Cryptonomicon

"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 trueor 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 yanking 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 itselfis not an integer!"

"But you can calculate the digitsof 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 move 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 [sigma], 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 nonprovable 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."

***