How metaphysical is "deepening the foundations"? - Hahn and Frank on Hilbert's axiomatic method
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
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.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Ivahn Smadja (2012). Local Axioms in Disguise: Hilbert on Minkowski Diagrams. Synthese 186 (1):315-370.
Volker Peckhaus (2003). The Pragmatism of Hilbert's Programme. Synthese 137 (1-2):141 - 156.
Richard Zach (2003). The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program. Synthese 137 (1-2):211 - 259.
Richard Zach, Hilbert's Program. Stanford Encyclopedia of Philosophy.
Volker Peckhaus (1995). Hilberts Logik. Von der Axiomatik zur Beweistheorie. NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 3 (1):65-86.
Kai F. Wehmeier (1997). Aspekte der frege–hilbert-korrespondenz. History and Philosophy of Logic 18 (4):201-209.
M. Stoltzner (2003). The Principle of Least Action as the Logical Empiricist's Shibboleth. Studies in History and Philosophy of Science Part B 34 (2):285-318.
Richard Zach (2006). Hilbert's Program Then and Now. In Dale Jacquette (ed.), Philosophy of Logic. North Holland. 5--411.
Michael Stöltzner (2003). The Principle of Least Action as the Logical Empiricist's Shibboleth. Studies in History and Philosophy of Science Part B 34 (2):285-318.
Ansten Klev (2011). Dedekind and Hilbert on the Foundations of the Deductive Sciences. Review of Symbolic Logic 4 (4):645-681.
Added to index2009-01-28
Total downloads5 ( #237,418 of 1,101,833 )
Recent downloads (6 months)0
How can I increase my downloads?