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 parts of the event and then to describe those sub-events as separate chronological events. It’s a clever illusion which poses that, if a single event can be described as a countless number of sub-events, those sub-events cannot be added back up to achieve the value of one. What is so clever about is that, by misdirecting the mind to consider something in a way that is beyond what can be perceived, it obscures the thing itself.
Second, it’s also a matter of space. The clearly defined physical distance between the chair and the door is as finite as it is real. And here again, it has the ultimate value of one. It can be divided up by any number of comparative measures (e.g. paces, feet, meters, etc…) but it cannot be divided into smaller and smaller fractional parts endlessly. Once the fractional division has been made down to the smallest sub-atomic particle known to exist, it has to stop. This is what gave the Greeks so much trouble.
The problem is infinity. It is an irresistible metaphysical idea which supposes that, beyond what can be known, there may be a greater sense of reality where there may be no limit to anything. As with all such metaphysical ideas, it arises from the desire to deal with the incomprehensible complexities of reality in a rational way. In attempting to do so, however, it refutes the very basis of reality. Without the limits of time and space, nothing can ever be shown to be real. Everything that is real, therefore, is necessarily finite.
What is so profound about the Paradox of Dichotomy is that it demonstrates the irrationality of trying to apply metaphysical ideas to physical problems. As such, it’s a warning, and yet, the world of modern thought continues to be infused with metaphysical ideas. Beyond the prolific art of science fiction, the discipline of theoretical physics tends to be stated more and more in metaphysical terms. In effect, it provides the Field of Science with the basis for evolving into the most irrational and intractable religion ever devised. More surprisingly, Zeno may have been able to predict this. After all, it was only a matter of time.