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Paradoxes

"How wonderful that we have met a paradox. Now we have some hope of making progress."

Niels Bohr, Danish physicist, in reference to the famous paradox present by Albert Einstein to challenge the Copenhagen interpretation of quantum reality, 1927.

Paradoxes are not only great fun and a challenge to solve or explain, they also often provide insights into new discoveries and understanding. A very early user of paradoxes was the logician Zeno of Elea:

Zeno of Elea

Elea was an ancient Magna Graecia (Greater Greece) town, which is now called Velia, located in southern Italy, near Salerno, 300km south of Rome. One of its famous philosophers was Zeno, who set this paradox about Achilles racing a tortoise:

Achilles and the tortoise

Zeno demonstrated that it is possible to argue that no matter how much faster Achilles is than the poor old tortoise, he will never catch it. The argument goes like this. the tortoise starts the race with a 100m headstart. When Achilles has this distance, the tortoise, although slow, will have covered some distance, say 1m. When Achilles covers this 1m, the tortoise has again moved on. Since no matter how quickly Achilles covers the distance that separates him from the tortoise, the tortoise will have moved on in the meantime, meaning that Achilles can never catch it.

A consequence of this rationale is that a bouncing ball, which loses a fixed percentage of its energy with each bounce, will bounce an infinite number of times.

The solution is that an infinite geometric sequence with ratio less than one can be summed, to show that Achilles does catch the tortoise.

Dichotomy Paradox

Similar to the Achilles and Tortoise paradox, this reductio ad absurdum argument proposes that locomation is not possible, since any journey may be divided into an infinite series of half journeys. A 16km journey first entails an 8km journey, which itself starts with a 4km journey, which starts with a 2km journey, etc., reducing to a minute first step, which itself requries an even minuter first step. Since an infinite series of journeys cannot be undertaken, the journey cannot even begin.

A little crazy, but a good excuse to stay at home.

Arrow Paradox

Movement is not possible because time may be divided into points, in each of which there is insufficient time for a body to move.

Toricelli's Horn of Mystery

Evangelista Toricelli, 1606-47, was an Italian physicist and student of Galileo. Without the benefit of calculus, which was not invented till the 1680s, Toricelli was still able to determine that a cylinder of finite volume can have infinite area!

A sheet of thin metal, $L^2$ long and $2π/L$ wide, is rolled into a cylinder. The circumference of the cylinder is: $2πr = 2π/L$, so the radius $r = 1/L$.

The volume of the cylinder is: $πr^2h = h{1}/{L^2}L^2 = π$.

As the length increases, the volume remains constant at π. Even if L → ∞, the volume remains constant.

How about that.

With the advent of calculus a generation later, the proof was extended to the rotation of a curve $f(x) = 1/x$ through 360° around the x-axis. This shape became known as Toricelli's Trumpet or Gabriel's Horn.

Content © Andrew Bone. All rights reserved. Created : January 18, 2015 Last updated :February 14, 2016

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