A MADMAN DREAMS OF TURING MACHINES

In 1931 he is a young man of twenty-five years, 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 has come to tell the circle that they are wrong, and he can prove it.

Introduction

The following message arrived from Janna Levin, Barnard physicist and writer:

"There have been a few recent articles in the press on the theme that "the novel is dead". Comments on Edge, on the other hand, have gone in the opposite direction, noting the widening umbrella of the third culture in terms of the work of accomplished novelists and playwrights who noodle around with scientific ideas like Ian McEwan in Saturday, Richard Powers in Galatea 2.2, Michael Frayne in Copenhagen, David Auburn in Proof – not to mention Mary Shelley in Frankenstein. Maybe these works hit some things more effectively than can be done in a straightforward popular science book. Conversely scientists have played with new forms of expression like Primo Levy in The Periodic Table and Alan Lightman in Einstein's Dreams.

"So let me throw this out there in the hopes that Edge readers will find the attached piece of interest — an early draft from a book I’ve been writing called A Madman Dreams of Turing Machines. This is a story. Does that make it fiction? It’s based on truth like all of our stories. It’s a story of coded secrets and psychotic delusions, mathematics and war. It’s a chronicle of the strange lives of Alan Turing and Kurt Gödel. These stories are so strange, so incredible, that they are totally unbelievable. Except they’re true. And fact is more extraordinary than fiction.

"This excerpt may be particularly relevant now given the recent Edge features on Gödel with Rebecca Goldstein and Verena Huber-Dyson."

— JB

JANNA LEVIN is a professor of physics at Barnard College of Columbia University and recently held a fellowship from NESTA (National Endowment for Science Technology and Arts) at the University of Oxford. She has worked on theories of the Early Universe, Chaos, and Black Holes. Her work tends to encompass the overlap of mathematics, general relativity, and astrophysics. She is the author of How The Universe Got Its Spots: Diary Of A Finite Time In A Finite Space.

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 over-stimulated. 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 honing 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 hatemysticism 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 oversized socialist. Most importantly, 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 discrete handful of 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 that 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 years, 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 has come to tell the circle 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 un-likeable. 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 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 importantly, on the vast intellectual world that lays 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 council 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 than the rest of us harbor, but his weaknesses all seem so extreme ­ hypochondria, paranoia, schizophrenia. They are even more pronounced when laid alongside his incredible mental strengths ­ 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's 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 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 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.

~~

The iron frame of Kurt's bed was a brutal conductor of the chill singeing his hand so sharply as he hoisted himself awake this morning that it might as well have left a burn and the cloud of condensation that escaped from his damp mouth could have been smoke. He prepared for his discussion with The Circle most the day and took care to present himself well. He applied layers of clothes like a dressing over a wound, carefully wrapping his limbs in strong woolen weaves. The third pair of pants buttoned easily over the inner two layers with just the right amount of resistance. He made sure the two pairs of trousers he wore closest were slightly short and stayed well hidden behind the cuff of the outer suit. A similar procedure was followed for his upper half ­ a series of shirts and vests created a padding five garments thick. Even then he looked lean although less alarmingly so.

Despite his detachment, his family's sophistication was not entirely lost on him and surfaced in the subtle choices he made, if not in the few kitsch objects he clashed against his mother's design in the interior of his large flat, then at least it showed in the many garments that he now used to flatter himself, a reference to the rich textiles manufactured in his father's factories. He applied the finely woven jacket that still hung loosely from the line connecting the points of his two shoulders and finally a handsome overcoat was draped over that.

Gödel loves these Thursday nights. The rest of the week is spent in near complete isolation, sometimes losing the sense of days. Comforted by the darkest hours when his loneliness is assured, he manipulates logical symbols into a flawless sequence, generating theorem after theorem in his notebooks. He fills the plain paper books with mathematical proofs that lead to new ideas that spawn new results. He can't always find a context for the proliferation of logical conclusions other than the pages themselves which are covered one-sided from left to right until the book is finished and he moves back through the volume covering the back of the pages from right to left. In these ordinary brown notebooks he builds a logical cosmos of his own in which the private ideas are nested, his secret gems. His most precious insights he transcribes in Garbelsberger, an obsolete form of German shorthand he was taught as a schoolboy and is sure no one else remembers.

While he often loses Monday easily and tries to find root in Tuesday, though Wednesday is a mere link between nights, he always knows Thursday. He likes to arrive early and choose the same place each time, a dark wooden chair near the wall, almost hidden behind the floral arm of an upholstered booth, not too close to the center but not too far out where it might become crowded, people pressing in to warm themselves against the heat of argument emanating from the core. Comfortably still, with an undisturbed tepid coffee he never intends to drink, he listens to the debates, the ideas, and the laughter, like a man marooned on an island tuning into a distant radio broadcast. Proof that there are others out there. Proof that he is not alone. Proof.

He usually disagrees with them. Still, The Circle gives him a clear form to relate to, an external setting for his private cosmos ­ solid rocks of reality appearing in a fog of ghosts.

This evening he is later than usual. Knocked unsteady as he has been by the recent turns. He has his latest notebook with him, pressed against his jacket. His knuckles protrude from the spine of the book like barbed wire lacings. The pages are nearly full, front and back covered, they must be read as a loop from the first page front to the last page back, then towards the first page again, a closed path, a broken triangle, and at the pointed tip a discovery. An incredible discovery. He is so impressed by the stream of symbols that accumulate particularly at the endpoint, where they began, that he feels lightheaded while his blood collects in pools about his boney knees.

He's in front of the glass doors of the Café Josephinum. Through the filter of the windowpanes the activity becomes an unreal smear of lights and colors. His hand on the door, it opens, that aroma, and he moves into the room. Through the filter of his eyes the activity persists, an unreal smear of lights and colors. Who here is real?

Pushing against a breeze of phantoms he moves towards the table, pressing into a chair. Amazing that he looks composed. His physical condition is fragile. His emotional condition is fragile. He hides the former behind thick textile weaves and a well-manicured façade. He hides the latter behind the pattern of reflected lights off his glasses. On this stage provided by the Café Josephinum, he looks at ease, as though he belongs. But the past few days have been irregular at best. For one thing, Adele almost poisoned him. He woke into the hardest cold this morning like breaking through the surface of a frozen lake and gasped for breath ­ the air shocking his nose and throat with brittle spikes of ice as his mind sucked in the progression of the past days. A terrible relief flooded his system and the relieved thoughts themselves confirmed to him that he was indeed alive. I think therefore I am, he thought. Both the thought and the condition of being alive amused him. While he has run the events over and over in his mind, they permute with each replay: An old woman, his death, then Adele who is kind until she dusts something into his stew. Then again: An old woman, his death, the rain, Adele manipulates his confession and blatantly builds a toxic pyre. An old woman. His death. The rain. Adele. Pretty, stained Adele. His heart aches with suspicion and the thick mucous of betrayal.

His heart also aches with disease. He is fatigued. His chest is sore. He has no breath. This very evening he coughed up blood. His heart has become stiff and scarred after a bout of rheumatic fever at the age of eight. A valve in his atrium fused and constricted over years. It took the disease a full decade to declare the specific threat intended. He is plagued by attacks. A backwash of his blood stretches the chambers, depriving his arteries. He lives in constant fear for his life. Every minute framed by panic. The flutter in his chest a warning of a potential blood clot, suffocation, or heart failure. He shouldn't be here with the smoky air, warm and virulent. But the relief that filled his limbs this morning gave him a feeling of urgency and ambition. And he needs to see Moritz.

The Circle doesn't take shape until Moritz Schlick arrives. He enters like a gale, his entrance embellished by a curl of eddies in his wake that flow around the door and into the room. He is the chair of natural philosophy at the University, a title that carries great prestige and authority. Moritz is always a gentleman, always gracious and earnest and admirable. As he rocks into a chair, hands are waved, more coffees are ordered and in the darkening room, darker than the ebbing day, they all begin to settle amid clanking dishes, knocking elbows, their collective weight leveraged inward. The table wobbles as cups rise and fall and a circle forms.

It's Moritz Schlick's Circle. Drawn together by his invitation and kept together by his soothing tones. They come here to orbit around truth, to throw off centuries of misguided faith, the shackles of religion, the hypnotism of metaphysics. They celebrate the heft of their own weight in a solid chair, the heat off the coffee, the sound their voices manufacture within the walls of the café. Some are delirious with the immediacy of this day because it is all that matters. There is nothing else. Everything true is summed up in the chair, the cup, the building. There is only gravity, heat and force. The world is all that is the case.

Moritz knows the greatness that can emerge from the members he has chosen by hand, so he smoothes the caustic edges between egos and makes out of them a collective, an eclectic orchestra out of dissonance. Moritz is the glue that holds together the communist, the mathematician, the empiricist. He selects each person here with care, patiently turning them over in his mind, studying them with his kind eyes. They are comforted by his self-assurance and are sincerely flattered by the invitation to Thursday's discussions, if they are ever fortunate enough to receive the summons. There are many for whom the hoped for invitation never comes.

Gödel blushed with either vanity or shyness, who can know for sure, when Moritz approached him in the room in the basement of the mathematics Institute and extended the invitation almost four years ago. Kurt was at the chalkboard organizing another student's thoughts in spare symbols, lovely dusty marks on a landscape of poorly erased predecessors. He always transcribes the skeleton in the pure notation of symbolic logic first and with such care before he begins to speak. Even though he was only a twenty-one year old student, the others watched with admiration for his ability to see through to the logical bones in their debates, like a chef skillfully removing the endoskeleton of a filleted fish without a morsel of clinging flesh. Moritz watched him too and moved by the lucidity of Gödel's resolution to a problem he himself had found distractingly difficult, he came to his final decision to extend to Kurt an invitation to his Circle on Thursday nights.

Moritz joined him at the board, quietly adding a fine comment on the infinite list of integers that might participate in the reasoning off the middle rib of the fish's spine. And in this smooth manner he eased Gödel into conversation. Everyone either knows by instinct or learns by plain experiment to meet Gödel with mathematics first. And so Moritz approached with the right words about infinity and integers and earned that look of gratitude and trust. As he shook Kurt's hand and his own head in grateful amazement, they talked:

"Herr Professor, I have been thinking about the Liar's Paradox where the liar says,this sentence is false."

"Ah, the antinomy of the liar. Yes, that liar who says, this sentence is false."

"The sentence cannot be false."

"Because if it is false as claimed, then it must be true. A contradiction." Studying his young student for a time Moritz stroked his lip dry and concluded his motion with the reply, "And it cannot be true. Because if it is true, then it is false which is again a contradiction. It is a paradox and an artifact of our careless use of language. Mathematics will never allow such a paradox. Mathematical propositions will either be true or false with no contradictions."

"What if mathematics is not free of such propositions?"

"It must be. Mathematics must be complete. There are no unsolvable problems."

Ever since that morning of the invitation and the antinomy of the liar, Gödel has found Moritz's very presence reassuring. If Kurt was different in character, more affectionate, less rigid, and if Moritz too were just a little different, more spontaneous, less reserved, Gödel could have come to love Moritz like a father. Instead he feels something more formal, more distant, more appropriate probably. He feels grateful. He keeps this feeling to himself and the sentiment has almost no outward manifestation beyond his attendance here at Moritz's discussions. He believes that Moritz is real, that he exists and it happened in the moment that Moritz shared the comment on the infinite list of numbers. With that insight, it was as though he uttered a code word. I am one of the real ones, his comment certified, and with that he crystallized from the cloud and took shape.