Cryptonomicon (3 page)

where
s
is a complex number.

Lawrence found this zeta function to be no more and no less interesting than any other math problem until his new friend assured him that it was frightfully important, and that some of the best mathematicians in the world had been gnawing on it for decades. The two of them ended up staying awake until three in the morning working out the solution to Lawrence’s sprocket problem. Lawrence presented the results proudly to his engineering professor, who snidely rejected it, on grounds of practicality, and gave him a poor grade for his troubles.

Lawrence finally remembered, after several more contacts, that the name of the friendly Brit was Al something-or-other. Because Al was a passionate cyclist, he and Al went on quite a few bicycle rides through the countryside of the Garden State. As they rode around New Jersey, they talked about math, and particularly about machines for taking the dull part of math off their hands.

But Al had been thinking about this subject for longer than Lawrence, and had figured out that computing machines were much more than just labor-saving devices. He’d been working on a radically different sort of computing mechanism that would work out any arithmetic problem whatsoever, as long as you knew how to write the problem down. From a pure logic standpoint, he had already figured out everything there was to know about this (as yet hypothetical) machine, though he had yet to build one. Lawrence gathered that actually building machinery was looked on as undignified at Cambridge (England, that is, where this Al character was based) or for that matter at Fine Hall. Al was thrilled to have found, in Lawrence, someone who did not share this view.

Al delicately asked him, one day, if Lawrence would terribly mind calling him by his full and proper name, which was Alan and not Al. Lawrence apologized and said he would try very hard to keep it in mind.

One day a couple of weeks later, as the two of them sat by a running stream in the woods above the Delaware Water Gap, Alan made some kind of an outlandish proposal to Lawrence involving penises. It required a great deal of methodical explanation, which Alan delivered with
lots of blushing and stuttering. He was ever so polite, and several times emphasized that he was acutely aware that not everyone in the world was interested in this sort of thing.

Lawrence decided that he was probably one of those people.

Alan seemed vastly impressed that Lawrence had paused to think about it
at all
and apologized for putting him out. They went directly back to a discussion of computing machines, and their friendship continued unchanged. But on their next bicycle ride—an overnight camping trip to the Pine Barrens—they were joined by a new fellow, a German named Rudy von something-or-other.

Alan and Rudy’s relationship seemed closer, or at least more multilayered, than Alan and Lawrence’s. Lawrence concluded that Alan’s penis scheme must have finally found a taker.

It got Lawrence to thinking. From an evolution standpoint, what was the point of having people around who were not inclined to have offspring? There must be some good, and fairly subtle, reason for it.

The only thing he could work out was that it was groups of people—societies—rather than individual creatures, who were now trying to out-reproduce and/or kill each other, and that, in a society, there was plenty of room for someone who didn’t have kids as long as he was up to something useful.

Alan and Rudy and Lawrence rode south, anyway, looking for the Pine Barrens. After a while the towns became very far apart, and the horse farms gave way to a low stubble of feeble, spiny trees that appeared to extend all the way to Florida—blocking their view, but not the headwind. “Where are the Pine Barrens I wonder?” Lawrence asked a couple of times. He even stopped at a gas station to ask someone that question. His companions began to make fun of him.

“Vere are ze Pine Barrens?” Rudy inquired, looking about quizzically.

“I should look for something rather barren-looking, with numerous pine trees,” Alan mused.

There was no other traffic and so they had spread out across the road to pedal three abreast, with Alan in the middle.

“A forest, as Kafka would imagine it,” Rudy muttered.

By this point Lawrence had figured out that they were, in fact, in the Pine Barrens. But he didn’t know who Kafka was. “A mathematician?” he guessed.

“
Zat
is a scary sing to sink of,” Rudy said.

“He is a writer,” Alan said. “Lawrence, please don’t be offended that I ask you this, but: do you recognize
any other people’s names
at all? Other than family and close friends, I mean.”

Lawrence must have looked baffled. “I’m trying to figure out whether it all comes from in here,” Alan said, reaching out to rap his knuckles on the side of Lawrence’s head, “or do you sometimes take in new ideas from other human beings?”

“When I was a little boy, I saw angels in a church in Virginia,” Lawrence said, “but I think that they came from inside my head.”

“Very well,” Alan said.

But later Alan had another go at it. They had reached the fire lookout tower and it had been a thunderous disappointment: just an alienated staircase leading nowhere, and a small cleared area below that was glittery with shards of liquor bottles. They pitched their tent by the side of a pond that turned out to be full of rust-colored algae that stuck to the hairs on their bodies. Then there was nothing left to do but drink schnapps and talk about math.

Alan said, “Look, it’s like this: Bertrand Russell and another chap named Whitehead wrote
Principia Mathematica
. . .”

“Now I know you’re pulling my leg,” Waterhouse said. “Even I know that Sir Isaac Newton wrote
that
.”

“Newton wrote a
different
book,
also
called
Principia Mathematica,
which isn’t
really
about mathematics at all; it’s about what we would
today
call physics.”

“Then why did he call it
Principia Mathematica?
”

“Because the distinction between mathematics and physics wasn’t especially clear in Newton’s day—”

“Or maybe even in zis day,” Rudy said.

“—which is directly relevant to what I’m talking about,” Alan continued. “I am talking about
Russell’s P.M.,
in which he and Whitehead started absolutely from
scratch,
I mean from
nothing,
and built it all up—all mathematics—from a small number of first principles. And why I am telling you this, Lawrence, is that—Lawrence! Pay attention!”

“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?”