Alan Turing: The Enigma (119 page)

(
3.11
) The brief logic paper was in J.
Symbolic Logic, 2
(1937). The work related to that of Baer was in
Compositio Math. 5
(1938). The other group theory paper was in
Ann. Math.
(
Princeton
)
39
(1938).

(
3.12
) A copy of von Neumann’s letter is held in AMT’s file at the Department of Mathematics, Princeton University. Formal recommendation of AMT came from the Vice-Chancellor of Cambridge University on 25 June.

(
3.13
) One letter from Bernays to AMT, dated 24 September 1937, is in
KCC.
AMT’s correction note appeared in
Proc. Lond. Math. Soc. (2) 43
(1937). There were other mistakes and inconsistencies in the specification of the universal machine, some of them detailed by Post in a 1947 paper (reprinted in
The Undecidable
, as note 2.31).

(
3.14
)
J. Symbolic Logic, 2
(1937).

(
3.15
) As
described in
The Undecidable
, page 71.

(
3.16
) J.B. Rosser,
J. Symbolic Logic 2
(1937).

(
3.17
) Letter from A.E. Ingham dated 1 June 1937 in
KCC.

(
3.18
) The following account draws heavily on H.H. Edwards,
Riemann’s Zeta Function
(Academic Press, New York, 1974), which also discusses AMT’s contributions.

(
3.19
) S. Skewes,
J. Lond. Math. Soc. 8
(1933). There is a letter in
KCC
from Skewes, dated 9 December 1937, with a brief expression of interest in AMT’s ideas.

(
3.20
) A.G.D. Watson, ‘Mathematics and its Foundations’, in
Mind 47
(1937).

(
3.21
) AMT was right. During the war Gerard Beuttell made important contributions to the design of instruments to estimate the visual range by measuring the scattering of light within a small enclosed space.
(J. Scientific Instruments, 26
(1949)). He died on a meteorological reconnaissance flight over the north Atlantic in early 1945.

(
3.22
) Letter to the author from Dr M. MacPhail, 17.12.77.

(
3.23
) It was in service until 1960, then being supplanted by a digital computer, and may now be seen in the Liverpool City Museum.

(
3.24
) Letter from E.C. Titchmarsh in
KCC.

(
3.25
) The original PhD thesis is held in the mathematics library at Princeton University; it was published as ‘Systems of Logic based on Ordinals’ in
Proc. Lond. Math. Soc. (2) 45
(1939), and reprinted in
The Undecidable.

(
3.26
) Letter to the author from Professor S. Ulam, 16.4.79.

(
3.27
) C. Andrew, ‘The British Secret Service and Anglo-Soviet Relations in the 1920s, Part I’ in the
Historical Journal, 20
(1977).

(
3.28
) Hinsley I (see note 3.31), page 10.

(
3.29
) Hinsley I, page 20.

(
3.30
) Administrative files relating to GC and CS are held at the Public Record Office in FO 366.

(
3.31
) F.H. Hinsley
et al., British Intelligence in the Second World War.
Volume I (1979), Volume II (1981). Published by HMSO as an official war history.

(
3.32
) FO 366/978.

(
3.33
) Hinsley I, page 54.

(
3.34
) As note 3.27.

(
3.35
) Hinsley I, page 53.

(
3.36
) Hinsley I, page 54.

(
3.37
) From records of the Faculty of Mathematics, Cambridge University.

(
3.38
) Parts of the revised
Encyklopädie
appeared in December 1939, but Scholz’s section on the foundations of mathematics, including the reference to AMT’s work, had to wait until August 1952.

(
3.39
) A transcript compiled from notes taken by others attending the lectures has been published as
Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge 1939
, ed. Cora Diamond (Harvester Press, 1976). The quoted dialogue comes from lectures 21 and 22. It is perhaps a pity that the most extensive verbatim record of AMT should be concerned with a discussion which was not central to his concerns, and where he was not in his
element. AMT sometimes liked to give the impression that he had scored off Wittgenstein at some point, but if so the evidence is not to be found in this transcript. In fact he showed a curious diffidence, one feature being that despite long discussions about the nature of a ‘rule’ in mathematics, AMT never offered a definition in terms of Turing machines.

(
3.40
) This is in
KCC.
It was corrected and completed by A.M. Cohen and M.J.E. Mayhew,
Proc. Lond. Math. Soc. (3)
18 (1968). Using AMT’s approach they reduced the ‘Skewes number’ to
. But in 1966 R.S. Lehman had by another method reduced the bound to the comparatively miniscule value of 1.65 X 10
¹¹⁶⁵
.

(
3.41
) His paper ‘A Method for the Calculation of the Zeta-function’ appeared only in 1943, in
Proc. Lond. Math. Soc. (2) 48.

(
3.42
) Quoting from a copy of part of the letter made by Mrs Turing and deposited in
KCC.
My guess is that she omitted some reference to the function of the proposed machine as a cipher generator, not knowing whether this would be a transgression of secrecy.

(
3.43
) Minutes of the Council of the Royal Society.

(
3.44
) The blueprint, initialled ‘D.C.M.’, is in
KCC.

(
3.45
) Hinsley I, page 51.

The Relay Race

(
4.1
) Letter and list of names in FO/366/1059, which contains no further reference to AMT.

(
4.2
) M. Muggeridge,
The Infernal Grove (
Collins, 1973).

(
4.3
) Pre-eminently H.F. Gaines,
Elementary Cryptanalysis,
1939. Only at the end of the 1970s did a serious technical discussion of specific modern cipher systems begin to appear.

(
4.4
) I am grateful to the staff of the National Archives, Washington, for bringing this material to my attention. In late 1940 the German raider
Komet
made several captures of British merchant ships and took this code and cipher material. This then found its way into German archives captured after the war.

(
4.5
) There is an account of Polish Enigma work in the appendix to J. Garlinski,
Intercept
(Dent, 1979). A fuller and better version is given by M. Rejewski, ‘How Polish Mathematicians Deciphered the Enigma’, in
Annals of the History of Computing 3
(1981). This would seem to be the definitive account, ending much confusion and speculation in earlier discussions.

(
4.6
) Hinsley I, page 490, itself quoting from the Polish claim at the time.

(
4.7
) Hinsley I, page 492.

(
4.8
) Letter to the author from Professor R.V. Jones, 7.2.78, expanding upon a passage of his
Most Secret War
(Hamish Hamilton, 1978).

(
4.9
) The following account of the Bombe is a simplified version of Gordon Welchman,
The Hut Six Story
(McGraw Hill, New York; Allen Lane, London, 1982). It is worth noting Welchman’s comment: ‘We thought very little, in these hectic days, of who should take credit for what.’ AMT would have thought least of all, though he did say that he thought Welchman’s idea had been the important one. Establishing priority and originality is hard
enough in open work, let alone when considering ideas kept secret for over forty years. I hope that the departure from truth, in this and other passages suffering from the same difficulty, is not too great. The more important point lies in the fact that pre-war cryptology, fossilised and isolated by secrecy, was transformed as soon as any contemporary mathematical mind was brought to bear on the subject.

(
4.10
) Hinsley I, page 493. The account in B. Johnson,
The Secret War
(BBC, London, 1978), identifies AMT as the ‘emissary’, following a statement made to BBC researchers by General Bertrand before his death. This seems rather unlikely as he was working on the Bombe, not the sheets, and as this was not really a job for a ‘man of the Professor type’. But it might be so – I have found no further evidence one way or the other.
EST
has a story concerning AMT being sent abroad, a mix-up over papers, and managing for a day with ‘a few francs’, but this could be taken to fit the 1945 mission (page 311).

(
4.11
) P. Beesly,
Very Special Intelligence
(Hamish Hamilton, 1977), which gives the Admiralty side of the story.

(
4.12
) Hinsley I, page 103.

(
4.13
) Hinsley I, page 336.

(
4.14
) Hinsley I, page 163.

(
4.15
) F.W. Winterbotham,
The Ultra Secret
, (Weidenfeld & Nicolson, 1974), which gives a secret service view.

(
4.16
) P. Beesly, as note 4.11.

(
4.17
) Hinsley I, page 109.

(
4.18
) Hinsley I, page 144.

(
4.19
) Hinsley I, page 336.

(
4.20
) I.J. Good, ‘Studies in the History of Probability and Statistics XXXVII. A.M. Turing’s Statistical Work in World War II’, in
Biometnka 66
(1979), which this description of AMT’s ideas follows closely. Further details are given in a note by Good appended to the article by M. Rejewski (note 4.5).

(
4.21
) Quoting from I.J. Good’s lecture at the National Physical Laboratory, 1976; this has since been published in slightly revised forms in several places, the most accessible being as a paper ‘Pioneering Work on Computers at Bletchley’ in the misleadingly entitled volume
A History of Computing in the Twentieth Century
, eds. N. Metropolis, J. Howlett and G.-C. Rota (Academic Press, New York, 1980).

(
4.22
) Quotation is from Beesly, as note 4.11, although I follow Hinsley in stating the capture to have been planned and not an accident.

(
4.23
) Messages as translated into English at the time, and taken from the first few pages of the gigantic PRO file DEFE 3/1.

(
4.24
) Hinsley I, page 337.

(
4.25
) Beesly, as note 4.11, pages 57, 97.

(
4.26
) Hinsley I, pages 273-4.

(
4.27
) Quoted in
EST.
There he appeared anonymously (presumably because working for GCHQ) as a colleague who later proved a ‘staunch friend’ – Mrs Turing’s only concession to the events of 1952.

(
4.28
) Hinsley I, page 296.

(
4.29
) R. Lewin,
Ultra Goes to War
(Hutchinson, 1978), page 183.

(
4.30
) Obituary of A.C. Pigou by D.G.
Champernowne in
Roy. Stats. J. A122
(1959).

(
4.31
) As note 4.2.

(
4.32
) Dorothy Sayers,
The Mind of the Maker
(Methuen, 1941). AMT referred to reading it in the first wartime letter to his mother, in August 1941 (see note 5.8), saying ‘You should read it when you come.’ The quoted passage is the one he himself quoted in 1948 (see page 377).

(
4.33
) Princeton records show that von Neumann gave a popular lecture on the game of poker on 19 March 1937. It would be very surprising if AMT had not attended it. He did not, in his discussions with Jack Good, draw a connection between his chess programs and game theory – nor indeed with the machines of
Computable Numbers.
But I have assumed that he had a general acquaintance with game theory, just as he could hardly have forgotten his own ‘machines’. I have also given space to game theory for another reason: AMT certainly showed an interest in it later, and often pointed out examples of strategies in everyday life.