Update to
Alan Turing: the Enigma

by Andrew Hodges

Part 3: New Men

Alan Turing: the Enigma

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Pages 116-117: There is a valuable passage on the history and culture of Princeton and its mathematics department in this period in Sylvia Nasar's A Beautiful Mind — her biography of the mathematician John F. Nash. The contrast of AMT with Nash is striking; indeed overwhelming in questions of personality. AMT was characterised by all who knew him as awkward but always harmless, genuine and honest; Nash was not. The way that AMT dealt with his sexuality at this period and later could again hardly be more different from what Nasar reports of Nash.

Page 117: It's not true that the tower of the Graduate College was an exact replica of the Magdalen College tower. This has been changed in the 2014 edition to say that it 'imitated' the Oxford feature. Also, I think the remark about the Ivory Tower was more the particular joke of a visitor from England than a usage enjoying general currency.

Page 123: 'Church reviewed it for the Journal of Symbolic Logic.' AMT might well have been disappointed by the lack of interest in his work on the part of the mathematical world in general, but Church was wholehearted in recommending and adopting AMT's definition of computability. Given that AMT was a young unknown outsider crashing into Church's field this was not something he could have taken for granted. As regards the tricky question of priority, Church wrote:

In an appendix, the author [i.e. AMT] sketches a proof of the equivalence of 'computability' in his sense and 'effective calculability; in the sense of the present author [i.e. Church's definition using the lambda-calculus.] The author's result concerning the existence of uncomputable sequences was also anticipated, in terms of effective calculability, in the cited paper [i.e. Church's paper]. His work was, however, done independently, being nearly complete and known in substance to a number of persons at the time that the paper appeared.
Newman would have been a primary witness here. In view of recent assertions made by the philosopher B. J. Copeland, it is also of interest to see how Church characterised the Turing machine definition; it was as follows:
The author [i.e. Turing] proposes as a criterion that an infinite sequence of the digits 0 and 1 be 'computable' that it shall be possible to devise a computing machine, occupying a finite space and with working parts of finite size, which will write down the sequence to any desired number of terms if allowed to run for a sufficiently long time. As a matter of convenience, certain further restrictions are imposed on the character of the machine, but these are of such a nature as obviously to cause no loss of generality — in particular, a human calculator, provided with a pencil and paper and explicit instructions, can be regarded as a kind of Turing machine.
So Church emphasised the idea of it being the most general possible kind of finitely defined machine, with the human calculator being just an example. As Church was writing this view while in close contact with AMT at Princeton, it is hard to believe that AMT seriously objected to this characterisation.

Now, the main assertion of Copeland's article for the Stanford Encyclopedia of Philosophy on 'The Church-Turing Thesis' is that both Church and AMT were careful never to confuse their statements with the thesis ('Thesis M') that anything performed by a finitely defined machine is computable. Yet this statement of Church's, which he repeated in a 1940 paper, is indistinguishable from 'Thesis M.'

As a further example of how Church positively endorsed AMT's definition, one sees on the very next page of the J.S.L. his review of Emil Post's paper. Church criticises Post for not giving a satisfactory definition of 'effectiveness' and says that 'To define effectiveness as computability by an arbitary machine, subject to restrictions of finiteness, would seem an adequate representation of the ordinary notion...' By this he means, of course, the Turing definition. The words 'arbitrary machine, subject to restrictions of finiteness,' are, again, tantamount to Thesis M. Surprisingly, Copeland's article quotes these words, and claims that Church meant 'a Turing machine, which has arbitrary aspects involved in its definition.' There is more comment in a paper by me available on this site.

Page 130: The Eisenharts' formal tea-parties have now been made famous by the physicist Richard Feynman (1918-1988) in his memoirs Surely you're Joking, Mr Feynman. Indeed Feynman's title comes from Mrs Eisenhart's words of reproach to his low-class New Yorker's culture. Coming to Princeton only in 1939, Feynman missed meeting AMT there; and as far as I know they never had any connection. But they were so alike in directness and irreverence that Feynman seems almost an (American, heterosexual) alter ego.

Page 145 and note 3.26: Ulam's letter to me about von Neumann can be read here.

Page 149 (original edition) page 188 (2014 edition): In August 1938, AMT joined the group of members from GC&CS and MI6 who viewed and approved Bletchley Park as the wartime headquarters. The visit is described on this page of the Bletchley Park Museum website.

There is further material in the Alan Turing Internet Scrapbook: Turing machines and Beyond the Turing machine.

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Andrew Hodges