The first part of Kant’s Prolegomena reiterates the questions from the Preamble and relates them specifically to mathematics. How is mathematics able to furnish us with synthetic a priori knowledge? Kant observes that, though mathematics offers us a priori knowledge about the world, this knowledge is obtained and confirmed through a specific kind of use of our visual faculties—namely, with numbers and shapes. These “visualizations” are different from our ordinary perceptions. In regular sensory perception, we can glean much information about an object—its color, its weight, its position. None of these judgments are necessary, however. They are true only for the specific object we are perceiving. In the visualizations of mathematics, however, we perceive only those aspects of an object that are necessarily and universally true. They are, in effect, a priori experiences.
But how is it possible to perceive (or intuit, in Kant’s terminology) something a priori? Isn’t that a contradiction in terms? Perception demands an object, but objects are only given in experience. To be sure, there are concepts, or, ways of ordering sensory experience, that we possess a priori: causation, for example, or quantity (that I perceive three lamps, for example, instead of one distinct object.) But these are distinct from the visualizations of mathematics.
The only way that a priori perception can be possible is if, in it, we see not objects themselves, but rather the forms of objects—a circle is a representation of the way our mind perceives all circular objects. It is a pure form that precedes an infinite number of sensory perceptions: a pizza, a bicycle wheel, a blueberry pie. The number three stands in for any three like objects in existence. Accordingly, mathematical visualizations are means by which we can intuit objects a priori—but only as they appear to us, and not as they are in themselves.
Now Kant explains this latter caveat. The “pure”—pure in the sense of abstract—intuitions, or forms, of mathematics are derived from the two pure intuitions of the human mind: space and time. If you take every sensory quality away from a sensory impression—e.g., its color, its weight, its texture—space and time would still remain. Like the forms of geometry, space and time are forms, rubrics that our mind imposes on our sensory impression in order to make things appear the way that they do. Geometry consists of pure intuitions of space, while arithmetic represents changes in quantity over time.
From these, mathematics is able to extract synthetic a priori knowledge not of things "themselves," but of things as they appear to us. That is apparent from the fact that many of the arguments of mathematics which appear to be true a priori in fact need empirical confirmation, or are derived from experience, e.g. that a line will extend infinitely in both directions, or that only three lines can intersect at a point to produce three right angles.
It follows, then, that space and time are not objective qualities of things themselves, but rather forms that human perception uses to arrange its various impressions. To prove his point, Kant asks the reader to imagine two objects that, in their internal composition, are identical down to the last detail: our hands, for example, or our ears. Despite the fact that they are identical, we still recognize one as the left and the other as the right. That is because space is produced not by an internal quality of objects, but by our mind.
So while mathematics cannot tell us anything about things beyond our perception, it is nonetheless objectively and universally true for all human minds. That does not mean, however, that the insights of mathematics are simply subjectives illusion—something randomly imposed by our minds on the "real" objects out there in the world. Experience comes from the understanding as well as the senses. While we each our have different sensory perceptions, our understanding—i.e., that part of our mind that arranges our sensory impressions—forms those perceptions in the same way. Error arises only when we try extend the judgments of mathematics to things in themselves.
Analysis
One striking aspect of the Prolegomena is the way that it rearranges and regroups the central discoveries of The Critique of Pure Reason. Each chapter, though devoted to a science, is in fact an exploration of a particular faculty of the human mind, and of that faculty’s capacity for producing synthetic a priori knowledge. Ostensibly devoted to mathematics, the first section in fact examines sensory perception, and the formal knowledge that we are able to derive from it.
It is also here that we encounter, for the first time, Kant’s “Copernican turn”: the discovery that many facets of the world as we perceive it are in fact imposed by the mind, particularly time and space. By understanding the objects of the perceived world not as “things in themselves,” but as structured and organized by the human mind, Kant believes we can think through impasses and errors of the philosophy that has come before him. It is this focus on the human mind, its capacities and its limits, rather than on speculative or religious questions, that marks Kant as a figure of the Enlightenment.
The central problem posed by mathematics, as Kant sees it, is its dual character. On the one hand, mathematics is entirely abstract, as evidenced by the fact that its proofs are universal. On the other hand, it relies on a system of visual representation, i.e. shapes and numbers. For Kant, the central question of mathematics is thus: what precisely our senses are doing when we do mathematics? How can visual impressions be a priori?
The answer is that the specific impressions of mathematics represent the form of our experience, the way our mind prefigures the information of our senses into a coherent whole, the way a glass gives form to water. A circle drawn on a chalkboard represents all circular objects we could possibly perceive; from it, we can learn something about those objects—that they can all be divided into four equal right angles, for example.
Following Hume, Kant insists that we can’t have any knowledge of things as they are beyond our perception. Therefore, we cannot know how space and time function, except as we perceive them. Why aren’t mathematics completely illusory, then? Kant’s answer is that though each person’s sensory apparatus may differ (a person who suffers from red-green color blindness might, for example, think that a red and a green object are the same color), our minds are hard-wired to organize space in the same way: we all arrange objects in space. Mathematics can derive facts from the forms of human perception, i.e., that the human mind is hard-wired in such a way that any object we perceive as circular will be, say, dividable into two equal half circles. This way, Kant is able to maintain that objective, universal truths are nonetheless possible.
Philosophers have disagreed about Kant’s understanding of space and time as forms. Hume held that space was created by objects themselves: to have a sense of a room, for example, we first need four walls to delimit it. Space does not precede objects, rather, it is created in the mind as the perceived distance between them. Later philosophers, like Henri Bergson, have observed that Kant fails to distinguish time from space—that for him, time is conceived as a line, and that therefore, he has simply reused the concept of space to describe the concept of time.
Another point of contention is Kant’s reliance on Euclidian geometry. Einstein’s Theory of General Relativity proved that space and time in fact act in ways that run directly counter to our sensory perception.
Still, Kant’s argument that we all share the same reflective faculties, if not the same sensory ones, provides a still-powerful answer to the problem of solipsism that plagues not only Hume, but all skeptics—in other words, those who ask, "if all I have are my sensory perceptions, then how can I be sure that anything else exists?" This problem, and Kant's attempt to answer it, will be explored further in the next chapter.
