Kant’s Mathematical World (KMW) presents a new interpretation of Kant’s philosophy of mathematics and its connection to his account of the world of experience. It argues that Kant’s mathematics is grounded in a theory of magnitudes, which has its roots in the Greek mathematical tradition and differs deeply from our contemporary way of thinking about mathematics. KMW argues that in Kant’s view, intuition is required to represent magnitudes and that mathematics and the world of experience are far more intertwined than has been appreciated, with implications for interpreting both his philosophy of mathematics and his account of human experience.

The main claims of Kant’s Mathematical World are that for Kant mathematics is a science of magnitudes, the world of experience is a world of magnitudes, and there is an essential role for intuition in representing magnitudes. This paper critically examines the argument for these main claims. It articulates an eight-step “Master Argument” that connects magnitude to mathematics through a series of intermediary concepts. Some of these steps are left unclear in Kant’s Mathematical World, and I argue that the book’s silence on syntheticity and the role of intuition in proof is a partial cause for this unclarity.

I begin by introducing Sutherland’s account of why Kant takes mathematics to require a non-conceptual source of knowledge of bare numerical difference. Given that objective space and time are wholes, however, I wonder why, according to Sutherland, such a source of knowledge can’t be circumvented by taking each spatial or temporal position to have conceptualizable features sufficient for distinguishing them. I then point to an issue with Sutherland’s appeals to “construction” and argue that if Sutherland more sharply distinguished between the syntheses involved in mathematical construction and empirical apprehension, then he could avoid some problems that his reading faces.

The main idea in Daniel Sutherland’s Kant’s Mathematical World is a strong claim about the immanence of mathematics in the world: Mathematics is literally true of objects of experience because it conditions the possibility of experience, and thereby the objects of experience. Sutherland grounds this claim on an insightful treatment of the general theory of magnitude that outlines constraints on magnitude’s representation. That representational task also gives rise to a regress problem, which Sutherland addresses by appeal to continuous successive synthesis. The result bears on recent debates between conceptualist and non-conceptualist readings of Kant’s theory of cognition.

This article contains Daniel Sutherland’s response to comments on Kant’s Mathematical World (KMW) by Lanier Anderson, Jeremy Heis, and Tyke Nunez. It locates KMW in relation to recent debates in Kant scholarship and metaphysical versus mathematical interpretations of Kant’s theoretical philosophy. It considers how best to approach a reconstruction of Kant’s view that mathematics is the science of magnitudes, and it explains why and how, in Kant’s view, mathematics requires intuition, providing a more fine-grained account by distinguishing the roles of intuition in mathematical concepts, judgments, and inferences. The article clarifies the relationship between empirical cognition and pure mathematical cognition, and between apprehension and construction, as well as a number of other issues.