“Hmmm?”
“Rudy—take this stick, here—that’s right—and keep a close eye on Lawrence, and when he gets that foggy look on his face, poke him with it!”
“Zis is not an English school, you can’t do zese kind of sing.”
“I’m listening,” Lawrence said.
“What came out of P.M., which was terrifically radical, was the ability to say that all of math, really, can be expressed as a certain ordering of symbols.”
“Leibniz said it a long time before zen!” protested Rudy.
“Er, Leibniz invented the notation we use for calculus, but—”
“I’m not talking about zat!”
“And he invented matrices, but—”
“I’m not talking about zat eezer!”
“And he did some work with binary arithmetic, but—”
“Zat is completely different!”
“Well, what the hell are you talking about, then, Rudy?”
“Leibniz invented ze basic alphabet—wrote down a set of symbols, for expressing statements about logic.”
“Well, I wasn’t aware that Herr Leibniz counted formal logic among his interests, but—”
“Of course! He wanted to do what Russell and Whitehead did, except not just with mathematics, but with everything in ze whole world!”
“Well, from the fact that you are the only man on the planet, Rudy, who seems to know about this undertaking of Leibniz’s, can we assume that he failed?”
“You can assume anything that pleases your fancy, Alan,” Rudy responded, “but I am a mathematician and I do not assume anything.”
Alan sighed woundedly, and gave Rudy a Significant Look which Waterhouse assumed meant that there would be trouble later. “If I may just make some headway, here,” he said, “all I’m really trying to get you to agree on, is that mathematics can be expressed as a series of symbols,” (he snatched the Lawrence-poking stick and began drawing things like + = 3) √-1π in the dirt) “and frankly I could not care less whether they happen to be Leibniz’s symbols, or Russell’s, or the hexagrams of the I Ching… .”
“Leibniz was fascinated by the I Ching!” Rudy began.
“Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.”
“All right, Alan.”
“Won’t take a minute if you will just stop interrupting.”
“But there is a locomotive too named Leibniz.”
“Is it that you don’t think I give enough credit to Germans? Because I am about to mention a fellow with an umlaut.”
“Oh, would it be Herr Türing?” Rudy said slyly.
“Herr Türing comes later. I was actually thinking of Gödel.”
“But he’s not German! He’s Austrian!”
“I’m afraid that it’s all the same now, isn’t it?”
“Ze Anschluss wasn’t my idea, you don’t have to look at me that way, I think Hitler is appalling.”
“I’ve heard of Gödel,” Waterhouse put in helpfully. “But could we back up just a sec?”
“Of course Lawrence.”
“Why bother? Why did Russell do it? Was there something wrong with math? I mean, two plus two equals four, right?”
Alan picked up two bottlecaps and set them down on the ground. “Two. One-two. Plus—” He set down two more. “Another two. One-two. Equals four. One-two-three-four.”
“What’s so bad about that?” Lawrence said.
“But Lawrence—when you really do math, in an abstract way, you’re not counting bottlecaps, are you?”
“I’m not counting anything.”
Rudy broke the following news: “Zat is a very modern position for you to take.”
“It is?”
Alan said, “There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps. That any mathematical operation you could do on paper, no matter how complicated, could be reduced—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real world.”
“But you can’t have two point one bottlecaps.”
“All right, all right, say we use bottlecaps for integers, and for real numbers like two point one, we use physical measurements, like the length of this stick.” Alan tossed the stick down next to the bottlecaps.
“Well what about pi, then? You can’t have a stick that’s exactly pi inches long.”
“Pi is from geometry—ze same story,” Rudy put in.
“Yes, it was believed that Euclid’s geometry was really a kind of physics, that his lines and so on represented properties of the physical world. But—you know Einstein?”
“I’m not very good with names.”
“That white-haired chap with the big mustache?”
“Oh, yeah,” Lawrence said dimly, “I tried to ask him my sprocket question. He claimed he was late for an appointment or something.”
“That fellow has come up with a general relativity theory, which is sort of a practical application, not of Euclid’s, but of Riemann’s geometry—”
“The same Riemann of your zeta function?”
“Same Riemann, different subject. Now let’s not get sidetracked here Lawrence—”
“Riemann showed you could have many many different geometries that were not the geometry of Euclid but that still made sense internally,” Rudy explained.
“All right, so back to P.M. then,” Lawrence said.
“Yes! Russell and Whitehead. It’s like this: when mathematicians began fooling around with things like the square root of negative one, and quaternions, then they were no longer dealing with things that you could translate into sticks and bottlecaps. And yet they were still getting sound results.”
“Or at least internally consistent results,” Rudy said.
“Okay. Meaning that math was more than a physics of bottlecaps.”
“It appeared that way, Lawrence, but this raised the question of was mathematics really true or was it just a game played with symbols? In other words—are we discovering Truth, or just wanking?”
“It has to be true because if you do physics with it, it all works out! I’ve heard of that general relativity thing, and I know they did experiments and figured out it was true.”
“Ze great majority of mathematics does not lend itself to experimental testing,” Rudy said.
“The whole idea of this project is to sever the ties to physics,” Alan said.
“And yet not to be vanking ourselves.”
“That’s what P.M. was trying to do?”
“Russell and Whitehead broke all mathematical concepts down into brutally simple things like sets. From there they got to integers, and so on.”
“But how can you break something like pi down into a set?”
“You can’t,” Alan said, “but you can express it as a long string of digits. Three point one four one five nine, and so on.”
“And digits are integers,” Rudy said.
“But no fair! Pi itself is not an integer!”
“But you can calculate the digits of pi, one at a time, by using certain formulas. And you can write down the formulas like so!” Alan scratched this in the dirt:
“I have used the Leibniz series in order to placate our friend. See, Lawrence? It is a string of symbols.”
“Okay. I see the string of symbols,” Lawrence said reluctantly.
“Can we mo
ve on? Gödel said, just a few years ago, ‘Say! If you buy into this business about mathematics being just strings of symbols, guess what?’ And he pointed out that any string of symbols—such as this very formula, here—can be translated into integers.”
“How?”
“Nothing fancy, Lawrence—it’s just simple encryption. Arbitrary. The number ‘538’ might be written down instead of this great ugly ∑, and so on.”
“Seems pretty close to wanking, now.”
“No, no. Because then Gödel sprang the trap! Formulas can act on numbers, right?”
“Sure. Like 2x.”
“Yes. You can substitute any number for x and the formula 2x will double it. But if another mathematical formula, such as this one right here, for calculating pi, can be encoded as a number, then you can have another formula act on it. Formulas acting on formulas!”
“Is that all?”
“No. Then he showed, really through a very simple argument, that if formulas really can refer to themselves, it’s possible to write one down saying ‘this statement cannot be proved.’ Which was tremendously startling to Hilbert and everyone else, who expected the opposite result.”
“Have you mentioned this Hilbert guy before?”
“No, he is new to this discussion, Lawrence.”
“Who is he?”
“A man who asks difficult questions. He asked a whole list of them once. Gödel answered one of them.”
“And Türing answered another,” Rudy said.
“Who’s that?”
“It’s me,” Alan said. “But Rudy’s joking. ‘Turing’ doesn’t really have an umlaut in it.”
“He’s going to have an umlaut in him later tonight,” Rudy said, looking at Alan in a way that, in retrospect, years later, Lawrence would understand to have been smoldering.
“Well, don’t keep me in suspense. Which one of his questions did you answer?”
“The Entscheidungsproblem,” Rudy said.
“Meaning?”
Alan explained, “Hilbert wanted to know whether any given statement could, in principle, be found true or false.”
“But after Gödel got finished, it changed,” Rudy pointed out.
“That’s true—after Gödel it became ‘Can we determine whether any given statement is provable or non-provable?’ In other words, is there some sort of mechanical process we could use to separate the provable statements from the non-provable ones?”
“ ‘Mechanical process’ is supposed to be a metaphor, Alan… .”
“Oh, stop it, Rudy! Lawrence and I are quite comfortable with machinery.”
“I get it,” Lawrence said.
“What do you mean, you get it?” Alan said.
“Your machine—not the zeta-function calculator, but the other one. The one we’ve been talking about building—”
“It is called Universal Turing Machine,” Rudy said.
“The whole point of that gizmo is to separate provable from nonprovable statements, isn’t it?”
“That’s why I came up with the basic idea for it,” Alan said. “So Hilbert’s question has been answered. Now I just want to actually build one so that I can beat Rudy at chess.”
“You haven’t told poor Lawrence the answer yet!” Rudy protested.
“Lawrence can figure it out,” Alan said. “It’ll give him something to do.”
Soon it became clear that Alan really meant: It’ll give him something to do while we’re fucking. Lawrence shoved a notebook into the waistband of his trousers and rode his bicycle a few hundred yards to the fire tower, then climbed up the stairs to the platform at the top and sat down, back to the setting sun, notebook propped up on his knees to catch the light.
He could not collect his thoughts, and then he was distracted by a false sunrise that lit up the clouds off to the northeast. He thought at first that some low clouds were bouncing fragments of the sunset back to him, but it was too concentrated and flickering for that. Then he thought it was lightning. But the color of the light was not blue enough. It fluctuated sharply, modulated by (one had to assume) great, startling events that were occulted by the horizon. As the sun went down on the opposite side of the world, the light on the New Jersey horizon focused to a steady, lambent core the color of a flashlight when you shine it through the palm of your hand under the bedsheets.
Lawrence climbed down the stairs and got on his bicycle and rode through the Pine Barrens. Before long he came to a road that led in the general direction of the light. Most of the time he could not see anything, not even the road, but after a couple of hours the glow bouncing off the low cloud layer lit up flat stones in the road, and turned the barrens’ wandering rivulets into glowing crevices.
The road began to tend in the wrong direction and so Lawrence cut directly into the woods, because he was very close now, and the light in the sky was strong enough that he could see it through the sparse carpet of scrubby pines—black sticks that appeared to have been burned, though they hadn’t. The ground had turned into sand, but it was damp and compacted, and his bicycle had fat tires that rode over it well. At one point he had to stop and throw the bike over a barbed-wire fence. Then he broke out of the sticks and onto a perfectly flat expanse of white sand, stitched down with tufts of beach grass, and just then he was dazzled by a low fence of quiet steady flames that ran across a part of the horizon about as wide as the harvest moon when it sinks into the sea. Its brightness made it difficult to see anything else—Lawrence kept riding into little ditches and creeks that meandered across the flats. He learned not to stare directly at the flames. Looking off to the sides was more interesting anyway: the table-land was marked at wide intervals by the largest buildings he had ever seen, cracker-box structures built by Pharaohs, and in the mile-wide plazas between them, gnomons of triangulated steel were planted in wide stances: the internal skeletons of pyramids. The largest of these pierced the center of a perfectly circular railway line a few hundred feet in diameter: two argent curves scored on the dull ground, interrupted in one place where the tower’s shadow, a stopped sundial, told the time. He rode by a building smaller than the others, with oval tanks standing next to it. Steam murmured from valves on the tops of the tanks, but instead of rising into the air it dribbled down the sides and struck the ground and spread out, coating the sea-grass with jackets of silver.
A thousand sailors in white were standing in a ring around the long flame. One of them held up his hand and waved Lawrence down. Lawrence came to a stop next to the sailor and planted one foot on the sand to steady himself. He and the sailor stared at each other for a moment and then Lawrence, who could not think of anything else, said, “I am in the Navy also.”Then the sailor seemed to make up his mind about something. He saluted Lawrence through, and pointed him towards a small building off to the side of the fire.
The building looked only like a wall glowing in the firelight, but sometimes a barrage of magnesium blue light made its windowframes jump out of the darkness, a rectangular lightning-bolt that echoed many times across the night. Lawrence started pedaling again and rode past that building: a spiraling flock of alert fedoras, prodding at slim terse notebooks with stately Ticonderogas, crab-walking photogs turning their huge chrome daisies, crisp rows of people sleeping with blankets over their faces, a sweating man with Brilliantined hair chalking umlauted names on a blackboard. Finally coming around this building he smelled hot fuel oil, felt the heat of the flames on his face and saw beach-grass curled toward it and desiccated.
He stared down upon the world’s globe, not the globe fleshed with continents and oceans but only its skeleton: a burst of meridians, curving backwards to cage an inner dome of orange flame. Against the light of the burning oil those longitudes were thin and crisp as a draftsman’s ink-strokes. But coming closer he saw them resolve into clever works of rings and struts, hollow as a bird’s bones. As they spread away from the pole they sooner or later began to wander, or split into bent parts, or just broke off and hung in the fire os
cillating like dry stalks. The perfect geometry was also mottled, here and there, by webs of cable and harnesses of electrical wiring. Lawrence almost rode over a broken wine bottle and decided he should now walk, to spare his bicycle’s tires, so he laid the bike down, the front wheel covering an aluminum vase that appeared to have been spun on a lathe, with a few charred roses hanging out of it. Some sailors had joined their hands to form a sort of throne, and were bearing along a human-shaped piece of charcoal dressed in a coverall of immaculate asbestos. As they walked the toes of their shoes caught in vast, ramified snarls of ropes and piano-wires, cables and wires, creative furtive movements in the grass and the sand dozens of yards every direction. Lawrence began planting his feet very thoughtfully one in front of the other, trying to measure the greatness of what he had come and seen. A rocket-shaped pod stuck askew from the sand, supporting an umbrella of bent-back propellers. The duralumin struts and catwalks rambled on above him for miles. There was a suitcase spilled open, with a pair of women’s shoes displayed as if in the window of a downtown store, and a menu that had been charred to an oval glow, and then some tousled wall-slabs, like a whole room that had dropped out of the sky—these were decorated, one with a giant map of the world, great circles arcing away from Berlin to pounce on cities near and far, and another with a photograph of a famous, fat German in a uniform, grinning on a flowered platform, the giant horizon of a new Zeppelin behind him.
After a while he stopped seeing new things. Then he got on his bicycle and rode back through the Pine Barrens. He got lost in the dark and so didn’t find his way back to the fire tower until dawn. But he didn’t mind being lost because while he rode around in the dark he thought about the Turing machine. Finally he came back to the shore of the pond where they had camped. The dawn-light shining on the saucer of calm reddish water made it look like a pool of blood. Alan Mathison Turing and Rudolf von Hacklheber were lying together like spoons on the shore, still smudged a little bit from their swim yesterday. Lawrence started a little fire and made some tea and they woke up eventually.