Between Russell and Hilbert: Behmann on the foundations of mathematics

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann’s doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann’s work and Hilbert’s foundational thought. §1. Logic and Foundations ofMathematics in Göttingen from 1910 to 1921. Recent work on Hilbert’s program has focused, among other things, on the development of logic in Hilbert’s school and on the philosophical underpinnings of the program. Sieg [30] and Moore [26] have investigated the development of first-order logic in Hilbert’s 1917–18 lectures, Zach [37] has given an in-depth analysis of the propositional calculus in Hilbert’s school from 1918 to 1928, and Mancosu [25] has investigated the philosophical context of Hilbert’s approach to the foundations of mathematics. TheHabilitationsschrift by Bernays [8] and Hilbert’s 1917–1918 lectures [19] represent the starting point of these important developments. However, these lectures were not the product of a sudden reawakening of interest in logic and the Received December 2, 1998; revised July 20, 1999. I would like to thank Volker Peckhaus, Christian Thiel, Peter Bernhard and Richard Zach for comments and for making it possible to access and reproduce some of the materials contained in the Behmann Archive in Erlangen. I am grateful to an anonymous referee for his comments, which helped me sharpen a number of issues raised in the paper. I am also grateful to the curators of the following collections for their help: Russell archive at McMaster University, Hamilton; Bernays Nachlaß, ETH Zürich; Hugo Dingler-Nachlaß, Aschaffenburg; Hilbert Nachlaß, Göttingen. I would finally like to thank theWissenschaftskolleg zu Berlin for having provided ideal conditions for work on the first draft of this paper during the academic year 1997–98. c © 1999, Association for Symbolic Logic 1079-8986/99/0503-0002/$3.80