9 - From Mereology to Mathematics

Summary

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.