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 ” (1936/1937) and Turing’s dissertation (1939), but in his exchanges with Wittgenstein in 1939 and in two later papers, “The Reform of Mathematical Phraseology and Notation” (1944/1945) and “Solvable and Unsolvable Problems” (1954).
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Notes
- 1.
- 2.
Hodges (1983/2012), p. 19.
- 3.
Hodges (1983/2012) pp. 63–64 and (1999) pp. 5–7 focus on the importance of Turing’s essay “The Nature of Spirit” (1932), where Turing discusses McTaggart, and his reading of Eddington’s (1929) before coming to Cambridge. Hodges also argues (1983/2012) that Eddington’s autumn 1933 course on the methodology of science also had a serious impact on Turing (p. 87), not least setting him on a course of thinking about the Central Limit Theorem, proved in his Fellowship essay. We shall contest none of this.
- 4.
In (1983/2012) Hodges weaves general aspects of Wittgenstein’s thought into his discussion of Turing’s work leading up to “On Computable Numbers”. In (1999), however, he stresses the absence of any direct positive evidence for Wittgenstein’s impact on Turing, arguing that Wittgenstein’s 1939 lectures (1976/1989) (LFM) “shed no light on Turing’s view of mind and machine” (pp. 22–24). In what follows we agree, but offer a different, though admittedly circumstantial, reconstruction of Turing’s development spelling out Hodges’ prior hints. As we argue, the nature of mind was not the driving issue behind Turing’s initial work in logic, whereas the nature of logic was.
- 5.
- 6.
Post (1944), p. 284 is looking backward here, summarizing the developments historically.
- 7.
Hodges (1983/2012), p. 96.
- 8.
- 9.
- 10.
The idea of “phraseology” occurs explicitly in The Blue Book (1965), p. 69; it appears in Wittgenstein’s manuscripts and lectures around differing conceptions of numbers and mathematics, cf. e.g. (1999) MS 121, p. 76 (1939), MS 126, p 131 (1942–1943); MS 127, p. 194 (1943) and (1976), pp. 18,91,98. It also occurs in Russell (1919), pp. 141–175, 192. Floyd and Mühlhölzer (unpublished) discuss this notion at length in the context of an analysis of Wittgenstein’s annotations to a 1941 edition Hardy’s Course of Pure Mathematics.
- 11.
I reconstruct the argument precisely in Floyd (2012b).
- 12.
Gandy (1988), p. 85.
- 13.
Davis (1982), p. 14.
- 14.
Gandy (1988), p. 86.
- 15.
The method of “comparisons” is evinced in The Blue Book (1965) (BB), pp. 5,70 and especially in The Brown Book BB §§13,10 (p. 140), §13 (p. 153), §15 (p. 158), §16 (pp. 162, 164), §22, p. 179).
- 16.
Wittgenstein’s use of “Vergleichsobjekte” in this particular context of the foundations of logic first appears in (1999) MS 157b, pp. 33–34, drafted in February 1937, just after he would have likely received “On Computable Numbers” from Turing. It reads:
Our exact language-games are not preparations for a future regimentation of our everyday language, like first approximations without friction and air resistance. This idea leads to Ungames (Unspiele) (Nicod & Russell [JF: who were pursuing the axiomatic approach to logic].) Much more, the language-games stand as objects of comparison, which, through similarities and differences should throw light on the relationships of our language.
For the clarity after which we strive is in any case a complete one. But this only means that the philosophical problems should completely disappear. (cf. (2009) (PI) §130).
- 17.
Until the late 1940s “computer” referred to a person, often a woman, who carried out calculations and computations in the setting of an office or research facility. Nowadays “computor” is used to make the human user explicit.
- 18.
Englemann (2013b) explains the importance of this anthropological element.
- 19.
Cf. Post (1936).
- 20.
Compare Post’s remarks about “psychological fidelity” (1936), p. 105, and Gödel (1972) in (1990), at p. 306, including the introductory note Webb (1990). The reader may contrast Hodges (1999), p. 22 and see Floyd (2012b) for further argumentation. On Gödel vs. Turing, see Copeland and Shagrir (2013), Sieg (2013a).
- 21.
- 22.
Carnap’s earlier program of “general axiomatics”, broached during Gödel’s student days with him in the late 1920s, had also been intended to make the proper analysis of symbolic logic (Carnap 2000).
- 23.
See Kripke (2013).
- 24.
- 25.
- 26.
Pace Grattan-Guinness, by 1931, when Turing arrived at Cambridge, Wittgenstein was an anti-“monist”, stressing, against his earlier Tractatus view, the importance of plurality and variety in systems of logic and grammar. Wittgenstein was hardly engaged in “revising” logicism in 1931–1935, having never embraced it in the higher-order style of Whitehead and Russell: for him logic consisted of tautologies, but mathematics did not; see Floyd (2001a), Weiss (2017).
- 27.
Russell’s (1919) had stimulated Kurt Gödel to turn toward logic from physics at the age of 19 (in a seminar of Moritz Schlick’s on the book in Vienna 1925–1926), and had been read by Stephen Kleene before he attended Church’s seminar on logic at Princeton in the fall of 1931–1932. Cf. Floyd and Kanamori (2016); Kleene (1981).
- 28.
- 29.
This identification of mathematical sentences with tautologies occurred as a logicist appropriation of Wittgenstein’s thought in the Vienna Circle , but also in Cambridge. On January 24, 1941, G. H. Hardy gave a talk to the Moral Sciences Club on “Mathematical Reality,” §§20–22 of his book A Mathematician’s Apology (1940), a book Wittgenstein would call “miserable”, probably because it took so little account of his philosophical criticisms (1999, 124, p. 35, from 1941). Mays (1967, p. 82) recalls: “Hardy mentioned that he did not accept Wittgenstein’s view that mathematics consisted of tautologies. Wittgenstein denied that he had ever said this, and pointed to himself saying in an incredulous tone of voice, ‘Who, I?”’ (cf. Klagge and Nordmann eds. (2003), p. 336.) For discussion of Wittgenstein’s resulting annotations to Hardy’s Course of Pure Mathematics (1941), see Floyd and Mühlhölzer (unpublished).
- 30.
Dreben and Floyd (1991).
- 31.
Wittgenstein discusses this Prefatorial Remark in his 162a, pp. 15–18.
- 32.
Turing Digital Archive, AMT/B/46, http://www.turingarchive.org/browse.php/B/46.
- 33.
- 34.
The Lecture was given on May 20, 1932, cf. Stern, Rogers, and Citron eds. (forthcoming); cf. Goodstein (1945), p. 407n, von Plato (2014), Marion and Okada (unpublished).
- 35.
- 36.
Wittgenstein dictated The Blue Book to Turing’s fellow mathematics students H.S.M. Coxeter, R.L. Goodstein, and Francis Skinner, along with Alice Ambrose (writing with Wittgenstein and Newman), as well as Margaret Masterman (later Braithwaite); cf. BB, Preface.
- 37.
Cf. BB and also the Francis Skinner archives, now at the Wren Library of Trinity College, Cambridge. These were donated by his close friend and colleague (and fellow mathematics student of Turing) R.L. Goodstein to the Mathematical Association of Great Britain, who held them until his death; Wittgenstein had given him the materials soon after Skinner’s own death. These contain more extensive and precise dictated material from the period 1933–1935, including a different version of The Yellow Book and some hitherto unpublished and unknown lecture notes on self-evidence in logic (cf. Gibson (2010)).
- 38.
- 39.
BB §41, p. 98. Wittgenstein remained unsatisfied with The Brown Book; after an attempted revision in the fall of 1936, it was abandoned. The beginning of Item 152, a notebook begun presumably by early 1937, revisits rule-following, but begins concretely, with a series of calculations from the theory of continued fractions.
- 40.
Russell’s remark is quoted without attribution in J.L. Austin’s “Plea for Excuses” (from 1956, in Austin (1979), p. 185; it was apparently widely known. Wittgenstein sent Russell The Blue Book, hoping for a response, in 1935 (McGuinness (2008), p. 250). As Cavell has suggested (2012, pp. 30ff), the opening of Wittgenstein’s Blue Book, with its truncated, cave-man-like language-game of builders, may be regarded as either a stimulus or a response to Russell’s remark about “the metaphysics of the Stone Age”.
- 41.
- 42.
- 43.
Hodges (1983/2012), p. 58; cf. the early editions (in 1933, 1938) of Hardy (1941); presumably Wittgenstein read out passages from the 1933 edition in his “Philosophy for Mathematicians” of 1932–1933 and fall 1933. Later Wittgenstein would annotate his copy of Hardy (1941) and copy these annotations into his manuscripts; cf. Floyd and Mühlhölzer (unpublished).
- 44.
After discussing these issues with Turing, both in 1937 and in his 1939 seminar, Wittgenstein wrote a long series of manuscript pages exploring the idea that a proof must be “surveyable” (Übersichtlich), i.e., “can be copied, in the manner of a picture” by a human being, and so “taken in”, communicated, archived, recognized, and acted upon; cf. Mühlhölzer (2006), Floyd (2015).
- 45.
Cf. Post (1941/1994), at n. 8 (p. 377); Post is referring to his initial work on “operational logic” for sequences done in 1924.
- 46.
On “pure propositions” as gaseous, see AWL, p. 55.
- 47.
Discussions of a variety of conceptions of “machines” and “mechanism” appear in Wittgenstein’s “Philosophy” lectures 1934–1935 and The Yellow Book; cf. AWL, pp. 52–53, 72, 80.
- 48.
In the Tractatus (1922) there had been posited an ultimate starting point, the “simple objects” of the final analysis. Wittgenstein had rejected these by 1929.
- 49.
The problem is evinced by his use of images of a spiral to gauge infinitary, rule or law-governed series at Wittgenstein AWL, p. 206. The image had been tethered to a particular origin in Philosophical Remarks, Wittgenstein’s fellowship submission (1980a) (PR) §§158, 189, 192, 197. But in the “Big Typescript” of 1932–1933 (2005), p. 379, as well as “Philosophy for Mathematicians” 1932–1933, AWL, p. 206 the issue of a point of origin and how to assign it in a single space has come to the fore. After this point, the image of the spiral is not used by Wittgenstein, but is replaced with that of a more localized, free-standing and “portable” table, or finite set, of commands.
- 50.
Though it is unclear exactly when, because after 1931 there was a hiatus in Wittgensteinstein’s attendance. Cf. Klagge and Nordmann (eds.) (2003), p. 377.
- 51.
See the archive of talks at https://www.srcf.ucam.org/tms/talks-archive/#earlier.
- 52.
Klagge and Nordman (eds.) (2003), p. 362.
- 53.
- 54.
Watson (1908–1982) entered King’s College in 1926, receiving firsts in both parts of the Mathematical Tripos; he was awarded Studentship prizes in 1929, 1930 and 1932. Though he failed in his first bid for a King’s Fellowship, “Chance and Determinism in Relation to Modern Physics” (1932) he succeeded in 1933, with a new thesis on “The Logic of the Quantum Theory”. He helped proofread Braithwaite’s edition of Ramsey’s papers (1931). A man of the left and a friend of Anthony Blunt, his alleged entanglement with the Cambridge Spy Ring (he never confessed) is discussed in Wright (1987).
- 55.
Hodges (1983/2012).
- 56.
Cf. McGuinness (2008), pp. 253, 280.
- 57.
Hodges (1983/2012), p. 109.
- 58.
Hodges (1983/2012), p. 109.
- 59.
Hodges (1983/2012), p. 112.
- 60.
Hodges (1983/2012), p. 109 tells us that Turing did discuss his ideas about machines with David Champernowne, who, along with Alister Watson, would later be relied on by Sraffa to check the mathematics of Sraffa (1960/1975). See Kurz and Salvadori (2001).We note that Yorick Smythies, Wittgenstein’s main amenuensis for his lectures 1938–1941 (cf. Munz and Ritter (eds.) (forthcoming)), took down notes of Max Newman’s 1935 logic course (now located at St. John’s College, Library, Cambridge). Cf. Copeland (2011), p. 152n3.
- 61.
Turing was disappointed at receiving only two requests for offprints, one from Braithwaite and one from Heinrich Scholz, who gave a seminar on it at Münster (Hodges (1983/2012), pp. 123–124.
- 62.
Turing to his mother February 11, 1937 AMT/K/1/54 in the Turing Digital Archive http://www.turingarchive.org/browse.php/K/1/54.
- 63.
Hodges (1983/2012), p. 136.
- 64.
Hodges tells us that Turing was “overoptimistic” at this time “in thinking he could re-write the foundations of analysis” ((1999), p. 19). It was just this question that interested Watson in his (1938) paper that they discussed in the summer of 1937 with Wittgenstein; our best guess is that the three urged one another on in conversation to ponder the question. Gandy (1988) p. 82, n26 reports that Turing initially planned a sequel to (1936/7) in which computable analysis would be developed. But he did not proceed, given the fact, pointed out to him by Bernays, that not all real numbers have unique representations as binary decimals (cf. Turing (1937)). Serving as an examiner of Goodstein’s PhD thesis in 1938, Turing spotted errors in the work that stemmed from underestimating this problem.
- 65.
- 66.
- 67.
Cf. Wittgenstein (1978) (RFM), Part II, written in 1938, analyzed in Floyd and Mühlhölzer (unpublished).
- 68.
- 69.
Cf. Floyd (2001b).
- 70.
Of course, conversations with many others went on, especially mathematicians. Newman himself went to Princeton in 1937–1938, joining Turing there, cf. Newman (2010).
- 71.
A brief history of the term is given in Church (1956), pp. 56–57 n. 125; Church traces the concerted use of “logistic” for mathematical logic back to the 1904 World Congress of philosophy. He notes that “sometimes ‘logistic’ has been used with special reference to the school of Russell or to the Frege-Russell doctrine that mathematics is a branch of logic”, but the “more common usage … attaches no such special meaning to this word”. For an overview of logos, logic vs. logistiké in connection with incompleteness and modern mathematics see Stein (1988).
- 72.
Except for one loose sheet Wittgenstein inserted into the very end of the manuscript of Philosophical Investigations (PI Part II/PPF xiv §372).
- 73.
Braithwaite (1934) regards Lewis and Langford (1932) as “eminently successful”, predicting that “it will probably not be superseded for some time as the standard work” on the subject of symbolic logic, though he objects to its treatment of the theory of types and finds it wanting as a text in the foundations of mathematics. He notes that Langford’s use of postulates is analogous to Hilbert’s Entscheidungsproblem. Langford, a student of H.M. Sheffer, spent 1925–1926 at Cambridge visiting from Harvard. He met Ramsey and indeed proved some of the earliest significant results on the completeness and decidability of first-order theories, pioneering the use of quantifier elimination (Urquhardt (2012))—another point of contact between Cambridge philosophy and the Enscheidungsproblem in the 1920s.
- 74.
Wisdom (1934), p. 101 rejects this on the ground that these rules “are principles of logic”. If Turing had read this, he might have demurred, taking the notion of “calculation” as basic, and, having analyzed it, applying it to the whole idea of “logic”.
- 75.
See Detlefsen (2005).
- 76.
- 77.
Carnap, Maund and Reeves (1934), p. 47.
- 78.
- 79.
- 80.
Cf. Putnam (2015).
- 81.
- 82.
Sieg (2013b).
- 83.
- 84.
- 85.
Watson uses the metaphor that the machine “gets stuck” ((1938), p. 445), but I have not found that metaphor either in Wittgenstein or Turing. In LFM the metaphor is criticized (LFM, pp. 178–179), as well as the idea that we have to fear contradictions more than empty commands.
- 86.
Martin Davis first gave this argument in 1952; see http://wikipedia.org/wiki/Halting_problem#History_of_the_halting_problem and Copeland (2004), p. 40 n 61.
- 87.
Copeland (2004), p. 38 nicely adapts H to a regress argument.
- 88.
Monk (1990), pp. 419–420.
- 89.
Turing himself worries about constructivism creeping in: cf.LFM, pp. 31, 67,105.
- 90.
This, at least, is one guess as to the contents of the lecture, which exists only in notes taken down by Smythies. See Munz and Ritter (eds.) (forthcoming).
- 91.
For more on Wittgenstein’s own remarks on Gödel and “phraseology”, see Floyd (2001b).
- 92.
Cf. Diamond (1996) for a defense of Wittgenstein’s “realism”.
- 93.
In (1936/1937) §8 Turing writes, of the more “direct proof” using a reductio, that “although [it is] perfectly sound, [it] has the disadvantage that it may leave the reader with a feeling that ‘there must be something wrong’”. His “Do-What-You-Do” argument offers a response to such a reader. In his corrections (1937), stimulated by Bernays, Turing develops this point with respect to intuitionism explicitly. See Floyd (2012b) for a discussion.
- 94.
Russell (1919), p. 6:
This point, that “0” and number and “successor” cannot be defined by means of Peano”s five axioms, but must be independently understood, is important. We want our numbers not merely to verify mathematical formula, but to apply in the right way to common objects. We want to have ten fingers and two eyes and one nose. A system in which “1” meant 100, and “2” meant 102, and so on, might be all right for pure mathematics, but would not suit daily life.
- 95.
- 96.
Our best guess is that Wittgenstein is alluding to the situation after Gödel 1931, in which both the Gödel sentence P and its negation, not-P, may each be added to the original system of arithmetic consistently, presenting a kind of branch of possible paths. Cf. Wittgenstein’s 1938 lecture on Gödel in Munz and Ritter (eds). (forthcoming).
- 97.
- 98.
It is just this idea to which Wittgenstein reverts when, probably reminiscing about these exchanges, he revisits the notion of an “oracle” much later. See RPP I §817 and (1974) (OC) §609, written just six days before his death.
- 99.
In (1944) Post criticized Turing for his “picturesque” use of the idea of an “oracle”, writing “the ‘if’ of mathematics is … more conductive to the development of a theory”, p. 311 n.23.
- 100.
Cf. Floyd (2013). Turing’s explorations of the logical tradition (Leibniz, Boole, Peano, etc.) continue in a notebook from c. 1942, “Notes on Notations”. See https://www.bonhams.com/auctions/22795/lot/1/.
- 101.
- 102.
See Davis and Sieg (2015).
- 103.
References
Abbreviations of Wittgenstein’s Works
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Braithwaite, R.B. 1934. Review of C.I. Lewis and C.H. Langford, Symbolic Logic (1932). The Mathematical Gazette 18(227): 57–58.
Carnap, R. 1929. Abriss der Logistik Mit Besonderer Berücksichtigung der Relationstheorie und Ihrer Anwendungen. Vienna, Julius Springer. English translation by Dirk Schlimm to appear in Collected Works of Rudolf Carnap, Vol. III: Pre-Syntax Logic 1927–1934, edited by Erich Reck, Georg Schiemer, and Dirk Schlimm.
———. 1931. Die logizistische Grundlegung der Mathematik. Erkenntnis 2 (2/3): 91–105.
———. 1934. Logische Syntax Der Sprache. Vienna: Julius Springer.
———. 1935. Philosophy and Logical Syntax. London: Kegan Paul, Trench, Trubner & Co.
———. 2000. Untersuchungen zur Allgemeinen Axiomatik. Edited from unpublished manuscript by T. Bonk and J. Mosterín. Darmstadt: Wissenschftliche Buchgesellschaft.
Carnap, R., C.A.M. Maund, and J.W. Reeves. 1934. Report of Lectures on Philosophy and Logical Syntax, Delivered on 8, 10 and 12 October at Bedford College in the University of London, by Professor Rudolf Carnap. Analysis 2(3), December: 42–48.
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———. 2001b. Prose versus Proof: Wittgenstein on Gödel, Tarski and Truth. Philosophia Mathematica 3 (9): 280–307.
———. 2012a. Wittgenstein, Carnap, and Turing: Contrasting Notions of Analysis. In Carnap’s Ideal of Explication and Naturalism, ed. P. Wagner, 34–46. Basingstoke: Palgrave MacMillan.
———. 2012b. Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing. In Epistemology versus Ontology, Logic, Epistemology, eds. P. Dybjer, S. Lindström, E. Palmgren, and G. Sundholm, 25–44. Dordrecht: Springer Science + Business Media.
———. 2013. Turing, Wittgenstein and Types: Philosophical Aspects of Turing’s ‘The Reform of Mathematical Notation’ (1944 –5). In eds. S.B. Cooper and J. van Leeuwen, 250–253.
———. 2015. Depth and Clarity: Critical Study, Felix Mühlhölzer. In Braucht die Mathematik eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen Über die Grundlagen der Mathematik (Vittorio Klostermann, Frankfurt am Main 2010). Philosophia Mathematica 23, Special Issue on Mathematical Depth (2): 255–277.
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———. 1946. Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics. In Gödel (1990), 150–154.
———. 1972. Some Remarks on the Undecidability Results. In Gödel (1990), 305–306.
———. 1986. In Kurt Gödel Collected Works Vol. I: Publications 1929–1936, eds. S. Feferman et al. New York: Oxford University Press.
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———. 1940. A Mathematician’s Apology. Cambridge, MA: Cambridge University Press.
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Acknowledgments
I am grateful to S. Barry Cooper, Akihiro Kanamori, and Mauro Engelmann for stimulating conversations about the ideas in this paper, as well as the audience at our Turing 100 conference at BU, November 11, 2012. In addition, the Stanhill Foundation provided me with generous funds for a visit to the Skinner archives at Trinity College Cambridge in June 2015. There, with the able and generous aid of Jonathan Smith, I was able to do a great deal of helpful and inspiring research. Thanks are due to Arthur Gibson, Ilyas Kahn, Susan Edwards McKie, and Jonathan Smith for their conversation during this visit. In the final stages, I also received very helpful feedback from Juliette Kennedy and Adriana Renero.
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Floyd, J. (2017). Turing on “Common Sense”: Cambridge Resonances. In: Floyd, J., Bokulich, A. (eds) Philosophical Explorations of the Legacy of Alan Turing. Boston Studies in the Philosophy and History of Science, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-319-53280-6_5
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