For neither in the representation 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 question; for in the case of an a.n.a.lytical proposition, the only point is whether I really cogitate the predicate in the representation of the subject.) But although the proposition is synthetical, it is nevertheless only a singular proposition. In so far as regard is here had merely to the synthesis of the h.o.m.ogeneous (the units), it cannot take place except in one manner, although our use of these numbers is afterwards general. If I say: "A triangle can be constructed with three lines, any two of which taken together are greater than the third," I 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 is possible only in one manner, and so is likewise 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 formulae.
This transcendental principle of the mathematics of phenomena greatly enlarges our a priori cognition. For it is by this principle alone that pure mathematics is rendered applicable 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 s.p.a.ce 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 s.p.a.ce (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 s.p.a.ce, and with it to all mathematics, objective validity, and no longer know wherefore, and how far, mathematics can be applied to phenomena. The synthesis of s.p.a.ces and times as the essential form of all intuition, is that which renders possible the apprehension of a phenomenon, and therefore every external experience, 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 erroneously 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 understanding. But in this case, no a priori synthetical cognition of them could be possible, consequently not through pure conceptions of s.p.a.ce and the science which determines these conceptions, that is to say, geometry, would itself be impossible.
2. ANTIc.i.p.aTIONS OF PERCEPTION.
The principle of these is: In all phenomena the Real, that which is an object of sensation, has Intensive Quant.i.ty, that is, has a Degree.
PROOF.
Perception is empirical consciousness, that is to say, a consciousness which contains an element of sensation. Phenomena as objects of perception are not pure, that is, merely formal intuitions, like s.p.a.ce and time, for they cannot be perceived in themselves.
[Footnote: They can be perceived only as phenomena, and some part of them must always belong to the non-ego; whereas pure intuitions are entirely the products of the mind itself, and as such are coguized IN THEMSELVES.--Tr]
They contain, then, over and above the intuition, the materials for an object (through which is represented something existing in s.p.a.ce or time), that is to say, they contain the real of sensation, as a representation merely subjective, which gives us merely the consciousness that the subject is affected, and which we refer to some external object. Now, a gradual transition from empirical consciousness to pure consciousness is possible, inasmuch as the real in this consciousness entirely vanishes, and there remains a merely formal consciousness (a priori) of the manifold in time and s.p.a.ce; consequently there is possible a synthesis also of the production of the quant.i.ty of a sensation from its commencement, that is, from the pure intuition = 0 onwards up to a certain quant.i.ty of the sensation. Now as sensation in itself is not an objective representation, and in it is to be found neither the intuition of s.p.a.ce nor of time, it cannot possess any extensive quant.i.ty, and yet there does belong to it a quant.i.ty (and that by means of its apprehension, in which empirical consciousness can within a certain time rise from nothing = 0 up to its given amount), consequently an intensive quant.i.ty. And thus we must ascribe intensive quant.i.ty, that is, a degree of influence on sense to all objects of perception, in so far as this perception contains sensation.
All cognition, by means of which I am enabled to cognize and determine a priori what belongs to empirical cognition, may be called an antic.i.p.ation; and without doubt this is the sense in which Epicurus employed his expression prholepsis. But as there is in phenomena something which is never cognized a priori, which on this account const.i.tutes the proper difference between pure and empirical cognition, that is to say, sensation (as the matter of perception), it follows, that sensation is just that element in cognition which cannot be at all antic.i.p.ated. On the other hand, we might very well term the pure determinations in s.p.a.ce and time, as well in regard to figure as to quant.i.ty, antic.i.p.ations of phenomena, because they represent a priori that which may always be given a posteriori 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 antic.i.p.ation 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 is, if I do not take into consideration a succession of many sensations.
As that in the phenomenon, the apprehension of which is not a successive synthesis advancing from parts to an entire representation, sensation has therefore no extensive quant.i.ty; the want of sensation in a moment of time would represent it as empty, consequently = 0. That which in the empirical intuition corresponds to sensation is reality (realitas phaenomenon); that which corresponds to the absence of it, negation = 0.
Now every sensation is capable of a diminution, so that it can decrease, and thus gradually disappear. Therefore, between reality in a phenomenon and negation, there exists a continuous concatenation of many possible intermediate sensations, the difference of which from each other is always smaller than that between the given sensation and zero, or complete negation. That is to say, the real in a phenomenon has always a quant.i.ty, which however is not discoverable in apprehension, inasmuch as apprehension take place by means of mere sensation in one instant, and not by the successive synthesis of many sensations, and therefore does not progress from parts to the whole. Consequently, it has a quant.i.ty, but not an extensive quant.i.ty.
[*Footnote: Apprehension is the Kantian word for preception, in the largest sense in which we employ that term. It is the genus which includes under i, as species, perception proper and sensation proper--Tr]
Now that quant.i.ty which is apprehended only as unity, and in which plurality can be represented only by approximation to negation = O, I term intensive quant.i.ty. Consequently, reality in a phenomenon has intensive quant.i.ty, that is, a degree. If we consider this 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 momentum, for example, the momentum of weight; and for this reason, that the degree only indicates that quant.i.ty the apprehension of which is not successive, but instantaneous. This, however, I touch upon only in pa.s.sing, 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 quant.i.ty, 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, etc.
This property of quant.i.ties, according to which no part of them is the smallest possible (no part simple), is called their continuity. s.p.a.ce and time are quanta continua, because no part of them can be given, without enclosing it within boundaries (points and moments), consequently, this given part is itself a s.p.a.ce or a time. s.p.a.ce, therefore, consists only of s.p.a.ces, and time of times. Points and moments are only boundaries, that is, the mere places or positions of their limitation. But places always presuppose intuitions which are to limit or determine them; and we cannot conceive either s.p.a.ce or time composed of const.i.tuent parts which are given before s.p.a.ce or time.
Such quant.i.ties may also be called flowing, because synthesis (of the productive imagination) in the production of these quant.i.ties is a progression in time, the continuity of which we are accustomed to indicate by the expression flowing.
All phenomena, then, are continuous quant.i.ties, in respect both to intuition and mere perception (sensation, and with it reality). In the former case they are extensive quant.i.ties; in the latter, intensive.
When the synthesis of the manifold of a phenomenon is interrupted, there results merely an aggregate of several phenomena, and not properly a phenomenon as a quant.i.ty, which is not produced by the mere continuation of the productive synthesis of a certain kind, but by the repet.i.tion of a synthesis always ceasing. For example, if I call thirteen dollars a sum or quant.i.ty of money, I employ the term quite correctly, inasmuch as I understand by thirteen dollars the value of a mark in standard silver, which is, to be sure, a continuous quant.i.ty, in which no part is the smallest, but every part might const.i.tute a piece of money, which would contain material for still smaller pieces. If, however, by the words thirteen dollars I understand so many coins (be their value in silver what it may), it would be quite erroneous to use the expression a quant.i.ty of dollars; on the contrary, I must call them aggregate, that is, a number of coins. And as in every number we must have unity as the foundation, so a phenomenon taken as unity is a quant.i.ty, and as such always a continuous quant.i.ty (quantum continuum).
Now, seeing all phenomena, whether considered as extensive or intensive, are continuous quant.i.ties, the proposition: "All change (transition of a thing from one state into another) is continuous," might be proved here easily, and with mathematical evidence, were it not that the causality of a change lies, entirely beyond the bounds of a transcendental philosophy, and presupposes empirical principles. For of the possibility of a cause which changes the condition of things, that is, which determines them to the contrary to a certain given state, the understanding gives us a priori no knowledge; not merely because it has no insight into the possibility of it (for such insight is absent in several a priori cognitions), but because the notion of change concerns only certain determinations of phenomena, which experience alone can acquaint us with, while their cause lies in the unchangeable. But seeing that we have nothing which we could here employ but the pure fundamental conceptions of all possible experience, among which of course nothing empirical can be admitted, we dare not, without injuring the unity of our system, antic.i.p.ate general physical science, which is built upon certain fundamental experiences.
Nevertheless, we are in no want of proofs of the great influence which the principle above developed exercises in the antic.i.p.ation of perceptions, and even in supplying the want of them, so far as to shield us against the false conclusions which otherwise we might rashly draw.
If all reality in perception has a degree, between which and negation there is an endless sequence of ever smaller degrees, and if, nevertheless, every sense must have a determinate degree of receptivity for sensations; no perception, and consequently no experience is possible, which can prove, either immediately or mediately, an entire absence of all reality in a phenomenon; in other words, it is impossible ever to draw from experience a proof of the existence of empty s.p.a.ce or of empty time. For in the first place, an entire absence of reality in a sensuous intuition cannot of course be an object of perception; secondly, such absence cannot be deduced from the contemplation of any single phenomenon, and the difference of the degrees in its reality; nor ought it ever to be admitted in explanation of any phenomenon. For if even the complete intuition of a determinate s.p.a.ce or time is thoroughly real, that is, if no part thereof is empty, yet because every reality has its degree, which, with the extensive quant.i.ty of the phenomenon unchanged, can diminish through endless gradations down to nothing (the void), there must be infinitely graduated degrees, with which s.p.a.ce or time is filled, and the intensive quant.i.ty in different phenomena may be smaller or greater, although the extensive quant.i.ty of the intuition remains equal and unaltered.
We shall give an example of this. Almost all natural philosophers, remarking a great difference in the quant.i.ty of the matter of different kinds in bodies with the same volume (partly on account of the momentum of gravity or weight, partly on account of the momentum of resistance to other bodies in motion), conclude unanimously that this volume (extensive quant.i.ty of the phenomenon) must be void in all bodies, although in different proportion. But who would suspect that these for the most part mathematical and mechanical inquirers into nature should ground this conclusion solely on a metaphysical hypothesis--a sort of hypothesis which they profess to disparage and avoid? Yet this they do, in a.s.suming that the real in s.p.a.ce (I must not here call it impenetrability or weight, because these are empirical conceptions) is always identical, and can only be distinguished according to its extensive quant.i.ty, that is, multiplicity. Now to this presupposition, for which they can have no ground in experience, and which consequently is merely metaphysical, I oppose a transcendental demonstration, which it is true will not explain the difference in the filling up of s.p.a.ces, but which nevertheless completely does away with the supposed necessity of the above-mentioned presupposition that we cannot explain the said difference otherwise than by the hypothesis of empty s.p.a.ces. This demonstration, moreover, has the merit of setting the understanding at liberty to conceive this distinction in a different manner, if the explanation of the fact requires any such hypothesis. For we perceive that although two equal s.p.a.ces may be completely filled by matters altogether different, so that in neither of them is there left a single point wherein matter is not present, nevertheless, every reality has its degree (of resistance or of weight), which, without diminution of the extensive quant.i.ty, can become less and less ad infinitum, before it pa.s.ses into nothingness and disappears. Thus an expansion which fills a s.p.a.ce--for example, caloric, or any other reality in the phenomenal world--can decrease in its degrees to infinity, yet without leaving the smallest part of the s.p.a.ce empty; on the contrary, filling it with those lesser degrees as completely as another phenomenon could with greater.
My intention here is by no means to maintain that this is really the case with the difference of matters, in regard to their specific gravity; I wish only to prove, from a principle of the pure understanding, that the nature of our perceptions makes such a mode of explanation possible, and that it is erroneous to regard the real in a phenomenon as equal quoad its degree, and different only quoad its aggregation and extensive quant.i.ty, and this, too, on the pretended authority of an a priori principle of the understanding.
Nevertheless, this principle of the antic.i.p.ation of perception must somewhat startle an inquirer whom initiation into transcendental philosophy has rendered cautious. We must naturally entertain some doubt whether or not the understanding can enounce any such synthetical proposition as that respecting the degree of all reality in phenomena, and consequently the possibility of the internal difference of sensation itself--abstraction being made of its empirical quality. Thus it is a question not unworthy of solution: "How the understanding can p.r.o.nounce synthetically and a priori respecting phenomena, and thus antic.i.p.ate these, even in that which is peculiarly and merely empirical, that, namely, which concerns sensation itself?"
The quality of sensation is in all cases merely empirical, and cannot be represented a priori (for example, colours, taste, etc.). But the real--that which corresponds to sensation--in opposition to negation = 0, only represents something the conception of which in itself contains a being (ein seyn), and signifies nothing but the synthesis in an empirical consciousness. That is to say, the empirical consciousness in the internal sense can be raised from 0 to every higher degree, so that the very same extensive quant.i.ty of intuition, an illuminated surface, for example, excites as great a sensation as an aggregate of many other surfaces less illuminated. We can therefore make complete abstraction of the extensive quant.i.ty of a phenomenon, and represent to ourselves in the mere sensation in a certain momentum, a synthesis of h.o.m.ogeneous ascension from 0 up to the given empirical consciousness, All sensations therefore as such are given only a posteriori, but this property thereof, namely, that they have a degree, can be known a priori. It is worthy of remark, that in respect to quant.i.ties in general, we can cognize a priori only a single quality, namely, continuity; but in respect to all quality (the real in phenomena), we cannot cognize a priori anything more than the intensive quant.i.ty thereof, namely, that they have a degree. All else is left to experience.
3. a.n.a.lOGIES OF EXPERIENCE.
The principle of these is: Experience is possible only through the representation of a necessary connection of Perceptions.
PROOF.
Experience is an empirical cognition; that is to say, a cognition which determines an object by means of perceptions. It is therefore a synthesis of perceptions, a synthesis which is not itself contained in perception, but which contains the synthetical unity of the manifold of perception in a consciousness; and this unity const.i.tutes the essential of our cognition of objects of the senses, that is, of experience (not merely of intuition or sensation). Now in experience our perceptions come together contingently, so that no character of necessity in their connection appears, or can appear from the perceptions themselves, because apprehension is only a placing together of the manifold of empirical intuition, and no representation of a necessity in the connected existence of the phenomena which apprehension brings together, is to be discovered therein. But as experience is a cognition of objects by means of perceptions, it follows that the relation of the existence of the existence of the manifold must be represented in experience not as it is put together in time, but as it is objectively in time. And as time itself cannot be perceived, the determination of the existence of objects in time can only take place by means of their connection in time in general, consequently only by means of a priori connecting conceptions. Now as these conceptions always possess the character of necessity, experience is possible only by means of a representation of the necessary connection of perception.
The three modi of time are permanence, succession, and coexistence.
Accordingly, there are three rules of all relations of time in phenomena, according to which the existence of every phenomenon is determined in respect of the unity of all time, and these antecede all experience and render it possible.
The general principle of all three a.n.a.logies rests on the necessary unity of apperception in relation to all possible empirical consciousness (perception) at every time, consequently, as this unity lies a priori at the foundation of all mental operations, the principle rests on the synthetical unity of all phenomena according to their relation in time. For the original apperception relates to our internal sense (the complex of all representations), and indeed relates a priori to its form, that is to say, the relation of the manifold empirical consciousness in time. Now this manifold must be combined in original apperception according to relations of time--a necessity imposed by the a priori transcendental unity of apperception, to which is subjected all that can belong to my (i.e., my own) cognition, and therefore all that can become an object for me. This synthetical and a priori determined unity in relation of perceptions in time is therefore the rule: "All empirical determinations of time must be subject to rules of the general determination of time"; and the a.n.a.logies of experience, of which we are now about to treat, must be rules of this nature.
These principles have this peculiarity, that they do not concern phenomena, and the synthesis of the empirical intuition thereof, but merely the existence of phenomena and their relation to each other in regard to this existence. Now the mode in which we apprehend a thing in a phenomenon can be determined a priori in such a manner that the rule of its synthesis can give, that is to say, can produce this a priori intuition in every empirical example. But the existence of phenomena cannot be known a priori, and although we could arrive by this path at a conclusion of the fact of some existence, we could not cognize that existence determinately, that is to say, we should be incapable of antic.i.p.ating in what respect the empirical intuition of it would be distinguishable from that of others.
The two principles above mentioned, which I called mathematical, in consideration of the fact of their authorizing the application of mathematic phenomena, relate to these phenomena only in regard to their possibility, and instruct us how phenomena, as far as regards their intuition or the real in their perception, can be generated according to the rules of a mathematical synthesis. Consequently, numerical quant.i.ties, and with them the determination of a phenomenon as a quant.i.ty, can be employed in the one case as well as in the other. Thus, for example, out of 200,000 illuminations by the moon, I might compose and give a priori, that is construct, the degree of our sensations of the sun-light.* We may therefore ent.i.tle these two principles const.i.tutive.
[*Footnote: Kant"s meaning is: The two principles enunciated under the heads of "Axioms of Intuition," and "Antic.i.p.ations of Perception,"
authorize the application to phenomena of determinations of size and number, that is of mathematic. For example, I may compute the light of the sun, and say that its quant.i.ty is a certain number of times greater than that of the moon. In the same way, heat is measured by the comparison of its different effects on water, &c., and on mercury in a thermometer.--Tr]
The case is very different with those principles whose province it is to subject the existence of phenomena to rules a priori. For as existence does not admit of being constructed, it is clear that they must only concern the relations of existence and be merely regulative principles.
In this case, therefore, neither axioms nor antic.i.p.ations are to be thought of. Thus, if a perception is given us, in a certain relation of time to other (although undetermined) perceptions, we cannot then say a priori, what and how great (in quant.i.ty) the other perception necessarily connected with the former is, but only how it is connected, quoad its existence, in this given modus of time. a.n.a.logies in philosophy mean something very different from that which they represent in mathematics. In the latter they are formulae, which enounce the equality of two relations of quant.i.ty, and are always const.i.tutive, so that if two terms of the proportion are given, the third is also given, that is, can be constructed by the aid of these formulae. But in philosophy, a.n.a.logy is not the equality of two quant.i.tative but of two qualitative relations. In this case, from three given terms, I can give a priori and cognize the relation to a fourth member, but not this fourth term itself, although I certainly possess a rule to guide me in the search for this fourth term in experience, and a mark to a.s.sist me in discovering it. An a.n.a.logy of experience is therefore only a rule according to which unity of experience must arise out of perceptions in respect to objects (phenomena) not as a const.i.tutive, but merely as a regulative principle. The same holds good also of the postulates of empirical thought in general, which relate to the synthesis of mere intuition (which concerns the form of phenomena), the synthesis of perception (which concerns the matter of phenomena), and the synthesis of experience (which concerns the relation of these perceptions). For they are only regulative principles, and clearly distinguishable from the mathematical, which are const.i.tutive, not indeed in regard to the certainty which both possess a priori, but in the mode of evidence thereof, consequently also in the manner of demonstration.
But what has been observed of all synthetical propositions, and must be particularly remarked in this place, is this, that these a.n.a.logies possess significance and validity, not as principles of the transcendental, but only as principles of the empirical use of the understanding, and their truth can therefore be proved only as such, and that consequently the phenomena must not be subjoined directly under the categories, but only under their schemata. For if the objects to which those principles must be applied were things in themselves, it would be quite impossible to cognize aught concerning them synthetically a priori. But they are nothing but phenomena; a complete knowledge of which--a knowledge to which all principles a priori must at last relate--is the only possible experience. It follows that these principles can have nothing else for their aim than the conditions of the empirical cognition in the unity of synthesis of phenomena. But this synthesis is cogitated only in the schema of the pure conception of the understanding, of whose unity, as that of a synthesis in general, the category contains the function unrestricted by any sensuous condition.
These principles will therefore authorize us to connect phenomena according to an a.n.a.logy, with the logical and universal unity of conceptions, and consequently to employ the categories in the principles themselves; but in the application of them to experience, we shall use only their schemata, as the key to their proper application, instead of the categories, or rather the latter as restricting conditions, under the t.i.tle of "formulae" of the former.
A. FIRST a.n.a.lOGY.
Principle of the Permanence of Substance.
In all changes of phenomena, substance is permanent, and the quantum thereof in nature is neither increased nor diminished.
PROOF.