Kant’s Critique of Pure Reason: Schematism (Transcendental Analytic Book II)

Contents:

1. Pure Concepts vs. Images
2. The Difficulty This Creates and the Need for Schemata
3. What is a Schema?
4. Schematism and the Categories

Following upon the two proofs of the necessity of the categories for knowledge (Transcendental Deductions A and B), Kant returns to focus more particularly upon the issue of how the categories may be related to our sense experience. He introduces a difficulty in terms that emerge from the conceptual background of Euclidean geometry: the impossibility of forming an image that exactly corresponds to the determination of a pure concept. This was a problem with which Plato was familiar. At a certain point, the reality of images must lead into a higher reality of pure Forms if Knowledge is to be attained.

Pure Concepts vs. Images

A classic example may serve to illuminate the difficulty. A Euclidean isosceles triangle is defined as a figure having at least two equal sides. Given the definition, it is a simple matter of deciding if, for example, a triangle without equal sides is isosceles or not. Or rather, it remains a simple matter in cases where that determination is a purely logical one. What if the question were left up to empirical verification? It would become a matter of measurement. In some cases, the determination would be an easy one, but in others it might be a matter of carrying out measurement with increasingly sophisticated and precise equipment–itself also material in nature and thus limited in its own accuracy. However, in principle, even a flawless measurement tool would not suffice to yield a true determination of whether the sides are exactly equal.

Suppose they are equal to the billionth of a millimeter. Is there any guarantee that they will continue to be equal at the next increment or at the trillionth? In principle, the process of verification must proceed infinitely, long past the point where the measurement would suffice for most practical purposes, but never quite far enough to suffice for the ideal determination that the two sides (or angles for that matter) are truly equal. In platonic terms, the “reality” of an isosceles triangle is an ideal one. Further examples might be drawn from the case of parallel lines or even the definition of a point, which, for the purposes of geometry, is considered to have no dimensions and no parts. This definition leads to the further difficulty of how such a basic component of geometrical objects can have any real existence at all. After all, lines and figures are made up of points!

The Difficulty This Creates and the Need for Schemata

The way out is simply to stop thinking of such objects as empirical objects at all, and rather get used to the idea that they are ideal objects that perhaps paradoxically represent the reality of all such objects insofar as definitions such as the Euclidean ones can be taken to represent anything at all. Kant enters this same kind of theoretical space in discussing the pure categories of thought. He takes his categories to be “ideal” in the same sense:

pure concepts of the understanding being quite heterogeneous from empirical intuitions, and indeed from all sensible intuitions, can never be met with in any intuition. For no one will say that a category, such as that of causality, can be intuited through sense and is itself contained in appearance.”

B176-177 (See also B180 where he discusses the concept of a triangle in general and B187)

This means that there is a gulf, as it were, a gap, between the ideality of the pure concepts and senses. It raises the question of how the pure concepts can ever be applied to images if they are to be considered so fundamentally different. As Kant puts it,

How is the subsumption of intuitions under pure concepts, the application of a category to appearances, possible?

B177

His solution is to suppose something intermediate between images and pure concepts; something that, in a sense, participates in both kinds of internal experience so as to connect the one with the other. Thus, we arrive at his notion of a schema.

What is a Schema?

In simple terms, a schema is the imaginative presentation of a rule for matching an image with a concept. It is important to notice that it corresponds to a genral rule for the formation or synthesis of particular images rather than a determinate image itself. It is, in a loose sense, a rule for the imagination that provides a guideline for the general “form” such an image should take if it is to correspond to a concept. Kant’s discussion of the schema of a triangle may prove illuminating here:

No image could ever be adequate to the concept of a triangle in general. It would never attain that universality of the concept which renders it valid of all triangles, whether right-angled, obtuse-angled, or acute-angled; it would always be limited to a part only of this sphere. The schema of a triangle can exist nowhere but in thought. It is a rule of synthesis of the imagination, in respect to pure figures in space.

B180

Kant also writes that a schema carries with it some “specific universal concept” that acts as a “rule for the determination of our intuition” (B180) and specifies that it does so “without limitation to any single determinate figure, such as experience…actually presents” (B180). His example shows that while the schema of a triangle should involve, for example, producing a three-sided image as a rule of its formation and determination, it cannot itself be associated with a determinate image.

It suggests, in other words, that while it is by means of an image that we become aware of our capacity to form a schema, the schema itself should be associated with the formal conditions under which it is possible for the imagination to form an image of a triangle at all. The capacity to schematize is intermediate in this sense. Kant provides us with the further example of a dog in B180. The rule whereby one might be able to “see” the concept “dogness” in an image must involve a determination of the imagination in accordance with a correct (even if partial) understanding of its general concept:

The concept ‘dog’ signifies a rule according to which my imagination can delineate the figure of a four-footed animal in a general manner, without limitation to any single determinate figure…. This schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze.

B180-181

Ultimately, the pure concepts of the understanding must find themselves expressed in such images, whether they are manifested as unity, plurality, totality, reality, negation, or limitation, or any others that might, for instance, be applied to in the imaginative “form” or schema of a dog (or triangle, or any other).

Schematism and the Categories

The categories simply present the conditions under which we may form concepts. That they may be “encoded” within images or objects makes them potential objects of contemplation themselves. It is only by means of such encoding in images that they have any objective reality for us and it is only because they appear in images by means of the schematizing function of the imagination that they can be taken up by the understanding at all. At the close of the section on schematism Kant writes,

The categories, therefore, without schemata, are merely functions of the understanding for concepts; and represent no object. This [objective] meaning they acquire from sensibility, which realises the understanding in the very process of restricting it.

B187

They would, in other words, be incapable of acting within us other than as a capacity to understand in a formal, abstract manner, “signifying only the bare unity of the representations” (B186). But, Kant writes, in such a case, “The pure concepts can find no object, and so can acquire no meaning which might yield a concept of some object.” Here Kant presents a case for supposing that conceptualization requires an image. The idea that all thought requires an object goes back to Aristotle, but Kant has given the idea much greater precision and has shown why such a requirement might be necessary.