Mathematics in Kant\u27s critical philosophy: Reflections on mathematical practice

Many recent attempts to analyze Kant\u27s philosophy of mathematics have proceeded from within the contextual confines of Kant\u27s own Critique of Pure Reason. I aim to give a new reading of some of Kant\u27s 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\u27s 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\u27s enterprise, and counter modern objections to Euclid\u27s 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\u27s 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\u27s critical claims about mathematics. A familiar geometric demonstration is used to clarify Kant\u27s 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\u27s theory of algebraic cognition is re-interpreted in order to demonstrate that for Kant all construction is ostensive