### The Map of Mathematics

The mathematics we learn in school doesn’t

quite do the field of mathematics justice. We only get a glimpse at one corner of it,

but the mathematics as a whole is a huge and wonderfully diverse subject. My aim with this video is to show you all

that amazing stuff. We’ll start back at the very beginning. The origin of mathematics lies in counting. In fact counting is not just a human trait,

other animals are able to count as well and e vidence for human counting goes back to

prehistoric times with check marks made in bones. There were several innovations over the years

with the Egyptians having the first equation, the ancient Greeks made strides in many areas

like geometry and numerology, and negative numbers were invented in China. And zero as a number was first used in India. Then in the Golden Age of Islam Persian mathematicians

made further strides and the first book on algebra was written. Then mathematics boomed in the renaissance

along with the sciences. Now there is a lot more to the history of

mathematics then what I have just said, but I’m gonna jump to the modern age and mathematics

as we know it now. Modern mathematics can be broadly be broken

down into two areas, pure maths: the study of mathematics for its own sake, and applied

maths: when you develop mathematics to help solve some real world problem. But there is a lot of crossover. In fact, many times in history someone’s

gone off into the mathematical wilderness motivated purely by curiosity and kind of

guided by a sense of aesthetics. And then they have created a whole bunch of

new mathematics which was nice and interesting but doesn’t really do anything useful. But then, say a hundred hears later, someone

will be working on some problem at the cutting edge of physics or computer science and they’ll

discover that this old theory in pure maths is exactly what they need to solve their real

world problems! Which is amazing, I think! And this kind of thing has happened so many

times over the last few centuries. It is interesting how often something so abstract

ends up being really useful. But I should also mention, pure mathematics

on its own is still a very valuable thing to do because it can be fascinating and on

its own can have a real beauty and elegance that almost becomes like art. Okay enough of this highfalutin, lets get

into it. Pure maths is made of several sections. The study of numbers starts with the natural

numbers and what you can do with them with arithmetic operations. And then it looks at other kinds of numbers

like integers, which contain negative numbers, rational numbers like fractions, real numbers

which include numbers like pi which go off to infinite decimal points, and then complex

numbers and a whole bunch of others. Some numbers have interesting properties like

Prime Numbers, or pi or the exponential. There are also properties of these number

systems, for example, even though there is an infinite amount of both integers and real

numbers, there are more real numbers than integers. So some infinities are bigger than others. The study of structures is where you start

taking numbers and putting them into equations in the form of variables. Algebra contains the rules of how you then

manipulate these equations. Here you will also find vectors and matrices

which are multi-dimensional numbers, and the rules of how they relate to each other are

captured in linear algebra. Number theory studies the features of everything

in the last section on numbers like the properties of prime numbers. Combinatorics looks at the properties of certain

structures like trees, graphs, and other things that are made of discreet chunks that you

can count. Group theory looks at objects that are related

to each other in, well, groups. A familiar example is a Rubik’s cube which

is an example of a permutation group. And order theory investigates how to arrange

objects following certain rules like, how something is a larger quantity than something

else. The natural numbers are an example of an ordered

set of objects, but anything with any two way relationship can be ordered. Another part of pure mathematics looks at

shapes and how they behave in spaces. The origin is in geometry which includes Pythagoras,

and is close to trigonometry, which we are all familiar with form school. Also there are fun things like fractal geometry

which are mathematical patterns which are scale invariant, which means you can zoom

into them forever and the always look kind of the same. Topology looks at different properties of

spaces where you are allowed to continuously deform them but not tear or glue them. For example a Möbius strip has only one surface

and one edge whatever you do to it. And coffee cups and donuts are the same thing

– topologically speaking. Measure theory is a way to assign values to

spaces or sets tying together numbers and spaces. And finally, differential geometry looks the

properties of shapes on curved surfaces, for example triangles have got different angles

on a curved surface, and brings us to the next section, which is changes. The study of changes contains calculus which

involves integrals and differentials which looks at area spanned out by functions or

the behaviour of gradients of functions. And vector calculus looks at the same things

for vectors. Here we also find a bunch of other areas like

dynamical systems which looks at systems that evolve in time from one state to another,

like fluid flows or things with feedback loops like ecosystems. And chaos theory which studies dynamical systems

that are very sensitive to initial conditions. Finally complex analysis looks at the properties

of functions with complex numbers. This brings us to applied mathematics. At this point it is worth mentioning that

everything here is a lot more interrelated than I have drawn. In reality this map should look like more

of a web tying together all the different subjects but you can only do so much on a

two dimensional plane so I have laid them out as best I can. Okay we’ll start with physics, which uses

just about everything on the left hand side to some degree. Mathematical and theoretical physics has a

very close relationship with pure maths. Mathematics is also used in the other natural

sciences with mathematical chemistry and biomathematics which look at loads of stuff from modelling

molecules to evolutionary biology. Mathematics is also used extensively in engineering,

building things has taken a lot of maths since Egyptian and Babylonian times. Very complex electrical systems like aeroplanes

or the power grid use methods in dynamical systems called control theory. Numerical analysis is a mathematical tool

commonly used in places where the mathematics becomes too complex to solve completely. So instead you use lots of simple approximations

and combine them all together to get good approximate answers. For example if you put a circle inside a square,

throw darts at it, and then compare the number of darts in the circle and square portions,

you can approximate the value of pi. But in the real world numerical analysis is

done on huge computers. Game theory looks at what the best choices

are given a set of rules and rational players and it’s used in economics when the players

can be intelligent, but not always, and other areas like psychology, and biology. Probability is the study of random events

like coin tosses or dice or humans, and statistics is the study of large collections of random

processes or the organisation and analysis of data. This is obviously related to mathematical

finance, where you want model financial systems and get an edge to win all those fat stacks. Related to this is optimisation, where you

are trying to calculate the best choice amongst a set of many different options or constraints,

which you can normally visualise as trying to find the highest or lowest point of a function. Optimisation problems are second nature to

us humans, we do them all the time: trying to get the best value for money, or trying

to maximise our happiness in some way. Another area that is very deeply related to

pure mathematics is computer science, and the rules of computer science were actually

derived in pure maths and is another example of something that was worked out way before

programmable computers were built. Machine learning: the creation of intelligent

computer systems uses many areas in mathematics like linear algebra, optimisation, dynamical

systems and probability. And finally the theory of cryptography is

very important to computation and uses a lot of pure maths like combinatorics and number

theory. So that covers the main sections of pure and

applied mathematics, but I can’t end without looking at the foundations of mathematics. This area tries to work out at the properties

of mathematics itself, and asks what the basis of all the rules of mathematics is. Is there a complete set of fundamental rules,

called axioms, which all of mathematics comes from? And can we prove that it is all consistent

with itself? Mathematical logic, set theory and category

theory try to answer this and a famous result in mathematical logic are Gödel’s incompleteness

theorems which, for most people, means that Mathematics does not have a complete and consistent

set of axioms, which mean that it is all kinda made up by us humans. Which is weird seeing as mathematics explains

so much stuff in the Universe so well. Why would a thing made up by humans be able

to do that? That is a deep mystery right there. Also we have the theory of computation which

looks at different models of computing and how efficiently they can solve problems and

contains complexity theory which looks at what is and isn’t computable and how much

memory and time you would need, which, for most interesting problems, is an insane amount. Ending

So that is the map of mathematics. Now the thing I have loved most about learning

maths is that feeling you get where something that seemed so confusing finally clicks in

your brain and everything makes sense: like an epiphany moment, kind of like seeing through

the matrix. In fact some of my most satisfying intellectual

moments have been understanding some part of mathematics and then feeling like I had

a glimpse at the fundamental nature of the Universe in all of its symmetrical wonder. It’s great, I love it. Ending

Making a map of mathematics was the most popular request I got, which I was really happy about

because I love maths and its great to see so much interest in it. So I hope you enjoyed it. Obviously there is only so much I can get

into this timeframe, but hopefully I have done the subject justice and you found it

useful. So there will be more videos coming from me

soon, here’s all the regular things and it was my pleasure se you next time.

True! But it's such a painful journey to get to the epiphany haha

thank you very much, thanks from Brazil

Thanks

Mathematics made up by us humans, so they say ? people who think that have no common sense – by the way, with all due respect.

ngl i still dont care and still hate maths

Really interesting, thank you.

Yaaaaaawwwwwnnnn. Where are the naked ladies? LOL

Would you mind doing maps on human sciences, like History, Linguistics, etc?

I have to take calc 1, calc 2, discrete structures, and linear algebra

my head started to hurt after hearing complex math name

lol i want the bead thats running in the background

Math will be much more easier to learn if you have a very good tutor. Let's find out via this link: https://www.youtube.com/watch?v=TaRoOJKutE8

very good.but you didn´t said that the first person who was using math matic is (Edris).And he was the first profed.

a very informative video.

my advice: please come to the front of the camera.

you will be able to convey your message better and you will also become a youtube celebrity.

You can have super affinity to others while being oblivious to others. For example I’m highly adept in manifolds and complex analysis but I’ll stupid to statistics

Keep doing these great things

The problem with me is I can understand the concepts of physics , but when it comes to theoritical

I'm busted 😔, coz of math.

Any idea how to over come this ??

I'm even poor at basic math 🤯

Once you've been bitten by the Mathsbug theres no going back.

Amazing

Before the invention of zero how people use to count things above 9

Because after 9 there is 10 which includes 0

Do they were having any other number system ?

Can anyone help??

Mathematicians would probably view "number systems" as foundations (in addition to others), "structures" as algebra, "spaces" as geometry and topology, and "changes" as analysis.

1 is not a prime number.

Mobius strip…..iron Man said something bout it in endgame

good and very interesting video, hopefully this is useful 👍

Is it possible to get theories of solving mathematical problems as in geometry as in algebra like example s=a2 or x-2=y so x= y+2

at 4:47 coffee cups and doughnuts are the same thing.

well, i thought about it and i realised : humans and doughnuts are the same thing too!

I believe that zero was first used in the arab world .. Not india

Is tensor calculus part of vector calculus?

I stick with frequencies https://youtu.be/W4ldx08KnLA

Başlık Türkçe video ingilizce hdjjdjc

Aswome work. I want to learn more.

Who all went searching for this video

Fucking music. Channel blocked. Bye.

PLS follow '3blue1brown' channel and 'welch labs'

Is there any print outs of this?

Okay why tf is this so interesting

zero was invented in the city of bagdad not in india

Wonderful explanation of mathematics mapping

After i watch entire video… I still doesn't like mathematics

10:13

My new wallpaper 🙂

hi from Russia, you opened my eyes

Hii

Iam

Sekhar. Request. Mathematics. Problems. Did. Not.. Multiple plaction. Sir.

India contributed 0 a starting of number is 0

🇮🇳🇮🇳🇮🇳

Awesome video,…. you know, what makes someone a better teacher and also makes the world a better place are those who lack hubris. You've taught me a lot with this video, thank you.

That was fantastic!!

We've come so far. what a time to live

I appreciate your effort and work! Brilliant! Like you, moments of epiphany here and there but Goedel comes up and says it's just a useful tool, not the ultimate truth!

Thank you so much

You jumped over lot of things by the way 0 is arabic also the way we write numbers as knowing now 1234567890 vs i ii ii iv v vi vii vii ix x….

Also the unknow values س "sin" named x

Lot of thnigs in math chimistry physics astronomy etc

Why the occident hate to tell the reality of islamic civilisationS

Why the spend lot of money to make islam and muslims monsters

Cause islam want equity and freedom the real freedom not slavery to any system

Probability is NOT considered as applied maths.

So this is how the largest student community hating a subject started.

like for his efforts to mapping everything and ours to understand that

2:37 zero is a natrual number

maybe u sould put more details

Maths are cool

But alike other subjects, school makes it suck

I gon subscribe cos u didnt ask for one and cos this vid is simply AMAZiNG

2:10 – "…that almost becomes like art." – like art? It is so MUCH more than art!

Mathematics is the father of Science😆💖👍👍👍

Now when i see a man who litterly lights my mind and reconnect me with math … I subscribe ♥️♥️♥️

Zero is stupid it's just nothing so no one needs it and negative numbers are less than nothing so those are even worse. Imaginary numbers do not even exist so those are nothing as well. The only real math uses real numbers.

is this video always available in german? that would be wonderful. <3

The Maya is too the civilitation that used a symbol for zero

The moment I realized Maths were cool as hell was the moment I wrapped my head around the phrase “Some infinities are bigger than other infinities”

Algebra was found by al-Khwarizmi and he is an Arabic mathematician not Persian.

3+1=4-1=3

Quick maths

It could be that I’m just too immature as a math student, but some examples given in the pure math section sounds very inaccurate to me. I thought it’s gonna be classified by analysis/algebra, or set theory, topology…etc.

math is fun when you understand

What software is this animation made with?

nobody:

domain of science: Renineeeeeeeeeyy-sonce

3:04 I like that pause lol

The video is excellent, thank you for your job!

I just have a remark about Gödel. Gödel's incompleteness says what you say but not what is written: there is no COMPLETE consistent set of axioms. This does not mean at all that maths is just made up by us: it simply means that we cannot capture it exhaustively by finitely many postulates (or postulates that can be generated by a finite program).

Besides, Gödel's completeness theorem gives a nice counterpart to incompleteness. Any time there is an undecidable proposition, it is for the best of all reasons: there are two worlds that satisfy our postulates, one where the proposition holds and one where it does not.

But apart from this detail, I find this video to be a cool contribution to making easy to feel the wholeness of mathematics, thanks again!

i love maths now

Map about electronic engineering !

i am very happy to see it because its very interesting and useful everyone……..

really nice video!

Is it possible that you could do a map of Economics or Finance?

Sure, Dolphins can count. What else is a creature who swims in circles all day going to do? "That was the hundredth circle today.. let's go for 150."

But hey, that's just a theory, a 7:19

Pictures are so cute

Very Well Done 🎁

Thank u

Mathematics🖤

Where u MAPUANS?

Universities should take a look and rearrange their math courses. My curriculum only says Analysis I, II, III and Algebra I, II, III …

Very informative and well designed presentation. Thank you.

u forgot googology.

nerd

Fuck. Fractals. My little head.

Friend now has the map of PSYCHOLOGY

I´ve always been more of a person of words, but in mathematics I've found a kind of a new language with a set of defined rules and everything. A languaje of order and caos, but most importantly a beautiful one.

http://engineering-and-science.com/mathematics-basic-formulas.html

Woe

Which all at the end is useless for the 99.99999999999999999999 of us lay people

1+1 = 3 OR MORE , IF you DON'T HAVE CONDOM

isnt cos a/h not o/a wise guy

.

The root of math is the minds capacity for abstraction. We use that capacity in many nonintuitive ways, but since mathematics is intuitive, it is the one we know best.

I have a poster if this in my maths room i knew i saw this before

Great video!