Neal Stephenson,
CRYPTONOMICON
"There is a remarkably close parallel between the problems of the physicist
and those of the cryptographer. The system on which a message is enciphered
corresponds to the laws of the universe, the intercepted messages to the
evidence available, the keys for a day or a message to important constants
which have to be determined. The correspondence is very close, but the
subject matter of cryptography is very easily dealt with by discrete
machinery, physics not so easily."
Alan Turing
This morning [Imelda Marcos] offered the latest in a series of explanations
of the billions of dollars that she and her husband, who died in 1989, are
believed to have stolen during his presidency.
"It so coincided that Marcos had money," she said. "After the Bretton Woods
agreement he started buying gold from Fort Knox. Three thousand tons, then
4,000 tons. I have documents for these: 7,000 tons. Marcos was so smart. He
had it all. It's funny; America didn't understand him."
The New York Times, Monday, 4 March, 1996
Prologue
Two tires fly. Two wail.
A bamboo grove, all chopped down
From it, warring songs.
...is the best that Corporal Bobby Shaftoe can do on short notice he's
standing on the running board, gripping his Springfield with one hand and
the rearview mirror with the other, so counting the syllables on his fingers
is out of the question. Is "tires" one syllable or two? How about "wail?"
The truck finally makes up its mind not to tip over, and thuds back onto
four wheels. The wail and the moment are lost. Bobby can still hear the
coolies singing, though, and now too there's the gunlike snicking of the
truck's clutch linkage as Private Wiley downshifts. Could Wiley be losing
his nerve? And, in the back, under the tarps, a ton and a half of file
cabinets clanking, code books slaloming, fuel spanking the tanks of Station
Alpha's electrical generator. The modern world's hell on haiku writers:
"Electrical generator" is, what, eight syllables? You couldn't even fit that
onto the second line!
"Are we allowed to run over people?" Private Wiley inquires, and then
mashes the horn button before Bobby Shaftoe can answer. A Sikh policeman
hurdles a night soil cart. Shaftoe's gut reaction is: Sure, what're they
going to do, declare war on us? but as the highest ranking man on this truck
he's probably supposed to be using his head or something, so he doesn't
blurt it out just yet. He takes stock of the situation:
Shanghai, 1645 hours, Friday, the 28th of November 1941. Bobby Shaftoe,
and the other half dozen Marines on his truck, are staring down the length
of Kiukiang Road, onto which they've just made this careening high speed
turn. Cathedral's going by to the right, so that means they are, what? two
blocks away from the Bund. A Yangtze River Patrol gunboat is tied up there,
waiting for the stuff they've got in the back of this truck. The only real
problem is that those particular two blocks are inhabited by about five
million Chinese people.
Now these Chinese are sophisticated urbanites, not suntanned yokels
who've never seen cars before they'll get out of your way if you drive fast
and honk your horn. And indeed many of them flee to one side of the street
or the other, producing the illusion that the truck is moving faster than
the forty three miles an hour shown on its speedometer.
But the bamboo grove in Bobby Shaftoe's haiku has not been added just
to put a little Oriental flavor into the poem and wow the folks back home in
Oconomowoc. There is a lot of heavy bamboo in front of this truck, dozens of
makeshift turnpikes blocking their path to the river, for the officers of
the U.S. Navy's Asiatic Fleet, and of the Fourth Marines, who dreamed up
this little operation forgot to take the Friday Afternoon factor into
account. As Bobby Shaftoe could've explained to them, if only they'd
bothered to ask a poor dumb jarhead, their route took them through the heart
of the banking district. Here you've got the Hong Kong and Shanghai Bank of
course, City Bank, Chase Manhattan, the Bank of America, and BBME and the
Agricultural Bank of China and any number of crappy little provincial banks,
and several of those banks have contracts with what's left of the Chinese
Government to print currency. It must be a cutthroat business because they
slash costs by printing it on old newspapers, and if you know how to read
Chinese, you can see last year's news stories and polo scores peeking
through the colored numbers and pictures that transform these pieces of
paper into legal tender.
As every chicken peddler and rickshaw operator in Shanghai knows, the
money printing contracts stipulate that all of the bills these banks print
have to be backed by such and such an amount of silver; i.e., anyone should
be able to walk into one of those banks at the end of Kiukiang Road and slap
down a pile of bills and (provided that those bills were printed by that
same bank) receive actual metallic silver in exchange.
Now if China weren't right in the middle of getting systematically
drawn and quartered by the Empire of Nippon, it would probably send official
bean counters around to keep tabs on how much silver was actually present in
these banks' vaults, and it would all be quiet and orderly. But as it
stands, the only thing keeping these banks honest is the other banks.
Here's how they do it: during the normal course of business, lots of
paper money will pass over the counters of (say) Chase Manhattan Bank.
They'll take it into a back room and sort it, throwing into money boxes (a
couple of feet square and a yard deep, with ropes on the four corners) all
of the bills that were printed by (say) Bank of America in one, all of the
City Bank bills into another. Then, on Friday afternoon they will bring in
coolies. Each coolie, or pair of coolies, will of course have his great big
long bamboo pole with him a coolie without his pole is like a China Marine
without his nickel plated bayonet and will poke their pole through the ropes
on the corners of the box. Then one coolie will get underneath each end of
the pole, hoisting the box into the air. They have to move in unison or else
the box begins flailing around and everything gets out of whack. So as they
head towards their destination whatever bank whose name is printed on the
bills in their box they sing to each other, and plant their feet on the
pavement in time to the music. The pole's pretty long, so they are that far
apart, and they have to sing loud to hear each other, and of course each
pair of coolies in the street is singing their own particular song, trying
to drown out all of the others so that they don't get out of step.
So ten minutes before closing time on Friday afternoon, the doors of
many banks burst open and numerous pairs of coolies march in singing, like
the curtain raiser on a fucking Broadway musical, slam their huge boxes of
tattered currency down, and demand silver in exchange. All of the banks do
this to each other. Sometimes, they'll all do it on the same Friday,
particularly at times like 28 November 1941, when even a grunt like Bobby
Shaftoe can understand that it's better to be holding silver than piles of
old cut up newspaper. And that is why, once the normal pedestrians and food
cart operators and furious Sikh cops have scurried out of the way, and
plastered themselves up against the clubs and shops and bordellos on
Kiukiang Road, Bobby Shaftoe and the other Marines on the truck still cannot
even see the gunboat that is their destination, because of this horizontal
forest of mighty bamboo poles. They cannot even hear the honking of their
own truck horn because of the wild throbbing pentatonic cacophony of coolies
singing. This ain't just your regular Friday P.M. Shanghai bank district
money rush. This is an ultimate settling of accounts before the whole
Eastern Hemisphere catches fire. The millions of promises printed on those
slips of bumwad will all be kept or broken in the next ten minutes; actual
pieces of silver and gold will move, or they won't. It is some kind of
fiduciary Judgment Day.
"Jesus Christ, I can't " Private Wiley hollers.
"The captain said don't stop for any reason whatsofuckinever," Shaftoe
reminds him. He's not telling Wiley to run over the coolies, he's reminding
Wiley that if he refrains from running over them, they will have some
explaining to do which will be complicated by the fact that the captain's
right behind them in a car stuffed with Tommy Gun toting China Marines. And
from the way the captain's been acting about this Station Alpha thing, it's
pretty clear that he already has a few preliminary strap marks on his ass,
courtesy of some admiral in Pearl Harbor or even (drumroll) Marine Barracks,
Eight and Eye Streets Southeast, Washington, D.C.
***
Shaftoe and the other Marines have always known Station Alpha as a
mysterious claque of pencil necked swabbies who hung out on the roof of a
building in the International Settlement in a shack of knot pocked cargo
pallet planks with antennas sticking out of it every which way. If you stood
there long enough you could see some of those antennas moving, zeroing in on
something out to sea. Shaftoe even wrote a haiku about it:
Antenna searches
Retriever's nose in the wind
Ether's far secrets
This was only his second haiku ever clearly not up to November 1941
standards and he cringes to remember it.
But in no way did any of the Marines comprehend what a big deal Station
Alpha was until today. Their job had turned out to involve wrapping a ton of
equipment and several tons of paper in tarps and moving it out of doors.
Then they spent Thursday tearing the shack apart, making it into a bonfire,
and burning certain books and papers.
"Sheeeyit!" Private Wiley hollers. Only a few of the coolies have
gotten out of the way, or even seen them. But then there is this fantastic
boom from the river, like the sound of a mile thick bamboo pole being
snapped over God's knee. Half a second later there're no coolies in the
street anymore just a lot of boxes with unmanned bamboo poles teeter
tottering on them, bonging into the streets like wind chimes. Above, a furry
mushroom of grey smoke rises from the gunboat. Wiley shifts up to high gear
and floors it. Shaftoe cringes against the truck's door and lowers his head,
hoping that his campy Great War doughboy helmet will be good for something.
Then money boxes start to rupture and explode as the truck rams through
them. Shaftoe peers up through a blizzard of notes and sees giant bamboo
poles soaring and bounding and windmilling toward the waterfront.
The leaves of Shanghai:
Pale doorways in a steel sky.
Winter has begun.
Chapter 1 BARRENS
Let's set the existence of God issue aside for a later volume, and just
stipulate that in some way, self replicating organisms came into existence
on this planet and immediately began trying to get rid of each other, either
by spamming their environments with rough copies of themselves, or by more
direct means which hardly need to be belabored. Most of them failed, and
their genetic legacy was erased from the universe forever, but a few found
some way to survive and to propagate. After about three billion years of
this sometimes zany, frequently tedious fugue of carnality and carnage,
Godfrey Waterhouse IV was born, in Murdo, South Dakota, to Blanche, the wife
of a Congregational preacher named Bunyan Waterhouse. Like every other
creature on the face of the earth, Godfrey was, by birthright, a stupendous
badass, albeit in the somewhat narrow technical sense that he could trace
his ancestry back up a long line of slightly less highly evolved stupendous
badasses to that first self replicating gizmo which, given the number and
variety of its descendants, might justifiably be described as the most
stupendous badass of all time. Everyone and everything that wasn't a
stupendous badass was dead.
As nightmarishly lethal, memetically programmed death machines went,
these were the nicest you could ever hope to meet. In the tradition of his
namesake (the Puritan writer John Bunyan, who spent much of his life in
jail, or trying to avoid it) the Rev. Waterhouse did not preach in any one
place for long. The church moved him from one small town in the Dakotas to
another every year or two. It is possible that Godfrey found the lifestyle
more than a little alienating, for, sometime during the course of his
studies at Fargo Congregational College, he bolted from the fold and, to the
enduring agony of his parents, fell into worldly pursuits, and ended up,
somehow, getting a Ph.D. in Classics from a small private university in
Ohio. Academics being no less nomadic than Congregational preachers, he took
work where he could find it. He became a Professor of Greek and Latin at
Bolger Christian College (enrollment 322) in West Point, Virginia, where the
Mattaponi and Pamunkey Rivers came together to form the estuarial James, and
the loathsome fumes of the big paper mill permeated every drawer, every
closet, even the interior pages of books. Godfrey's young bride, nee Alice
Pritchard, who had grown up following her itinerant preacher father across
the vastnesses of eastern Montana where air smelt of snow and sage threw up
for three months. Six months later she gave birth to Lawrence Pritchard
Waterhouse.
The boy had a peculiar relationship with sound. When a fire engine
passed, he was not troubled by the siren's howl or the bell's clang. But
when a hornet got into the house and swung across the ceiling in a broad
Lissajous, droning almost inaudibly, he cried in pain at the noise. And if
he saw or smelled something that scared him, he would clap his hands over
his ears.
One noise that troubled him not at all was the pipe organ in the chapel
at Bolger Christian College. The chapel itself was nothing worth mentioning,
but the organ had been endowed by the paper mill family and would have
sufficed for a church four times the size. It nicely complemented the
organist, a retired high school math teacher who felt that certain
attributes of the Lord (violence and capriciousness in the Old Testament,
majesty and triumph in the New) could be directly conveyed into the souls of
the enpewed sinners through a kind of frontal sonic impregnation. That he
ran the risk of blowing out the stained glass windows was of no consequence
since no one liked them anyway, and the paper mill fumes were gnawing at the
interstitial lead. But after one little old lady too many staggered down the
aisle after a service, reeling from tinnitus, and made a barbed comment to
the minister about the exceedingly dramatic music, the organist was
replaced.
Nevertheless, he continued to give lessons on the instrument. Students
were not allowed to touch the organ until they were proficient at the piano,
and when this was explained to Lawrence Pritchard Waterhouse, he taught
himself in three weeks, how to play a Bach fugue, and signed up for organ
lessons. Since he was only five years old at the time, he was unable to
reach both the manuals and the pedals, and had to play standing or rather
strolling, from pedal to pedal.
When Lawrence was twelve, the organ broke down. That paper mill family
had not left any endowment for maintenance, so the math teacher decided to
have a crack at it. He was in poor health and required a nimble assistant:
Lawrence, who helped him open up the hood of the thing. For the first time
in all those years, the boy saw what had been happening when he had been
pressing those keys.
For each stop each timbre, or type of sound, that the organ could make
(viz. blockflöte, trumpet, piccolo) there was a separate row of pipes,
arranged in a line from long to short. Long pipes made low notes, short
high. The tops of the pipes defined a graph: not a straight line but an
upward tending curve. The organist/math teacher sat down with a few loose
pipes, a pencil, and paper, and helped Lawrence figure out why. When
Lawrence understood, it was as if the math teacher had suddenly played the
good part of Bach's Fantasia and Fugue in G Minor on a pipe organ the size
of the Spiral Nebula in Andromeda the part where Uncle Johann dissects the
architecture of the Universe in one merciless descending ever mutating
chord, as if his foot is thrusting through skidding layers of garbage until
it finally strikes bedrock. In particular, the final steps of the organist's
explanation were like a falcon's dive through layer after layer of pretense
and illusion, thrilling or sickening or confusing depending on what you
were. The heavens were riven open. Lawrence glimpsed choirs of angels
ranking off into geometrical infinity.
The pipes sprouted in parallel ranks from a broad flat box of
compressed air. All of the pipes for a given note but belonging to different
stops lined up with each other along one axis. All of the pipes for a given
stop but tuned at different pitches lined up with each other along the
other, perpendicular axis. Down there in the flat box of air, then, was a
mechanism that got air to the right pipes at the right times. When a key or
pedal was depressed, all of the pipes capable of sounding the corresponding
note would speak, as long as their stops were pulled out.
Mechanically, all of this was handled in a fashion that was perfectly
clear, simple, and logical. Lawrence had supposed that the machine must be
at least as complicated as the most intricate fugue that could be played on
it. Now he had learned that a machine, simple in its design, could produce
results of infinite complexity.
Stops were rarely used alone. They tended to be piled on top of each
other in combinations that were designed to take advantage of the available
harmonics (more tasty mathematics here!). Certain combinations in particular
were used over and over again. Lots of blockflötes, in varying lengths, for
the quiet Offertory, for example. The organ included an ingenious mechanism
called the preset, which enabled the organist to select a particular
combination of stops stops he himself had chosen instantly. He would punch a
button and several stops would bolt out from the console, driven by
pneumatic pressure, and in that instant the organ would become a different
instrument with entirely new timbres.
The next summer both Lawrence and Alice, his mother, were colonized by
a distant cousin a stupendous badass of a virus. Lawrence escaped from it
with an almost imperceptible tendency to drag one of his feet. Alice wound
up in an iron lung. Later, unable to cough effectively, she got pneumonia
and died.
Lawrence's father, Godfrey, freely confessed that he was not equal to
the burdens now laid on his shoulders. He resigned from his position at the
small college in Virginia and moved, with his son, to a small house in
Moorhead, Minnesota, next door to where Bunyan and Blanche had settled.
Later he got a job teaching at a nearby normal school.
At this point, all of the responsible adults in Lawrence's life seemed
to arrive at a tacit agreement that the best way to raise him certainly the
easiest was to leave him alone. On the rare occasions when Lawrence
requested adult intervention in his life, he was usually asking questions
that no one could answer. At the age of sixteen, having found nothing in the
local school system to challenge him, Lawrence Pritchard Waterhouse went off
to college. He matriculated at Iowa State College, which among other things
was the site of a Naval ROTC installation in which he was forcibly enrolled.
The Iowa State Naval ROTC had a band, and was delighted to hear that
Lawrence had an interest in music. Since it was hard to drill on the deck of
a dreadnought while playing a pipe organ, they issued him a glockenspiel and
a couple of little dingers.
When not marching back and forth on the flood plain of the Skunk River
making loud dinging noises, Lawrence was majoring in mechanical engineering.
He ended up doing poorly in this area because he had fallen in with a
Bulgarian professor named John Vincent Atanasoff and his graduate student,
Clifford Berry, who were building a machine that was intended to automate
the solution of some especially tedious differential equations.
The basic problem for Lawrence was that he was lazy. He had figured out
that everything was much simpler if, like Superman with his X ray vision,
you just stared through the cosmetic distractions and saw the underlying
mathematical skeleton. Once you found the math in a thing, you knew
everything about it, and you could manipulate it to your heart's content
with nothing more than a pencil and a napkin. He saw it in the curve of the
silver bars on his glockenspiel, saw it in the catenary arch of a bridge and
in the capacitor studded drum of Atanasoff and Berry's computing machine.
Actually pounding on the glockenspiel, riveting the bridge together, or
trying to figure out why the computing machine wasn't working were not as
interesting to him.
Consequently he got poor grades. From time to time, though, he would
perform some stunt on the blackboard that would leave his professor weak in
the knees and the other students baffled and hostile. Word got around.
At the same time, his grandmother Blanche was invoking her extensive
Congregational connections, working the angles on Lawrence's behalf, totally
unbeknownst to him. Her efforts culminated in triumph when Lawrence was
awarded an obscure scholarship, endowed by a St. Paul oat processing heir,
whose purpose was to send Midwestern Congregationalists to the Ivy League
for one year, which (evidently) was deemed a long enough period of time to
raise their IQs by a few crucial points but not long enough to debauch them.
So Lawrence got to be a sophomore in Princeton.
Now Princeton was an august school and going there was a great honor,
but no one got around to mentioning either of these facts to Lawrence, who
had no way of knowing. This had bad and good consequences. He accepted the
scholarship with a faintness of gratitude that infuriated the oat lord. On
the other hand, he adjusted to Princeton easily because it was just another
place . It reminded him of the nicer bits of Virginia, and there were some
nice pipe organs in town, though he was not all that happy with his
engineering homework of bridge designing and sprocket cutting problems. As
always, these eventually came down to math, most of which he could handle
easily. From time to time he would get stuck, though, which led him to the
Fine Hall: the headquarters of the Math Department.
There was a motley assortment of fellows wandering around in Fine Hall,
many sporting British or European accents. Administratively speaking, many
of these fellows were not members of the Math Department at all, but a
separate thing called IAS, which stood for Institute for Advanced something
or other. But they were all in the same building and they all knew a thing
or two about math, so the distinction didn't exist for Lawrence.
Quite a few of these men would pretend shyness when Lawrence sought
their advice, but others were at least willing to hear him out. For example:
he had come up with a way to solve a difficult sprocket tooth shape problem
that, as normally solved by engineers, would require any number of perfectly
reasonable but aesthetically displeasing approximations. Lawrence's solution
would provide exact results. The only draw back was that it would require a
quintillion slide rule operators a quintillion years to solve. Lawrence was
working on a radically different approach that, if it worked, would bring
those figures down to a trillion and a trillion respectively. Unfortunately,
Lawrence was unable to interest anyone at Fine Hall in anything as prosaic
as gears, until all of a sudden he made friends with an energetic British
fellow, whose name he promptly forgot, but who had been doing a lot of
literal sprocket making himself lately. This fellow was trying to build, of
all things, a mechanical calculating machine specifically a machine to
calculate certain values of the Riemann Zeta Function
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 head wind. "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)
[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 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 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 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 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."
***
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