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.
