第 25 页

existence of the manifold. This, again, may be divided into the

physical synthesis, of the phenomena divided among each other, and the

metaphysical synthesis, or the connection of phenomena a priori in the

faculty of cognition.

1. AXIOMS OF INTUITION.

The principle of these is: All Intuitions are Extensive

Quantities.

PROOF.

All phenomena contain, as regards their form, an intuition in

space and time, which lies a priori at the foundation of all without

exception. Phenomena, therefore, cannot be apprehended, that is,

received into empirical consciousness otherwise than through the

synthesis of a manifold, through which the representations of a

determinate space or time are generated; that is to say, through the

composition of the homogeneous and the consciousness of the

synthetical unity of this manifold (homogeneous). Now the

consciousness of a homogeneous manifold in intuition, in so far as

thereby the representation of an object is rendered possible, is the

conception of a quantity (quanti). Consequently, even the perception

of an object as phenomenon is 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 manifold

in the conception of a quantity is cogitated; that is to say, all

phenomena are quantities, and extensive quantities, because as

intuitions in space or time they must be represented by means of the

same synthesis through which space and time themselves are determined.

An extensive quantity I call that wherein the representation of

the parts renders possible (and therefore necessarily antecedes) the

representation of the whole. I cannot represent to myself any line,

however small, without drawing it in thought, that is, without

generating from a point all its parts one after another, and in this

way alone producing this intuition. Precisely the same is the case

with every, even the smallest, portion of time. I cogitate therein

only the successive progress from one moment to another, and hence, by

means of the different portions of time and the addition of them, a

determinate quantity of time is produced. As the pure intuition in all

phenomena is either time or space, so is every phenomenon in its

character of intuition an extensive quantity, inasmuch as it can

only be cognized in our apprehension by successive synthesis (from

part to part). All phenomena are, accordingly, to be considered as

aggregates, that is, as a collection of previously given parts;

which is not the case with every sort of quantities, but only with

those which are represented and apprehended by us as extensive.

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, "be tween two

points only one straight line is possible," "two straight lines cannot

enclose a space," etc. 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 various propositions

synthetical and immediately certain (indemonstrabilia); 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 production of the one quantity with the production of the other;

whereas axioms must be a priori synthetical propositions. On the other

hand, the self-evident propositions as to the relation of numbers, are

certainly synthetical but not universal, like those of geometry, and

for this reason cannot be called axioms, but numerical formulae.

That 7 + 5 = 12 is not an analytical proposition. 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 analytical 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 homogeneous (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 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 can

be applied to phenomena. The synthesis of spaces 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 space and the

science which determines these conceptions, that is to say,

geometry, would itself be impossible.

2. ANTICIPATIONS OF PERCEPTION.

The principle of these is: In all phenomena the Real, that

which is an object of sensation, has Intensive Quantity,

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 space and time, for they cannot be perceived in themselves.

They contain, then, over and above the intuition, the materials for an

object (through which is represented something existing in space 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

space; consequently there is possible a synthesis also of the

production of the quantity of a sensation from its commencement,

that is, from the pure intuition = 0 onwards up to a certain

quantity of the sensation. Now as sensation in itself is not an

objective representation, and in it is to be found neither the

intuition of space nor of time, it cannot possess any extensive

quantity, and yet there does belong to it a quantity (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 quantity. And thus we must ascribe intensive

quantity, 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 anticipation; 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 constitutes 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 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 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 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 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 quantity; the

want of sensation in a moment of time would represent it as empty,

consequently = O. That which in the empirical intuition corresponds to

sensation is reality (realitas phaenomenon); that which corresponds to

the absence of it, negation = O. 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

quantity, 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 quantity, but not an extensive quantity.

Now that quantity which is apprehended only as unity, and in which

plurality can be represented only by approximation to negation = O,

I term intensive quantity. Consequently, reality in a phenomenon has

intensive quantity, 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

quantity the apprehension 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 quantity, 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 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 boundaries (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 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 space or

time composed of constituent parts which are given before space or

time. Such quantities may also be called flowing, because synthesis

(of the productive imagination) in the production of these

quantities is a progression in time, the continuity of which we are

accustomed to indicate by the expression flowing.

All phenomena, then, are continuous quantities, in respect both to

intuition and mere perception (sensation, and with it reality). In the

former case they are extensive quantities; in the latter, intensive.

When the synthesis of the manifold of a phenomenon is interrupted,