## Between Russell And Hilbert: Behmann On The Foundations Of Mathematics (1999)

Venue: | Bulletin of Symbolic Logic |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Mancosu99betweenrussell,

author = {Paolo Mancosu},

title = {Between Russell And Hilbert: Behmann On The Foundations Of Mathematics},

journal = {Bulletin of Symbolic Logic},

year = {1999},

volume = {5},

pages = {303--330}

}

### Years of Citing Articles

### OpenURL

### Abstract

. After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in G ottingen 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 Auflosung 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 of Mathematics in G ...

### Citations

99 |
Principia Mathematica
- Whitehead, Russell
- 1957
(Show Context)
Citation Context ...Schsonfinkel [13] goes back to work by Schsonfinkel in 1922 (see [13], p. 350). We know little about Boskovitz but he is, together with Behmann, thanked in the second edition of Principia Mathematica =-=[36]-=- (henceforth PM) for his help in correcting various oversights found in the first edition. 2 It is during the period from 1918 to 1922 that the major ideas and some results about metatheoretical inves... |

20 |
Randbemerkungen zu Hauptproblemen der Mathematik, II, Fundamentalsatz der Algebra and Grundlagen der
- Weyl
(Show Context)
Citation Context ... of the above. 37. The metaphor of the chess game to speak about formalized mathematics had great fortune in the twenties. However, it is usually attributed to Weyl who formulated it in print in 1924 =-=[33, 34]. 38.-=- " Bekanntlich lsat sich die symbolische Logik axiomatisieren, d. h. auf ein System verh altnism aig weniger Grundformeln und Grundregeln zursuckfsuhren, soda auch das Beweisen von Satzen nunmehr... |

12 | Hilbert’s programs: 1917–1922
- Sieg
- 1999
(Show Context)
Citation Context ...tingen 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 1... |

10 |
Hilbertprogramm und kritische Philosophie, Göttingen: Vandenhoeck
- Peckhaus
- 1991
(Show Context)
Citation Context ...angen). The normal form procedure is found in [18]. Sieg [30, p. 8] gives additional evidence from Hilbert's lecture course given in 1910 "The quadrature of the circle and related problems."=-= Peckhaus [28]-=- also provides very relevant information for the period in question. However, Behmann does not appear in Peckhaus' account. 4. Sieg [30, pp. 37--38] has edited the postcard exchange between Russell an... |

9 |
Completeness before Post: Bernays, Hilbert, and the development of propositional logic
- Zach
- 1999
(Show Context)
Citation Context ...f 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 foundation... |

7 |
Grundbegriffe der Mengenlehre. Abhandlungen der Fries’schen Schule n. s
- Hessenberg
- 1906
(Show Context)
Citation Context ...the Entscheidungsproblem. One should not confuse Behmann's introduction of the Entscheidungsproblem either with the Entscheidungsproblem formulated, in connection to Grelling's paradox, in Hessenberg =-=[16]-=- or with the Kroneckerian requirement of decidability for the introduction of concepts in mathematics (for the latter see Pasch [27]). Sieg [30, p. 19] tentatively claims that the first occurrence of ... |

5 |
Die Philosophie des Als
- Vaihinger
- 1927
(Show Context)
Citation Context .... The concept of fiction was of course much discussed during this period. One of the major philosophical developments of the decade had been the publication of Hans Vaihinger's Philosophie des Als Ob =-=[31]-=-. However, the book had not been received favourably by mathematicians. Behmann found the dismissal of the book to be explained by a number of blunders and infelicities in Vaihinger's discussion of is... |

5 | Heijenoort (editor), From Frege to G odel: A source book in mathematical logic, 1879--1931 - van - 1967 |

2 |
zur Algebra der Logik, insbesondere zum Entscheidungsproblem
- Beitrage
- 1922
(Show Context)
Citation Context ...neuen arithmetischen Theorien von Weyl und Brouwer February 21, 22, D. Hilbert, Eine neue Grundlegung des Zahlbegri#es [21] May 10, 1921, H. Behmann, Das Entscheidungsproblem der mathematischen Logik =-=[4]-=- December 6, 1921, P. Bernays and M. Schsonfinkel, Das Entscheidungsproblem im Logikkalksul Even a cursory look at the above list of talks should convince the reader that Behmann was a central player ... |

2 | Hilberts Gedanken zur Grundlegung der Arithmetik, Jahresbericht der Deutschen - Uber - 1922 |

2 |
and Moses Sch onfinkel, Zum Entscheidungsproblem der mathematischen Logik
- Bernays
- 1928
(Show Context)
Citation Context ...el is well known for his article "ber die Bausteine der mathematischen Logik," all of whose contents go back to a talk he gave in Gsottingen in 1920. 1 Moreover, the paper by Bernays and Sch=-=sonfinkel [13]-=- goes back to work by Schsonfinkel in 1922 (see [13], p. 350). We know little about Boskovitz but he is, together with Behmann, thanked in the second edition of Principia Mathematica [36] (henceforth ... |

2 |
der Mathematik, Lecture notes by Paul Bernays. WinterSemester 1917--18. Unpublished typescript
- Prinzipien
(Show Context)
Citation Context ...to 1928, and Mancosu [25] has investigated the philosophical context of Hilbert's approach to the foundations of mathematics. The Habilitationsschriftsby 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... |

1 |
mathematische Logik, unpublished manuscript, dated December 1
- Behmann, Uber
- 1914
(Show Context)
Citation Context ...dea of the increasing interest in logic and foundations in Gsottingen from 1914 onwards. February 17, 1914, D. Hilbert, Axiome der ganzen Zahlen December 1, 1914, H. Behmann,sUber mathematische Logik =-=[1] Dece-=-mber 18, 1914, F. Berstein and K. Grelling,sUber mathematische Logik. Erg anzungen und genauere Ausfsuhrungen zum Referat vom 1. Dezember February 16, 1915, F. Bernstein,sUber J. Ksonigs "Neue Gr... |

1 |
Antinomie der transfiniten Zahl und ihre Auflosung durch die Theorie von Russell und Whitehead, Dissertation, Universit at G ottingen
- Die
- 1918
(Show Context)
Citation Context ...thmetik und Mengenlehre" November 7, 1916, E. Zermelo, Uber einige neuere Ergebnisse in der Theorie der Wohlordnung November 14, 1916, H. Behmann, Die Russell-Whiteheadsche Theorie und die Parado=-=xien [2]-=- July 3, 1917, H. Behmann, Die Russell-Whiteheadsche Theorie und die Grundlagen der Arithmetik [2] July 10, 1917, H. Behmann, Die Russell-Whiteheadsche Theorie und die Grundlagen der Arithmetik (Schlu... |

1 | und Algebra der Logik, unpublished manuscript, dated May 10 - Entscheidungsproblem - 1921 |

1 |
Kritik der Analysis, Hilbert Nachla, Nieders achsiche Staats- und Universit atsbibliothek
- Weyls
- 1917
(Show Context)
Citation Context ..., F. Bernstein, Geschichte der Mengenlehre (Schlu) July 31, 1917, D. Hilbert, Referatsuber seine Vorlesungensuber Mengenlehre [17] November 20, 1917, P. Bernays, Weylsuber die Grundlagen der Analysis =-=[7]-=- November 27, 1917, D. Hilbert,sUber axiomatisches Denken [20] December 7, 1920, M. Schsonfinkel, Elemente der Logik [29] February 1 and 8, 1921, R. Courant and P. Bernays,sUber die neuen arithmetisch... |

1 |
zur axiomatischen Behandlung des Logik-Kalkuls, Habilitationsschrift, Universit at G ottingen
- Beitrage
- 1918
(Show Context)
Citation Context ...alculus 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. The Habilitationsschriftsby 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 ... |

1 | of: Aloys Muller, "Der Gegenstand der Mathematik - Review - 1923 |

1 |
Nachla Heinrich Behmanns (1891--1970): Gesamtverzeichnis, Aachener Schriften zur Wissenschaftstheorie, Logik und Logikgeschichte
- Haas, Stemmler, et al.
- 1981
(Show Context)
Citation Context ... curiosity can be satisfied. The lecture is extant in its entirety in the Behmann Nachla although it does not appear in the list of the documents contained in the Nachla compiled by Haas and Stemmler =-=[15]-=-. I will here stress only two aspects of this lecture that seem to be quite relevant for later developments in Hilbert's program. Let me also remark that on December 18, 1914 Bernstein and Grelling ga... |

1 |
Lecture notes by Margarethe Loeb. Sommer-Semester 1917. Unpublished manuscript
- Hilbert, Mengenlehre
(Show Context)
Citation Context ... 305 July 17, 1917, F. Bernstein, Geschichte der Mengenlehre July 24, 1917, F. Bernstein, Geschichte der Mengenlehre (Schlu) July 31, 1917, D. Hilbert, Referatsuber seine Vorlesungensuber Mengenlehre =-=[17]-=- November 20, 1917, P. Bernays, Weylsuber die Grundlagen der Analysis [7] November 27, 1917, D. Hilbert,sUber axiomatisches Denken [20] December 7, 1920, M. Schsonfinkel, Elemente der Logik [29] Febru... |

1 |
Principien des mathematischen Denkens, Vorlesung, SommerSemester 1905. Lecture notes by Ernst Hellinger. Unpublished manuscript. Bibliothek, Mathematisches Institut, Universit at G ottingen
- Logische
(Show Context)
Citation Context ...Darstellung der Logik, und zwar insbesondere der Aussagenlogik, besch aftigte, kennen gelernt" (Behmann to Scholz, December 29, 1927, Behmann Archive, Erlangen). The normal form procedure is foun=-=d in [18]. Sieg [30-=-, p. 8] gives additional evidence from Hilbert's lecture course given in 1910 "The quadrature of the circle and related problems." Peckhaus [28] also provides very relevant information for t... |

1 |
pp. 405--15, Lecture given at the Swiss
- Denken
- 1918
(Show Context)
Citation Context ...917, D. Hilbert, Referatsuber seine Vorlesungensuber Mengenlehre [17] November 20, 1917, P. Bernays, Weylsuber die Grundlagen der Analysis [7] November 27, 1917, D. Hilbert,sUber axiomatisches Denken =-=[20]-=- December 7, 1920, M. Schsonfinkel, Elemente der Logik [29] February 1 and 8, 1921, R. Courant and P. Bernays,sUber die neuen arithmetischen Theorien von Weyl und Brouwer February 21, 22, D. Hilbert, ... |

1 |
und mathematisches Erkennen: Vorlesungen, gehalten 1919--1920
- Natur
- 1992
(Show Context)
Citation Context ...arly by Behmann in 1914 and treated at length in his dissertation of 1918. Hilbert's detailed treatment of ideal elements is first found in the 1919--1920 lectures "Natur und mathematisches Erken=-=nen" [23]-=-. I should emphasize that I am not making here a priority claim on behalf of Behmann. Rather, I am simply interested in pointing out that within Hilbert's circle extensive discussion of ideal elements... |

1 |
Bernays on metamathematics
- Hilbert
(Show Context)
Citation Context ...stigated 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. The Habilitationsschriftsby Bernays [8] and Hilbert's 1917--1918 lectures [19] represent the starti... |

1 |
and the emergence of modern mathematical logic
- Moore, Hilbert
- 1997
(Show Context)
Citation Context ...0 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, an... |

1 |
Die Forderung der Entscheidbarkeit, Jahresbericht der Deutschen
- Pasch
- 1918
(Show Context)
Citation Context ...blem formulated, in connection to Grelling's paradox, in Hessenberg [16] or with the Kroneckerian requirement of decidability for the introduction of concepts in mathematics (for the latter see Pasch =-=[27]-=-). Sieg [30, p. 19] tentatively claims that the first occurrence of the word Entscheidungsproblem is found in lectures of Hilbert from 1922--23. The assertion should be corrected in light of the above... |

1 |
Sch onfinkel, Uber die Bausteine der mathematischen Logik
- Moses
- 1924
(Show Context)
Citation Context ...lehre [17] November 20, 1917, P. Bernays, Weylsuber die Grundlagen der Analysis [7] November 27, 1917, D. Hilbert,sUber axiomatisches Denken [20] December 7, 1920, M. Schsonfinkel, Elemente der Logik =-=[29]-=- February 1 and 8, 1921, R. Courant and P. Bernays,sUber die neuen arithmetischen Theorien von Weyl und Brouwer February 21, 22, D. Hilbert, Eine neue Grundlegung des Zahlbegri#es [21] May 10, 1921, H... |

1 |
heutige Erkenntnislage in der Mathematik
- Die
- 1925
(Show Context)
Citation Context ... of the above. 37. The metaphor of the chess game to speak about formalized mathematics had great fortune in the twenties. However, it is usually attributed to Weyl who formulated it in print in 1924 =-=[33, 34]. 38.-=- " Bekanntlich lsat sich die symbolische Logik axiomatisieren, d. h. auf ein System verh altnism aig weniger Grundformeln und Grundregeln zursuckfsuhren, soda auch das Beweisen von Satzen nunmehr... |