Chapter 8 focuses on Kant’s reaction to the metaphysics of quantity found in Leibnizian and Wolffian rationalism. Leibniz had broad ambitions for a unified theory of all knowledge that subsumed mathematics under metaphysics. Leibniz accordingly sought metaphysical definitions of quality and quantity that in turn supported metaphysical definitions of similarity and equality as identity of quality and quantity, respectively. A criterion of success was that these definitions corresponded to Euclid’s geometrical notions of similarity and equality. This chapter examines Leibniz, Wolff, and Baumgarten’s views of quality and quantity and the contrast between them, which was closely tied to the conditions of their representation and distinct cognition. Kant adopts some of their understanding of the metaphysics of quantity, such as the definitions of similarity and equality as identity of quality and quantity, respectively. At the same time, he radically reforms it. Kant distinguishes between two notions of quantity, quanta and quantitas, and hence draws two contrasts with two corresponding notions of quality: quality versus quantum, and quality versus quantitas. Most importantly, Kant holds that quanta require intuition for their representation. This preserves the general framework of the Leibnizian and Wolffian metaphysics of quantity while radically reforming it at its foundation.

The previous chapter argued that intuition allows us to indeterminately represent a continuous manifold of space. On the other hand, this possibility appears to be inconsistent with Kant’s characterization of intuitions. He contrasts them to concepts by stating that the former are singular and immediate representations. Singularity seems to commit Kant to the view that, by its nature, intuition must represent an individual object, and many have understood him in this way. That would directly contradict the previous chapter. Chapter 5 addresses this problem. It argues against a quick solution to this problem and for a deeper account. Examining the generality of concepts suggests a distinction between representing and represented, and the singularity of intuition is explained as a mode of representing singularly. The chapter argues that representing singularly is compatible with the indeterminate representation of a continuous manifold; moreover, it is what makes possible the cognition of singulars in intuition. This new reading of the singularity of intuition solves the extensive magnitude regress and also has important implications for understanding mathematical cognition as well as the current Kantian nonconceptualist debate. It also allows us to give a clear account of Kant’s views of concreteness and abstractness.

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.