Thursday, September 1, 2022

How Athena Punished Pythagoras

How Athena Punished Pythagoras

Sing through me, O Muse, of a sage’s high hubris, and a goddess’s elegant revenge! Sing through me of Pythagoras the would-be number-lord, and Athena’s diagonal retort!

Pythagoras was a sage of ancient times. He and his cult of followers lived together in a cave near Athens. You’d expect them to forbid eating beans, which they did, but not for the reasons that you’d expect; for Pythagoras preached reincarnation, and he feared that some deceased friend might reincarnate as a bean.

Once he was visiting the Athenian marketplace, and he witnessed a man beating a dog. Pythagoras said, “Stop! I recognize his voice.” For Pythagoras, irrationality times irrationality often yielded rationality.

Pythagoras discovered that two strings under the same tension produce harmonious chords when plucked, if their lengths are in simple numerical ratios. He also discovered the theorem now named after him: in a right triangle, the sum of the squares of the lengths of the sides equals the square of the length of the diagonal. Therefore, for instance, the 3-4-5 triangle, for 3^2 + 4^2 = 5^2. Likewise the 5-12-13 triangle, and the 20-21-29 triangle, and many more.

Emboldened by this triumph of arithmetic over both music and geometry, Pythagoras preached that number rules the world. He and his followers researched the properties of numbers, so that they may rule numbers, and by those numbers, rule the world.

When wise Athena, goddess of Athens, heard about this, she was incensed. “He, a mere mortal, would rule the world by number? How dare he! Measures must be taken.”

Now, you’d expect that a wrathful wisdom-goddess bent on punishing an over-reaching geometer would send him a diagram that drives him mad. That is true; she did send him such a diagram. But you’d also expect that diagram to be an eldritch abomination of Lovecraftian complexity; and you would be wrong!

The diagram that Athena sent to Pythagoras was plain and simple: nothing but a square with unit sides and a diagonal. Then Athena taunted Pythagoras: “If you are so wise, O numerologist, then tell me; how long is this diagonal?”

Pythagoras deduced that its length was the square root of two; but when he tried to reduce that to a ratio of whole numbers, he failed! All of his followers tried, and they failed too. Then they and Pythagoras learned a terrible truth; that the square root of two is not a ratio of whole numbers. That simple diagram displays, plain as day, a length that defies Pythagoras’s teachings!

Terrified, they called such a number ‘irrational’, but it was the Pythagoreans themselves who became irrational. The cult swore each other to secrecy; but one of them threatened to tell the world, and they threw him off a ship at sea. Few nowadays show such devotion to mathematics! The murder stirred up mass passion against the cult. A mob pursued Pythagoras. He might have outrun them, but he ran up against a field of beans. He feared even to touch bean plants, so the mob caught him and killed him.

Thus died an over-reaching arithmetician. Since then wise Athena has negotiated a truce with Pythagoras’s followers. We mortals may swiftly approximate the square root of two, to any finite number of digits desired; but its infinity of digits is known all at once only upon Mount Olympus. So now we are reformed Pythagoreans. We believe that infinitely deep number rules the world, of which our knowledge is finite.

Honor Wisdom!