Chapter

Hilbert and Bernays on Metamathematics

Paolo Mancosu

in The Adventure of Reason

Published in print November 2010 | ISBN: 9780199546534
Published online May 2011 | e-ISBN: 9780191594939 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199546534.003.0002
Hilbert and Bernays on Metamathematics

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One the most influential foundational programs in the twentieth century is that championed by Hilbert. Hilbert had proposed a foundational approach for showing the reliability of classical mathematics that consisted in the formalization of classical mathematics followed by a consistency proof given by means of “finitist” considerations. This chapter provides a presentation of Hilbert’s program that is more sensitive to the diachronic changes in the program than has hitherto been the case and also sets Hilbert’s program in the context of the philosophical exchanges Hilbert and Bernays had with contemporary philosophers (Müller, Nelson) on topics such as the nature of ideal elements and the epistemic status of “finitist” intuition. The addendum to Chapter 2 discusses the evidence in favor of the claim that around 1922/23 Hilbert and Bernays changed position of the nature of “finitist” evidence abandoning an empiricist position in favor of a transcendentalist position.

Keywords: Bernays; Hilbert’s program; ideal elements; finitist intuition; Müller; Nelson

Chapter.  17681 words. 

Subjects: Philosophy of Mathematics and Logic

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