I originally posted this article on the Imzy community around Christmas. I planned to post it here after I edited the Christmas stuff out of it, but I think it’s more fun this way. Brand-new paradox coming this Wednesday!
Zeno’s paradoxes are some of the oldest in the book; they were written around 450 bc. Traditionally, the motion paradoxes have been presented with a tortoise and/or Achilles running a race. But, because this is Christmas Eve, I’m going to present it with a kid running to their Christmas tree!
At 6:00 a.m., the kid wakes up and jumps out of bed. Let’s call him Jimmy. Let’s also say he lives in a really big house, and it will take him 10 seconds to get to the Christmas tree, because that makes the math easy.
Before Jimmy gets to his tree, he needs to get to the halfway point, obviously. The teleporter kit is under the tree- he doesn’t have it yet. In this thought experiment, Jimmy gets halfway to his Christmas tree, no problem. After five seconds, he is just starting to go down the stairs to the living room. But, before he gets to his Christmas tree, now Jimmy has to get 3/4th of the way to his Christmas tree. Obviously. So, 2.5 seconds later, Jimmy is stepping off the stairs. Now, he’s got to get 7/8th of the way to the tree. 1.25 seconds later, he can see the tree; he’s mostly through the living room. Now, he just has to get 15/16ths of the way… then 35/36ths… then 71/72ths… forever. There will always be another fraction to go: there is no limit to how small you can make a step. And if there is always another step, how could he ever get to the tree? You can’t do infinite things in a limited amount of time, right? It would have to take eternity to move anywhere… or so the Greeks thought. For millennia, this paradox stood, an awkwardly unsolved guest at the logic party. But then, an unlikely hero stepped forward: integral calculus.
(Sidebar: I’ve known about this paradox since middle school, and for years, I couldn’t understand the solution. Then, last year, I took BC Calculus. When I realized that the problem on the board was the solution to Zeno’s paradox, I literally gasped. My classmates thought I was weird when I tried to explain, but it was worth it.)
Now, I’m not here to explain Calculus to you. Check out Khan Academy for that. All you need to know is that integrals allow you to add up an infinite number of things, and if they behave themselves and converge, you can get a finite answer. Luckily for anything that wants to move, Zeno’s sequence of half, then half again, then half again, forever, converges quite nicely. In our case, it converges to a 10 second journey. Jimmy gets to his present with ease, unaware that, theoretically, he just did infinite things.
The story could stop there. Math had stepped in and given philosophy a helping hand, with the bonus of making every journey an infinite one. Sadly, physics showed up and ruined everything. Well, not really. But it is a bit of a bummer. Turns out, you can’t keep dividing distances forever, like I said you could. Sorry. Turns out, there is a ‘smallest distance’, called the Planck length. If you try to divide a Planck length in half, it straight-up doesn’t work.
This paradox is about as resolved as it gets. Both math and physics have beaten it to a pulp. Still, though, I’ve found that Zeno’s paradoxes are the most fun to explain to non-mathematicians, because they are the only paradoxes that allow you to move around a bit, adding some drama. If you describe this paradox to someone unfamiliar with its solutions, you can make every walk across the room a little magical. And in this season especially, we can all do with a little more magic.