Kant's Critique of Pure Reason is divided into two sections, the “Transcendental Doctrine of Elements” and the “Transcendental Doctrine of Method”, the former of which is further divided into two parts, the “Transcendental Aesthetic” and the “Transcendental Logic.” Although it is comparatively very short, the Transcendental Aesthetic is a crucially important component of Kant's work, its stated aim being to present a “science of all principles of a priori sensibility” (A 21/B 35). Here, Kant articulates a theory of pure sensible intuition, and deploys arguments in support of the transcendental ideality of space and time. Taken together, the Transcendental Aesthetic and the Transcendental Logic (“which contains the principles of pure thinking”) are meant to provide an account of human cognition and judgment according to which sensibility and understanding - our capacities for being affected by and for thinking about objects, respectively - each play ineliminable roles. In what follows, I will identify and explain the terminology that Kant introduces in the Aesthetic; present and discuss the arguments Kant offers in the Metaphysical and Transcendental Expositions of Space and Time; and show how (and why) Kant concludes from these “expositions” that space and time are transcendentally ideal.

In his Critique of Pure Reason, Kant proposes to investigate the sources and boundaries of pure reason by, in particular, uncovering the ground of the possibility of synthetic a priori judgments: “The real problem of pure reason is now contained in the question: How are synthetic judgments a priori possible?” (Pure Reason, B 19). In the course of answering this guiding question, Kant defends the claim that all properly mathematical judgments are synthetic a priori, the central thesis of his account of mathematical cognition, and provides an explanation for the possibility of such mathematical judgments.

In what follows I aim to explicate Kant's account of mathematical cognition, which will require taking up two distinct issues. First, in sections 2 and 3, I will articulate Kant's philosophy of mathematics. That is, I will identify the conception of mathematical reasoning and practice that provides Kant with evidence for his claim that all mathematical judgments are synthetic a priori, and I will examine in detail the philosophical arguments he gives in support of this claim. Second, in section 4, I will explain the role that Kant's philosophy of mathematics - and, in particular, his claim that mathematical judgments are synthetic a priori - plays in his critical (transcendental) philosophy.