[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.79,0:00:04.60,Default,,0000,0000,0000,,"The laws of nature are but the mathematical thoughts of God."
Dialogue: 0,0:00:04.86,0:00:07.52,Default,,0000,0000,0000,,And this is a quote by Euclid of Alexandria.
Dialogue: 0,0:00:07.52,0:00:12.66,Default,,0000,0000,0000,,He was a Greek mathematician and philosopher who lived about 300 years before Christ
Dialogue: 0,0:00:12.66,0:00:19.69,Default,,0000,0000,0000,,And the reason why I include this quote is because Euclid is considered to be the father of geometry.
Dialogue: 0,0:00:19.69,0:00:22.66,Default,,0000,0000,0000,,And it is a neat quote, regardless of your views of God.
Dialogue: 0,0:00:22.66,0:00:25.05,Default,,0000,0000,0000,,Whether or not God exists or the nature of God.
Dialogue: 0,0:00:25.05,0:00:27.52,Default,,0000,0000,0000,,It says something very fundamental about nature.
Dialogue: 0,0:00:27.52,0:00:31.65,Default,,0000,0000,0000,,The laws of nature are but the mathematical thoughts of God.
Dialogue: 0,0:00:31.65,0:00:35.02,Default,,0000,0000,0000,,That math underpins all of the laws of nature.
Dialogue: 0,0:00:35.02,0:00:37.80,Default,,0000,0000,0000,,And the word "geometry" itself has Greek roots.
Dialogue: 0,0:00:37.80,0:00:40.98,Default,,0000,0000,0000,,"Geo" comes from Greek for "Earth".
Dialogue: 0,0:00:40.98,0:00:44.21,Default,,0000,0000,0000,,"Metry" comes from Greek for "measurement".
Dialogue: 0,0:00:44.21,0:00:47.18,Default,,0000,0000,0000,,You're probably used to something like the "metric" system.
Dialogue: 0,0:00:47.18,0:00:50.13,Default,,0000,0000,0000,,And Euclid is considered to be the father of geometry.
Dialogue: 0,0:00:50.13,0:00:52.80,Default,,0000,0000,0000,,(not because he was the first person who studied geometry),
Dialogue: 0,0:00:52.80,0:00:56.28,Default,,0000,0000,0000,,you could imagine the very first humans might have studied geometry.
Dialogue: 0,0:00:56.56,0:01:00.02,Default,,0000,0000,0000,,They might have looked at two twigs on the ground that looked something like that.
Dialogue: 0,0:01:00.02,0:01:02.46,Default,,0000,0000,0000,,And they might have looked at another pair of twigs that looked like that.
Dialogue: 0,0:01:02.46,0:01:05.18,Default,,0000,0000,0000,,And said "This is a bigger opening. What is the relationship here?"
Dialogue: 0,0:01:05.18,0:01:13.65,Default,,0000,0000,0000,,Or they might have looked at a tree that had a branch that came off like that.
Dialogue: 0,0:01:13.65,0:01:18.27,Default,,0000,0000,0000,,And they said, "Well, there's something similar about this opening here and this opening here."
Dialogue: 0,0:01:18.27,0:01:19.74,Default,,0000,0000,0000,,Or they might have asked themselves,
Dialogue: 0,0:01:19.74,0:01:26.12,Default,,0000,0000,0000,,"What is the ratio or what is the relationship between the distance around a circle and the distance across it?
Dialogue: 0,0:01:26.12,0:01:28.35,Default,,0000,0000,0000,,And is that the same for all circles?
Dialogue: 0,0:01:28.35,0:01:31.81,Default,,0000,0000,0000,,And is there a way for us to feel really good that that is definitely true?"
Dialogue: 0,0:01:31.81,0:01:34.41,Default,,0000,0000,0000,,And then once you got to the early Greeks,
Dialogue: 0,0:01:34.41,0:01:39.01,Default,,0000,0000,0000,,they started to get even more thoughtful about geometric things.
Dialogue: 0,0:01:39.01,0:01:43.26,Default,,0000,0000,0000,,When you talk about Greek mathematicians like Pythagoras
Dialogue: 0,0:01:43.26,0:01:45.54,Default,,0000,0000,0000,,(who came before Euclid).
Dialogue: 0,0:01:45.54,0:01:54.51,Default,,0000,0000,0000,,The reason why people often talk about "Euclidean geometry" is around 300 B.C.
Dialogue: 0,0:01:54.51,0:01:59.83,Default,,0000,0000,0000,,(and this over here is a picture of Euclid painted by Raphael, and no one really knows what Euclid looked like
Dialogue: 0,0:01:59.83,0:02:05.79,Default,,0000,0000,0000,,or even when he was born or when he died, so this is just Raphael's impression of what Euclid might have looked like
Dialogue: 0,0:02:05.79,0:02:08.38,Default,,0000,0000,0000,,while he was teaching in Alexandria).
Dialogue: 0,0:02:08.38,0:02:14.40,Default,,0000,0000,0000,,But what made Euclid the "Father of Geometry" is really his writing of "Euclid's Elements".
Dialogue: 0,0:02:14.40,0:02:21.26,Default,,0000,0000,0000,,And, "Euclid's Elements" was essentially a 13-volume textbook
Dialogue: 0,0:02:21.26,0:02:24.77,Default,,0000,0000,0000,,(and arguably the most famous textbook of all time).
Dialogue: 0,0:02:24.77,0:02:31.44,Default,,0000,0000,0000,,And what he did in those thirteen volumes was a rigorous, thoughtful, logical march
Dialogue: 0,0:02:31.44,0:02:37.52,Default,,0000,0000,0000,,through geometry, number theory and solid geometry (geometry in three-dimensions).
Dialogue: 0,0:02:37.52,0:02:40.68,Default,,0000,0000,0000,,And this right over here is the frontispiece of the English version---
Dialogue: 0,0:02:40.68,0:02:44.96,Default,,0000,0000,0000,,or the first translation of the English version---of "Euclid's Elements".
Dialogue: 0,0:02:44.96,0:02:47.53,Default,,0000,0000,0000,,This was done in 1570.
Dialogue: 0,0:02:47.53,0:02:51.85,Default,,0000,0000,0000,,But it was obviously first written in Greek, and, during the Middle Ages,
Dialogue: 0,0:02:51.85,0:02:55.33,Default,,0000,0000,0000,,that knowledge was shepherded by the Arabs and it was translated into Arabic.
Dialogue: 0,0:02:55.33,0:03:02.39,Default,,0000,0000,0000,,And then eventually the late Middle Ages translated it into Latin and then eventually English.
Dialogue: 0,0:03:02.39,0:03:05.81,Default,,0000,0000,0000,,And when I say that he did a "rigiorous march", Euclid didn't just say,
Dialogue: 0,0:03:05.81,0:03:14.37,Default,,0000,0000,0000,,"the square of the length of two sides of a right triangle is going to be the same as the square of
Dialogue: 0,0:03:14.37,0:03:18.18,Default,,0000,0000,0000,,the hypotenuse..." and all these other things (and we'll go into depth about what all these mean).
Dialogue: 0,0:03:18.18,0:03:24.48,Default,,0000,0000,0000,,He says, "I don't want to feel good that it's probably true. I want to prove to myself that it's true."
Dialogue: 0,0:03:24.48,0:03:29.72,Default,,0000,0000,0000,,And what he did in "Elements" (especially the six volumes concerned with planar geometry),
Dialogue: 0,0:03:33.22,0:03:37.72,Default,,0000,0000,0000,,was he started with basic assumptions.
Dialogue: 0,0:03:37.72,0:03:43.75,Default,,0000,0000,0000,,And those basic assumptions in "geometric speak" are called "axioms" or "postulates".
Dialogue: 0,0:03:43.75,0:03:51.55,Default,,0000,0000,0000,,And from them he proved, he deduced other statements or "propositions" (these are sometimes called "theorems").
Dialogue: 0,0:03:51.55,0:03:55.73,Default,,0000,0000,0000,,And then he says, "Now, I know. If this is true and this is true, this must be true."
Dialogue: 0,0:03:55.73,0:03:58.49,Default,,0000,0000,0000,,And he could also prove that other things cannot be true.
Dialogue: 0,0:03:58.49,0:04:01.26,Default,,0000,0000,0000,,So then he could prove that this is not going to be the truth.
Dialogue: 0,0:04:01.26,0:04:04.04,Default,,0000,0000,0000,,He didn't just say, "Well, every circle I've sat in has this property."
Dialogue: 0,0:04:04.04,0:04:06.16,Default,,0000,0000,0000,,He said, "I've now proven that this is true".
Dialogue: 0,0:04:06.16,0:04:11.40,Default,,0000,0000,0000,,And then, from there, he could go and deduce other propositions or "theorems"
Dialogue: 0,0:04:11.40,0:04:14.10,Default,,0000,0000,0000,,(and we can use some of our original "axioms" to do that).
Dialogue: 0,0:04:14.10,0:04:17.07,Default,,0000,0000,0000,,And what's special about that is no one had really done that before.
Dialogue: 0,0:04:17.07,0:04:23.48,Default,,0000,0000,0000,,Rigorously proven beyond a shadow of a doubt across a whole, broad sweep of knowledge.
Dialogue: 0,0:04:23.48,0:04:30.10,Default,,0000,0000,0000,,So not just one proof here or there. He did that for an entire "set" of knowledge.
Dialogue: 0,0:04:30.88,0:04:39.69,Default,,0000,0000,0000,,A rigorous "march" through a subject so that he could build this scaffold of "axioms" and "postulates" and "theorems" and "propositions"
Dialogue: 0,0:04:39.69,0:04:42.02,Default,,0000,0000,0000,,(and theorems and propositions are essentially the same thing).
Dialogue: 0,0:04:43.07,0:04:47.88,Default,,0000,0000,0000,,And for about 2,000 years after Euclid (so this is an unbelievable shelf life for a textbook!),
Dialogue: 0,0:04:47.88,0:04:55.43,Default,,0000,0000,0000,,people didn't view you as educated if you had not read and understood Euclid's "Elements".
Dialogue: 0,0:04:55.43,0:04:59.86,Default,,0000,0000,0000,,And "Euclid's Elements" (the book itself) was the second-most printed book in the Western World
Dialogue: 0,0:04:59.86,0:05:01.58,Default,,0000,0000,0000,,after the Bible.
Dialogue: 0,0:05:01.58,0:05:04.34,Default,,0000,0000,0000,,This is a math textbook second only to the Bible.
Dialogue: 0,0:05:04.34,0:05:07.94,Default,,0000,0000,0000,,When the first printing presses came out they said "Okay, let's print the bible. What next?"
Dialogue: 0,0:05:07.94,0:05:09.94,Default,,0000,0000,0000,,"Let's print 'Euclid's Elements'".
Dialogue: 0,0:05:10.52,0:05:16.61,Default,,0000,0000,0000,,And to show that this is relevant into the fairly recent past (although it may depend whether or not you argue that
Dialogue: 0,0:05:16.61,0:05:19.42,Default,,0000,0000,0000,,150-160 years ago is a recent past),
Dialogue: 0,0:05:19.82,0:05:23.78,Default,,0000,0000,0000,,this right here is a direct quote from Abraham Lincoln (obviously one of the great
Dialogue: 0,0:05:23.78,0:05:26.61,Default,,0000,0000,0000,,American Presidents). I like this picture of Abraham Lincoln.
Dialogue: 0,0:05:26.61,0:05:29.75,Default,,0000,0000,0000,,This is actually a photograph of Lincoln in his late-30s.
Dialogue: 0,0:05:29.75,0:05:35.90,Default,,0000,0000,0000,,But he was a huge fan of "Euclid's Elements". He would actually use it to "fine-tune" his mind.
Dialogue: 0,0:05:35.90,0:05:38.87,Default,,0000,0000,0000,,While he was riding his horse he would read "Euclid's Elements". While he was in the
Dialogue: 0,0:05:38.87,0:05:40.78,Default,,0000,0000,0000,,White House he would read "Euclid's Elements".
Dialogue: 0,0:05:41.21,0:05:43.80,Default,,0000,0000,0000,,But this is a direct quote from Lincoln,
Dialogue: 0,0:05:43.80,0:05:48.42,Default,,0000,0000,0000,,"In the course of my law reading, I constantly came upon the word 'demonstrate'.
Dialogue: 0,0:05:48.42,0:05:53.45,Default,,0000,0000,0000,,I thought at first that I understood its meaning, but soon became satisfied that I did not.
Dialogue: 0,0:05:53.45,0:05:59.38,Default,,0000,0000,0000,,I said to myself, what do I do when I demonstrate more than when I reason or prove?
Dialogue: 0,0:05:59.38,0:06:02.58,Default,,0000,0000,0000,,How does 'demonstration' differ from any other proof..."
Dialogue: 0,0:06:02.58,0:06:08.45,Default,,0000,0000,0000,,So, Lincoln is saying there is this word "demonstration" that means proving beyond doubt.
Dialogue: 0,0:06:08.45,0:06:13.31,Default,,0000,0000,0000,,Something more rigorous---more than just simple feeling good about something or reasoning through it.
Dialogue: 0,0:06:13.31,0:06:17.100,Default,,0000,0000,0000,,"...I consulted Webster's Dictionary..." (so Webster's dictionary was around even in Lincoln's era)
Dialogue: 0,0:06:17.100,0:06:23.06,Default,,0000,0000,0000,,"...they told of certain proof---proof beyond the possibility of doubt. But I could
Dialogue: 0,0:06:23.06,0:06:28.00,Default,,0000,0000,0000,,form no idea of what sort of proof that was. I thought a great many things were proven beyond
Dialogue: 0,0:06:28.00,0:06:32.65,Default,,0000,0000,0000,,the possibility of doubt without recourse to any such extraordinary process of reasoning
Dialogue: 0,0:06:32.65,0:06:35.67,Default,,0000,0000,0000,,as I understood 'demonstration' to be.
Dialogue: 0,0:06:35.67,0:06:41.24,Default,,0000,0000,0000,,I consulted all the dictionaries and books of reference I could find but with no better results.
Dialogue: 0,0:06:41.24,0:06:45.68,Default,,0000,0000,0000,,You might as well have defined 'blue' to a blind-man.
Dialogue: 0,0:06:45.68,0:06:55.15,Default,,0000,0000,0000,,At last I said, 'Lincoln, you never can make a lawyer if you do not understand what 'demonstrate' means.
Dialogue: 0,0:06:55.15,0:07:00.47,Default,,0000,0000,0000,,And I left my situation in Springfield, went home to my father's house, and stayed there until
Dialogue: 0,0:07:00.47,0:07:04.34,Default,,0000,0000,0000,,I could give any proposition in the six books of Euclid at sight."
Dialogue: 0,0:07:04.34,0:07:06.81,Default,,0000,0000,0000,,(This refers to the six books concerned with planar geometry.)
Dialogue: 0,0:07:06.81,0:07:11.87,Default,,0000,0000,0000,,"...I then found out what 'demonstrate' means and went back to my law study."
Dialogue: 0,0:07:11.87,0:07:17.35,Default,,0000,0000,0000,,So one of the greatest American Presidents of all time felt that, in order to be a great lawyer,
Dialogue: 0,0:07:17.35,0:07:24.13,Default,,0000,0000,0000,,he had to understood---be able to prove any proposition in the six books of "Euclid's Elements"
Dialogue: 0,0:07:24.13,0:07:30.88,Default,,0000,0000,0000,,at sight. And also, once he was in the White House he continued to do this to "fine-tune" his mind
Dialogue: 0,0:07:30.88,0:07:32.95,Default,,0000,0000,0000,,to become a great President.
Dialogue: 0,0:07:33.45,0:07:36.92,Default,,0000,0000,0000,,And so, what we're going to be doing in the geometry playlist is essentially that.
Dialogue: 0,0:07:36.92,0:07:42.81,Default,,0000,0000,0000,,What we're going to study---we're going to think about how do we "rigorously" prove things?
Dialogue: 0,0:07:42.87,0:07:49.62,Default,,0000,0000,0000,,We're essentially going to be---in a more modern form---studying what Euclid studied 2,300 years ago.
Dialogue: 0,0:07:49.62,0:07:59.81,Default,,0000,0000,0000,,To really tighten our reasoning of different statements and be sure that when we say something,
Dialogue: 0,0:07:59.81,0:08:01.97,Default,,0000,0000,0000,,we can really prove what we're saying.
Dialogue: 0,0:08:01.97,0:08:06.39,Default,,0000,0000,0000,,This is really some of the most fundamental, "real" mathematics that you will do.
Dialogue: 0,0:08:06.39,0:08:08.52,Default,,0000,0000,0000,,Arithmetic was really just computation.
Dialogue: 0,0:08:08.52,0:08:12.82,Default,,0000,0000,0000,,Now, in geometry, (and what we'll be doing is Euclidean geometry)
Dialogue: 0,0:08:12.82,0:08:17.00,Default,,0000,0000,0000,,this is really what math is about.
Dialogue: 0,0:08:17.00,0:08:21.39,Default,,0000,0000,0000,,Making some assumptions and then deducing other things from those assumptions.