Zeno of Elea

1. Zeno of Elea • 490 – 430 B.C.

2. Life • Not much known about life • Actual birth and death dates are not know, but estimated from Parmenides, which was written by Plato and contains most biographical information on Zeno • Said to be son of Teleutagoras • Pupil and friend of Parmenides, and studied in Elea • Influenced by Parmenides philosophy of monism, which claimed that many things which appear to exist are merely a single eternal reality called Being, and that change is impossible • According to Parmenides, Zeno traveled to Athens with Parmenides around 450 B.C. and the two discussed philosophy with Socrates

3. Philosophical Work • None of Zeno’s original work has survived • Most likely wrote only one book • This book contained 40 paradoxes, 4 of which were very influential in the development of mathematics • His arguments were the first examples of reductio ad absurdum, or proof by contradiction

4. Zeno’s Paradoxes: Achilles and the Tortoise • In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. –Aristotle • Basically, this paradox says that if Achilles and the Tortoise were racing, we are assuming Achilles would give the Tortoise a 100 foot start. When the race starts, after some finite time Achilles will have reached the place where the tortoise started. But, during this time the tortoise will have moved forward, maybe a foot. It will then take another amount of time for Achilles to go this distance, while the tortoise will move further. Thus, since Achilles always has further to go, he can never catch up to the tortoise.

5. Zeno’s Paradoxes: The Dichotomy • That which is in locomotion must arrive at the half-way stage before it arrives at the goal. – Aristotle • Basically states that to get to a point, first you must get halfway there. Before you get halfway there, you must get ¼ of the way there, and before that 1/8 of the way there, and before that 1/16 of the way there, and so-on. • This requires one to complete an infinite number of tasks, which Zeno says is impossible. He uses this to say that movement must then be impossible

6. Zeno’s Paradoxes: The Arrow • If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. – Aristotle • This paradox asks that you divide the flight of an arrow into a series of indivisible “nows” or “moments”. The arrow occupies an exact location at any of these moments. But, there can’t be movement in the past and future if there is not movement in the present. So, Zeno says that the arrow is always at rest, and because of this that movement is impossible.

7. Zeno’s Paradox: The Stadium • This paradox, generally thought of as the most controversial, sets forth this situation. If someone is running at me from the west at the maximum possible speed, and someone else is running at me from the east at the maximum possible speed, then they are approaching each other at double the maximum possible speed.

8. The Stadium Cont’d. • Another way of looking at this paradox is as if you are on a stationary train (A), and one train (B) is coming one way as fast as possible, and another (C) coming the other way equally as fast.

9. The Stadium Cont’d. • Since B and C are moving at equal speeds, so at some time later, the situation looks like this. • The simplest way to look at this situation and draw a conclusion is like this. An instant from the perspective of a moving train to that of the stationary train is double that of an instant from the perspective of a moving train to another moving train. Therefore, half a given time is equal to double that given time, which is impossible.