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What was up with Pythagoras?
Ok, so I've been learning about Pythagoras and the dirt on him is just too good. You've probably heard of the Pythagorean Theorem but not the part where Pythagoras was a crazy cult leader who thought he'd made a deal with a god thousands of years ago and could remember all of his past lives. Oh, and he killed a guy. I mean maybe it was a long time ago and he was afraid of beans. As in beans, they just like freaked him out or something, I don't know. But mostly I want to talk about the murdery part. See, Pythagoras and his cult of Pythagoreans had this cool-kids club where they'd talk about proportions all day. They'd be like, "Hey, I drew a two by three rectangle using a straight-edge and compass. Isn't that awesome?" And then someone would be like, "Hey guys, I have a box that's two by three and a half?" And the cool kids would be like, "Three and a half? That's not a number! Get out of our club!" And then they'd make the units half the length and call it four by seven, and everything was okay. Even if your box is 2.718 by 6.28 you can just divide your units into thousandths and you'd get a box that's a nice, even 2,718 by 6,280. It's not a simple proportion but hey, the box still has a whole number proportion, so Pythagoras is happy. Unless it's a box of beans, then he freaks out. I'd like to imagine what it would be like to think of numbers the way they did. Maybe you think of numbers as being on a line. Numbers one way, zero, and negative numbers the other, and there are numbers between them: fractions, rationals, filling in the gaps. But Pythagoras didn't think about numbers like this at all. They weren't points in a continuum they were each their own, separate being, which was still pretty modern because before that people only thought of numbers as adjectives, numbers of. In Pythagoras's world, there is no number between seven and eight, and there is no number three over two so much as a relationship between three and two, a proportion. Six to four has the same relationship because the numbers share this evenness, which when accounted for makes it three to two. The universe to Pythagoras was made up of these relationships. Mathematics wasn't numbers, mathematics was between the numbers. Though while people admire how much Pythagoras loved proportions, there's a dark flip side to that obsession. How far was he willing to go to protect the proportions he loved? Would he kill for them? Would he die for them? And the answer was he'd go pretty far until beans got involved. It's time for Time Line Time. Mathematicians like lines. I want the context, because in school today if you bring out the ruler and compass and are like, "Let's do some geometry! Let's draw two lines at 90 degree angles using a straight-edge and compass! Here's a happy square!" Then you've probably had years of math class already and think of geometry as being harder than adding big numbers together. You probably think that zero is a simple, easy concept and have heard of decimals too. Well, here's now, 2012. Here's Einstein, Euler, Newton and Da Vinci - - that sure was a while ago! Now let's go all the way back to when Arabic numerals were invented and brought to the West by Fibbonacci. Before that, arithmetic was nightmarishly hard, so if you can multiply multi-digit numbers together you can go back in time and impress the beans out of Pythagoras. And before that there was no concept of zero, except in India where zero was discovered around here. And if you keep going back you get to the year one, (there's no year zero, of course, because zero hadn't been invented) and back a bit more you get to folk like Aristotle, Euclid, Archimedes and then finally Pythagoras, all the way back in 6th century BCE. Point is, you can do some pretty cool mathematics without having a good handle on arithmetic and people did for a long time. And in school when they tell you you need to memorize your multiplication table and graph a parabola before you can learn real mathematics, they are lying to you. In Pythagoras's time there were no variables, no equations or formulas like we see today, Pythagoras's theorem wasn't 'a squared plus b squared equals c squared,' it was 'The squares of the legs of a right triangle have the same area as the square of the hypotenuse,' all written out. And when he said 'square' he meant 'square.' One leg's square plus the other leg's square equals hypotenuse's square. Three literally squared plus four made into a square. Those two squares have the same area as a five by five square. You can cut out the nine squares here and the sixteen here and fit them together where these 25 squares are, and in the same way, you can cut out the 25 hypotenuse squares and fit them into the two leg squares. Pythagoras thought you could do this trick with any right triangle, that it was just a matter of figuring out how many pieces to cut each side into. There was a relationship between the length of one side and the length of another and he wanted to find it on this map. But the trouble began with the simplest right triangle one where both the legs are the same length, one where both the legs' squares are equal. If the legs are both one then the hypotenuse is something that, when squared, gives two. So what's the square root of two and how do we make it into a whole number ratio? Square root two is very close to 1.4 which would be a whole number ratio of 10:14 but 10 squared plus 10 squared is definitely not 14 squared, and a ratio of 1,000 to 1,414 is even closer, and a ratio of 100,000,000 to 141,421,356 is very close indeed but still not exact, so what is it? Pythagoras wanted to find the perfect ratio he knew it must exist, but meanwhile someone from his very own Pythagorean brotherhood proved there wasn't a ratio, the square root of two is irrational, that in decimal notation (once decimal notation was invented) the digits go on forever. Usually this proof is given algebraically, something like this, which is pretty simple and beautiful if you know algebra, but the Pythagoreans didn't. So I like to imagine how they thought of this proof, no algebra required. Okay, so Pythagoras is all like, "There's totally a ratio, you can make this with whole numbers." And this guy's like,"Is not!" "Is too!" "Is not!" "Is too!" "Fine have it your way. So, there's a whole number ratio in simplest form, where this square plus this square equals this square." "Yeah, that's the Pythagorean theorem, I made it." "Yeah, though for this triangle you don't even need the full theorem. it's easy to see that it's the same area by cutting each part into four triangles." "But I don't want to divide the squares up into triangles, I want unit squares." "So kind of like this, where this square is divided into units and so is this one and they all fit perfectly into this one and vice versa, but not like this. It almost works, but you start dividing this square evenly to fill up the two equal other squares, and you've got this one odd one out. There's an odd number of squares to begin with, so you can't divide them evenly between the two squares." "That's not even a right triangle, what's your point?" "Just that you know an odd number like seven isn't gonna be it without even trying. An odd number times itself gives an odd number of squares, so whatever this number is, it can't be seven, it has to be even." "Ok, so the hypotenuse is even, that's fine." "So what if I proved the leg is even too?" "Then it's not in simplest form." "Any ratio where both are even you divide by two until you can't divide anymore, because one of them is odd and then that ratio is the best. I thought we assumed we were talking about the simplest form ratio." "We are. If there's a ratio in simlest form at least one of the numbers is odd and since the hypotenuse has to literally be divisible by two, then the leg must be the odd one. So what if I proved the leg had to be even?" "You just proved it's not. It can't be both." "Unless it doesn't exist! What you forget Pythagoras is that if this is a square then the two sides are the same. Just as this is divsible right down the center so too is it divisible the other way! And the number squares on this side, which are the number of squares in just one leg is an even number. And for a number of squares to be even what does the number have to be, Pythagoras, oh my brother?" "If leg squared is even then the leg is even. But it can't be even, because it's already odd." "Unless it doesn't exist." "But if they're both even you can divide both by two and start again, but this still has to be even which means this still has to be even, which means you can divide by two again, but then it has to be even so everything is even forever and you never find the perfect ratio. Aww, beans" He had a vision, a beautiful vision of a world made up of relationships between numbers. If this wasn't a whole number ratio, then what was it? The Pythagoreans still believed, wanted to believe that irrationality was somehow false and the world was as they wanted it. So this proof stayed secret. Until someone spilled the beans. According to some, it was all a guy named Hippasus and Pythagoras threw him of a boat to drown him as punishment for ruining what had been perfect. Or maybe it was someone else who discovered it or Hippasus or someone else who was killed by the Pythagoreans long after Pythagoras was dead or maybe they just got exiled, who knows? And how did Pythagoras die? Well, according to one guy some guys got mad because they didn't get into the cool kids' club. So they set Pythagoras' house on fire. And Pythagoras was running away and they were chasing him, but then they came upon a field and not just any field, but a field of beans. And Pythagoras turned around to face his pursuers and proclaimed: "Better to be slaughetered by enemies than to trample on beans!" And he was. Others say he ran off and starved himself to death. Or just got caught by his enemies because he ran around the bean field instead of through it or who knows what happened. People claim Pythagoras didn't like beans because he thought they were bad for digestion, or gave you bad dreams or reminded him of male genitalia or because he didn't want a clubhouse full of flatulating mathematicians or he just didn't like them metaphorically. He and his followers were or weren't vegetarian did or didn't sacrifice animals possibly were only allowed to eat certain colors of birds I mean he definitely had a lot of rules to follow but just what they were and what they meant is lost to history. I'd like to give you a colorful story about exactly what happened with Pythagoras, but somehow that kind of truth doesn't last. What I do know is that the square root of two is irrational, that there's no way to have the length of a side of a square and of the square's diagonal both be whole numbers. Mathematical truth is truth that indures. This proof is just as good now as it was 2500 years ago, I mean it's awesome and it shows that there's more to the world than whole numbers and shame on the Pythagoreans who didn't have the beans to admit it.