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Who do you think would win a race between the mighty Greek warrior Achilles and a very clever tortoise? Obviously, Achilles would clobber the tortoise … or would he? The answer might surprise you.

By
Jason Marshall, PhD,
October 16, 2015
Episode #261

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Have you ever woken up to a blaring alarm and the feeling that some cosmic committee held a meeting during the night to decide whether or not the idea of motion is a good one … and that they decided that no, it’s terrible—so terrible, in fact, that walking across the room should forevermore be impossible. Of course, we’ve all been there—and we’ve all eventually conceded that motion is still possible and that sometimes we have to drag ourselves out of bed.

But what if I told you that I could prove that you were actually right … that motion itself is indeed impossible? Would you believe me? Or would you think that I was trying to pull a fast one on you? According to a well-known tale penned by the Greek philosopher Zeno, this is very similar to the situation that the famous Greek warrior Achilles faced when a very clever tortoise asked him if he’d like to race.

Can I prove that motion is impossible? What does this have to do with Zeno's paradox? And how is it related to the idea of a limit that we talked about last time? Let’s find out.

How to Prove that Motion Is Impossible

I know you think you’ve been moving your whole life, but what I’m about to tell you might force you to contemplate how that's even possible. Here’s why: in order to walk across your bedroom in the morning, you first have to walk across the first half of the room. But in order to walk across that first half of the room, you have to walk across the first half of that first half. And in order to walk across that first half of the first half, you have to walk across the first half of the first half of the first half. And so on … forever.

To walk across your bedroom, you have to walk across an infinite number of pieces of your bedroom.

If you think about it, you’ll see that this presents us with a bit of a conundrum. Namely, in order to walk across your bedroom, you have to walk across an infinite number of pieces of your bedroom. Surely, it must be impossible to walk across an infinite number of anything … right? Which means it must be absolutely impossible to make any progress across your room … or even to walk at all! Except, of course, that you can walk across your bedroom. So something must be wrong with this argument. Before we dig any deeper into this, let’s take a look at another version of the story which paints an even more convincing picture.

The Story of Achilles and the Tortoise

According to the Greek philosopher Zeno, long ago the hero of the whole Iliad saga, Achilles, was confronted by a tortoise. We don’t know what the tortoise’s name was, so we’ll just call him Mr. T. As the story goes, Mr. T told Achilles that he’d like to challenge him to a good old-fashioned race. Being a noble soul, Achilles was reluctant to accept a challenge from an obviously inferior competitor. Mr. T scoffed at this chivalrous act and claimed that so long as Achilles gave him just a little head start, he (the tortoise) would absolutely win. At this point, Mr. T’s bravado began to rub Achilles the wrong way, so he agreed to race.

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