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.