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 the same.”
“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?”
They went round
...and round so many times that Laurie lost count. Just as she was about to give up, the next turn round put her on a different road.
This road was neat and straight, and seemed to stretch on forever. A man in a Greek helmet was sitting on a large, Round, Green animal with a Shell. They were moving slowly and steadily away.
“Hey! Wait!” Laurie shouted, running up to the pair.
“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.”
“I don’t have a tail,” she said.
“So it got away from you, did it?”
“What? I don’t—”
“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.
“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!” the 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.”
“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 little 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 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.
Scribble. “Yes, 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-inch away from the limit, and you’ve only done 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!” Scribble. “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. There is only one-thirty-second inch remaining, and an infinity of strings to go! There will be PLENTY of rope left over to hang yourself with!” said Achilles.
“Please add a sixty-fourth inch piece,” said Tortoise, “then a one-hundred-twenty-eighth inch piece, and a two-hundred-and-fifty-sixth inch piece, and then a five-hundred-twelveth inch piece of string, 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-twelveth 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, it 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:
+1/512 +1/1,024 +1/2,048 +1/4,096 +1/8,192 +1/16,384 +1/32,768 +1/65,536 +1/131,072 +1/262,144...
“THAT should keep him busy. 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,” said Tortoise. “In fact, it is infinite.”
“Oh, no! How do I get to the end?”
“That can be done in two simple steps.”
“How do you think? A step with your right foot, then a step with your left foot. Your point of view is what’s important. It’s integral.”
Laurie closed her eyes and took a deep breath. When she opened her eyes, Achilles and Tortoise were gone. The infinite road was a tiny short thing now, hardly a sidewalk. She stepped forward with her right foot. Then she stepped again with her left foot...