Syntheticity, Intuition and Symbolic Construction in Kant's Philosophy of Arithmetic
Dissertation, Columbia University (1997)
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
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