Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: (...) genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation. (shrink)

This review of contemporary discussions of Kantian philosophy of mathematics is timed for the publication of the essay Kant’s Philosophy of Mathematics. Volume 1: The Critical Philosophy and Its Roots (2020) edited by Carl Posy and Ofra Rechter. The main discussions and comments are based on the texts contained in this collection. I first examine the more general questions which have to do not only with the philosophy of mathematics, but also with related areas of Kant’s philosophy, e. g. the (...) question: What is intuition and singular term? Then I look at more specific questions, e. g.: What is the subject of arithmetic and what is the significance of diagrams in mathematical reasoning? As a result, the reader is presented with a fairly complete overview of modern discussions which can be used as an introduction to the problem field of Kant’s philosophy of mathematics. (shrink)

In A Theory of Justice (1971), John Rawls introduces the concept of “reflective equilibrium.” Although there are innumerable references to and discussions of this concept in the literature, there is, to the present author's knowledge, no discussion of the most important question: Why reflective equilibrium? In particular, the question arises: Is the method of reflective equilibrium applicable to the choice of this method itself? Rawls's drawing of parallels between Kant's moral theory and his own suggests that his concept of “reflective (...) equilibrium” is on a par with Kant's concept of “transcendental deduction.” Treating these two approaches to justification as paradigmatic, I consider their respective merits in meeting the reflexive challenge, i.e., in offering a justification for choice of mode of justification. My enquiry into this topic comprises three parts. In the first part (Eng 2014a), I raised the issue of the reflexivity of justification and questioned whether the reflexive challenge can be met within the framework of A Theory of Justice. In this second part, I shall outline a Kantian approach that represents a paradigmatic alternative to Rawls. (shrink)

The role of intuition in Kant's theory of mathematics is similar to instantiation rules in first-order logic according to Jaakko Hintikka. This paper is a critical examination of Hintikka's interpretation and reconstruction of Kant's theory. It is argued that Kant's position is question-begging on this interpretation.