The Man Who Knew Too Much: Alan Turing and the Invention of the Computer (Great Discoveries)

by David Leavitt

  Chance, in other words, has in the age of quantum mechanics supplanted “spirit” as the guiding principle that must underlie any effort to understand the universe—or has it? Turing is clearly ambivalent on this point. Although the atoms, in their action, “seem to be regulated by pure chance” (italics mine), in fact they are “probably” subject to the same “will” by means of which we as human beings are able to control at least a small portion of our brains. Thus “the remote effects of spirit” have not, in fact, been banished.

  What Turing appears to be struggling to reconcile here are his dedication to scientific rigor (a dedication partially instilled in him by Christopher Morcom) and his longing to preserve some link with Christopher’s spirit after his death. Indeed, at this point the essay becomes much more personal, and though Christopher’s name is never mentioned, his ghost hovers in the white spaces:

  Personally I think that spirit is really eternally connected with matter but certainly not always by the same kind of body. I did believe it possible for a spirit at death to go to a universe entirely separate from our own, but now I consider that matter and spirit are so connected that this would be a contradiction in terms. It is possible however but unlikely that such universes may exist.

  Does the spirit survive the body? If so, how and where? The question is religious in nature, yet in discussing it, Turing makes a point of never lapsing into the language of mysticism, or sacrificing his objective “scientific” outlook. Certainly it would have been a comfort for him to imagine that Christopher Morcom’s spirit, in some sense, had not just outlived his body but remained in the same “universe” as Turing:

  Then as regards the actual connection between spirit and body I consider that the body by reason of being a living body can “attract” and hold on to a “spirit,” whilst the body is alive and awake the two are firmly connected. When the body is asleep I cannot guess what happens but when the body dies the “mechanism” of the body, holding the spirit is gone and the spirit finds a new body sooner or later perhaps immediately.

  As regards the question of why we have bodies at all; why we cannot live free as spirits and communicate as such, we probably could do so but there would be nothing whatever to do. The body provides something for the spirit to look after and use.

  There is something intensely intimate and moving about this passage, written to comfort the mother of a boy whom Turing adored, and with whose “spirit” he hopes to remain “connected” by means of his body. More than two decades earlier, Forster had prefaced Howards End with the words “Only connect. . . .” The phrase reappears in chapter 22 of the novel, where Forster writes, “Only connect the prose and the passion, and both will be exalted, and human love will be seen at its highest. Live in fragments no longer.” Forster’s exhortation can be read as a call to connect the body and the spirit, and as such it puts him at odds with Clive Durham, who is able to sustain his relationship with Maurice only so long as sex plays no role in it; what scandalizes Clive, at novel’s end, is not just the discovery that Maurice is about to run off with Clive’s gamekeeper, Alec Scudder, but the announcement that Maurice and Alec have “shared.” Turing, by contrast, concludes his essay with a passionate affirmation of the physical that suggests the possibility of Forster’s having had an indirect influence on him. Were it not for the body, which “provides something for the spirit to look after and use,” the spirit, presumably, will languish. Without the body to give it expression, the spirit remains an abstraction from which in the long run no sustenance can be derived.

  As Hodges has noted, “Nature of Spirit” prefigures the investigations into the question of free will and determinism—the degree to which the spirit controls the body, and vice versa—that later provided the backbone for “Computable Numbers.” After all, the two points of view with which Turing is concerned here exactly parallel the “state of mind” and “instruction note” arguments from “Computable Numbers.” Yet the idea of investigating a “state of mind” might also have come from King’s College. Indeed, John Maynard Keynes, in describing the “religion” of G. E. Moore’s Principia Ethica, had also made use of the term “state of mind,” writing that for him and his fellow Apostles, “states of mind were not associated with action or achievement or with consequences. They consisted in timeless, passionate states of contemplation and communion, largely unattached to ‘before’ and ‘after.’ ” The value of these states of mind, Keynes continued,

  depended, in accordance with the principle of organic unity, on the state of affairs as a whole which could not be usefully analysed into parts. For example, the value of the state of mind of being in love did not depend merely on the nature of one’s own emotions, but also on the worth of their object and on the reciprocity and nature of the object’s emotions; but it did not depend, if I remember rightly, or did not depend much, on what happened, or how one felt about it. . . .

  At first glance, Keynes’s world of “passionate contemplation” and romantic friendship seems remote from Turing’s, with its human computers performing algorithmic operations. Yet there is a common element. Both men were driven by an impulse to analyze the mental apprehension of experience, to break it down into discrete units: in one case “timeless, passionate states of contemplation and communion, largely unattached to ‘before’ and ‘after,’ ” and in the other the “moments”—the m-configurations—into which a computational procedure can be subdivided. Also, under the surface of both analyses of the mental, there lies a tacit understanding that such states must take physical form if they are to have meaning. The body is implicit in Keynes’s essay, just as the unspoken possibility that an a-machine might actually be built resonates in Turing’s hypothetical, even abstract use of the machine analogy.

  2.

  Turing finished his first draft of “Computable Numbers” in the spring of 1936. It is hard to guess whether at this point he was aware of just how far-reaching its ramifications would be. In many ways he was as improbable a maverick as Gödel, from whose results he went to great pains to distinguish his own:

  It should probably be remarked that what I shall prove is quite different from the well-known results of Gödel. Gödel has shown that (in the formalism of Principia Mathematica) there are propositions U such that neither U nor –U is provable. As a consequence of this, it is shown that no proof of consistency of Principia Mathematica (or of K) can be given within that formalism. On the other hand, I shall show that there is no general method which tells whether a given formula U is provable in K, or, what comes to the same, whether the system consisting of K with –U adjoined as an extra axiom is consistent.

  Turing’s result emphasized process. True, his paper took as its starting point the Gödelian idea that mathematical operations involving numbers could be expressed as numbers. But then his fascination with the mind took Turing in a different direction from Gödel, a self-proclaimed “Platonist” and “mathematical realist” who once attempted an ontological proof of the existence of God. In many ways Gödel was an antiformalist. As he wrote in a letter to Hao Wang (December 7, 1967),

  I may add that my objectivistic conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my other work in logic.

  How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to consist of meaningless symbols which acquire some substitute of meaning only through metamathematics?

  Unlike Turing’s, Gödel’s preoccupations were not of a sort that would inevitably lead him to the Entscheidungsproblem. Yet Turing’s result, once he got there, had a distinctly Gödelian flavor: that is to say, his answer, in some sense, was no answer at all, since in effect all he had shown was that the decision problem itself was an example of an undecidable problem. On the other hand, Turing’s paper was immensely constructive in that it set out a clear-cut theory of computability while giving specific examples of large classes of compu
table numbers. The paper also put into circulation the first really usable model of a computing machine.* No matter that this machine, at least at first, was hypothetical; its simplicity was in many ways its greatest virtue.

  In April 1936 Turing gave his draft of “Computable Numbers” to Newman. At first, according to Solomon Feferman, Newman was “skeptical of Turing’s analysis, thinking that nothing so straightforward in its basic conception as the Turing machines could be used to answer this outstanding problem,” but he soon came around and encouraged Turing to publish the paper. Turing was naturally elated. At twenty-four, he stood on the brink of making a major contribution to his discipline, the sort that would secure his position at Cambridge and lead to an increase on his rather paltry £300 per annum stipend. Everything seemed to be going swimmingly. And then a difficulty arose.

  That May, Newman received by post an offprint of an article by the Princeton mathematician Alonzo Church entitled “An Unsolvable Problem of Elementary Number Theory.” The paper introduced a system called the lambda calculus, developed by Church in conjunction with his students Stephen Kleene and John Barkley Rosser, and then used that system to propose a definition of “λ-definability” that was in effect synonymous with Turing’s of computability. Worse for Turing, in a second paper Church used the concept of λ-definability to show that the Entscheidungsproblem was insoluble. The first paper, though it had been presented before the American Mathematical Society on April 19, 1935, had taken a year to cross the pond. The second had appeared in the Journal of Symbolic Logic just as Turing was finishing the first draft of “Computable Numbers.”

  Newman shared the news of Church’s papers with Turing, for whom it came as a shock. Once again—as at Sherborne, as with Sierpinski, as with his dissertation—history had pipped him at the post. Church, he explained to his mother, was “doing the same things in a different way.” But did this mean that his own paper was unpublishable? Newman, to Turing’s great relief, thought not. On the contrary, he told his mother, “Mr. Newman and I have decided that the method is sufficiently different [from Church’s] to warrant publication of my paper too.” Newman even suggested that Turing go to Princeton in order to study with Church, and toward that end wrote Church a letter outlining the situation:

  An offprint which you kindly sent me recently of your paper in which you define “calculable numbers,” and shew that the Entscheidungsproblem for Hilbert logic is insoluble, had a rather painful interest for a young man, A. M. Turing, here, who was just about to send in for publication a paper in which he had used a definition of “Computable Numbers” for the same purpose. His treatment—which consists in describing a machine which will grind out any computable sequence—is rather different from yours, but seems to be of great merit, and I think it of great importance that he should come and work with you next year if that is at all possible.

  Newman was worried, even at this early stage in the game, that Turing’s habit of working in isolation might end up hurting his career. Afterward, in his memoir, Newman wrote that his former student’s “strong preference for working everything out from first principles instead of borrowing from others . . . gave freshness and independence to his work, but also undoubtedly slowed him down, and later on made him a difficult author to read.” As a rule, most mathematicians labor alone, their only tools a pencil and a pad (or a blackboard and a piece of chalk); even so, in mathematical circles, an excess of self-imposed isolation tends to be frowned upon. Working on his own, as Turing did, involved a trade-off. On the one hand, as Gandy would later argue, “it is almost true to say that Turing succeeded in his analysis because he was not familiar with the work of others. . . . The approach is novel, the style refreshing in its directness and simplicity. . . . Let us praise the uncluttered mind.” On the other, Turing’s ignorance of what his contemporaries were up to meant that Church’s paper took him by surprise. As it happened, concern with the question of computability was very much in the mathematical air in the mid-1930s. Not only Church, Kleene, and Rosser but also Gödel, Jacques Herbrand, and Emil Post had been at work on the problem, which each described by means of his own terminology: Herbrand’s “effective calculability” was equivalent to Church’s “λ-definability,” Gödel’s concept of the “recursive function,”* and Turing’s of the computable number, just as Post’s formulation for a “finite-1 process” (worked out in cognizance of Church’s work, though not of Turing’s, and also published in the Journal of Symbolic Logic in 1936) bore a startling resemblance to Turing’s a-machine:

  In the following formulation . . . two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.

  Rather than using the metaphor of the machine, Post envisioned a sort of factory divided up into “boxes” in which “the problem solver or worker is to move and work . . . being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.” Although Post was American and taught at City College in New York, his frame of reference was the same mass-manufacturing ethos that informs The Man in the White Suit.* Yet his sequence of boxes is literally analogous to Turing’s tape, just as his “worker” is analogous to the a-machine. Indeed, the exactitude of the parallel between Post’s formulation and Turing’s machine lends credence to the Platonic conception of mathematics as a process of discovery rather than invention. It was as if an idea were issuing forth from nature itself, avid to find expression. And though Church had the very real advantage of being the first out of the starting gate, it was not yet clear whether his lambda calculus would, in the end, prove to be the most usable, the most pragmatic, or the most compelling of the approaches in circulation.

  3.

  Church himself was a decidedly odd figure. Born in Washing-ton, D.C., in 1903, he had spent virtually his entire adult life at Princeton, earning his bachelor’s degree, master’s degree, and Ph.D. from the university before joining the faculty in 1929. The only period of time he spent away from Princeton was in 1927 and 1928, when, as a National Research Fellow, he studied at Harvard, in Göttingen with Hilbert, and in Amsterdam with Brouwer. In a memoir of his years at Princeton, the Italian mathematician Gian-Carlo Rota recalls Church as looking “like a cross between a panda and a large owl. He spoke slowly in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a talking machine. When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument.”

  Church was famous for working all night, leaving his notes—carefully marked with colored pencils—for the mathematics department secretary to find and type up in the morning. He contributed little to the department aside from teaching and editing the reviews section of the Journal of Symbolic Logic, which he helped found in 1936; indeed, his failure to show up for faculty meetings often raised eyebrows and may have been part of the reason why he was not promoted to the rank of full professor until 1947—eighteen years after he joined the faculty.

  Stories circulated about Church’s remoteness. His colleague Albert Tucker recalled being told by the dean of the faculty at Princeton “that he often met Church crossing the campus, and he would speak to Church and Church would not speak in reply.” When afternoon tea was served in the departmental common room, Church would arrive “toward the end of the tea session, and he would take any milk or cream that was left in the pitchers there and dump that into one of the almost-empty teapots and drink this mixture of milk and tea. Then he would depart for his office, where he would work through the night.” His lecture style was pedantic and painstaking to a fault, leading to the witticism “If Church said it’s obvious, then everybody saw it half an hour ago.”*


  His behavior tended toward the compulsive. For example, Rota recalls, he had a vast collection of science-fiction novels, each of which was marked cryptically with either a circle or a cross. In some cases he had made corrections to wrong numberings in the margins of the table of contents. His lectures, Rota continues, invariably “began with a ten-minute ceremony of erasing the blackboard until it was absolutely spotless. We tried to save him the effort by erasing the board before his arrival, but to no avail. The ritual could not be disposed of; often it required water, soap, and brush, and was followed by another ten minutes of total silence while the blackboard was drying.” The lectures themselves required nothing in the way of preparation, since they consisted of literal recitations of typewritten texts prepared over the course of twenty years and kept in Fine Hall library. On those rare occasions when Church felt obliged to diverge from the prepared text, he would warn his students in advance.

  To some degree, Church’s punctiliousness was part and parcel of his talent as a logician; as Tucker notes, “he was completely oblivious of everything that went on in the world except in mathematical logic.” According to Rota, Church would never make such a simple statement as “It is raining,” because “such a statement, taken in isolation, makes no sense. . . . He would say instead: ‘I must postpone my departure for Nassau Street, inasmuch as it is raining, a fact I can verify by looking out the window.’ ” The lambda calculus was similarly flawless in its precision; in Kleene’s words, it boasted “the remarkable feature that it [was] all contained in a very simple and almost inevitable formulation, arising in a natural connection with no prethought of the result.” Still, Kleene argued, “for rendering the identification with effective calculability the most plausible—indeed, I believe compelling, Turing computability [had] the advantage of aiming directly at the goal. . . .” Church’s “λ-definitions” might, as Turing himself modestly put it, be more “convenient,” but the a-machine was “possibly more convincing.”