Zeno’s Paradoxes
Episode #3 of the course Brain-twisting paradoxes by John Robin
I hope you’re ready for more paradoxes because today’s tour will stop on one of the most famous of them all. It’s also very old. This paradox was conceived of by Zeno of Elea, a Greek philosopher who lived 490-430 B.C.
How Long It Takes to Get Nowhere …
You might be familiar with the story of the tortoise and the hare. This paradox involves a tortoise, and just like that famous fable, the tortoise is going to achieve a seemingly impossible feat: He’s going to beat the mighty Greek hero Achilles at a race. And no, Achilles won’t be stopping to sleep on the side of the road while the tortoise passes him!
The race will go something like this:
• The tortoise gets a head start.
• Achilles will catch up to him.
• While Achilles dashes ahead, the tortoise will have crept forward just a bit more.
• Once Achilles catches up, he’ll dash that small distance the tortoise crept.
• But the tortoise will have crept forward just a bit more in that short time.
• Achilles will dash forward that small amount.
• But even in that time, the tortoise will have crept forward a smaller small amount.
• Achilles will catch up to that …
Can you see where this is going? For each dash Achilles makes to catch up, the tortoise will always have crept ahead. It will always be by less and less, but he will still be in the lead, and Achilles will never pass him.
But surely this is ludicrous. Achilles must pass him eventually, right?
Take a moment to think about what’s happening here.
What’s your answer? How is it that Achilles doesn’t win this race? What’s the secret behind this brain-twister?
Motion Is an Illusion
The secret to understanding this paradox is in changing the way you think about movement. It’s not intuitive. But as these sorts of paradoxes reveal, the laws of the world often do not follow what our intuition would tell us is true.
Take the way the tortoise is moving, for example.
He is always moving forward. So is Achilles, but he is moving always close to the tortoise. Approaching something that is always moving just a little bit more means you have to approach it a bit more still. It will always be ahead of you, no matter how quickly you catch up.
The overall point behind this isn’t that Achilles will never pass the tortoise. The point is that if we try to find the actual point at which they cross, we never will, because they will always get closer and closer together. It will always seem like Achilles is almost about to pass the tortoise.
What this reveals about the universe is that motion is an illusion. In order to move, everything around you must move, and if it moves with you, you will never catch up; in effect, you will move nowhere. What this implies, most profoundly: Space cannot be made of discrete points!
This paradox has more practical applications, though:
• It shows you can always break up something finite into smaller (“infinitesimal”) increments; no matter how small it is, it’s still finite and can be broken up even more.
• It’s used in calculating mortgage and loan repayments.
And that’s a wrap for today’s tour. Your main takeaway should be that movement isn’t what you think it is. Stay tuned for tomorrow, when we will explore how the best way to save money is to spend it.
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Recommended reading
Find out more about the use of the geometric series (Zeno’s paradox in mathematical form) in finance.
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