In the preface to CPR/B Kant cites Thales who, from the figure of one isosceles triangle, in order to discover the properties of all isosceles triangles, does not follow step by step what he sees, but has to produce, to construct the isosceles triangle in general.
The schema is not an image, because the image is a product of the reproductive imagination, while the schema of sensible concepts (and also of figures in space) is a product of the pure a priori capacity to imagine, “a monogram, so to say” (CPR/B: 136). If anything it could be said that the Kantian schema, more than what we usually refer to with the term “mental image” (which evokes the idea of a photograph) is similar to Wittgenstein’s Bild, a proposition that has the same form as the fact that it represents, in the same sense in which we speak of an iconic relation for an algebraic formula, or a model in a technical-scientific sense.
Perhaps, to better grasp the concept of a schema, we could appeal to the idea of the flowchart, used in computer programming. The machine is capable of “thinking” in terms of if … then go to, but a logical system like this is too abstract, since it can be used either to make a calculation or to design a geometrical figure. The flowchart clarifies the steps that the machine must perform and that we must order it to perform: given an operation, a possible alternative is produced at a certain juncture; and, depending on the answer that appears, a choice must be made; depending on the new response, we must go back to a higher node of the flowchart, or proceed further; and so on. The flowchart has something that can be intuited in spatial terms, but at the same time it is substantially based on a temporal progression (the flow), in the same sense in which Kant reminds us that the schemata are fundamentally based on time.
The idea of the flowchart seems to provide a good explanation what Kant means by the schematic rule that presides over the conceptual construction of geometrical figures. No image of a triangle that we find in experience—the face of a pyramid, for example—can ever be adequate to the concept of the triangle in general, which must be valid for every triangle, whether it be right-angled, isosceles, and scalene (CPR/B: 136). The schema is proposed as a rule for constructing in any situation a figure having the general properties triangles have (without resorting to strict mathematical terminology if we have, say, three toothpicks on the table, one of the steps that the schema would prescribe would be not to go looking for a fourth toothpick, but simply to close up the triangular figure with the three available).
Kant reminds us that we cannot think of a line without tracing it in our mind; we cannot think of a circle without describing it (in order to describe a circle, we must have a rule that tells us that all points of the line describing the circle must be equidistant from the center). We cannot represent the three dimensions of space without placing three lines perpendicular to each other. We cannot even represent time without drawing a straight line (CPR/B: 120, 21 ff.). At this point, what we had initially defined as Kant’s implicit semiotics has been radically modified, because thinking is not just applying pure concepts derived from a preceding verbalization, it is also entertaining diagrammatical representations, for example, flowcharts.
In the construction of these diagrammatical representations, not only is time relevant, but memory too. In the first edition of the first Critique (CPR/A: 78–79), Kant says that if, while counting, we forget that the units we presently have in mind have been added gradually, we cannot know the production of plurality through successive addition, and therefore we cannot even know the number. If we were to trace a line with our thought, or if we wished to think of the time between one noon and the next, but in the process of addition we always lost the preceding representations (the first parts of the line, the preceding parts in time) we would never have a complete representation.
Look how schematism works, for example, in the anticipations of perception, a truly fundamental principle because it implies that observable reality is a segmentable continuum. How can we anticipate what we have not yet intuited with our senses? We must work as though degrees could be introduced into experience (as if one could digitize the continuous), though without our digitization excluding infinite other intermediate degrees. As Cassirer (1918: 215) points out, “Were we to admit that at instant a a body presents itself in state x and at instant b it presents itself in state x′ without having travelled through the intermediate values between these two, then we would conclude that it is no longer the ‘same’ body. Rather, we would assert that the body at state x disappeared at instant a, and that at instant b another body in state x′ appeared. It results that the assumption of the continuity of physical changes is not a single result from observation but a presupposition of the knowledge of nature in general,” and therefore this is one of those principles presiding over the construction of the schemata.
13.4. Does the Dog Schema Exist in Kant?
So much for the schemata of the pure concepts of the intellect. But it so happens that it is in the very same chapter on schematism that Kant introduces examples that concern empirical concepts. It is not simply a question of understanding how the schema allows us to homogenize the concepts of unity, reality, inherence, subsistence, possibility, and so on, with the manifold of the intuition. There also exists the schema of the dog: “the concept of a dog indicates a rule, according to which my for imaginative capacity can universally trace the figure of a four-legged animal, without being restricted to either a unique particular figure supplied by experience, or to any possible image that I am able to portray in concrete” (CPR/B:136).
Right after this example, a few lines further on, Kant writes the famous sentence stating that this schematism of our intellect, which also concerns the simple form of appearances, is an art hidden in the depths of the human soul. Schematism is an art, a procedure, a task, a construction, but we know very little about how it works. Because it is clear that our analogy of the flowchart, which was useful in understanding how the schematic construction of the triangle takes place, doesn’t work as well for the dog.
What is certain is that a computer is able to construct the image of a dog, if it is provided with the appropriate algorithms. But if someone who had never seen a dog were to study the flowchart to see how it was constructed, they would have trouble forming a mental image of it (whatever a mental image may be). We would find ourselves once more faced with a lack of homogeneity between categories and intuition, and the fact that the schema of the dog can be verbalized as a four-legged animal only brings us back to the extreme abstractness of every predication by genus and specific differentia, without helping us distinguish a dog from a horse.
Deleuze (1963:73) reminds us that “the schema does not consist in an image, but in spatiotemporal relations that incarnate or realize some purely conceptual relations” (my emphasis), and this seems right as far as the schemata of the concepts of the pure intellect go. But it doesn’t seem sufficient when it comes to empirical concepts, since Kant was the first to tell us that to think of a plate we must resort to the image of the circle. While the schema of the circle is not an image but a rule to follow in constructing the image, nevertheless in the empirical concept of plate the constructability of its form should find a place somehow, and precisely in a visual sense.
We can only conclude that when Kant thinks of the schema of the dog he is thinking of something very close to what Marr and Nishishara (1978), in the field of modern cognitive sciences, called a “3D Model,” which is nothing but a three-dimensional schematization (through the composition and articulation of more elementary forms) of various objects that we are able to recognize. To put it plainly, the 3D model of a human being—thinking of it only in the form of cylindrical elements—is composed of a smaller cylinder attached to a longer cylinder, from which cylindrical joints branch off, corresponding to the upper and lower limbs, including the elbows and knees.
In the perceptual judgment the 3D model is applied to the manifold of experience, and an x is distinguished as a man and not as a dog. This should demonstrate how a perceptual judgment is not necessarily resolved into a verbal statement. In point of fact, it is based on the application of a structural diagram to the manifold of sensation. The fact that further judgments are required to determine the concept of man with all his possible characteristics is something else entirely (and, as is the case for all empirical concepts, the task appears to be infinite, and never fully realized). With a 3D model, we could even mistake a man for a primate and vice versa—which is exactly what sometimes happens, although it is unlikely that a man would be
