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The Essential Turing Reviewed by Andrew Hodges Book Review The Essential Turing Reviewed by Andrew Hodges The Essential Turing the Second World War, with particular responsibility A. M. Turing, B. Jack Copeland, ed. for the German Enigma-enciphered naval commu- Oxford University Press, 2004 nications, though this work remained secret until US$29.95, 662 pages the 1970s and only in the 1990s were documents ISBN 0198250800 from the time made public. Many mathematicians would see Turing as a The Essential Turing is a selection of writings of the hero for the 1936 work alone. But he also makes a British mathematician Alan M. Turing (1912–1954). striking exemplar of mathematical achievement in Extended introductions and annotations are added his breadth of attack. He made no distinction be- by the editor, the New Zealand philosopher B. Jack tween “pure” and “applied” and tackled every kind Copeland, and there are some supplementary pa- of problem from group theory to biology, from ar- pers and commentaries by other authors. Alan Tur- guing with Wittgenstein to analysing electronic ing was the founder of the theory of computabil- component characteristics—a strategy diametri- ity, with his paper “On Computable numbers, with cally opposite to today’s narrow research training. an application to the Entscheidungsproblem” (Tur- The fact that few had ever heard of him when he ing 1936). This, a classic breakthrough of twenti- died mysteriously in 1954, and that his work in de- eth century mathematics, was written when he was feating Nazi Germany remained unknown for so twenty-three. In the course of this work in mathe- long, typifies the unsung creative power of math- matical logic, he defined the concept of the uni- ematics which the public—indeed our own stu- versal machine. As he himself put it, digital com- dents and our colleagues in the sciences—should puters are practical versions of this concept; and understand much better. he himself created an original detailed design for Many therefore will welcome this new edition and an electronic computer in 1945–46. His 1936 analy- the increased availability of Turing’s work. But the sis of mental rule-based operations was the start- foregoing remarks should make it clear that defin- ing point for his later advocacy of what is now ing the Turing oeuvre is not straightforward. There called Artificial Intelligence. His paper “Computing is no default option of reproducing published pa- machinery and intelligence” (Turing 1950a) is one pers and compiling them under a new cover. There of the most cited in modern philosophical litera- is a spectrum ranging from formal publication to ture. His paper in mathematical biology (Turing reports, talks, unpublished papers, unfinished work, 1952) then inaugurated a new field in nonlinear ap- letters, and several areas where other people de- plied mathematics. But he was also the leading sci- veloped work that he had inspired. Choices here are entific figure in the British codebreaking effort of not easy. Nor it is straightforward to define a genre or field in which to place his work, and the usual Andrew Hodges is lecturer in mathematics at Wadham Col- criteria of important papers in leading journals lege, University of Oxford. His email address is are of no use. Turing ignored conventional classi- [email protected]. fications, and created work which would now be 1190 NOTICES OF THE AMS VOLUME 53, NUMBER 10 described as at the foundations of computer sci- to explain the relationship of his ence or the cognitive sciences, areas which in his collection to the existing edition, time had no clear names of their own. and to refer readers to it for other On top of this, there is the difficulty that his war (presumably “inessential”) Turing work was never written for publication and exists works. Indeed, Copeland could only in operational reports which have been re- justly claim to have advanced upon leased in a chaotic fashion. Indeed, even since this the Collected Works in some areas— collection was published, Turing’s report on his ad- as he does when detailing tran- vanced electronic speech scrambler, with the won- scription errors in (Turing 1948a). derful date 6 June 1944, has emerged (Turing 1944). He has included more technical In the last decade the World Wide Web has trans- Enigma material than the Collected formed access to original Turing documents. In Works did, as well as Turing’s late particular, the collection of Turing’s papers at talks on machine intelligence, which King’s College, Cambridge University, is now ac- Copeland finds particularly impor- cessible at http://www.turingarchive.org. tant. But the inclusions and exclu- Those interested in original material are now much sions are nowhere systematically less dependent on the work of editors and pub- listed. lishers. Even so, a printed source-book will be val- Copeland’s edition has all the pa- ued by many to whom Turing is a somewhat leg- pers reset in a uniform typography: some readers endary figure, often cited but not easy to look up will always prefer to see the original format and this and quote. Such a book has enormous potential to decision means the loss of original page refer- educate and to inspire. ences. But Copeland is certainly no slouch when it comes to textual detail. For example, he devotes Copeland’s Anthology nearly a page to discussion of the spelling of the We now come to Copeland’s own editorial choices. word “program”. This review will consider points of interest in an More problematic is the central question of what order roughly corresponding to the order of is “essential”. To illustrate how differently the “es- Copeland’s anthology, which in turn reflects Tur- sential” Turing may appear in different eyes, it is ing’s chronology. But one point should be made worth recalling the survey of the topologist clear at the outset. Oxford University Press bills this M. H. A. (Max) Newman, written for the Biographi- as “the first purchasable book by Turing”, but cal Memoirs of the Royal Society after Turing’s Copeland’s volume is not the first edition of Tur- death (Newman 1955). In some ways Turing’s men- ing’s papers and not the most complete. A four- tor and father figure, Newman interestingly de- volume Collected Works of A. M. Turing was pub- fined him as “at heart more of an applied than a lished by Elsevier (Turing 1992, 2001). This work, pure mathematician” and devoted serious attention totalling some 938 pages, resulted from the pro- only to his mathematical papers. Of course, Tur- tracted collaboration of distinguished mathemati- ing’s war work was then totally secret, but even so cians and computer scientists. Particularly notable Newman’s characterization of it as a cruel loss to is the volume where John L. Britton, as editor, an- science was somewhat severe. Computer design notated Turing’s pure-mathematical work in line- and Artificial Intelligence received the briefest of by-line detail. mentions. This was too narrow a mathematical Unfortunately, little effort was made to pro- viewpoint, but it did reflect, perhaps, that sub mote the Collected Works, and the high price guar- specie aeternitatis aspect of mathematics in which anteed it few sales, even to university libraries. For Turing shared: he threw himself into the war effort this reason, Copeland’s new collection, offered at (as did Newman) but never, even in its darkest a paperback price, makes a good part of Turing’s days, forgot that he was a serious mathematician. work much more accessible in practice. It is still In contrast to Newman, Copeland highlights the odd, however, that Copeland virtually ignores the Enigma cipher machine as the subject of his sec- Collected Works. Indeed, an inattentive reader, miss- ond main section, the third focal point being Arti- ing the small print on pages 409, 510, and 581, ficial Intelligence and the Turing Test. would remain ignorant of it. This omission is compounded by another: Computability and Logic Copeland does not list Turing’s works, so the reader In one respect, however, Copeland is entirely in uni- cannot even guess how complete his selection is. son with Newman, and that is on the topic of com- There are quotations from some of Turing’s papers putability, which forms the first main section of his which have not been included in the work, but volume. The Essential Turing includes not only “On there is no overview of Turing’s output, nor any ex- Computable Numbers”, but also part of a paper by planation of what is considered “essential” and Emil Post which gives some corrections, and another why. Yet it would have been simple for Copeland technical commentary by Donald Davies. (For NOVEMBER 2006 NOTICES OF THE AMS 1191 A page from the typescript of Alan Turing’s classic 1936 paper, “On computable numbers, with an ap- plication to the Entscheidungs- problem”. This text appeared on page 256 of Turing’s published paper, with some very minor textual changes. The typewriting is not his, but the in- serted mathematical expressions are in his own hand. The lines drawn through the material may mean that it had been retyped, or merely that it was finished with and could be re- cycled as scrap paper (see below). The results on this page show that “computable numbers” include all the real numbers that normally arise in mathematics through limit definitions. Although Turing’s paper is usually thought of as concerned with the discrete world of mathe- matical logic, Turing wanted to con- nect computability with the main- stream of continuous analysis. In fact his opening remarks rather rashly asserted that he would soon give a theory of real functions based on the concept of computable num- bers.
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