Chapter 7 returns to Kant for a deeper analysis of his views and their relation to the Euclidean mathematical tradition. Chapter 6 revealed that Euclid defined neither magnitude nor homogeneity, so that these notions are at best implicitly defined by the Euclidean-Eudoxian theory of proportions. Kant reworks the Euclidean theory of magnitudes, defining magnitude in terms of his own understanding of homogeneity, which admits of no qualitative difference of the manifold and which I call strict homogeneity. Most importantly, he thinks that intuition is required to represent either a continuous or a discrete manifold without qualitative difference. This role for intuition in Kant’s philosophy of mathematics and experience has not been appreciated, but finds support in various texts, especially Kant’s lectures on metaphysics and his criticisms of Leibniz’s views in the Amphiboly. Given Kant’s understanding of qualities, differences in dimension correspond to qualitative differences, so that Kant’s account corresponds to Euclid’s understanding of homogeneous magnitudes. Understanding the role of intuition allows us to appreciate the role of the categories of quantity and intuition in part–whole relations and the composition of magnitudes. The chapter closes with clarifying the sense in which intuition is required for the representation of magnitudes.

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.