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The pursuit of rigor : David Hilbert’s early philosophy of mathematics Ogawa, Yoshinori

Abstract

The present study attempts to provide a new approach to Hilbert's philosophy of mathematics by going back to the origin of his foundational investigation and clearly describing the Problematik within which it was framed and developed. In so doing, its main objective is to identify and highlight the general intellectual tendencies invariably and continuously motivating Hilbert's research program throughout his long career. The study consists of two parts. In the first half, special emphasis is laid upon Hilbert's axiomatic method and his accompanying view of axioms and definitions. It is argued there that Hilbert's goal with his axiomatization program is to demonstrate the objectivity of mathematical judgment and inference and to systematize and thereby to increase our understanding of mathematics. The present study attempts to support this claim by embedding Hilbert's project in the context of the late nineteenth century movement of the "rigorization" of mathematics and by understanding it as a development of the methodological standpoint represented by Dedekind. On the interpretation presented here, then, Hilbert's foundational investigation was not, as is often claimed, motivated by the philosophical concerns for the absolute certainty and a prioricity of our mathematical knowledge and, indeed, it combated against the intrusion of such concerns by relegating framework-independent elements through the "new" methodological turn in the conception of axiomatics. In the second half of the study, this non-standard reading is extended to Hilbert's consistency program, and his first attempt of a direct consistency proof and Poincare's criticism of it are considered in this light. Hilbert's answer to Poincare came with the remarkable idea of proof-theory and the formulation of finitary mathematics as the framework for proof - theoretic considerations. But, seen from a philosophical viewpoint, this methodological move meant the re-introduction of the notions of truth and existence taken in the absolute sense and, as a result, a motivation for adopting (mathematical) instrumentalism as the philosophy of Hilbert's program arose. But even after this "epistemological" turn, the earlier view continued to be operative in Hilbert's thought, and this, I shall argue, explains the "tension" found in the philosophy behind Hilbert's program.

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