Completions, Constructions, and Corollaries

In H. Pulte, G. Hanna & H.-J. Jahnke (eds.), Explanation and Proof in Mathematics: Philosophical and Educational Perspectives. Springer (2009)
According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim of this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively.
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