Lauren Ipsum: A Story About Computer Science and Other Improbable Things (3 page)

Laurie took a left turn at the sign marked “Recursion Junction.” After cresting a
little hill, she ended up at . . . Recursion Junction!

“Is this the same place, Xor?” Laurie asked. “It looks like
it.”

“Try a right turn,” he said.

She did, but after a short while they were back where they started. When she tried a second,
and a third, and twenty-seventh time, they always came back to Recursion Junction.

“It seems as though I am going somewhere else, but we always come back to the same
place. What’s going on?” Laurie wondered.

Just as she was about to give up, there was a gigantic smashing sound, like a stack of plates
falling to the floor.

They both jumped in fright, and for a moment Xor accidentally turned the same color as
Laurie’s shirt. They looked around for the source of the noise but couldn’t find it. To
their surprise, the next turn put them on a different road.

This road was neat and straight, and it seemed to stretch on forever. Up ahead, a man in a
Greek helmet was riding on a large Round Green animal with a Shell. (Eponymous might have called the
creature a
turtle
, though she wouldn’t have been quite right.) The two
were moving slowly and steadily away.

“Hey! Wait!” Laurie shouted, running up to them.

“At last, someone has caught up to us!” said the animal.

“I thought that was
impossible
,” said the man.

“Hello!” the animal said to Laurie. “I am Tortoise, a humble
tortoise.” He was much too large to be a mere
turtle
. “This is my
esteemed companion, Achilles the Logician.”

“At your service, miss!” said Achilles, bowing to her from his perch atop
Tortoise.

“Um, hello. My name is Lauren Ipsum.” She attempted a curtsey.

“How did you get here, Miss Ipsum?” asked Tortoise.

“I don’t really know,” Laurie said. “We were following the path to
Symbol, but I got turned around at Recursion Junction.”

“That often happens. You spent quite a bit of time chasing your tail, I
imagine.”

“But I don’t have a tail,” she said.

“So it got away from you, did it?”

“What?” Laurie asked. “No, I don’t—”

“Or perhaps it was optimized away,” said Tortoise. “No matter. Most of you
made it through, and that’s the important thing. You can help us resolve a
question.”

“Well, I can try,” Laurie said, not sure that she and Tortoise were having the
same conversation.

“Splendid!” said Tortoise. “The question my dear friend Achilles and I are
considering is this: how long is an infinite piece of string?”

“An infinite string? Infinite means it’s really, really, really, really, really,
really
long. Really,” said Laurie. Really.

“Ah! So you agree with
me
,” Achilles said. “That means
the burden of proof must be borne by the other side.”

“The burden of Achilles on my
back
is more than enough!”
Tortoise grumbled.

“Friend Tortoise is wise about many things,” said Achilles. “But he is
clearly wrong this time. He says that an infinite string can be exactly
two
inches
long!”

“But how can an infinite string be two inches long?” Laurie asked.

“His claim is preposterous and indiscrete,” said Achilles. “We are in
continuous disagreement about it.”

“I never disagree,” said Tortoise. “I only discuss, especially with an
intellect such as yours, Achilles. Your understanding has no limit.”

“You are too kind, dear Tortoise.”

“I mean every word,” said Tortoise. “Allow me to suggest a way to settle the
question by Experiment.”

“Please, suggest away,” said Achilles.

“Let us build—hypothetically, of course!—an infinitely long piece of string
and then measure it. Miss Ipsum can be our impartial judge.”

“I accept. Experiment is always better than mere Theory,” Achilles said.
“And an impartial judge sounds wonderful, especially when she already agrees with
me!”

“Excellent,” said Tortoise. “Miss Ipsum, imagine you have an infinite number
of pieces of string. If you laid them all end-to-end, would that be infinitely long?
Hypothetically?”

“Yes, it must be,” said Laurie.

“Infinity is infinity,” said Achilles. “It’s only
logical.”

“I wonder. Suppose we start with a piece of string
one
inch
long,” Tortoise said. “Then add a second piece of string that is
one-half
inch long. How long are they together?”

“One and a half inches,” Laurie said.

“And that is shorter than two inches?” Tortoise asked.

“One-half inch shorter. Unmistakably,” Achilles said.

“We all agree thus far,” said Tortoise. “Perhaps we shall converge upon the
same conclusion.”

“I doubt that!” said Achilles. Laurie wasn’t sure what Tortoise was getting
at, but she doubted it, too.

“Achilles, would you please keep count of our hypothetical string? I want to add a third
piece
one-quarter
of an inch long,” said Tortoise. “Is our string
now one and three-quarters inches long?”

Achilles retrieved a much-used notebook from under his helmet and scribbled some figures.
“It seems so,” he said.

“With one-quarter inch to spare?” asked Tortoise.

“Yes,” replied Achilles. “Only one-quarter inch! You are a finger’s
width away from defeat!”

“Add an
eighth
-inch piece,” Tortoise said. “Do I still
have some space left over?”

“Yes, but I’ll have beaten you soon!” the Greek Logician crowed. “Your
string is an eighth of an inch away from the limit, and you’ve done only
four
pieces!”

“Your arithmetic is correct as always, Achilles. But in the interest of science, let us
continue until the bitter end,” Tortoise said.

“It won’t be long,” said Achilles. “What is your next
move?”

“I would like to add another piece of string, this time one-sixteenth inch
long.”

“Done!” Achilles scribbled away. “Only one-sixteenth inch left, old
friend!”

“Only that much?” said Tortoise. “Then for the next one, I would like to add
a piece of string one thirty-second inch long.”

“As you wish, poor Tortoise. One thirty-second of an inch added. There is only one
thirty-second inch remaining, and an infinity of strings to go! There will be
plenty
of rope left to trip yourself with!” said Achilles.

“Please add a sixty-fourth-inch piece of string,” said Tortoise, “then a
one-hundred-twenty-eighth-inch piece, and a two-hundred-and-fifty-sixth-inch piece, and then a
five-hundred-twelfth-inch piece, and then—”

“Slow down, Tortoise! You are going too fast,” Achilles said. “And those are
very big—no, very
small
numbers.” He figured and scribbled for a
minute. “There is only a five-hundred-and-twelfth inch remaining. It’s too bad
we’re not splitting
hairs
, or you could have gotten a little farther! Do
you give up now?”

“
Oh.
I see!” exclaimed Laurie. “Achilles, Tortoise is
right.”

“What? Don’t change your mind
now
when we are so close to
victory!” Achilles cried.

“No, I’m sure Tortoise is right,” said Laurie. “Don’t you see?
Every piece he adds is
half
as long as the one before. That leaves just enough
room left over. Even if he adds an infinite number of pieces, the string will
never
reach two inches.”

“Well, hardly ever,” said Tortoise.

Achilles grimaced. “It appears you’ve proven the impossible again, Tortoise. But
just to make sure, I will check the arithmetic
myself
.” He continued to
scribble in his notebook:

“
That
should keep him busy. If anyone has the patience to actually
count to infinity, it’s Achilles. Thank you for your assistance, Miss Ipsum.”

“You’re welcome, Mister Tortoise,” said Laurie. “I didn’t know
something so big could be so small.”

“That’s the Power of Two,” said Tortoise. “If you cut a number into
two halves, then cut it in two again, and so on, very soon it will be too small to see. But there
will always be something left over.”

“Mister Tortoise, do you know how long this road is? It feels like it goes on forever.
I’m trying to get to Symbol.”

“This road is quite long,” he replied. “In fact, it is
infinite.”

“Oh, no! How do I get to the end?”

“You can do it in two simple steps.”

“How?”

“How do you think? A step with your right foot, then a step with your left foot,”
said Tortoise. “Your point of view is what’s important. It’s
integral.”

Of course! If an infinite string could be less than two inches long, then an infinite road
could certainly be less than two steps, if you looked at it in the right way. Laurie closed her eyes
and took a deep breath. She tried to imagine an infinite road. That was a little too much to handle,
so instead she imagined a really, really, really, really long road. Then she imagined folding it in
half. Then she folded it in half again, and again, and again, and again . . .

When Laurie opened her eyes, Achilles and Tortoise were gone. The infinite road was a tiny,
short thing now, hardly more than a stepping stone. She stepped forward with her right foot. Then
she stepped again with her left foot. In front of her was a road sign that read . . .

The town of Symbol was surrounded by high, perfectly smooth walls. A large stream split into
two and flowed around either side of the town.

“Do you know anything about this place?” Laurie asked.

“The people are kind of strange,” said Xor.

“What does that mean?”

“You’ll see.”

The path led to a gateway with a turnstile. It looked like this: ╞

Two boys about Laurie’s age were guarding the gateway. One wore a bright white suit and
a black shirt. The other had a bright black suit and a white shirt.

Maybe they’ll ask me a riddle
, Laurie thought.
Or
I’ll need to figure out which one is a liar and which one tells the truth!
Laurie
had read a lot of fairy tales, you see. She believed she was figuring out how this place
worked.

“Name?” asked one.

“Laurie Ipsum.”

“Password?” asked the other.

“Why, I don’t know the password,” Laurie said.

“Then you may not enter!” the boys said in unison.

“That’s not much of a riddle.”

“A riddle? Ha ha, no, begging your pardon, miss. That’s not how it works. There
are no riddles, no bets, no liars, and no truth-tellers. We’ve read those books too,
haven’t we, Tollens?”

“Yes, we have, Ponens. There’s none of that mythic fairy-tale stuff in our System,
miss. Word games and clever riddles, ha! That’s just bad security!”

“But that’s not fair!” cried Laurie. “How do I get
inside?”

“It’s very simple,” said Ponens. “If you have a password, then that
means you can pass through the semantic turnstile.”

“If you can’t pass through the turnstile, then that means you don’t have a
password. It’s only logical,” Tollens said. “Do you have a password?”

“No, I don’t know what it is,” Laurie said.

“Then you may not enter!” they said again, together.

“Can you give me a hint?” Laurie was sure she could guess the password, given some
kind of clue.

“Yes, hints are part of the System,” said Ponens.

“Oh, good.”

“Did you set up a hint with your account?” asked Tollens.

“Well, no,” she said. “This is my first time here.”

“Then it’s hard luck for you, miss,” said Ponens. “Once you are
inside, do make sure to set up a hint for the next time.”

“And remember to change your password to something memorable, but hard to guess,”
said Tollens. “It’s just good security.”

“But I’m trying to
get
inside! I don’t know what to
do!”

“It’s very simple,” said Ponens. “If you have a password, then that
means you can pass through the semantic turnstile.”

“And if you can’t pass through the turnstile, then that means you don’t have
a password. It’s only logical,” Tollens said.

“That’s not completely true, is it?” Laurie said. “What if the
turnstile is broken? I wouldn’t be able to enter even if I had a password.”

“Um . . .” Tollens looked a little unsure of himself.

“Or what if I fooled you into
believing
I had the password, even if
I didn’t? Then I could enter without it.”

“Hmm.” Ponens considered Laurie’s argument, trying to find a flaw.

Laurie rushed on. “Or what if I had the password, but I didn’t want to give it to
you?”

“No, you still couldn’t encroach on our premises,” Ponens said with more
confidence. “You have to give us a password that matches the name you give us.”

“Is it ‘Laurie’?” she guessed.

“No!” they shouted together.

“Is it ‘November First’?” That was Laurie’s birthday.

“No!” they shouted again.

“Only one try left, miss,” said Tollens.

“Oh, no! Really?”

“You can try only so many times. It’s just good security.”

“Do I even have an account?” she wondered.

“We can’t confirm or deny that,” answered Ponens. “That would be bad
security.”

“Do I . . .
not
have an account?”

“We can’t deny or confirm that, either!” answered Tollens.

“It’s very simple—” began Ponens.

“—yes, yes,” Laurie interrupted. “If I have it, I know it, but if I
don’t, you can’t tell me. And you can give me a hint only if I set it up earlier!”
This certainly was very good security.
Think, Laurie, think!

“Okay, let’s start over,” she said.

“Very well, miss,” said Tollens. “Name?”

“I told you, it’s Lau—” she stopped herself. “Actually . . . my
name is Eponymous Bach.”

“Password?” said Ponens.

It couldn’t be that simple, could it?

“Bach’s Password.”

“Welcome!” said Ponens and Tollens, waving her through the turnstile.

Of course Bach would name her password after herself!