Zeno’s Paradox Solved By Calculus

Zeno is a Greek philosopher who lived around the time of 490 to 430 BC. His full name is Zeno of Elea. Sometimes, some people spell Zeno with an X as in Xeno.

He actually came up with many various paradoxes. So there is not just one “Zeno Paradox”, but “Zeno Paradoxes”.

The three of the well known one are …

  • Dichotomy paradox
  • Achilles and the tortoise paradox
  • Arrow paradox

Dichotomy Paradox

Perhaps the one that is most commonly provided as an example of one of Zeno’s paradox is the dichotomy paradox. And it goes like this…

Let’s say that I need to travel a distance of one mile. In order to get there, I must get to half-way. But in order to get to half-way, I must get to half of that (or 1/4 of mile). But in order to get to 1/4 of a mile, I must first get to 1/8 of a mile. Before that I need to go 1/16 of a mile. And before that, 1/32. Then 1/64 and 1/128 and on and on and on for an infinity. There is an infinite number of steps. So how in the world will I ever get to where I want to go?

Generalizing the problem… In order to travel a distance d, one must travel d/2. And before that, one must travel d/4. And d/8, etc.

The solution is resolved via calculus. In effect we have an infinite sum of a “geometric series”. In particular, we are summing (1/2)i as i goes from 1 to infinity. The answer to that sum converges to 1 and can be proven via calculus.

In short, it mean that the sum of an infinite number of “half-step” is finite. Therefore, you will get to where you will be going.

Math Joke

There is a math joke that is based off of Zeno’s paradox.

A group of boys line up at one wall at one end of the ballroom. A group of girls on the opposite wall. The two group walks towards each other. When will they meet at the center of the ballroom?

The mathematician says never, because it involves an infinite number of steps. The physicist says that they would meet when time equals infinity. And the engineer says that within one minute, they are close enough for all practical purposes. Hand it to the engineer for being practical.

Achilles and Tortoise Paradox

Achilles and tortoise are in a race. Since Achilles is a faster runner, he gives tortoise an 100 meter head start.

Therefore, it will take some time before Archilles reaches the tortoise starting point. But by that time, the tortoise (although slow) has since moved ahead. In order to catch up, Archilles must reach the spot where tortoise has already been. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Because in the time it take for Achilles to reach there, the tortoise has progressed further.

So how in the world does Archilles catch up with the tortoise?

In the below video, watch as a math teacher explains this paradox…

Paradox Solved by Calculus

The solution is similar to the one before. The infinite number of catching up that Archilles has to do is counterbalanced the infinitely small time it takes for the subsequent steps. And therefore Archilles is able to catch up to the tortoise.

Calculus comes to the rescue by saying that it is possible to add an infinite number of steps. In fact, Calculus is the subject of adding, comparing, and manipulating infinities. But Calculus was not invented yet in the time of Zeno. That is why they were perplexed.

Arrow Paradox

In the book Physics written by Aristotle, the Arrow Paradox goes like this …

“if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless”

Confusing? Okay, here it is stated in another way by Wolfram Math World

“An arrow in flight has an instantaneous position at a given instant of time. At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived?”

Basically, Zeno is saying that the flying arrow is motionless.

Of course that is ridiculous. And it is refuted by Aristotle when Aristotle writes …

“This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.”