As has been pointed out,[1] the aim of the second part of the Analytic of Principles is to determine the a priori principles involved in the use of the categories under the necessary sensuous conditions. These principles Kant divides into four classes, corresponding to the four groups of categories, and he calls them respectively 'axioms of perception', 'anticipations of sense-perception', 'analogies of experience', and 'postulates of empirical thought'. The first two and the last two classes are grouped together as 'mathematical' and 'dynamical' respectively, on the ground that the former group concerns the perception of objects, i. e. their nature apprehended in perception, while the latter group concerns their existence, and that consequently, since assertions concerning the existence of objects presuppose the realization of empirical conditions which assertions concerning their nature do not, only the former possesses an absolute necessity and an immediate evidence such as is found in mathematics.[2] These two groups of principles are not, as their names might suggest, principles within mathematics and physics, but presuppositions of mathematics and physics respectively. Kant also claims appropriateness for the special terms used of each minor group to indicate the kind of principles in question, viz. 'axioms', 'anticipations', 'analogies', 'postulates'. But it may be noted as an indication of the artificiality of the scheme that each of the first two groups contains only one principle, although Kant refers to them in the plural as axioms and anticipations respectively, and although the existence of three categories corresponding to each group would suggest the existence of three principles.

The axiom of perception is that 'All perceptions are extensive quantities'. The proof of it runs thus:

"An extensive quantity I call that in which the representation of the parts renders possible the representation of the whole (and therefore necessarily precedes it). I cannot represent to myself any line, however small it may be, without drawing it in thought, that is, without generating from a point all its parts one after another, and thereby first drawing this perception. Precisely the same is the case with every, even the smallest, time.... Since the pure perception in all phenomena is either time or space, every phenomenon as a perception is an extensive quantity, because it can be known in apprehension only by a successive synthesis (of part with part). All phenomena, therefore, are already perceived as aggregates (groups of previously given parts), which is not the case with quantities of every kind, but only with those which are represented and apprehended by us as extensive."[3]

Kant opposes an extensive quantity to an intensive quantity or a quantity which has a degree. "That quantity which is apprehended only as unity and in which plurality can be represented only by approximation to negation = 0, I call intensive quantity."[4] The aspect of this ultimate distinction which underlies Kant's mode of stating it is that only an extensive quantity is a whole, i. e. something made up of parts. Thus a mile can be said to be made up of two half-miles, but a velocity of one foot per second, though comparable with a velocity of half a foot per second, cannot be said to be made up of two such velocities; it is essentially one and indivisible. Hence, from Kant's point of view, it follows that it is only an extensive magnitude which can, and indeed must, be apprehended through a successive synthesis of the parts. The proof of the axiom seems to be simply this: 'All phenomena as objects of perception are subject to the forms of perception, space and time. Space and time are [homogeneous manifolds, and therefore] extensive quantities, only to be apprehended by a successive synthesis of the parts. Hence phenomena, or objects of experience, must also be extensive quantities, to be similarly apprehended.' And Kant goes on to add that it is for this reason that geometry and pure mathematics generally apply to objects of experience.

We need only draw attention to three points. Firstly, no justification is given of the term 'axiom'. Secondly, the argument does not really appeal to the doctrine of the categories, but only to the character of space and time as forms of perception. Thirdly, it need not appeal to space and time as forms of perception in the proper sense of ways in which we apprehend objects, but only in the sense of ways in which objects are related[5]; in other words, it need not appeal to Kant's theory of knowledge. The conclusion follows simply from the nature of objects as spatially and temporally related, whether they are phenomena or not. It may be objected that Kant's thesis is that all objects of perception are extensive quantities, and that unless space and time are allowed to be ways in which we must perceive objects, we cannot say that all objects will be spatially and temporally related, and so extensive quantities. But to this it may be replied that it is only true that all objects of perception are extensive quantities if the term 'object of perception' be restricted to parts of the physical world, i. e. to just those realities which Kant is thinking of as spatially and temporally related,[6] and that this restriction is not justified, since a sensation or a pain which has only intensive quantity is just as much entitled to be called an object of perception.

The anticipation of sense-perception consists in the principle that 'In all phenomena, the real, which is an object of sensation, has intensive magnitude, i. e. a degree'. The proof is stated thus:

"Apprehension merely by means of sensation fills only one moment (that is, if I do not take into consideration the succession of many sensations). Sensation, therefore, as that in the phenomenon the apprehension of which is not a successive synthesis advancing from parts to a complete representation, has no extensive quantity; the lack of sensation in one and the same moment would represent it as empty, consequently = 0. Now that which in the empirical perception corresponds to sensation is reality (realitas phaenomenon); that which corresponds to the lack of it is negation = 0. But every sensation is capable of a diminution, so that it can decrease and thus gradually vanish. Therefore, between reality in the phenomenon and negation there exists a continuous connexion 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 the phenomenon has always a quantity, which, however, is not found in apprehension, since apprehension takes place by means of mere sensation in one moment and not by a successive synthesis of many sensations, and therefore does not proceed 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 = 0, I call an intensive quantity. Every reality, therefore, in a phenomenon has intensive quantity, that is, a degree."[7]

In other words, 'We can lay down a priori that all sensations have a certain degree of intensity, and that between a sensation of a given intensity and the total absence of sensation there is possible an infinite number of sensations varying in intensity from nothing to that degree of intensity. Therefore the real, which corresponds to sensation, can also be said a priori to admit of an infinite variety of degree.'

Though the principle established is of little intrinsic importance, the account of it is noticeable for two reasons. In the first place, although Kant clearly means by the 'real corresponding to sensation' a body in space, and regards it as a phenomenon, it is impossible to see how he can avoid the charge that he in fact treats it as a thing in itself.[8] For the correspondence must consist in the fact that the real causes or excites sensation in us, and therefore the real, i. e. a body in space, is implied to be a thing in itself. In fact, Kant himself speaks of considering the real in the phenomenon as the cause of sensation,[9] and, in a passage added in the second edition, after proving that sensation must have an intensive quantity, he says that, corresponding to the intensive quantity of sensation, an intensive quantity, i. e. a degree of influence on sense, must be attributed to all objects of sense-perception.[10] The difficulty of consistently maintaining that the real, which corresponds to sensation, is a phenomenon is, of course, due to the impossibility of distinguishing between reality and appearance within phenomena.[11]

In the second place, Kant expressly allows that in this anticipation we succeed in discovering a priori a characteristic of sensation, although sensation constitutes that empirical element in phenomena, which on Kant's general view cannot be apprehended a priori. "Nevertheless, this anticipation of sense-perception must always be somewhat surprising to an inquirer who is used to transcendental reflection, and is thereby rendered cautious. It leads us to feel some misgiving as to whether the understanding can anticipate such a synthetic proposition as that respecting the degree of all that is real in phenomena, and consequently respecting the possibility of the internal distinction of sensation itself, if we abstract from its empirical quality. There remains, therefore, a problem not unworthy of solution, viz. 'How can the understanding pronounce synthetically and a priori upon phenomena in this respect, and thus anticipate phenomena even in that which is specially and merely empirical, viz. that which concerns sensations?'"[12] But although Kant recognizes that the anticipation is surprising, he is not led to revise his general theory, as being inconsistent with the existence of the anticipation. He indeed makes an attempt[13] to deal with the difficulty; but his solution consists not in showing that the anticipation is consistent with his general theory—as he should have done, if the theory was to be retained—but in showing that, in the case of the degree of sensation, we do apprehend the nature of sensation a priori.

Strangely enough, Hume finds himself face to face with what is in principle the same difficulty, and treats it in a not dissimilar way. "There is, however, one contradictory phenomenon, which may prove, that 'tis not absolutely impossible for ideas to go before their correspondent impressions. I believe it will readily be allow'd, that the several distinct ideas of colours, which enter by the eyes, or those of sounds, which are convey'd by the hearing, are really different from each other, tho' at the same time resembling. Now if this be true of different colours, it must be no less so of the different shades of the same colour, that each of them produces a distinct idea, independent of the rest. For if this shou'd be deny'd, 'tis possible, by the continual gradation of shades, to run a colour insensibly into what is most remote from it; and if you will not allow any of the means to be different, you cannot without absurdity deny the extremes to be the same. Suppose therefore a person to have enjoyed his sight for thirty years, and to have become perfectly well acquainted with colours of all kinds, excepting one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be plac'd before him, descending gradually from the deepest to the lightest; 'tis plain that he will perceive a blank, where that shade is wanting, and will be sensible, that there is a greater distance in that place betwixt the contiguous colours, than in any other. Now I ask, whether 'tis possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, tho' it had never been conveyed to him by his senses? I believe there are few but will be of opinion that he can; and this may serve as a proof, that the simple ideas are not always derived from the correspondent impressions; tho' the instance is so particular and singular, that 'tis scarce worth our observing, and does not merit that for it alone we should alter our general maxim."[14]