Mathematics in Kant's critical philosophy: Reflections on mathematical practice
Abstract
Many recent attempts to analyze Kant's philosophy of mathematics have proceeded from within the contextual confines of Kant's own Critique of Pure Reason. I aim to give a new reading of some of Kant's most important claims about mathematical cognition by examining them within the context of the eighteenth century mathematical practice with which he was engaged. First, I investigate Euclid's reasoning in the Elements and show that the Euclidean diagram serves a valid demonstrative role in the proofs of Euclidean propositions. I thereby re-evaluate the "axiomatic" nature of Euclid's enterprise, and counter modern objections to Euclid's reasoning made on the basis of subsequently developed standards of proof. Second, I assess the state of early modern elementary mathematics by using Christian Wolff's Elementa Matheseos Universae as a tool for revealing how Wolff and his contemporaries reformulated the "elements" of pure mathematics. In particular, I analyze the method of "constructing equations" and conclude that algebra was not conceived as an independent discipline with its own object of investigation, but rather was a method of reasoning about the constructible objects of arithmetic and geometry. Finally, I provide a new reading of Kant's critical claims about mathematics. A familiar geometric demonstration is used to clarify Kant's distinction between pure and empirical intuition and to locate the source of the synthetic a priority of mathematical judgments; the "schematism" of mathematical concepts is shown both to provide the rules that we follow for the construction of those concepts and to confer universality on our mathematical judgments; and Kant's theory of algebraic cognition is re-interpreted in order to demonstrate that for Kant all construction is ostensive.
Recommended Citation
Lisa A Shabel,
"Mathematics in Kant's critical philosophy: Reflections on mathematical practice"
(January 1, 1998).
Dissertations available from ProQuest.
Paper AAI9829986.
http://repository.upenn.edu/dissertations/AAI9829986
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