Zeno’s Paradoxes
By Josh Peete

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Zeno, who lived from about 495 and 480 BC, was a disciple of Parmenides of Elea. Parmenides went around telling people that reality was an absolute, unchanging whole, and that therefore many things we take for granted, such as motion, space, and plurality, were simply illusions. At this period of time, motion and space were major philosophical questions. A philosophical question is a real question, which cannot be answered by the conceptual resources at the time the question is posed. Zeno decided to contradict Parmenides’s notion of motion and space by creating philosophical paradoxes. These paradoxes stumped philosophers for a millennia and paved the way for new philosophical thought.

During this time in Greek philosophy, Parmenides of Elea realized that people believed in too many fictional concepts. He is quoted in saying, “for thou couldst not know that which is not nor utter it, for the same thing exists for thinking and being.” (Pyne) Fictional concepts like holes, space, time, motion are all ideas that do not refer to anything and therefore are illusions. His perception of space, motion, and time as fictional concepts could be interpreted in two different ways. Parmenides could be expressing that there is no space, time, or motion because reality is spacesless, timeless, and changless. Or, our ordinary concepts are in a state of grave disorder. We need to clarify the concepts that we have and replace them with new ones. During this period in history, philosophers are looking to understand how space, time, and motion work.

Space, at this point in Greek philosophy, was thought to be either discrete or continuous. If space is discrete then there can be smaller units of it. These smaller units of space are called “space atoms.” (Pyne) If space is continuous then it is infinitely divisible. Something that is infinitely divisible can be divided up and infinite amount of times. Zeno disproves the ideas of discrete and continuous space though his paradoxes.

Zeno’s Arrow paradox disproves how space can be discrete. Imagine a bow and arrow where the arrow is pointing at a target. An arrow is either in motion or at rest in one “space atom”. An arrow cannot move, because for motion to occur, the arrow would have to be in one “space atom” at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because by definition, instants are indivisible. Hence, the arrow is always at rest in one of the “space atom”. (Pyne) Space therefore cannot be thought of as discrete, so then space must be continuous.

The paradox of Achilles and the tortoise disproves how space can be continuous. Imagine a racetrack where Achilles is at the starting block of the race and a tortoise is at the halfway point. Achilles can run 10 times faster than the tortoise but he will never catch up to win the race. By the time Achilles reaches the halfway point, the tortoise will be ¾ done with the race. By the time Achilles gets ¾ done with the race the tortoise will be at 7/8 done with the race, and so on. This process of looking at where the tortoise will be when Achilles catches up to where he was can be repeated indefinitely. (Pyne) Therefore, Achilles, even though he runs ten times as fast as the tortoise, will never catch up. Space is therefore not continuous and philosophers are left pondering upon the correct notion of it.

Understanding that an infinitely long series equals a limit solves this paradox. An equation that will explain how a convergent series converges to a limit is as follows: “½ + ¼ + 1/8 + 1/16… = 1.” (Pyne) The end of the race is represented by “1” instead of infinity, so when you add up this series it equals it’s limit.

The major contribution Zeno makes in current philosophy is how he teaches the possibility of drawing absurd conclusions from apparently reasonable assumptions. Zeno knew that people finished races and arrows hit targets, but his paradoxes showed how the assumed knowledge people had about space and motion at the time was not necessarily correct. Zeno was the first to bring upon this type of reduction to absurdity, which is used in all types of logic and philosophical research today.

Space and motion were major philosophical issues in early Greek philosophy. Zeno used paradoxes to prove that the assumptions that space was either discrete or continuous were incorrect. He disproved the theory of “space atoms” in the Arrow paradox and the notion of an infinitely divisible space was disproved by the Achilles paradox. He contributes to philosophy today by showing how to take reasonable assumptions and create absurd conclusions.

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