Editor's Remarks

This relies heavily on the account by Martin Davis, along with a primary source (Turing's own writings) and the very useful account of Körner, which gave us good insight into what went on at Bletchley Park as far as the direct decoding of Enigma with the aid of a crib is concerned. He gives less information on the machines involved; Welchman would be good for that. It turns out that the "bombes" were less sophisticated than one might think, because of the simplicity of the logic involved. There are some hints in Welchman and Moreau that one might want to look into the Colossus project in which Turing was also involved, if the information is public.

The problem of decoding Enigma in the presence of a plugboard, and designing the Turing bombe, are subtle matters, and it takes Körner three chapters to lay out these stories in some detail, without going into any of the electronic details. The account given here seems to me to be clear enough (though probably a challenge to anyone who has never encountered a fuller description of the Enigma) and an excellent job of boiling down a complex situation in relatively few lines.

Finding a manageable theme, and finding the right references, turned out to be the major challenges here, along with avoiding some of the excessive intricacies of 20th century mathematics. After looking into Welchman and Moreau I feel our thesis was stretched to the breaking point, but it seemed worth looking into. Moreau offers some evidence that suggests to me that Turing's theoretical model may actually have been influenced by the earlier practical work on computing devices. All of this remains fairly murky (to me, at least) down to the present time. Nobody seems to know what von Neumann and Turing actually talked about. Also, Welchman is highly critical of the postwar development effort in Britain, feeling that only the Americans really responded to the potential.

Incidentally, I see a connection between this very modern topic and the history of algebra as well. To Nesselmann's three categories of rhetorical, syncopated, and symbolic algebra (concerning which, see Orlansky's essay) we now need to add a fourth, which still has not reached textbooks on the history of mathematics: electronic algebra. Complex algebraic computations are no longer made using a "literal" symbolism, but are re-coded into the binary structure of a computer calculation, finally releasing the full power of Viite's creatio n. Alan Turing, following on the heels of a number of visionary thinkers (Leibniz and Babbage, notably), played a key role in bringing this about. See also Pattanayak's essay on Leibniz, which touches on these points.

The 1950 quotation from Turing, with its reference to "digital computers", is very striking. It seems very early to be using such terminology at all.

While there are a limited number of references they are of good quality. Martin Davis' book is on the popular side, but useful. Moreau's book is interesting, and was not known to me. Turing's ideas, and his presentation of those ideas, are elegant and clear, though technical. And the Körner is a gem. The relationship to our main theme is a bit strained but the material is certainly interesting.

It took us a while to decide on a theme here. With a bit more time to work things out, we would have wanted to develop the post-war history, and take the work of von Neumann into account.

Back Back to the paper or the index