Abstract
On 30 July 1947 Wittgenstein penned a series of remarks that have become well-known to those interested in his writings on mathematics. It begins with the remark “Turings ‘machines’: these machines are humans who calculate. And one might express what he says also in the form of games”. Though most of the extant literature interprets the remark as a criticism of Turing’s philosophy of mind (that is, a criticism of forms of computationalist or functionalist behaviorism, reductionism and/or mechanism often associated with Turing), its content applies directly to the foundations of mathematics. For immediately after mentioning Turing, Wittgenstein frames what he calls a “variant” of Cantor’s diagonal proof. We present and assess Wittgenstein’s variant, contending that it forms a distinctive form of proof, and an elaboration rather than a rejection of Turing or Cantor.
Thanks are due to Per Martin-Löf and the organizers of the Swedish Collegium for Advanced Studies (SCAS) conference in his honor in Uppsala, May 2009. The audience, especially the editors of the present volume, created a stimulating occasion without which this essay would not have been written. Helpful remarks were given to me there by Göran Sundholm, Sören Stenlund, Anders Öberg, Wilfried Sieg, Kim Solin, Simo Säätelä, and Gisela Bengtsson. My understanding of the significance of Wittgenstein’s Diagonal Argument was enhanced during my stay as a fellow 2009–2010 at the Lichtenberg-Kolleg, Georg August Universität Göttingen, especially in conversations with Felix Mühlhölzer and Akihiro Kanamori. Wolfgang Kienzler offered helpful comments before and during my presentation of some of these ideas at the Collegium Philosophicum, Friedrich Schiller Universität, Jena, April 2010. The final draft was much improved in light of comments provided by Sten Lindström, Sören Stenlund and William Tait.
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Notes
- 1.
This part of the remark is printed as §1097 of Wittgenstein et al. (1980). See footnote 21 below for the manuscript contexts.
- 2.
I have not been able to identify with certainty what this game is. I presume that Wittgenstein is thinking of a board game in which cards are drawn, or dice thrown, and pieces are moved in a kind of race. See below for specifics.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
Malcolm queried by letter (3 November 1950, now lost) whether Wittgenstein had read “Computing Machinery and Intelligence”, asking whether the whole thing was a “leg pull”. Wittgenstein answered (1 December 1950) that “I haven’t read it but I imagine it’s no} leg-pull”. (Wittgenstein (2004)).
- 9.
- 10.
The Entscheidungsproblem asks, e.g., for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and determine whether or not the statement is provable in the system (or: whether or not a first-order formula of the predicate calculus is or is not valid) in a finite number of steps. Turing 1936 offered a proof that there is no such algorithm, as had, albeit with a different proof, the earlier Church (1936).
- 11.
As Turing writes (1937a, p. 231), “the justification lies in the fact that the human memory is necessarily limited”; cf. §9.
- 12.
- 13.
See the note Gödel added to his “Some remarks on the undecidability results” (1972a), in Gödel (1990), p. 304, and Webb (1990). Gödel (somewhat unfairly) accuses Turing of a “philosophical error” in failing to admit that “mind, in its use, is not static, but constantly developing”, as if the appropriateness of Turing’s analysis turns on denying that mental states might form a continuous series.
- 14.
- 15.
- 16.
Turing’s argument in 1937a in §8 is not formulated as a halting problem; this was done later, probably by Martin Davis in a lecture of 1952. For further details on historical priority, see http://en.wikipedia.org/wiki/ Halting_problem#History_of_the_halting_problem and Copeland (2004), p. 40 n 61.
- 17.
Cf. Bernays to Turing 24 September 1937. The corrections using Brouwer’s notion of an overlapping sequence are explained in Petzold (2008), pp. 310ff. Petzold conjectures that conversations with Church at Princeton (or with Weyl) may have stimulated Turing’s interest in recasting his proof, though he suspects that “Turing’s work and his conclusions are so unusual that … he wasn’t working within anyone’s prescribed philosophical view of mathematics” (2008, p. 308). I agree. But in terms of possible influences on Turing, Bernays should be mentioned, and Wittgenstein should be added to the mix. The idea of expressing a rule as a table-cum-calculating device read off by a human being was prevalent in Wittgenstein’s philosophy from the beginning, forming part of the distinctive flavor in the air of Cambridge in the early 1930s, and discussed explicitly in his Wittgenstein (1980) [DL].
- 18.
Gödel, concerned with his own notion of general recursiveness when formulating the absoluteness property (in 1936) later noted the importance of this notion in connection with the independence of Turing’s analysis from any particular choice of formalism. He remarked that with Turing’s analysis of computability “one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen” (Gödel here means a formal system of the relevant (recursively axiomatizeable, finitary language) kind). See Gödel’s 1946 “Remarks before the Princeton bicentennial conference on problems in mathematics”, in Gödel (1990), pp. 150–153; Compare his Postscriptum to his 1936a essay “On the Length of Proofs”, Ibid., p. 399. See footnote 28, and Sieg (2006a, b), especially pp. 472ff.
- 19.
Compare the discussion in Dreben and Floyd (1991).
- 20.
Wittgenstein 1947a (MS 135) p. 118; the square brackets indicate a passage later deleted when the remark made its way into Wittgenstein 1947b (TS 229) §1764, published at RPP I §1096-7. (At Z §695 only the second remark concerning the proof is published, thereby separating it from the mention of Turing and Watson (Wittgenstein (1970).) The argument as written here occurs here with “F” replacing the original “ϕ”, following the typescript. For the manuscripts, see Wittgenstein (1999).
- 21.
See also Stenius (1970) for another general approach to the antinomies distinguishing between contradictory rules (that cannot be followed) and contradictory concepts (e.g., “the round square”) that is explicitly based on a reading of Wittgenstein (in this case, the Tractatus).
- 22.
- 23.
Recall that in the earlier 1938 remarks on the Cantor diagonal argument Wittgenstein is preoccupied with the idea that the proof might be thought to depend upon interpreting a particular kind of picture or diagram in a certain way. Wittgenstein (1978). There are many problematic parts of these remarks, and I hope to discuss them in another essay. For now I remark only that they are much earlier than the 1947 remarks I am discussing here, written down in the immediate wake of his summer 1937 discussions with Watson and Turing.
- 24.
Though Turing himself would write that “these [limitative] results, and some other results of mathematical logic, may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of ‘reason’ unsupported by common sense”. Turing (1954), p. 23.
- 25.
Watson uses the metaphor that the machine “gets stuck” (Watson 1937, p. 445), but I have not found that metaphor either in Wittgenstein or Turing: it is rather ambiguous, and does not distinguish Turing’s General Argument from that of the Pointerless Machine. Both Watson and Turing attended Wittgenstein’s 1939 lectures at Cambridge; see LFM p. 179, where Wittgenstein criticizes the metaphor of a contradiction “jamming”, or “getting stuck”: I assume this is in response to a worry about the way of expressing things found in Watson 1937. He worries that the machine metaphor may bring out a perspective on logic that is either too psychologistic, or too experimental. He emphasizes, characteristically, that instead what matters if we face a contradiction is that we do not recognize any action to be the fulfillment of a particular order, we say, e.g., that it “makes no sense”. As he writes in the 1947 remarks considered here, “an order only makes sense in certain positions”. Recall Z s. 689: “Why is a contradiction to be more feared than a tautology”?
- 26.
- 27.
In a note added in 1963 to a reprinting of his famous 1931 incompleteness paper, Gödel called Turing’s analysis “a precise and unquestionably adequate definition of the general notion of formal system”, allowing a “completely general version” of his theorems to be proved. See Gödel (1986), p. 195. On the subject of “formalism freeness” in relation to Gödel see Kennedy (unpublished). Compare footnote 19.
- 28.
Hodges (1998).
- 29.
Fogelin (1987).
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Floyd, J. (2012). Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_2
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