Math, music and mysticism: Pythagoreans believed all things are made of numbers

Ask any high school student to name one theorem, and chances are they’ll name the Pythagorean Theorem. So we may justly regard its namesake, Pythagoras, as the most famous mathematician. But who was he?

Pythagoras was born in the Greek isle of Samos, off the coast of Asia Minor, in 585 BC. His teacher, Thales, was the first mathematician in history. Thales had gone down to Egypt to learn their practical art of geometry, which at that time was simply part of land-surveying (literally “geometria,” a Greek word meaning “earth measurement”). He had then used logic to prove theorems, the first person to ever do so.

The prophet Ezekiel was born at about the same time as Thales, if that gives you any idea of how long ago this was. During his lifetime, King Nebuchadnezzar built the famous Hanging Gardens of Babylon and destroyed the Temple of Jerusalem.

Pythagoras himself went on to found his own school in Croton, a Greek colony in Italy. He and his followers, the Pythagoreans, got involved in politics there, siding with the old aristocrats against those who wanted a democracy. The people rose up against them, and Pythagoras was murdered sometime around 497 BC.

In using the word “school,” we mean something a little different from what we think of nowadays. At that time, a school wasn’t so much a place as a group of people gathered around a teacher, in this case, Pythagoras. The members were something like students but also something like religious disciples.

Pythagorean mathematics was just one part of their worldview and way of life. We don’t know much about their beliefs, for they left behind no writings. They had a reputation for secrecy. According to tradition, one member, Hippasus, was either expelled for publishing their teachings, or thrown off a ship and drowned for revealing a mathematical “secret.”

It is known that the Pythagoreans believed in metempsychosis: the soul survives after death, they said, and is reincarnated as another person or an animal. Because of this they were vegetarians. They also were forbidden to eat beans or lentils. One ancient author (possibly joking) said this was because Pythagoras believed that a person loses some of their soul every time they pass gas.

The Greeks were the first to try to find the basis of physical reality. Thales believed that all substances are made of water. His student Anaximander believed that the apeiron, a boundless cosmic reality, produces new materials perpetually. Anaximander’s student Anaximenes believed that all substances are made of air. Later philosophers said that all things are made of tiny indivisible particles. They called this building block of nature the “atom,” a Greek word meaning “uncuttable.”

The Pythagoreans had their own ideas. They believed that all things are made of numbers. Not atoms, not some kind of universal substance or principle of material creation, but, literally, numbers. For them, nature itself was a kind of mathematical harmony.

This idea might have originated in Pythagoras’ discovery that musical harmonies are governed by numerical ratios. The octave corresponds to the ratio 2:1, the fifth to 3:2, and the fourth to 4:3. Pythagoras made this discovery while standing outside a blacksmith’s shop, listening to the hammers strike the anvils. The Pythagorean scale, devised by Pythagoras and still in use today, is based on the ratio 3:2.

As a school of philosophy, the Pythagoreans endured for hundreds of years, well after the time of Christ, but their serious mathematical work was done before 400 BC. They are credited with showing that the three angles in a triangle sum to two right angles. But of course their main claim to fame is the Pythagorean Theorem, which states that the sum of the squares of two legs of a right triangle is equal to the square of the hypotenuse.

Stay tuned next month for the story of this beautiful and amazing theorem.

MICHAEL ORTIZ is an associate professor of mathematics at Sul Ross State University-Rio Grande College in Uvalde. He contributes a monthly math/science-related column to the Uvalde Leader-News.



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