SECTION I. Space

SPACE

METAPHYSICAL EXPOSITION OF THE CONCEPTION OF SPACE

=Space: First Argument.=--“Space is not an empirical concept (Begriff) which has been abstracted from outer experiences. For in order that certain sensations be related to something outside me (i.e. to something in another region of space from that in which I find myself), and similarly in order that I may be able to represent them as outside [and alongside] one another, and accordingly as not only [qualitatively] different but as in different places, the representation of space must be presupposed (muss schon zum Grunde liegen). The representation of space cannot, therefore, be empirically obtained at second-hand from the relations of outer appearance. This outer experience is itself possible at all only through that representation.”

The first sentence states the thesis of the argument: space is not an empirical concept abstracted from outer experiences. The use of the term Begriff in the title of the section, and also in this sentence, is an instance of the looseness with which Kant employs his terms. It is here synonymous with the term representation (Vorstellung), which covers intuitions as well as general or discursive concepts. Consequently, the contradiction is only verbal, not real, when Kant proceeds to prove that the concept of space is an intuition, not a concept. But this double employment of the term is none the less misleading. When Kant employs it in a strict sense, it signifies solely the general class concept. All true concepts are for Kant of that single type. He has not re-defined the term concept in any manner which would render it applicable to the relational categories. For unfortunately, and very strangely, he never seems to have raised the question whether categories are not also concepts. The application to the forms of understanding of the separate title categories seems to have contented him. Much that is obscure and even contradictory in his teaching might have been prevented had he recognised that the term concept is a generic title which includes, as its sub-species, both general notions and relational categories.

Kant’s limitation of the term concept to the merely generic, and his consequent equating of the categorical proposition with the assertion of the substance-attribute relation, would seem in large part to be traceable to his desire to preserve for himself, in the pioneer labours of his Critical enquiries, the guiding clues of the distinctions drawn in the traditional logic. Kant insists on holding to them, at least in outward appearance, at whatever sacrifice of strict consistency. Critical doctrine is made to conform to the exigencies of an artificial framework, with which its own tenets are only in very imperfect harmony. Appreciation of the ramifying influence, and, as regards the detail of exposition, of the far-reaching consequences, of this desire to conform to the time-honoured rubrics, is indeed an indispensable preliminary to any adequate estimate whether of the strength or of the defects of the Critical doctrines. As a separate and ever-present influence in the determining of Kant’s teaching, this factor may conveniently and compendiously be entitled Kant’s logical architectonic. We shall have frequent occasion to observe its effects.

The second sentence gives expression to the fact through which Kant proves his thesis. Certain sensations, those of the special senses as distinguished from the organic sensations, are related to something which stands in a different region of space from the embodied self, and consequently are apprehended as differing from one another not only in quality but also in spatial position. As is proved later in the Analytic, thought plays an indispensable part in constituting this reference of sensations to objects. Kant here, however, makes no mention of this further complication. He postulates, as he may legitimately do at this stage, the fact that our sensations are thus objectively interpreted, and limits his enquiry to the spatial factor. Now the argument, as Vaihinger justly points out, hinges upon the assumption which Kant has already embodied in his definition of the “form” of sense, viz. that sensations are non-spatial, purely qualitative. Though this is an assumption of which Kant nowhere attempts to give proof, it serves none the less as an unquestioned premiss from which he draws all-important conclusions. This first argument on space derives its force entirely from it.

The proof that the representation of space is non-empirical may therefore be explicitly stated as follows. As sensations are non-spatial and differ only qualitatively, the representation of space must have been added to them. And not being supplied by the given sensations, it must, as the only alternative, have been contributed by the mind. The representation of space, so far from being derived from external experience, is what first renders it possible. As a subjective form that lies ready in the mind, it precedes experience and co-operates in generating it. This proof of the apriority of space is thus proof of the priority of the representation of space to every empirical perception.

In thus interpreting Kant’s argument as proving more than the thesis of the first sentence claims, we are certainly reading into the proof more than Kant has himself given full expression to. But, as is clearly shown by the argument of the next section, we are only stating what Kant actually takes the argument as having proved, namely, that the representation of space is not only non-empirical but is likewise of subjective origin and precedes experience in temporal fashion.

The point of view which underlies and inspires the argument can be defined even more precisely. Kant’s conclusion may be interpreted in either of two ways. The form of space may precede experience only as a potentiality. Existing as a power of co-ordination, it will come to consciousness only indirectly through the addition which it makes to the given sensations. Though subjective in origin, it will be revealed to the mind only in and through experience. This view may indeed be reconciled with the terms of the proof. But a strictly literal interpretation of its actual wording is more in keeping with what, as we shall find, is the general trend of the Aesthetic as a whole. We are then confronted by a very different and extremely paradoxical view, which may well seem too naive to be accepted by the modern reader, but which we seem forced, none the less, to regard as the view actually presented in the text before us. Kant here asserts, in the most explicit manner, that the mind, in order to construe sensations in spatial terms, must already be in possession of a representation of space, and that it is in the light of this representation that it apprehends sensations. The conscious representation of space precedes in time external experience. Such, then, would seem to be Kant’s first argument on space. It seeks to establish a negative conclusion, viz. that space is not derived from experience. But, in so doing, it also yields a positive psychological explanation of its origin.

Those commentators who refuse to recognise that Kant’s problem is in any degree psychological, or that Kant himself so regards it, and who consequently seek to interpret the Aesthetic from the point of view of certain portions of the Analytic, give a very different statement of this first argument. They state it in purely logical terms. Its problem, they claim, is not that of determining the origin of our representation of space, but only its logical relation to our specific sense-experiences. The notion of space in general precedes, as an indispensable logical presupposition, all particular specification of the space relation. Consciousness of space as a whole is not constructed from consciousness of partial spaces; on the contrary, the latter is only possible in and through the former.

Such an argument does of course represent a valuable truth; and it alone harmonises with much in Kant’s maturer teaching; but we must not therefore conclude that it is also the teaching of the Aesthetic. The Critique contains too great a variety of tendencies, too rich a complexity of issues, to allow of such simplification. It loses more than it gains by such rigorous pruning of the luxuriant secondary tendencies of its exposition and thought. And above all, this procedure involves the adoption by the commentator of impossible responsibilities, those of deciding what is essential and valuable in Kant’s thought and what is irrelevant. The value and suggestiveness of Kant’s philosophy largely consist in his sincere appreciation of conflicting tendencies, and in his persistent attempt to reduce them to unity with the least possible sacrifice. But in any case the logical interpretation misrepresents this particular argument. Kant is not here distinguishing between space in general and its specific modifications. He is maintaining that no space relation can be revealed in sensation. It is not only that the apprehension of any limited space presupposes the representation of space as a whole. Both partial and infinite space are of mental origin; sensation, as such, is non-spatial, purely subjective. And lastly, the fact that Kant means to assert that space is not only logically presupposed but is subjectively generated, is sufficiently borne out by his frequent employment elsewhere in the Aesthetic of such phrases as “the subjective condition of sensibility,” “lying ready in our minds,” and “necessarily preceding [as the form of the subject’s receptivity] all intuitions of objects.”

=Second Argument.=--Having proved by the first argument that the representation of space is not of empirical origin, Kant in the second argument proceeds to establish the positive conclusion that it is a priori. The proof, when all its assumptions are rendered explicit, runs as follows. Thesis: Space is a necessary representation, and consequently is a priori. Proof: It is impossible to imagine the absence of space, though it is possible to imagine it as existing without objects to fill it. A representation which it is impossible for the mind to be without is a necessary representation. But necessity is one of the two criteria of the a priori. The proof of the necessary character of space is therefore also a proof of its being a priori.

The argument, more freely stated, is that what is empirically given from without can be thought away, and that since space cannot be thus eliminated, it must be grounded in our subjective organisation, i.e. must be psychologically a priori. The argument, as stated by Kant, emphasises the apriority, not the subjectivity, of space, but none the less the asserted apriority is psychological, not logical in character. For the criterion employed is not the impossibility of thinking otherwise, but our incapacity to represent this specific element as absent. The ground upon which the whole argument is made to rest is the merely brute fact (asserted by Kant) of our incapacity to think except in terms of space.

The argument is, however, complicated by the drawing of a further consequence, which follows as a corollary from the main conclusion. From the subjective necessity of space follows its objective necessity. Space being necessary a priori, objects can only be apprehended in and through it. Consequently it is not dependent upon the objects apprehended, but itself underlies outer appearances as the condition of their possibility. This corollary is closely akin to the first argument on space, and differs from it only in orientation. The first argument has a psychological purpose. It maintains that the representation of space precedes external experience, causally conditioning it. The corollary has a more objective aim. It concludes that space is a necessary constituent of the external experience thus generated. The one proves that space is a necessary subjective antecedent; the other that it is a necessary objective ingredient.

To consider the proof in detail. The exact words which Kant employs in stating the nervus probandi of the argument are that we can never represent (eine Vorstellung davon machen) space as non-existent, though we can very well think (denken) it as being empty of objects. The terms Vorstellung and denken are vague and misleading. Kant himself recognises that it is possible to conceive that there are beings who intuit objects in some other manner than in space. He cannot therefore mean that we are unable to think or conceive space as non-existent. He must mean that we cannot in imagination intuit it as absent. It is the necessary form of all our intuitions, and therefore also of imagination, which is intuitive in character. Our consciousness is dependent upon given intuitions for its whole content, and to that extent space is a form with which the mind can never by any possibility dispense. Pure thought enables it to realise this de facto limitation, but not to break free from it. Even in admitting the possibility of other beings who are not thus constituted, the mind still recognises its own ineluctable limitations.

Kant offers no proof of his assertion that space can be intuited in image as empty of all sensible content; and as a matter of fact the assertion is false. Doubtless the use of the vague term Vorstellung is in great part responsible for Kant’s mistaken position. So long as imagination and thought are not clearly distinguished, the assertion is correspondingly indefinite. Pure space may possibly be conceived, but it can also be conceived as altogether non-existent. If, on the other hand, our imaginative power is alone in question, the asserted fact must be categorically denied. With the elimination of all sensible content space itself ceases to be a possible image. Kant’s proof thus rests upon a misstatement of fact.

In a second respect Kant’s proof is open to criticism. He takes the impossibility of imagining space as absent as proof that it originates from within. The argument is valid only if no other psychological explanation can be given of this necessity, as for instance through indissoluble association or through its being an invariable element in the given sensations. Kant’s ignoring of these possibilities is due to his unquestioning belief that sensations are non-spatial, purely qualitative. That is a presupposition whose truth is necessary to the cogency of the argument.

=Third Argument.=--This argument, which was omitted in the second edition, will be considered in its connection with the transcendental exposition into which it was then merged.

=Fourth (in second edition, Third) Argument.=--The next two arguments seek to show that space is not a discursive or general concept but an intuition. The first proof falls into two parts, (a) We can represent only a single space. For though we speak of many spaces, we mean only parts of one and the same single space. Space must therefore be an intuition. For only intuition is thus directly related to a single individual. A concept always refers indirectly, per notas communes, to a plurality of individuals. (b) The parts of space cannot precede the one all-comprehensive space. They can be thought only in and through it. They arise through limitation of it. Now the parts (i.e. the attributes) which compose a concept precede it in thought. Through combination of them the concept is formed. Space cannot, therefore, be a concept. Consequently it must, as the only remaining alternative, be an intuition. Only in an intuition does the whole precede the parts. In a concept the parts always precede the whole. Intuition stands for multiplicity in unity, conception for unity in multiplicity.

The first part of the argument refers to the extension, the second part to the intension of the space representation. In both aspects it appears as intuitional.

Kant, in repeating his thesis as a conclusion from the above grounds, confuses the reader by an addition which is not strictly relevant to the argument, viz. by the statement that this intuition must be non-empirical and a priori. This is simply a recapitulation of what has been established in the preceding proofs. It is not, as might at first sight appear, part of the conclusion established by the argument under consideration. The reader is the more apt to be misled owing to the fact that very obviously arguments for the non-empirical and for the a priori character of space can be derived from proof (b). That space is non-empirical would follow from the fact that representation of space as a whole is necessary for the apprehension of any part of it. Empirical intuition can only yield the apprehension of a limited space. The apprehension of the comprehensive space within which it falls must therefore be non-empirical.

“As we intuitively apprehend (anschauend erkennen) not only the space of the object which affects our senses, but the whole space, space cannot arise out of the actual affection of the senses, but must precede it in time (vor ihr vorhergehen).”

But in spite of its forcibleness this argument is nowhere presented in the Critique.

Similarly, in so far as particular spaces can be conceived only in and through space as a whole, and in so far as the former are limitations of the one antecedent space, the intuition which underlies all external perception must be a priori. This is in essentials a stronger and more cogent mode of formulating the second argument on space. But again, and very strangely, it is nowhere employed by Kant in this form.

The concluding sentence, ambiguously introduced by the words so werden auch, is tacked on to the preceding argument. Interpreted in the light of § 15 C of the Dissertation, and of the corresponding fourth argument on time, it may be taken as offering further proof that space is an intuition. The concepts of line and triangle, however attentively contemplated, will never reveal the proposition that in every triangle two sides taken together are greater than the third. An a priori intuition will alone account for such apodictic knowledge. This concluding sentence thus really belongs to the transcendental exposition; and as such ought, like the third argument, to have been omitted in the second edition.

Kant’s proof rests on the assumption that there are only two kinds of representation, intuitions and concepts, and also in equal degree upon the further assumption that all concepts are of one and the same type. Intuition is, for Kant, the apprehension of an individual. Conception is always the representation of a class or genus. Intuition is immediately related to the individual. Conception is reflective or discursive; it apprehends a plurality of objects indirectly through the representation of those marks which are common to them all. Intuition and conception having been defined in this manner, the proof that space is single or individual, and that in it the whole precedes the parts, is proof conclusive that it is an intuition, not a conception. Owing, however, to the narrowness of the field assigned to conception, the realm occupied by intuition is proportionately wide, and the conclusion is not as definite and as important as might at first sight appear. By itself, it amounts merely to the statement, which no one need challenge, that space is not a generic class concept. Incidentally certain unique characteristics of space are, indeed, forcibly illustrated; but the implied conclusion that space on account of these characteristics must belong to receptivity, not to understanding, does not by any means follow. It has not, for instance, been proved that space and time are radically distinct from the categories, i.e. from the relational forms of understanding.

In 1770, while Kant still held to the metaphysical validity of the pure forms of thought, the many difficulties which result from the ascription of independent reality to space and time were, doubtless, a sufficient reason for regarding the latter as subjective and sensuous. But upon adoption of the Critical standpoint such argument is no longer valid. If all our forms of thought may be subjective, the existence of antinomies has no real bearing upon the question whether space and time do or do not have a different constitution and a different mental origin from the categories. The antinomies, that is to say, may perhaps suffice to prove that space and time are subjective; they certainly do not establish their sensuous character.

But though persistence of the older, un-Critical opposition between the intellectual and the sensuous was partly responsible for Kant’s readiness to regard as radical the very obvious differences between a category such as that of substance and attribute and the visual or tactual extendedness with which objects are endowed, it can hardly be viewed as the really decisive influence. That would rather seem to be traceable to Kant’s conviction that mathematical knowledge is unique both in fruitfulness and in certainty, and to his further belief that it owes this distinction to the content character of the a priori forms upon which it rests. For though the categories of the physical sciences are likewise a priori, they are exclusively relational, and serve only to organise a material that is empirically given. To account for the superiority of mathematical knowledge Kant accordingly felt constrained to regard space and time as not merely forms in terms of which we interpret the matter of sense, but as also themselves intuited objects, and as therefore possessing a character altogether different from anything which can be ascribed to the pure understanding. The opposition between forms of sense and categories of the understanding, in the strict Kantian mode of envisaging that opposition, is thus inseparably bound up with Kant’s doctrine of space and time as being not only forms of intuition, but as also in their purity and independence themselves intuitions. Even the sensuous subject matter of pure mathematics--so Kant would seem to contend--is a priori in nature. If this latter view be questioned--and to the modern reader it is indeed a stone of stumbling--much of the teaching of the Aesthetic will have to be modified or at least restated.

=Fifth (in second edition, Fourth) Argument.=--This argument is quite differently stated in the two editions of the Critique, though the purpose of the argument is again in both cases to prove that space is an intuition, not a general concept. In the first edition this is proved by reference to the fact that space is given as an infinite magnitude. This characteristic of our space representation cannot be accounted for so long as it is regarded as a concept. A general conception of space which would abstract out those properties and relations which are common to all spaces, to a foot as well as to an ell, could not possibly determine anything in regard to magnitude. For since spaces differ in magnitude, any one magnitude cannot be a common quality. Space is, however, given us as determined in magnitude, namely, as being of infinite magnitude; and if a general conception of space relations cannot determine magnitude, still less can it determine infinite magnitude. Such infinity must be derived from limitlessness in the progression of intuition. Our conceptual representations of infinite magnitude must be derivative products, acquired from this intuitive source.

In the argument of the second edition the thesis is again established by reference to the infinity of space. But in all other respects the argument differs from that of the first edition. A general conception, which abstracts out common qualities from a plurality of particulars, contains an infinite number of possible different representations under it; but it cannot be thought as containing an infinite number of representations in it. Space must, however, be thought in this latter manner, for it contains an infinite number of coexisting parts. Since, then, space cannot be a concept, it must be an intuition.

The definiteness of this conclusion is somewhat obscured by the further characterisation of the intuition of space as a priori, and by the statement that it is the original (ursprüngliche) representation which is of this intuitive nature. The first addition must here, again, just as in the fourth argument, be regarded as merely a recapitulation of what has already been established, not a conclusion from the present argument. The introduction of the word ‘original’ seems to be part of Kant’s reply to the objections which had already been made to his admission in the first edition that there is a conception as well as an intuition of space. It is the original given intuition of space which renders such reflective conception possible.

The chief difficulty of these proofs arises out of the assertion which they seem to involve that space is given as actually infinite. There are apparently, on this point, two views in Kant, which were retained up to the very last, and which are closely connected with his two representations of space, on the one hand as a formal intuition given in its purity and in its completeness, and on the other hand as the form of intuition, which exists only so far as it is constructed, and which is dependent for its content upon given matter.

=Third Argument, and Transcendental Exposition of Space.=--The distinction between the metaphysical and the transcendental expositions, introduced in the second edition of the Critique, is one which Kant seems to have first made clear to himself in the process of writing the Prolegomena. It is a genuine improvement, marking an important distinction. It separates out two comparatively independent lines of argument. The terms in which the distinction is stated are not, however, felicitous. Kant’s reason for adopting the title metaphysical is indicated in the Prolegomena:

“As concerns the sources of metaphysical cognition, its very concept implies that they cannot be empirical.... For it must not be physical but metaphysical knowledge, i.e. knowledge lying beyond experience.... It is therefore a priori knowledge, coming from pure understanding and pure Reason.”

The metaphysical exposition, it would therefore seem, is so entitled because it professes to prove that space is a priori, not empirical, and to do so by analysis of its concept. Now by Kant’s own definition of the term transcendental, as the theory of the a priori, this exposition might equally well have been named the transcendental exposition. In any case it is an essential and chief part of the Transcendental Aesthetic. Such division of the Transcendental Aesthetic into a metaphysical and a transcendental part involves a twofold use, wider and narrower, of one and the same term. Only as descriptive of the whole Aesthetic is transcendental employed in the sense defined.

Exposition (Erörterung, Lat. expositio) is Kant’s substitute for the more ordinary term definition. Definition is the term which we should naturally have expected; but as Kant holds that no =given= concept, whether a priori or empirical, can be defined in the strict sense, the substitutes the term exposition, using it to signify such definition of the nature of space as is possible to us. To complete the parallelism Kant speaks of the transcendental enquiry as also an exposition. It is, however, in no sense a definition. Kant’s terms here, as so often elsewhere, are employed in a more or less arbitrary and extremely inexact manner.

The distinction between the two expositions is taken by Kant as follows. The metaphysical exposition determines the nature of the concept of space, and shows it to be a given a priori intuition. The transcendental exposition shows how space, when viewed in this manner, renders comprehensible the possibility of synthetic a priori knowledge.

The omission of the third argument on space from the second edition, and its incorporation into the new transcendental exposition, is certainly an improvement. In its location in the first edition, it breaks in upon the continuity of Kant’s argument without in any way contributing to the further definition of the concept of space. Also, in emphasising that mathematical knowledge depends upon the construction of concepts, Kant presupposes that space is intuitional; and that has not yet been established.

The argument follows the strict, rigorous, synthetic method. From the already demonstrated a priori character of space, Kant deduces the apodictic certainty of all geometrical principles. But though the paragraph thus expounds a consequence that follows from the a priori character of space, not an argument in support of it, something in the nature of an argument is none the less implied. The fact that this view of the representation of space alone renders mathematical science possible can be taken as confirming this interpretation of its nature. Such an argument, though circular, is none the less cogent. Consideration of Kant’s further statements, that were space known in a merely empirical manner we could not be sure that in all cases only one straight line is possible between two points, or that space will always be found to have three dimensions, must meantime be deferred.

In the new transcendental exposition Kant adopts the analytic method of the Prolegomena, and accordingly presents his argument in independence of the results already established. He starts from the assumption of the admitted validity of geometry, as being a body of synthetic a priori knowledge. Yet this, as we have already noted, does not invalidate the argument; in both the first and the last paragraphs it is implied that the a priori and intuitive characteristics of space have already been proved. From the synthetic character of geometrical propositions Kant argues that space must be an intuition. Through pure concepts no synthetic knowledge is possible. Then from the apodictic character of geometry he infers that space exists in us as pure and a priori; no experience can ever reveal necessity. But geometry also exists as an applied science; and to account for our power of anticipating experience, we must view space as existing only in the perceiving subject as the form of its sensibility. If it precedes objects as the necessary subjective condition of their apprehension, we can to that extent predetermine the conditions of their existence.

In the concluding paragraph Kant says that this is the only explanation which can be given of the possibility of geometry. He does not distinguish between pure and applied geometry, though the proof which he has given of each differs in a fundamental respect. Pure geometry presupposes only that space is an a priori intuition; applied geometry demands that space be conceived as the a priori form of external sense. Only in reference to applied geometry does the Critical problem arise:--viz. how we can form synthetic judgments a priori which yet are valid of objects; or, in other words, how judgments based upon a subjective form can be objectively valid. But any attempt, at this point, to define the nature and possibility of applied geometry must anticipate a result which is first established in Conclusion b. Though, therefore, the substitution of this transcendental exposition for the third space argument is a decided improvement, Kant, in extending it so as to cover applied as well as pure mathematics, overlooks the real sequence of his argument in the first edition. The employment of the analytic method, breaking in, as it does, upon the synthetic development of Kant’s original argument, is a further irregularity.

It may be noted that in the third paragraph Kant takes the fact that geometry can be applied to objects as proof of the subjectivity of space. He refuses to recognise the possibility that space may be subjective as a form of receptivity, and yet also be a mode in which things in themselves exist. This, as regards its conclusion, though not as regards its argument, is therefore an anticipation of Conclusion a. In the last paragraph Kant is probably referring to the views both of Leibniz and of Berkeley.

CONCLUSIONS FROM THE ABOVE CONCEPTS

=Conclusion a.=--Thesis: Space is not a property of things in themselves, nor a relation of them to one another. Proof: The properties of things in themselves can never be intuited prior to their existence, i.e. a priori. Space, as already proved, is intuited in this manner. In other words, the apriority of space is by itself sufficient proof of its subjectivity.

This argument has been the subject of a prolonged controversy between Trendelenburg and Kuno Fischer. Trendelenburg was able to prove his main point, namely, that the above argument is quite inconclusive. Kant recognises only two alternatives, either space as objective is known a posteriori, or being an a priori representation it is subjective in origin. There exists a third alternative, namely, that though our intuition of space is subjective in origin, space is itself an inherent property of things in themselves. The central thesis of the rationalist philosophy of the Enlightenment was, indeed, that the independently real can be known by a priori thinking. Even granting the validity of Kant’s later conclusion, first drawn in the next paragraph, that space is the subjective form of all external intuition, that would only prove that it does not belong to appearances, prior to our apprehension of them; nothing is thereby proved in regard to the character of things in themselves. We anticipate by a priori reasoning only the nature of appearances, never the constitution of things in themselves. Therefore space, even though a priori, may belong to the independently real. The above argument cannot prove the given thesis.

Vaihinger contends that the reason why Kant does not even attempt to argue in support of the principle, that the a priori must be purely subjective, is that he accepts it as self-evident. This explanation does not, however, seem satisfactory. But Vaihinger supplies the data for modification of his own assertion. It was, it would seem, the existence of the antinomies which first and chiefly led Kant to assert the subjectivity of space and time. For as he then believed that a satisfactory solution of the antinomies is possible only on the assumption of the subjectivity of space and time, he regarded their subjectivity as being conclusively established, and accordingly failed to examine with sufficient care the validity of his additional proof from their apriority. This would seem to be confirmed by the fact that when later, in reply to criticisms of the arguments of the first edition, he so far modified his position as to offer reasons in support of the above general principle, even then he nowhere discussed the principle in reference to the forms of sense. All his discussions concern only the possible independent reality of the forms of thought. To the very last Kant would seem to have regarded the above argument as an independent, and by itself a sufficient, proof of the subjectivity of space.

The refutation of Trendelenburg’s argument which is offered by Caird is inconclusive. Caird assumes the chief point at issue, first by ignoring the possibility that space may be known a priori in reference to appearances and yet at the same time be transcendently real; and secondly by ignoring the fact that to deny spatial properties to things in themselves is as great a violation of Critical principles as to assert them. One point, however, in Caird’s reply to Trendelenburg calls for special consideration, viz. Caird’s contention that Kant did actually take account of the third alternative, rejecting it as involving the “absurd” hypothesis of a pre-established harmony. Undoubtedly Kant did so. But the contention has no relevancy to the point before us. The doctrine of pre-established harmony is a metaphysical theory which presupposes the possibility of gaining knowledge of things in themselves. For that reason alone Kant was bound to reject it. A metaphysical proof of the validity of metaphysical judgments is, from the Critical point of view, a contradiction in terms. As the validity of all speculations is in doubt, a proof which is speculative cannot meet our difficulties. And also, as Kant himself further points out, the pre-established harmony, even if granted, can afford no solution of the Critical problem how a priori judgments can be passed upon the independently real. The judgments, thus guaranteed, could only possess de facto validity; we could never be assured of their necessity. It is chiefly in these two inabilities that Kant locates the “absurdity” of a theory of pre-established harmony. The refutation of that theory does not, therefore, amount to a disproof of the possibility which we are here considering.

=Conclusion b.=--The next paragraph maintains two theses: (a) that space is the form of all outer intuition; (b) that this fact explains what is otherwise entirely inexplicable and paradoxical, namely, that we can make a priori judgments which yet apply to the objects experienced. The first thesis, that the pure intuition of space is only conceivable as the form of appearances of outer sense, is propounded in the opening sentence without argument and even without citation of grounds. The statement thus suddenly made is not anticipated save by the opening sentences of the section on space. It is an essentially new doctrine. Hitherto Kant has spoken of space only as an a priori intuition. The further assertion that as such it must necessarily be conceived as the form of outer sense (i.e. not only as a formal intuition but also as a form of intuition), calls for the most definite and explicit proof. None, however, is given. It is really a conclusion from points all too briefly cited by Kant in the general Introduction, namely, from his distinction between the matter and the form of sense. The assertions there made, in a somewhat casual manner, are here, without notification to the reader, employed as premisses to ground the above assertion. His thesis is not, therefore, as by its face value it would seem to profess to be, an inference from the points established in the preceding expositions. It interprets these conclusions in the light of points considered in the Introduction; and thereby arrives at a new and all-important interpretation of the nature of the a priori intuition of space.

The second thesis employs the first to explain how prior to all experience we can determine the relations of objects. Since (a) space is merely the form of outer sense, and (b) accordingly exists in the mind prior to all empirical intuition, all appearances must exist in space, and we can predetermine them from the pure intuition of space that is given to us a priori. Space, when thus viewed as the a priori form of outer sense, renders comprehensible the validity of applied mathematics.

As we have already noted, Kant in the second edition obscures the sequence of his argument by offering in the new transcendental exposition a justification of applied as well as of pure geometry. In so doing he anticipates the conclusion which is first drawn in this later paragraph. This would have been avoided had Kant given two separate transcendental expositions. First, an exposition of pure mathematics, placed immediately after the metaphysical exposition; for pure mathematics is exclusively based upon the results of the metaphysical exposition. And secondly, an exposition of applied mathematics, introduced after Conclusion b. The explanation of applied geometry is really the more essential and central of the two, as it alone involves the truly Critical problem, how judgments formed a priori can yet apply to objects. Conclusion b constitutes, as Vaihinger rightly insists, the very heart of the Aesthetic. The arrangement of Kant’s argument diverts the reader’s attention from where it ought properly to centre.

The use which Kant makes of the Prolegomena in his statement of the new transcendental exposition is one cause of the confusion. The exposition is a brief summary of the corresponding Prolegomena sections. In introducing this summary into the Critique Kant overlooked the fact that in referring to applied mathematics he is anticipating a point first established in Conclusion b. The real cause, however, of the trouble is common to both editions, namely Kant’s failure clearly to appreciate the fundamental distinction between the view that space is an a priori intuition and the view that it is the a priori form of all external intuition, i.e. of outer sense. He does not seem to have fully realised how very different are those two views. In consequence of this he fails to distinguish between the transcendental expositions of pure and applied geometry.

=Third paragraph.=--Kant proceeds to develop the subjectivist conclusions which follow from a and b.

“We may say that space contains all things which can appear to us externally, but not all things in themselves, whether intuited or not, nor again all things intuited by any and every subject.”

This sentence makes two assertions: (a) space does not belong to things in and by themselves; (b) space is not a necessary form of intuition for all subjects whatsoever.

The grounds for the former assertion are not here considered, and that is doubtless the reason why the oder nicht is excised in Kant’s private copy of the Critique. As we have seen, Kant does not anywhere in the Aesthetic even attempt to offer argument in support of this assertion. In defence of (a) Kant propounds for the first time the view of sensibility as a limitation. Space is a limiting condition to which human intuition is subject. Whether the intuitions of other thinking beings are subject to the same limitation, we have no means of deciding. But for all human beings, Kant implies, the same conditions must hold universally.

In the phrase “transcendental ideality of space” Kant, it may be noted, takes the term ideality as signifying subjectivity, and the term transcendental as equivalent to transcendent. He is stating that judged from a transcendent point of view, i.e. from the point of view of the thing in itself, space has a merely subjective or “empirical” reality. This is an instance of Kant’s careless use of the term transcendental. Space is empirically real, but taken transcendently, is merely ideal.

KANT’S ATTITUDE TO THE PROBLEMS OF MODERN GEOMETRY

This is an appropriate point at which to consider the consistency of Kant’s teaching with modern developments in geometry. Kant’s attitude has very frequently been misrepresented. As he here states, he is willing to recognise that the forms of intuition possessed by other races of finite beings may not coincide with those of the human species. But in so doing he does not mean to assert the possibility of other spatial forms, i.e. of spaces that are non-Euclidean. In his pre-Critical period Kant had indeed attempted to deduce the three-dimensional character of space as a consequence of the law of gravitation; and recognising that that law is in itself arbitrary, he concluded that God might, by establishing different relations of gravitation, have given rise to spaces of different properties and dimensions.

“A science of all these possible kinds of space would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry.”

But from the time of Kant’s adoption, in 1770, of the Critical view of space as being the universal form of our outer sense, he seems to have definitely rejected all such possibilities. Space, to be space at all, must be Euclidean; the uniformity of space is a presupposition of the a priori certainty of geometrical science. One of the criticisms which in the Dissertation he passes upon the empirical view of mathematical science is that it would leave open the possibility that “a space may some time be discovered endowed with other fundamental properties, or even perhaps that we may happen upon a two-sided rectilinear figure.” This is the argument which reappears in the third argument on space in the first edition of the Critique. The same examples are employed with a somewhat different wording.

“It would not even be necessary that there should be only one straight line between two points, though experience invariably shows this to be so. What is derived from experience has only comparative universality, namely, that which is obtained through induction. We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions.”

But that Kant should have failed to recognise the possibility of other spaces does not by itself point to any serious defect in his position. There is no essential difficulty in reconciling the recognition of such spaces with his fundamental teaching. He admits that other races of finite beings may perhaps intuit through non-spatial forms of sensibility; he might quite well have recognised that those other forms of intuition, though not Euclidean, are still spatial. It is in another and more vital respect that Kant’s teaching lies open to criticism. Kant is convinced that space is given to us in intuition as being definitely and irrevocably Euclidean in character. Both our intuition and our thinking, when we reflect upon space, are, he implies, bound down to, and limited by, the conditions of Euclidean space. And it is in this positive assumption, and not merely in his ignoring of the possibility of other spaces, that he comes into conflict with the teaching of modern geometry. For in making the above assumption Kant is asserting that we definitely know physical space to be three-dimensional, and that by no elaboration of concepts can we so remodel it in thought that the axiom of parallels will cease to hold. Euclidean space, Kant implies, is given to us as an unyielding form that rigidly resists all attempts at conceptual reconstruction. Being quite independent of thought and being given as complete, it has no inchoate plasticity of which thought might take advantage. The modern geometer is not, however, prepared to admit that intuitional space has any definiteness or preciseness of nature apart from the concepts through which it is apprehended; and he therefore allows, as at least possible, that upon clarification of our concepts space may be discovered to be radically different from what it at first sight appears to be. In any case, the perfecting of the concepts must have some effect upon their object. But even--as the modern geometer further maintains--should our space be definitely proved, upon analytic and empirical investigation, to be Euclidean in character, other possibilities will still remain open for speculative thought. For though the nature of our intuitional data may constrain us to interpret them through one set of concepts rather than through another, the competing sets of alternative concepts will represent genuine possibilities beyond what the actual is found to embody.

Thus the defect of Kant’s teaching, in regard to space, as judged in the light of the later teaching of geometrical science, is closely bound up with his untenable isolation of the a priori of sensibility from the a priori of understanding. Space, being thus viewed as independent of thought, has to be regarded as limiting and restricting thought by the unalterable nature of its initial presentation. And unfortunately this is a position which Kant continued to hold, despite his increasing recognition of the part which concepts must play in the various mathematical sciences. In the deduction of the first edition we find him stating that synthesis of apprehension is necessary to all representation of space and time. He further recognises that all arithmetical processes are syntheses according to concepts. And in the Prolegomena there occurs the following significant passage.

“Do these laws of nature lie in space, and does the understanding learn them by merely endeavouring to find out the fruitful meaning that lies in space; or do they inhere in the understanding and in the way in which it determines space according to the conditions of the synthetical unity towards which its concepts are all directed? Space is something so uniform and as to all particular properties so indeterminate, that we should certainly not seek a store of laws of nature in it. That which determines space to the form of a circle or to the figures of a cone or a sphere, is, on the contrary, the understanding, so far as it contains the ground of the unity of these constructions. The mere universal form of intuition, called space, must therefore be the substratum of all intuitions determinable to particular objects, and in it, of course, the condition of the possibility and of the variety of these intuitions lies. But the unity of the objects is solely determined by the understanding, and indeed in accordance with conditions which are proper to the nature of the understanding....”

Obviously Kant is being driven by the spontaneous development of his own thinking towards a position much more consistent with present-day teaching, and completely at variance with the hard and fast severance between sensibility and understanding which he had formulated in the Dissertation and has retained in the Aesthetic. In the above Prolegomena passage a plasticity is being allowed to space, sufficient to permit of essential modification in the conceptual processes through which it is articulated. But, as I have just stated, that did not lead Kant to disavow the conclusions which he had drawn from his previous teaching.

This defect in Kant’s doctrine of space, as expounded in the Aesthetic, indicates a further imperfection in his argument. He asserts that the form of space cannot vary from one human being to another, and that for this reason the judgments which express it are universally valid. Now, in so far as Kant’s initial datum is consciousness of time, he is entirely justified in assuming that everything which can be shown to be a necessary condition of such consciousness must be uniform for all human minds. But as his argument is not that consciousness of Euclidean space is necessary to consciousness of time, but only that consciousness of the permanent in space is a required condition, he has not succeeded in showing the necessary uniformity of the human mind as regards the specific mode in which it intuits space. The permanent might still be apprehended as permanent, and therefore as yielding a possible basis for consciousness of sequence, even if it were apprehended in some four-dimensional form.

* * * * *

=Fourth Paragraph.=--The next paragraph raises one of the central problems of the Critique, namely, the question as to the kind of reality possessed by appearances. Are they subjective, like taste or colour? Or have they a reality at least relatively independent of the individual percipient? In other words, is Kant’s position subjectivism or phenomenalism? Kant here alternates between these positions. This fourth paragraph is coloured by his phenomenalism, whereas in the immediately following fifth paragraph his subjectivism gains the upper hand. The taste of wine, he there states, is purely subjective, because dependent upon the particular constitution of the gustatory organ on which the wine acts. Similarly, colours are not properties of the objects which cause them.

“They are only modifications of the sense of sight which is affected in a certain manner by the light.... They are connected with the appearances only as effects accidentally added by the particular constitution of the sense organs.”

Space, on the other hand, is a necessary constituent of the outer objects. In contrast to the subjective sensations of taste and colour, it possesses objectivity. This mode of distinguishing between space and the matter of sense implies that extended objects are not mere ideas, but are sufficiently independent to be capable of acting upon the sense organs, and of thereby generating the sensations of the secondary qualities.

Kant, it must be observed, refers only to taste and colour. He says nothing in regard to weight, impenetrability, and the like. These are revealed through sensation, and therefore on his view ought to be in exactly the same position as taste or colour. But if so, the relative independence of the extended object can hardly be maintained. Kant’s distinction between space and the sense qualities cannot, indeed, be made to coincide with the Cartesian distinction between primary and secondary qualities.

A second difference, from Kant’s point of view, between space and the sense qualities is that the former can be represented a priori, in complete separation from everything empirical, whereas the latter can only be known a posteriori. This, as we have seen, is a very questionable assertion. The further statement that all determinations of space can be represented in the same a priori fashion is even more questionable. At most the difference is only between a homogeneous subjective form yielded by outer sense and the endlessly varied and consequently unpredictable contents revealed by the special senses. The contention that the former can be known apart from the latter implies the existence of a pure manifold additional to the manifold of sense.

=Fifth Paragraph.=--In the next paragraph Kant emphasises the distinction between the empirical and the transcendental meanings of the term appearance. A rose, viewed empirically, as a thing with an intrinsic independent nature, may appear of different colour to different observers.

“The transcendental conception of appearances in space, on the other hand, is a Critical reminder that nothing intuited in space is a thing in itself, that space is not a form inhering in things in themselves ... and that what we call outer objects are nothing but mere representations of our sensibility, the form of which is space.”

In other words, the distinction drawn in the preceding paragraph between colour as a subjective effect and space as an objective existence is no longer maintained. Kant, when thus developing his position on subjectivist lines, allows no kind of independent existence to anything in the known world. Objects as known are mere Ideas (blosse Vorstellungen unserer Sinnlichkeit), the sole correlate of which is the unknowable thing in itself. But even in this paragraph both tendencies find expression. “Colour, taste, etc., must not rightly be regarded as properties of things, but only as changes in the subject.” This implies a threefold distinction between subjective sensations, empirical objects in space, and the thing in itself. The material world, investigated by science, is recognised as possessing a relatively independent mode of existence.

=Substituted Fourth Paragraph of second edition.=--In preparing the second edition Kant himself evidently felt the awkwardness of this abrupt juxtaposition of the two very different points of view; and he accordingly adopts a non-committal attitude, substituting a logical distinction for the ontological. Space yields synthetic judgments a priori; the sense qualities do not. Only in the concluding sentence does there emerge any definite phenomenalist implication. The sense qualities, “as they are mere sensations and not intuitions, in themselves reveal no object, least of all [an object] a priori.” The assertion that the secondary qualities have no ideality implies a new and stricter use of the term ideal than we find anywhere in the first edition--a use which runs counter to Kant’s own constant employment of the term. On this interpretation it is made to signify what though subjective is also a priori. Here, as in many of the alterations of the second edition, Kant is influenced by the desire to emphasise the points which distinguish his idealism from that of Berkeley.

THE TRANSCENDENTAL AESTHETIC