Chapter 4 completes the new interpretation of the Axioms of Intuition by reconstructing the argument that all appearances and intuitions are not just magnitudes, but extensive magnitudes. It also examines the Anticipations of Perception, which concerns intensive magnitudes, and clarifies Kant’s distinction between extensive and intensive magnitude, which depends on their mereology. An infinite regress appears to threaten Kant’s mereological definition of extensive magnitude, since the representation of a whole extensive magnitude presupposes the representation of its parts, ad infinitum. The regress is avoided by the indeterminate representation of parts that are not themselves extensive magnitudes. There are several different accounts of how the parts of space can be indeterminately represented. The chapter examines and rejects two of them and argues for a third: they are indeterminately represented through a continuous successive synthesis of space in the generation of a representation of an extensive magnitude. This interpretation is supported by Kant’s explanation of continuity and he references to Newton’s theory of fluxions and fluents. The chapter concludes by arguing that the manifold of a continuous quantum can be indeterminately cognized under the category of plurality, while all three categories of quantity are required for the cognition of quantitas.

Chapter 2 explains the place of mathematics in Kant’s Critique of Pure Reason. The Transcendental Aesthetic and the Axioms of intuition both make claims about mathematics that overlap and also appear inconsistent concerning the possibility and the applicability of mathematics. This has baffled prior commentators, leading them either to claim that Kant was inconsistent or to introduce distinctions to render him consistent. Two of those distinctions are between pure and applied mathematics and between general “topological” mathematics and a mathematics that includes a metric. Both these interpretations are supported by the fact that the Axioms of Intuition makes important claims about the applicability of mathematics, but the interpretations cannot be sustained. A consideration of the treatment of space and time in the Transcendental Aesthetic and the Axioms of Intuition faces parallel tensions. Examining the latter tensions points to a distinction between indeterminate space and time and determinate spaces and times as a solution, which also solves the tension concerning mathematics. This resolution allows us to see that the Axioms of Intuition concerns not just applied mathematics and not just the introduction of a metric, but all mathematics, topological or otherwise, and both pure and applied.

Chapter 3 provides a new interpretation of the Axioms of Intuition, which argues that all appearances and all intuitions are extensive magnitudes. Previous commentators have failed to understand the structure of the argument, which splits into twos: an argument that appearances and intuitions are magnitudes, and an argument that they are extensive magnitudes. The content of the first argument has also been misunderstood. It pivots on the definition of magnitude, which a prominent Kant scholar emended to help clarify the argument. Unfortunately, the emendation is misleading and obscures Kant’s views. This chapter provides a new analysis of Kant’s argument and his definition of magnitude that clarifies the relationship between the Axioms of Intuition and the categories of quantity and his understanding of magnitude. It reveals that Kant makes substantive claims about pure as well as applied mathematics and that he directly connects our cognition of magnitudes in pure mathematics to our cognition of the world. It also improves our understanding of two sorts of magnitude, quanta and quantitas. Most importantly, the improved interpretation points to a previously unrecognized role for intuition in representing magnitudes.