• Schematism and Subsumption
• The Nature of the Transcendental Schematism
• The Categories and Their Schema: The Problem of Schema Judgments
  • The Problem Defined
  • The Modal Categories and Their Schemata
  • Substance, Causality and Their Schemata
• The Schemata and the Principles

In Schematism of the Pure Concepts of the Understanding Kant explicitly deals with the products of the transcendental synthesis of the imagination, which unites the pure concepts with pure intuition. This chapter explains what is transcendental schema and to delineate the function of schema in the overall argument of Transcendental Analytic. Also the chapter aims to explain the principle of synthetic judgments:

All synthetic judgments of theoretical knowledge are only possible through the relation of a given concept to an intuition. If the synthetic judgment is experiential, the underlying intuition must be empirical; if it is a judgment a priori, the intuition must be pure.

Schematism and Subsumption

The official task of Schematism is to explain how the categories can apply to what is sensibly given to the mind, the appearances. Kant states that the Schematism accomplishes this task by providing the “sensible conditions under which alone pure concepts of the understanding can be employed”s (A136/B175). The basic idea here is that apart from such conditions (schemata) the pure concepts of the understanding have a “logical use” but not a “real use”.

The specify these conditions is also to specify what is being claimed about the phenomenal world when it is claimed that particular categories apply to it.

Kant is using “subsumption” as a synonym for “application”.

The problem for categories and appearances is the heterogeneity of the two elements to be brought into connection. This heterogeneity is due to the fact that the pure concepts of the understanding, in contrast to “pure sensible” are derived from the very nature of the understanding. So they have no direct relation to intuition. Yet as Transcendental Deduction demonstrates, they do relate to intuition and therefore to appearances. In the present case, where the pure concepts of the understanding are the universal rules, there is a need for some analogue of the conditions of the rule, under which appearances can be “subsumed”. This analogue will turn out to be the transcendental schema, the infamous “third thing”, which makes possible the mediation between category and appearance.

When Kant posed the question how synthetic judgments are possible a priori in the introduction to Critique, he alluded mysteriously to an “unknown = x” (B13), which is needed to ground the connection between the concepts that is asserted in the judgment. When he returned to the Transcendental Analytic, this “unknown = x” is more precisely characterized as a transcendental schema.

The Nature of the Transcendental Schematism

What then is transcendental schema? There are variety of answers, but we may find them characterized in following ways:

1. As a “third thing” or “mediating representation” which is “homogeneous on the one hand with the category and on the other hand with the appearance, and which makes the application of the former to later possible”. (A138/B177)

2. As a “transcendental determination of time”, which is homogeneous with both category and appearance, and therefore “mediates the subsumption of the appearances under the category” (A139/B178)

3. As the “formal and pure condition of sensibility to which the concept of understanding is restricted” (A140/B179)

4. As the “representation of a universal procedure of imagination in providing an image of a concept” (A140/B179-80)

5. As “simply the pure synthesis, determined by a rule of that unity, in accordance with concepts, to which the category gives expression” It is further described as a “transcendental product of imagination, a product which concerns the determination of inner sense in general according to conditions of its form in respect of all representations, so far as these representations are to be connected a priori in one concept in conformity with the unity of apperception” (A142/B181)

6. As “the true and sole conditions under which these concepts obtain relation to objects and so posses significance” (A146/B185)

7. As “nothing but a priori determination of time in accordance with rules” (A145/B184)

8. As “only the phenomenon, or sensible concept, of an object in agreement with the category” (A146/B186)

We can argue that a transcendental schema is to be construed as a pure intuition, and that this is compatible with all of the formulations with the possible exception of the fourth. Kant did this in Critique of Practical Reason and Critique of Judgment.

There are two senses of pure intuition discussed in chapter 5. Space is a mere form of intuition or sensibility and an actual representation of space which is a formal intuition. Only the first one can be said to be purely sensible.

The conception of a determinate pure intuition is as central to Kant’s thought as the doctrine of transcendental synthesis of the imagination, from which it is in fact inseparable.

The pure(formal) intuition that is produced by such an activity is both sensible and intellectual, and it’s both universal and particular.

Therefore we can take transcendental schema as a pure intuition in the second sense (formal intuition). Most of the other characterizations are equivalent to this one.

We must now consider whether it is compatible with second one “transcendental determination of time”. “Transcendental determine an intuition” would simply be one that is governed by an a priori concept, which is precisely what Kant indicates in the passage. Transcendental determination of time must be a conceptualization of time in accordance with an a priori concepts, which refers time to an object while also providing objective reality for the concept involved. Transcendental determination of time, as products of the transcendental synthesis of the imagination, are universal and necessary (a priori) characteristics of objective temporal order. This would explain their homogeneity with appearances, since all appearances are contained in such an order and with the pure concepts, since these concepts provide the rules whereby this order is determined. The situation is complicated by the fact that time (and space) is not itself an object and cannot be perceived, therefore a transcendental determination of time must be regarded as a universal and necessary characteristics of things in time. For example, the schema of the concept of substance is “permanence of the real in time” (A143/B183). Kant argues in the First Analogy that it is only by reference to something permanent that change, and with it the passage of time, can be intuited.

But does this undermine the claim that transcendental schema is a pure intuition? How can things in time be termed a pure intuition? To continue the substance example, permanence is a concept which refers to any number of possible objects, this is why transcendental schemata are construed as concepts and equated with the schematized categories.

Transcendental schema is a sensible concept. Although Kant begins with the radical separation of sensibility an understanding, the very heart of his account of knowledge consists in the claim that any cognition contains both elements. The intuitive element must be located in the irreducibly sensible component of the representation. To think of something permanent is just to think of it as lasting through time, so the concept presupposes the intuition of time, which it determines.

The Categories and Their Schema: The Problem of Schema Judgments

Kant also provides a catalogue of the particular schemata that are connected with the various categories. This catalogue contains a set of claims of “schema judgments”, a judgment that asserts that a certain schema pertains to a certain category. For example, the schema of substance is permanence of the real in time.

The Problem Defined

Our present concern is with the nature and justification of the schema judgments which asserts this connection. The heterogeneity of the intellectual and sensible rules out the possibility that the transcendental schemata is analytic. And other possibilities are ruled out as well, and we are left with the only alternative that schema judgment is both synthetic and a priori. We can also see positively that schema judgments must be classified as synthetic a priori.

The synthetic a priori character of schema judgments indicates that they require a “deduction” or justification. However Kant does not offer a trace of justification for it. We thus find ourselves led to the paradoxical result that the whole problem of the synthetic a priori breaks out within the doctrine of the Schematism, even though this doctrine is intended as an essential step in the resolution of this problem.

An initial plausible way of dealing with this problem is through the familiar distinction between pure and schematized categories. One might argue that since the pure concepts stand in no connection with time, no schemata can be supplied for them; but since the schematized categories already stand in connected with time, the connected between these categories and their schemata can be determined analytically. For instance, if we define the pure category of substance as “the concept of the synthesis of subject and predicate” and the schematized category as “the concept of the synthesis of the permanent and the changing in time”, then while it would be impossible to provide a schema for the former, it would be a trivial matter to do so for the latter.

The problem is that it only pushes the problem back on step, to the connection between the pure and the schematized categories. It is easy to show that the connection must be both synthetic and a priori, so we come back to our original question.

We must find the “ground” of the synthetic a priori judgment connecting category and schema. In order to do that, we must go beyond what Kant explicitly tell us. Two points are of particular relevance: (1) that the categories, as rules for the transcendental synthesis of the imagination, serve to determine time; and (2) that the schema is in each case the product of such determination. Assuming these premises, it is possible to construct a plausible case for the connection between at least some of the categories and their designated schemata.

The Modal Categories and Their Schemata

The modal categories and their respective schemata are possibility (“the agreement of the synthesis of different representations of time in general”), actuality (“existence in some determinate time”) and necessity (“existence of an object at all times”). Kant comments that “The schema of modality and of its categories is time itself as the correlate of the determination whether and how and object belongs to time” (A145/B185). They tell us how and in what specific terms, categorical thinking must proceed. They can be regarded as rules for the application to appearances of rules. It is more accurate to characterize them as “transcendental products” that can serve or function as rules.

Real possibility is defined in terms of the agreement of the thought of an object with the “conditions of time in general”, to be really possible means to be possible in or over a period of time.

At this point the question naturally arises, what has happened to space? Transcendental determination of time are properties not of time itself, but of things in time, so it does not preclude the things being in space.

The connection between actuality and its schema (existence at a determinate time) seems obvious. The point is simply that for anything to be regarded as actual in the “real” or empirical sense, it must be assignable a determinate location in time.

More serious problem arise concerning the schema of necessity, which Kant defines as “the existence of an object at all times”. First, it seems false that when we say of some object or state of affairs that it is necessary, what we really mean or presuppose is that it exists at all times. Second, existence at all times is the schema of substance and in the Postulates Kant says explicitly that “it is not the existence of things that we can know to be necessary, but only the existence of their state” (A227/B279). Finally, in connection with the same point Kant links the “material necessity in existence”, which he contrasts with the “merely logical necessity in the connection of concepts”. These considerations suggests that Kant does not mean quite what he appears to mean. We can follow Paton, who characterizes the schema of necessity as “existence in relation to the whole of time”. Given this, we can take Kant as claiming that it is the product of a causal chain, which, since it can have no first member, must itself exists throughout all of time.

Substance, Causality and Their Schemata

The “deduction” of the schema of the relational categories is based upon the same principle that was used in modal categories: the schema must provide a translation into temporal terms of the purely judgmental or conceptual sense that pertains to the pure concept.

We have already seen that pure concept of substance is the concept of something that must always be considered as subject and never as predicate of something else. Our first concern is to determine how such a thought can be specified in temporal terms.

We begin by asking for the necessary condition under which we can say of something temporal that it is real subject or owner of properties rather than a merely logical subject of predicates. The most obvious candidate for such a necessary condition for such a necessary condition is re-identifiability; only something that is re-identifiable throughout a change of states can be distinguished from one or more of these states and considered to be their real subject. In order to be re-identifiable, the subject must continue to exist throughout that period. So at least a relative permanence is a necessary condition for real subject.

However, Kant assigns real permanence, not relative permanence to the concept of substance. The problem is to see if it is possible to justify the stronger claim. The key lies in the distinction between the strictly judgmental concept of something that is fixed as the subject of a given judgment and the “pure” ontological concept of substance, which is the concept of something that for every judgmental context must be conceived always as subject and never as predicates. The schema of substance is required for the conception in temporal terms of the latter (the ontological concept), but not the former. The argument here consists of the extension of the line of reasoning sketched in the preceding paragraph. Just as re-identifiability throughout a certain period of time, and thus relative permanence, is a condition that must be met by anything temporal that is to serve as “real subject” to which properties are attached, so re-identifiability throughout all time, and thus absolute permanence, is a condition that must be met by anything temporal that is always to be conceived as subject and never as property of anything else.

Kant equates the pure concept of causality with the relation of ground and consequent. This is the concept of the logical sequence of thoughts in a judgment, and it is connected with the hypothetical form of judgment. It is a rule for the sequential ordering of thoughts that are connected together in a hypothetical judgment. The schema of causality is defined as “the succession of the manifold, insofar as that succession is subject to a rule” (A144/B187). Simply put, the schema is rule-governed succession.

The essential point here is that the pure concept serves as an ordering rule; it determines the sequence of thoughts in a judgment as a necessary sequence. The schema must provide a representation of a temporal sequence which exhibits the same or an analogous necessity. But to represent a sequence of states of affairs or events in time as necessary is simply to think of it as governed by a rule of the form: if A at t1 then B at t2. Equivalently, it is to think of the order as irreversible. Rule-governed succession or irreversibility is the schema of the pure concept of causality. As we will see in chapter 10, Kant’s answer to Hume consists in the demonstration of the claim that this schema is also the condition under which alone we can experience objective succession.

The Schemata and the Principles

The transcendental schemata are not only the sensible conditions which give “real” significance to, and restrict the scope of, the pure concepts, they are also conditions of the determination of appearances in time, and thus of the possibility of experience. It is precisely because of this dual function as “condition” that they can be said to mediate between the pure concepts and appearances.

The first sense of “condition” is the main focus of the Schematism chapter. The second sense of “condition” is operative in the Principles of Pure Understanding. With the exception of the modal Principles, each of these Principles can be characterized as a synthetic a priori judgment which asserts that a particular schema functions as a necessary condition of the possibility of experience. Kant emphasizes this in the end of his general discussion of Analogies of Experience, he suggests that these Analogies appearances are subsumed “not simply under the categories, but under their schemata” (A181/B224).

B224 also mentions “we are justified in combining appearances only according to what is no more than an analogy with the logical and universal unity of concepts”. It is been frequently recognized that here Kant is introducing a second sense of the term “analogy”. In this sense, the reference of the term is not limited to the Analogies of Experience.

The first sense of “analogy” is equivalent to the mathematical terms “ratio” and “proportion”. Kant justifies the choice of the term on the grounds that the schemata involved in these Principles correspond to the relational categories and that the specific function of these Principles is to determine the relation of appearances to one another in a single time.

The cryptic discussion of second sense seems like a mere afterthought on Kant’s part. Nevertheless it is of considerable significance for the understanding of the synthetic a priri character of the Principles. The basic analogy that Kant has in mind is between the pure concepts and their schemata. It means the schemata provide translations into temporal terms of what is thought in the pure concepts. The ensuing analogy between category and principle is thus attributable to the fact that the Principles all make use of the schemata: namely, they subsume appearances under them. This is precisely what makes thees judgments both synthetic and a priori.

Category and schema (and therefore Principle) are merely “analogous” rather than being identical is a direct consequence of Kant’s transcendental distinction between sensibility and understanding. To deny the transcendental nature of the distinction is to deny the basis for any real distinction between the pure concept and its sensible counterpart (the schema). This denial underlies the conflation, typical of rationalistic philosophy, of the temporal relation of cause and effect with the logical relation of ground and consequent. More generally, it also leads to the “transcendental illusion” by means of which these pure concepts are seen as themselves the source of metaphysical principles that are applicable to “real objects”. Since such principles would be based on nothing more than an analysis of what is logically necessary for the unity of thought, they would be analytic. An example of such an analytic principle is the Leibnizian principle of sufficient reason.

Equally significant for Kant’s position is the claim that because of the “analogy” between pure concept and schema, there is an “analogy” between pure concept and Principle.

Given this interpretation of the Principles, there can be little remaining doubt concerning their synthetic a priori character. I have also tried to argue the possibility of these judgments, and therefore the possibility of a “metaphysic of experience” rests upon the priori possibility of specifying the temporal “analogues” of the categorial rules provided by the pure concepts of the understanding. In his letter to Reinhold, Kant suggests that it is here that we find the real beginning of the account of synthetic a priori knowledge.

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