But how does Kant prove that mathematics is constituted of synthetic a priori statements and propositions? And how is it possible? Kant deals with these questions both in his Prolegomena and The Critique. Since the necessity of mathematics does not and cannot be derived from mere experience, then its origin must be somewhere other than experience. Kant’s answer is that the certainty of mathematics is subjective in origin. That is, we supply the certainty, and we do this by means of what Kant calls “pure intuitions.” What Kant means by “pure” is empty of matter. In other words, a pure intuition is only the form of intuition. We have two of these pure intuitions and only two — space and time. Space is the special condition of all outer experiences and time is that of inner ones. However, time is also necessity for outer experiences, as well. These two pure intuitions, space and time, are formal conditions of experience and without them we cannot have experience. Furthermore, the formal conditions impart necessity to experience.
As I have mentioned, space and time form respectively the outer and the inner sense of the mind. Space is the form of all appearances of our outer sense, while time is a formal a priori condition of sensibility. The notions of space and time for Kant are very important in the context of their problemization by Newton and Leibniz. The former claimed that space was the necessary pre-condition for the existence of all physical objects, so it is prior to them. Leibniz, on the other hand, believed that space was a kind of relation among physical objects and, since relations are mind-dependent, it follows that space is in a way dependent on the physical objects. He contended, in other words, that first there must exist things and then the spatial relation among them.
Kant tries to find a middle path between these two assertions. He agrees to some extent with Leibniz in that space is a kind of relation. But he also accepts Newton’s claim that space is a “real” part of the apparatus of the mind and prior to objects. Thus Kant approaches the inference that space is a subjective formal working of the mind. It is a pure intuition, as is also time, which belongs a priori to the sensibility and is not acquired a posteriori. According to Kant we cannot imagine anything existing abstractly from space and time. That is why those two pure intuitions are absolutely necessary and indispensable for the formal workings of the mind. Expressed in a more loose fashion — we are in space and space is in us.
However, these formal conditions do have a limitation. They are limited to what Kant calls appearances. That is, appearances are that which is given to the sensibility. Therefore, they are the matter which will be “formed” by the formal conditions of space and time. Appearances compose that which we can sense and know. But we can sense and know things only as they appear to us. This is in contrast to things as they really are. We cannot have knowledge of those things because they are beyond our faculty of sensibility. And, we cannot know them because, as Kant repeatedly insists, we can know things only when we have the combination of sensibility and understanding. Having discusses sensibility, and having shown that only on the assumption that space and time are subjective, and are purely formal conditions of the mind, Kant turns to the other faculty: understanding.
Just as space and time are the formal conditions of sensibility that make mathematics possible, what Kant calls the categories are the forms of the understanding that make natural science possible. The notion of the categories dates back to Aristotle. Kant, however, insists that his categories are markedly improved over his predecessor. What gives him reason to believe so is that Aristotle’s categories are arrived at haphazardly and nothing that is haphazard can serve as a foundation for knowledge. Kant is again stressing universality and necessity. Kant finds the key to this by looking at the science of logic. Logic concerns only the formal conditions of thinking; it does not care for the content of thought. Its rules apply necessarily and universally to all thinking.
There are, according to Kant, a total of twelve categories, divided into four sections, with each section having three sub-sections in it. The most important of these categories are substance, community and causality, and the latter, beyond doubt, is the most crucial. These twelve categories of logic have their counterparts in thinking. These counterparts are a special type of concept. Just as space and time are pure intuitions, the categories are pure concepts. These categories, or pure concepts of the understanding, are formal conditions of the understanding. Along with space and time, they are what is necessary for us to have knowledge. However, Kant needs to prove that these categories have justification. He provides it by what he calls a deduction. In our lives we use many terms; the question is: are they entitled to be used? Terms such as fate and fortune do not have any legitimate use, at least compared to terms such as desk and chair, because the legitimacy of the latter terms is derived from experience. Experience, however, as Kant continuously reminds, can yield only relative or comparative necessity. What Kant is seeking here is strict necessity for the basic terms and concepts of science. What makes science possible is not experience because science has strict necessity. It is based instead upon the pure concepts of the understanding. Kant’s twelve categories are subjective forms of the understanding; the necessity then is imparted by us. The final task of the understanding, as Kant puts it in the Kritik, is to “confer upon appearances their conformity to law, and so make experience possible.”
To get back to the question of synthetic a priori statements, Hume believed that he had shown that analytical statements possess strict necessity. Furthermore, he had demonstrated that the statement “it has always happened like that” does not necessarily imply that “it will, or must, happen like that.” That is because, in Kantian terms, this is a synthetic judgment, not analytic. Before Kant, no one seemed to believe that there could be synthetic judgments known a priori. Kant’s category (or pure concept) of cause and effect is just such a case of synthetic judgments that are known before experience. From whence comes the necessity of accompanying the a priori? — from the formal conditions of the mind. Our mind is constituted in such a way that whenever we see A happen, we are led by the formal conditions of the mind to expect that B will follow. Thus, according to Kant, the strict necessity is guaranteed by the formal workings of the mind. Finally, to express it in the style of the Kritik, in order to have pure concepts, we need the synthetic unity of apperception which eventually equals reason.
What role does imagination play in the process of cognition? Kant is clear — imagination produces immediating representation or intuition of an object that is not itself present. In conjunction with reason , imagination forms understanding.
The unity of apperception in relation to the synthesis of imagination is the understanding; and this same unity, with reference to the transcendental synthesis of the imagination, is the pure understanding… A pure imagination, which conditions all a priori knowledge, is thus one of the fundamental faculties of the human soul… (CPR, pp. 143-146)
Thus, the two parts of the mind — sensibility and understanding — are not abysmally remote because imaginations serves as a bridge between them. The two extremes are intimately close through the “mediation of the transcendental function of imagination.” Without imagination, sensibility would account for no objects of empirical knowledge. This fact consequently entails no experience although the latter produces appearances. Thus, by means of imagination “we bring the manifold of intuition on the one side, into connection with the condition of the necessary unity of pure apperception on the other.” (CPR,
P.146)
Having discussed the two types of formal conditions of the mind, pure intuitions and pure concepts, and the role of imagination, Kant makes it explicitly clear — all three faculties are necessary; without them there can be no knowledge. Thus, in terms of employment for knowledge, the categories must be conjoined with the matter and form of sensibility. If one attempts to use the pure concepts without sensation then one is using these in a transcendent sense, i.e. they are being used beyond the bounds of sense. An important point here is the distinction between “transcendent” (or illegitimate) as opposed to “transcendental” (or legitimate). While the first can never yield knowledge it still can have a “regulative” use of the categories. The notion of regulative use leads to Kant’s practical or moral philosophy.
The regulative use is also bound up with Kant’s distinction between appearances (or the world of phenomena) and the things in themselves (or the noumenal realm). The strict necessity of causality is limited to the phenomenal realm: all things that are appearances are determined. The rule “if A happens, then B will follow” applies with no exception.
Kant’s Critique of Pure Reason is a metaphysics of experience; he delineates the formal conditions of the mind that make experience possible. Reason and sensibility are the two stems of knowledge; they are simultaneously the formal conditions and the limitations of knowledge.
In my paper I have tried to examine and justify the challenging endeavor that Kant undertakes, his so-called “Copernican revolution.” The revolutionary element is present in the radical change of perception of the traditional philosophic concepts — knowledge, causality and reason — that Kant sets out to make. However, insofar as things can be knowable by what we input in them, they still remain ontologically only things by themselves, i.e. unknowable to us. Thus Kant’s Kritik der Reinen Vernuft is an ontology of the mind; it is an examination of its potentialities of having knowledge and, at the same time, of its limitations. Kant’s revolutionizing of the traditional philosophical understandings is a great achievement and, beyond doubt, paves the way for a radically new era — that of the transcendental self determining its own experiences.