Paradoxes• According to Webster’s College Dictionary, a paradox

is defined as “a seemingly contradictory or absurd statement that expresses a possible truth”.

Zeno• Lived during the 5th

century B.C.E.• Most well known for his

paradox of Achilles and the Tortoise. – Achilles can never catch

the tortoise as long as the tortoise is given a head start…

• Argues that if something is divisible then it is infinitely divisible.

Divide……Infinitely

The number 1 can be divided in half. ½ can be divided in ½ again which = ¼. ¼ ÷ 2 = 1/8 ÷ 2 = 1/16 ÷ 2 = 1/32 ÷ 2 = 1/64 ÷ 2 = 1/128… This process can be carried out an infinite number of times, which is the basis for Zeno’s paradox. In order to reach ½, you must reach ¼, but before ¼, you must reach 1/8, and because that can be done forever, reaching the number 1 is impossible.

Albert Einstein’s The Twins Paradox

• We have 2 twins (with obviously the same age).

• One stays home, the other takes a trip on a spaceship capable of traveling very near the speed of light.

• Time passes slower for the child in space, due to the speed of travel.

• This “paradox” / theory has been proven; however, I believe it is called a paradox because it is hard to comprehend with or without mathematical data.

• Einstein's Demo• Hyperlink

Theseus’ ShipTheseus’ Ship• Theseus is the man in Greek Mythology that killed the Minotaur – a half man half bull, that Theseus is the man in Greek Mythology that killed the Minotaur – a half man half bull, that

lived in the Labyrinth on the island of Crete.lived in the Labyrinth on the island of Crete.

• According to the story, the ship that Theseus sailed According to the story, the ship that Theseus sailed back to Athens in was preserved for years, and as back to Athens in was preserved for years, and as the planks on the ship decayed, they were the planks on the ship decayed, they were replaced.replaced.

• As the paradox goes, suppose that after hundreds of As the paradox goes, suppose that after hundreds of years, every board had to be years, every board had to be replaced and there was not a single original board. Is it replaced and there was not a single original board. Is it still Theseus’ ship? Or is it a copy? If it is still Theseus’ ship? Or is it a copy? If it is considered a copy, at what point did it become a considered a copy, at what point did it become a copy? Half way through replacement? copy? Half way through replacement? Three quarters?Three quarters?

• It makes sense to think that if only one was replaced it’s still the same ship, and even if two It makes sense to think that if only one was replaced it’s still the same ship, and even if two were replaced it’s still the same; therefore, with this reasoning, it would still be Theseus’ even were replaced it’s still the same; therefore, with this reasoning, it would still be Theseus’ even if all the boards were replaced.if all the boards were replaced.

• What if all of the old boards were stored somewhere? And what if, with all the original boards, What if all of the old boards were stored somewhere? And what if, with all the original boards, intact, the ship was rebuilt, would there be, could there be, intact, the ship was rebuilt, would there be, could there be, twotwo of Theseus’ ships? of Theseus’ ships?

Irresistible ForceIrresistible Force vs.vs. Immovable Immovable ObjectObject

A classic paradox: What happens when an irresistible force meets an immovable object?

The Unexpected The Unexpected HangingHangingThere was a man who had committed a serious

murder. He was convicted, in spite of a high priced lawyer, and came before the judge for sentencing. Now the judge was known to be a man of his word, and when the convict came before him, he said, "You are hereby sentenced to death by hanging. You will be hung on one of the next seven days and on a day that you do not expect."

The sentence was given on a Saturday and the possible days for the execution were, therefore, Sunday through the following Saturday.

The man returned to his cell in dismay. But his lawyer followed him with a big smile on his face.

"What are you smiling about?", asked the convict."Relax, kid.", said the lawyer, "The judge cannot

possibly hang you.""Why not?"

Other Classic Paradoxes

The Barber The Heap Hotel Infinity The Liar The Paradox of the Stone The Two Envelopes