Rabu, 19 April 2017

asking alexandria koko

asking alexandria koko

"the laws of nature are but themathematical thoughts of god." and this is a quote byeuclid of alexandria, who was a greek mathematicianand philosopher who lived about 300years before christ. and the reason whyi include this quote is because euclid is consideredto be the father of geometry. and it is a neat quote. regardless of your views ofgod, whether or not god exists or the nature of god,it says something

very fundamental about nature. the laws of nature are but themathematical thoughts of god. that math underpins allof the laws of nature. and the word geometryitself has greek roots. geo comes from greek for earth. metry comes fromgreek for measurement. you're probablyused to something like the metric system. and euclid is considered tobe the father of geometry

not because he was the firstperson who studied geometry. you could imagine the veryfirst humans might have studied geometry. they might have looked attwo twigs on the ground that looked somethinglike that and they might have looked atanother pair of twigs that looked like that andsaid, this is a bigger opening. what is the relationship here? or they might havelooked at a tree that

had a branch thatcame off it like that. and they said, ohthere's something similar about this opening hereand this opening here. or they might have askedthemselves, what is the ratio? or what is the relationshipbetween the distance around a circle andthe distance across it? and is that the samefor all circles? and is there a way forus to feel really good that that is definitely true?

and then once you gotto the early greeks, they started to get evenmore thoughtful essentially about geometric things when youtalk about greek mathematicians like pythagoras, whocame before euclid. but the reason whyeuclid is considered to be the father ofgeometry, and why we often talk about euclideangeometry, is around 300 bc-- and this right overhere is a picture of euclid painted by raphael.

and no one really knowswhat euclid looked like, even when he wasborn or when he died. so this is justraphael's impression of what euclid mighthave looked like when he was teaching in alexandria. but what made euclidthe father of geometry is really his writingof euclid's elements. and what the elementswere were essentially a 13 volume textbook.

and arguably the mostfamous textbook of all time. and what he did inthose 13 volumes is he essentially did arigorous, thoughtful, logical march through geometryand number theory, and then also solid geometry. so geometry in three dimensions. and this right over hereis the frontispiece piece for the english version,or the first translation of the english versionof euclid's elements.

and this was done in 1570. but it was obviouslyfirst written in greek. and then during muchof the middle ages, that knowledge wasshepherded by the arabs and it was translatedinto arabic. and then eventually in thelate middle ages, translated into latin, and thenobviously eventually english. and when i say that he did arigorous march, what euclid did is he didn't justsay, oh well, i

think if you take the lengthof one side of a right triangle and the length of the otherside of the right triangle, it's going to be the same as thesquare of the hypotenuse, all of these other things. and we'll go into depth aboutwhat all of these things mean. he says, i don't wantto just feel good that it's probably true. i want to prove tomyself that it is true. and so what he did in elements,especially the six books that

are concerned withplanar geometry, in fact, he did all of them,but from a geometrical point of view, he startedwith basic assumptions. so he started withbasic assumptions and those basic assumptionsin geometric speak are called axioms or postulates. and from them, he proved,he deduced other statements or propositions. or these are sometimescalled theorems.

and then he says, now i know ifthis is true and this is true, this must be true. and he could also prove thatother things cannot be true. so then he could prove that thisis not going to be the truth. he didn't just say,well, every circle i've said has this property. he says, i've now proventhat this is true. and then from there, we can goand deduce other propositions or theorems, and we can usesome of our original axioms

to do that. and what's specialabout that is no one had really done thatbefore, rigorously proven beyond a shadow of a doubtacross a whole broad sweep of knowledge. so not just oneproof here or there. he did it for anentire set of knowledge that we're talking about. a rigorous marchthrough a subject

so that he could buildthis scaffold of axioms and postulates andtheorems and propositions. and theorems and propositionsare essentially the same thing. and essentially forabout 2,000 years after euclid-- so thisis unbelievable shelf life for a textbook-- peopledidn't view you as educated if you did not read andunderstand euclid's elements. and euclid's elements,the book itself, was the second most printedbook in the western world

after the bible. this is a math textbook. it was second only to the bible. when the first printingpresses came out, they said ok, let'sprint the bible. what do we print next? let's print euclid's elements. and to show that this isrelevant into the fairly recent past-- althoughwhether or not you

argue that about150, 160 years ago is the recent past--this right here is a direct quote fromabraham lincoln, obviously one of the greatamerican presidents. i like this pictureof abraham lincoln. this is actually a photographof lincoln in his late 30s. but he was a huge fanof euclid's elements. he would actually use itto fine tune his mind. while he was ridinghis horse, he

would read euclid's elements. while was in the white house,he would read euclid's elements. but this is a directquote from lincoln. "in the course ofmy law reading, i constantly came uponthe word demonstrate. i thought at first thati understood its meaning, but soon becamesatisfied that i did not. i said to myself, what doi do when i demonstrate more than when ireason or prove?

how does demonstrationdiffer from any other proof?" so lincoln's saying, there'sthis word demonstration that means something more. proving beyond doubt. something more rigorous. more than just simplefeeling good about something or reasoning through it. "i consultedwebster's dictionary." so webster's dictionarywas around even

when lincoln was around. "they told of certain proof. proof beyond thepossibility of doubt. but i could form no idea ofwhat sort of proof that was. "i thought a greatmany things were proved beyond the possibilityof doubt, without recourse to any such extraordinaryprocess of reasoning as i understooddemonstration to be. i consulted all the dictionariesand books of reference

i could find, but withno better results. you might as well havedefined blue to a blind man. "at last i said, lincoln--"he's talking to himself. "at last i said,lincoln, you never can make a lawyer ifyou do not understand what demonstrate means. and i left my situationin springfield, went home to my father'shouse, and stayed there till i could give anyproposition in the six

books of euclid at sight." so the six books concernedwith planar geometry. "i then found out whatdemonstrate means and went back to my law studies." so one of the greatestamerican presidents of all time felt that in orderto be a great lawyer, he had to understand,be able to prove any proposition in the six booksof euclid's elements at sight. and also once he wasin the white house,

he continued to do thisto make him, in his mind, to fine tune his mind tobecome a great president. and so what we're going to bedoing in the geometry play list is essentially that. what we're going to study iswe're going to think about how do we really tightly,rigorously prove things? we're essentially going to be,in a slightly more modern form, studying what euclidstudied 2,300 years ago. really tighten our reasoningof different statements

and being able to make surethat when we say something, we can really provewhat we're saying. and this is really someof the most fundamental, real mathematicsthat you will do. arithmetic was reallyjust computation. now in geometry-- and whatwe're going to be doing is really euclideangeometry-- this is really what math is about. making some assumptions andthen deducing other things

from those assumptions.