The Pythagorean Theorem is named, ironically, for Pythagoras, a Greek Philosopher and Mathematician. Pythagoras was certainly a smart guy, but his biggest blunder is one we should all learn from. There is some evidence to suggest it resulted in a murder to prevent its discovery.
Pythagoras went on a journey to Babylonia and then to Egypt through much of his life, learning everything he could in every temple he found along the way. When he came back to his hometown, he found too few people interested in his philosophies and theorems, and decided to settle down in a Greek colony in Italy, called Croton. There, he formed a “school” (cult), which revered the integers as either spirits or gods. Yes, that’s right – the integers themselves were worshiped, or at least prayed to.
Everything in the Universe, in the Pythagorean school of thought, could be understood as either an integer (a pure incarnation of one of these spirits) or as a ratio of integers (a mixture or a child of two of them). Thus everything had some divine essence, and that essence could be understood by learning about the ratios involved in it. This spiritual system was so well developed that it presented Pythagoras with some difficulty when he really started looking at the theorem we now call the Pythagorean Theorem.
It may very well be true that Pythagoras brought the Pythagorean Theorem home to the Greeks for the first time, though it had been known to the Babylonians for at least a millennium by the time he did. But the problem Pythagoras faced is the problem this theorem gives us when we ask the question “What is the ratio of the side of a square to the diagonal of the square?”
According to the Pythagorean Theorem, this ratio is √2. But wait… we now know √2 is NOT a ratio. Indeed, the proof of the fact that there is no ratio of integers which is equal to √2 is now well-known and doesn’t even take up a full page. Students of Mathematics are now regularly asked to prove this fact as part of an introduction to the art of proof-writing.
But in Greece at this time, mathematical proofs were pictures, not paragraphs. A proof like that of the true irrationality of √2 was troublesome to produce, but it is now believed this was done by a certain Pythagorean, named Hippasus. In fact, by some accounts, even Pythagoras knew of the existence of irrational numbers like this before Hippasus, and it is claimed by some historians that Pythagoras himself called irrationality “unutterable” – indeed, irrationality does call to question his entire philosophy.
What would have happened had Pythagoras acknowledged this new understanding instead of trying to cover it up? After all, we now know Pythagoras’s favorite number, φ, was irrational, and one would think trying to understand irrationality would be the goal of a devout Pythagorean even if just for this reason. Even to this day, we still call φ the “golden ratio” – a misnomer because the number is not rational… is it just natural to try and think like a Pythagorean about these things? Truth be told, there is no golden ratio! Perhaps the fear of this fact motivated Pythagoras to cover up the existence of irrationals.
There were many mathematical secrets which were unavailable to any Pythagorean except those in his inner circle, who were sworn to secrecy, and it seems clear the existence of irrationals was one of them. Poor Hippasus, who was a Pythagorean in this inner circle (albeit quite some time after the death of Pythagoras himself), who did nothing more than prove the irrationality of √2, was punished by death. By some accounts, the other Pythagoreans drove him out into the ocean, tied a rock to his feet, and threw him overboard. Of course, if this happened, nobody would really know about it except the Pythagoreans, right?
One historian named Iamblichus, who was himself a Pythagorean, perjures himself several times when he gives conflicting accounts of the death of Hippasus. In one story, it is said that the gods punished Hippasus for revealing and claiming credit for a (less dangerous) secret of the inner circle regarding the inscription of a dodecahedron within a sphere. The punishment the gods meted out for this impiety? Drowning. Go figure. In another story told by the same historian, it is said that Hippasus claimed there were irrational numbers and for this heresy he was banished from the Pythagorean group – and that he later died, by drowning. Yet another story written by the same historian (recall this historian was himself a Pythagorean) tells us that there was once a Pythagorean who angered the gods when he claimed the existence of irrationals (while on a boat), who was punished by drowning in the sea. It seems Iamblichus is trying to cover up a telltale heart but his guilty conscience can’t help but tell us the instrument of Hippasus’s murder.
This is certainly not enough to convict the Pythagoreans of murder in court, but it is suggestive. And it can serve to teach us a lesson: when faced with new facts which undermine our beliefs, the correct response is to change our beliefs so the conflict is no longer a problem, if possible. Pythagoras’s inappropriate response of attempting to make the facts secret was irresponsible and may have led to the murder of Hippasus.
And so, let this story pop into your head whenever you hear the phrase “Pythagorean Theorem” – remember that when we use this name for the theorem, we are really thumbing our nose at Pythagoras! For Pythagoras, the theorem would be a constant reminder of his biggest blunder, the sin of being unwilling to learn for want of the will to change his beliefs. And let it remind you not to commit the same sin yourself. Let reason and evidence, not belief, guide you towards the truth!
And if you ever get the chance to show someone how to motivate irrational numbers, go ahead and start from the Pythagorean Theorem – that would really piss off Pythagoras!!!
Neato Math! 