Pythagoras

so of all of those nature philosophers that you read about what was the only one whose name was familiar to you beforehand Pythagoras and why do you all know Pythagoras because the Pythagorean theorem can everybody recite the Pythagorean theorem let's do it a squared plus B squared equals C squared exactly the holy theorem of Pythagoras the one thing I can still count on high school graduates still knowing what is the Pythagorean theorem what is it good for why is it useful some about triangles any kind of triangle some about right triangles what about them right if you got two sides you can use the Pythagorean theorem to find the length of another side so let's say we've got a triangle all right and we know that this side is three and this side is four we're trying to find out this side so what's the Pythagorean theorem again we have a squared plus B squared equals C squared so if a is three squared plus B 4 squared equals C squared was 3 squared 9 what's 4 squared was it okay equals C square we got to get C by itself how do we get that rittle doohickey off there you take square root of both sides all right so this gets C by itself and then 9 plus 16 was 25 the square root of 25 is 5 so our hypotenuse equals 5 boom very good you've remembered how to apply the Pythagorean theorem why does this work no no works because your ninth grade geometry teacher told you would work right and you'll dutifully like good students memorize the theorem and you know how to apply it let me ask you something how many all hate math several of y'all hate math you know why you hate math I can tell you why my very first job teaching was when I was a student at Valencia like you and I tutored at the West Campus tutoring center and I'm mostly tutored algebra and geometry students and I quickly figured out why it is that so many people hate math you know why because you don't know what the hell you're doing and I mean that very literally very directly you know how to memorize things and how to process them down but you don't know what you're actually doing when you do this you're just going through a series of memorized steps now I was fortunate because I had mrs. booth for eighth grade math class and mrs. booth did not let us memorize our way and play number voodoo to try to pop out some mathematical answers she made sure we actually understood what we were doing and I can still remember the day that she explained the Pythagorean theorem first of all what you're seeing right here this is not reality that is symbolism reality is an actual triangle now we said that three and four and square and do all what are we really doing here first of all we're saying that okay if a is 3 then 3 squared why do we say squared why not 3 to the little what is 3 to the little to mean multiplied by itself so why don't we just say 3 self times E why do we say squared because we're making a square we're taking a two-dimensional link that we're squaring it we're taking it we're taking it from one to the second dimension ok so let's just start with the basics here 3 that is not 3 that is a symbol for 3 this is three or this is three one two three an extension of line so when we say this sides square we say three squared what we're doing is taking this one-dimensional line and stretching it out into the second dimension or making a square out of it one two three four five six seven eight nine if my length is three when I square it I make nine squares that's why three squared is nine that's physical reality that is symbolism expressing that reality think about Greek mathematics is that they did not have a very good symbolic system for describing what they were doing the Greeks were very good at geometry but it was direct applied geometry they didn't have these convenient little symbols that we have most of that stuff comes from Arabic mathematicians in the Middle Ages or beyond that at the time of you know Newton so they were wrestling with actual physical lines they were using their letters of their alphabet to express mathematical ratios one of which is still very commonly known today we still use the Greek letter pi to describe one of those problematic ratios so they're wrestling with it but we're going to use our modern mathematical symbolism to finish describing this we said that this was for and so when we square this side we end up with how many squares four squared is sixteen now when we say a squared plus B squared equals C squared what we're saying really is that a squared all those squares plus all of these squares are going to equal all the squares on the other side so we've got nine so far let's keep counting 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 squares in this side if we square it five 10 15 20 25 a squared plus the B squares add up to the same as the seaside squares yeah how many of you had your geometry teacher explained the Pythagorean theorem with that diagram none of them did I'm sorry I didn't have mrs. booth you should have this is reality and this is symbolism more modern symbolism that expresses that reality the trick to doing well in math and actually enjoying math is to not just stop at the surface level of the symbolism and just do number Voter two things what you need to do is always keep in mind what these symbols what these equations literally and physically refer to what you're actually doing what these words really mean and if you keep yourself attached to that and realize this is just a convenient symbolic shorthand you'll not only understand math better you'll actually enjoy doing math Pythagoras was one of many founders of philosophical schools in the classical Greek world following the tradition of fail ease that was exploring the nature of the physical world around it and trying to describe it in rational ways without recourse to gods and goddesses and supernatural elements and Pythagoras his school shrouded in mystery we know that they were very interested in mathematics we know he's interested in astronomy I mean he was very interested in music the Pythagorean school is largely known to us through the descriptions of people that were outsiders to the school or by later descendants of the school very little about Pythagoras is known directly these are mostly legendary tales but they seem to have been mystics in a lot of ways Pythagoras believed in reincarnation and the transmigration of souls he thought that the souls of corrupted humans apparently became animals or beans lesser things like that and there's this whole weird thing but why the Pythagorean and how Pythagoras did not like beans he was like had a phobia about beans would not eat them I mean we thought his flatulence was you know corrupt souls coming out of him or something this well could be just the the mockery of of his enemies it's hard to know exactly what was his but the Pythagorean school developed by Pythagoras left their home town and went to the southern coast of Italy which was controlled by the Greeks at the time and there he set up the school for which he would be famous and Pythagoras his accomplishments are many he's the first one that we know of to teach that the world is round and everybody that's gone to school have been educated since the days of Pythagoras for the last 2,500 years has learned that the world is round people didn't really think that Columbus was gonna sail off a flat earth that was a later historical invention he is the one that gave the name planets to the heavenly bodies that are distinct from the Stars because they move on independent paths through the sky the word planet is great for wonderer so we distinguished clearly between planets and stars but his most significant contribution in the long run has to do with this study of music of sound and the mathematical structure of it and his studies begin as he's playing around with something called a mono chord a mono chord is a one stringed musical device it's not really used for playing songs on it's more used for tuning instruments and he uses it to study sound any mono chord has two bridges in one string stretched across a bass and it has a movable bridge that can slide back and forth in either direction and as you move this thing around the point at which this context it means that there are you can change easily the length of string that you're vibrating when you pluck it so when it's very near to this side you're plucking a very tiny bit of string and then you can move it this way and pluck a longer and longer piece music majors out there as the string gets longer does the pitch go up or down the pitch goes down so when it's very short like this is things very high and he's moving longer hmm the pitch drops okay so it was understood already that there was a relationship some soar between the length of string under various tensions and the pitch that I would make but pythagoras is studying this very carefully and he makes his most profound discovery if you pluck a string that is say 24 inches long makes a toe oh god see if you take a length of string exactly half that length 12 inches under the same tension and you pluck that note if this one was a see what tone does the half length make see but an octave higher oh the pitch is going to be doubled and when you play those two tones together did they sound good or bad to your ears they sound good they make a pleasant combination that we call harmony harmony is when two tones are played that make you smile because they sound good together why they sounded good together was the mystery and it doesn't stop there if you have a length of string that's 16 inches long they're also going to sound harmonic together and if you have a length of string that is 18 inches long and you played against the 24 inch string they're going to sound good together they're gonna make harmonies Pythagoras was looking at these in his own Greek measurements and he realizes that there is a ratio a relationship between these because it doesn't really matter how many units you call this it can be inches or feet or quatloos or whatever you want to call them it doesn't really matter the point is the relationship or the ratio between them you learn this as fractions if you have 24 compared to 12 what does that reduce to 2 over 1 and if you have 24 compared to 16 does that reduce to find the greatest common factor 3 over 2 and if you have 24 over 18 what does that reduce to 4 over 3 these are all very simple fractions they're easily reduced you can find that there is a common number a common length that will go into both of these evenly and simplify as showing that they have a simple relationship this is called a ratio you call them fractions in school and Pythagoras realized that there is something uncanny about the way that strings that have simple ratios to each other always sound harmonic but if you had some other length of string that violated this ratio if you had something that was say you know 17 and 1/3 inches long and you play those together it's not going to make harmony it's gonna sound nasty and it's gonna make an experience called dissonance dissonance is when two tones make you frown because they sound bad together now to demonstrate this for the people that didn't have piano lessons or guitar class I brought in mr. guitar the Greeks had always had this sort of mystical idea about music they knew that there's something mysterious about the way a song can just change your mood influenced the the feelings in your soul they didn't know why but when you play what we would call today major chords sounds uplifting it's not triumphant but change one little tone in that chord make what we call a minor chord how does that change the mood comes somber him sadder it kind of kind of brings you down so they gave their own names to these different modes and there were what we would call scales or keys and if you were wanting something that was uplifting and proper and triumph of the state you'd blame the Dorian or Ionian mode if you wanted some kind of wild you know dancing and that ecstatic on a rock-and-roll feeling you know if you're at the festival of Dionysus you play songs in the Phrygian mode since people into wild madness and stuff they even had a God for music the god Orpheus as a sort of a demigod who had the the mystical power of music and he played so beautifully that you know wild animals laid tame at his feet and stones would weep and so forth so they've always had this mystical association and just like everything else in their world that was described through mythology the nature philosophers are trying to explain it rationally and so Pythagoras finds a rational explanation for these different things ok so a length of string on a guitar from the bridge to the nut is typically 24 inches so here is 24 inches of string playing ok this is called the octave fret when I push down right here that little piece of metal connects right there and that is exactly at the 12 inch point from end to end so I push it here and I can do what he does on the mono chord and play half the length of string 24 inches 12 now I need to play them at the same time that you can hear the harmony the tone that I play here I can also play right there here it's the same tone right so now I can play them both at the same time let's see how it sounds it's pleasant enough right now if I drop it or raise it a little bit and make it not quite evenly divisible see your eyebrows scrunching up that sounds nasty or a little bit too high so your stomach just curdling up your stomach relax okay so you're responding to harmony and dissonance and most of you are not music majors you didn't have extensive music lessons and yet it didn't matter you naturally respond to harmony and dissonance in similar ways it seems to be built into you in what Pythagoras really found here is that it is actual evidence that he was on the right track that ratios and proportions were really at the basis of the structure of the universe ratio is his answer to that question what is the essence of being and it's not something material like water or air or theoretical like a piron's it was the observation of mathematical ratios proportions ratio is at the base of it all and if this vibrating length of catgut is obeying precise mathematical laws in creating harmony then it stands to reason that everything in the universe is behaving according to these mathematical laws it's has a ratio and it sounds beautiful therefore it sounds beautiful because of the ratios and everything in the universe must be this way it doesn't look like that at a glance you've grown up with the idea that math is you know the queen of the sciences the central science and all that but when you just look out at nature and you see trees and rocks and mountains and river you don't see straight lines and 90-degree angles and you know perfect geometrical forms you see a lot of wild chaotic looking stuff so this idea that these mathematical concepts lie at the basis of all the things we observed was fairly radical thinking in his day we've just had twenty five centuries they get used to the idea and it continues with other ratios as well this was the halfway point between the two this is the two thirds of the way point between the two and this is the 3/4 point between the two in these tones always sound good together this tone is also there the stone is here any combination of those I make always sounds good so there was a mathematical structure of that lies underneath beauty and it is something that is still with us today Pythagoras is the one that divided our musical scale in the western world into the seven tones that we still have it right we have ABCD efg there's no key of H right after G we start over so we have a division into seven tones and then the process starts over again that's also Pythagoras is work in the same divisions that he based ancient Greek music on is what we still base it on today we use modern terminology for music derive from Yan Sebastian Bach who improved a little bit tweaked the system but the same basic structure is there you still listen to it in popular music that you hear every day this basic progression one four five is at the basis of half the songs you've ever heard on the radio in your life that progression can also be played and you can slide up and down the neck and play it anywhere see if y'all know the song it's a little before your time I know right a little before your time fine let's go to the 60 same chord same 1 4 5 division Wow thing yeah you got it ok good ok still a little before your time early eighties we're number one get nice truck and I'm so strong help not working either okay all right little Violent Femmes fans in the backyard excellent okay I need something that's gonna be universally understandable and enjoyable to you all got it I'm gonna have to switch to minor seventh chords here but the progression is still the same it's gonna be spider-man spider-man does whatever a spider can spins a web any size yeah just thieves just like my eyes look out it comes spider-man yeah everybody okay all right so what Pythagoras has demonstrated what he's done that is quite rare for these nature philosophers is that he actually has found evidence that his theories are on the right track as you recall most of these theories of the nature philosophers the early scientific theories were intuitive based on the observations of the behavior of nature and then abstracting and extrapolating from that theories that might explain it what they rarely ever did in the ancient world is get their hands dirty with experimentation and try to demonstrate that there is all right that's going to wait till the Scientific Revolution so Pythagoras is one of the very very few people in the ancient world that actually found physical evidence that he was on the right track again if vibrating strings are obeying mathematical laws probably everything is obeying mathematical laws and there are two major ways two major traditions that spring from this from this influence one is in the sciences since Pythagoras was able to demonstrate this carrying on bailey's ideas of the development of abstract math is a body of interrelate theorems from pythagoras on everybody's been pretty convinced in the Western world that math is the key to understanding the universe we describe math today as the universal language of science scientists in different fields chemistry astronomy they can all communicate with each other because they all speak the same mathematical language by which we describe the behavior of the universe but the other influence of pythagoras is in the arts because he found that wood links of string have simple ratios they sound beautiful to our ears it stands to reason that those same ratios would look beautiful to the eyes so greek architecture tends to follow this tradition and you'll find that the length to the width of their temples and the ratio of the height to the width etc are also rooted in these simple pythagorean harmonies these simple ratios in their statues will look at sculpture soon and we'll see that their statuary also embodies a sense of harmony and balance and proportion as they depict the human form in an idealized way not how very many humans actually look but according to how they would ideally look if they were the perfect representations of the human form the sculpture in the architecture especially takes imparts these Pythagorean mathematical qualities and so Pythagoras was essentially on the verge of showing that the world was rational and the word rational is literally rooted in the word ratio a rational world is measurable a rational world is appealing to the senses and the qualities that Pythagoras imparted to the ideas of nature are still with us today even though we've largely forgotten what these words really technically mean so Pythagoras was pretty excited about all this and then came a problem in measuring just the simplest of mathematical forms he found that there were some things that were really hard to reduce to a ratio if you take a square and then take the length of its diagonal and try to figure out the ratio between the diagonal to the side of a square in other words what unit will evenly divide into each of them you can ever figure it out no matter how small you divide out no matter how microscopic you make those links of string you cannot find one unit that will go into both of those you cannot show that they are commensurate they have a ratio between them in the same thing with the circle the diameter to the circumference of a circle is also something that you can't reduce and find a basic unit that's going to fit into both very frustrating the Greeks eventually developed a shorthand we still use today how do we express the incommensurability of the circumference to the diameter pi and what does pi equal 3.14159265 while y'all must have done a lot of geometry to have reduce it down to something that precise when you hit that on your calculator that's what pops out right okay all right well this number goes on forever and it never terminates it never repeats it doesn't make a pattern it just keeps generating number after number expressing the fact that matter how tiny you make it that will never go into both of these these kinds of relationships what are they called what kind of a number is that that is in ear rational number and for Pythagoras this is something that is just philosophically repugnant to him he was right on the verge of demonstrating that the world is rational it's beautiful it's harmonic and then he finds that these simplest of forms have these irrational relationships within them they cannot be expressed there's a legend that there was a member of the school who were all told don't tell anybody about this this was a secret of the school they didn't want people realizing that there were flaws in this beautiful mystical rational system and somebody was blabbing about it too much publicly and according to legend they took him out on a boat ride and drowned him and got rid of this guy do not spill our mathematical secrets and scandals Pythagoras is going to be the mathematical thinker that will have the long term influence from the ancient world his colleague Parmenides took a completely different tack on this per minute e's decided following the ideas that everything that our senses tell us is untrustworthy water looks blue in the ocean it's clear when we scoop it up all the things that our senses lie to us about and decided that only rational thought could be trusted and that the world itself was all just illusion and he had a student named Zeno that tried to prove his master's theories with a series of paradoxes the most famous of which is the idea of Achilles in The Tortoise kilise famous for being a runner racing a tortoise the tortoise gets a head start and the idea is that if the tortoise gets a head start on the way to the goal Achilles can never catch him because by the time Achilles gets here the tortoise will have gone a little further by the time Achilles catches up to where the tortoise is now he'll have gone a little further and so on so Achilles can never catch up to the tortoise he's always catching up to where he was so that's also a mathematical form of reasoning but I guarantee if me and Reggie race and even if he gives me a head start he's eventually gonna blow right by me and get to the finish line my senses are gonna tell me that he races right by me but mathematically I've just shown that he can never catch up to me so should we trust the rational mind or the illusory world of Sense experience Parmenides argue only the mind can be trusted and everything that you see is most likely an illusion and I see some of you got the point of the question what movie did he think he lived in the matrix basically Parmenides thought that everything in the physical world is illusion being sent to our senses in some way and that none of it was trustworthy what he doesn't realize is that space is not infinitely divisible and eventually he's going to make that distance and pass right by him but this was the world of mathematics remember you have grown up with this you're used to the idea that math explains all the world around us they were just trying to wrestle with this 25 centuries ago in classical Greece and pythagoras to his credit found the one solid piece of evidence that of all those nature philosophers his theories were in fact on the right track he will continue to influence us in other ways combining all of these things together music mathematics and astronomy Pythagoras presents a picture of the world that lasted for a couple of thousand years he taught that the earth was round and in the center of the universe is everybody almost everybody thought the ancient world in that the planets went in perfect circles around us and if you're looking at the universe as if the earth is the center point of it it looks like this would be the order of them the moon Mercury Venus the Sun strange to us imagine the Sun in the middle of all this Mars Jupiter and Saturn boom Mercury Venus Sun Mars Jupiter Saturn with an outer sphere of stars that tied it all together when Pythagoras realized that between the earth and the stars which all keep their same relative positions that there were only seven lights up there that were moving independently and seemingly intelligently he decided that seven must be a special number and that's why your musical scale got divided into seven tones he believed that each of these heavenly bodies was sounding one of those tones as they moved through the sky and if you could go up into the heavens and listen to him it would make a beautiful celestial symphony called the harmony of the spheres an idea that has influenced music for many many centuries also particularly up in the Renaissance and Baroque period they were trying to recapture something of the harmony of the spheres when they composed music these seven heavenly bodies all of these would have been called planets we'd of course call that a star and that a satellite today but all of these wonders in the sky gave us our division into seven the Babylonians who discovered the same thing decided to have a day of worship for each of these gods in the sky and they created the seven-day week which the Hebrews absorbed during Babylonian captivity and you still live with that today if you look at the heavenly bodies and look at the names of them you realize that there's a relationship between them in the days of the week there's your seven planets under their Latin names through which we still know them today and we realize that hey this one is still called Sunday and we still have moon day no way Monday and we still have Saturn's day don't we now these four in the middle aren't working so well in English but you Spanish speakers are recognizing the root words aren't you any French speakers out there let me speak Italian okay so in the romance languages that many of you speak and Romance language means it's derived from the Romans or from Latin the days are still there so what is the Spanish for Tuesday Martis and in French it's modern day right Mardi Gras mercury America Ellis I took French so for the French up here melody nice Jupiters other name is Jove which makes it closer with us and Venus thing is fooling no one reckless and Venus Venus and in French of Andra D so the French the Spanish all the names right from the Latin still carry these now what happened to these four but in the middle is that when Christianity moved to the north of Europe and they began converting Germanic speaking people's one way of sort of easing people into the new religion is to take things that we're already familiar to them and borrow it and sort of Christianize it and that way it sort of eases their movement in there and so what we end up with are some equivalents the Germanic peoples have a God that's similar to Mars what's got them with Mars the God of War and they have a war God named Q a mercury the messenger of the gods is also known for his wisdom cleverness and who's the Viking God who is the all-knowing all father Odin also called Wilton Jupiter what's his weapon same guys Zeus the lightning bolt and who's the god of thunder and lightning for the Vikings he saw his cool movies right Thor in Venus is the goddess of love and fertility and the Vikings have a tiller the goddess named Freya and that's why in Germania Germanic speaking regions which includes English which is a dialect of German we end up with Tuesday weapons day Thursday and Friday Tuesday Wednesday Thursday Friday but they're still linked to the equivalent deities that take us back to the Latin names of these planets the seven-day week seven tones of the musical scale the whole idea of seven being this special lucky magical number the way we make lists of seven things seven wonders of the world seven virtues seven vices seven deadly sins this habit of ours of finding seven to be special is all rooted in the observation that of all those lights in the sky seven of them seem to be doing their own unique thing guided by their own intelligence there are seven special Wanderers up there in the sky looking down on us and so both the ancient Greeks with Pythagoras and Babylonians observe this have each contributed something to your modern understanding of the world so even when your name in the days of the week you're linking into ancient mystical astronomy and mathematical fascination of the nature philosophers

Channel: George Brooks

Views: 22,069

Rating: 4.9212599 out of 5

Keywords: Brooks, Humanities, Pythagoras, Valencia, Nature Philosophy

Id: AgCRcGfHSPc

Length: 37min 47sec (2267 seconds)

Published: Tue Jan 26 2016