The transcendental schema is a product of the imagination. Let us set aside for now the discrepancy that exists between the first and the second editions of the Critique of Pure Reason, as a consequence of which in the first edition the Imagination is one of the three faculties of the soul, together with Sense (which empirically represents appearances in perception) and Apperception, while in the second edition, Imagination becomes simply a capacity of the Intellect, an effect that the intellect produces on the sensibility. For many of Kant’s interpreters, like Heidegger, this transformation is immensely relevant, so much so in fact as to oblige us to return to the first edition, overlooking the changes in the second. From our point of view, however, the issue is of minor importance. Let us admit, then, that the Imagination, whatever type of faculty or activity it may be, provides a schema to the intellect, so that it can apply it to the intuition. Imagination is the capacity to represent an object even without its being present in the intuition (but in this sense it is “reproductive,” in the sense we have called “imagining1”), or it is a synthesis speciosa, “productive” imagination, the capacity for “figuring.”
This synthesis speciosa is what allows us to think the empirical concept of a plate, through the pure geometrical concept of a circle, “because rotundity, which is thought in the first, can be intuited in the second” (CPR/B: 134). In spite of this example, the schema is still not an image; and it therefore becomes apparent why we preferred “figure” to “imagine.” For instance, the schema of number is not a quantitative image, as if we were to imagine the number 5 in the form of five dots placed one after the other as in the following example: •••••. It is evident that in such a way we could never imagine the number 1,000, to say nothing of even 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” (CRP/2: 135), so that Peano’s five axioms could be understood as the elements of a schema for representing numbers. Zero is a number; the successor to every number is a number; there are no numbers with the same successor; zero is not the successor of any number; every property belonging to zero, and the successor to every number sharing this property, belongs to all numbers. Thus any series x0, x1, x2, x3 … xn is a series of numbers, under the following assumptions: it is infinite, does not contain repetitions, has a beginning; and, in a finite number of passages, does not contain terms that are unreachable starting from the first.
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.