CHAPTER I. The Schematism of Pure Concepts of Understanding[1103]

THE SCHEMATISM OF PURE CONCEPTS OF UNDERSTANDING

The more artificial aspect of Kant’s argument again appears in the reason which he assigns for the existence of a problem of schematism, namely, that pure concepts, and the sensuous intuitions which have to be subsumed under them, are completely opposite in nature. No such explanation can be accepted. For if category and sensuous intuition are really heterogeneous, no subsumption is possible; and if they are not really heterogeneous, no such problem as Kant here refers to will exist. The heterogeneity which Kant here asserts is merely that difference of nature which follows from the diversity of their functions. The category is formal and determines structure; intuition yields the content which is thereby organised. Accordingly, the “third thing,” which Kant postulates as required to bring category and intuition together, is not properly so describable; it is simply the two co-operating in the manner required for the possibility of experience. Kant’s method of stating the problem of schematism is, however, so completely misleading, that before we can profitably proceed, the various strands in his highly artificial argument must be further disentangled. This is an ungrateful task, but has at least the compensating interest of admirably illustrating the kind of influence which Kant’s logical architectonic is constantly exercising upon his statement of Critical principles.

The architectonic has in this connection two very unfortunate consequences. It leads Kant to describe schematism as a process of subsumption, and to speak of the transcendental schema as “a third thing.” Neither assertion is legitimate. Schematism, properly understood, is not a process of subsumption, but, as Kant has already recognised in A 124, of synthetic interpretation. Creative synthesis, whereby contents are apprehended in terms of functional relations, not subsumption of particulars under universals that are homogeneous with them, is what Kant must ultimately mean by the schematism of the pure forms of understanding. A category, that is to say, may not be viewed as a predicate of a possible judgment, and as being applied to a subject independently apprehended; its function is to articulate the judgment as a whole. The category of substance and attribute, for instance, is the form of the categorical judgment, and may not be equated with any one of its single parts.

Thus the criticisms which we have already passed upon Kant’s mode of formulating the distinction between formal and transcendental logic, are no less applicable to the sections now before us. The terminology which Kant is here employing is borrowed from the traditional logic, and is out of harmony with his Critical principles.

Kant’s description of the schema as a third thing, additional to category and intuition, and intermediate between them, is also a result of his misleading mode of formulating his problem. What Kant professes to do is to interpret the relation of the categories to the intuitional material as analogous to that holding between a class concept and the particulars which can be subsumed under it. This is implied in his use of the plate and circle illustration. But as the relation holding between categories and the material of sense is that of form and matter, structure and content, the analogy is thoroughly misleading. As all content, strictly so called, falls on the side of the intuitional material, there is no content, i.e. no quality or attribute, which is common to both. And thus it happens that the inappropriateness of the analogy which Kant is seeking to enforce is ultimately the sole ground which he is able to offer in support of his description of the schema as “a third thing.”

“Now it is clear [!] that there must be a third thing, which is homogeneous on the one hand with the category and on the other with the appearance, and which thus makes the application of the one to the other possible.”

On the contrary, the true Critical teaching is that category and intuition, that is to say, form and content, mutually condition one another, and that the so-called schema is simply a name for the latter as apprehended in terms of the former.

But there is a further complication. Kant, as we have already observed, defines judgment as being

“...the faculty of subsuming under rules, i.e. of distinguishing whether something does or does not stand under a given rule (casus datae legis).”

Now this view of judgment really connects with the syllogism, not with the proposition. As Kant states in his Logic, there are

“...three essential elements in all inference: (1) a universal rule which is entitled the major premiss; (2) the proposition which subsumes a cognition under the condition of the universal rule, and which is entitled the minor premiss; and lastly, (3) the conclusion, the proposition which asserts or denies of the subsumed cognition the predicate of the rule.”

Regarded in this way, as the application of a rule, subsumption is more broadly viewed and becomes a more appropriate analogy for the relation of category to content. And obviously it is this comparison that Kant has chiefly in mind in these introductory sections. For only when the subsumption is that of a particular instance under a universal rule, can the necessity of a mediating condition be allowed.

Such, then, are the straits to which Kant is reduced in the endeavour to hold loyally to his architectonic. He has to identify the two very different kinds of subsumption which find expression in the proposition and in the syllogism respectively; and when his analogy between logical subsumption, thus loosely interpreted, and synthetic interpretation, proves inapplicable, he uses the failure of the analogy as an argument to prove the necessity of “a third thing.” On his own Critical teaching, as elsewhere expounded, no such third thing need be postulated. Even the definitions which he proceeds to give of the various schemata do not really support this description of them.

But though Kant’s method of introducing and expounding the argument of this chapter is thus misleading, the contents themselves are of intrinsic value, and have a threefold bearing: (a) on the doctrine of productive imagination; (b) on the relation holding between image and concept; and (c) on the nature of the categories in their distinction from the pure forms of understanding.

(a) Kant gives definite and precise expression to the two chief characteristics of the productive imagination, namely, that it deals with an a priori manifold of pure intuition and that it exercises a “hidden art in the depths of the human soul.” Kant’s description of the schema as “a third thing,” at once intellectual and sensuous, seems to be in large part due to the transference to it of predicates already applied to the faculty which is supposed to be its source. The distinction between the transcendental schema and the particularised image is also given as analogous to that between the pure and the empirical faculties of imagination. In A 141-2 = B 180-1, Kant speaks of the empirical faculty of productive imagination, and so is led, to the great confusion of his exposition, though also to the enrichment of his teaching, to allow of empirical as well as of transcendental schemata, and thus contrary to his own real position to recognise schemata of such empirical objects as dog or horse--a view which empirical psychology has since adopted in its doctrine of the schematic image. This passage was doubtless written at the time when he was inclining to the view that the empirical processes run parallel with the transcendental. Kant’s final view is that empirical imagination is always reproductive. This brings us, however, to our second main point.

(b) Kant makes a statement which serves as a valuable corrective of his looser assertions in other parts of the Critique. Five points set after one another, thus,....., form an image of the number five. The schema of the number five is, however, of very different nature, and must not be identified with any such image. It is

“...rather the representation of a method whereby a multiplicity [in this case five] may be represented in an image in accordance with a certain concept, than this image itself....”

This becomes more evident in the case of large numbers, such as a thousand. The thought or schema of the number remains just as clear and definite as in the case of smaller numbers, but cannot be so adequately embodied and surveyed in a concrete image.

“This representation of a general procedure of imagination in providing its image for a concept, I name the schema to this concept.”

But even in the simplest cases an image can never be completely adequate to the concept. The image of a triangle, for instance, is always some particular triangle, and therefore represents only a part of the total connotation. As the schema represents a universal rule of production in accordance with a concept, it resembles the concept in its incapacity to subsist in an objective form. Images become possible only through and in accordance with schemata, but can never themselves be identified with them. Schemata, therefore, and not images--such is the implied conclusion--form the true subject-matter of the mathematical sciences. Images are always particular; schemata are always universal. Images represent existences; schemata represent methods of construction.

There are three criticisms which must be passed upon this position. In the first place, the selection of the triangle as an illustration tends to obscure the main point of Kant’s argument. As there are three very different species of triangle, the concept triangle is a class concept in a degree and manner which is not to be found in the concepts, say, of the circle or of the number five. So that while Kant may seem to be chiefly insisting upon the inadequacy of the image to represent more than a part of the connotation of the corresponding concept, his real intention is to emphasise that the schema expresses the conceptual rule whereby, even in images that cover the whole connotation, the true meaning of the image can alone be determined.

Secondly, the above definition of the schema as being “the representation of a general procedure of imagination in providing an image for a concept” is obviously bound up with Kant’s view of it as “a third thing,” additional to the concept, and as intermediate between it and the image. But as we have already found occasion to note, in discussing Kant’s doctrine of the “construction” of mathematical concepts, this threefold distinction is out of harmony with his Critical principles. It results from his retention of the traditional view of the concept as in all cases a mere concept, i.e. an abstracted or class concept. In defining the schema Kant is defining the true nature of the concept as against the false interpretation of it in the traditional class-theory; he misrepresents the logic of his own standpoint when he interpolates a third kind of representation intermediate between the concept and the image. The concept ‘triangle,’ as a concept, is (to employ Kant’s own not very satisfactory terms) the representation of the method of constructing a certain type of object; and the only other mode of representing this kind of object is the image. There may, indeed, as Kant has himself suggested, be a species of image that may be entitled schematic; but if that be identified with a blurred or indeterminate or merely symbolic form of representation, it can have nothing in common with the transcendental or conceptual schema, save the name.

Thirdly, the entire discussion of the nature of the schemata of “sensuous concepts” and of their relation to the sense image, is out of order in this chapter; and however valuable in itself, bewilders the reader who very properly assumes for it a relevancy which it does not possess. The pure concepts of the understanding, whose schemata Kant is endeavouring to define, are altogether different in nature from sensuous representations, and can never be reduced in any form or degree to an image. They are wholly transcendental, representing pure syntheses unified through categories in accordance with the form of inner sense. This, however, brings us to our last main point.

(c) Kant’s manner of employing the term category is a typical example of his characteristic carelessness in the use of his technical terms. Sometimes it signifies the pure forms of understanding. But more frequently it stands for what he now, for the first time, entitles schemata, namely, the pure conceptual forms as modified through relation to time. To take as examples the two chief categories of relation. The first category of relation, viewed as a form of the pure understanding, is the merely logical conception of that which is always a subject and never a predicate. The corresponding schema is the conception of that which has permanent existence in time; it is not the logical notion of subject, but the transcendental conception of substance. The pure logical conception of ground and consequence is similarly distinguished from the transcendental schema of cause and effect.

This contrast is of supreme importance in the Critical philosophy, and ought therefore to have been marked by a careful distinction of terms. Had Kant restricted the term category to denote the pure forms, and invariably employed the term schemata to signify their more concrete counterparts, many ambiguities and confusions would have been prevented. The table of categories, in its distinction from the table of logical forms, would then have been named the table of schemata, and the definitions given in this chapter would have been appended to it, as the proper supplement to the metaphysical deduction, completing it by a careful definition of each separate schema. For what Kant usually means when he speaks of the categories are the schemata; and the chapter before us therefore contains their delayed definitions. As Kant has constantly been insisting, and as he again so emphatically teaches in this chapter, the pure forms of understanding, taken in and by themselves, apart from the forms of intuition, have no relation to any object, and are mere logical functions without content or determinate meaning.

From this point of view the misleading influence of Kant’s architectonic may again be noted. It forces him to preface his argument by introductory remarks which run entirely counter to the very point which he is chiefly concerned to illustrate and enforce, namely, the inseparability of conception and intuition in all experience and knowledge. He does, indeed, draw attention to the fact that the conditions which serve to realise the pure concepts of understanding also at the same time restrict them, but it is with their empirical employment that he is here chiefly concerned.

Caird’s mode of expounding Kant’s doctrine of schematism may serve as an example of the misleading influence of Kant’s artificial method of introducing his argument. As Caird accepts Kant’s initial statements at their face value, he is led to read the entire chapter in accordance with them, and so to interpret it as being a virtual recantation of the assumptions which underlie the statement of its problem. The truer view would rather seem to be that the introduction is demanded by the exigencies of Kant’s architectonic, and therefore yields no true account either of the essential purpose of the chapter or of its actual contents. Cohen not unjustly remarks that

“...recent writers are guilty of a very strange misreading of Kant when they maintain, as if in opposition to him, a thought to which his doctrine of schematism gives profound expression, namely, that intuition and conception do not function independently, and that thought, and still more knowledge, is and must always be intuitive.”

Cohen fails, however, to draw attention to the cause of the misunderstanding for which Kant must certainly share the blame. Riehl, while adopting a somewhat similar view to that here given, traces Kant’s misleading mode of stating the problem to his holding a false view of the universality of the concept. Such criticism of Kant, like that passed by Caird, is in many respects justified, but the occasion upon which the admonition is made to follow would none the less seem to be ill-chosen.

It may be asked why Kant in this chapter so completely ignores space. No really satisfactory answer seems to present itself. It is true that time is the one universal form of all intuition, of outer as well as of inner experience. It is also true that, as Kant elsewhere shows, consciousness of time presupposes consciousness of space for its own possibility, and so to that extent may be regarded as including the latter form of consciousness within itself. Nevertheless Kant’s concentration on the temporal aspect of experience is exceedingly arbitrary, and results in certain unfortunate consequences. Owing to the manner in which Kant envisages his problem he is bound, indeed, to lay the greater emphasis upon time, but that need not have involved so exclusive a recognition of its field and function. Possibly Kant’s very natural preoccupation with his new and revolutionary doctrines of inner sense and productive imagination has something to do with the matter.

Though the definitions given of the various schemata, especially of those of reality and existence, raise many difficulties, consideration of them must be deferred. They can be properly discussed only in connection with the principles which Kant bases upon them. Only one further point calls for present remark. Kant does not give a schema for each of the categories. In the first two groups of pure conceptual forms, those of quantity and of quality, he gives a schema only for the third category in each case. Number is strictly not the schema of quantity as such, but of totality. The schema of quality is a definition only of limitation. This departure from the demands of strict architectonic is made without comment or explanation of any kind. Kant delights to insist upon the confirmation given to his teaching by the fulfilment of architectonic requirements; he is for the most part silent when they fail to correspond. This architectonic was a hobby sufficiently serious to yield him keen pleasure in its elaboration, but was not so vital to his main purposes as to call for stronger measures when shortcomings occurred.

In concluding this chapter Kant draws attention to the fact that the sensuous conditions which serve to realise the pure concepts also at the same time restrict their meaning. Their wider meaning is, however, of merely logical character. Their function, as pure concepts, lies solely in establishing unity of representation; they do not therefore suffice to yield knowledge of any object. Objective application “comes to them solely from sensibility.” In these statements Kant expounds one of his fundamental doctrines, but in a manner which does less than justice to the independent value of pure thought. As he elsewhere teaches, it is not sense that sets limits to understanding; it is the pure forms of thought that enable the mind to appreciate the limited and merely phenomenal character of the world experienced.