*All combination (conjunctio) is either composition (compositio)or connection (nexus).The former is the synthesis of a manifold, the parts of which do not necessarily belong to each other.For example, the two triangles into which a square is divided by a diagonal, do not necessarily belong to each other, and of this kind is the synthesis of the homogeneous in everything that can be mathematically considered.This synthesis can be divided into those of aggregation and coalition, the former of which is applied to extensive, the latter to intensive quantities.The second sort of combination (nexus) is the synthesis of a manifold, in so far as its parts do belong necessarily to each other; for example, the accident to a substance, or the effect to the cause.Consequently it is a synthesis of that which though heterogeneous, is represented as connected a priori.This combination- not an arbitrary one- Ientitle dynamical because it concerns the connection of the 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.
