How Metaphysical is "Deepening the Foundations"? - Hahn and Frank on Hilbert's Axiomatic Method

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Stoeltzner, Michael
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Only recently, David Hilbert's program to axiomatize the sciences according to the pattern of geometry has left the shade of his formalist program in the foundations of mathematics. This relative neglect - which is surprising in view of the enormous efforts Hilbert had himself devoted to it - was certainly influenced by Logical Empiricists' almost exclusively focusing on his contributions to the foundational debates. The present paper investigates the stand of two core members of the Vienna Circle who had studied with Hilbert at Göttingen, the mathematician Hans Hahn and the theoretical physicist Philipp Frank. At bottom of their neglect of Hilbert's axiomatic method stands their conviction that reconciling Ernst Mach's empiricist heritage with modern mathematics required to draw a rigid boundary between mathematics and physics and to subscribe to logicism, according to which mathematics consisted in tautologous logical transformations. In this way, they missed the substantial difference between the logical structure of a particular axiom system and the axiomatic method as a critical study of arbitrary axiom systems. If this distinction is not properly observed - and admittedly Hilbert himself did deliberately obscure it at places - a core concept of the axiomatic method, "deepening the foundations" (Tieferlegung), becomes metaphysical because it might appear as an ontological reduction of basic physical concepts to mathematical ones rather than - as Hilbert intended - an epistemological reduction availing itself of the unity of mathematical knowledge. To be sure, Logical Empiricists considered the goal of axiomatizing the sciences as an important task, but in the way how they set it up axiomatization became much closer tied to a success of the foundationalist program for all mathematics than Hilbert's axiomatic method ever was.