This synthesis is that whereby the empirical concept of plate can be thought by means of the pure geometric concept of the circle, because «the roundness which is conceived in the first» forms an intuition in the second (CPR/B: 134). Despite this example, the schema is not an image; and therefore it becomes clear here why I preferred «figure» to «imagine.» For example, the schema of number is not a quantitative image, as if I imagined the number 5 in the form of five dots lined up one after the other, like this: It is evident that in such a way I could never imagine the number 1,000, not to mention greater numbers. The schema of number is «rather the representation of a method of representing in one image a certain quantity … according to a certain concept» (CPR/B: 135), so that one could understand Peano’s five axioms as the elements of a schema for the representation of numbers: zero is a number; the successor of every number is a number; there are no numbers with the same successor; zero is not the successor of any number; every property of the zero, and the successor of any number with those properties, belongs to all numbers—so that any series x0, x1, X2, x3 … xn, which is infinite, contains no repetitions, has a beginning, and does not contain terms that cannot be reached starting from the first, in a finite number of passages, is a series of numbers.
In the preface to the second edition of the first Critique, Kant mentions Thales, who from the figure of one isosceles triangle, in order to discover the properties of all isosceles triangles, did not follow step by step what he saw, but had 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 (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 one should say that the Kantian schema, more than what is commonly understood as a «mental image» (which evokes the idea of a photograph), is like Wittgenstein’s Bild, a proposition that has the same form as the fact it represents, in the same sense in which we talk of an «iconic» relation for an algebraic formula, or of a «model» in the technical-scientific sense.
In order to gain a better understanding of the concept of schema, perhaps we need to consider what computer operators call a flowchart. The machine is capable of «thinking» in terms of IF … THEN GOTO, but this is an overly abstract logical device, given that it can serve us both for making a calculation and for drawing a geometrical figure. The flowchart shows us the steps that the machine must perform and that we must order it to perform. Given one operation, at a certain juncture in the process a possible alternative is produced, and, depending on the answer that appears, a choice needs to be made; depending on the new answer, it is necessary to return to a higher node of the chart, or proceed beyond, and so on. The chart has something that can be intuited in spatial terms, but at the same time it is substantially based on a temporal course (the flow), in the same way as Kant observes that the schemata are based fundamentally on time.
This idea of the flowchart seems to explain rather well what Kant meant by the schematic rule that governs the conceptual construction of geometrical figures. No image of a triangle, which I find in experience—the face of a pyramid, for example—can ever provide adequate cover for the concept of triangle in general, which must hold good for all triangles, be they right-angled, isosceles, or scalene (CPR/B: 136, 1–10). The schema is proposed as a rule for the construction in any situation of a figure having the general properties of triangles (let us say, even without talking in strictly mathematical terms, that one of the prescribed steps the schema obliges me to take is that, if I have arranged three toothpicks on the table, I must not seek a fourth but must for the time being close the figure with the three toothpicks available).16
Kant reminds us that we cannot think of a line without tracing it in our thoughts; we cannot think of a circle without describing it (I believe that in order to describe it I must have a rule that tells me that all the points of the circle must be equidistant from the center). We cannot represent the three spatial dimensions without putting three lines perpendicular to one another. We cannot even represent time without tracing a straight line (CPR/B: 120, 21 ff.). Note that at this point we have radically modified what we defined at the beginning as Kant’s implicit semiotics, because thinking is not just the application of pure concepts deriving from a previous verbalization, it is also the entertaining of diagrammatic representations.
As well as time, memory comes into the construction of these diagrammatic representations: in the first edition of the Critique (CPR/A: 78–79), Kant says that if while counting I forget that the units now present to my senses have been added gradually, I cannot know the production of pluralities through successive addition, and therefore I cannot even know the number. If in thought I were to trace a line, or if I wished to think of the time between one noon and the next, but in the process of addition I always lost the preceding representations (the first parts of the line, the preceding parts in time), I would never have a complete representation.
We can see how schematism works in the anticipations of perception, a really fundamental principle because it implies that observable reality is a segmentable continuum. How can we anticipate what we have not yet sensibly intuited? We must work as if degrees might be inserted into experience (as if one could digitize the continuous) without this causing our digitization to exclude infinite other intermediate degrees. Cassirer points out that if we were to admit that in the instant a a body manifests itself in the state x and in the instant b it manifests itself in the state x, without having passed through the intermediate values between these two, then we would conclude that we were not dealing with the «same» body: we would assert that the body that was in the state x in the moment a, had disappeared, and that in the moment b another body appeared in the state x. The upshot is that the assumption of the continuity of physical changes is not a singular result of observation but a presupposition of knowledge of nature in general, and therefore it is one of those principles that govern the construction of the schemata (Cassirer 1918, III, 3).
2.6 And the Dog?
So much for the schemata of the pure concepts of the intellect. But it so happens that it is precisely in the chapter on schematism that Kant introduces examples concerning empirical concepts. It is not only a matter of seeing how the schema allows us to homogenize the concepts of unity and reality, inherence and subsistence, possibility and so on with the manifold of the intuition. There is also the schema of the dog: «[T]he concept of dog means a rule, according to which my imagination can always draw a general outline of the figure of a four-footed animal, without being restricted to any particular figure supplied by experience or to any possible image which I may draw in the concrete» (CPR/B: 136).
It is no accident that after this example, a few lines later, Kant wrote the renowned phrase according to which this schematism of our intellect, which also concerns the simple form of appearances, is an art concealed in the depths of the human soul. It is an art, a procedure, a task, a construction, but we know very little of how it works. Because it is clear that our nice little analogy with the flow-chart, which might help us understand how the schematic construction of the triangle proceeds, works far less well in the case of the dog.
A computer can certainly construct the image of a dog, provided it is given suitable algorithms: but it is not by examining the flow-chart for the construction of the dog that a person who has never seen a dog can have a mental image (whatever a mental image may be) of one. Once more we find ourselves faced with a lack of homogeneity between categories and intuition, and the fact that the schema of the dog can be verbalized as «quadruped animal» brings us back only to the extreme abstractness of every predication by genus and differentia, but it does not allow us to distinguish a dog from a horse.
Deleuze (1963) observes that the schema consists not of an image but of spatiotemporal relations that embody or realize some purely conceptual relations, and this seems exact as far as the schemata of concepts of pure intellect are concerned. But it does not seem to be sufficient when it comes to empirical concepts, since Kant was the first to tell us that in order to think of a plate, I must resort to the image of the circle. While the schema of the circle is not an image but a rule for constructing the image if necessary, the empirical concept of the plate should nonetheless
