Alan Turing: The Enigma (118 page)

(
1.23
) The quoted passages come from the letters and notes that AMT wrote for Mrs Morcom in 1930 and 1931 (see page 53, and note 1.26).

(
1.24
) This report is held together
with the school reports at Sherborne. Mrs Turing has annotated it as being either 1929 or 1930, and my placing it in 1929 is only a matter of guesswork.

(
1.25
) A.H.T. Ross (note 1.18 makes specific mention of the danger of accepting holiday invitations from boys in other houses. Curiously enough he was writing his remarks on ‘Problems’ and ‘Tone’ in the spring and summer of 1954, with the result that they are interspersed with comments on the impact of the Montagu trials and news of AMT’s death.

(
1.26
) AMT kept the letters he had from Christopher Morcom, and other souvenirs (see page 47). In 1931 Mrs Morcom copied out the letters, and then the originals, which AMT kept all his life, were returned after AMT’s death. The Morcom family also kept the letters written by AMT just before and then after Christopher’s death. I am deeply indebted to Mr Rupert Morcom for making all these and other family documents available.

(
1.27
) There are no letters in
KCC
between those of May 1926 and this one. The paraphrase is of a passage on page 215 of Sir Arthur Eddington,
The Nature of the Physical World
(Cambridge University Press, 1928). This is good evidence that by this time he had absorbed Eddington’s account of relativity, which comes well before this discussion of the new quantum mechanical picture of matter.

The Spint of Truth

(
2.1
) This letter is not in
KCC.
Another loss is that of the letters AMT had both from his mother and his father at this time. According to
EST
he also kept these all his life. This deprives us in particular of a glimpse of the relationship between father and son. Mrs Turing had her say later, but in this as in so many other ways Mr Turing’s part has just been wiped out.

(
2.2
) Quoted in
EST
from a letter written to Mrs Turing by A.J.P. Andrews after AMT’s death.

(
2.3
) Letter to the author from Major L. Knoop, 24.1.79.

(
2.4
) No diary has survived from this or any other part of AMT’s life.

(
2.5
) Letter to the author from Mr Patrick Barnes, 12.2.79.

(
2.6
) This was the 1922 edition of
Mathematical Recreations and Essays
(Macmillan).

(
2.7
) A short biography of Alfred W. Beuttell (1880-1965) was commissioned and published privately by Victor Beuttell in 1971, under the title
The Man Who Made Linolite.

(
2.8
)
The Shirburnian, 36,
page 113.

(
2.9
) Here and elsewhere I have drawn upon C. Reid,
Hilbert
(George Allen & Unwin; Springer Verlag, 1970), for quotations.

(
2.10
) As note 3.3.

(
2.11
) The paper was in
J. Lond. Math. Soc. 8
(1933). Champernowne’s result concerned ‘normal numbers’, a fairly light-hearted application of the study of the real number’ system as it had developed since the late nineteenth century. A ‘normal’ number was defined as one whose decimal expansion contained the ten digits equally and evenly distributed in a certain precise sense. It was already known that if a real number were picked ‘at random’,
then there was a probability of one hundred per cent that it would be ‘normal’. Yet no actual example of a ‘normal number’ was known until Champernowne produced one. AMT took some interest in the question later. There was a connection with his interest in randomness, but also a similarity to the concept of computability. For a ‘random’ real number has a probability of one hundred per cent of being uncomputable, but it requires some effort to produce, as he did, an example of an uncomputable number.
KCC
contains a letter from G.H. Hardy to AMT on ‘normal numbers’, undated but presumably of the later 1930s.

(
2.12
) It was undated, but written out on Clock House notepaper. This places it as composed on one of his visits. Mr Rupert Morcom writes that he believes it was written before 1933, and the handwriting style would support this belief. My guess is that 1930 is too early for the McTaggert reference, and that the style is more consistent with AMT’s wider-ranging intellectual life at Cambridge. These considerations all point to 1932. But certainly AMT could have thought in terms very like these at any time since 1929 or so, and the date of this piece of writing is not too significant.

(
2.13
) Quoting from the English translation of Laplace’s
Essai sur les probabilités
(Dover edition, 1951).

(
2.14
) In his obituary of AMT in the
Shirburnian,
1954.

(
2.15
) Quoted in
EST
from a letter written to her by Geoffrey O’Hanlon.

(
2.16
) A.W. Beutteil, ‘An Analytical Basis for a Lighting Code’, in
The Journal of Good Lighting
, January 1934.

(
2.17
) I am grateful to Professor W.T. Jones for bringing this passage to my attention in describing the impression AMT made on him in 1937. (See page 137). Keynes’ talk on
My Early Beliefs
, given in 1938, was published after his death as one of
Two Memoirs
(Rupert Hart-Davis, 1949).

(
2.18
)
The Autobiography of G. Lowes Dickinson,
published posthumously (Duckworth, 1973).

(
2.19
)
New Statesman and Nation,
4 February 1933. The progressive journal here used the medical model for homosexuality.

(
2.20
) J.S. Mill,
On Liberty
(1859). I owe to Robin Gandy the identification of AMT as ‘a J.S. Mill man’. In fact I have chosen to set AMT against less business-like and competitive libertarians, but certainly this essay contains many points of contact with AMT’s outlook and convictions.

(
2.21
)
Maurice,
written in 1913, was published after E.M. Forster’s death in 1971.

(
2.22
) The passage quoted was actually written by Shaw in 1944, but it only condensed the comment in Shaw’s Preface to
Back to Methuselah
of 1920.

(
2.23
) Bertrand Russell’s
Introduction to Mathematical Philosophy
(George Allen & Unwin, 1919) did not deal with the background in geometry, but started with the problem of giving meaning to the Peano axioms. However, I have included mention of Hilbert at this point, in order to lend greater unity to the discussion.

(
2.24
) The minutes are held in the University Library, Cambridge.

(
2.25
)
The Times
, 10 November 1933. But if the mathematicians had delivered a politically advantageous formula, they had surrendered little in private content. The phrase ‘a mixture of logic and intuition’ was unexceptionable (compare AMT’s remarks
apropos
of the ordinal logics in 1938); and the
work of Gödel had just recently served to delineate the limitations of deductive logic.

(
2.26
) The standard work for this course was Whittaker and Robinson,
The Calculus of Observations
, 1924.

(
2.27
) Lindeberg,
Math. Zeitschrift 15
(1922).

(
2.28
) AMT would have offered about six advanced courses for the Schedule B examination. Unfortunately the records of the Faculty of Mathematics do not seem to show what these were.

(
2.29
) AMT’s fellowship dissertation,
On the Gaussian Error Function
, remained unpublished. The original typescript is held in
KCC.

(
2.30
) As note 2.9.

(
2.31
) An English translation of Gödel’s paper is in
The Undecidable
, ed. Martin Davis (Raven Press, New York, 1965).

(
2.32
) This was Hardy’s Rouse Ball Lecture for 1928, published in
Mind,
1929, as ‘Mathematical Proof’.

(
2.33
) AMT’s paper was ‘Equivalence of Left and Right Almost Periodicity’,
J. Land. Math. Soc. 10
(1935).

(
2.34
) J. von Neumann,
Trans. Amer. Math. Soc. 36
(1934).

(
2.35
) For a modern biographical study, with many points of contact with this book, see Steve J. Heims,
John von Neumann and Norbert Wiener
(MIT Press, 1980).

(
2.36
) AMT also corresponded with von Neumann. In
KCC
there is an isolated letter from von Neumann to ‘My dear Mr Turing’, dated ‘December 6’ without year. It concerns a theorem about topological groups proposed to him by AMT. The year is most likely 1935; von Neumann’s letter contains a reference to the mailboat, so this could not be 1936 or 1937. By 1938 AMT’s research interests had moved away from this field. My search through the von Neumann papers in the Library of Congress did not reveal any more of this correspondence.

(
2.37
) AMT’s great paper, quoted here, was ‘On Computable Numbers, with an Application to the Entscheidungs problem’,
Proc. Land. Math. Soc.
(2),
42
(1937). It is reprinted in
The Undecidable
(as note 2.31).

(
2.38
) Did AMT think in terms of constructing a universal machine at this stage? There is not a shred of direct evidence, nor was the design as described in his paper in any way influenced by practical considerations. Yet in his obituary of AMT in
The Times
, Newman wrote: ‘The description that he then gave of a “universal” computing machine was entirely theoretical in purpose, but Turing’s strong interest in all kinds of practical experiment made him
even then
interested in the possibility of actually constructing a machine on these lines.’ (My italics.) Newman did not repeat this claim in his Royal Society memoir, in which the practical side was so much played down, although there he commented on how bold an innovation it had been to bring ‘paper tape’ into symbolic logic. Both comments reflected the impact made by AMT’s concreteness upon a classical pure mathematician, but like the other obituary writers, Newman was concerned to delineate AMT’s mental unorthodoxy, rather than to document anything in the history of technology. We have nothing more to go on. My own belief is that the ‘interest’ must have been at the back of his mind all the time after 1936, and quite possibly
motivated some of his eagerness to learn about engineering techniques. But as he never said or wrote anything to this effect, the question must be left to tantalise the imagination.

New Men

(
3.1
) A. Church, ‘A Note on the Entscheidungs problem’, in
J. Symbolic Logic, 1
(1936), reprinted in
The Undecidable
(note 2.31). The paper of a year earlier was ‘An Unsolvable Problem of Elementary Number Theory’,
Amer. J. Math. 58
(1936), presented 19 April 1935.

(
3.2
) The first letter in what was to be a more copious flow of correspondence during the Princeton period, in which AMT managed while away to think of something to say every three weeks or so. There are only eighteen letters in
KCC
for the five academic years 1931 to 1936, but twenty-eight for the two Princeton years. This frequency was never resumed, a total of nine letters home representing the remaining sixteen years of his life.

(
3.3
) G.H.
Hardy, AMathematician ‘s Apology
(Cambridge University Press, 1940).

(
3.4
) When Mrs Turing came to write her biography, she found herself better informed about AMT’s environment at Princeton than anywhere else, thanks to his letters. Though largely transcribing the information in these, she added one story which does not derive from a
KCC
letter: ‘Though prepared to find democracy in full flower, the familiarity of the tradespeople surprised him; he cited as an extreme case the laundry vanman who, while explaining what he would do in response to some request of Alan’s, put his arm along Alan’s shoulder. “It would be just incredible in England.’” Perhaps there was something of an ‘alas!’ in AMT’s remark, which would not have fitted in with Mrs Turing’s ideas about tradesmen.

(
3.5
) Two postcards from Scholz, dated 11 February and 15 March 1937, are in
KCC.

(
3.6
) A remark quoted in the review of von Neumann’s contributions to the ‘Theory of Games and Mathematical Economics’, by H.W. Kuhn and A.W. Tucker,
Bull. Amer. Math. Soc. 64
(1958).

(
3.7
) Published posthumously in
The Undecidable
(note 2.31).

(
3.8
) Post’s paper is reprinted in
The Undecidable.

(
3.9
) Letter to the author from Dr A.V. Martin, 26.1.78.

(
3.10
) See note 8.67.