Kant's Theory of Knowledge

by H. A. Prichard

It begins thus: "Mathematics carries with it thoroughly apodeictic certainty, that is, absolute necessity, and, therefore, rests on no empirical grounds, and consequently is a pure product of reason, and, besides, is thoroughly synthetical. How, then, is it possible for human reason to accomplish such knowledge entirely a priori?... But we find that all mathematical knowledge has this peculiarity, that it must represent its conception previously in perception, and indeed a priori, consequently in a perception which is not empirical but pure, and that otherwise it cannot take a single step. Hence its judgements are always intuitive.... This observation on the nature of mathematics at once gives us a clue to the first and highest condition of its possibility, viz. that there must underlie it a pure perception in which it can exhibit or, as we say, construct all its conceptions in the concrete and yet a priori. If we can discover this pure perception and its possibility, we may thence easily explain how a priori synthetical propositions in pure mathematics are possible, and consequently also how the science itself is possible. For just as empirical perception enables us without difficulty to enlarge synthetically in experience the conception which we frame of an object of perception through new predicates which perception itself offers us, so pure perception also will do the same, only with the difference that in this case the synthetical judgement will be a priori certain and apodeictic, while in the former case it will be only a posteriori and empirically certain; for the latter [i. e. the empirical perception on which the a posteriori synthetic judgement is based] contains only that which is to be found in contingent empirical perception, while the former [i. e. the pure perception on which the a priori synthetic judgement is based] contains that which is bound to be found in pure perception, since, as a priori perception, it is inseparably connected with the conception before all experience or individual sense-perception."

This passage is evidently based upon the account which Kant gives in the Doctrine of Method of the method of geometry.[35] According to this account, in order to apprehend, for instance, that a three-sided figure must have three angles, we must draw in imagination or on paper an individual figure corresponding to the conception of a three-sided figure. We then see that the very nature of the act of construction involves that the figure constructed must possess three angles as well as three sides. Hence, perception being that by which we apprehend the individual, a perception is involved in the act by which we form a geometrical judgement, and the perception can be called a priori, in that it is guided by our a priori apprehension of the necessary nature of the act of construction, and therefore of the figure constructed.

The account in the Prolegomena, however, differs from that of the Doctrine of Method in one important respect. It asserts that the perception involved in a mathematical judgement not only may, but must, be pure, i. e. must be a perception in which no spatial object is present, and it implies that the perception must take place before all experience of actual objects.[36] Hence a priori, applied to perception, has here primarily, if not exclusively, the temporal meaning that the perception takes place antecedently to all experience.[37]

The thought of the passage quoted from the Prolegomena can be stated thus: 'A mathematical judgement implies the perception of an individual figure antecedently to all experience. This may be said to be the first condition of the possibility of mathematical judgements which is revealed by reflection. There is, however, a prior or higher condition. The perception of an individual figure involves as its basis another pure perception. For we can only construct and therefore perceive an individual figure in empty space. Space is that in which it must be constructed and perceived. A perception[38] of empty space is, therefore, necessary. If, then, we can discover how this perception is possible, we shall be able to explain the possibility of a priori synthetical judgements of mathematics.'

Kant continues as follows: "But with this step the difficulty seems to increase rather than to lessen. For henceforward the question is 'How is it possible to perceive anything a priori?' A perception is such a representation as would immediately depend upon the presence of the object. Hence it seems impossible originally to perceive a priori, because perception would in that case have to take place without an object to which it might refer, present either formerly or at the moment, and accordingly could not be perception.... How can perception of the object precede the object itself?"[39] Kant here finds himself face to face with the difficulty created by the preceding section. Perception, as such, involves the actual presence of an object; yet the pure perception of space involved by geometry—which, as pure, is the perception of empty space, and which, as the perception of empty space, is a priori in the sense of temporally prior to the perception of actual objects—presupposes that an object is not actually present.

The solution is given in the next section. "Were our perception necessarily of such a kind as to represent things as they are in themselves, no perception would take place a priori, but would always be empirical. For I can only know what is contained in the object in itself, if it is present and given to me. No doubt it is even then unintelligible how the perception of a present thing should make me know it as it is in itself, since its qualities cannot migrate over into my faculty of representation; but, even granting this possibility, such a perception would not occur a priori, i. e. before the object was presented to me; for without this presentation, no basis of the relation between my representation and the object can be imagined; the relation would then have to rest upon inspiration. It is therefore possible only in one way for my perception to precede the actuality of the object and to take place as a priori knowledge, viz. if it contains nothing but the form of the sensibility, which precedes in me, the subject, all actual impressions through which I am affected by objects. For I can know a priori that objects of the senses can only be perceived in accordance with this form of the sensibility. Hence it follows that propositions which concern merely this form of sensuous perception will be possible and valid for objects of the senses, and in the same way, conversely, that perceptions which are possible a priori can never concern any things other than objects of our senses."

This section clearly constitutes the turning-point in Kant's argument, and primarily expresses, in an expanded form, the central doctrine of § 3 of the Aesthetic, that an external perception anterior to objects themselves, and in which our conceptions of objects can be determined a priori, is possible, if, and only if, it has its seat in the subject as its formal nature of being affected by objects, and consequently as the form of the external sense in general. It argues that, since this is true, and since geometrical judgements involve such a perception anterior to objects, space must be only the[40] form of sensibility.

Now why does Kant think that this conclusion follows? Before we can answer this question we must remove an initial difficulty. In this passage Kant unquestionably identifies a form of perception with an actual perception. It is at once an actual perception and a capacity of perceiving. This is evident from the words, "It is possible only in one way for my perception to precede the actuality of the object ... viz. if it contains nothing but the form of the sensibility."[41] The identification becomes more explicit a little later. "A pure perception (of space and time) can underlie the empirical perception of objects, because it is nothing but the mere form of the sensibility, which precedes the actual appearance of the objects, in that it in fact first makes them possible. Yet this faculty of perceiving a priori affects not the matter of the phenomenon, i. e. that in it which is sensation, for this constitutes that which is empirical, but only its form, viz. space and time."[42] His argument, however, can be successfully stated without this identification. It is only necessary to re-write his cardinal assertion in the form 'the perception of space must be nothing but the manifestation of the form of the sensibility'. Given this modification, the question becomes, 'Why does Kant think that the perception of empty space, involved by geometrical judgements, can be only a manifestation of our perceiving nature, and not in any way the apprehension of a real quality of objects?' The answer must be that it is because he thinks that, while in empirical perception a real object is present, in the perception of empty space a real object is not present. He regards this as proving that the latter perception is only of something subjective or mental. "Space and time, by being pure a priori perceptions, prove that they are mere forms of our sensibility which must precede all empirical perception, i. e. sense-perception of actual objects."[43] His main conclusion now follows easily enough. If in perceiving empty space we are only apprehending a manifestation of our perceiving nature, what we apprehend in a geometrical judgement is really a law of our perceiving nature, and therefore, while it must apply to our perceptions of objects or to objects as perceived, it cannot apply to objects apart from our perception, or, at least, there is no ground for holding that it does so.

If, however, this fairly represents Kant's thought, it must be allowed that the conclusion which he should have drawn is different, and even that the conclusion which he does draw is in reality incompatible with his starting-point.

His starting-point is the view that the truth of geometrical judgements presupposes a perception of empty space, in virtue of which we can discover rules of spatial relation which must apply to all spatial objects subsequently perceived. His problem is to discover the presupposition of this presupposition. The proper answer must be, not that space is a form of sensibility or a way in which objects appear to us, but that space is the form of all objects, i. e. that all objects are spatial.[44] For in that case they must be subject to the laws of space, and therefore if we can discover these laws by a study of empty space, the only condition to be satisfied, if the objects of subsequent perception are to conform to the laws which we discover, is that all objects should be spatial. Nothing is implied which enables us to decide whether the objects are objects as they are in themselves or objects as perceived; for in either case the required result follows. If in empirical perception we apprehend things only as they appear to us, and if space is the form of them as they appear to us, it will no doubt be true that the laws of spatial relation which we discover must apply to things as they appear to us. But on the other hand, if in empirical perception we apprehend things as they are, and if space is their form, i. e. if things are spatial, it will be equally true that the laws discovered by geometry must apply to things as they are.