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What the Tortoise Said to Laurie

Laurie took a left turn at the sign marked “Recursion Junction”. After a cresting a little hill she ended up at... Recursion Junction! (Excerpted from Lauren Ipsum, a book about computer science, programming, and other strange fantasies.)

“Is this the same place?” Laurie asked herself. “It certainly looks the same.”

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

“Every time I take a turn, it seems as though I am going somewhere else, but I always end up in the same place. What’s going on?”

She went round

...and round

...and 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.

The road was neat and straight, and seemed to stretch on forever. Ahead of her, a man in a Greek helmet was sitting on a large green round animal with a shell. They were moving slowly and steadily away.

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

“Ah, at last someone has caught up to us!” the animal said. “I thought it was impossible.”

“Don’t start THAT old argument again!” said the man.

“Hello, I am Tortoise, a humble tortoise,” said the animal. “And this is my esteemed companion, Achilles the Logician.”

At your service, miss!” said Achilles, bowing grandly from his perch atop the Tortoise.

“Um, Hello. My name is Lauren Ipsum.” Laurie attempted a curtsey.

“‘Lauren Ipsum’. That’s quite a GENERIC name, isn’t it?” observed Achilles.

“Never mind that. How did you get here?” asked Tortoise.

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

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

“But I don’t have a tail!”

“Got away from you, did it? Well, it should turn up again,” said Tortoise. “Or perhaps it was optimized away. But seeing as MOST of you is present, perhaps you can help us resolve a discussion.”

“Well, I can try.”

“Splendid. The question I was posing to my dear friend Achilles 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.

“Ah! You agree with ME,” Achilles said, “and so the burden of proof must be borne by the other side.”

“The burden of Achilles on my BACK is more than enough!” said Tortoise.

“My colleague the Tortoise is wise in many matters,” Achilles explained. “But he is clearly wrong this time. He maintains that an infinite string can be less than TWO INCHES long!”

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

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

“I never disagree,” said Tortoise. “I only discuss, especially with a formidable intellect such as yours, Achilles.” Achilles preened at the Tortoise’s praise. “Allow me to suggest a way to settle the matter.”

“Please, suggest away,” said Achilles.

“Let us build —hypothetically, of course— an infinite piece of string, and then measure it. Laurie can be our impartial judge.”

“I accept. Experiment always beats Theory,” Achilles said. “And an impartial judge sounds wonderful, especially if she already agrees with me!”

“Excellent,” said Tortoise. “Let us begin. If you had an infinite number of pieces of string, and laid them end-to-end, would that be infintely long? Hypothetically?”

“Yes, it must be.” said Laurie.

“No matter how long or short each individual piece is?” asked Tortoise.

“Surely,” said Achilles. “Infinity is infinity.”

“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 answered.


“That sounds right.”

“We all agree thus far,” said Tortoise. “Perhaps we’ll converge on the same conclusion.”

“I doubt that!” said Achilles. Laurie wasn’t sure where Tortoise was going, 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 continued. “Do I still have some space left over?”

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

“You may well prove right, Achilles, but honor demands that we continue until the bitter end,” said Tortoise.

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

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

“Done!” Scribble. “You have only one-sixteenth inch left, old friend!”

“My word,” said Tortoise. “I should be careful with my allotted space. For the next one, I would like to add a piece of string one-thirty-second inches long.”

“As you wish, poor Tortoise, one-thirty-second inch added. 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.

“Put on a sixty-fourth inch piece, a one-hundred-and-twenty-eighth inch piece, then a two-hundred- and-fifty-sixth inch piece, and a five-hundred-and-twelveth inch piece of string,” said Tortoise.

“Hold on, those are very big —no, SMALL— numbers!” Achilles figured and scribbled for a minute. “Ah! 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, wait, I see!” exclaimed Laurie. “Achilles, Tortoise is right.”

“What? Don’t change your mind NOW, when I am 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. Then he repeats again and again with shorter and shorter strings. Even if he adds an infinite number of pieces, it will NEVER reach two inches.”

“Well, hardly ever,” said Tortoise.

Achilles grimaced. “It seems you’ve done it again, Tortoise. But just to make sure, I will check the arithmetic MYSELF.” He proceeded to scribble in his notebook: one-thousand-twenty-fourth inches, plus two-thousand-forty-eighth inches, plus four-thousand-ninety-sixth inches, plus...

“THAT should keep him busy. Thank you for your assistance, Laurie.”

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

“Or that something so small can also be so big,” Tortoise said.

“Mr Tortoise, do you know how long this road is?” asked Laurie. “It feels like it goes on forever. I’m trying to get to the next town.”

“It’s 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.”

“Really? What steps?”

“A step with your right foot, then a step with your left foot. But the attitude is important. It’s integral. Close your eyes and picture the road as only two steps long, like the string.”

Laurie closed her eyes, took a deep breath, and stepped forward with her right foot. Then she stepped again with her left foot. When she opened her eyes, Achilles and the Tortoise were gone. In front of her was a signpost: