# 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.

Thank you!

Great video!! But shouldn't probability and statistics be classified under pure mathematics!!???

Some Senku shii right here.

Computer science: the application of pure mathematics in the most explicit and complete encompassment of all possible permutations of the problem

Which type of maths explains why 2300 idiots clicked thumbs down on this?

Sounds interesting, I would like to really like and be good at maths, but just survive my classes

The Mayans used 0 first, not the indians

“A Rubik’s Cube is an example of a permutation group”

Cayley: “Yes”

1:53 is the trippiest thing I've ever seen

This is all really cool, but I only hate math even fucking more now

Addition is cool

Some infinities are bigger than others? That's just wrong. Litteraly incorrect

I wish all videos on YT were as informative as this one. You just earned a sub, sir!

Mathmatics is the queen of science..

hi dos i’m dos

2=2

8:00 Optimization is at the heart of Artificial Intelligence and it's fairly simple

Seeking the magic eye in math where it all pops into focus

https://www.youtube.com/results?search_query=magic+eye

are you gonna do one on economics?

tan theta = o/a not cos theta

pa que hijueputas ponen el titulo en español?

Combine all of those maths and add chemistry to it and you might be God in making “Man”.

But hey that's just a theory, a game theory

Could u do it in spanish? 🙁

It's funny the foundation part, where idiots argue that math is a universal language to aliens. NOPE. As to question, if God created us in his image, then our derived math makes sense to explain the creator's math.

???? bravo . liked

How come I’ve only done calculus the last 5 years in school?

When you stated, "the epiphany moment when you glimpse the fundamental nature of mathematics". I can't agree more it's an incredible feeling, it's an ecstasy that lasts for weeks or months. I've had the opportunity to experience it twice for unrelated maths research.

You missed trigonometry!!!!!!!

Youtube : this is mathmatics video for u

Me : …

Youtube : algebra?

Me : …

Youtube : statistics?

Me : …

Youtube : look so clever

Me : …

Youtube : r u korean?

Me : ok ok i'm watching it

I always got low marks in maths because I didn't understood the question properly.

Weren’t the Inca the first to use the number 0? Certainly not the civilization that shared it with the world, but the first to use it

Where is category theory ? Homotopy theory ?

mathematicians are a pain in the ass, for they just create more problems than they solve

Errrrrrr…yes

2:55 when you said some infinity’s are greater than other I understand what you mean but one infinity cannot be greater than another infinity think of it like this there are as many even numbers as even and odd numbers combined. No matter what there is no greater infinity. Otherwise I loved the video

That is not a donut, it is a Torus.

In part of the video it sounds like someone is playing a baseline from The Killers a few rooms over very faintly.

Wow this made me what to lrean math thank you

I want this map printed on a wall of my bedroom

what! you have a poster of this too. dam if i could get one. maybe ill remember in a few months

Correction : Algebra was given by Indian mathematician

"Some infinities are bigger than others"

I love maths, i really do but why am i watching this at 2am i need to sleep smh

"Math is fun but lemme first inject fear directly into your core first"

@1:30 Why is the

realandimaginaryaxis labels swapped?So probability and statistics is not part of pure mathematics?

I love math… but I now know I have a long way to go:P

Dude Hindus have been using mathematics since your great grand father was learning how and where to shit… get your facts straight Maths was first used by Ancient Hindu Astronomer, which later on muslims and christians defiled our peaceful land and took away all the Historical maths with them.. instant dislike…

0:30 seconds in.

So 50,000 bce humans are scratching on bones..

And in the years 3000 bce a group of people designed, engineered and built the most amazing building in the world..

To which the people of 2019 ad still cannot duplicate or fully understand.

Now the real quesion is Why Americans uses pound,miles while everyone around the world uses S.I. units !!

You forgot commercial mathematics

cos = o/a thats wrong tf?

By the way, I discovered a new, previously completely unknown formula for the area of a right triangle. I call it "NordaVinci's Formula." Here it is R(R + H) = X(H – X). Here's a link to my "NordaVinci's Cat" poster with the Cat thinking about the derivation of NordaVinci's Formula, that you can purchase for $20 – the poster, not the Cat. Be one of the first in the world to know something about the Right Triangle that is not in the Elements of Euclid Math Book or any other Math Book in Human History! Be one of the first to be able to brag to your trigonometry teacher that you know something that he or she doesn't know about the area of a right triangle! Sell the poster to your teacher for only $30. Profits are for a non-profit group involving math, science and technology!

https://www.flickr.com/photos/nordavinci/49183141018/in/dateposted-public/

Now I understand why I hate math. My teachers didn't understand shit.

Can you fix your dice drawing?

5:10 hmmmmm baby i dont understand this

You have half and you made video out of it . First book of algebra is not written in Persia it was a translation from original text which reached Persia by trade

Wonderful job. So insightful. So well articulated

When you don't mind any tree but changes

Dam you calculus for stopping me from being stupidly good at something.

Changes is by far the hardest tree in my opinion when you look at math like this

Where's mensuration

mostly Kabbalah. Comunication between mathematicians and rest of the world would be much easier if mathematicians have recognised the math as fundamentaly creative activity and not some superior striving for the Truth

godel's teorem (left for the very end of the presentation) denied the whole left part of the map

Pq le titre et en français alors qui parle englais ?!

Et

Qui a rien compris vu que c'est en englais? ??

amazing video

Wrong the zero was invented by the Mayans before India

Very good summary . After 21st century .Pure Maths should upgrade to Virtual Maths ? And Apply Maths should change to Real Maths ? All these should classified as EARTH Maths ! Do we have UFO/Universe Maths ????

i consider myself decent at math but the second u throw me any applied mathematics i blank so fast and feel so so so dumb

Sooooo… If I want to study maths let's say from 0 scratch… WHERE THE FUCK SHOULD I STARTTTT!!!!

Excellent ?

Thanks

Whiat is the right path which a uni student should take to cover all the map

AMAZING thanks for sharing!! niceeeee

2:49 1 is not prime!

Sir negative numbers were first used in India

You can check it on Wikipedia

the map of headache lol

I gave up me renewed interest in maths when the guy said, some infintes are bigger than others. Let me just accept my limited brain capacity and continue browsing youtube.

You left out Fourier Analysis! 🙁

Great video man

10:06 OMG YES. It's like feeling a new type of orgasm every new time you open a new door.

Thanks for that, Dom! Great explanation.

A teacher in Highschool said that mathematics is not the answer, is the lenguaje in which you ask and speak to the universe; you just need to do the right question

I keep getting math videos recommended to me. God must have something prepared for me

that made dizzy

FANTASTIC

AMAZING

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

I adore Mathematics so much but pity myself that I don't put any effort in studying it. Sometimes I start studying and hit the rock bottom of my understanding, then everything starts to become more and more difficult, my mind breaks into pieces with my confidence and there ends my interest for that time.

OK, I'm stuck at 1+1=2.

An artifical intelligence developed by Google learned the all information accumulated in the history of humanity in 3000 years in 40 days

When mathematics you learn in school wasn't enough to punish you

Thank you for this video. It clarified why I love Mathematica.

What about proofs? I guess that would kind of fall under fundamentals

it is awesome !! thank you

majored in applied math in college and now friends just ask me to calculate tip or split the bill because that's what they think i did in my classes…

Ultimate presentation

…why do i love theoretical math?

unfortunately here in Brazil we dont have access to an education of quality, only whom has money paid for them. and the poor people continues to get jobs in situation of semi-slavery

Anyone get recommendation of this video after started learning datascience and Machine learning

01:23 I'm so proud I recognised Leibniz

Awesome video!

Hey! Which app was used to create theses wonderfully animations?

By the way, the π is not a Real Number — vice versa — it is Transcedental!