not
learned," the condition "at the same time" must be added, for he who at one time ignorant, may at another be learned. But say "No ignorant man learned man," the pro position analytical, because the characteristic ignorance now constituent part of the conception of the subject and in this case the negative proposition evident immediately
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117
from the proposition of contradiction, without the necessity of adding the condition " at the same time. " --This is the reason why I have altered the formula of this principle, --an alteration which shows very clearly the nature of an analytical proposition.
TnE System of the Pbincifi-es O1 the Pubs Uhdeb- STAlTDING.
SECTION SECOND.
Of the Supreme Principle of all Synthetical Judgments.
The explanation of the possibility of synthetical judgments is a task with which general Logic has nothing to do ; indeed she needs not even be acquainted with its name. But in trans cendental Logic it is the most important matter to be dealt with, -- indeed the only one, if the question is of the possibility of synthetical judgments a priori, the conditions and extent of their validity. For when this question is fully decided, it can reach its aim with perfect ease, the determination, to wit, of the extent and limits of the pure understanding.
In an analytical judgment I do not go beyond the given conception, in order to arrive at some decision respecting it.
If the judgment is affirmative, I predicate of the conception only that which was already cogitated in it ; if negative, I merely exclude from the conception its contrary. But in syn thetical judgments, I must go beyond the given conception, in order to cogitate, in relation with something quite dif ferent from that which was cogitated in relation which consequently never one either of identity or contradiction, and
means of which the truth or error of the judgment cannot be discerned merely from the judgment itself.
Granted then, that we must go out beyond given concep tion, in order to compare synthetically with another, third thing necessary, in which alone the synthesis of two con ceptions can originate. Now what this tertium quid, that
to be the medium of all synthetical judgments It only complex,* in which all our representations are contained,
? the internal sense to wit, and its form priori, Time*
The synthesis of our representations rests upon the imagi nation their synthetical unity (which requisite to judg
ment), upon the unity of apperception. In this, therefore, u lnbcgriff
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to be sought the possibility of synthetical judgments, and at all three contain the sources of a priori representations, the
possibility of pure synthetical judgments also ; nay, they are necessary upon these grounds, if we are to possess a know ledge of objects, which rests solely upon the synthesis of re presentations.
If a cognition is to have objective reality, that to relate to an object, and possess sense and meaning in respect to
necessary that the object be given in some way or ano ther. Without this, our conceptions are empty, and we may indeed have thought bymeans of them, but such thinking, we have not, in fact, cognized anything, we have merely played with representation. To give an object, this expression be understood in the sense of to present the object, not mediately but immediately in intuition, means nothing else than to apply the representation of to experience, be that experience real or only possible. Space and time themselves, pure as these conceptions are from all that empirical, and certain as
that they are represented fully priori in the mind, would be completely without objective validity, and without sense and significance, their necessary use in the objects of experi ence were not shewn. Nay, the representation of them
mere schema, that always relates to the reproductive imagina tion, which calls up the objects of experience, without which they have no meaning. And so with all conceptions without distinction.
The possibility 'experience then, that which gives objective reality to all our priori cognitions. Now experience depends upon the synthetical unity of phsenomena, that upon synthesis according to conceptions of the object of pheno mena in general, synthesis without which experience never could become knowledge, but would be merely rhapsody of perceptions, never fitting together into any connected text, according to rules of thoroughly united (possible) conscious ness, and therefore never subjected to the transcendental and necessary unity of apperception. Experience has therefore for foundation, priori principles of its form, that to say, general rules of unity in the synthesis of phenomena, the objective reality of which rules, as necessary conditions -- even of the possibility of experience --can always shewn experience. But apart from this relation, prion synthetical
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propositions are absolutely impossible, because they have no third term, that no pure object, in which the synthetical
unity can exhibit the objective reality of its conceptions. Although, then, respecting space, or the forms which pro
ductive imagination describes therein, we do cognize much priori in synthetical judgments, and are really in no need
of experience for this purpose, such knowledge would never theless amount to nothing but a busy trifling with mere chimera, were not space to be considered as the condition of the phsenomena which constitute the material of ex ternal experience. Hence those pure synthetical judgments do relate, though but mediately, to possible experience, or rather to the possibility of experience, and upon that alone
founded the objective validity of their synthesis.
While then, on the one hand, experience, as empirical syn thesis, the only possible mode of cognition which gives reality to all other synthesis on the other hand, this latter synthesis, as cognition priori, possesses truth, that ac cordance with its object, only in so far as contains nothing more than what necessary to the synthetical unity of ex perience.
Accordingly, the supreme principle of all synthetical judg ments Every object subject to the necessary conditions of the synthetical unity of the manifold of intuition in possible experience.
A priori synthetical judgments are possible, when we ap ply the formal conditions of the priori intuition, the synthe sis of the imagination, and the necessary unity of that syn thesis in transcendental apperception, to possible cognition of experience, and say The conditions of the possibility of ex perience in general, are at the same time conditions of the pos sibility of the objects of experience, and have, for that reason, objective validity in an priori synthetical judgment.
Mental synthesis. --Jr.
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STSTEM O1 THE PRINCIPLES OF THE PrjBB UhDIR- ST>>Nnrso,
SECTION THIRD.
Systematic Representation of all Synthetical Principles thereof.
That principles exist at all is to be ascribed solely to the pure understanding, which is not only the faculty of rules in regard to that which happens, but is even the source of principles ac cording to which every thing that can be presented to us as an object is necessarily subject to rules, because without such rules we never could attain to cognition of an object. Even the laws of nature, if they are contemplated as principles of the empirical use of the understanding, possess also a charac teristic of necessity, and we may therefore at least expect thera to be determined upon grounds which are valid a priori and antecedent to all experience. But all laws of nature, without distinction, are subject to higher principles of the under standing, inasmuch as the former are merely applications of the latter to particular cases of experience. These higher principles alone therefore give the conception, which contains the necessary condition, and, as it were, the exponent of a rule ; experience, on the other hand, gives the case which comes under the rule.
There is no danger of our mistaking merely empirical prin ciples for principles of the pure understanding, or conversely ; for the character of necessity, according to conceptions which distinguishes the latter, and the absence of this in every em pirical proposition, how extensively valid soever it may be, is a perfect safeguard against confounding them. There are, how ever, pure principles a priori, which nevertheless I should not ascribe to the pure understanding -- for this reason, that they are not derived from pure conceptions, but (although by the mediation of the understanding) from pure intuitions. But understanding is the faculty of conceptions. Such principles mathematical science possesses, but their application to ex
? their objective validity, nay the possi bility of such a priori synthetical cognitions (the deduction
thereof) rests entirely upon the pure understanding.
On this uccoint, I shall not reckon among my principl. ?
perience, consequently
? ? ? PRINCIPLES OF PUItE UNDERSTANDING. 121
? . hose of mathematics ; though I shall include those upon the possibility and objective validity a priori, of principles of the mathematical science, which, consequently, are to be looked upon as the principle of these, and which proceed from con ceptions to intuition, and not from intuition to conceptions.
In the application of the pure conceptions of the under standing to possible experience, the employment of their syn thesis is either mathematical or dynamical, for it is directed partly on the intuition alone, partly on the existence of a pheno menon. But the d priori conditions of intuition are in relation
to a possible experience absolutely necessary, those of the ex istence of objects of a possible empirical intuition are in them selves contingent. Hence the principles of the mathematical use of the categories will possess a character of absolute necessity, that will be apodeictic those, on the other hand, of the dynamical use, the character of an priori necessity indeed, but only under the condition of empirical thought in nn experience, therefore only mediately and indirectly. Con sequently they will not possess that immediate evidence which peculiar to the former, although their application to experience does not, for that reason, lose its truth and certitude. But of this point we shall be better able to judge at the conclusion of this system of principles.
The table of the categories naturally our guide to the table of principles, because these are nothing else than rules
? for the objective employment of the former. all principles of the pure understanding are--
Axioms of Intuition.
Accordingly,
3. Analogies
of Experience.
have chosen advisedly, in order that we might not lose sight of the distinctions in respect of tht
2. Anticipations
of Perception.
These appellations
4. Postulates of Empirical Thought in general.
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? 122 TRANSCENDENTAL DOCTRINE.
evidence and the employment of these principles. It will, however, soon nppear that --a fact which concerns bcth the evidence of these principles, and the a priori determination of phenomena -- according to the categories of Quantity and Quality (if we attend merely to the form of these), the prin ciples of these categories are distinguishable from those of the two others, inasmuch as the former are possessed of an intui tive, but the latter of a merely discursive, though in both instances a complete certitude. I shall therefore call the former mathematical,* and the latter dynamical principles. f It must be observed, however, that by these terms I mean,
just as little in the one case the principles of mathematics, as those of general (physical) dynamics, in the other. I have
here in view merely the principles of the pure understanding, in their application to the internal sense, (without distinction of the representations given therein), by means of which the sciences of mathematics and dynamics become possible. Ac cordingly, I have named these principles rather with reference to their application, than their content ; and I shall now pro ceed to consider them in the order in which they stand in the table.
I.
Axioms of Intuition.
The principle of these "All Intuitions are Extensive Quantities. "
Mathematically, in the Kantian sense. -- 7V.
All combination (conjunctio) either composition {composilio) or connection (nexus). The former the synthesis of manifold, the parts of which do not necessarily belong to each other. For example, the two triangles into which square divided by diagonal, do not necessarily belong to each other, and of this kind the synthesis of the homogeneous in every thing that can be mathematically considered. This synthesis can be divided into those of aggregation and coalition, the former of which
applied to extensive, the latter to intensive quantities. The second sort of combination {nexus) the synthesis of manifold, in so far as its parti do belong necessarily to each other for example, the accident to sub stance, or the effect to the causa. Consequently synthesis of that which, though heterogineous, represented as connected, priori. This combination -- not an arbitrary one -- entitle dynamical, because con cerns the connection of the existence of the manifold. This, again, may be divided into the physical synthesis of the phenomena among each other, and the mitaphysical synthesis, or the connection of phenomena a priori in the faculty of cognition.
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All phenomena contain, as regards their form, an intuition in space and time, which lies a priori at the foundation of all with out exception. Phenomena, therefore, cannot be apprehended, that received into empirical consciousness otherwise than through the synthesis of manifold, through which the repre sentations of determinate space or time are generated that
to say, through the composition
the consciousness of the synthetical unity of this manifold
(homogeneous). Now the consciousness of homogeneous manifold in intuition, so far as thereby the representation of an object rendered possible, the conception of quan tity (quanti). Consequently, even the perception of an object as phsenomenon possible only through the same synthetical unity of the manifold of the given sensuous intuition, through which the unity of the composition of the homogeneous mani fold the conception of quantity cogitated that to
say, all phenomena are quantities, and extensive quantities, because as intuitions in space or time, they must be repre sented by means of the same synthesis, through which space and time themselves are determined.
An extensive quantity call that wherein the representa tion of the parts renders possible (and therefore necessarily antecedes) the representation of the whole. cannot repre sent to myself any line, however small, without drawing in thought, that is, without generating from point all its parts one after another, and in this way alone producing this intui tion. Precisely the same the case with every, even the smallest portion of time. cogitate therein only the succes sive progress from one moment to another, and hence, by means of the different portions of time and the addition of them, determinate quantity of time produced. As the pure intuition in all phenomena either time or space, so every phenomenon in its character of intuition an extensive
? inasmuch as can only be cognized in our appre hension by successive synthesis (from part to part). All phenomena are, accordingly, to be considered as aggregates, that as collection of previously given parts which not the case with every sort of quantities, but only with those
which are represented and apprehended by us as extensive.
quantity,
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? 124 TRAXSCEWDKJTTAL DOCTBITTE.
On this successive synthesis of the productive imagination, in the generation of figures, is founded the mathematics of extension, or geometry, with its axioms, which express the conditions of sensuous intuition a priori, under which alone the schema of a pure conception of external intuition can exist ; for example, " between two points only one straight line is possible," " two straight lines cannot enclose a space," &c. These are the axioms which properly relate only to quantities (quanta) as such.
But, as regards the quantity of a thing (quantitas), that is to say, the answer to the question, How large is this or that object ? although, in respect to this question, we have vari ous propositions synthetical and immediately certain (inde- monstrabilia) ; we have, in the proper sense of the term, no axioms. For example, the propositions, " If equals be added to equals, the wholes are equal ;" " If equals be taken from equals, the remainders are equal ;" are analytical, because I am immediately conscious of the identity of the produc tion of the one quantity with the production of the other ; whereas axioms must be h priori synthetical propositions. On the other hand, the self-evident propositions as to the relation of numbers, are certainly synthetical, but not uni versal, like those of geometry, and for this reason cannot be called axioms, but numerical formula? . That 7+5 = 12, is not an analytical proposition. For neither in the represen tation of seven, nor of five, nor of the composition of the two numbers, do I cogitate the number twelve. (Whether I cogitate the number in the addition of both, is not at present the ques tion ; for in the case of an analytical proposition, the only point whether really cogitate the predicate in the repre sentation of the subject. ) But although the proposition synthetical, nevertheless only singular proposition. In so far as regard here had merely to the synthesis of the homogeneous (the units), cannot take place except in onn manner, although our use of these numbers afterwards ge neral. If say, " triangle can be constructed with three lines, any two of which taken together are greater than the third," exercise merely the pure function of the productive imagination, which may draw the lines longer or shorter, and construct the angles at its pleasure. On the contrary, the number seven possible only in one manner, and so like
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? ANTICIPATIONS OF PERCEPTION. 123
wise the number twelve, which results from the synthesis of seven and five. Such propositions, then, cannot be termed axioms (for in that case we should have an infinity of these), but numerical formule.
This transcendental principle of the mathematics of phe nomena greatly enlarges our & priori cognition. For it is by this principle alone that pure mathematics is rendered appli cable in all its precision to objects of experience, and without it the validity of this application would not be so self-evident ; on the contrary, contradictions and confusions have often arisen on this very point. Phenomena are not things in themselves. Empirical intuition is possible only through pure intuition
(of space and time) ; consequently, what geometry affirms of the latter, is indisputably valid of the former. All evasions, such as the statement that objects of sense do not conform to the rules of construction in space (for example, to the rule of the infinite divisibility of lines or angles), must fall to the ground.
For, if these objections hold good, we deny to space, and with it to all mathematics, objective validity, and no longer know wherefore, and how far, mathematics cnn be applied to phe nomena. The synthesis of spaces and times as the essential form of all intuition, is that which renders possible the appre hension of a phenomenon, and therefore every external expe rience, consequently all cognition of the objects of experience ; and whatever mathematics in its pure use proves of the former, must necessarily hold good of the latter. All objections are but the chicaneries of an ill-instructed reason, which errone ously thinks to liberate the objects of sense from the formal
conditions of our sensibility, and represents these, although mere phenomena, as things in themselves, presented as such to our understandings. But in this case, no a prion syn thetical cognition of them could be possible, consequently not through pure conceptions of space, and the science which
determines these conceptions, that is to say, geometry, would
itself be impossible.
II. Anticipations of Perception.
The principle of these is, "In all phamomena the Real, thai tvnich is an object of sensation, has Intensive Quantity, that has
Degree. "
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Proof.
Perception is empirical consciousness, that is to say, u con
sciousness, which contains an element of sensation. Phsenomena
as objects of perception are not pure, that merely formal in tuitions, like space and time, for they cannot be perceived in themselves. * They contain, then, over and above the intui tion, the materials for an object (through which represented something existing in space or time), that to say, they con tain the real of sensation, as representation merely subjec tive, which gives us merely the consciousness that the subject
affected, and which we refer to some external object. Now, gradual transition from empirical consciousness to pure con
sciousness possible, inasmuch as the real in this conscious ness entirely evanishes, and there remains merely formal consciousness priori) of the manifold in time and space consequently there possible synthesis also of the production of the quantity of sensation from its commencement, that from the pure intuition = onwards, up to certain quantity of the sensation. Now as sensation in itself not an objective
? and in to be found neither the intuition of space nor of time, cannot possess any extensive quantity, and yet there does belong to quantity (and that means of its
apprehension, in which empirical consciousne*is can within certain time rise from nothing = up to its given amount), consequently an intensive quantity. And thus we must ascribe intensive quantity, that degree of influence on sense to all objects of perception, in so far as this perception contains sensation.
All cognition, by means of which am enabled to cognize and determine a priori what belongs to empirical cognition, may be called an Anticipation and without doubt this the sense in which Epicurus employed his expression vpoy. ^i;. But as there in phsenomena something which never cog nized priori, which on this account constitutes the proper difference between pure and empirical cognition, that to say, sensation (as the matter of perception), follows, that
sensation just that element in cognition which cannot be at
They can be perceived only as phsenomena, and some part of them
must always belong to the non-ego whereas pure intuitions are entirely the orodncts of the mind itself, and as such arc cognized in thenuehn. -- Tr
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ANTICIPATIONS OP PKBCEPTIOJT.
127
? 11 anticipated. On the other hand, we might very well term the pure determinations in space and time, as well in regard to figure as to Quantity, anticipations of phenomena, because they represent a priori that which may always be given a pos teriori in experience. But suppose that in every sensation, as sensation in general, without any particular sensation being thought of, there existed something which could be cognized a priori, this would deserve to be called anticipation in a special sense --special, because it may seem surprising to forestall experience, in that which concerns the matter of experience, and which we can only derive from itself. Yet such really is the case here.
Apprehension,* by means of sensation alone, fills only one moment, that do not take into consideration succes sion of many sensations. As that in the phenomenon, the apprehension of which not successive synthesis advancing from parts to an entire representation, sensation has therefore no extensive quantity the want of sensation in moment of time would represent as empty, consequently = 0. That which in the empirical intuition corresponds to sensation rea
? lity (real as phenomenon) absence of negation = 0. of diminution, so that
that which corresponds to the
Now every sensation capable can decrease, and thus gradually
Therefore, between reality phenomenon and negation, there exists continuous concatenation of many pos sible intermediate sensations, the difference of which from each other always smaller than that between the given sen sation and zero, or complete negation. ' That to say, the realiia phenomenon has always quantity, which however not discoverable in Apprehension, inasmuch as Apprehension takes place means of mere sensation in one instant, and not the successive synthesis of many sensations, and there fore does not progress from parts to the whole.
has quantity, but not an extensive quantity.
Now that quantity which apprehended only as unity, and in which plurality can be represented only by approximation
to negation = term intensive quantity. Consequently, rea lity in phenomenon has intensive quantity, that degree,
Apprehension tbs Kantian word for perception, in the largest ? cast in which we employ that term. the genus which includes under
M species, perception proper and sensation proper. -- Tr.
Consequently,
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? 128 TRANSCENDENTAL iJOCTltlNX.
If we consider tins reality as cause (be it of sensation or of another reality in the phenomenon, for example, a change) ;
we call the degree of reality in its character of cause a momen tum, for example, the momentum of weight; and for this reason, that the degree only indicates that quantity the appre hension of which is not successive, but instantaneous. This, however, I touch upon only in passing, for with Causality I have at present nothing to do.
Accordingly, every sensation, consequently every reality in phenomena, however small it may be, has a degree, that is, an intensive quautity, which may always be lessened, and between reality and negation there exists a continuous connection of possible realities, and possible smaller perceptions. Every colour -- for example, red -- has a degree, which, be it ever so small, is never the smallest, and so is it always with heat, the momentum of weight, &c.
This property of quantities, according to which no part of them is the smallest possible (no part simple*), is called their continuity. Space and time are quanta continua, because no part of them can be given, without enclosing it within bound aries (points and moments), consequently, this given part is itself a space or a time. Space, therefore, consists only of spaces, and time of times. Points and moments are only boundaries, that the mere places or positions of their limi tation. But places always presuppose intuitions which are to limit or determine them and we cannot conceive either space or time composed of constituent parts which are given
before space or time.