WEBVTT 00:00:00.789 --> 00:00:04.600 "The laws of nature are but the mathematical thoughts of God." 00:00:04.862 --> 00:00:07.523 And this is a quote by Euclid of Alexandria. 00:00:07.523 --> 00:00:12.655 He was a Greek mathematician and philosopher who lived about 300 years before Christ 00:00:12.655 --> 00:00:19.691 And the reason why I include this quote is because Euclid is considered to be the father of geometry. 00:00:19.691 --> 00:00:22.663 And it is a neat quote, regardless of your views of God. 00:00:22.663 --> 00:00:25.054 Whether or not God exists or the nature of God. 00:00:25.054 --> 00:00:27.516 It says something very fundamental about nature. 00:00:27.516 --> 00:00:31.649 The laws of nature are but the mathematical thoughts of God. 00:00:31.649 --> 00:00:35.016 That math underpins all of the laws of nature. 00:00:35.016 --> 00:00:37.802 And the word "geometry" itself has Greek roots. 00:00:37.802 --> 00:00:40.983 "Geo" comes from Greek for "Earth". 00:00:40.983 --> 00:00:44.211 "Metry" comes from Greek for "measurement". 00:00:44.211 --> 00:00:47.183 You're probably used to something like the "metric" system. 00:00:47.183 --> 00:00:50.132 And Euclid is considered to be the father of geometry. 00:00:50.132 --> 00:00:52.802 (not because he was the first person who studied geometry), 00:00:52.802 --> 00:00:56.285 you could imagine the very first humans might have studied geometry. 00:00:56.562 --> 00:01:00.024 They might have looked at two twigs on the ground that looked something like that. 00:01:00.024 --> 00:01:02.462 And they might have looked at another pair of twigs that looked like that. 00:01:02.462 --> 00:01:05.178 And said "This is a bigger opening. What is the relationship here?" 00:01:05.178 --> 00:01:13.654 Or they might have looked at a tree that had a branch that came off like that. 00:01:13.654 --> 00:01:18.274 And they said, "Well, there's something similar about this opening here and this opening here." 00:01:18.274 --> 00:01:19.737 Or they might have asked themselves, 00:01:19.737 --> 00:01:26.123 "What is the ratio or what is the relationship between the distance around a circle and the distance across it? 00:01:26.123 --> 00:01:28.352 And is that the same for all circles? 00:01:28.352 --> 00:01:31.812 And is there a way for us to feel really good that that is definitely true?" 00:01:31.812 --> 00:01:34.412 And then once you got to the early Greeks, 00:01:34.412 --> 00:01:39.010 they started to get even more thoughtful about geometric things. 00:01:39.010 --> 00:01:43.259 When you talk about Greek mathematicians like Pythagoras 00:01:43.259 --> 00:01:45.535 (who came before Euclid). 00:01:45.535 --> 00:01:54.511 The reason why people often talk about "Euclidean geometry" is around 300 B.C. 00:01:54.511 --> 00:01:59.832 (and this over here is a picture of Euclid painted by Raphael, and no one really knows what Euclid looked like 00:01:59.832 --> 00:02:05.793 or even when he was born or when he died, so this is just Raphael's impression of what Euclid might have looked like 00:02:05.793 --> 00:02:08.383 while he was teaching in Alexandria). 00:02:08.383 --> 00:02:14.397 But what made Euclid the "Father of Geometry" is really his writing of "Euclid's Elements". 00:02:14.397 --> 00:02:21.263 And, "Euclid's Elements" was essentially a 13-volume textbook 00:02:21.263 --> 00:02:24.773 (and arguably the most famous textbook of all time). 00:02:24.773 --> 00:02:31.441 And what he did in those thirteen volumes was a rigorous, thoughtful, logical march 00:02:31.441 --> 00:02:37.524 through geometry, number theory and solid geometry (geometry in three-dimensions). 00:02:37.524 --> 00:02:40.682 And this right over here is the frontispiece of the English version--- 00:02:40.682 --> 00:02:44.955 or the first translation of the English version---of "Euclid's Elements". 00:02:44.955 --> 00:02:47.532 This was done in 1570. 00:02:47.532 --> 00:02:51.851 But it was obviously first written in Greek, and, during the Middle Ages, 00:02:51.851 --> 00:02:55.334 that knowledge was shepherded by the Arabs and it was translated into Arabic. 00:02:55.334 --> 00:03:02.393 And then eventually the late Middle Ages translated it into Latin and then eventually English. 00:03:02.393 --> 00:03:05.806 And when I say that he did a "rigiorous march", Euclid didn't just say, 00:03:05.806 --> 00:03:14.374 "the square of the length of two sides of a right triangle is going to be the same as the square of 00:03:14.374 --> 00:03:18.182 the hypotenuse..." and all these other things (and we'll go into depth about what all these mean). 00:03:18.182 --> 00:03:24.475 He says, "I don't want to feel good that it's probably true. I want to prove to myself that it's true." 00:03:24.475 --> 00:03:29.723 And what he did in "Elements" (especially the six volumes concerned with planar geometry), 00:03:33.215 --> 00:03:37.721 was he started with basic assumptions. 00:03:37.721 --> 00:03:43.747 And those basic assumptions in "geometric speak" are called "axioms" or "postulates". 00:03:43.747 --> 00:03:51.549 And from them he proved, he deduced other statements or "propositions" (these are sometimes called "theorems"). 00:03:51.549 --> 00:03:55.729 And then he says, "Now, I know. If this is true and this is true, this must be true." 00:03:55.729 --> 00:03:58.492 And he could also prove that other things cannot be true. 00:03:58.492 --> 00:04:01.255 So then he could prove that this is not going to be the truth. 00:04:01.255 --> 00:04:04.042 He didn't just say, "Well, every circle I've sat in has this property." 00:04:04.042 --> 00:04:06.155 He said, "I've now proven that this is true". 00:04:06.155 --> 00:04:11.402 And then, from there, he could go and deduce other propositions or "theorems" 00:04:11.402 --> 00:04:14.096 (and we can use some of our original "axioms" to do that). 00:04:14.096 --> 00:04:17.068 And what's special about that is no one had really done that before. 00:04:17.068 --> 00:04:23.477 Rigorously proven beyond a shadow of a doubt across a whole, broad sweep of knowledge. 00:04:23.477 --> 00:04:30.095 So not just one proof here or there. He did that for an entire "set" of knowledge. 00:04:30.884 --> 00:04:39.692 A rigorous "march" through a subject so that he could build this scaffold of "axioms" and "postulates" and "theorems" and "propositions" 00:04:39.692 --> 00:04:42.022 (and theorems and propositions are essentially the same thing). 00:04:43.069 --> 00:04:47.881 And for about 2,000 years after Euclid (so this is an unbelievable shelf life for a textbook!), 00:04:47.881 --> 00:04:55.427 people didn't view you as educated if you had not read and understood Euclid's "Elements". 00:04:55.427 --> 00:04:59.862 And "Euclid's Elements" (the book itself) was the second-most printed book in the Western World 00:04:59.862 --> 00:05:01.581 after the Bible. 00:05:01.581 --> 00:05:04.344 This is a math textbook second only to the Bible. 00:05:04.344 --> 00:05:07.943 When the first printing presses came out they said "Okay, let's print the bible. What next?" 00:05:07.943 --> 00:05:09.940 "Let's print 'Euclid's Elements'". 00:05:10.525 --> 00:05:16.606 And to show that this is relevant into the fairly recent past (although it may depend whether or not you argue that 00:05:16.606 --> 00:05:19.416 150-160 years ago is a recent past), 00:05:19.816 --> 00:05:23.779 this right here is a direct quote from Abraham Lincoln (obviously one of the great 00:05:23.779 --> 00:05:26.612 American Presidents). I like this picture of Abraham Lincoln. 00:05:26.612 --> 00:05:29.747 This is actually a photograph of Lincoln in his late-30s. 00:05:29.747 --> 00:05:35.900 But he was a huge fan of "Euclid's Elements". He would actually use it to "fine-tune" his mind. 00:05:35.900 --> 00:05:38.872 While he was riding his horse he would read "Euclid's Elements". While he was in the 00:05:38.872 --> 00:05:40.777 White House he would read "Euclid's Elements". 00:05:41.207 --> 00:05:43.795 But this is a direct quote from Lincoln, 00:05:43.795 --> 00:05:48.415 "In the course of my law reading, I constantly came upon the word 'demonstrate'. 00:05:48.415 --> 00:05:53.454 I thought at first that I understood its meaning, but soon became satisfied that I did not. 00:05:53.454 --> 00:05:59.375 I said to myself, what do I do when I demonstrate more than when I reason or prove? 00:05:59.375 --> 00:06:02.580 How does 'demonstration' differ from any other proof..." 00:06:02.580 --> 00:06:08.454 So, Lincoln is saying there is this word "demonstration" that means proving beyond doubt. 00:06:08.454 --> 00:06:13.307 Something more rigorous---more than just simple feeling good about something or reasoning through it. 00:06:13.307 --> 00:06:17.998 "...I consulted Webster's Dictionary..." (so Webster's dictionary was around even in Lincoln's era) 00:06:17.998 --> 00:06:23.060 "...they told of certain proof---proof beyond the possibility of doubt. But I could 00:06:23.060 --> 00:06:28.005 form no idea of what sort of proof that was. I thought a great many things were proven beyond 00:06:28.005 --> 00:06:32.649 the possibility of doubt without recourse to any such extraordinary process of reasoning 00:06:32.649 --> 00:06:35.668 as I understood 'demonstration' to be. 00:06:35.668 --> 00:06:41.241 I consulted all the dictionaries and books of reference I could find but with no better results. 00:06:41.241 --> 00:06:45.676 You might as well have defined 'blue' to a blind-man. 00:06:45.676 --> 00:06:55.150 At last I said, 'Lincoln, you never can make a lawyer if you do not understand what 'demonstrate' means. 00:06:55.150 --> 00:07:00.467 And I left my situation in Springfield, went home to my father's house, and stayed there until 00:07:00.467 --> 00:07:04.345 I could give any proposition in the six books of Euclid at sight." 00:07:04.345 --> 00:07:06.806 (This refers to the six books concerned with planar geometry.) 00:07:06.806 --> 00:07:11.868 "...I then found out what 'demonstrate' means and went back to my law study." 00:07:11.868 --> 00:07:17.348 So one of the greatest American Presidents of all time felt that, in order to be a great lawyer, 00:07:17.348 --> 00:07:24.128 he had to understood---be able to prove any proposition in the six books of "Euclid's Elements" 00:07:24.128 --> 00:07:30.885 at sight. And also, once he was in the White House he continued to do this to "fine-tune" his mind 00:07:30.885 --> 00:07:32.954 to become a great President. 00:07:33.447 --> 00:07:36.922 And so, what we're going to be doing in the geometry playlist is essentially that. 00:07:36.922 --> 00:07:42.806 What we're going to study---we're going to think about how do we "rigorously" prove things? 00:07:42.868 --> 00:07:49.624 We're essentially going to be---in a more modern form---studying what Euclid studied 2,300 years ago. 00:07:49.624 --> 00:07:59.812 To really tighten our reasoning of different statements and be sure that when we say something, 00:07:59.812 --> 00:08:01.972 we can really prove what we're saying. 00:08:01.972 --> 00:08:06.388 This is really some of the most fundamental, "real" mathematics that you will do. 00:08:06.388 --> 00:08:08.525 Arithmetic was really just computation. 00:08:08.525 --> 00:08:12.820 Now, in geometry, (and what we'll be doing is Euclidean geometry) 00:08:12.820 --> 00:08:17.000 this is really what math is about. 00:08:17.000 --> 00:08:21.388 Making some assumptions and then deducing other things from those assumptions.