Zeno was a Greek philosopher who was an admirer of Parmenides’ doctrine which states that plurality and change are just illusions and motion does not exist in reality. Plato, in his book “Parmenides”, points out that Zeno came up with these paradoxes as a reply to other philosophers who had come up with paradoxes against the Parmenidian view of reality.
Zeno stated that the purpose of the paradoxes “is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one.” (Parmenides 128d).
His arguments are possibly the first examples of ‘Reductio ad absurdum’ or proof by contradiction.
Three of the stoutest and most famous of Zeno’s paradoxes — Achilles and the tortoise, the Dichotomy argument, and an arrow in flight — are presented in detail below.
In this paradox, Achilles is in a foot race with a tortoise. Achilles gives the tortoise a head start and they both start running at the same instant. It takes a limited amount of time, T2 minus T1, for Achilles to reach the point where the turtle started running from. But because some time has elapsed, the tortoise would have gone further to the next point. When Achilles reaches this point then the tortoise would have gone few steps further again.
Zeno argued that every time Achilles reached the tortoise’s position at an earlier instant, the tortoise would have advanced just a little bit further. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” – Aristotle, Physics VI:9, 239b10.
Another one of Zeno’s famous paradoxes, which is quite like the first one but provides a better insight to his views and opinions about reality.
Suppose you want to walk to the nearest supermarket to grab some groceries. In order to reach there you must first reach halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be written as:
So, to go to the super market you need to do an infinite number of tasks which Zeno maintains is an impossibility.
Also, there is no first distance to start walking when you start walking because any possible finite step can be broken down in halves again. Therefore the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. It contains some of the same elements as the ‘Achilles and the Tortoise’ paradox, but with a more apparent conclusion of motionlessness.
“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, Physics VI:9, 239b5.
In this paradox, Zeno states that for motion to occur, an object must change the position which it occupies.
He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. If time passes in small duration-less instants then in any such instant an arrow cannot move to wherever it is, because it is already there. And it cannot move to some other spot because no time has elapsed.
In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.