Achilles and the tortoise – Zeno’s paradox revisited

The Big Idea by Elena Butti

You have a new unit-mate, an exchange student from the Spatial University of Saturn (UCU’s fame has shifted from international to interplanetary). His name is Pollus and he has sneaked into your room and stolen your precious Little Brown Handbook (LBH), which you cunningly kept hidden under your pillow together with a picture of the Dean in Sinterklaas clothes.

Without your LBH, your whole UC career is ruined – there is no way you can survive without it.

You immediately start chasing Pollus across the Quad. You’re the chair of MarathonCo, while Pollus is very lazy – he has no chance of running away from you. You’ll catch him in the blink of an eye.  Or so you think.

Let’s say that when you saw Pollus running away with your LBH, he was already 100 metres away from your room. It takes you 10 seconds to run 100 metres. “In 10 seconds he’s mine,” you think.

Exactly 10 seconds later, you are there, where you have just seen Pollus running away. But, alas, Pollus isn’t there any more. Although you run much faster than him, he’s still running. During those 10 seconds, he has managed to move 20 metres and he’s now 120 metres from your room. This means Pollus is now 20 metres ahead of you (since you’re now 100 metres from your room). “20 metres are nothing,” you think. “I’ll catch him in 2 seconds.” 2 seconds later, you’re there, 120 metres from your room.

But guess what? Once again, Pollus isn’t there. In those 2 seconds in which you’ve moved 20 metres, he didn’t stay still. He’s slower than you, but he managed to run 4 metres – which means that he’s now 124 metres away from your room.

Only 4 metres between you and Pollus left. Not even half a second (0.4 seconds, to be precise) and you’ll catch him. But 0.4 seconds later you’ve reached his previous position, and he’s not there – he’s once again a little bit further. The game goes on like this forever.

I know, this is becoming frustrating. How is it possible that you, the pride of MarathonCo, can’t manage to catch this malefactor who has stolen the thing you’ve valued most since your UC career started?

You can’t find an answer.

If it makes you feel better, even the Greeks could not find one.

Your state of frustration right now must have been very similar to the one Achilles experienced when, no matter how hard he tried, he realised he couldn’t reach a tortoise that had started moving 100 metres ahead of him.

Do you find all this paradoxical?

That’s normal, since we’re dealing with the well-known Zeno’s paradox. Real life experience shows that Achilles does in fact reach the tortoise. Nevertheless, the fact that mathematicians have discussed a convincing mathematical confutation of this paradox for millennia shows the strength of Zeno’s argument.

“Who cares about the Greeks,” you think. “In reality I can reach him.”

Well, yes, in reality. But not in our story. Surprise: all this was nothing but a nightmare caused by indigestion of Dining Hall broccoli last night. You wake up in the middle of the night all sweaty from the strenuous run. You immediately slide your hand under your pillow. Your Little Brown Handbook is there, safe and sound. The time hasn’t yet come for you two to part.

Zeno’s paradox: Achilles and the tortoise 

Achilles is in a footrace with a tortoise. Achilles runs ten times faster than the tortoise. The tortoise starts the race 100 metres ahead of Achilles. Although the tortoise is much slower than Achilles, Achilles will never reach the tortoise. In fact, before Achilles can reach the tortoise, he will have to reach the tortoise’s starting point. But by that time, the tortoise will have advanced a bit farther. Achilles will then have to reach the new tortoise’s position. But again, by that time, the tortoise will have moved a bit. Whenever Achilles reaches somewhere the tortoise has been, he still needs to go farther. This process goes on infinitely. Therefore, he can never overtake the tortoise.

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