A Madman Dreams of Turing Machines
There is no beginning. I’ve tried to invent one but it was a lie and I don’t want to be a liar. This story will end where it began, in the middle. A triangle or a circle. A closed loop with three points.
At one apex is a paranoid lunatic, at another is a lonesome outcast: Kurt Gödel, the greatest logician of many centuries; and Alan Turing, the brilliant code breaker and mathematician. Their genius is a testament to our own worth, an antidote to insignificance; and their bounteous flaws are luckless but seemingly natural complements, as though greatness can be doled out only with an equal measure of weakness.
These two people converge in history and diverge in belief. They act out lives that are only tangentially related and deaths that are written for each other, inverted reflections. They are both brilliantly original and outsiders. They are both loyal to reason and to truth. They are both besotted with mathematics. But for all their devotion, mathematics is indifferent, unaltered by any of their dramas—Gödel’s psychotic delusions, Turing’s sexuality. One plus one will always be two. Their broken lives are mere anecdotes in the margins of their discoveries. But then their discoveries are evidence of our purpose, and their lives are parables on free will. Against indifference, I want to tell their stories.
Don’t our stories matter?
I shouldn’t even be here but some things you can only get to in the most awkward ways. Even if I tried to hide it, to lie, the truth is it’s still me telling this story. The unsorted catalogue of biographical facts provides nothing without stories with their dents and omissions and sometimes outright lies to create meaning that just won’t emerge from the debris of unassembled facts. Because some truths can never be proven by adhering to the rules. So this whole story about Truth is a lie. The liar says, This is a lie.
I am that liar, the third and final point on the triangle, the weak link, the wobbling hinge, the misaligned vertex. I am meant to carry on from the previous point and give over to the next. But I don’t know where to begin. I am standing on a street, in a city. I am going to catch a train. There are people streaming in all directions and one old woman strolling. Will any of us be remembered? Do any of us matter?
This story about truth and logic leads to atheism and mysticism. To despair and suicide. To the future, our past. To the present. So there’s a starting point as arbitrary as any because logic does not unfold in time. It exists forever into the past, dictating how the universe began; and forever into the future, our fate already written by the inescapable rules of logic. We can enforce chronology because the linearity of time helps; it gives us roles in the script, a place to start and to end. So here I start. In Vienna. It is the year 1931. This is as good a place, this is as good a time, as any.
Vienna, Austria. 1931
The scene is a coffeehouse. The Café Josephinum is a smell first, a stinging smell of roasted Turkish beans too heavy to waft on air and so waiting instead for the more powerful current of steam blown off the surface of boiling saucers fomenting to coffee. By merely snorting the vapors out of the air, patrons become overstimulated. The café appears in the brain as this delicious, muddy scent first, awaking a memory of the shifting room of mirrors second—the memory nearly as energetic as the actual sight of the room, which appears in the mind only third. The coffee is a fuel to power ideas. A fuel for the anxious hope that the harvest of art and words and logic will be the richest ever because only the most fecund season will see them through the siege of this terrible winter and the siege of that terrible war. Names are made and forgotten. Famous lines are penned, along with not so famous lines. Artists pay their debt with work that colors some walls while other walls fall into an appealing decrepitude. Outside, Vienna deteriorates and rejuvenates in swatches, a motley, poorly tended garden. From out here, the windows of the coffeehouse seem to protect the crowd inside from the elements and the tedium of any given day. Inside, they laugh and smoke and shout and argue and stare and whistle as the milky brew hardens to lace along the lip of their cups.
A group of scientists from the university begin to meet and throw their ideas into the mix with those of artists and novelists and visionaries who rebounded with mania from the depression that follows a nation’s defeat. The few grow in number through invitation only. Slowly their members accumulate and concepts clump from the soup of ideas and take shape until the soup deserves a name, so they are called around Europe, and even as far as the United States, the Vienna Circle.
At the center of the Circle is a circle: a clean, round, white marble tabletop. They select the Café Josephinum precisely for this table. A pen is passed counterclockwise. The first mark is made, an equation applied directly to the tabletop, a slash of black ink across the marble, a mathematical sentence amid the splatters. They all read the equation, homing in on the meaning amid the disordered drops. Mathematics is visual not auditory. They argue with their voices but more pointedly with their pens. They stain the marble with rays of symbolic logic in juicy black pigment that very nearly washes away.
They collect here every Thursday evening to distill their ideas—to distinguish science from superstition. At stake is Everything. Reality. Meaning. Their lives. They have lost any tolerance for ineffectual and embroidered attitudes, for mysticism or metaphysics. That is putting it too dispassionately. They hate mysticism and metaphysics, religion and faith. They loathe them. They want to separate out truth. They feel, I imagine, the near hysteria of sensing it just there, just beyond the nub of their fingers at the end of arms stretched to their limits.
I’m standing there, looking three hundred and sixty degrees around the table. Some of them stand out brighter than the others. They press forward and announce themselves. The mathematician Olga Hahn-Neurath is here. She has a small but valuable part to play in this script as does her husband, Otto Neurath, the over- sized socialist. Most important, Moritz Schlick is here to form the acme and source of the Circle. Olga, whose blindness descended with the conclusion of an infection, smokes her cigar while Otto drinks lethal doses of caffeine and Moritz settles himself with a brush of his lapels. The participation of the others present today is less imperative. A circle can be approximated by a handful of discrete points and the others will not be counted. There are perhaps more significant members of the Circle over the years, but these are the people who glow in color against my grainy black-and-white image of history. A grainy, worn, poorly resolved, monochromatic picture of a still scene. I can make out details if I look the shot over carefully. Outside, a wind frozen in time burns the blurred faces of incidental pedestrians. Men pin their hats to their heads with hands gloved by wind-worn skin. Inside a grand mirror traps the window’s images, a chunk of animated glass.
In a plain, dark wooden chair near the wall, almost hidden behind the floral arm of an upholstered booth, caught in the energy and enthusiasm of that hopeful time as though caught in a sandstorm, is Kurt Gödel.
In 1931 he is a young man of twenty-five, his sharpest edges still hidden beneath the soft pulp of youth. He has just discovered his theorems. With pride and anxiety he brings with him this discovery. His almost, not-quite paradox, his twisted loop of reason, will be his assurance of immortality. An immortality of his soul or just his name? This question will be the subject of his madness. Can I assert that suprahuman longevity will apply only to his name? And barely even that. Even now that we live under the shadow of his discovery, his name is hardly known. His appellation denotes a theorem; he’s an initial, not a man. Only here he is, a man in defense of his soul, in defense of truth, ready to alter the view of reality his friends have formulated on this marble table. He joins the Circle to tell the members that they are wrong, and he can prove it.
Gödel is taciturn, alone even in a crowd, back against the wall, looking out as though in the dark at the cinema. He is reticent but not unlikable. The attention with which his smooth hair, brushed back over his head away from his face, is creamed and tended hints at his strongest interest next to mathematics, namely women. His efforts often come to fruition, only adding to his mystery for a great many of the mathematicians around him. And while he has been known to show off a girlfriend or two, he keeps his real love a secret. His bruised apple, his sweet Adele.
There is something sweet about his face too, hidden as it is behind thick-rimmed goggle glasses, inverted binoculars, so that those who are drawn into a discussion of mathematics with him feel as though they are peering into a blurry distant horizon. The completely round black frames with a thick nosepiece have the effect of accentuating his eyes or replacing them with cartoon orbs—a physical manifestation of great metaphorical vision. They leave the suggestion, with anyone looking in, that all emphasis should be placed there on those sad windows or, more important, on the vast intellectual world that lies just beyond the focus of the binocular lenses.
He speaks only when spoken to and then only about mathematics. But his responses are stark and beautiful and the very few able to connect with him feel they have discovered an invaluable treasure. His sparse counsel is sought after and esteemed. This is a youth of impressive talent and intimidating strength. This is also a youth of impressive strangeness and intimidating weakness. Maybe he has no more weaknesses than the rest of us harbor, but his all seem so extreme—hypochondria, paranoia, schizophrenia. They are even more pronounced when laid alongside his incredible mental strengths. They appear as huge black voids, chunks taken out of an intensely shining star.
He is still all potential. The potential to be great, the potential to be mad. He will achieve both magnificently.
Everyone gathered on this Thursday, the rotating numbers accounting for some three dozen, believe in their very hearts that mathematics is unassailable. Gödel has come tonight to shatter their belief until all that is left are convincing pieces that when assembled erect a powerful monument to mathematics, but not an unassailable one—or at least not a complete one. Gödel will prove that some truths live outside of logic and that we can’t get there from here. Some people—people who probably distrust mathematics—are quick to claim that they knew all along that some truths are beyond mathematics. But they just didn’t. They didn’t know it. They didn’t prove it.
Gödel didn’t believe that truth would elude us. He proved that it would. He didn’t invent a myth to conform to his prejudice of the world—at least not when it came to mathematics. He discovered his theorem as surely as if it was a rock he had dug up from the ground. He could pass it around the table and it would be as real as that rock. If anyone cared to, they could dig it up where he buried it and find it just the same. Look for it and you’ll find it where he said it is, just off center from where you’re staring. There are faint stars in the night sky that you can see, but only if you look to the side of where they shine. They burn too weakly or are too far away to be seen directly, even if you stare. But you can see them out of the corner of your eye because the cells on the periphery of your retina are more sensitive to light. Maybe truth is just like that. You can see it, but only out of the corner of your eye.
Dorset, England. 1928
The scene is Sherborne boarding school for boys in Dorset. A sixteen-year-old boy lies in the grit and sawdust against the rough unfinished foundation below the floorboards of the dayroom at Westcott House. The boy under the floor is Alan Turing.
He isn’t there by choice. It has already been a while since the sudden shoving and scrabbling and it went dark. The wooden boards slotted into place. The clean ominous clink. The weighty, fitted lid of a makeshift coffin. Turing buried alive. It’s a transparent observation, but children always under estimate the menace of their weapons.
There is a window in time, no longer than it takes for a few wrought puffs of air to loosen dust clinging awkwardly to the underside of the floorboard, when the wood shifts a millimeter in response to his struggling. The force of several tangled arms pushing downward minus the force of one boy pushing upward. He isn’t entirely weak, Turing. He is wiry and strong, with a sinewy musculature. He strains and shoves until desperation bellows out of him in a deeper register than his voice can ordinarily reach. While the sound, a bit of a horror really, makes some of them anxious, no one of the boys has the authority to stop the others and so, with a small pit of dread planted deep in the lining at the tops of their stomachs, they collaborate to drag the heavy old oak table into place, two legs on the loose board, two well anchored on firmer slats.
Maybe this is an initiation ceremony. He grasps at the possibility optimistically. He means no irony. He does not mean a symbolic initiation of a boy into manhood. He neither uses nor understands metaphors. He means literally. There is a precedent for this miscomprehension. During his first week at Sherborne in 1925, there was an official ceremony that culminated in the older boys roughly collecting his arms and legs so that he was disconnected from the ground and folded up until he fit snugly into a barrel. He was then kicked down the hall. Although the blows were delivered indirectly, the humiliation was delivered rather more directly. Not knowing what else to do, he gave himself over to the incident. He bounced along—quite good-naturedly all in all—with eyes focused on a sliver of space between buckled wooden bands. But he thought it was stupid. He thought it was probably really stupid.
This is stupid too. He knows it is. He wants to shriek, to curse them, but the epithets catch in his throat, rough as sticklebacks. The boards brutally press his face to the right, mashing his cheek, his arms, and his thighs, so every arrested move is agony. The blackness an additional defeat.
He panics. He chokes on his own saliva. He twists. He knots his toes. He clamps his legs. He shrieks—a squeal that reaches up high in pitch as it mounts into a waterless sob. Struggling is making it worse. This is claustrophobia. This is delirium. Dementia. Even as he gags and spits vomit, he wants to yield. But he can’t relax, soften, surrender to the incident, be kicked down the hall in a wooden barrel. While he knows they are stupid, while he knows they are phonies, he also admits there are many things in the world he does not understand and, on occasion, it is better to confess this lack of knowledge of the world. On these occasions it is better to accept being trapped beneath the loose boards of Westcott House dayroom. But he can’t.
His knees grind. His toes kink. His back seizes and he recoils with a hyena’s laugh—not altogether uncharacteristic for Turing—such strange speech deserves a complementary laugh—but exaggerated, a high and serrated scream of a laugh. Alan’s entire vocal instrumentation is unmanageable. It is a kind of aural deformity whose source is no doubt the extreme brain chemistry of the highly functioning autistic. His voice booms and bangs and halts and stalls. He modulates the pitch but never the rhythm or the tone. The language of intonation is missing. The cadence that makes a voice bearable and creates meaning is missing. His speech is the grating clatter of a child’s spoon on a hard surface.
His overzealous volume and peculiar elocution sometimes come off as cheerful. This malfunction, along with the tendency to withdraw, leaves the masters of the school conflicted as to whether or not he is happy. When they correspond with Mrs. Turing, they try to structure their observations of her son into a poignant conclusion but bungle the job. Unable to temper their frustration they ultimately blurt, “Undeniably, he is not a normal boy.” This much she knew.
Above the floorboards, the room becomes a cavity trapping his shrieking laughter until it dies off ominously. The boys try to show off their callousness by ignoring the cackling vibrato slicing their ears. While they continue to play an anomalously quiet game, the pit of dread is jostled and falls deep into the fertile gastrointestinal soil where it begins its life cycle. Will it fester as an ulcer, or blossom into rancid abnormal cells? That depends on how each chooses to tend that messy garden.
Turing can’t see through the darkness. The blackness coats his open eyes, as do large salty dust grains so that his eyes burn and water but technically it doesn’t amount to crying. That comes later, when his ordeal is over. Or at least this one skirmish in his lifelong ordeal—a life of loneliness, persecution, and depression, lightly salted with childish bouts of happiness. But it won’t be a bad life all told and it will come to be laden with achievement and significance so great that even he never recognizes it.
This isn’t an initiation ceremony. It is punishment. Punishment for looking right through them, for looking right inside them.
There are bolts of luminescence in the world. Hard, brilliant candies that crackle like jewels, fanning pointed rays of gold through an otherwise gray landscape. Sometimes Alan can see these splendors unaided. He finds them in the woods or sees them in the sky. Sometimes he has to distill them from ordinary rock with clumsy chemical experiments he executes poorly in his room but pursues with the devotion of an alchemist until they yield gold, or just iodine, or some other element with a rightful place in the periodic table. Sometimes he discovers them with his mind like the inverse trigonometric function that he managed to express as an infinite series of simpler algebraic forms. These are the best, these dazzling gems of his brain’s relentless, systematic expeditions. The smells of his chemicals and their incendiary threat don’t deter him, but others are irritated by the mess and complain. His pure mathematical discoveries have the advantage of privacy—splinters of truth broken through the skin of confusion, framed by pools of crimson that only he can see.
To Alan, this is the world: luminous boulders, a string of precious stones. He jumps from one to the next just barely able to balance above the murky sea of fakes and phonies. Their world of facial signals and false strides, social norms. Their indirect, dull universe of dishonesty and contests and artifice.
He is baffled and lost in the tangled, witty exchanges the other boys are learning—mechanisms that evade candor with the flourish of the middle class. Through the swirl of debris of human behavior, he gropes for truth; and if he finds it, he stares at truth’s sparkle with a fidelity that offends the others. Annoyed, resentful, they kick the rocks up in his face, not knowing why he stares.
His room has never been clean, much to his mother’s frustration. She tries to show him, to demonstrate: “Look, Alan. Arrange your shirts in an orderly fashion.” And there are instructions too. “Clean carefully at the join of the wall and the floor or neglect will yield a rich harvest of furry dust.” She pats his jackets down in search of filthy handkerchiefs with the desperation of a mother searching her child’s pockets for contraband. There are always so many dirty cloths that ever she belches with disgust as she reaches in for those buried deepest, hence oldest. With a lick of her palm, she tries to train his hair and wipe down his dark complexion. Alan is her cause and she endures her frustration with patience and a confidence that falters only occasionally.
Either he doesn’t understand how to clean or he doesn’t care or, most flattering of all interpretations, he is too preoccupied with deep and complex distractions, the sanctity and purity of mathematics, the profound truth so completely immune to human stains that even his clumsy approach can’t tarnish its luster. Probably he just does not understand what it means to be clean, both for personal hygiene and for social success. And so the dusty taste of the dayroom floor is familiar to him, if more intense in concentration.
Turing reached sixteen this summer and recently discovered hormones. The hormones do nothing for his high voice, filling it with adult strength but he’s unable to retune it to a lower register. So his odd, hesitant, not-quite-stammer gains power instead of abating. The onset of his adolescent biochemistry heightens his sensitivity to the bristle of stiff hairs that climb in dark corners of his armpits. As if to embarrass him and reveal at a glance what lurks in his most private of areas, the hairs sparsely grass his greasy jaw—something dirty and sexual right there on his face. But he no more knows how to groom and organize this new manly growth than he knows how to groom the hair atop his head that stubbornly flops forward, a deep black fringe, or than he knows how to align the buttons with the buttonholes on his perpetually askew coat that covers his partially untucked shirt and loose off-kilter tie, a man-child’s poorly tied noose around an ink-stained collar. He is a mess. A dirty mess.
Dirtiness hangs about him like a slightly sharp smell rising up people’s noses until the work “dirty” rises in their minds—as it simultaneously rises in his own—a reminder of the adolescent sexuality that the others are already busy forgetting.
The sexual tension of Sherborne School for boys is manageable. Smutty comments and explosive wrestling are valves to regulate that tension and they worked well enough—not perfectly, but fine, just fine. The boys get through. Most of them yearn for girls, long for soft downy bodies with the basest for of urgency they are ever to experience, and the desire will never be quenched in their lifetimes. They can never satiate that adolescent lust since they can never again conjure up quite the same desperation. But they manage, even if they are tightly wound. Then there’s Turing. He never understands how to stand, how to talk, where to look, what to say. His innuendoes, his leering stares. Bloody Turing. Like someone who leans over to whisper and when you lean in, licks your ear instead. That’s Turing. Forever trying to lick your ear.
He blames the broccoli for today’s predicament. The broccoli on his plate put him in a state of great agitation in the dining hall. It was pungent and fractal with the texture of a small bush. He likes his food smooth. Lumpiness or random texture plunge him into anxiety. Plus the sour yellow green was very similar to the color of the pea soup. The soup was okay because it was smooth and thin as tea—the country’s culinary habits have been inspired by the stringent rationing imposed during the Great War. This suits him. He likes his food diluted and weak. Uniform in flavor and limited in variety. Color confusion is an especially foul trick. If foods are to have the same color, they must taste the same. But the broccoli tasted nothing like the soup, though they were both bland. The broccoli was soft and mushy but shapely. The very thought revolting.
What he really wanted was an apple. The thin tea-pea soup followed by an apple. AN apple is not smooth, but it is red, nowhere near sour yellow green, and it has this in its favor. An apple is also round. Some apples are as round as globes, lovely and geometric, the symmetry pierced by the stem, orienting the fruit to the north. He loves apples.
He could not invent a happy resolution to the broccoli. It sat unwanted on his plate, cold as fish. Discarding the boiled veg uneaten wasn’t easy. The masters were intolerant of wastefulness and the prefects would surely notice. So he was already at a great disadvantage when he sat down to dinner.
He never knew where to accompany his tray. He preferred to arrive early or late. If he was early then he could likely arrange the food on his plate in private. Sorting the hues and consistencies and hiding any real offenders to be disposed of later. If anyone joined him he would do so by choice and he would probably be an all right dinner companion. Not everyone hates him. Nobody hates him. Some of the boys just can’t bear him, which is not at all the same thing. They often privately regret their assaults on old Turing but still find the urge hard to resist the next time around. If he arrived late, there would be less food on offer, his plate automatically streamlined. With less sorting required, he could eat quickly and alone.
Today there were few seats to choose from and none of those were isolated. It was a game of roulette. Which seat to choose? Sometimes all the boys looked the same, each wearing a rationed fleshy-pink face. This happened only when he panicked—he couldn’t put face to name to character. And so he tossed himself into an empty seat as though tossing a shilling, leaving it to chance.
It wasn’t too bad a first. The table was noisy with chatter as he concentrated on his tray and continued to deal with the broccoli crisis. In Alan’s unique inner world, he imagined himself to be invisible, entirely unnoticeable. But to the boys at the table, it was impossible not to focus on him. He fretted over his vegetables with jerky movements while his own dandruff salted his soup. One by one they stopped talking and stopped smiling. For a full minute the few boys at the table put down their implements and watched Alan’s oblivious preparations for the accident about to happen. Later, none of them would be able to remember which one of them first stabbed Alan with Chris’s name. But just like that, the collision began as Alan was forced to abandon the solution to the food groups and search for their eyes.
It’s not exactly that Turing didn’t know where to look. He didn’t know how to look. He gawped. He stared. His eyes dilated and seemed to draw some essence out of his classmates. It felt as though he could look right through them. It felt as though he could look right into them. Even more offended by his blue-eyed gaze than they were by his obliviousness, they mocked him with the name. It was a sing-song accusation, a whine and a taunt: Christopher Morcom.
Chris is Alan’s first, greatest, purest, entirely unrequited love. He is a class ahead of Alan in school and from another house. Although a year older he is light of frame, with thin, delicate features that Alan loves to study furtively, praying Chris won’t notice. This slender prettiness can be blamed, at least in part, on a frailty induced by chronic illness, induced in turn by a heavy white glass of thick cow’s milk, a satisfying half-pint of milk crowned by a lovely warm froth, seemingly ordinary were it not for the contamination of bovine tuberculosis. There was nothing special about the day the milk was poured. There was nothing special about the glass his hands encircled, except a mildly unpleasant chill from a layer of condensation that struck his palms. The chill nowhere near shocking enough to ward him off the poisoned drink, and down his throat it slid, coating his upper lip innocently and his bowels rather less innocently. The cough developed quickly, adding timbre to his young voice and plaguing him with a coarse production of phlegm. Since then, the bacteria have procrastinated, sustaining the host out of simple laziness. Chris keeps his illness secret, leaving Alan to speculate as to the cause of his small frame and delicate appearance.
Chris offers Alan the only respite from his loneliness. A pretty blond boy, he is everything Turing is not. An easy success, naturally winning, polished, and lovely. Alan could be jealous of his scientific aptitude, his adept physical chemistry experiments, his knowledge of astronomy, his superior telescope, and his family’s cultivation of scientific literacy as well as a sophistication in the arts. But instead he takes the only other route available—he loves him. He never confesses his love outright, although Chris must know that this odd classmate loves him. But then Chris is easy to love and he must know that too.
There was Alan, pushing at the fibrous trees of broccoli, unsure as to the identity of his dinner companions, unable to resolve their faces, not knowing who was with him of who was against him. When the taunt landed, he didn’t protest, didn’t defend himself, didn’t resent the implication. He grinned. A grin of indulgence at the dirty allusion. Bloody Turing.
Although the shattering of his plate shocked him, he was relieved to see the broccoli go.
Under the floorboards, he is given time to consider his gaffe over dinner. As the minutes pass, powdery dust clumps in the steam of his nose and mouth. Although he can identify the sequence of events—the boiled veg, the chair, the name, the grin, the broken plate—he’s still not quite sure what went wrong.