Monsters of Reason

Francisco de Goya’s 1797 etching Found at Wikicommons.

Part 1:

In 1797 de Goya produced this famous etching “The Sleep of Reason Produces Monsters,” little did he know that in the next 200 years reason itself would be pushed far beyond the realms of common sense and would produce incredibly counter intuitive results. In this series of posts I will highlight some examples of the truly wierd and strange mathematics that has come directly from reason.

The first topic I will discuss is the square root of 2. This is a good topic to begin with as it is one of which almost everyone is familiar with and has some understanding. Today all of us know about the irrational numbers, but when math was young the Pythagoreans did not have that luxury; the idea of numbers that could not be expressed as a ratio of two whole numbers filled them with dread.Pythagoras is a man mostly lost to legend, however the results attributed to him lay the foundation of our conception of mathematical truth.

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Bust of Pythagoras of Samos in the Capitoline Museums, Rome. Found at Wikicommons.

Pythagoras became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Aristotle and others’ accounts, some ancients believed that he had the ability to travel through space and time, and to communicate with animals and plants.

But was this golden-femured time traveler as benevolent as Plato and his students would have us believe? Or was there something monstrous in him waiting to be triggered by the right circumstances?

Part 2: The Cult of Pythagoras and the Irrationality of Root 2.

Our first monster, is not really much of a monster anymore. However legend has it that a man died over $latex \sqrt{2}$. The man we credit with the discovery of the irrationality of \sqrt{2} was called Hippasus of Metapontum. Most of the facts of his life is lost to the mysteries of time, but legend has it that he was a Pythagorean who was able to show that \sqrt{2} was not the ratio of two whole numbers. This was a colossal feat. At the time there was no algebra. There was no idea of using symbols to represent numbers, so our standard way of proving irrationality just would not fly. The Greeks did a lot with pebbles and lines in the sand. So a proof of a concept like irrationality is rather incredible. Sadly how he did it is lost, however the story of his feud with the other Pythagoreans is legendary. Supposedly he was out sailing on the Mediterranean Sea when he made his discovery. When he announced his discovery to his fellow travelers, the traveling Pythagoreans went into a frenzy, and threw poor Hippasus overboard.

Yard with Lunatics by de Goya (1794)

Yard with Lunatics by de Goya (1794)

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