Achilles and Tortoise

David Cowles

Oct 15, 2022

If Zeno can defeat his teacher, the whole class wins! Zeno today, me tomorrow!

The students in Mrs. Dooley’s 5th grade math class are scratching their heads. According to Mrs. Dooley, someone has challenged the great Achilles to a foot race. But what makes this particularly puzzling is that the challenge is coming from a tortoise.


Now, Achilles is the fastest runner in all of Greece and tortoises are, well, tortoises. This can’t end well for Tortoise, can it?


Fortunately, though, Achilles is a generous man. He’s agreed to give Tortoise a head start. Achilles will begin the race at the start line, but Tortoise will begin halfway to the finish line.


Still, Tortoise has a problem. As soon as the race begins, ESPN measures the speed of both runners, and Achilles is running four times as fast as Tortoise. Remember, though, Tortoise has a head start. Achilles must cover twice as much ground as Tortoise, but then again, Achilles is running four times as fast.


Mrs. Dooley asks the class, “Do you think Achilles can catch up to Tortoise and win the race?”

All the students nod their heads. All except one, Zeno!


“You don’t agree, Zeno?” Mrs. Dooley asks, as the class groans impolitely. Zeno has a reputation for coming up with contrary answers and weird arguments. One time the banter between Mrs. Dooley and Zeno became so intense that the class was two minutes late for recess. Zeno’s classmates were not pleased!


But Mrs. Dooley is always polite. “Ok, Zeno, how do you see it?”


“Well,” the boy began. “Achilles can’t win the race unless he can catch up to Tortoise. He must catch Tortoise before he can pass him. Right?”


“Ok,” Mrs. Dooley agreed, warily.


“But obviously, Achilles can never catch up to Tortoise!” Zeno asserted confidently, as though their argument was over and the matter settled.


Expecting more, but not getting it, Mrs. Dooley kept going, “Why not, Zeno? Why can’t Achilles catch Tortoise. After all, Achilles is running four times as fast and only has twice as far to go.”


Mrs. Dooley was sure she’d got the better of Zeno this time, but Zeno was just annoyed. After all, Mrs. Dooley was supposed to be his teacher, and now he was going to have to teach her.


With a slight air of condescension, Zeno continued. “Before Achilles can pass Tortoise, he has to catch up to Tortoise, right?”


“Yes.”


“So, the first thing Achilles has to do is get to the halfway marker where Tortoise began the race,” Zeno explained.


“Ok,” Mrs. Dooley replied, still not understanding where the argument was headed.


Zeno, by now a little exasperated, continued, “Well, by the time Achilles gets to the halfway mark, Tortoise isn’t there anymore, is he?”


“No, I don’t suppose he is.”


“Well, Achilles can’t beat Tortoise unless he can catch up to him and so far, he hasn’t done that, has he?”


“No, but the race isn’t over yet, Zeno. Achilles will catch Tortoise later on.”


“No, he won’t, Mrs. Dooley: the race is over. Achilles can never catch Tortoise.”


Mrs. Dooley has a puzzled look but, strangely, Zeno’s classmates are suddenly listening intently. If Zeno can defeat his teacher, the whole class wins! Zeno today, me tomorrow! And this time, Zeno seems to be on to something.


“Whenever Achilles gets where Tortoise was, Tortoise has moved on again.”

“Of course.”

“Well, Mrs. Dooley,” Zeno concluded, trying, unsuccessfully, to remain as respectful as possible. “Don’t you see that this same process will repeat over and over again? No matter how many times Achilles gets to where Tortoise was, Tortoise won’t be there.”


“So, Zeno,” asked an exasperated Mrs. Dooley. “Who wins the race?”


At that moment, the bell rang for lunch, but guess what? Something magical happened. The whole class sat motionless. Their attention was focused on Mrs. Dooley…and Zeno.


Next to recess, lunch was the high point of the day, but believe it or not, the students in Mrs. Dooley’s class were willing to give up lunch if they could get to the bottom of what their whispers were already calling “Zeno’s Paradox.”


Hungry herself, Mrs. Dooley thought to ‘tap out.' “So, then Tortoise wins the race, right Zeno?”


Well, Zeno almost fell out of his seat, laughing. He couldn’t stop himself, “Of course, Tortoise didn’t win the race; everyone knows Achilles smoked him. Tortoise ate dust!”


The class exploded in laughter, and Mrs. Dooley could not have been more annoyed. Speaking sternly, she pressed Zeno, “Explain yourself, Zeno. First you said Tortoise won the race, now you say Achilles. Are you just messin’ with us?”


Mrs. Dooley didn’t often use slang with her students. Yup, she was angry; Zeno would soon find out just how angry! Sensing disaster, Zeno made one last futile effort to repair the situation, but, of course, it only made matters worse.


“Mrs. Dooley, I never said that Tortoise won the race. I said Achilles couldn’t win the race…not according to the rules of arithmetic anyway. We all know that Achilles won the race. The point is that according to the rules of arithmetic, he couldn’t have won, could he? So, the rules of arithmetic must be wrong.”

If only Mrs. Dooley had left it there, but she couldn’t help herself. “So if the rules of arithmetic are wrong, what are the right rules, Zeno?”


And of course, Zeno couldn’t help himself either. “Aren’t you the teacher?”


Class dismissed…for an abbreviated lunch, but now Zeno had become the class hero. Kids even offered him their hard-won desserts. And what of Zeno? He’s home now, writing 100 times, “I must not disrespect my teacher,” and he has never been happier in his whole life.”


 

Image: Achilles during the Trojan War, polychromatic pottery painting, 300 BC. Public Domain


 

David Cowles is the founder and editor-in-chief of Aletheia Today Magazine. He lives with his family in Massachusetts where he studies and writes about philosophy, science, theology, and scripture. He can be reached at david@aletheiatoday.com.


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