The Turing Guide

NOTES TO PAGES 60–65 | 489

13. Turing (1936), Section 9.


14. Stephen Kleene, a member of Church’s Princeton group, came a step closer. In 1936 he published his
    first example of what he later called a ‘universal function’, but did not connect this idea with comput-
    ing machinery; see S. C. Kleene, ‘General recursive functions of natural numbers’, Mathematische
    Annalen, 112 (1936), 727–42 (reprinted in Davis (Note 3)), and S. C. Kleene, Introduction to
    Metamathematics, North-Holland (1952), on p. 289.


15. Gödel (Note 3).


16. See H. Wang, Reflections on Kurt Gödel, MIT Press (1987), on p. 171.


17. Turing (1936), Sections 8 and 11.


18. D. Hilbert, ‘The new grounding of mathematics: first report’ [Neubegründung der Mathematik.
    Erste Mitteilung] (1922), in W. B. Ewald, From Kant to Hilbert: a Source Book in the Foundations of
    Mathematics, Vol. 2, Clarendon Press, Oxford (1996), p. 1119.


19. Hilbert, p. 1119 (Note 18).


20. Hilbert, p. 1120 (Note 18). Italics added.


21. Hilbert, p. 1121 (Note 18). Here Hilbert was talking about analysis.


22. Hilbert, p. 1119 (Note 18).


23. Hilbert, p. 1119 (Note 18).


24. Turing (1936), p. 84.


25. Hilbert, p. 1132 (Note 18).


26. Hilbert, D. and Ackermann, W. Grundzüge der Theoretischen Logik [Principles of Mathematical Logic],
    Julius Springer (1928), p. 76.


27. Hilbert and Ackermann, p. 77 (Note 26).


28. Hilbert and Ackermann, p. 77 (Note 26).


29. Hilbert and Ackermann, p. 81 (Note 26).


30. Turing (1936) p. 84. (Turing’s German ‘A’, which follows ‘given formula’, has been omitted from the
    quotation, together with his ‘–A’ preceding ‘adjoined’.)


31. See S. C. Kleene, ‘Origins of recursive function theory’, Annals of the History of Computing, 3 (1981),
    52–67, on pp. 59, 61.


32. K. Gödel, ‘Some basic theorems on the foundations of mathematics and their implications’ (1951), in
    S. Feferman et al. (eds), Collected Works, Vol. 3, Oxford University Press (1995), pp. 304–5.


33. Turing (1937), p. 153.


34. Turing’s doctoral dissertation, ‘Systems of logic based on ordinals’, completed in 1938, was published
    as Turing (1939). For an introduction to Turing’s project in this thesis, see The Essential Turing,
    pp. 135–44, and Turing (Copeland 2012), Chapter 3.


35. Hilbert, D. ‘On the infinite’ [Über das Unendliche], 1925, in P. Benacerraf and H. Putnam (eds),
    Philosophy of Mathematics: Selected Readings, 2nd edn, Cambridge University Press (1983), p. 196.


36. Turing (1939), p. 192.


37. Hilbert, p. 201 (Note 35). I am indebted to Carl Posy for discussions of Hilbert’s views on intuition.


38. Hilbert, pp. 1121–2 (Note 18).


39. Hilbert, p. 1126 (Note 18).


40. Hilbert, pp. 1121, 1124 (Note 18). See also P. Bernays, ‘On Hilbert’s thoughts concerning the ground-
    ing of arithmetic’ [Über Hilberts Gedanken zur Grundlegung der Arithmetik], 1921, in P. Mancosu
    (ed.) From Brouwer to Hilbert: the Debate on the Foundations of Mathematics in the 1920s, Oxford
    University Press (1998).


41. Hilbert, p. 192 (Note 35).


42. Hilbert, p. 198 (Note 35).


43. Hilbert, D. ‘Probleme der Grundlegung der Mathematik’ [Problems concerning the foundation of
    mathematics], Mathematische Annalen, 102 (1930), 1–9, on p. 9.


44. Hilbert, p. 198 (Note 35).


45. Turing (1939), pp. 192–3.


46. Hilbert, p. 9 (Note 43).


47. Hilbert and Ackermann, p. 72 (Note 26). I am grateful to Giovanni Sommaruga for his translation of
    the German. The full sentence is: ‘After the logical formalism has been fixed, it is to be expected that