The Dichotomy Paradox

Dichotomy means division into 2 equal parts. The concept of the paradox is this:

Let’s take a man, John for example, who needs to travel the distance of 1km from his house to Walmart, because he noticed there’s no milk at home. To reach 1 km he needs to travel the first half (½ km) of the given distance. But to reach the center he first needs to travel ¼ km. To reach ¼ km John needs to walk ⅛ km first, and so on.

The distance can be divided infinitely into infinitely small equal half distances, meaning John will never be able to exit his home and buy some milk. Also, the divided distances cannot be equal to 0 (units of space), because the infinite amount of zeros is just zero, which is not true, since the distance is 1 km.

Logically, there seems to be no fallacy in Zeno’s argument, but however, John easily moves in real life and completes the 1 km journey to Walmart. What is happening then in actuality? Zeno said that the movement we perceive is just an illusion of some sort, while in actuality there is no movement taking place.

The Achilles and the Tortoise Paradox 

This paradox is basically the same as the first one, but more fun!

Achilles (a very fast runner) and the Tortoise decide to have a running competition. Tortoise starts at 1 m distance from the starting point, while Achilles starts from 0 m, on the starting line. Achilles is faster than the tortoise, and both have a constant speed throughout the whole race. When they both begin to run, Achilles reaches 1 meter, but the tortoise would have already moved a little further, let’s say the tortoise is at 1.3 m from the start. Now Achilles needs to run another 0.3 m to catch up to the tortoise, but he can’t! The tortoise has moved another 0.09 m, but Achilles doesn’t give up and he believes he will catch up to the tortoise soon. But when Achilles reaches 1.309 m the tortoise is at 1.336 m (1.309 + 0.027). And it goes this way forever.

There’s an infinite amount of finite distances Achilles needs to complete to catch up to the tortoise. The paradox is that an infinity of finite distances equals infinity, which is impossible, since the race has a real finite distance. Another option is that there is an infinite amount of distances equal to 0 (units of space), but then the overall distance is 0, which is also wrong.

Both of these paradoxes are considered to be solved with the help of modern calculus and the infinite summation. To not make this article overwhelming with math, basically, calculus proves that the infinite summation of finite numbers equals to some finite number. In the example with Achilles and the tortoise, the number is 1.5 m:

1 + 1/3 + 1/9 + 1/27 + … = 1.5

So, at the distance of 1.5 m Achilles catches up to the tortoise, considering the tortoise starts at 1 m and Achilles at 0 m.

However, I still can’t quite comprehend how the infinite sum can be applied to the real world and real distances. How can an infinity of finite distances be equal to a finite distance, and not an infinite distance? Or how a finite distance can contain an infinity within itself? Perhaps, there’s some deeper explanation that could be provided with the help of sophisticated physical theories.

The Arrow Paradox

To continue with the last paradox, which discusses the idea of motion not in just space but also in time, Zeno introduces a flying arrow and the distance the arrow completes before hitting the target. Visually we can easily imagine that the arrow is moving in space and time and hitting the target, but is it really so?

Zeno considers that time is just a collection of infinite “instants”, still moments. Therefore, we can divide the time into infinite instants or nows, as Aristotle puts it in his Physics, and look at our arrow. Now is equal to 0 seconds, and in each now the arrow is at some point in space without motion. But if we take another now, suddenly the arrow is in some other place. It looks as if the arrow moved, but how? Each now contains 0 seconds and therefore 0 motion, but every now has a different location for the arrow. When or where is the moment that contains an arrow actually moving? If we take a period of 1 second, we can clearly see how an arrow moves from one point to another, but this 1 second period still can be divided into infinite nows.

This paradox concludes that time also doesn’t exist, just like space.

I cannot say Zeno was right, but I cannot say that he was wrong too, especially in a logical sense. If there is some physical theory that can solve this paradox, can it really disprove its logic? Maybe, using geometrical points and lines to build the arguments behind the paradoxes is inherently wrong, and there is only a certain extent to which geometry and math can describe the physical world?