Zeno of Elea (5th century BC) produced numerous ingenious paradoxes relating to movement and divisibility. This particular paradox is known as ‘the dichotomy’, since it involves the halving of spatial distances. It begins by highlighting something we would all readily accept about movement: that to travel between any two points in space, no matter how far apart those points are, we need to move half way between those two points before we cross the whole distance between them.
The next step in the argument asks us to consider the nature of infinities. An infinite series, like the natural numbers 1, 2, 3, 4…, is a succession of things that has no last (and/or) no first member. A task containing infinitely many subtasks (like counting every natural number), then, is one that could not be completed. For by definition, there is no last task in the series to complete, so no matter how much time we have, we cannot reach the end point. Now, while both of these principles seem obviously true, when we put them together, we reach the somewhat unsettling conclusion that movement is impossible. For take any two points in space: A and B. To fully move between A and B, we (by the first principle) first need to get half way from A to B (‘½ AB’). However (again by the first principle), to get between A and ½ AB, we first need to get to ¼ AB, and to get between A and ¼ AB we first need to get to… and so on. Travelling between any two points, then, no matter how close together they are, thus requires us to do infinitely many things: traverse all the points that space divides into. And since (by the second principle) it is impossible to do infinitely many things, it is impossible to travel between any two points! In fact, even beginning the journey seems futile, for what is the first point we must reach in order to begin our journey? It is not ½ AB, it’s not 1/32 AB… Finding the first point to reach is equivalent to vainly searching for the first (smallest) fraction to count! | |||