Juliet Floyd
Boston University, United States of America | jfloyd@bu.edu
Received: 20-December-2018 | Accepted: 3-April-2019 | Published: 30-June-2019
Disputatio [Jun. 2019], Vol. 8, No. 9, pp. 00-00 | DOI: 10.5281/zenodo.3568216
Article | [EN] | Full Text | Statistics | Copyright Notice [es] | Vol. 8 No. 9
How to cite this article:
Floyd, Juliet (2019). «Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing / El argumento diagonal de Wittgenstein: una variación sobre Cantor y Turing». Disputatio. Philosophical Research Bulletin 8, no. 9: pp. 00–00.
Abstract | Turing was a philosopher of logic and mathematics, as well as a mathematician. His work throughout his life owed much to the Cambridge milieu in which he was educated and to which he returned throughout his life. A rich and distinctive tradition discussing how the notion of “common sense” relates to the foundations of logic was being developed during Turing’s undergraduate days, most intensively by Wittgenstein, whose exchanges with Russell, Ramsey, Sraffa, Hardy, Littlewood and others formed part of the backdrop which shaped Turing’s work. Beginning with a Moral Sciences Club talk in 1933, Turing developed an “anthropological” approach to the foundations of logic, influenced by Wittgenstein, in which “common sense” plays a foundational role. This may be seen not only in “On Computable Numbers” (1936/7) and Turing’s dissertation (written 1938, see (1939)), but in his exchanges with Wittgenstein in 1939 and in two later papers, “The Reform of Mathematical Phraseology and Notation” (1944/5) and “Solvable and Unsolvable Problems” (1954).
Keywords | Common Sense · Formal System · Turing Machine · Ordinary Language · Philosophical Discussion.
El argumento diagonal de Wittgenstein: una variación sobre Cantor y Turing
Resumen | Turing era un filósofo de la lógica y matemática, y también era un matemático. Su trabajo a lo largo de toda su vida estaba muy en deuda con el ambiente de Cambridge en que fue educado y al cual volvió toda su vida. Se desarrolló una tradición rica y distintiva de la noción como el “sentido común” se relaciona con los fundamentos de la lógica durante los días de estudiante de Turing, de manera sumamente intensiva por Wittgenstein. Los intercambios de éste con Russell, Ramsey, Sraffa, Hardy, Littlewood y otros era parte del trasfondo que moldeó el trabajo de Turing. A partir de un discurso en el Moral Sciences Club in 1933, Turing desarrolló un planteamiento “antropológico” a la fundación de la lógica, influenciado por Wittgenstein, en el cual el “sentido común” juega un papel fundacional. Esto se puede no sólo en “On Computable Numbers” (1936/1937) y la disertación de Turing (redactada en 1938, véase (1939)), sino en sus intercambios con Wittgenstein en 1939 y en dos trabajos posterioes, “The Reform of Mathematical Phraseology and Notation” (1944/1945) y “Solvable and Unsolvable Problems” (1954).
Palabras Clave | Sentido Común · Sistemas Formales · Máquina de Turing · Lenguaje Común · Discusión Filosófica.
References
Church, Alonzo (1936). “An unsolvable problem of elementary number theory”. American Joumal of Mathematics, vol. 58, no. 2: 345–363. doi: https://doi.org/10.2307/2371045
Copeland, Brian J. (ed.). (2004). The Essential Turing: The ideas that gave birth to the computer age. Oxford: Clarendon Press.
Dreben, Burton, y Juliet Floyd. (1991). “Tautology: How not to use a word”. Synthese vol. 87, no. 1: pp. 23–50. doi: https://doi.org/10.1007/BF00485329
Floyd, Juliet (2001). Prose versus proof: “Wittgenstein on Gödel, Tarski and Truth”. Philosophia Mathematica, vol. 9, no. 3: pp. 901–928. doi: https://doi.org/10.1093/philmat/9.3.280
Fogelin, Robert J. (1987). Wittgenstein. London/New York: Routledge & K. Paul.
Gandy, Robin O. (1988). “The confluence of ideas in 1936”. In: The universal Turing machine: A halfcentury survey, ed. R. Herken, pp. 55–112. New York: Oxford University Press.
Gefwert, Christoffer (1998). Wittgenstein on mathematics, minds and mental machines. Burlington: Ashgate Publishing.
Gödel, Kurt (1986). Kurt Gödel collected works. Volume I: Publications 1929–1936. New York: Oxford University Press.
Gödel, Kurt (1990). Kurt Gödel collected works. Volume II: Publications 1938–1974. New York: Oxford University Press.
Hodges, Andrew (1983). Alan Turing the enigma of intelligence. New York: Touchstone.
Hodges, Wilfrid (1998). “An editor recalls some hopeless papers”. Bulletin of Symbolic Logic, vol. 4, no. 1 : pp. 1–16. doi: https://doi.org/10.2307/421003
Kennedy, Juliette (unpublished). Gödel’s quest for decidability: The method of formal systems; The method of informal rigor.
Kreisel, Georg (1950). “Note on arithmetic models for consistent formulae of the predicate calculus”. Fundamenta Mathematicae 37: pp. 265–285. doi: https://doi.org/10.4064/fm-37-1-265-285
Kripke, Saul A. (1982). Wittgenstein on rules and private language: An elementary exposition. Cambridge: Harvard University Press.
Marion, Mathieu (2011). “Wittgenstein on the surveyability of proofs”. In The Oxford handbook to Wittgenstein, ed. M. McGinn. New York/Oxford: Oxford University Press. doi: https://doi.org/10.1093/oxfordhb/9780199287505.003.0008
Martin–Löf, Per (1984). Intuitionistic type theory. Napoli: Bibliopolis.
Martin–Löf, Per (1996). “On the meanings of the logical constants and the justifications of the logical laws”. Nordic Journal of Philosophical Logic, vol. 1 no. 1: pp. 11–60.
McGuinness, Brian (ed.). (2008). Wittgenstein in Cambridge: Letters and documents, 1911–1951. Malden/Oxford: Blackwell. doi: https://doi.org/10.1002/9781444301243
Mühlhölzer, Felix (2010). Braucht die Mathematik eine Grundlegung? Ein Kommentar des Teils Ill von Wittgensteins Bemerkungen iiber die Grundlagen der Mathematik. Frankfurt am Main: Vittorio Klostermann.
Petzold, Charles (2008). The annotated Turing: A guided tour through Alan Turing’s historic paper on computability and the Turing machine. Indianapolis: Wiley Publishing, Inc.
Quine, Willard V. (1937). “New foundations for mathematical logic”. American Mathematical Monthly 44: pp. 70–80. doi: https://doi.org/10.1080/00029890.1937.11987928
Quine, Willard V. (1953, 1980). From a logical point of view. Cambridge: Harvard University Press.
Shanker, Stuart G. (1987). “Wittgenstein versus Turing on the nature of Church’s thesis”. Notre Dame Journal of Formal Logic, vol. 28, no. 4: pp. 615–649. doi: https://doi.org/10.1305/ndjfl/1093637650
Shanker, Stuart G. (1998). Wittgenstein’s remarks on the foundations of AI. New York: Routledge.
Sieg, Wilfried (1994). “Mechanical procedures and mathematical experience”. In Mathematics and mind, ed. A. George, 91–117. New York/Oxford: Oxford University Press.
Sieg, Wilfried (2006a). “Gödel on computability”. Philosophia Mathematica, vol. 14, no. 2: pp. 189–207. doi: https://doi.org/10.1093/philmat/nkj005
Sieg, Wilfried (2006b). “Step by recursive step: Church’s analysis of effective calculability”. In Church’s thesis after 70 years, ed. A. Olszewski, J. Wolenski and R. Janusz, pp. 456–485. Frankfurt/Paris/Ebikon/Lancaster/New Brunswick: Ontos Verlag.
Sieg, Wilfried (2008). “On computability”. In Handbook of the philosophy of science: Philosophy of mathematics, ed. A. Irvine pp. 535-630. Amsterdam: Elsevier BY. doi: https://doi.org/10.1016/B978-0-444-51555-1.50017-1
Stenius, Erik (1970). “Semantic antinomies and the theory of well–formed rules”. Theoria, vol. 36, no. 2: pp. 142–160. doi: https://doi.org/10.1111/j.1755-2567.1970.tb00416.x
Turing, A.M. (1937a). “On calculable numbers, with an application to the Entscheidungsproblem”. Proceedings of the London Mathematical Society vol. s2– 42, no. 1: pp. 230–265. doi: https://doi.org/10.1112/plms/s2-42.1.230
Turing, A.M. (1937b). “On calculable numbers, with an application to the Entscheidungsproblem. A correction”. Proceedings of the London Mathematical Society vol. s2-43, no. 1: pp. 544–546. doi: https://doi.org/10.1112/plms/s2-43.6.544
Turing, A.M. (1937c). Letter to Ethel Sarah Turing. Cambridge, U.K.: King’s College Archives, K/1/54, February 11, 1937.
Turing, A.M. (1937d). Correspondence with Paul Bernays. Zürich: Eidgenossische Technische Hochschule Zürich/Swiss Federal Institute of Technology Zürich, Bibliothek.
Turing, A.M. (1950). “Computing machinery and intelligence”. Mind, vol. 59, no. 236: pp. 433–460. doi: https://doi.org/10.1093/mind/LIX.236.433
Turing, A.M. (1954). “Solvable and unsolvable problems”. Science News, vol. 20, no. 31: pp. 7–23.
Watson, A.G.D. (1938). “Mathematics and its foundations”. Mind, vol. 47, no. 188: pp. 440–451. doi: https://doi.org/10.1093/mind/XLVII.188.440
Webb, Judson C. (1990) “Remark 3, introductory note to Gödel (1972a)”. In Kurt Gödel collected works. Volume Il: Publications 1938–1974, eds. S. Feferman, et al., pp. 281–304. New York: Oxford University Press.
Wittgenstein, Ludwig (1970). Zettel [Z]. Berkeley: University of California Press.
Wittgenstein, Ludwig (1978). Remarks on the foundations of mathematics. Cambridge: MIT Press.
Wittgenstein, Ludwig (1980). Wittgenstein’s lectures, Cambridge 1930–32, from the notes of John King and Desmond Lee [DL]. Oxford: Blackwell.
Wittgenstein, Ludwig (1999). The published works of Ludwig Wittgenstein [CD–Rom], Charlottesville, VA/Oxford: Intelex Corporation. Oxford University Press.
Wittgenstein, Ludwig (2004). Ludwig Wittgenstein: Briefwechsel [CD–Rom, Innsbrucker elektronische Ausgabe], ed. M. Seekircher, B. McGuinness, A. Unterkircher, A. Janik and W. Methlagl. Charlottesville, VA: Intelex Corporation.
Wittgenstein, Ludwig, and G.H.Y. Wright et al. (1980). Remarks on the philosophy of psychology, vol. 1[RPP I]. Chicago/Oxford: University of Chicago Press/B. Blackwell.
Wittgenstein, Ludwig (1989). Wittgenstein’s lectures on the foundations of mathematics: Cambridge, 1939, ed. C. Diamond. Chicago: University of Chicago Press. Wright, C. (2001). Rails to infinity: Essays on themes from Wittgenstein’s philosophical investigations. Cambridge: Harvard University Press. doi: https://doi.org/10.7208/chicago/9780226308609.001.0001
© The author(s) 2019. This work, published by Disputatio [www.disputatio.eu], is an Open Access article distributed under the terms of the Creative Commons License [BY–NC–ND]. The copy, distribution and public communication of this work will be according to the copyright notice. For inquiries and permissions, please email: boletin@disputatio.eu.