Anyone who's learned even the most rudimentary geometry has heard his name. Yes, in his own way, Pythagoras achieved the Greek ideal of fame that causes him to live forever, even if he is relegated to the pages of High School textbooks. But there is more to this mathematician than a simple ratio. Among other things, he believed himself to be the reincarnation of a Trojan hero, and refused to eat beans. His mathematekoi brotherhood was thought to be the most well-learned in the ancient world, but they did not hesitate to murder one man who would expose their most embarrassing secret: irrational numbers.
Though he seems harmless in his textbook depictions, Pythagoras was considered by many in the ancient world to be a dangerous cult leader, and a malcontent. He lived during that necessary twilight between oral tradition and written history, and thus his life and work is shrouded in an unfortunate cloud of mystery and myth. It is said that he traveled all throughout the world to gain knowledge of mathematics, science, medicine, astronomy, astrology, and mysticism from whomever would teach him. I think it likely that he traveled to Egypt, home of the famous ancient mathematician Thales, who accurately calculated the height of a pyramid by measuring its shadow. The Greek philosophy of math and science was that it was attainable, that they could actually observe and learn from what they saw to predict or manipulate future behavior. However, it was also steeped in Pagan mysticism, something which taints their learning somewhat and caused many, especially Pythagoras, to go off the rails.
Eccentric though he was, even by ancient weirdo standards, he built a school around this central tenet: All is number. He believed that numbers could be used not only to define all things, but could even give them greater meaning. His disciples took this motto to heart, and immediately began measuring angles and lengths to find the hidden constant ratios between them. In fact, most of the really boring parts of Geometry today (constructions, proofs) were what the Pythagoreans discovered when they were just playing around. They would challenge each other with number riddles like, “can you form a right triangle if given two points?”
One story claims that Pythagoras discovered the mathematical value of music. He was passing by a smithy one day, and the ringing of the anvils was sticking in his head. He noticed the relationship between their individual pitches, and examined three of them to discover that the middle one was one-third bigger than the smallest, and that the largest was one third bigger than the middle. Through experiments that the school performed on strings, bells, and other instruments, they created the octave as a means of dividing the musical notes, something we still do in Western music today.
As clever as these stories make him out to be, my belief is that Pythagoras was just the charismatic leader of some very bright young Greeks. I believe his school as a group made much of the discoveries that he is given credit for, just like professors will occasionally take credit for their students' findings today. In either case, he should at least be honored for cultivating an environment in which learning and discovery could take place.
However, this was long before the days of public education, and knowledge of every sort was a tightly guarded secret, particularly mathematics. As we will see in coming weeks, mathematics can kill people, and it often does so in great quantities with a minimal effort. For the Pythagoreans, the biggest secrets that they kept were the ones that they hated and couldn't explain.
Pythagoras and his followers were so convinced that everything could be expressed as a ratio of whole numbers that when they discovered ratios that didn't work out to nice, neat, clean whole numbers they were thrown into a crisis of faith. You see, this wasn't “just math” to them; this was their religion. Everything they held to be true, yes, the very order of the universe was at stake, and if it was just all random, irrational events, then knowledge itself was a fools game, a mirage in the desert of unpredictability.
Try this to see what I mean: draw yourself a square. Go ahead, draw one. Now, assign each side the number 1. Doesn't matter how big you've made it, just pretend you've created your own unit of measurement, name it after yourself if you like. Now that you have your square, draw a diagonal line from one corner to the other. Great. Now, using the Pythagorean Theorem, a^2 + b^2 = c^2, calculate the length of that diagonal. You should get 1^2 + 1^2 = c^2, and c^2 is equal to 2. So how long is that diagonal? The number √2, if you bother to plug it in, should make your calculator go crazy with an endless stream of numbers. However, it's a decimal answer, and therefore not a whole number. The Pythagoreans probably convinced themselves that there must be some sort of mistake in all of this, but their equations would continue to come out irrational, that is, being unable to express in a ratio of whole numbers. And since the square was one of their 'sacred shapes,' there's no doubt that they wanted to keep this a secret until they could find a way to make it work.
Well, enter Hippasus, a Pythagorean disciple who couldn't keep this cat inside its loosely constructed bag. He blew the whistle to the general public and alerted everyone in the Mediterranean that Pythagoras and his school were just a bunch of frauds. Little is truly known about this character as well, though some believe he created a rival school of mathematics. In either case, it is believed that he was murdered by Pythagorean zealots while trying to leave town. Yes, that's right, he was murdered for mathematics.
Pythagoras himself was killed during a political uprising in Croton in which he found himself on the wrong side. Supposedly, he ran from his assassins and was gaining ground until he came to a field of beans. He stopped and declared that he would not cross a field of beans, at which point I like to imagine his attackers stopping for a second to tilt their heads in unison before promptly dispatching this mathematical primadonna.
Pax vobiscum
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