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 Long ago I came across something called Zeno’s paradox. Strictly speaking it’s not a paradox, it is an antinomy.  A paradox is an apparent contradiction. An antinomy is a real, genuine unsolvable contradiction.*  You may have come across Zeno’s Paradox yourself, and It is a puzzle.

Zeno said think about Achilles and a tortoise. If they are in a race, Achilles can never win.  He can’t even end in a tie because he can never catch up.  Sounds absurd, of course, but let us suppose that both Achilles and the tortoise decided to enter the dash in the Athenian Olympics. They were lined up together on the starting line. Achilles looked down at the turtle and said, “What are you doing here, Little Legs? You couldn’t beat a snail.” “Yeah, yeah,” said the turtle, “ I know you’ve got some big wins from Troy, but this will be different,” Achilles replied. “Your mouth is bigger than your shell. I’ll be finished by the time you move a couple of feet.” “Cool it, Big Guy. Give me a couple of feet head start, and you’ll see,”  said the turtle. Achilles replied, “Start from the half way mark if you want to. Let me know when you get there. I’ll finish before you can say, ‘Spartans are sissies.” “You’re on,”  said the turtle and started off. He eventually got to the half way mark and called back, “Ready.”  

The stadium was empty by then because the crowd had gone home. Then the gun went off and the race started. Achilles sauntered to the halfway mark in a few moments, but when he got there, the turtle had moved on a little so now there was a new halfway mark between Achilles and the turtle. Achilles kept on, but so did the turtle, and when Achilles reached where the turtle had been, the turtle had moved again so Achilles still had some space to cover. This kept up. Achilles would get half way to where the turtle had been and when he got there, the turtle had moved on. There was always the other half to go. Then the same thing again and again. The remaining half may have become  a short distance, a fraction of a fraction, but it was always there, and even Achilles couldn’t make there into not there, something into nothing.

Zeno was in the stands watching and he soon realized he watching an unsolvable contradiction between logic and experience, an antinomy. He tried to explain to his friends, but he got so excited that he made a mistake and called the antinomy a paradox.