Between Russell and Hilbert: Behmann on the foundations of mathematics. (English) Zbl 0942.03002

Mancosu, Paolo

Bull. Symb. Log. 5, No. 3, 303-330 (1999).

This paper, based on unpublished and archival sources, gives an important contribution to the prehistory of D. Hilbert’s metamathematical program. It focuses on the role of Heinrich Behmann (1891-1970) who defended his dissertation “Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead” in Göttingen 1918. Only a small part of it was published [H. Behmann, Jahrbuch Math.-Naturwiss. Fak. Göttingen 23, 55-64 (1922)].
The author is most interested in Behmann’s philosophical considerations concerning the foundations of arithmetic which might be read as an attempt to make use of the theory of B. Russell and A. N. Whitehead’s “Principia Mathematica” [3 vols., Cambridge UP, Cambridge (1910-13); partial reprint of vol. 1 see Zbl 0877.01042)] for the foundational efforts in Hilbert’s circle. For Behmann, arithmetic is justified as a science by its function of investigating certain features of concrete objects of experienced reality. Justification by applicability seems to entail an aposteriori character of mathematical knowledge. Not for Behmann. He opts for the aprioricity of arithmetic.
The author gives good evidence for his assumption “that through Behmann’s dissertation and his earlier lectures it is possible to see some elements of continuity between the logicist approach [of Russell and Whitehead] and Hilbert’s program as it was presented in the early twenties” (p.317).

Reviewer: V.Peckhaus (Erlangen)

MSC:

03-03	History of mathematical logic and foundations
01A60	History of mathematics in the 20th century
03A05	Philosophical and critical aspects of logic and foundations

Keywords:

Hilbert’s program; foundations; type theory; axiom of reducibility; Cantor’s paradox; context principle; individuals; propositional functions; classes; aprioricity; fictions; metamathematics; Entscheidungsproblem; no-class theory

Citations:

Zbl 0877.01042

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