Prompted by Hume's skepticism, Kant addresses the question of whether and how metaphysics is possible. Metaphysicians have yet to agree on one definite proposition, or even to establish a basis for agreement upon judgments.
Kant distinguishes between a priori and a posteriori cognitions and between analytic and synthetic judgments. Knowledge we gain from experience is a posteriori, and what we can know independent of experience is a priori. A synthetic judgment is one whose predicate contains information not contained in the subject, and an analytic judgment is one whose predicate is a mere analysis of the subject. Kant claims that mathematics, natural science, and metaphysics all lay claim to synthetic a priori propositions—propositions that are necessarily but not trivially true, and can be known prior to experience. Since mathematics and pure natural science are well-established fields, he proposes to examine how their synthetic truths are possible a priori in the hope that this examination will shed light on the possibility of metaphysics as a science.
Mathematics is possible, Kant suggests, thanks to the pure intuitions of our faculty of sensibility. Space and time are not things in themselves that we meet with in experience; rather, they are pure intuitions that help us structure our sensations. Geometry comes from our pure intuition of space, and mathematics comes from our pure intuition of time—our concept of numbers is built from the successive moments in our concept of time.
Pure natural science is possible thanks to the pure concepts of our faculty of understanding. Kant distinguishes between "judgments of perception," which are based on subjective sensations, and "judgments of experience," which try to draw objective, necessary truths from experience. Science, as an objective body of knowledge, is only possible if we can consider nature as according itself with objective, regular laws. These laws—like "every effect has a cause"—are concepts of our understanding just as space and time are intuitions of our sensibility. We cannot know anything about things in themselves, but the appearances that constitute our experience follow these laws. Kant constructs a complex table of categories to show how the pure concepts of the understanding structure experience.
Metaphysics relies on the faculty of reason, which has nothing to do with experience. In its drive for completeness, reason aspires to know about things in themselves, and mistakenly applies concepts of the understanding to matters outside experience. Kant classifies the "ideas of reason" into three types: psychological, which deals with our idea of substance and of a soul, cosmological, which gives rise to four sets of "antinomies" based on causal reasoning, and theological, which deals with our idea of God. In each case, Kant argues, reason oversteps its bounds and tries to make claims about things in themselves, often confusing these with appearances.
Metaphysics is unlike math or science in that its reach exceeds its grasp. It aspires to know what it cannot know. In finding itself bounded, however, reason also explores the full extent and possibility of human knowledge. While reason cannot tell us anything about things in themselves, it can be used to examine our own faculties. Kant redefines metaphysics as a "critique," an attempt to examine how knowledge is structured and justified.
One gathers the impression from what is here, that Kant's not terribly compelling, or plausible, whatever his historical importance. I think this is debatable.
About the claim that is made here, that Frege was the first to point out that geometry is not synthetic a priori. Well, this implies that indeed, geometry is not synthetic a priori. However, Some believe that Frege was wrong. I'll note that I'm also reading, here, about how this position 'was given a boost by Einstein's relativity..' That is to say, that Einstein's relativity<... Read more→