### BibTeX

@MISC{Sieg_gödelon,

author = {Wilfried Sieg},

title = {Gödel on computability },

year = {}

}

### OpenURL

### Abstract

Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic

### Citations

1227 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...ote A philosophical error in Turing’s work, both of which had not been published yet. The former section contains remarks of Gödel’s on general recursive functions as well as a detailed discussion of =-=Turing 1936-=-; however, from Wang’s text it is not at all clear whether that discussion reflects Gödel’s understanding. 13 In the letter to Nagel, Gödel points out that Turing and Post clarified the general concep... |

404 | Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme,’ Monatshefte für Mathematik und Physik - Gödel - 1931 |

280 | An Unsolvable Problem of Elementary Number Theory - Church - 1936 |

108 | A set of postulates for the foundation of logic - Church - 1932 |

102 | Correction to a Note on the Entscheidungsproblem - Church - 1936 |

63 | Church’s thesis and principles for mechanics - Gandy - 1980 |

24 | Formal definitions in the theory of ordinal numbers - Church - 1936 |

24 | 2004] The Essential Turing - Copeland |

23 |
From Mathematics to Philosophy
- Wang
- 1974
(Show Context)
Citation Context ...eszen, and Neil Tennant have improved this essay; my sincere thanks to all. 1 I am in particular referring to Gödel 1951 and Turing 1954, but also to Gödel 1947 and Turing 1950. 2 Gödel 1951, p. 312, =-=Wang 1974-=-, pp. 324-6, and Turing 1948 pp.120-1. 1feature of such a machine, has to be supplemented by “initiative”, and it is a central scientific task “to discover the nature of this residue as it occurs in ... |

20 | Begrtlndung des "Tertium non datur' mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen 93 - Ackermann - 1925 |

14 | Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem - Behmann - 1922 |

10 | Mathematische probleme. Archiv der Mathematik und Physik - Hilbert - 1901 |

8 | Grundlagen der Mathematik, vols - Hilbert, Bernays |

7 | K-Graph machines: generalizing Turing’s machines and arguments - Sieg, Byrnes - 1996 |

7 | Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft - Bernays - 1976 |

5 |
Turing in the Land of O(z
- Feferman
- 1988
(Show Context)
Citation Context ...rs are operating on all kinds of finite discrete configurations, then I locate a deeply problematic appeal to intuition at the precise point where one has to claim that the operations of computors 18 =-=Feferman 1988-=- examines the use of intuition and ingenuity in the context of Turing’s work on ordinal logics; see, pp. 129-132. 17satisfying boundedness and locality conditions can be carried out by “string machin... |

5 | The history and concept of computability
- Soare
- 1999
(Show Context)
Citation Context ...as Church’s or Turing’s Thesis. In early 1934 Church had proposed in a conversation with Gödel to 4 This part of the history of computability has been widely discussed, for example, in Gandy 1988 and =-=Soare 1999-=-; contributions to that discussion, with a distinctive perspective on Gödel’s and Turing’s impact, are found in my papers 1994, 1997, and 2005. 3identify effective calculability of number-theoretic f... |

4 | On the consistency of arithmetic - Herbrand - 1931 |

4 | Mechanical procedures and mathematical experience. Mathematics and mind - Sieg - 1994 |

4 | 2000) Address at the Princeton University Bicentennial Conference on problems of mathematics (December 17-19, 19460, by Alfred Tarski - Sinaceur |

4 | Über Hilberts Gedanken zur Grundlegung der Arithmetik. Jahresbericht der Deutschen - Bernays - 1922 |

4 | Axiomatische Untersuchung des Aussagen-Kalkuls der “Principia mathematica - Bernays - 1926 |

4 | A set of postulates for the foundation of logic II - Church - 1933 |

3 | Beiträge zur axiomatischen Behandlung des Logik-Kalküls - Bernays - 1918 |

3 | Sur le concept de nombre en mathématique’. Cours inédit de Leopold Kronecker à Berlin (1891).” Revue d’Histoire des Mathématiques 7(2):207–275 - Boniface, Schappacher - 2001 |

3 | The Collected Works of - Büchi - 1990 |

2 | The Richard Paradox - Church - 1934 |

2 | Review of [Turing - Church - 1936 |

2 | Intelligent Machinery; Turing’s Ideas and Their Relation to the Work of Newell and Simon, unpublished M.S. Dissertation - Colvin - 1997 |

2 | Clockwork or Turing universe? — remarks on causal determinism and computability - Cooper - 1999 |

1 | An unsolvable problem of elementary number theory - Math - 1936 |

1 |
The Undecidable; New York 1982 Why Gödel did not have Church’s Thesis
- Davis
- 1965
(Show Context)
Citation Context ...ed purely in terms 3 This observation is formulated as a Bemerkung added in proof to Gödel 1936, and as a Remark in the paper’s first translation that was published on p. 83 of The Undecidable, i.e., =-=Davis 1965-=-. The significance of this discovery was described by Gödel in a letter to Kreisel of 1 May 1968: “That my [incompleteness] results were valid for all possible formal systems began to be plausible for... |

1 | About logics and logicians – A palimpsest of essays by - Odifreddi - 1990 |

1 | Emil 1947 Recursive unsolvability of a problem of Thue - Post |

1 | Introductory Note to Remark 3 - Webb - 1990 |

1 | Zum Hilbertschen Aufbau der rellen Zahlen - Ackermann - 1928 |

1 | Review of [Wolfram, 2002 - Baldwin - 2004 |

1 | A graphical presentation of Gandy’s parallel machines (Abstract - Byrnes, Sieg - 1996 |

1 | Die logizistische Grundlegung der Mathematik - Carnap - 1931 |

1 | Review of [Post - Church - 1936 |