Kant’s reworking of the Euclidean theory of magnitudes and reformation of the Leibnizian-Wolffian metaphysics of quantity are in service of his project of explaining the foundations of mathematical cognition and the mathematical character of experience. The previous chapters revealed that Kant’s account is fundamentally mereological. The categories of quantity allow us to represent the part–whole relations among magnitudes, and Kant’s understanding of the role of the categories of quantity, the nature of composition, and the definitions of extensive and intensive magnitudes are all mereological. This introduces a gap between Kant’s mereological account of magnitudes and Euclid’s notion of magnitude for the latter is implicitly defined by its role in the theory of proportions – a richer, mathematical notion of magnitude. This prompts a closer look at what makes Euclid’s understanding of magnitudes mathematical. This chapter argues that Euclid’s geometry presupposes a tacit theory of measurement that is general, pure, and concrete, a theory that crucially depends on the relation of equality. It traces these presuppositions through the Euclidean tradition. It then argues that Kant also tacitly assumed the theory of measurement, but that he was aware of the crucial role that equality plays in bridging the gap between mereology and mathematics.

Chapter 7 analyzes the Appendix to the Transcendental Analytic entitled “On the Amphiboly of the Concepts of Reflection.” Using Leibniz’s monadology as a prism, Kant here seeks to account for the ultimate premises of his critique and intended reform of metaphysics. More specifically, the chapter conceives of this critique as a variety of transcendental reflection that is guided by four pairs of concepts, including sameness and difference. In order to contextualize this account, the chapter briefly discusses Wolff’s and Baumgarten’s treatment of these concepts. Commentators generally assume that the activity called transcendental reflection is carried out in the Critique of Pure Reason alone. The chapter argues, by contrast, that Kant distinguishes the version of transcendental reflection that informs the ontology of his predecessors from the critical version enacted in the Critique. On this basis, it outlines Kant’s understanding of the difference between a Leibnizian employment of the concepts of reflection and his own.