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The article explores the relationship between the philosopher and historian of mathematics Jacob Klein's account of the transformation of the concept of number coincident with the invention of algebra and Husserl's early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein's research is shown to present the historical context for Husserl's twin failures in the Philosophy of Arithmetic, (1) to provide a psychological foundation for the proper concept of number and (2) to show how this concept of number functions as the mathematical foundation of universal (symbolic) arithmetic. The argument is advanced that one significant result of bringing together Klein's and Husserl's thought on these issues is the need to fine-tune Husserl's Crisis project of desedimenting the mathematization of nature. Specifically, Klein's research shows that "a "sedimented' understanding of numbers" "is superposed upon the first stratum of "sedimented' geometrical "evidences'" uncovered by Husserl's fragmentary analyses of geometry in the Crisis. In addition, then, to the task of "the intentional-historical reactivation of the origin of geometry" recognized by Husserl as intrinsic to the reactivation of the origin of mathematical physics, Klein discloses a second task, that of "the reactivation" of the "complicated network of sedimented significances" that "underlies the "arithmetical' understanding of geometry."

DOI: 10.1007/978-94-024-1132-4_17

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