Cryptonomicon

"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?"

"Youcan assumeanything that pleases your fancy,Alan," Rudy responded, "but Iam 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) [square root of -1][pi] 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 claimedhe 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'sgeometry--"

"The same Riemann of your zeta function?"

"Same Riemann, different subject. Now let's not get sidetracked here Lawrence--"