Book Review

Alan Turing: HisWork and ImpactReviewed by Jeremy Avigad

Alan Turing: His Work and ImpactS. Barry Cooper and Jan van Leeuwen, eds.Elsevier Science, 944 pages,English, US$74.95ISBN-13: 978-0123869807

The year 2012 marked the centennial of Alan Tur-ing’s birth, with conferences, articles, broadcasts,and celebrations in his honor. The popular mediahas embraced Turing, and even in staid academiccircles his status borders on sainthood. In an essayin the collection under review, Jonathan Swintonsums up the state of affairs nicely:

Propelled by a brilliant biography (Hodges,1992), the awareness of Turing as aniconic, field-defining figure passed fromlogicians and philosophers, through gayrights activists, applied mathematicians,cryptographers, artificial lifers, mainstreamcomputer scientists, and into popular sci-entific culture, where he is now firmlyestablished as a boffin of fiction, stage, andscreen.

Some may wonder whether the level of praisethat has been lavished on Turing is justified. Yes,the 1936 paper in which he defined what arenow known as Turing machines is a landmarkin the history of science, but other logiciansat the time were involved in the development offormal models of computation, including Herbrand,Gödel, Post, and Church. And, although Turingplayed an important part in the development ofthe modern electronic computer, his role should

Jeremy Avigad is professor of philosophy and mathematicalsciences at Carnegie Mellon University. His email address isavigad@cmu.edu.

DOI: http://dx.doi.org/10.1090/noti1160

not be overstated: a number of early computingdevices were built from the late 1930s to mid-1940s with little or no influence of Turing’searly foundational work, and the real historyinvolves a complex mix of theoretical and practicalinsights, with contributions from a substantialcast of researchers and engineers alongside Turing.Finally, although the hideous abuses that Turingwas subjected to as a homosexual in postwarBritain are made even more ironic by the role heplayed in helping the Allies win the war, thoseabuses are no more morally offensive than thosesuffered by countless others whose lives are leftuncelebrated. With all this in mind, it is reasonableto ask: Was all the hoopla really called for?

One need only sit down with Turing’s workfor an hour or two, however, to find such doubtspushed aside. His papers are full of technical detail,but are surprisingly easy to read. His writing iscareful and methodical, but even after all theseyears the work feels creatively fresh and original.He dealt with weighty topics and raised importantquestions, but relied on technical mastery and agood deal of hard work to back up his answers.When we read his original papers, we get a senseof Turing as an accomplished mathematician andscientist, one whose philosophical disposition,talent, and drive are well worth celebrating.

One of many publications designed to mark thecentennial is Alan Turing: His Work and Impact,a collection that surrounds some of Turing’smost important works with a mixture of scholarlycommentary, introductory background, historicaland scientific surveys, and personal reflections.These supporting essays are written by notableresearchers in mathematics, computer science,logic, biology, cognitive science, and philosophy,as well as a few who knew Turing personally.

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It may be helpful to compare Work and Impactto the Collected Works of A. M. Turing, the fourvolumes of which appeared between 1992 and2001. Whereas the Collected Works was more thanforty years in the making, Work and Impact wascompleted in less than three. And whereas theCollected Works is regimented and carefully edited,Work and Impact is a much more heterogeneoussprawl, ranging from expositions of Turing’s workitself to personal accounts of the effects thatTuring’s contributions have had on some of theessayists. According to the introduction, Work andImpact encourages “a visceral engagement” withTuring’s work, and it reflects “a personal organicinvolvement” on the part of many of the authors.Reading the work of a seminal and visionary thinkerinevitably encourages one to try one’s own handat speculation, and there is no shortage of that,either. For all it offers, the price is quite reasonable:you can have all 944 pages of Work and Impactadorning your coffee table, in hardcover, for under$75.00. (It is worth mentioning that there is alsoa smaller, more focused collection published byClarendon Press: The Essential Turing features aselection of Turing’s most important works, editedby B. Jack Copeland with helpful introductorynotes.)

Turing is best known for his work on com-putability, the “Turing test,” and his contributionsto the British war effort in decrypting messagesencoded by the German Enigma machine. But hiscollected works contain contributions to a muchbroader array of topics, both pure and applied.The Collected Works is divided into four volumes:

(1) Mathematical logic(2) Pure mathematics(3) Mechanical intelligence(4) Morphogenesis

The four parts of Work and Impact roughly mir-ror these divisions, but with more colorful anddescriptive labels:

(1) How do we compute? What can we prove?(2) Hiding and unhiding information: cryptog-

raphy, complexity, and number theory(3) Building a brain: intelligent machines,

practice, and theory(4) The mathematics of emergence: the mys-

teries of morphogenesis

In the collection, more than 80 surveys andcommentaries supplement two dozen or so ofTuring’s original works. Space does not allow me todiscuss all of the supporting essays, but I will try toconvey a sense of the range of topics they addressand pass along some of the observations andcomments that caught my attention. Some of themore speculative pieces stand in stark contrast toTuring’s deliberate, measured style of writing, and I

generally preferred the more conservative analysesas well as the essays that help situate Turing’swork within our current scientific understanding.But these preferences should not be interpretedas a value judgment: like the seven-season runof Mad Men, everyone spending time with Workand Impact will have their favorite parts, and it isoften more interesting to talk about the generalplot lines rather than the individual scenes.

Part I contains Turing’s work on logic andcomputability. It begins with the 1936 classic,“On computable numbers, with an applicationto the Entscheidungsproblem,” which introducedwhat are now known as Turing machines. This isfollowed by a 1937 paper, “Computability and λ-definability,” which demonstrates the equivalenceof Turing computability with Church’s notion ofcomputability byλ-terms, and the Herbrand–Gödel–Kleene notion of a general recursive function. Inthe years after World War II, these papers played akey role in laying the foundations for a theory ofcomputation, which was needed to make sense ofand to support the technology that has transformedour lives so dramatically. These early works areso iconic that providing informative commentaryis no easy task, but a number of essays in thecollection rise to the occasion.

It is interesting to note, however, that Tur-ing’s early work dealt with the foundations ofmathematics as much as with the foundation ofcomputing. In the wake of Gödel’s discovery of theincompleteness phenomena, Turing wrote paperson truth and ordinal logics, which provide ways ofextending a formal system with stronger axioms.These are helpfully explained in articles by MichaelRathjen, Solomon Feferman, and Philip Welch.Turing was interested not only in logical strengthbut in the expressivity of formal languages, as ismade clear in his essays “Practical forms of typetheory” and “The reform of mathematical notationand phraseology.” The latter begins amusingly asfollows:

It has long been recognised that mathe-matics and logic are virtually the sameand that they may be expected to mergeimperceptibly into one another. Actuallythis merging process has not gone all thatfar, and mathematics has profited verylittle from researches in symbolic logic.The chief reasons for this seem to be alack of liaison between the logician andthe mathematician-in-the-street. Symboliclogic is a very alarming mouthful for mostmathematicians, and the logicians are notvery much interested in making it morepalatable. It seems however that symboliclogic has a number of small lessons forthe mathematician which may be taught

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without it being necessary for him to learnvery much of symbolic logic.

The particular small lessons Turing discussesin this essay involve ways that Church’s simpletype theory provides a good framework to encodeordinary mathematical assertions and make themprecise. In his associated commentary, StephenWolfram underscores the importance and difficultyof making sense of informal notation, and herelates the challenge to the design of computeralgebra systems.

Part II, which surveys the range of Turing’s math-ematical contributions outside of logic and thetheory of computability, offers more surprises tothose only casually familiar with Turing’s work. Forexample, few may be aware that, as an undergrad-uate, Turing did substantial work in probability.After attending a series of lectures in 1933 in whichthe astrophysicist Sir Arthur Stanley Eddingtonnoted that repeated experimental measurementsare often normally distributed, Turing set himselfthe task of providing a rigorous explanation. Withina few months, he had proved a general form ofthe central limit theorem for independent but notnecessarily identically distributed variables. Whathe had discovered was a version of a theoremthat the Finnish mathematician Lindeberg hadproved twelve years earlier, as well as part of aconverse, due to Feller and Lévy, which had notyet been published at the time. These results arenow textbook standards, and are fundamental tomodern probability. And Turing discovered themas an undergraduate.

Others will be pleased to learn that Turing hada longstanding interest in the Riemann hypothesis,an interest he shared with Littlewood’s student,Stanley Skewes, a regular rowing partner of hisin Cambridge. Turing introduced a number ofimportant ideas, including algorithmic methodsfor computing the zeros of the zeta function thatare now standard in computational branches ofanalytic number theory. Denis Hejhal and AndrewOdlyzko do a fine job of explaining the intricaciesof Turing’s methods, but the take-away messageis that Turing was no mere dabbler: when hepicked up a problem, he threw himself into it, withingenuity and hard work.

Part II contains excerpts from, and commentarieson, Turing’s 1940 paper on the Enigma. It alsocontains Turing’s 1944 notes on the design of asystem, “Delilah,” for voice encryption. A few yearslater Turing turned his attention to the burgeoningfield of electronic computers, whose early days arecolorfully described in an article by Toby Howard.The first digital stored-program computer wasdesigned and built at the University of Manchester,and known as the “Baby.” Its first program ran onJune 21, 1948, and was designed to take an input,

N, and compute the largest proper divisor. Withina few weeks, the program was able to computethe largest factor of N = 218. It took the Baby,however, 52 minutes to complete the task.

Turing had accepted a post in Manchester inMay 1948, but he did not arrive until October 1948,having written programs for the Baby in the interim.Work and Impact contains a remarkable reportwritten by Turing, “Checking a large routine,” whichdiscusses strategies for verifying the correctnessof a computer program. As Cliff B. Jones correctlypoints out in his essay, this foreshadows the studyof methods of program verification, an ongoingand active research area in computer science.Part II also devotes seven pages to excerpts fromTuring’s ninety-page programmer’s handbook fora successor to the Baby, known as the FerrantiMark I. Though the machine code was rudimentary,the excerpt reads much like any contemporaryintroduction to programming, with advice onhow to break a problem down to componentsand modularize the task through the use ofsubroutines.

Part II also contains Turing’s 1949 paper thatshows that the word problem in semi-groups withcancellation is undecidable. Having shown in 1936that the halting problem is algorithmically unsolv-able, here he was concerned with exploring theextent to which unsolvability arises in traditionalmathematical questions. The work improves onthe prior result, due to Post and Markov indepen-dently, that the word problem for semi-groupsis unsolvable. The result was later extended togroups by Novikov and Boone independently, inthe mid-1950s. There is now, of course, a well-developed theory of undecidability phenomena inmathematics. One of the crowning achievementsis the result of Matiyasevich, Robinson, Davis, andPutnam, who showed that there is no algorithm todetermine whether a given Diophantine equationhas a solution in the integers, thereby providing anegative answer to Hilbert’s 10th problem.

The last work by Turing presented in this sectionis a paper called “Rounding off errors in matrixprocesses.” As Lenore Blum explains, this paperintroduces the notion of the condition number ofa matrix, a quantity which measures the extent towhich calculations with linear systems of equationsare subject to rounding errors. Both the problemof managing rounding errors and the notion ofcondition number itself are now central to numericcomputation.

Part III turns to Turing’s work on mechanizedintelligence, with such landmark papers as “In-telligent machinery,” “Computing machinery andintelligence,” and “Digital computers applied togames.” Discussions of the Turing test are now com-monplace in undergraduate philosophy classes,

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but the essays make clear that the idea was notjust a passing fancy for Turing. These writings areamong his most speculative and most prescient.“Intelligent machinery” in and of itself inauguratedthe field of artificial intelligence, distinguishedclearly between discrete and continuous modelsof computation, and even foreshadowed the studyof neural networks and machine learning.

The essays are also among his most provocative.“Computing machinery and intelligence,” whichproposes the Turing test, addresses a wide rangeof objections to the thesis that machines canexhibit intelligent behavior, including theological,philosophical, and mathematical arguments. Re-sponding to a skeptical 1947 lecture, “The Mindof Mechanical Man,” by the British neurologistand neurosurgeon Geoffrey Jefferson, Turing chal-lenges us to imagine how we might react to amachine that could not only compose a sonnet,but also respond to an interrogator’s queries aboutthe choice of words and the sentiments expressed.But even when Turing was being provocative, hewas making a clear scientific point. Recognizingthat questions as to whether machines can thinkare not only vague but also loaded with social andreligious implications, Turing deftly shifted focusto the much more scientifically fruitful questionas to whether machines can exhibit intelligentbehavior. (The Turing test was designed to framethat question in even more pointed observationalterms.) The essays moreover provide a thoroughand thoughtful exploration of what it means toexhibit intelligent behavior, ranging from carryingout mathematical calculations and playing gamesto composing sonnets and even engaging in friv-olous banter. The first step in developing a newscience is to gain clarity as to the phenomena oneis trying to model and explain, and Turing did anadmirable job in mapping out the landscape.

But Turing went further than that, with concreteproposals as to how various types of intelligentmachine behavior might be obtained. For example,“Digital computers applied to games” presents anexplicit two-move lookahead algorithm for playingchess and a “run” of this algorithm against anovice player, with Turing playing the role ofthe computer and simulating the algorithm byhand. In this way, Turing’s essays on machinebehavior are as scientifically substantive as theyare entertaining. As a result, it is fitting that almosttwo dozen supplementary essays in Section III aredevoted to exploring the significance of this work.

The historical data presented in Part II makesthe essays in Part III even more poignant. It is onething for us, today, to take a break from shuttlingbetween Google and Siri to speculate on the natureof machine intelligence. It is yet another thingentirely to spend as much time as Turing did

writing machine code to compute basic numericoperations and to come away from the experiencewith the conviction that computers are capable ofexhibiting intelligent behavior.

Part IV deals with Turing’s work in biology,centered on the single long paper “The chemicalbasis of morphogenesis.” The paper was publishedin 1952, just after Turing was tried and convictedfor homosexual activities, and it is reproduced infull. It addresses a fundamental question: Howdo structured and complex organisms, like plants,animals, and humans, develop from a single cell?Somehow, the information and mechanisms storedin the original cell have to govern a reproductiveprocess in which future cells are created andassume radically different roles.

One might speculate about the path that ledTuring to this work. Although Crick and Watsondid not discover the structure of DNA until 1953,the paradigm of classical genetics—based on theMendelian notion of a gene—was firmly in place inthe early 1950s, when Turing took up the project.What was called for, it would seem, is an analysis ofgenes as a method of encoding computer programspowerful enough to orchestrate the growth process.

But this is emphatically not the approach thatTuring took. He was not one to use a tool or methodjust because it was near to hand. He approachedevery problem afresh, distilled it down to whathe took to be the essentials, and adopted themethods that he felt were most appropriate. Inthe case at hand, Turing was more interested inthe physical and chemical processes that couldgive rise to such complex behavior, and he foundthe language of dynamical systems and partialdifferential equations more appropriate to theanalysis.

Turing’s approach to the problem is intricate,but his exposition is, as usual, clear and methodical.The supplementary essays in Part IV, especially thesurveys by Henri Berestycki and Hans Meinhardt,James D. Murray, Peter Saunders, and JonathanSwinton do a good job of explaining the scientificsubstance of Turing’s work and its contemporaryrelevance. We do not often think of Turing asan influential biologist, but, as Saunders pointsout, the paper has had a tremendous impact, withwell over 3,000 citations listed on the ISI Web ofScience. And although the general consensus isthat the processes that Turing described cannotbe the entire story, the mechanisms that underliemorphogenesis are still poorly understood today.Which is to say, more than sixty years later, it isstill too early to fully assess the importance ofTuring’s contributions to biology.

The editors of Work and Impact, S. Barry Cooperand Jan van Leeuwen, are right to emphasizethe rewards of a personal engagement with Turing’s

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work. The sheer magnitude of Turing’s accomplish-ments is impressive. As Rodney A. Brooks notes inhis essay:

It is humbling to read Alan Turing’s papers.He thought of it all. First.

But reading Turing is as encouraging as it ishumbling. When we spend time with his work,we become keenly aware that Turing was not acomic-book superhero or a demigod; he was apure and applied mathematician, practicing hiscraft, like the rest of us. He was exceedinglyclever and talented, but he was also careful,methodical, thoughtful, and hard working. Hesought out questions and problems that he feltwere interesting and important, and he pursuedanswers with energy, focus, creativity, and élan.He was not afraid to entertain big ideas, but bigideas by themselves are light and airy things, andTuring did the work that was needed to make themsubstantial. There is a lot to be learned from hisoutlook and temperament, and Alan Turing: HisWork and Impact can serve as both an instructionaland an inspirational text.

AcknowledgmentsI am grateful to Thomas Hales and Robert Lewisfor very helpful comments.

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