Cryptonomicon

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 oscillating 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 cat walks 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 down town 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.

"Did you solve the problem?" Alan asked him.

"Well you can turn that Universal Turing Machine of yours into any machine by changing the presets--"

"Presets?"

"Sorry, Alan, I think of your U.T.M. as being kind of like a pipe organ."

"Oh."

"Once you've done that, anyway, you can do any calculation you please, if the tape is long enough. But gosh, Alan, making a tape that's long enough, and that you can write symbols on, and erase them, is going to be sort of tricky--Atanasoffs capacitor drum would only work up to a certain size--you'd have to--"

"This is a digression," Alan said gently.

"Yeah, okay, well--if you had a machine like that, then any given preset could be represented by a number--a string of symbols. And the tape that you would feed into it to start the calculation would contain another string of symbols. So it's Gödel's proof all over again--if any possible combination of machine and data can be represented by a string of numbers, then you can just arrange all of the possible strings of numbers into a big table, and then it turns into a Cantor diagonal type of argument, and the answer is that there must be some numbers that cannot be computed."

"And ze Entscheidungsproblem?" Rudy reminded him.

"Proving or disproving a formula--once you've encrypted the formula into numbers, that is--is just a calculation on that number. So it means that the answer to the question is, no! Some formulas cannot be proved or disproved by any mechanical process! So I guess there's some point in being human after all!"

Alan looked pleased until Lawrence said this last thing, and then his face collapsed. "Now there you go making unwarranted assumptions."

"Don't listen to him, Lawrence!" Rudy said. "He's going to tell you that our brains are Turing machines."

"Thank you, Rudy," Alan said patiently. "Lawrence, I submit that our brains are Turing machines."

"But you proved that there's a whole lot of formulas that a Turing machine can't process!"

"And you have proved it too, Lawrence."

"But don't you think that we can do some things that a Turing machine couldn't?"

"Gödel agrees with you, Lawrence," Rudy put in, "and so does Hardy."

"Give me one example," Alan said.