Abstract
The emergence of the mathematical concept of computability in the 1930 s was marked by an interesting shift of perspective, from viewing the intuitive concept, “human calculability following a fixed routine” in terms of calculability in a logic, to viewing the concept as more adequately expressed by Turing’s model.
Juliette Kennedy: I thank Liesbeth de Mol for the invitation to address a special session of CiE2017, and to contribute this extended abstract to its proceedings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Gandy quote. See below.
- 2.
See, e.g. Sieg [3], p. 387.
- 3.
Some of the text below is adapted from the author’s [4].
- 4.
[1], Sect. 14.8. Following Gandy we use the term “effectively computable,” or just “effective,”to mean “intuitively computable.”.
- 5.
- 6.
The phrase “Church’s Thesis” was coined by Kleene in 1943. See his [10].
- 7.
- 8.
Church, letter to Kleene November 29, 1935. Quoted in Sieg, op. cit., and in Davis [11].
- 9.
Church, op.cit. In this talk and elsewhere we distinguish the axiomatic from the logical method, viewing the former as an informal notion, and the latter as involving a formalism.
- 10.
[13], p. 357.
- 11.
In detail, recursive enumerability would be guaranteed here by the so-called step-by-step argument: if each step is recursive then f will be recursive; and three conditions: (i) each rule must be a recursive operation, (ii) the set of rules and axioms must be recursively enumerable, (iii) the relation between a positive integer and the expression which stands for it must be recursive. Conditions i-iii are Sieg’s formulation of Church’s conditions. See Sieg, [12], p. 165. For Gandy’s formulation of the step-by-step argument, see [1], p. 77.
- 12.
See also Sieg’s discussion of the “semi-circularity” of the step-by-step argument in his [12].
- 13.
[1], p. 71. Of course this shift presaged a much more dramatic shift of perspective in 1936, inaugurated by the work of Turing together with Post’s earlier work.
- 14.
See Gandy’s [1], p. 72.
- 15.
Gödel was careful not to claim complete generality for the Second Incompleteness Theorem in his 1931 paper:
For this [formalist JK] viewpoint presupposes only the existence of a consistency proof in which nothing but finitary means of proof is used, and it is conceivable that there exist finitary proofs that cannot be expressed in the formalism of P (or of M and A).Footnote 16
- 16.
[15], p. 195. P is a variant of Principia Mathematica.
- 17.
Sieg [16], p. 554.
- 18.
[15], p. 346 and 349, resp. Emphasis added.
- 19.
As Gödel would later write to Martin Davis, “...I was, at the time of these lectures, not at all convinced that my concept of recursion comprises all possible recursions.” Quoted in [15], p. 341.
- 20.
[15], p. 361.
- 21.
As Gödel wrote to Kreisel in 1965:
That my [incompleteness] results were valid for all possible formal systems began to be plausible for me (that is since 1935) only because of the Remark printed on p. 83 of ‘The Undecidable’ ...But I was completely convinced only by Turing’s paper.Footnote 22
- 22.
Quoted in Sieg [17], in turn quoting from an unpublished manuscript of Odifreddi, p. 65.
- 23.
Or quintuples, in Turing’s original presentation.
- 24.
As Sieg puts it [14], “The work of Gödel, Church, Kleene, and Hilbert and Bernays had intimate historical connections and is still of deep interest. It explicated calculability of functions by exactly one core notion, namely calculability of their values in logical calculi via (a finite number of) elementary steps. But no one gave convincing and non-circular reasons for the proposed rigorous restrictions on the steps that are permitted in calculations.”.
- 25.
Remark to Hao Wang, [18], p. 203. Emphasis added.
- 26.
[2], p. 80.
- 27.
ibid, p. 85.
- 28.
See Gödel’s letter to van Heijenoort of February 22, 1964, in [20].
References
Gandy, R.: The confluence of ideas in 1936. In: The universal turing machine: A half-century survey, pp. 55–111. Oxford Science Publications, Oxford University Press, New York (1988)
Kripke, S.: The church-turing “Thesis” as a special corollary of gödel’s completeness theorem. In: Copeland, B.J., Posy, C.J., Shagrir, O. (eds.) Computability: Gödel, Church, and Beyond. MIT Press, Cambridge (2013)
Sieg, W.: Hilbert’s Programs and Beyond. Oxford University Press, Oxford (2013)
Kennedy, J.: Turing, Gödel and the “Bright Abyss”. In: Philosophical Explorations of the Legacy of Alan Turing, vol. 324 of Boston Studies in Philosophy (Springer)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys. 38, 173–198 (1931)
Church, A.: A set of postulates for the foundation of logic I, II. Ann. Math. 33(2), 346–366 (1932)
Kleene, S.C., Rosser, J.B.: The inconsistency of certain formal logics. Ann. Math. 36(2), 630–636 (1935)
Church, A.: The richard paradox. Amer. Math. Monthly 41, 356–361 (1934)
Kleene, S.C.: Origins of recursive function theory. Ann. Hist. Comput. 3, 52–67 (1981)
Kleene, S.C.: Recursive predicates and quantifiers. Trans. Am. Math. Soc. 53, 41–73 (1943)
Davis, M.: Why Gödel didn’t have church’s thesis. Inf. Control 54, 3–24 (1982)
Sieg, W.: Step by recursive step: Church’s analysis of effective calculability. Bull. Symbolic Log. 3, 154–180 (1997)
Church, A.: A note on the Entscheidungsproblem. J. Symbolic Log. 1(1), 40–41 (1936). (Correction 1:101–102)
Sieg, W.: Church without dogma: Axioms for computability. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 139–152. Springer, New York (2008)
Gödel, K.: Collected works. Vol. I. The University Press, New York (1986). Publications 1929–1936, Edited and with a preface by Solomon Feferman. Clarendon Press, Oxford
Sieg, W.: On computability. In: Philosophy of Mathematics. Handbook of the Philosophy of Science, pp. 535–630. Elsevier/North-Holland, Amsterdam (2009)
Sieg, W.: Gödel on computability. Philos. Math. 14(3), 189–207 (2006)
Wang, H.: A Logical Journey. Representation and Mind. MIT Press, Cambridge (1996)
Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284–316 (1944)
Gödel, K.: Collected Works. V: Correspondence H-Z. Oxford University Press, Oxford (2003). Feferman, S., et al. (eds.)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kennedy, J. (2017). Gödel’s Reception of Turing’s Model of Computability: The “Shift of Perception” in 1934. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-58741-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58740-0
Online ISBN: 978-3-319-58741-7
eBook Packages: Computer ScienceComputer Science (R0)