How to Resolve Zeno’s Paradox
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.
Jason Marshall, PhD
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How to Resolve Zeno’s Paradox
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.
When Achilles asked how much of a head start he wanted, Mr. T said it really didn’t matter, but a 100 meter head start would do nicely. Achilles knew he could run much, much faster than Mr. T, so he knew he could cover these 100 meters quickly. Mr. T agreed with this, but also noted that by the time Achilles ran these 100 meters, he’d have moved—which means that Achilles won’t yet have caught up with him. Achilles agreed with this, but was unconcerned since he could again make up this extra distance that Mr. T traveled in a jiffy. Again, Mr. T agreed, but noted that he would once again travel a bit farther while Achilles was busy making up this extra ground … so Achilles still won’t have caught him.
Achilles can never quite catch up to the tortoise.
And on and on the story goes. No matter how often Achilles makes up the little bit of extra distance, Mr. T travels a bit further. And, apparently, Achilles can never quite catch up to the tortoise. According to the story, this argument convinced Achilles that he couldn’t win, so he conceded the race to Mr. T before it ever began. And thus, the tortoise managed to beat the mighty Achilles—but it wasn’t with his physical ability, it was with his mind. Because although Mr. T’s argument seems pretty convincing, it’s actually based upon a bit of mathematical slight of hand.
How Limits Resolve Zeno’s Paradox
So how can Achilles ever catch the tortoise (which surely he must)? And how can you ever make it across your bedroom (which surely you do)? The answer to both of these questions is related to the idea of a limit that we talked about last time.
To see why, let’s start by thinking about the problem of walking across your bedroom. Let’s say you know it takes 1 second to complete the trek. Assuming you walk at a constant speed, it must take 1/2 second to complete the journey across the second half of the room, 1/4 second for the previous quarter of the room, 1/8 second for the previous eighth, 1/16 second for the previous sixteenth, and so on for each of the infinitely tinier and tinier pieces of the room. The key thing to notice here is that it takes increasingly tiny amounts of time to cross those increasingly tiny pieces of the room. And, in the limit that those pieces become infinitely small, it take an infinitely small amount of time to cross them. So, rather amazingly, it takes a finite amount of time to walk across those infinitely many pieces (because by the “end” we’re essentially adding zero time on for each infinitely small piece). In fact, the total time for the trip is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … = 1 second (no big surprise since that’s what we started with).
The total time for the trip is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … = 1 second.
Something similar is happening to Achilles as he chases down Mr. T. Each time Achilles catches up to Mr. T’s previous location, the tortoise has moved a bit more. But that distance that Mr. T is moving each time is getting tinier and tinier. In the limit that this distance gets infinitely tiny (which it eventually must), it takes Achilles an infinitely tiny amount of time to make up. If you were to add up the total elapsed time for the infinitely many time intervals it takes Achilles to catch up, you’ll find that it takes a finite amount of time (just as it did to cross the room). As we look closer and closer at these increasingly tiny time intervals, we’re actually zooming in on the moment that Achilles finally catches up to Mr. T. And a bit of math will show you that it doesn’t take an infinite amount of time for him to do so.
Which means that if Achilles had been as clever as Mr. T, he would have won the race. But I’m glad he wasn’t, because now we get to enjoy Zeno’s famous paradox.
Wrap Up
OK, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Tortoise image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.