Ofra Rechter

The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context of his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant.

In their recent paper “Do Accelerating Turing Machines Compute the Uncomputable?” Copeland and Shagrir draw a distinction between a purist conception of Turing machines, according to which these machines are purely abstract, and Turing machine realism according to which Turing machines are spatio-temporal and causal “notional" machines. In the present response to that paper we concede the realistic aspects of Turing’s own presentation of his machines, pointed out by Copeland and Shagrir, but argue that Turing's treatment of symbols in the course of that presentation opens the door for later purist conceptions. Also, we argue that a purist conception of Turing machines plays an important role not only in the analysis of the computational properties of Turing machines, but also in the philosophical debates over the nature of their realization.

Kant notably holds that arithmetic is synthetic a priori and has to do with the pure intuition of time. This seems to run against our conception of arithmetic as universal and topic neutral. Moreover, trained in the tradition constituting the aftermath of W.V. Quine's attack on the the a priori and on the analytic/synthetic distinction, the modern philosopher of arithmetic is likely to consider Kant's position a nonstarter, and leave settling the question of what Kant's philosophy of arithmetic is exclusively to the Kant scholar and the historian of the philosophy of arithmetic. I argue that this conclusion is misguided because it rests on the unfounded supposition that the pure intuition of time is the basis for Kant's syntheticity and a priority theses. I recover Kant's grounds for holding those theses and their significance to contemporary philosophy of arithmetic. ;I consider and reject Friedman's eliminativist attempt at making Kant palatable to the contemporary philosopher. I argue that Kant's ideas about the mathematical method in 1763, before he explicitly draws the analytic/synthetic distinction, inform the appreciation of Kant's mature view. The idea of construction in intuition is a key to Kant's Critical position that explains the relation between the intellectual and the sensible aspects in Kant's thought. I show that Kant employs a distinct notion of pure formal intuition that is associated with arithmetical necessity construed as peculiarly mathematical; irreducible to logical or sensible modality. Kant's claim is not that the intuition of time serves to justify arithmetical judgments, I argue, but that we cannot represent time as we do unless we think of it arithmetically. According to Kant, arithmetic is not reducible to logic but it is nonetheless just as fundamental to thought in general. The singularity numerical judgments in relation to the category of quantity is shown to involve a notion of a form of an object that is primary with respect to the concept of an object in general. Finally, I reconstruct Kant's notion of symbolic construction and explicates Kant's conception of a constructive procedure. I argue that a Kantian view of the ontology of arithmetic takes the numbers to be nominalizations of construction procedures for intuitable symbolic types