The pre-Socratic philosopher Parmenides (c.500 BCE) had a disciple named Zeno who worked at proving Parmenides’ theories to be true. He offered four paradoxes with which to prove that the idea of change and sense perception are logical fictions. His main contention was that the idea of motion is illogical. See what you think of his reasoning.
1. Motion is illogical – because for it to exist there would have to be an infinite amount of time for a runner, for instance, to traverse a given amount of space. But no matter how long one has the given distance to be covered would have to be subdivided an infinite number of times, thus taking an infinite amount of time to do so. Thus motion itself is an illogical notion.
2. Similarly, if we give a very slow tortoise a tiny head start his opponent, a rabbit, no matter how long the race course and how fast the rabbit is, the rabbit would never be able to catch the tortoise because it would first have to cover half the distance between himself and the tortoise while the tortoise would have time to inch ahead a tiny amount, ad infinitum! So, the idea of motion is illogical.
3. Again, if an archer shoots an arrow into the air no matter how fast the arrow goes it will never be able to reach the target because it would first have to traverse half of the distance to the target, and each inch of this distance is subdividable into logically dividable parts, thus requiring an infinite amount of time for the arrow to cover those parts. Thus, the idea of motion is itself contradictory.
4. Finally, the idea of motion is self-contradictory because it requires an infinite amount of both time and distance for any movement to take place. So, motion itself is impossible because would require an infinite amount of space and time in order to be real.
Clearly, these puzzles posed by Zeno somehow miss the
point since we all know that time and space are in fact divisible and motion is neither impossible or self-contradictory. So wherein lies the solution to Zeno’s paradoxes? Both time and space are real yet these paradoxes seem to call them into question.
Surely our senses do sometimes deceive us concerning how far away something is, or how fast something is moving. How, then, are these puzzles to be explained? One way to explain them is to point out that these puzzles trade on the idea that the concepts of space and time are always relative to some system or other, and thus do not exist in and of themselves as abstractions.
Zeno’s paradoxes trade back and forth on the difference between the abstract concepts of space and time, on the one hand, and their actual use in everyday life. One might turn the tables on him by suggesting that his own spoken words must be meaningless because they are divisible since they are spoken one at a time and not all at once.