Update to
Alan Turing: the Enigma

by Andrew Hodges

Part 2: The Spirit of Truth

Alan Turing: the Enigma

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Page 59:  That AMT bathed naked, I know from a photograph of him taken on the rocky coast of Sark. You will see it on this page of my Alan Turing Internet Scrapbook.

Page 67: I did not labour the obvious fact that AMT took it, as a matter of course, that he should read von Neumann's book in German. Reading German was, in 1932, an essential part of serious mathematical work. (The Gödel paper that was his main reference in Computable Numbers was also in German, and Princeton Ph.D. involved a German language qualification.) On the same page 67, it is noted that AMT visited Germany with his father; he was in Austria skiing in March 1934, and back in Germany again in June 1934 (pages 88-90). The incidents reported there mean that he had quite adequate conversational German. He would not have had the slightest difficulty with any German language questions arising in the cryptanalytic work after 1938.

Page 90 and note 2.26 The leading statistician R. A. Fisher was the referee for AMT's dissertation. His correspondence with King's College can be seen here.

Page 92: It is impossible to give a conventional nationality for Gödel. He was born into the Austro-Hungarian Empire, in a German-speaking part of what is now the Czech Republic. But he would not have identified with the Czechoslovak state of the 1930s. The indication of 'Vienna' would have been more appropriate. The question of Gödel's national identity illustrates vividly the difficulty of applying the principle of 'self-determination' as embodied in the Versailles treaty.

Page 94:  I suspect that the clerihew about AMT's alluring qualities did not actually circulate until the period of 1948 when he was back in residence at King's.

Pages 98-99:  The idea of the Turing machine is much better illutstrated by computer simulation, rather than by the printed page. Some examples, implemented in JavaScript code, are available on this page of the Update area. There is a more extensive version of this simulator, due to Paul Rendell, here.

Pages 104-109 and note 2.37: The publication year for AT's paper is now generally given as 1936, but some references may give it as 1937. The reason is that volume 42 of the Proceedings of the London Mathematical Society was published in parts. The parts containing AT's paper bore the dates of 23 November and 23 December 1936. However, the complete volume counts as a 1937 volume (see the LMS page, from which a facsimile copy can be downloaded.

My original text gave the publication date of the paper as 1937 for this reason, but I have changed it to 1936 in the 2014 editions for conformity with the general usage. In any case, the writing and submission of the paper all fell within 1936.

Page 105:  The philosopher B. J. Copeland criticises me for crediting Turing with the 'brave suggestion' that states of mind are finite in number. (A quibble: actually, I don't; my remark refers to the states of mind being countable rather than finite.) In his 1992 Artificial Intelligence: a Philosophical Introduction  (page 280, as a note to his argument on page 238 about the possibly infinite number of brain states) Copeland argues against the significance of Turing's restriction to finitely many states of mind by saying '...Turing's point seems to be rather that a Turing machine can simulate a device with an infinite number of internal states by using symbols to represent the states.' These are not however Turing's words, which were: 'We will suppose that the number of states of mind which need be taken into account is finite... If we admitted an infinity of states of mind, some of them will be 'arbitrarily close' and will be confused... the use of more complicated states of mind can be avoided by writing more symbols on the tape.'

Turing's remarks are not entirely clear, but I suggest that what he was referring to here is more like the trade-off between states and symbols that the Universal machine demonstrates. A machine with a limited and indeed quite small number of machine states can simulate the effect of a Turing machine with any number of states, because any machine can be encoded as a program on the tape (i.e. as a sufficiently long sequence of symbols). This is not the same thing at all as simulating a device with infinitely many states.

It still seems to me a brave suggestion of Turing's that non-observable entities such as states of mind could be countable, let alone finite in number. But I do not really wish to argue about Turing's bravery or lack of it, more concerned to clarify the finiteness conditions in his definition. It is crucial to the definition of computability and the application to the Entscheidungsproblem that any particular Turing machine has a finite number of states. (An infinite-state 'machine' could store the answers to all mathematical questions and so 'solve' the Entscheidungsproblem trivially.) This restriction is blurred by Copeland's gloss. (Also, Copeland's comment might suggest to some readers that a Turing machine could use an infinite number of different symbols to represent an infinite number of states, but Copeland could not possibly have meant this.)

Page 109:  My reference to Babbage's 'projected universal machine' recurs at various points in the book. I now think this understates AMT's achievement, by making it appear that he had merely rediscovered Babbage's idea. In fact he did much more. Turing himself led me into this, by claiming in his 1950 paper that Babbage's Analytical Engine would be a universal machine, but I need not have followed AMT's modest self-effacement. And others generally, awestruck at Babbage's farsightedness, have tended to assume that Babbage had all the ideas. It would be more accurate to say that Turing's exact account of a 'universal machine' principle gave a precise mathematical definition to Babbage's hazier concept. Furthermore I think it is unlikely that the Analytical Engine actually allowed for the unlimited number of conditional branches that would be necessary for universality.

A more subtle point is that Babbage did not have the modern concept of a program. His tables were more like logs of what the machine would be doing. In contrast, Turing's 'tables of instructions' are programs in the modern sense, being logical entities in their own right. In particular, it is essential to Turing's analysis that tables of instructions can be seen as sets of data for the machine itself to process. Babbage does not seem to have had any idea like this, but it is vital to the concept of the modern computer.

Page 109:  I wrote the sentence 'Alan had proved that there was no 'miraculous machine' that could solve all mathematical problems, but in the process he had discovered something almost equally miraculous, the idea of a universal machine thst could take over the work of any  machine.' This sentence is picked out by the reviewer for Amazon.com as going to the heart of Turing's story. And rightly so. The sentence has however also been picked out by B. J. Copeland for criticism in the pages of the Times Literary Supplement.

Copeland says it is wrong because it overlooks the 'oracle' concept Turing defined in 1938, (as appears in chapter 3 of the book). The oracle has mathematical properties that cannot be performed by the universal Turing machine. But Turing wrote that the nature of the 'oracle' is that it cannot be a machine. In fact the word 'oracle' is there to convey much the same sense as 'miraculous'. It is defined precisely to allow the mathematical exploration of what cannot  be done by machines: non-mechanical steps of mathematical proof.

My usage of the word 'machine' in this sentence simply echoes Turing's own language. In his 1938 work, he made his own summary of what had been achieved here by identifying the concepts of 'effectively calculable', 'purely mechanical process', and 'what can be carried out by a machine.' More comment and references are available in web-page form here.

There is further material in the Alan Turing Internet Scrapbook: The Inspiration of Life and Death and Turing machines.

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