There can hardly be any danger of our mistaking purely empirical principles for principles of the pure understanding or vice versa; for the character of necessity which distinguishes the concepts of the pure understanding, and the absence of which can easily be perceived in every empirical proposition, however general it may seem, will always prevent their confusion. There are, however, pure principles a priori which I should not like to ascribe to the pure understanding, because they are derived, not from pure concepts, but from pure intuitions (although by means of the understanding); the understanding being the faculty of the concepts. We find such principles in mathematics, but their application to experience, and therefore their objective validity, nay, even the possibility of such synthetical knowledge a priori (the deduction thereof) rests always on the pure understanding.
Hence my principles will not include the principles of mathematics, but they will include those on which the possibility and objective validity a priori of those mathematical principles are founded, and which consequently are to be looked upon as the source of those principles, proceeding from concepts to intuitions, and not from intuitions to concepts.
When the pure concepts of the understanding are applied to every possible experience, their synthesis is either mathematical or dynamical, for it is directed partly to the intuition of a phenomenon only, partly to its existence. The conditions a priori of intuition are absolutely necessary with regard to every possible experience, while the conditions of the existence of the object of a possible empirical intuition are in themselves accidental only. The principles of the mathematical use of the categories will therefore be absolutely necessary, that is apodictic, while those of their dynamical use, though likewise possessing the character of necessity a priori, can possess such a character subject only to the condition of empirical thought in experience, that is mediately and indirectly, and cannot therefore claim that immediate evidence which belongs to the former, although their certainty with regard to experience in general remains unaffected by this. Of this we shall be better qualified to judge at the conclusion of this system of principles.
Our table of categories gives us naturally the best instructions for drawing up a table of principles, because these are nothing but rules for the objective use of the former.
All principles of the pure understanding are therefore,
I
Axioms of Intuition.
II
III
Anticipations of
Analogies of
Perception.
Experience.
IV
Postulates of Empirical
Thought in General.
I have chosen these names not unadvisedly, so that the difference with regard to the evidence and the application of those principles should not be overlooked. We shall soon see that, both with regard to the evidence and the a priori determination of phenomena according to the categories of quantity and quality (if we attend to the form of them only) their principles differ considerably from those of the other two classes, inasmuch as the former are capable of an intuitive, the latter of a merely discursive, though both of a complete certainty. I shall therefore call the former mathematical, the latter dynamical principles.2 It should be observed, however, that I do not speak here either of the principles of mathematics, or of those of general physical dynamics, but only of the principles of the pure understanding in relation to the internal sense (without any regard to the actual representations given in it). It is these through which the former become possible, and I have given them their name, more on account of their application than of their contents. I shall now proceed to consider them in the same order in which they stand in the table.
I
[Of the Axioms of Intuition3
Principle of the Pure Understanding
'All Phenomena are, with reference to their intuition, extensive quantities']
I call an extensive quantity that in which the representation of the whole is rendered possible by the representation of its parts, and therefore necessarily preceded by it. I cannot represent to myself any line, however small it may be, without drawing it in thought, that is, without producing all its parts one after the other, starting from a given point, and thus, first of all, drawing its intuition. The same applies to every, even the smallest portion of time. I can only think in it the successive progress from one moment to another, thus producing in the end, by all portions of time and their addition, a definite quantity of time. As in all phenomena pure intuition is either space or time, every phenomenon, as an intuition, must be an extensive quantity, because it can be known in apprehension by a successive synthesis only (of part with part). All phenomena therefore, when perceived in intuition, are aggregates (collections) of previously given parts, which is not the case with every kind of quantities, but with those only which are represented to us and apprehended as extensive.
On this successive synthesis of productive imagination in elaborating figures are founded the mathematics of extension with their axioms (geometry), containing the conditions of sensuous intuition a priori, under which alone the schema of a pure concept of an external phenomenal appearance can be produced; for instance, between two points one straight line only is possible, or two straight lines cannot enclose a space, etc. These are the axioms which properly relate only to quantities (quanta) as such.
But with regard to quantity (quantitas), that is, with regard to the answer to the question, how large something may be, there are no axioms, in the proper sense of the word, though several of the propositions referring to it possess synthetical and immediate certainty (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 really analytical, because I am conscious immediately of the identity of my producing the one quantity with my producing the other; axioms on the contrary must be synthetical propositions a priori. The self-evident propositions on numerical relation again are no doubt synthetical, but they are not general, like those of geometry, and therefore cannot be called axioms, but numerical formulas only. That 7 + 5=12 is not an analytical proposition. For neither in the representation of 7, nor in that of 5, nor in that of the combination of both, do I think the number 12. (That I am meant to think it in the addition of the two, is not the question here, for in every analytical proposition all depends on this, whether the predicate is really thought in the representation of the subject.) Although the proposition is synthetical, it is a singular proposition only. If in this case we consider only the synthesis of the homogeneous unities, then the synthesis can here take place in one way only, although afterwards the use of these numbers becomes general. If I say, a triangle can be constructed with three lines, two of which together are greater than the third, I have before me the mere function of productive imagination, which may draw the lines greater or smaller, and bring them together at various angles. The number 7, on the contrary, is possible in one way only, and so likewise the number 12, which is produced by the synthesis of the former with 5. Such propositions therefore must not be called axioms (for their number would be endless) but numerical formulas.
This transcendental principle of phenomenal mathematics adds considerably to our knowledge a priori. Through it alone it becomes possible to make pure mathematics in their full precision applicable to objects of experience, which without that principle would by no means be self-evident, nay, has actually provoked much contradiction. Phenomena are not things in themselves. Empirical intuition is possible only through pure intuition (of space and time), and whatever geometry says of the latter is valid without contradiction of the former. All evasions, as if objects of the senses should not conform to the rules of construction in space (for instance, to the rule of the infinite divisibility of lines or angles) must cease, for one would thus deny all objective validity to space and with it to all mathematics, and would no longer know why and how far mathematics can be applied to phenomena. The synthesis of spaces
and times, as the synthesis of the essential form of all intuition, is that which renders possible at the same time the apprehension of phenomena, that is, every external experience, and therefore also all knowledge of its objects, and whatever mathematics, in their pure use prove of that synthesis is valid necessarily also of this knowledge. All objections to this are only the chicaneries of a falsely guided reason, which wrongly imagines that it can separate the objects of the senses from the formal conditions of our sensibility, and represents them, though they are phenomena only, as objects by themselves, given to the understanding. In this case, however, nothing could be known of them a priori, nothing could be known synthetically through pure concepts of space, and the science which determines those concepts, namely, geometry, would itself become impossible.
II
[Anticipations of Perception
The principle which anticipates all perceptions as such, is this: In all phenomena sensation, and the Real which corresponds to it in the object (realitas phenomenon), has an intensive quantity, that is, a degree4]
All knowledge by means of which I may know and determine a priori whatever belongs to empirical knowledge, may be called an anticipation, and it is no doubt in this sense that Epicurus used the expression. But as there is always in phenomena something which can never be known a priori, and constitutes the real difference between empirical and a priori knowledge, namely, sensation (as matter of perception), it follows that this can never be anticipated. The pure determinations, on the contrary, in space and time, as regards both figure and quantity, may be called anticipations of phenomena, because they represent a priori, whatever may be given a posteriori in experience.
If, however, there should be something in every sensation that could be known a priori as sensation in general, even if no particular sensation be given, this would, in a very special sense, deserve to be called anticipation, because it seems extraordinary that we should anticipate experience in that which concerns the matter of experience and can be derived from experience only. Yet such is really the case.
Apprehension, by means of sensation only, fills no more than one moment (if we do not take into account the succession of many sensations). Sensation, therefore, being that in the phenomenon the apprehension of which does not form a successive synthesis progressing from parts to a complete representation, is without any extensive quantity, and the absence of sensation in one and the same moment would represent it as empty, therefore = 0. What corresponds in every empirical intuition to sensation is reality (realitas phenomenon), what corresponds to its absence is negation =0. Every sensation, however, is capable of diminution, so that it may decrease, and gradually vanish. There is therefore a continuous connection between reality in phenomena and negation, by means of many possible intermediate sensations, the difference between which is always smaller than the difference between the given sensation and zero or complete negation. It thus follows that the real in each phenomenon has always a quantity, though it is not perceived in apprehension, because apprehension takes place by a momentary sensation, not by a successive synthesis of many sensations; it does not advance from the parts to the whole, and though it has a quantity, it has not an extensive quantity.
That quantity which can be apprehended as unity only, and in which plurality can be represented by approximation only to negation = 0, I call intensive quantity. Every reality therefore in a phenomenon has intensive quantity, that is, a degree. If this reality is considered as a cause (whether of sensation, or of any other reality in the phenomenon, for instance, of change) the degree of that reality as a cause we call a momentum, for instance, the momentum of gravity: and this because the degree indicates that quantity only, the apprehension of which is not successive, but momentary. This I mention here in passing, because we have not yet come to consider causality.
Every sensation, therefore, and every reality in phenomena, however small it may be, has a degree, that is, an intensive quantity which can always be diminished, and there is between reality and negation a continuous connection of possible realities, and of possible smaller perceptions. Every colour, red, for instance, has a degree, which, however small, is never the smallest; and the same applies to heat, the momentum of gravity, etc.
This peculiar property of quantities that no part of them is the smallest possible part (no part indivisible) is called continuity. Time and space are quanta continua, because there is no part of them that is not enclosed between limits (points and moments), no part that is not itself again a space or a time. Space consists of spaces only, time of times. Points and moments are only limits, mere places of limitation, and as places presupposing always those intuitions which they are meant to limit or to determine. Mere places or parts that might be given before space or time, could never be compounded into space or time. Such quantities can also be called flowing, because the synthesis of the productive imagination which creates them is a progression in time, the continuity of which we are wont to express by the name of flowing, or passing away.
All phenomena are therefore continuous quantities, whether according to their intuition as extensive, or according to mere perception (sensation and therefore reality) as intensive quantities. When there is a break in the synthesis of the manifold of phenomena, we get only an aggregate of many phenomena, not a phenomenon, as a real quantum; for aggregate is called that what is produced, not by the mere continuation of productive synthesis of a certain kind, but by the repetition of a synthesis (beginning and) ending at every moment. If I call thirteen thalers a quantum of money, I am right, provided I understand by it the value of a mark of fine silver. This is a continuous quantity in which no part is the smallest, but every part may constitute a coin which contains material for still smaller coins. But if I understand by it thirteen round thalers, that is, so many coins (whatever their value in silver may be), then I should be wrong in speaking of a quantum of thalers, but should call it an aggregate, that is a number of coins. As every number must be founded on some unity, every phenomenon, as a unity, is a quantum, and, as such, a continuum.
If then all phenomena, whether considered as extensive or intensive, are continuous quantities, it might seem easy to prove with mathematical evidence that all change also (transition of a thing from one state into another) must be continuous, if the causality of the change did not lie quite outside the limits of transcendental philosophy, and presupposed empirical principles. For the understanding a priori tells us nothing of the possibility of a cause which changes the state of things, that is, determines them to the opposite of a given state, and this not only because it does not perceive the possibility of it (for such a perception is denied to us in several kinds of knowledge a priori), but because the changeability relates to certain determinations of phenomena to be taught by experience only, while their cause must lie in that which is unchangeable. But as the only materials which we may use at present are the pure fundamental concepts of every possible experience, from which all that is empirical is excluded, we cannot here, without injuring the unity of our system, anticipate general physical science which is based upon certain fundamental experiences.
Nevertheless, there is no lack of evidence of the great influence which our fundamental principle exercises in anticipating perceptions, nay, even in making up for their deficiency, in so far as it (that principle) stops any false conclusions that might be drawn from this deficiency.
If therefore all reality in perception has a certain degree, between which and negation there is an infinite succession of ever smaller degrees, and if every sense must have a definite degree of receptivity of sensations, it follows that no perception, and therefore no experience, is possible, that could prove, directly or indirectly, by any roundabout syllogisms, a complete absence of all reality in a phenomenon. We see therefore that experience can never supply a proof of empty space or empty time, because the total absence of reality in a sensuous intuition can itself never be perceived, neither can it be deduced from any phenomenon whatsoev
er and from the difference of degree in its reality; nor ought it ever to be admitted in explanation of it. For although the total intuition of a certain space or time is real all through, no part of it being empty, yet as every reality has its degree which, while the extensive quality of the phenomenon remains unchanged, may diminish by infinite degrees down to the nothing or void, there must be infinitely differing degrees in which space and time are filled, and the intensive quantity in phenomena may be smaller or greater, although the extensive quantity as given in intuition remains the same.
We shall give an example. Almost all natural philosophers, perceiving partly by means of the momentum of gravity or weight, partly by means of the momentum of resistance against other matter in motion, that there is a great difference in the quantity of various kinds of matter though their volume is the same, conclude unanimously that this volume (the extensive quantity of phenomena) must in all of them, though in different degrees, contain a certain amount of empty space. Who could have thought that these mathematical and mechanical philosophers should have based such a conclusion on a purely metaphysical hypothesis, which they always profess to avoid, by assuming that the real in space (I do not wish here to call it impenetrability or weight, because these are empirical concepts) must always be the same, and can differ only by its extensive quantity, that is, by the number of parts. I meet this hypothesis, for which they could find no ground in experience, and which therefore is purely metaphysical, by a transcendental demonstration, which, though it is not intended to explain the difference in the filling of spaces, will nevertheless entirely remove the imagined necessity of their hypothesis which tries to explain that difference by the admission of empty spaces, and which thus restores, at least to the understanding, its liberty to explain to itself that difference in a different way, if any such hypothesis be wanted in natural philosophy.
Critique of Pure Reason Page 18
