There were at least four self contractictory arguments concerning time and space that circulated around the philosophical schools in Athens during the fifth century B.C. The one that was most closely associated with Zeno of Elea is called the Paradox of Dichotomy. It's one of those rare gems of reasoning from the ancient world that can make you smile. It's as clever as it is simple and it's as profound as it is nonsensical. Further, it does not require any complex mathematical description to be able to solve because the solution has less to do with what you know than it does with how you think.

The Dichotomy Paradox states that the simple act of getting up from a chair in a room and walking to the door, which leads outside the room, can be demonstrated to be theoretically impossible. Here's how it works. In order to get to the door, a person must first walk half of the distance to the door. From that point, the person must walk half of the remaining distance. Then, the person must walk half of the distance again. This process of halving the remaining distance is repeated, over and over again with smaller and smaller distances being crossed, without ever ending. And since the process never ends, the person can never really reach the end or ever be expected to get to the door. It's a simple as that.

While this theory appears entirely logical, the person invariably does get to the door. In most cases, the person should be able to reach the door in less time than it takes to state the paradox fully (unless, of course, it happens to be a very large room). So, what went wrong with the supposed theoretical impossibility? Believe it or not, the Greeks had a tough time with this one. It took decades for them to resolve the paradox because there are basically two things wrong with it.

First and foremost, it's a matter of time and, no matter how you look at it, it will always come out the same. Time either can be described as the duration of an action that is called an event or can be described as the interval between two events where the events are defined as particular sets of circumstances (e.g. at the chair and at the door). Either way, as one event or as one interval, it is still only one thing that is being measured. That's why no math formula is applicable (i.e. there is no form of computation required merely to express the value of one).

What the paradox does is to split the single event into smaller and smaller fractional

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