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"The laws of nature are but the mathematical thoughts of God."

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And this is a quote by Euclid of Alexandria.

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He was a Greek mathematician and philosopher who lived about 300 years before Christ

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And the reason why I include this quote is because Euclid is considered to be the father of geometry.

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And it is a neat quote, regardless of your views of God.

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Whether or not God exists or the nature of God.

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It says something very fundamental about nature.

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The laws of nature are but the mathematical thoughts of God.

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That math underpins all of the laws of nature.

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And the word "geometry" itself has Greek roots.

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"Geo" comes from Greek for "Earth".

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"Metry" comes from Greek for "measurement".

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You're probably used to something like the "metric" system.

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And Euclid is considered to be the father of geometry.

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(not because he was the first person who studied geometry),

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you could imagine the very first humans might have studied geometry.

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They might have looked at two twigs on the ground that looked something like that.

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And they might have looked at another pair of twigs that looked like that.

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And said "This is a bigger opening. What is the relationship here?"

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Or they might have looked at a tree that had a branch that came off like that.

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And they said, "Well, there's something similar about this opening here and this opening here."

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Or they might have asked themselves,

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"What is the ratio or what is the relationship between the distance around a circle and the distance across it?

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And is that the same for all circles?

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And is there a way for us to feel really good that that is definitely true?"

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And then once you got to the early Greeks,

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they started to get even more thoughtful about geometric things.

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When you talk about Greek mathematicians like Pythagoras

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(who came before Euclid).

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The reason why people often talk about "Euclidean geometry" is around 300 B.C.

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(and this over here is a picture of Euclid painted by Raphael, and no one really knows what Euclid looked like

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or even when he was born or when he died, so this is just Raphael's impression of what Euclid might have looked like

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while he was teaching in Alexandria).

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But what made Euclid the "Father of Geometry" is really his writing of "Euclid's Elements".

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And, "Euclid's Elements" was essentially a 13-volume textbook

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(and arguably the most famous textbook of all time).

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And what he did in those thirteen volumes was a rigorous, thoughtful, logical march

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through geometry, number theory and solid geometry (geometry in three-dimensions).

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And this right over here is the frontispiece of the English version---

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or the first translation of the English version---of "Euclid's Elements".

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This was done in 1570.

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But it was obviously first written in Greek, and, during the Middle Ages,

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that knowledge was shepherded by the Arabs and it was translated into Arabic.

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And then eventually the late Middle Ages translated it into Latin and then eventually English.

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And when I say that he did a "rigiorous march", Euclid didn't just say,

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"the square of the length of two sides of a right triangle is going to be the same as the square of

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the hypotenuse..." and all these other things (and we'll go into depth about what all these mean).

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He says, "I don't want to feel good that it's probably true. I want to prove to myself that it's true."

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And what he did in "Elements" (especially the six volumes concerned with planar geometry),

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was he started with basic assumptions.

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And those basic assumptions in "geometric speak" are called "axioms" or "postulates".

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And from them he proved, he deduced other statements or "propositions" (these are sometimes called "theorems").

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And then he says, "Now, I know. If this is true and this is true, this must be true."

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And he could also prove that other things cannot be true.

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So then he could prove that this is not going to be the truth.

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He didn't just say, "Well, every circle I've sat in has this property."

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He said, "I've now proven that this is true".

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And then, from there, he could go and deduce other propositions or "theorems"

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(and we can use some of our original "axioms" to do that).

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And what's special about that is no one had really done that before.

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Rigorously proven beyond a shadow of a doubt across a whole, broad sweep of knowledge.

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So not just one proof here or there. He did that for an entire "set" of knowledge.

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A rigorous "march" through a subject so that he could build this scaffold of "axioms" and "postulates" and "theorems" and "propositions"

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(and theorems and propositions are essentially the same thing).

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And for about 2,000 years after Euclid (so this is an unbelievable shelf life for a textbook!),

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people didn't view you as educated if you had not read and understood Euclid's "Elements".

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And "Euclid's Elements" (the book itself) was the second-most printed book in the Western World

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after the Bible.

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This is a math textbook second only to the Bible.

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When the first printing presses came out they said "Okay, let's print the bible. What next?"

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"Let's print 'Euclid's Elements'".

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And to show that this is relevant into the fairly recent past (although it may depend whether or not you argue that

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150-160 years ago is a recent past),

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this right here is a direct quote from Abraham Lincoln (obviously one of the great

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American Presidents). I like this picture of Abraham Lincoln.

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This is actually a photograph of Lincoln in his late-30s.

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But he was a huge fan of "Euclid's Elements". He would actually use it to "fine-tune" his mind.

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While he was riding his horse he would read "Euclid's Elements". While he was in the

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White House he would read "Euclid's Elements".

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But this is a direct quote from Lincoln,

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"In the course of my law reading, I constantly came upon the word 'demonstrate'.

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I thought at first that I understood its meaning, but soon became satisfied that I did not.

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I said to myself, what do I do when I demonstrate more than when I reason or prove?

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How does 'demonstration' differ from any other proof..."

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So, Lincoln is saying there is this word "demonstration" that means proving beyond doubt.

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Something more rigorous---more than just simple feeling good about something or reasoning through it.

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"...I consulted Webster's Dictionary..." (so Webster's dictionary was around even in Lincoln's era)

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"...they told of certain proof---proof beyond the possibility of doubt. But I could

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form no idea of what sort of proof that was. I thought a great many things were proven beyond

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the possibility of doubt without recourse to any such extraordinary process of reasoning

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as I understood 'demonstration' to be.

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I consulted all the dictionaries and books of reference I could find but with no better results.

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You might as well have defined 'blue' to a blind-man.

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At last I said, 'Lincoln, you never can make a lawyer if you do not understand what 'demonstrate' means.

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And I left my situation in Springfield, went home to my father's house, and stayed there until

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I could give any proposition in the six books of Euclid at sight."

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(This refers to the six books concerned with planar geometry.)

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"...I then found out what 'demonstrate' means and went back to my law study."

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So one of the greatest American Presidents of all time felt that, in order to be a great lawyer,

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he had to understood---be able to prove any proposition in the six books of "Euclid's Elements"

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at sight. And also, once he was in the White House he continued to do this to "fine-tune" his mind

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to become a great President.

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And so, what we're going to be doing in the geometry playlist is essentially that.

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What we're going to study---we're going to think about how do we "rigorously" prove things?

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We're essentially going to be---in a more modern form---studying what Euclid studied 2,300 years ago.

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To really tighten our reasoning of different statements and be sure that when we say something,

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we can really prove what we're saying.

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This is really some of the most fundamental, "real" mathematics that you will do.

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Arithmetic was really just computation.

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Now, in geometry, (and what we'll be doing is Euclidean geometry)

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this is really what math is about.

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Making some assumptions and then deducing other things from those assumptions.