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 const.i.tutes 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, s.p.a.ce must be only the[40] form of sensibility.

[40] _The_ and not _a_, because, for the moment, time is ignored.

Now why does Kant think that this conclusion follows? Before we can answer this question we must remove an initial difficulty. In this pa.s.sage 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 s.p.a.ce 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 const.i.tutes that which is empirical, but only its form, viz. s.p.a.ce and time."[42] His argument, however, can be successfully stated without this identification. It is only necessary to re-write his cardinal a.s.sertion in the form 'the perception of s.p.a.ce 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 s.p.a.ce, 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 s.p.a.ce a real object is not present. He regards this as proving that the latter perception is only of something subjective or mental. "s.p.a.ce 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 s.p.a.ce 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.

[41] _Prol._, -- 9.

[42] _Prol._, -- 11.

[43] _Prol._, -- 10.

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 s.p.a.ce, 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 s.p.a.ce is a form of sensibility or a way in which objects appear to us, but that s.p.a.ce 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 s.p.a.ce, and therefore if we can discover these laws by a study of empty s.p.a.ce, 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 s.p.a.ce 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 s.p.a.ce 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.

[44] Kant expresses the a.s.sertion that s.p.a.ce is the form of all objects by saying that s.p.a.ce is the form of _phenomena_.

This of course renders easy an unconscious transition from the thesis that s.p.a.ce is the form of objects to the quite different thesis that s.p.a.ce is the form of sensibility; cf.

p. 39.

Again, Kant's starting-point really commits him to the view that s.p.a.ce is a characteristic of things as they are. For--paradoxical though it may be--his problem is to explain the possibility of _perceiving a priori_, i. e. of _perceiving_ the characteristics of an object anterior to the actual presence of the object in perception.[45] This implies that _empirical_ perception, which involves the actual presence of the object, involves no difficulty; in other words, it is implied that empirical perception is of objects as they are. And we find Kant admitting this to the extent of allowing _for the sake of argument_ that the perception of a present thing can make us know the thing as it is in itself.[46] But if empirical perception gives us things as they are, and if, as is the case, and as Kant really presupposes, the objects of empirical perception are spatial, then, since s.p.a.ce is their form, the judgements of geometry must relate to things as they are. It is true that on this view Kant's first presupposition of geometrical judgements has to be stated by saying that we are able to perceive a real characteristic of things in s.p.a.ce, before we perceive the things; and, no doubt, Kant thinks this impossible. According to him, when we perceive empty s.p.a.ce no object is present, and therefore what is before the mind must be merely mental. But no greater difficulty is involved than that involved in the corresponding supposition required by Kant's own view. It is really just as difficult to hold that we can perceive a characteristic of things as they appear to us _before_ they appear, as to hold that we can perceive a characteristic of them as they are in themselves _before_ we perceive them.

[45] Cf. _Prol._, Section 8.

[46] _Prol._, -- 9 (cf. p. 55).

The fact is that the real difficulty with which Kant is grappling in the _Prolegomena_ arises, not from the supposition that spatial bodies are things in themselves, but from the supposed presupposition of geometry that we must be able to perceive empty s.p.a.ce before we perceive bodies in it. It is, of course, impossible to defend the perception of empty s.p.a.ce, but _if_ it be maintained, the s.p.a.ce perceived must be conceded to be not, as Kant thinks, something mental or subjective, but a real characteristic of things. For, as has been pointed out, the paradox of pure perception is reached solely through the consideration that, while in empirical perception we perceive objects, in pure perception we do not, and since the objects of empirical perception are spatial, s.p.a.ce must be a real characteristic of them.

The general result of the preceding criticism is that Kant's conclusion does not follow from the premises by which he supports it.

It should therefore be asked whether it is not possible to take advantage of this hiatus by presenting the argument for the merely phenomenal character of s.p.a.ce without any appeal to the possibility of perceiving empty s.p.a.ce. For it is clear that what was primarily before Kant, in writing the _Critique_, was the _a priori_ character of geometrical judgements themselves, and not the existence of a perception of empty s.p.a.ce which they were held to presuppose.[47]

[47] The difficulty with which Kant is struggling in the _Prolegomena_, ---- 6-11, can be stated from a rather different point of view by saying that the thought that geometrical judgements imply a perception of empty s.p.a.ce led him to apply the term '_a priori_' to perception as well as to judgement.

The term, _a priori_, applied to judgements has a valid meaning; it means, not that the judgement is made prior to all experience, but that it is not based upon experience, being originated by the mind in virtue of its own powers of thinking. Applied to perception, however, '_a priori_' must mean prior to all experience, and, since the object of perception is essentially individual (cf. B. 741, M. 435), this use of the term gives rise to the impossible task of explaining how a perception can take place prior to the actual experience of an individual in perception (cf.

_Prol._, -- 8).

If, then, the conclusion that s.p.a.ce is only the form of sensibility can be connected with the _a priori_ character of geometrical judgements without presupposing the existence of a perception of empty s.p.a.ce, his position will be rendered more plausible.

This can be done as follows. The essential characteristic of a geometrical judgement is not that it takes place prior to experience, but that it is not based upon experience. Thus a judgement, arrived at by an activity of the mind in which it remains within itself and does not appeal to actual experience of the objects to which the judgement relates, is implied to hold good of those objects. If the objects were things as they are in themselves, the validity of the judgement could not be justified, for it would involve the gratuitous a.s.sumption that a necessity of thought is binding on things which _ex hypothesi_ are independent of the nature of the mind. If, however, the objects in question are things as perceived, they will be through and through conditioned by the mind's perceiving nature; and, consequently, if a geometrical rule, e. g. that a three-sided figure must have three angles, is really a law of the mind's perceiving nature, all individual perceptions, i. e. all objects as perceived by us, will necessarily conform to the law. Therefore, in the latter case, and in that only, will the universal validity of geometrical judgements be justified. Since, then, geometrical judgements are universally valid, s.p.a.ce, which is that of which geometrical laws are the laws, must be merely a form of perception or a characteristic of objects as perceived by us.

This appears to be the best form in which the substance of Kant's argument, stripped of unessentials, can be stated. It will be necessary to consider both the argument and its conclusion.

The argument, so stated, is undeniably plausible. Nevertheless, examination of it reveals two fatal defects. In the first place, its starting-point is false. To Kant the paradox of geometrical judgements lies in the fact that they are not based upon an appeal to experience of the things to which they relate. It is implied, therefore, that judgements which are based on experience involve no paradox, and for the reason that in experience we apprehend things as they are.[48] In contrast with this, it is implied that in geometrical judgements the connexion which we apprehend is not real, i. e. does not relate to things as they are. Otherwise, there would be no difficulty; if in geometry we apprehended rules of connexion relating to things as they are, we could allow without difficulty that the things must conform to them. No such distinction, however, can be drawn between _a priori_ and empirical judgements. For the necessity of connexion, e. g.

between being a three-sided figure and being a three-angled figure, is as much a characteristic of things as the empirically-observed shape of an individual body, e. g. a table. Geometrical judgements, therefore, cannot be distinguished from empirical judgements on the ground that in the former the mind remains within itself, and does not immediately apprehend fact or a real characteristic of reality.[49]

Moreover, since in a geometrical judgement we do in fact think that we are apprehending a real connexion, i. e. a connexion which applies to things and to things as they are in themselves, to question the reality of the connexion is to question the validity of thinking altogether, and to do this is implicitly to question the validity of our thought about the nature of our own mind, as well as the validity of our thought about things independent of the mind. Yet Kant's argument, in the form in which it has just been stated, presupposes that our thought is valid at any rate when it is concerned with our perceptions of things, even if it is not valid when concerned with the things as they are in themselves.

[48] Cf. p. 17.

[49] For the reasons which led Kant to draw this distinction between empirical and _a priori_ judgements, cf. pp. 21-2.

This consideration leads to the second criticism. The supposition that s.p.a.ce is only a form of perception, even if it be true, _in no way a.s.sists_ the explanation of the universal validity of geometrical judgements. Kant's argument really confuses a _necessity_ of relation with the _consciousness of a necessity_ of relation. No doubt, if it be a law of our perceiving nature that, whenever we perceive an object as a three-sided figure, the object as perceived contains three angles, it follows that any object as perceived will conform to this law; just as if it be a law of things as they are in themselves that three-sided figures contain three angles, all three-sided figures will in themselves have three angles. But what has to be explained is the universal applicability, not of a law, but of a judgement about a law.

For Kant's real problem is to explain why _our judgement_ that a three-sided figure must contain three angles must apply to all three-sided figures. Of course, if it be granted that in the judgement we apprehend the true law, the problem may be regarded as solved. But how are we to know that what we judge _is_ the true law? The answer is in no way facilitated by the supposition that the judgement relates to our perceiving nature. It can just as well be urged that what we think to be a necessity of our perceiving nature is not a necessity of it, as that what we think to be a necessity of things as they are in themselves is not a necessity of them. The best, or rather the only possible, answer is simply that that of which we apprehend the necessity must be true, or, in other words, that we _must_ accept the validity of thought. Hence nothing is gained by the supposition that s.p.a.ce is a form of sensibility. If what we judge to be necessary is, as such, valid, a judgement relating to things in themselves will be as valid as a judgement relating to our perceiving nature.[50]

[50] The same criticism can be urged against Kant's appeal to the necessity of _constructing_ geometrical figures. The conclusion drawn from the necessity of construction is stated thus: "If the object (the triangle) were something in itself without relation to you the subject, how could you say that that which lies necessarily in your subjective conditions of constructing a triangle must also necessarily belong to the triangle in itself?" (B. 65, M. 39). Kant's thought is that the laws of the mind's constructing nature must apply to objects, if, and only if, the objects are the mind's own construction. Hence it is open to the above criticism if, in the criticism, 'construct' be subst.i.tuted for 'perceive'.

This difficulty is concealed from Kant by his insistence on the _perception_ of s.p.a.ce involved in geometrical judgements. This leads him at times to identify the judgement and the perception, and, therefore, to speak of the judgement as a perception. Thus we find him saying that mathematical judgements are always _perceptive_,[51] and that "It is only possible for my perception to precede the actuality of the object and take place as _a priori_ knowledge, if &c."[52]

Hence, if, in addition, a geometrical judgement, as being a judgement about a necessity, be identified with a necessity of judging, the conformity of things to these universal judgements will become the conformity of things to rules or necessities of our judging, i. e. of our perceiving nature, and Kant's conclusion will at once follow.[53]

Unfortunately for Kant, a geometrical judgement, however closely related to a perception, must itself, as the apprehension of what is necessary and universal, be an act of thought rather than of perception, and therefore the original problem of the conformity of things to our mind can be forced upon him again, even after he thinks that he has solved it, in the new form of that of the conformity within the mind of perceiving to thinking.

[51] _Prol._, -- 7.

[52] _Prol._, -- 9.

[53] Cf. (_Introduction_, B. xvii, M. xxix): "But if the object (as object of the senses) conforms to the nature of our faculty of perception, I can quite well represent to myself the possibility of _a priori_ knowledge of it [i. e.

mathematical knowledge]."

The fact is simply that the universal validity of geometrical judgements can in no way be 'explained'. It is not in the least explained or made easier to accept by the supposition that objects are 'phenomena'. These judgements must be accepted as being what we presuppose them to be in making them, viz. the direct apprehension of necessities of relation between real characteristics of real things.

To explain them by reference to the phenomenal character of what is known is really--though contrary to Kant's intention--to throw doubt upon their validity; otherwise, they would not need explanation. As a matter of fact, it is _impossible_ to question their validity. In the act of judging, doubt is impossible. Doubt can arise only when we subsequently reflect and temporarily lose our hold upon the consciousness of necessity in judging.[54] The doubt, however, since it is non-existent in our geometrical consciousness, is really groundless,[55] and, therefore, the problem to which it gives rise is unreal. Moreover if, _per impossibile_, doubt could be raised, it could not be set at rest. No vindication of a judgement in which we are conscious of a necessity could do more than take the problem a stage further back, by basing it upon some other consciousness of a necessity; and since this latter judgement could be questioned for precisely the same reason, we should only be embarking upon an infinite process.

[54] Cf. Descartes, _Princ. Phil._ i. -- 13, and _Medit._ v sub fin.

[55] The view that kinds of s.p.a.ce other than that with which we are acquainted are possible, though usually held and discussed by mathematicians, belongs to them _qua_ metaphysicians, and not _qua_ mathematicians.

We may now consider Kant's conclusion in abstraction from the arguments by which he reaches it. It raises three main difficulties.

In the first place, it is not the conclusion to be expected from Kant's own standpoint. The phenomenal character of s.p.a.ce is inferred, not from the fact that we make judgements at all, but from the fact that we make judgements of a particular kind, viz. _a priori_ judgements. From this point of view empirical judgements present no difficulty. It should, therefore, be expected that the qualities which we attribute to things in empirical judgements are not phenomenal, but belong to things as they are. Kant himself implies this in drawing his conclusion concerning the nature of s.p.a.ce. "s.p.a.ce does not represent any quality of things in themselves or things in relation to one another; that is, it does not represent any determination of things which would attach to the objects themselves and would remain, even though we abstracted from all subjective conditions of perception. For neither absolute nor relative[56] determinations of objects can be perceived prior to the existence of the things to which they belong, and therefore not _a priori_."[57] It is, of course, implied that in experience, where we do not discover determinations of objects prior to the existence of the objects, we do apprehend determinations of things as they are in themselves, and not as they are in relation to us. Thus we should expect the conclusion to be, not that all that we know is phenomenal--which is Kant's real position--but that spatial (and temporal) relations alone are phenomenal, i. e. that they alone are the result of a trans.m.u.tation due to the nature of our perceiving faculties.[58] This conclusion would, of course, be absurd, for what Kant considers to be the empirically known qualities of objects disappear, if the spatial character of objects is removed. Moreover, Kant is prevented by his theory of perception from seeing that this is the real solution of his problem, absurd though it may be. Since perception is held to arise through the origination of sensations by things in themselves, empirical knowledge is naturally thought of as knowledge about sensations, and since sensations are palpably within the mind, and are held to be due to things in themselves, knowledge about sensations can be regarded as phenomenal.

[56] The first sentence shows that 'relative determinations'

means, not 'determinations of objects in relation to us', but 'determinations of objects in relation to one another.' Cf.

B. 37, M. 23; and B. 66 fin., 67 init., M. 40 (where these meanings are confused).

[57] B. 42, M. 26.

[58] This conclusion is also to be expected because, inconsistently with his real view, Kant is here (B. 41-2, M.

25-6) under the influence of the presupposition of our ordinary consciousness that in perception we are confronted by things in themselves, known to be spatial, and not by appearances produced by unknown things in themselves. Cf. (B.

41, M. 25) "and thereby of obtaining immediate representation of them [i. e. objects];" and (B. 42, M. 26) "the receptivity of the subject to be affected by objects necessarily precedes all perceptions of these objects." These sentences identify things in themselves and bodies in s.p.a.ce, and thereby imply that in empirical perception we perceive things in themselves and as they _are_.

On the other hand, if we consider Kant's conclusion from the point of view, not of the problem which originates it, but of the distinction in terms of which he states it, viz. that between things as they are in themselves and things as perceived by us, we are led to expect the contrary result. Since perception is the being affected by things, and since the nature of the affection depends upon the nature of our capacity of being affected, in _all_ perception the object will become distorted or transformed, as it were, by our capacity of being affected. The conclusion, therefore, should be that in all judgements, empirical as well as _a priori_, we apprehend things only as perceived. The reason why Kant does not draw this conclusion is probably that given above, viz. that by the time Kant reaches the solution of his problem empirical knowledge has come to relate to sensation only; consequently, it has ceased to occur to him that empirical judgements could possibly give us knowledge of things as they are. Nevertheless, Kant should not have retained in his formulation of the problem a distinction irreconcilable with his solution of it; and if he had realized that he was doing so he might have been compelled to modify his whole view.

The second difficulty is more serious. If the truth of geometrical judgements presupposes that s.p.a.ce is only a property of objects as perceived by us, it is a paradox that geometricians should be convinced, as they are, of the truth of their judgements. They undoubtedly think that their judgements apply to things as they are in themselves, and not merely as they appear to us. They certainly do not think that the relations which they discover apply to objects only as perceived. Not only, therefore, do they not think that bodies in s.p.a.ce are phenomena, but they do not even leave it an open question whether bodies are phenomena or not. Hence, if Kant be right, they are really in a state of illusion, for on his view the true geometrical judgement should include in itself the phenomenal character of spatial relations; it should be ill.u.s.trated by expressing Euclid I. 5 in the form that the equality of the angles at the base of an isosceles triangle belongs to objects as perceived. Kant himself lays this down.

"The proposition 'all objects are beside one another in s.p.a.ce'

is valid under[59] the limitation that these things are taken as objects of our sensuous perception. If I join the condition to the perception, and say 'all things, as external phenomena, are beside one another in s.p.a.ce', the rule is valid universally, and without limitation."[60] Kant, then, is in effect allowing that it is possible for geometricians to make judgements, of the necessity of which they are convinced, and yet to be wrong; and that, therefore, the apprehension of the necessity of a judgement is no ground of its truth. It follows that the truth of geometrical judgements can no longer be accepted as a starting-point of discussion, and, therefore, as a ground for inferring the phenomenal character of s.p.a.ce.

[59] A. reads 'only under'

[60] B. 43, M. 27.