Completions, constructions, and corollaries
Kant argued that pure intuition is an indispensable ingredient of mathematical proofs. However, Kant's thesis has been considered obsolete since the advent of modern relational logic at the end of the nineteenth century. Against this logicist orthodoxy, Cassirer's "critical idealism" insisted that formal logic alone could not make sense of the conceptual coevolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, play a fundamental role in the formation of the mathematical and empirical concepts. This paper outlines the basics of Cassirer's idealizational account and points out some interesting similarities to Kant's and Peirce's philosophies of mathematics, which are based on the key notions of pure intuition and theorematic reasoning, respectively.
Mormann, T. (2010)., Completions, constructions, and corollaries, in G. Hanna, H. N. Jahnke & H. Pulte (eds.), Explanation and proof in mathematics, Dordrecht, Springer, pp. 59-70.
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