>>–always wanted

to do this, and when you get ready to

do something like this, after you get ready

for a while, you begin to wonder if

it was such a good idea. (audience laughing)

Because, I mean, the “50 minutes” thing has

got me really nervous. There’s probably about 50 weeks’

worth of material here, which will now be distilled

in 50 minutes or less. The reason for the 50 minutes

is because I’d like to have a few minutes at

the end for questions. If you have a question that you

just can’t– you can’t survive unless you ask it immediately,

you may certainly do so, and I may give an answer,

or I may say, “That’s a great question–

I’ll answer it at the end,” Or I may give the answer

that I give most of the time, now that I am department head,

which is, “I have no idea.” (audience laughing) Um, when you’re gonna

give a talk like this, you gotta figure out

where you’re gonna start. And… okay, so, mathematics–

basic mathematics– counting or measuring things,

measuring areas, measuring lengths. Somebody had to do that

first, but we don’t know who, we don’t know where,

we don’t know when. And you know, there are

some extremely deep, fundamental ideas here that

we’re probably never gonna know the answer to. I mean, it was genius

for someone to realize that if you had this,

and if you had… those, and

if you had… these… that somehow, they were

all related by something that we now

call “five.” And to actually realize that

there was that uniting theme that needed a name and

a notation, that’s deep. And we have no

idea who did that. And anthropologists

might have theories, but that’s way out

of my realms. So, we don’t know

who did that, so we’ll start

someplace where we have some

idea of what went on. When you talk about

early civilizations– Egypt, uh, Mesopotamia,

the Babylonians, the present-day Iraq,

Iran, Syria– that area– the Tigris and

Euphrates… the cradle of

civilization. Uh, China and India

come to mind. Now, these all had

certain things in common. They all dealt with

arithmetic and geometry– in some cases,

fairly advanced. Babylonian geometry actually

got pretty intricate. They knew

the Pythagorean Theorem a long time before

Pythagoras was around. China was using

the equivalent of our decimal

fractions– not with our notation,

of course– um, several hundred

years BC, which is about

1,500 years before they were

used in Europe. And India– the numerals

that we use are called “Hindu-Arabic numerals” because they began

in India probably around

the 3rd century BC. So, they worked

with arithmetic, they did some amazing

things with geometry, considering

the era, but one thing that is notable

about all of the mathematics from this era is there is not one instance,

anywhere, of anything that would be considered

a formal proof, or even a discussion

of a demonstration. “Here’s why something works”–

they never did that. Never. It was, “Here’s

a problem. “Do this, do this, do this,

do this, do this,” and they often ended

with something like, “And you will see

that it is right.” They never

explained why. It was just something you

were supposed to “see.”>>That’s not

mathematics, is it? I’m sorry…

(audience laughing)>>There’s a heckler

in every crowd. (audience laughing)

Um… and the mathematics that they

did was often very practical– I mean, commerce and finding

areas for land and what not, but it was often

recreational. You find problems that

were clearly posed merely to be puzzles or to maybe see if you

could stump someone else– you know, like the word problems

we give our algebra students. Now, for those of us who

are a little more refined in our tastes and feel that

you don’t have mathematics unless you have

formal proof, then we have proof by

deductive reasoning, which was started

by the Greeks… And Thales, 600 BC,

the flyer– the person in the upper

left-hand corner of the flyer– is some artist’s rendition

of who Thales was. He is the one

who is credited with first giving

logical proofs. Now, a logical proof in

mathematics is airtight if done correctly. You start with axioms,

postulates– you start with statements that

everybody believes to be true. Given two points, there is

exactly one straight line that goes through them–

that is an axiom. You start with

definitions… “A circle is the set

of all points in the plane “a fixed distance

from any given point.” And then, you use logic to

deduce other propositions from those definitions

and axioms. That approach is–

I mean, the idea that you have

to prove

things based on previously

established results or previous definitions

and axioms– that is still

used today. The Greeks set

the standard for that. And the geometry

that they did– most of their work

was in geometry. They did very little

with number theory and even less

with algebra… uh, a little bit

with trig. I wanna say something

about that in a moment, but their

geometry– all the geometry that you

learned in high school and if you had any

geometry prior to some

non-Euclidean stuff– all that geometry, the Greeks

knew 2,500 years ago. They knew all of that

and a lot more. The trigonometry is interesting

because it was astronomers who started the study

of trigonometry– it was essentially

an outgrowth of– an extension

of geometry, and they were trying to

figure out how the earth moved around

the sun. Now, this is

3rd century BC. Greeks knew– or at least

some of them knew– that the earth moved. The earth moved

around the sun. It was not something

that was born in Europe in the 16th

or 15th century. It’s actually much

older than that. In terms of who was doing these

things– well, Pythagoras. We all know

that name. It’s the Pythagorean Theorem,

named after him. He may have proved it–

no one knows for sure. It’s fairly certain that

the school of philosophy that he founded– someone

in that school proved it. That school was also the first

to come up with the idea that there is such a thing

as an irrational number– a number that is not simply

the ratio of two integers. As far as the Pythagoreans

were concerned, that was a

huge finding. Euclid is not necessarily

as common a name, but his “Elements” is the most influential

mathematics text ever. It’s been in print

for 2,400 years. It basically– if you

studied mathematics from the time of Euclid until

into the 19th century, when you started studying

serious mathematics, you studied Euclid. Lincoln

studied Euclid. Newton

studied Euclid. All the big names

studied Euclid. And Euclid based his geometry

on that axiomatic method with deductive reasoning, and it was considered to

be such an excellent work that it was viewed

as infallible. It was almost as the same

level– in the Christian era, it was almost at the

same level as the Bible. You know, people

who know the Bible and will quote it

by chapter and verse? People who

knew Euclid– which was anyone

versed in mathematics in medieval Europe

and later– they would quote

Euclid, as well. “In Book I, Proposition 47,

we find–” and by the way, that’s

the Pythagorean Theorem. (audience laughing)

It was revered. Well, somebody has

to know this stuff. Archimedes–

we’ll probably say a little

bit more about him later, but for right now, you

have to say that Archimedes was 2,000 years

ahead of his time, in terms of what he was

able to do with mathematics. Not only did geometry

and number theory– he did some very amazing things

with some calculus ideas. One of the greatest

mathematicians of all time. His ideas predated those of

Europe by about 2,000 years. And if you’ve ever

studied conic sections– those got brought

up recently. Yeah, conic sections–

uh, ellipse, circle, parabola, hyperbola. The Greeks

studied those, and Apollonius did

significant treatise on those in the 2nd century BC. Conic sections played a

very important role later with astronomy and the orbits

of planets around the sun. Meanwhile, from 400

to 1200 AD in Europe, nothing was happening. (audience giggling)

Nothing. I mean, they call it

the “Dark Ages” for a reason. If you’re looking for anything

of significance in Europe between 400 and 1200 AD,

there’s nothing there. You’re looking in

the wrong place. Maybe go looking

in the Middle East. Mesopotamia,

the Babylonians– the prophet Muhammad founded

the religion of Islam in the 7th century AD. By around 700,

the followers of Islam were basically heeding

the call of that religion which, at that time,

was very strong on “We want to

seek knowledge, “we want to seek truth,”

and so, there was a… a tremendous calling

among scholars to learn for the sake

of learning. And so, because of that, um–

(cell phone ringing) Middle Eastern–

I won’t do what I do in class, but– Middle Eastern

mathematicians acquired ancient geometric

texts from the Greeks, translated them

into Arabic. They took the numerals from

India– the Hindu numerals– and now, we call them

“Hindu-Arabic,” and created what

we would now call “arithmetic with

Hindu-Arabic numerals.” You know, adding, subtracting,

multiplying, dividing, like we do on a

normal basis. That was done on a regular basis

in the Middle East by 1000 AD. This word

right here… uh, “The Condensed Book of

Calculation of al-Jabr.” al-Jabr…

“algebra.” That’s where

“algebra” came from. The word “algebra”

comes from this work by al-Khwarizmi

in 825. A tremendous amount

of algebra was being done in the Middle East

at that time. Now, you wouldn’t

recognize it as such because it was very

geometric in nature, in the sense that you base

things on geometric ideas, and they didn’t have

the notation we had. But if you look carefully,

a lot of what we do today, a lot of the factoring we do–

you know, squaring binomials, cubing binomials,

all that stuff that our students sometimes

don’t do so well with… they were

doing that. And if you are of

a literary bent and you’re familiar with

“The Rubaiyat of Omar Khayyam,” that is the same

Omar Khayyam– he was also

an excellent mathematician and did some work with third-degree

polynomial equations– cubic equations– that was not surpassed in

Europe until about 16th century. Interesting thing about this

is there is a huge amount about Middle Eastern

mathematics from this time that is virtually

untouched. There are mosques and

palaces and basements filled with manuscripts that

have never been looked at. And so, the contribution

from Middle East at present is somewhat known, but not even

close to known completely. Well, okay,

so, 1200 AD– eventually, Europe

started to wake up. The Hindu-Arabic

numerals that we use started making their

way into Europe by around the

10th century, and if you’ve ever heard of

the Fibonacci Sequence… let’s see, how

does that go again? 1, 1, 2–

what’s the next one?>>Three.

>>And then?>>Five.

>>Just keep adding…>>Eight.

>>Okay. Fibonacci is known for that,

and that’s wonderful, but that’s really,

in a sense, trivial. What he’s really to be

honored for is the fact that he was a very

vocal advocate of Hindu-Arabic numerals

being used in Europe. Uh, printed the

“Libre Abaci” in 1228 AD. And it only took about 300 years

for Hindu-Arabic numerals to become widespread

in Europe. I mean, if you

think about that, you look at the numbers

we use on a regular basis, you’d think, “Why wouldn’t

you use something like that?” They were using Roman

numerals prior to that. And Roman numerals

are really cumbersome, hard to read, hard to

compute with, okay? They basically used Roman

numerals to record things with, in terms of the

mercantile class. But 300 years it took to

actually get universal adoption of Hindu-Arabic numerals

across Europe. And just to give

you a sense of how advanced Hindu-Arabic

numerals were considered, I’m gonna read you

something here. This is from

about 1450. “A German merchant had

a son whom he desired “to give an advanced

commercial education. “He appealed to a prominent

professor of a university “for advice as to where

he should send his son. “The reply was that if

the mathematical curriculum “of the young was to be confined

to adding and subtracting, “he could go to

a German university,” but if he wanted to learn

how to multiply and divide, the only country where he

could learn that was Italy. It was high-powered

stuff back in 1450. (audience giggling)

So, I don’t know if some of you wanna share that

with your 095 classes or not… (audience laughing) So, Europe begins

to awaken. Fractions had been around,

essentially forever. I mean,

in recorded history, fractions have

always been present. The Egyptians were using

fractions 5,000 years ago. Um, now, fractions

gave way to fractions that were a little

more predictable. Base 60 fractions. Babylonians used

base 60 numbers– sexagesimal

number system. Not decimal–

sexagesimal. They used base 60, and they

used base 60 fractions. And we still

do today, right? That’s what

this is. This is 57-60ths of a degree,

and then this is 48 seconds, but there are 3,600 seconds

in 1 degree. And so, this is 48– these

are sexagesimal fractions. So, it’s not like you’ve

never seen ’em before. But those were used pretty

extensively in Europe and the Middle East well

into the 17th century. Decimal fractions

as we know them– um, Simon Stevin,

Belgium, advocated base 10

fractions using, um… a notation that we

might find a bit odd. We would use

this notation. Stevin’s “Art of Tenths” from

1585 used this notation… Which, if you look

at it the right way, you’re really talking about

negative exponents, aren’t ya? ‘Cause this is

one-tenth, right? 10 to the -1. This is 4/100ths,

10 to the -2. It’s not a horrible notation,

but it’s kinda cumbersome. But the important thing

was two things– he published a book saying,

“We should all be using “these decimal fractions

and here’s how they work.” And at least

he had a notation. So, it was a step in

the right direction. Um, did not get much attention

for quite some time. It was one of those books

that got published and hardly anybody

read it… but then, along

came logarithms. Now, this will warm the hearts

of certain people in here who are of a

certain age– I’m seeing some

smiles from some and others just blank looks

’cause you’re too young. (audience chuckling)

Um, John Napier, Scottish, had the brilliant idea that you

could transform multiplication into addition. And the theory

is he got that idea from trigonometric

identities. There are trigonometric

identities that allow you to transform products

to the sums… and he basically figured, “Well,

you oughta be able to do that “for just plain

old numbers,” and so he came up

with an idea that allowed you to transform

products into sums, quotients into differences,

powers into multiplication, and roots– like cube root–

into division. And for those of you

old enough to remember, it wasn’t a lot of fun but

it was a heck of a lot easier than doing it

by hand, okay? The problem with Napier, though,

was that he used a base that– well, okay. This gets

complicated. He essentially used

logarithms of base 1-over-e. It wasn’t exactly that,

and “e” hadn’t even been

discovered yet, okay? But that’s basically

what he had. They’re very

difficult to use. With logarithms of that base,

as the numbers got bigger, the logarithms got smaller–

it was just weird. So, Napier and

Briggs met, and Briggs had the idea that

we’d use base 10 logarithms– the common logarithms

that were in the tables that some of us used, and you have a “log” key

on your calculator– L-O-G will find

log-base 10 logarithms. Briggs probably goes

down in history as one of the most

heroic figures in the era of computational

mathematics. In order to create “log” tables

that he knew would be accurate, he began by taking

the number 10 and extracting 54 consecutive

square-roots of it, by hand, to 30 decimal places. It took him a couple

of months to do that. And then, he built

his logarithm tables meticulously to

14 decimal places… and I say “decimal places”–

if you’re dealing with “log”s– they’re base 10 “log”s, you

gotta use base 10 fractions, you want to use

decimal numbers. Simon Stevin’s decimal numbers

really didn’t catch on that fast because there wasn’t

a need for it, but as soon as you had

logarithms in play, it was like, “Oh, my gosh–

we need decimal points,” and that’s when decimal

numbers really took off. Now, symbolic algebra– that’s

the algebra we all know, okay? Algebra was not like

that until very recently. Algebra started out

being rhetorical. Rhetorical algebra was common

in Europe and the Middle East for a very

long time. Here is a statement in

rhetorical algebra, okay? Some of you can try this on

in 095, 6, 7, or 8, all right? “In the rule of three,

argument, fruit, “and requisition are

the names of the terms. “The first and last

terms must be similar. “Requisition

multiplied by fruit “and divided by

argument is produce.” (audience giggling)

Okay? “The first and last

must be similar.” Argument… requisition. Requisition multiplied by

fruit produces produce– you’re talking

about a proportion. This times this,

divided by that… will give you “P.” It was called

the “Rule of Three.” It was called

the “Golden Rule.” Merchants use that all the time

to figure out how much– you know, if it’s this

much for how many, how much it’ll be for this

many– they use that a lot. But that was

rhetorical algebra. It was prosaic but it wasn’t

easy to do anything with. Syncopated algebra

was a step up. Instead of using words, you used

abbreviations, okay, like this. This is from

Italy, 1494. Um, my knowledge of

Italian is vast… It’s non-existent, okay? This stands for “cubo”–

this is X-cubed. “meno,” “less”

means “minus.” This is X-cubed minus–

“censo” is for X-squared. You got X-cubed

minus X-squared. This is for “plus.” Uh, this is the most

interesting one. This is for “cosa.” “Cosa” translates

into “thing.” “Thing.” What is the “thing”

you are solving for? That’s the

variable X. So, X-cubed minus X-squared

plus X equals– that’s “(indistinct),”

I think– zero. That’s a

cubic equation. Okay? Um, what makes this

interesting is that “cosa”– algebra became known in Europe–

see, this is what, 1494– yeah, 1400s,

1500s, 1600s– algebra was known

as the “cosic art.” “Cosic art,” because

you work with “cosa,” you work with “things,”

you try to find “things.” You try to solve

for the unknown. But, you know, this

is not our algebra. Our algebra–

the symbolic algebra that we’re

familiar with– really began taking off

in the 16th century, and by the middle

of the 17th century, it was pretty

much standard. Not completely, but

you would recognize it. If you picked up something

from the 1650s in algebra and read it,

you’d recognize it. Um, it’s hard to say

who did what, when, because so much was going

on then, so we’ll just say, “Many people in

Europe did this” between 16th and

middle of 17th century. Now that you’ve got algebra–

okay, so you got algebra, right? You got things like X-cubed

minus X-squared plus X equals zero. Well, then, you’ve

had geometry, right? Remember– Greeks, geometry,

Euclid, revered, everyone– anyone who studied

mathematics knew geometry. Now you’ve

got algebra. So, what’s

the next step? Unite them. Now, I have to mention Oresme–

Nicole Oresme– because he was so far ahead of

his time that it’s almost scary. 1350– that was the era of

the Black Death in Europe, and here is Oresme studying,

among other things, velocity-time graphs. Are you familiar with this at

all from anything in physics?>>Say it again.

>>Uh, velocity-time graphs.>>Sure.

>>But from this era.>>No.

>>He was doing

velocity-time graphs around 1350, which is about

300 years ahead of anybody else. And then,

he died young and people pretty much

forgot what he did. The two people for

whom we give credit for establishing

modern-day coordinate

system are right here– Fermat and

Descartes. Both French. Both came up with their

coordinate systems at virtually

the same time. Fermat was a little

ahead of Descartes… there’s probably a joke

in there someplace. (audience chuckling)

But I won’t do it. I’m not Pruis. (audience laughing)

Um… it’s an “in” joke. Fermat was first, but Fermat

was an amateur mathematician. He was a lawyer by trade,

and from what I’ve read, he wasn’t a very

good lawyer. (audience laughing)

And you know why? He spent so much

time doing math. So, obviously, he’s one

of my personal heroes. Um, he actually came

up with his idea first, but he didn’t

publish anything. He just

didn’t care. He didn’t publish

a thing. He circulated

a few manuscripts. And actually,

Descartes got a hold of

one of those manuscripts, and he was just putting the

finishing touches on his method and published this right away

because he wanted credit. And okay, fine,

I won’t get into that. I mean,

he published– they both came over

essentially the same time. They had very different

takes on things. Um, Fermat essentially

was interested– “Give me an equation, let’s see

what we can find out about it. “Let’s study the equation

geometrically.” You know, like we do with

a graphing calculator. You wanna figure out

where that– you know that equation

up there earlier– X-cubed minus X-squared

plus X equals zero– what are its

solutions? Graph it and look at

the X-intercepts. That’s roughly what Fermat

was interested in doing. Descartes was more

interested in… “Give me a curve–

I wanna find its equation, “then study the curve

algebraically.” His intent was “I wanna see if

we can study Euclid’s geometry “from an algebraic

point of view.” Okay, that

was the method. A very important idea

that underlies all this is that variables,

now actually varied, there is a huge difference

between this… and this. Equations like this had been

worked on in various forms. Babylonians did this

2,000 years BC. Find a number that you

can subtract 2 from in order

to get 10. This acts as a placeholder

and nothing else. That’s all it is. It just takes the spot of

something you don’t know. This– within this context,

this has a continuous graph. It’s a line. You know, it looks

something like this… I hope it looks

something like that. It looks something–

it’s continuous. I mean, you can go from

point to point to point in a continuous manner, which means these variables

can vary continuously, and that was

a brand new idea. That was huge because that

actually gave rise to something really significant

in just a few years. The stage is set. In 1650 in Europe,

Greek geometry was known, symbolic algebra was

essentially as we have it today, Hindu-Arabic numerals, people were very

comfortable with them, logarithms were being used,

and we had a coordinate system. So, now, we

are ready for… enter “the calculus.” Now, “the calculus”– there

is a distinction to be made between this

and this, and I think the easiest way

I can explain it is simple. I can do this. I’m just a guy. High school students around

the country do this every year. You don’t have to be

a genius to do this. I’m living proof

of that. But if you’re

going to do this, you had to be

extraordinarily clever, because this is a collection of

rules, notation, and procedures that creates a system by

which you can solve problems in an organized

manner. “The calculus” is

a tool that applies to a wide variety

of situations, and you don’t have

to be a genius to use it. These– the people

who did this had to be

extraordinarily clever– I mean, Archimedes was

a genius of the first rank, and to do what he did,

he had to be. Nobody else could do what

he did in 3rd century BC. He was pretty

much alone… because he would take

some really fundamental

calculus ideas and apply them in

extraordinarily clever ways, mostly geometric for him–

but for some of these people, it was largely geometric

and algebraic– and come up

with results. These people down here–

these are all– they did their work from

1600 to around 1660… and impressive results. Kepler, who

came up with his “Three Laws of

Planetary Motion”– one of which is all

the planets orbit the sun in the shape of an ellipse,

with the sun at one focus. And in order to

deal with that, he also dealt with some

problems involving area. He used calculus techniques

to work with those areas. Fermat, who we met earlier, was

working tangent line problems. You know your calc

problems where you wanna find maximums and minimums while looking for a tangent line

that’s horizontal? Fermat was doing

that in 1629… but he had to treat

each problem as if it was

a brand new problem. All these people did a lot

with calculus ideas but they didn’t get

to “the calculus” until someone who may

have been sitting in on one

of Barrow’s classes. Isaac Barrow was at Cambridge

and, in 1664 to 1665, he gave lectures

in which he demonstrated the geometric

connection between derivatives

and intervals. And for those of you that

don’t know any calculus, he demonstrated geometrically

that the tangent line problem was the inverse of

the area problem. He demonstrated that, and he moved on because

he didn’t see any significance. That was left to

one of his pupils. Isaac Newton… I mean, what

can you say? I mean,

there are geniuses and then there are

transcendent geniuses. His work in

the calculus– his discovery of calculus took

place over a very brief period. He published,

essentially, none of that. He published

none of that. Some of that work was not

published till the mid-1900s. It’s true. And some of you will glare

at me when I say this– he was a top-notch

mathematician. One of the

greatest ever. I would not say he

is the greatest ever. He was the greatest

physicist ever. His fame, in my eyes,

lies right here. He published

this book, which is the most influential

book in the history of science– no question. It transformed

science. Science became mathematical with

the publication of this book. Newton demonstrated “If you

wanna understand the world, “if you wanna understand

the universe, “I have some

fundamental ideas. “I’m not gonna tell

you why they work. “I feign no

hypothesis. “I don’t know why gravity works,

but here’s how it works.” I mean, if you use these

ideas together with calculus, you can do a lot. And that transformed

science. But he didn’t

publish much… and he wasn’t a lousy writer but

he wasn’t the greatest writer, and he used

lousy notation. His notation

was poor. He didn’t care. If he wasn’t writing

for the masses, he was writing for scholars

and for himself. So, Newton developed

his calculus, Leibniz developed his

a few years later. It is Leibniz’s calculus

that we use today. Every time

you write this… which is just beautiful–

that’s Leibniz’s. October 1675–

I forget the exact date. Copious notes. I mean, you can

look at his notes. You can find out exactly when

he came up with this notation, exactly when he came

up with this notation. The common notations

used in calculus, those are Leibniz’s

notation. And he was very–

you know, “the calculus”– Newton really wasn’t

so concerned about creating

“the calculus” that other people

could use. He wanted to use it but

he didn’t really care too much if other people

could use it. Leibniz wanted other people to

know how to use “the calculus,” and so, he actually

created a journal– he founded a

journal in 1684 for the sole purpose of

publishing his calculus results. It’s one way

to get attention. It was before

the internet. You couldn’t just post

this stuff online yet– had to have

a journal, okay? Imagine sending

a 140-character tweet, and “Guess what?

Today, I founded calculus.” (audience laughing)

Um, now, there is– if you know anything

at all about

the history of mathematics, you’ve probably heard

of the priority dispute between Newton’s followers

and Leibniz’s followers, and I wanna emphasize

their followers. Neither Newton nor Leibniz

started the priority dispute. Neither one did. But they were both

drawn into it. It got ugly. It’s just– it’s a

sad state of affairs in the history

of mathematics. It really hurt English

mathematics because the English

mathematicians– I mean, they were ardent

followers of Newton, and Newton was

harder to understand and had lousy

notation, and so, mathematical work

in England sort of stagnated for about 100 years. Meanwhile, in Europe,

things were really heating up. Now, I had

to include this. This is something I inserted

a couple of days ago– we have to pause

for just a moment. 17th century in Europe is known

as the “Heroic Century “in Mathematics,” and the best way

to describe that is if you were to compare

mathematics in 1600 to mathematics

in 700… a vast change

had taken place. European mathematics

in 1600, aside from the language

differences, look pretty much like

the mathematics of Greece. There was a little more

algebra, but remember, there wasn’t much

algebraic notation. So, it was basically not much

different than Greek mathematics in 200 BC. By 1700,

everything had changed. You had the algebraic symbolism,

you had logarithms, you had Hindu-Arabic numerals

being used everywhere, you had calculus,

you had graphs. The emphasis

had changed. 1600– mathematics was

almost all geometrics. 1700– geometry was

still important, but it was definitely

in the background. Symbolic algebra,

symbolic calculus had taken the

foreground, and it was the

exponents of Leibniz who took the charge

on that. The Bernoulli

family… Jacob and Johann Bernoulli–

and there were others– learned from Leibniz. Leibniz

taught them, the Bernoullis taught

L’Hospital and Leonhard Euler, another giant titan

of mathematics– greatest mathematician

of the 18th century, no question

about that. Um, brilliant and

scarily intuitive. He just sort of said,

“Oh, this’ll work.” And then, he’d work out

a bunch of stuff and he was convinced it

was right, and he moved on. We need to come back

to that in a moment. But he was definitely

the chief proponent of “We’ve got this powerful

tool called ‘calculus.’ “We’re just gonna

keep using it “because it’s giving us

these amazing results.” And the amazing results were

sometimes mathematical and sometimes they were

in the applied world– physics, astronomy,

engineering. One of my favorite episodes

from this era is the discovery of

the planet Neptune, which was done not by taking

a whole bunch of telescopes and probing the night sky

for months and years. It was done mathematically,

using the physics of Newton, as improved by Laplace…

celestial mechanics. And the mathematics of the era,

which was calculus– they did what’s called

“inverse perturbation theory,” and they essentially said,

“There’s got to be a planet “out there that is

screwing up the orbit “of the planet

Uranus… “and we have

done the math, “and the planet will

be there at 9 o’clock “on this certain night”

in 1846, and they pointed their

telescopes and there it was. It’s just–

it’s astounding. So, the mathematics was

working marvelously well, and Euler was

their leader, okay? Now, that’s great. If you’re going to make

a lot of discoveries, you want to just

try stuff. I tell my students

that all the time. “Just try stuff!” Okay? But every once in a while, it’s

important that you can prove that what you’re

doing is correct. And after a giddy 100 years

of just flying with calculus to see what it would do,

we get to the 19th century, which was kind of a tumultuous

century in mathematics. Uh, it began by–

I don’t know, “began” is– this is not necessarily

purely chronological. Challenging truth. Truth. In this slide,

“truth” means Euclid. Remember, Euclid

was infallible. Euclid’s geometry

was the geometry. If you’re gonna do geometry–

it’s Euclid. That’s it, there’s nothing

else to discuss, move on. Well… the parallel postulate

is one of those things– remember, now it’s a postulate,

it’s an axiom, it’s one of those things that’s

supposed to be really obvious. You know, like, “Here’s a point,

here’s a point– “there’s

exactly one line.” Euclid’s parallel

postulate is very wordy, and you look at it

and you think, “Oh, this is something

we could prove.” This isn’t

an axiom. This is something you

state without proof. We can prove this,

and mathematicians

for over 2,000 years were convinced that they

could prove Euclid’s postulate from other

postulates… that it was

actually a theorem. And they tried and they failed,

and they tried and they failed, and after about 2,000 years

of trying and failing, people began to get

the idea that, “Well, maybe

we can’t do it.” And some interesting

things began to happen. If you have Euclid’s

parallel postulate, then one of the things you can

prove is that in any triangle– when you add up all the

angles in any triangle, what do you get?>>180 degrees.

>>180. Now, that’s Euclidean

geometry– “E.G.”– okay? Gauss– who more has to be said

of– when he was 15 years old– he’s 15 years old and he begins looking at

the parallel postulate and saying, “There’s something

about this that bugs me.” And by 1870, which was

quite a few years later– but when he was

quite young, actually, he had convinced himself that

Euclid’s parallel postulate couldn’t be proven and, in fact,

wasn’t even necessary. That you could have other

parallel postulates besides Euclid’s. He didn’t publish

any of this, because it wasn’t

good enough for him. Gauss had probably the highest

standards of any mathematician well into the

20th century. But other people were thinking

about the same things, and hyperbolic

geometry, um– here, hyperbolic geometry,

unpublished was Gauss. Published–

Lobachevski and Bolyai. They came up with

a parallel postulate which said that it’s possible

to have a triangle with a sum of the angles

instead of equal to 180, less than 180. Now, if you’re saying,

“That’s not a triangle,” I can understand that, but it all depends on

what space you’re in. If you’re in Euclidean space,

this isn’t a triangle. But if you’re in

hyperbolic space, it is. And you say, “There’s no such

thing as hyperbolic space…” Einstein found it

very useful when he was coming up

with the mathematics of general relativity. The space-time continuum–

curved space– it’s hyperbolic space. And then, Riemann

came up with… Fat triangles. Some of the angles are

greater than 180 degrees. That’s Riemannian geometry

or elliptic geometry. Now, the point of

all this was… prior to the 1860s and earlier,

there was one geometry– it was Euclid’s. When these came out, first, they were viewed as

nothing more than curiosities. It’s like, “Oh, you have

a very interesting geometry. “Isn’t that nice?” And then,

people moved on. But by around 1870,

it finally hit home– this geometry

and this geometry, as weird as they

might have seemed, were absolutely as valid,

as true, as this one. Mathematically,

there was– if there was something

wrong with this, there was something

equally wrong with this or something equally

wrong with this. Or to put it

in another way, if this was invalid

and this was invalid, then so was this. So, either accept them all or

you don’t accept any of ’em. And that became apparent

by around 1870. So, “Euclid equals truth”–

gone. Gone. Gauss… another candidate for greatest

mathematician of all time. Did not publish

nearly as much as many other

mathematicians did, but his motto was,

“Few but ripe.” He didn’t publish a lot,

but what he published was essentially

perfect. Rigorous, correct in every

detail, high standards of rigor. If you’ve ever done a fitting

of curves to a set of data– least-squares analysis–

Gauss came up with that. Differential geometry, which folks in Calc 3 are

gonna start studying soon– differential geometry–

that was Gauss’s. Hyperbolic geometry– Gauss didn’t publish

but he knew about that. Number theory–

Gauss did tremendous work

in number theory. Definitely the

greatest mathematician in the 19th century–

no question about that. And believe me, the 19th century

was quite a century for mathematicians. Algebra got modified

quite a bit. You know that cubic equation

we looked at earlier? Are you aware of the fact

that, in Europe, in Italy, in the 15th– er, excuse me,

16th century Italy, there were actually publicly

held equation-solving contests? I kid you not. That is not hard to find online

and in many other sources. But they only worked

with equations up to the

fourth-degree… because after

the fourth-degree, things just didn’t

work very well. And Galois established what is

now known as “Galois theory”– I mean, he didn’t call

it that himself, okay? (audience chuckling)

Um, Galois theory, which essentially proved that

if it’s fifth-degree or higher, there’s not gonna be

a formula for solving it. You’re just gonna have

to use ad hoc techniques. Um, something

a little closer to home, something we can all relate to–

Hamilton and Cayley. Okay– oh, and

if you’re in Calc 3 or if you are in

linear algebra and taking any matrix theory–

with real numbers, we all know this is true,

and what is that called?>>(all) Commutative property.

>>Commutative property. The order in which you

multiply doesn’t matter, right? Both Hamilton and Cayley came up

with noncommutative algebras, both as they were looking for

ways to describe rotations in three-dimensional

space. The idea that this was not true

for certain areas of mathematics was…

stunning. It was like,

“You could do that?” And it didn’t take long

for mathematicians to go, “Yeah, we can

do that!” And then, they just discovered

all sorts of other algebras. You know, you give

mathematicians an inch, they’ll take a kilometer.

(audience chuckling) Because you have to

convert the units. (audience laughing)

Um… Euclid didn’t equal

truth anymore. The parallel postulate was

shown to be unessential. You could use other

postulates which– now, look,

for 2,000 years, mathematicians thought

that Euclid was infallible, and it turned out that

there were other geometries. And so, the whole idea

of, “Well, what about

other mathematics?” Well, what about

calculus? Is the foundation of

calculus infallible? Well, Joseph Fourier did

some work with heat transfer that involved

trigonometric series, and the short story on that

was what he came up with did really weird stuff

that nobody could explain. It was like, “What–

what are you doing?” And he said, “Well, it works,

so I’m gonna keep doing it.” (audience chuckling)

There were some real cracks in the foundation

of calculus. In the mid-1800s, the definite

integral had never been defined, limits had never been

carefully defined, there was confusion

about the relationship between continuous and

differential functions. People didn’t know

what they were doing. And so, the arithmetization

of analysis took place. Very quickly what happened

was if you’re gonna do limits, you gotta base ’em

on real numbers. Real numbers can be

based on the rationals, rationals on the integers,

integers on natural numbers, and so… Peano came up with his

axioms for establishing the entire set

of natural numbers. Dedekind came up with

his version of Dedekind cuts, which defined real numbers

in terms of rationals, which then could be brought back

down to the natural numbers. Weierstrass gave the final

perfect definition of what a

“limit” is, and if you’ve ever done

epsilon-delta proofs of limits, this is the guy. Riemann defined

the definite integral. Cauchy– not as

carefully as Weierstrass, but Cauchy worked

with continuity and differentiability. And then,

of course, Gauss– always the most

careful of anyone– way back when had worked

with convergence of series. By 1900, there was,

supposedly, a firm, unshakable

foundation for all of analysis

for calculus and everything

related to calculus. That takes us

to 1900. So, now–

oh, I almost forgot. How could

I forget this? To– okay, I was tempted

to infinity and beyond– (audience laughing) that is the Hebrew letter

“aleph,” subscript is zero, and that is pronounced

“aleph-null”… which is a symbol for

a certain level of infinity. So, I could have said,

“To infinity and beyond,” which I believe was a very

popular phrase in a movie of some time ago, okay,

but I typed that up. Um, Cantor– Cantor developed

his theory of sets to help bring solid

structure to this edifice that we call

“calculus.” He overthrew Greek thought

of more than 2,000 years, which was “potential

infinity is fine, “but there’s no such thing

as actual infinity.” The Greeks, with maybe

a few exceptions– there is some evidence

that Archimedes might have accepted

actual infinity, but that’s tenuous

even now. For the most part, Greeks felt

that infinity was potential. You know, one, two, three, four,

five, six– how far can you go? “Well, you can go

forever,” you know? “I can name a number

bigger than you.” “Oh, yeah?” “Yeah– a million.”

“A million and one.” “A billion.”

“A billion and one.” You can keep going…

but that was potential. They accepted that. What they did not

accept is, okay, all of the natural numbers–

there is this set that contains all the natural numbers

and here it is. There’s the set of

all natural numbers. It can be thought of as

an actual collection. You need to give that

back when you’re done. (audience laughing)

There’s an actual collection that they’d, “No,

there’s no such thing.” They do not believe

that was possible. Cantor not only embraced

this but ran with it, and if you want to learn

more about this idea that some infinities

are bigger than others– in fact, some are a lot

bigger than others– our October seminar, Kelly Rozin

will be speaking to us on infinity. So, I have a feeling Cantor’s

name might come up again. So, now, okay,

as I was saying, it was around 1900 and now we are at essentially

the modern, modern era– um, you know, mathematics

of the last 100 years. That first bullet-point

is a gross understatement. Much, much, much more

mathematics has been discovered or created in the last 100 years

that in the previous 5,000. It is just mind-boggling

how much has been done. And so, you know, summarizing

that in two minutes– no. But also the

problem is… most of it is pretty deep,

difficult, abstract stuff. And it’s hard to say which of

that is really gonna be seen as important 100 years from now,

but since it’s my talk… (audience laughing) I will now tell you

two things that– when I’m thinking about

“How am I gonna end this?” I thought,

“You know what? “I’m gonna think

of a couple things “that I think will

still be significant “maybe, you know,

200 years from now.” And I thought, “Oh, the first

two things that came to mind”– well, here’s

the first one. This is actually abstract

and fairly deep, but the gist of it,

a little bit simplified, is not hard

to grasp, okay? Kurt Godel– in my opinion,

greatest logician of all time. Brilliant enough to be able

to walk around the grounds of the advanced– the Institute for Advanced Study

at Princeton with Einstein and carry on conversations

at Einstein’s level. They were very

good friends. I wish, I wish, I wish someone

had wandered with them every once in a while

and taken notes, ’cause it would have

been interesting. You know, maybe they did

talk about the weather. Maybe they talked

about other things, but I guess

we’ll never know. Godel, in 1931,

essentially said, “You know these solid

foundations that you

think you’ve got? “It’s an illusion.” ‘Cause here’s what

he was able to prove– if you take any formal

axiomatic system that’s rich enough to contain

the natural numbers– so, let’s just use the natural

numbers as an example– you’ve got

your axioms, you’ve got

your definitions. So, you create this system

for the natural numbers. It is impossible

to prove that that– it is impossible to prove

within that system that that system

is complete. In other words, Godel showed that there

will always be statements in that system that you

can’t prove within the system. Furthermore, he showed

that one of those statements that you can’t prove

within the system is the statement that

“this system is consistent.” If you want to establish

that the system of all natural numbers doesn’t

contain contradictions, you can’t do it

within the system. Godel proved that. So, the dream of creating

rock-solid foundations that are infallible–

it may be true… we can’t prove it. Some of the people

in this room who are not in

the math department like to remind me of things

like this every once in a while, just to try

and keep me honest, and I tell them to go away.

(audience laughing) And then, this is almost

diametrically opposite. This is not

abstract. This is practical. You’ve got cell phones,

you’re on Facebook, you use the internet–

heck, I use the internet. I don’t know what

a cell phone is. I’ve seen them.

(audience chuckling) YouTube– all that stuff

that involves communication from one place

to another– that all depends on

transferring information

using ones and zeros. Ones and zeros. It’s all about

ones and zeros. Someone had to come up with

the mathematical formulation which said, “You can do it,

and here’s how you do it,” and that was

Claude Shannon. Now, Claude Shannon is native–

he was born in Gaylord, went to the University

of Michigan, engineering and mathematics

Bachelor’s degrees from

U of M, Master’s in engineering

from MIT, PhD in mathematics

from MIT. The engineers claim him

as their own, but he has a PhD in mathematics,

so he belongs to us. (audience laughing)

And this paper– this is– this is, I mean,

a ground-breaking paper. Well, you’re gonna use

your cell phone later? A couple of you have

been checking it in here while I’ve been talking–

that’s okay. I’m not

glaring at you. Um, you’re doing that

based on this mathematics. That made

it possible. So, whether you love the

digital era that we’re in or you loathe it,

he’s responsible. Without that mathematics,

you wouldn’t have any of it. Um… that’s about it in terms

of what I need to say. Now, if you want

to study further, these are all good– um, my

favorite is probably this one. This is good but

it’s higher level. This is good but

it’s lower level. This is fantastic but it’s

a tough read sometimes. If you want to know

anything about the history

of mathematical notations, you cannot do any

better than this. This is the source. You go online and you look

up “history of the plus sign,” they took it from Cajori– maybe

with or without attribution. And if you actually want a site

online that you can go to for history

of mathematics, that’s out of the University

of St. Andrews in Scotland– that’s a good place. Uh, do you have any

questions about anything? Well, then we’re done.

(audience chuckling) If you do have any questions and

wanna hang around, that’s fine. Thanks for coming. (applause)

Europeans actually cheated everyone after dark age ..they coped every basic things from middle east …..that is a necked truth if u know the prefabricated story……but now middle east behave like a grand father of Europeans.

they r not interested in anything …haha

My this was a fascinating talk

9:28 bullshit, islam wasn't designed for seeking knowledge…

52:55 "i'm kNot glaring at you"

<:> But pay attention 'cause i CAN pawn you mathematically from dimensions you may knot kNow <:>

John Dersch is -2SIN666 Living Proof 🙂

Can someone explain the title and description of this video: Catastrophic Electrical Damage took place – let’s check the Math to see why!? Am I missing something?

"There's two ways of doing anything, the smart way and the dumb way. When you do it the smart was, that's mathematics."

I didn't see anything about catastrophic electrical damage. Also, the sound is muffled, and some of the words are not easy to hear.

this is ABSOLUTE STUPID CRAP….math is 0,1,2,3,4,5,6,7,8,an 9 added together or subtracted…all this other stuff uses math to find what your looking for…..dumbass earthlings

@8:45, to see a distinguished intellectual like Professor Dersch regurgitate the "Renaissance"-"Illuminist" smear against the Medieval Period is highly disappointing.

fix your audio

If manuscripts found in Middle East and Sahel haven't been fully examined, we can't assume 20th century Maths have contributed more than the 5,000 years before it. Moreover it's harder to conceive a concept than to improve on it.

history of mathematics: R^c 1`k… 1 + 1 = … advanced calculus… mathematics…Mathematics

TIL Mohammed was more important to mathematics than Sir Isaac Newton.

ch

Turing > Shannon

Is there a link to that seminar on infinity?

October 29 33:00

The amazing constructions of the ancient Egyptian Pyramids clearly demonstrate an advanced state of mathematics however either that knowledge was 'lost', destroyed or maybe never was written down limits what 'proof' we can find today. Only the stones survive

One correction, the dating of mathematics in India seems unrealistic. It should be before Egypt or at least on same time line of Egypt and Babylonia. Archaeologists have discovered some weight and measurement devices at Harrapa (Indus Valley) which are dated more than 5K years ago.

How many years is that?

thank you professor ! it was an awesome lecture 😀

I watched the whole thing in one go and I think I have A.D.D

Totally ignorant video. Title should be 'a brief history of Europe's Mathematics'

I can't believe he doesn't mention the Kerala school of Mathematics scholars who discovered the infinite series and Calculus. Piece if shit..

WoW ~ This lecturer John Dersch is SO wonderful … He has the knack/gift for hiolding attention and imparting knowledge in a very engaging, interesting and personable way, ie with the sensitivity and depth of charisma~ I LOVE his style. Maybe the subject and his apparant appreciation for the history of mathematics is what makes his lecture so intriguing …

Awesome that's all there is to say

Leibniz.

Modern math origins from north africa and andalucia…stolen by force and massacres and all original books burned by the churche of hate and lied for for centuries as europe has a complex of its own history as the comments section show. (btw not only math but also music system, music, mapping of the world, navigation, architecture, ….).

I love this stuff!

Superb presentation, loved it. Only thing missing, for me, was the invention (or discovery) of imaginary numbers.

Proof by deducting reasoning was started in asia by the chinese and the indians, that was a very racist statement to say.

Most of white people have racism imprinted into their culture, just like romans had slavery inprinted into their culture.

They just dont realize when they do something racist.

(and btw racism is a human trait not a white man trait, so chinese and blacks can also be racists)

The called in the Dark Ages for a reason.

Just think of where we might be if that hadn't sucked out centuries of scientific advancement.

Europeans went from nomads to walking on the moon in 384 years? ( @16:22 1585 to 1969) and all these other civilizations had a couple thousand years head start … kudos white people

This app is a good challenge to see how fast you are in mathematics

https://play.google.com/store/apps/details?id=comp.successpioneers.aliofrei.mathchallenge

ALWAYS on concluding REMARKS it is WHISPERING!: VERY BAD HABIT!

You could have ten lectures like this in succession and say after each one, "and then it gets more interesting ". Thanks.

http://www-history.mcs.st-and.ac.uk/

35:25 cool

Can you sum this video up as that early math was involved functions or to applied? Given the facts were based on some concrete/tangible evidences i.e. architecture that survived the destruction of time?

I can do a whole piece of paper on the pyramid and the mathematics involved? it is investigative?

The audio quality of the video is not very good. Just needs the be E.Q.`d a little to be more natural sounding. Easy fix really.

There can be no discussion of Mathematics without mentioning the Bakshail Manuscript and the preceding knowledge. This is just to start with, there numerous other numerous other manuscripts in the Vedas and related texts that expouses previous studies in Mathematics and Astronomy. This lecture is incomplete and lacks substance in the absence of history

nothing about the mezoamerican math like the maya?

11:01 Rubaiyat is one of my favourite reads and memorizations. Omar was a mathematician of considerable penetration who also gave expression in poetry to Persia's freest thought.

Brilliant lecture… But he was unbelievably brutal to that one student who asked the question he didn't like… Very demeaning… Spoiled the whole thing for me…

Dear Prof John Dersch, this is great history, perspective on mathematics. Can you please do one as a sequel to this one on the 20th century mathematics development so we can tie all the new developments and new branches of development together?

50:42 to 51:20 "You can't prove any system is consistent. Godel proved that." In which system did he prove that?

Fargen brill. It is so good when lecturers know their stuff and are prepared to part with it.

So the west went & TOOK ALL THE SOLUTIONS to problems developed by the east, & pretend that those who created the solutions to problems that You can't imagine EVEN TODAY… the problems you claim couldn't even EXIST as per yPur christian assumptions, were STUPID ANIMALS just trying to get to the piece of cheese? They were able to find & calculate not only the rotation & revolution of the earth, but EVEN THE WOBBLE, are all fools, & YOU are the ones who PATENT THEIR KNOWLEDGE to be "refined?" You refined blokes can DO NOTHING BUT STEAL, appropriate, & lie about the real person who did/created it. No wonder you christians & muslims put together have been able to bring the planet & almost all its liv=fe forms to the brink of extinction within 2000 years of BEING CREATED AS identity….

ALL the science that happened in europe happened 200 years AFTER THEY CAME INTO INDIA… even the MONEY & WEALTH you acquired FROM ROBBING INDIA, including your chemistry, medicine, botany…

That one girls laugh 😍😍 you know who I’m talking about.

war…

Omar Khayyam was also a great poet.

Algebra and calculous were used in India from ancient times for navigation, agriculture and art-Thru Agama in temple constructions etc. Jesuit priests from 1500 onwards translated the Indian texts and sent to Europe. Their it was not understood by them properly. Newtons laws failed miserably due to incomplete understanding of calculous.https://www.youtube.com/watch?v=IaodCGDjqzs

i always think mathemathicians are failed scientists or dropout physicists, like they couldn't take the heavy learning of physics or engineering,which require not just a very well knowledge of math but of other branches of science…so after 4 years of struggle or so they just went '' oh fuck this physic shit ,am just gonna teach algebra '''

bullshit,the arabs invented nothing

What a wonderful lecture. Thank you for posting this.

GREAT OFFER ,EVERYONE SHOULD TRY THIS ,ONLY 50 RUPEES http://imojo.in/2k734m This book tells about origin of math , numbers,different types of primes, some conjectures, some remarkable biography of mathematicians.

Very Interesting!

How can I get to the point?

Very dated. To the point of being just plain wrong.

It is utterly ignorant of this lecturer to claim mathematics used in Mesopotamia, India, Egypt and China purely existed without any proof! He needs to first educate himself at the very least about those regions to learn that all branches of science including mathematics, astronomy, chemistry, and so on are rooted in those regions. They experimented their theories and generated proof and therefore were able to develop it further. The Romans only expand it from that point on.

It is disturbing to see uneducated people such as John Dersch are lecturing our children!!

Pity about the sound quality, fascinating though.

No mention of any women? Emmy Noether? No wonder math was so slow to develop. Let's not make that mistake ever again.

Speaker I will urge you to listen to Prof CK Raju

The students there are very lucky.

Such a beautiful presentation. Great job.

Sumerian cuneiform tablets : 3600 bc . 100 000 bc : cave men were drawing abstract symbols on the walls & they were most probably able to count what they were hunting . They were also counting the number of full moons ( we have discovered very old bones on which stripes were made ) , this to figure out when would start the hunting season … Counting is VERY old …

Another westerner sucking up to the pedophile prophet.

They love to throw Hypatia in , to say it was not only about men. Is that really necessary?

the fun part of history of mathematics is that it has actually very little mathematics and more anecdotes and curiosities. It gives you the feeling that you are smart……

My volume is 100% and still low.

Virtually no scientific papers come from the Islamic world now. Centuries of breeding your cousins will lower your IQ.

Well this class was worth my pound of flesh 😂😂😂

Hard to imagine a world without calculus and algebra. I guess that's what they all thought about geometry and counting numbers.

Thx god finally a math video with non-hindu accent

He does an excellent job of conveying the, gravity, of Principia and its influence on just about everything in Science

perfect!!

Love this guy

I don't really understand what he means with Calculus vs The Calculus? Anybody reading this who can explain?

If you are talking about history of mathematics, then you must properly study history. And study history with curiosity, without prejudice. If you had done that, you would have mentioned what Indian mathematicians were doing.

One who knows history and sees the millennium from a distance, he can only but laugh. How the narrative has changed. People who couldn't fix a calendar properly are claiming to have discovered nearly everything in mathematics. This only serves them, not mathematics.

Logic involves infinite axioms, but mathematical rules/law/algorithm involves finite axioms. Logic cannot be proved within the propositions but mathematics can be proved.

Quantum computation provides superposition of bits called qubits that provide an insight into multi dimensional systems.

The whole universe is a quantum computer-Maldecena. Life is a quantum computer-Penrose/Hagelin/SMNH.

Boole is the greatest mathematician.

Ramanujan proves man and god are entangled and reads the 'mind of god'.

It's still Unicorns and Fairies are valid math axioms.

finally logarithms make sense

TURINGG 🙂

Science is only between Europe and the Middle East.

Stupid blow hard

53:24 which book is his favourite?? I can't see what he's pointing at!!!

https://www.goodreads.com/book/show/17993513-euclid-and-jesus <–

India –> Arabs –> Greeks, not the other way around; Al Jabr was actually copied from the Hindus, as were the numerals, as was geometry. Al Khwarizmi gives credit to the Hindus, later they simply left out that detail for some ridiculous reason.

Fibonacci numbers are in fact Hemachandra numbers

Stunning. I like his lecturing style. Much better than what I experienced most of the time…

Fantastic lecture

there's a place called gaylord? lmao

that's Tom Hardy lol

Good practice for my alpha theya

52:26 "Claude Shannon is native – he was born in Gaylord".

We Indians already established our maths initial surviving proof as veda. In yajurvedha we already have counting in 10 base, edited/organised date is around 3762 bce as of now…

This actually lessened my unknown algebraic symbol fear …it is not that a different symbol means a whole knew different knowledge of mathematics which is way way different than everything…it is not!

Euclid is a Myth. Most of the European knowledge are appropriated due to Vatican Inquisition. Open your mind, read history and don't conclude, keep read all views.

https://www.youtube.com/watch?v=MvpuC7Dg4e0