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.
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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.
OK, that’s all the math we have time for today.
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Tortoise image from Shutterstock.