Gödel had not met Turing, and never would; in 1937 he was living in the Josefstadt, the doctors’ district of Vienna, and would return to Princeton only in the fall of 1938, by which point Turing had gone back to England. Nonetheless, he was soon making it clear how much he preferred Turing’s formulation to Church’s.* The universality and directness of the a-machine—the fact that, in Kleene’s words, it “aimed directly at the goal”—appealed to the Platonist in Gödel, who in 1946 applauded Turing for
giving an absolute definition of an interesting epistemological notion, i.e. one not depending on the formalism chosen. In all other cases treated previously, such as demonstrability or definability, one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for.
By “definability,” Gödel was no doubt alluding to Church’s λ-definability, which he considered so “unsatisfactory.” On the other hand, Gödel later wrote that “due to A. M. Turing’s work, a precise and unquestionably adequate definition of the general concept of formal system can now be given. . . .” As Feferman points out, Gödel never voiced publicly his dissatisfaction with Church’s thesis. Even so, he made it known how he felt. For Church, receiving a failing grade from the greatest figure in his field must have stung.
Whether Gödel’s obvious preference for Turing’s approach inclined Church to keep his distance from his ostensible student remains a matter of speculation. In certain ways they were very much alike—both eccentric, solitary, slightly out of step with ordinary social intercourse. Whereas Church, however, was genuinely antisocial—indeed, almost Aspergerian in his rigidity and disconnection—Turing voiced without shame a hunger for friendship and love, for Forsterian connection, that only his sexual anxiety stymied. He was also deeply pragmatic; he wanted to understand how daisies grew, and to invent patent inks, and to build typewriters and computers. Church, by contrast, lived almost entirely in his own head. Daisies and typewriters were to him what horned sheep and white sheep were to Boole: figurines in the game of logic. Earlier in the century, Bertrand Russell had asserted that “pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true.” Church subscribed to the formalist tradition, insisting that mathematical symbols be rigorously emptied of any semantic content that might inhere in them. Such a perspective was deeply troubling to Gödel, given that he made a point of “distanc[ing] himself from . . . formal-syntactic interpretations of science and mathematics.” In the end, it is no less surprising that Turing should have found working with Church a limiting and, finally, frustrating experience than that Gödel—the self-proclaimed realist—should have chafed at the systematic exclusion of semantic “meaning” that was in Church’s own view the lambda calculus’s great virtue.
* * *
*Hodges correctly urges caution in using the term “Turing machine,” pointing out that the phrase “is analogous to ‘the printed book’ in referring to a class of potentially infinitely many examples. . . . Again, although we speak of ‘the’ universal Turing machine, there are infinitely many designs with this property.”
*Jacques Herbrand (1908–1931) coined the term “effectively calculable” just before his death, in a skiing accident, at the age of twenty-three. According to Church, the work that he himself conducted in conjunction with Kleene and Rosser had its origin in some lectures that Gödel gave at Princeton in 1934, for which Kleene and Rosser took the notes. Casti and DePauli give a good definition of the recursive function, calling it “a function for which there is a mechanical rule for computing the values of the function from previous values, one after the other, starting with some initial value.”
*Post, who had lost an arm in an accident as a child, was probably bipolar. He died in 1954, at the age of fifty-seven, after suffering a heart attack during an electroshock therapy session.
*The joke—taking in other prominent Princeton mathematicians—continues, “If Weyl says it’s obvious, von Neumann can prove it. If Lefschetz says it’s obvious, it’s false.”
*Interestingly, both names remain in circulation, though the thesis is more commonly known today as the “Church-Turing thesis.”
*Maurice Pryce (1913–2003), physicist and professor at Oxford.
*Although George David Birkhoff (1884–1944) was considered one of the greatest mathematicians of his time, he was also, in Einstein’s words, “one of the world’s greatest anti-Semites,” consistently keeping Jews out of his department at Harvard.
*A complex number is defined as the combination of a real number and a so-called imaginary number. An imaginary number is, quite simply, the square root of a negative number—imaginary because any number, positive or negative, when multiplied by itself, gives a positive result. Hence cannot exist, and it is “imaginary.” the basis of all the imaginary numbers—is referred to as i. By the same token, is 2i, is 3i, etc. Because they combine real with imaginary numbers, complex numbers are expressed as 3 + 6i, – 2.547 – 1.34i, etc.
*According to Kleene, “only after Turing’s formulation appeared did Gödel accept Church’s thesis, which had then become the Church-Turing thesis.”
5
The Tender Peel
1.
It was at this point that Alan Turing’s intellectual life began to diverge from the course on which it appeared set. Having turned down an offer to work as John von Neumann’s assistant at Princeton (at a salary of $1,500 a year), he returned in fall 1938 to England, where he was recruited to join a course on cryptography and encipherment sponsored by the Government Code and Cipher School in London. Somehow word of his interest in codes and code breaking—not to mention his talent at mathematics—had found its way to Commander Alastair Denniston, the school’s director. Although far from a rabid patriot, Turing had no qualms about lending his services to the government; war appeared likely, and he was gravely distrustful of Hitler. The decision to take the course marked the beginning of a long association with the British government that would, in Hodges’ words, have “fateful” repercussions, in that it would require Turing, for the first time, to surrender “a part of his mind, with a promise to keep the government’s secrets.”
Nineteen thirty-eight also saw the British premiere of the Disney version of Snow White and the Seven Dwarfs—a film with which, curiously enough, both Turing and Gödel became fascinated. Turing went to see Snow White with his friend David Champernowne and took an especially keen pleasure in the scene where the Wicked Queen immerses the apple in her poisonous brew. “Dip the apple in the brew, / Let the sleeping death seep through,” she chants, then cackles to her sidekick, a raven, as the poison forms a skull on the surface of the apple. “Look at the skin,” the Queen continues.
A symbol of what lies within.
Now turn red to tempt Snow White,
To make her hunger for a bite.
The Queen offers the apple to the raven, who flaps wildly, trying to escape. “It’s not for you, it’s for Snow White,” the Queen says, laughing.
When she breaks the tender peel,
To taste the apple from my hand,
Her breath will still, her blood congeal,
Then I’ll be the fairest in the land!
The scene captivated Turing to such a degree that he took to chanting the Queen’s verses over and over. Perhaps what appealed to him was its morbid eroticism, not to mention the rather blatant allusion to the biblical myth: this apple tempts Snow White as another tempted Eve. Yet whereas Eve bites into the apple of her own free will (she could, after all, have resisted the serpent’s blandishments), Snow White is the victim of a well-rehearsed campaign of deception on the part of the Queen, who dresses as a hag, and tries out a variety of ploys to persuade Snow White to taste the apple; indeed, she is in the end able to convince her nemesis only by telling her that the apple is “magic” and that it will grant Snow White her wish. Only by appealing to her passion for the P
rince can the Queen break down Snow White’s resistance and persuade her to break “the tender peel,” in so doing losing her psychic virginity and consigning herself to a “sleeping death.” Which aspect of the film’s fraught psychosexual architecture appealed to Turing, however, he never let on.
Over the Christmas break, Turing attended another training course sponsored by the Government Code and Cipher School, after which he visited the school every two to three weeks to assist in the work being done there. But he had not forgotten his Princeton project to build a machine to calculate the zeros of the Riemann zeta function, and from Cambridge he applied for a grant from the Royal Society to pay for the machine’s construction, naming Hardy and Titchmarsh as references. On the application form, he wrote,
Apparatus would be of little permanent value. It could be added to for the purpose of carrying out similar calculations for a wider range of t and might be used for some other investigations connected with the zeta-function. I cannot think of any applications that would not be connected with the zeta-function.
The emphasis Turing put on the machine’s lack of “permanent value” underscored its remoteness from the universal machine. That machine, after all, could perform any algorithm presented to it, whereas the machine Turing wanted the money to build was so single-purpose as to defy any effort to think up further uses for it. So desultory, in fact, is Turing’s statement that one wonders if perhaps he hoped his proposal would be turned down. For better or worse, though, the Royal Society granted him £40, and he settled down in earnest to constructing the device by means of which he hoped to prove, once and for all, the falsity of the Riemann hypothesis.
His initial plan was to mimic the design of the Liverpool machine, which used a system of strings and pulleys to simulate the periodic sine waves corresponding to the motions of the tides and then added up their values; an answer could be obtained by measuring the lengths of string as they wrapped around the pulleys. But after consulting with Donald MacPhail, a student of mechanical engineering at Cambridge and the brother of Turing’s Princeton friend Malcolm MacPhail, he changed his mind and decided that instead of strings and pulleys, he would replicate the circular motions of the zeta function by means of gear wheels that meshed together. Like the Liverpool machine, this would be an analog machine, replicating motions in order to measure them. By contrast, digital machines (of which the early computers are examples) work by manipulating symbols and can therefore be put to much more general use. That the machine Turing envisioned was analog, by definition limited its applicability.
The money was there, though, so MacPhail drafted a blueprint (now in the Turing archive at King’s College), and for a time the floors of Turing’s rooms were strewn with precision-cut gear wheels awaiting their eventual incorporation into the machine. But the project was destined, once again, to be left unfinished. The war intervened. In fact, it was not until 1950 that Alan Turing was finally able to use a machine to test the zeros of the Riemann zeta function. This one would be digital.
2.
The spring semester of 1939 at Cambridge, Turing was involved in two courses, both entitled “The Foundations of Mathematics.” The first, an investigation into the history of mathematical logic, he taught himself. It had fourteen students, though Turing told his mother that he suspected attendance would drop off as the term advanced. The second, an investigation into the philosophical basis of mathematics, was taught by the Austrian philosopher Ludwig Wittgenstein. The participants included, in addition to Turing, several people with remarkable names: R. G. Bosanquet, Yorick Smythies, Rush Rhees, Marya Lutman-Kokoszynska, and John Wisdom. Notes were taken by Bosanquet, Rhees, Smythies, and Norman Malcolm, an American graduate student in philosophy who later wrote a remarkable memoir of the philosopher.
Wittgenstein was an eccentric figure. The scion of a wealthy as well as intellectually gifted family (his father was an engineer and something of a titan in the Austrian steel and iron industry), he counted Russell and Frege among his intellectual mentors and was close friends with Keynes and Hardy. Because he had been educated at Trinity College, Cambridge, he spoke flawless English, and in 1937 had assumed the chair in philosophy at Cambridge previously held by G. E. Moore. He had inherited a fortune upon the death of his father in 1912, but had given away most of his money, much of it in the form of a large anonymous grant for the promotion of literature. He also acted as something of a benefactor for the poet Rainer Maria Rilke.
Like Turing, Wittgenstein was a “confirmed solitary,” frequently retreating to a farmhouse in rural Norway where he could write and think in seclusion. His desire to flout the values of the world in which he had been raised led him, for a time, to work as a schoolmaster. During the First World War he joined the Austrian army as a volunteer, eventually being captured by the Italians, who held him as a prisoner of war for the better part of a year. The manuscript of one of his seminal works, the Tractatus Logico-Philosophicus, was in his rucksack at the time. Fortunately he was able to send copies to Russell and Frege, thanks chiefly to the intervention of John Maynard Keynes.
In 1939 Wittgenstein was fifty. Malcolm, who got to know him around this time, writes that when he first met the author of the Tractatus Logico-Philosophicus, he expected him
to be an elderly man, whereas this man looked young—perhaps about thirty-five. His face was lean and brown, his profile was aquiline and strikingly beautiful, his head was covered with a curly mass of brown hair. I observed the respectful attention that everyone in the room paid to him. . . . His look was concentrated, he made striking gestures with his hands as if he were discoursing. All the others maintained an intent and expectant silence. I witnessed this phenomenon countless times thereafter and came to regard it as entirely natural.
In sharp contrast to Alonzo Church’s lectures, which never varied from the prepared text in the library at Fine Hall, Wittgenstein’s “were given without preparation and without notes.” He told Malcolm “that once he had tried to lecture from notes but was disgusted with the result; the thoughts that came out were ‘stale,’ or, as he put it to another friend, the words looked like ‘corpses’ when he began to read them.” Though he spoke “with the accent of an educated Englishman,” his dress was astonishingly informal, given the time and the place. “He always wore light grey flannel trousers, a flannel shirt open at the throat, a woolen lumber jacket or a leather jacket. . . . One could not imagine Wittgenstein in a suit, necktie or hat.”
Wittgenstein taught his class in a highly unconventional manner. For one thing, the course was long, thirty-one hours divided into twice weekly sessions over the span of two terms. Generally speaking, the meetings took place in his own rooms in Whewell’s Court, Trinity College, with the students sitting on the floor or bringing in chairs. The rooms
were austerely furnished. There was no easy chair or reading lamp. There were no ornaments, paintings, or photographs. The walls were bare. In his living-room were two canvas chairs and a plain wooden chair, and in his bedroom a canvas cot. . . . There was a metal safe in which he kept his manuscripts, and a card table on which he did his writing. The rooms were always scrupulously clean.
Certain rules governed attendance. Although Wittgenstein placed no limitations on who could participate, prospective students had to submit first to an interview with the philosopher over tea, during which there would be long, intimidating silences, as Wittgenstein loathed small talk. Students also had to commit to attending the course continuously, not just for one or two meetings. Late arrivals were frowned on. “One had to be brave to enter after the lecture had begun,” Mal-colm recalls, “and some would go away rather than face Wittgenstein’s glare.” Though Wittgenstein could be hard on his students—once telling Yorick Smythies, “I might as well talk to the stove!”—he could be equally hard on himself, punctuating his lectures with exclamations such as “I’m a fool,” “You have a dreadful teacher,” or “I’m just stupid today.” According to Malcolm, “lecture” was hardly the rig
ht term to describe the classes, since they consisted principally of exchanges between Wittgenstein and the participants, dialogues broken only by the prolonged silences to which those in attendance soon became acclimated.
The subject of the course was the relationship of mathematics to what Wittgenstein called “ordinary” language and, by extension, “ordinary” life. As John Casti and Werner DePauli explain, “Wittgenstein applied his own principle that ‘the boundaries of my language signify the boundaries of my world’ to mathematics. In other words, the objects of mathematics were understood as being limited to those entities that could be formulated in mathematical language.” The approach was made clear at the first lecture, in which—perhaps because Turing was the lone mathematician among the participants—Wittgenstein used him as an example:
The Man Who Knew Too Much: Alan Turing and the Invention of the Computer (Great Discoveries) Page 12