## 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.

## LESLIE TURNER

Jul 7, 2019, 1:16 pmI was terrible at mathematics & computers until I started learning how to operate Microsoft Excel via YouTube tutorials for various business and Financial reconciliation, compilation and consolidation. The beauty of numbers is in everything and they're confirmation of the existence of God. The numbers can help you see how short and precious life is and that accounting for dollars and things over people is why we're miserable as a whole. If we invested more time in accounting for people and what positive attributes we share, we would be able to predict a more positive future outcome by unlocking the individual's true potential. The key is to drown out the material noise and distractions in our physical environments in order to recognize this constant flow that is rooted in all of us across all times and places. If someone offered me a trillion dollars if I could count the trillion dollars, I would decline because the time it would take would exceed the time I'm willing to physically & mentally sacrifice to execute the job. We've been gifted with mathematics in order to extend and optimize life as long as that is the intention.

## _ hdj07_

Jul 7, 2019, 7:02 pmI thought zero was created by Arab countries

## كوتشيكي روكيا

Jul 7, 2019, 12:39 pmشكرا على هذا الفيديو حقا رائع تستحق أكبر لايك?????❤

## Reza Kian

Jul 7, 2019, 4:21 pmAm I missing something? Where does real analysis fit in?

## Akshika Wijesundara

Jul 7, 2019, 4:26 pmThanks man! This is quite useful.

## Martin Nolte

Jul 7, 2019, 6:02 pmWe have learned half of that in school

## No no

Jul 7, 2019, 6:04 pmgreat job thank you!

## EDUARDO SANTOS

Jul 7, 2019, 2:14 am?

## Jonathan Davis

Jul 7, 2019, 3:26 pmPYTHON

## ianthespaceguy

Jul 7, 2019, 7:00 pmThis was awesome, thanks.

Phat stacks lol

Physics is Phun:)

## Ale xis

Jul 7, 2019, 9:52 pmThank you!!! I was clarifying to me

## KMS

Jul 7, 2019, 12:11 amI have a hard time with math but enjoy others talking about it. It’s all so interesting.

## Potato Tomato

Jul 7, 2019, 12:13 am7:59?

## Mario Abarca

Jul 7, 2019, 5:34 amI have never liked the idea of separating the so-called “pure” mathematics from applied mathematics. This categorisation si vague: optimisation and game theory can be very abstract, and also theory of computation is actually used in a lot of applications.

## gilad klos

Jul 7, 2019, 8:13 amI think differencial equations should be in the middle between theory and applecation since most of them are used in order to describe physics

## sakshi mehta

Jul 7, 2019, 1:39 pmNegative numbers are also invented in india , if you want you can check.

## СВЭП

Jul 7, 2019, 2:05 pmClassifying general statistics (not financial or game theory) to applied math seems questionable to me, especially now, when due to quantum physics we know that universe in depth has pure random nature, as numbers themselfs (as far as I concerned) have random properties, for instance, distribution of primes among natural numbers.

## Emma Add

Jul 7, 2019, 2:50 pmComplexe numbers is the best subject in maths ??

## Atul Sarmah

Jul 7, 2019, 2:57 pmI learned all of this

## N J G

Jul 7, 2019, 12:46 amWhenever people say "maths" it just makes me think of what they might actually be saying: "mathematics-es" lol

## Dissolute Void

Jul 7, 2019, 4:43 amQuaternians and octonians were forgotten

## john louie abendan

Jul 7, 2019, 1:51 pmIt would take me 5 billion years to learn all of this

## Sami Loul

Jul 7, 2019, 2:14 pmThank you ❤️

## ArachnosMusic

Jul 7, 2019, 7:08 pmFor a second I thought you'd forgotten Set Theory

## Kameron Briggs

Jul 7, 2019, 7:25 pmNow how does graphics programming tie into this? ??? Lol

## Angelo Angelo

Jul 7, 2019, 11:43 am"Some infinities are bigger than others."

Ah yes, I see you're a fan of The Smiths too.

## so _weird

Jul 7, 2019, 1:40 pmMan,i wish i was Omniscience?

## Lucas Oliveira

Jul 7, 2019, 3:04 pm"Mathematics is the language with which God has written the universe"

## Skylar Demsworth

Jul 7, 2019, 5:03 pmThank you

## Joel Doxtator

Jul 7, 2019, 7:04 pm1 is not a prime number.

## Renan Dias Rennó

Jul 7, 2019, 9:28 pmWhat about induction proof?

## Cameron Oliver

Jul 7, 2019, 9:36 pmDislike for using BCE

## Cameron Oliver

Jul 7, 2019, 9:42 pmIt's very naive to say that Godel's incompleteness theorem means that mathematics is made up by humans. The same fundamental relationships can be seen through different sets of axioms: the relationships themselves are objective, and there is ultimately not that much freedom over which ones you choose; but even if there was, mathematics is all discovered given the set of axioms commonly chosen, which could be chosen by any aliens.

## Ricardo Espinel

Jul 7, 2019, 1:10 amThis is one of the most beautiful videos Ive had the fortune to come across, thanks very much, I might buy it and print it on my wall to keep the inspiration every morning

## Demitri Kennelkoff

Jul 7, 2019, 2:26 amBecause we think we made it up, but rather we were allowed to discover it when we were ready.

## Hello HELLO!!

Jul 7, 2019, 7:03 amI hate maths.

## Ben Lee

Jul 7, 2019, 12:26 pm1:56 thats trippy…

## Afayanl

Jul 7, 2019, 4:52 pmSo …I can blame Iran for Algebra?

## Afayanl

Jul 7, 2019, 4:53 pmI think Math has a lot of problems and should see a therapist ASAP.

## Daisy C

Jul 7, 2019, 6:44 pm……………… pure maths makes me wanna die

## Recho Rides

Jul 7, 2019, 8:34 pmroses are red

violets are blue

the titel is German

so why aren't you

## John Concha

Jul 7, 2019, 5:25 amGood video. All of your dice, save one, are incorrect, however. Opposite sides of a regulation die add to seven.

## Human Being

Jul 7, 2019, 12:33 pmYou are so cute ❤

## Leweson

Jul 7, 2019, 1:58 pmAt 1:40 the coordinate system is wrong. The imaginary part ist the vertical line!

## Ayoub AÏT

Jul 7, 2019, 6:10 pmGreat work. Much Appreciated.

## james thomson

Jul 7, 2019, 1:40 amThis video helps me to sleep

Thank you dude!!

## Bobby

Jul 7, 2019, 2:14 pmThank you, that was fun.

## Sharif Azmi

Jul 7, 2019, 4:58 pmI learned engineering mathematics and you nailed it

## Abi Mannan

Jul 7, 2019, 5:33 pmFor the sectors with given m

## Jo Kah

Jul 7, 2019, 12:16 pm10:10 This is exactly why i love science: it is hard but it is rewarding with nirvana

## SpangeBab

Jul 7, 2019, 11:01 amWhat's the point of learning something and stressing about it when we will never use it in the future

## Kathy S.

Jul 7, 2019, 12:57 pm3:00 ouch…

## KayLow

Jul 7, 2019, 5:50 pmThis is the best video I've seen in a long long time

## Sibghatullah Sherazi

Jul 7, 2019, 8:26 pm❤

## No Idol

Jul 7, 2019, 9:21 amWhere do Discrete Mathematics and Abstract Algebra fit in?

## Vincent FreiWild

Jul 7, 2019, 2:14 pmAs far as I know one can not deduce out of Gödel's Incompleteness Theorems that maths is made up by humans

1. The impossibility to proof in a system of axioms that this system is coherent and non-contradictory does not imply that maths is made up neither that one is not abel to proof this for example in another system of axioms.

2. This Theorem refers to maths which is expressed within symbolic logic. It does not say that the contentual (formal) logic, which lays behind this, can not be fully coherent, it is just the symbolism, which is limited in its expressiveness. (Finsler is analysing this in his Theorems about Gödels work)

## active285

Jul 7, 2019, 3:24 pmWe are people still so ignorant about younger disciplines of mathematics? Modern probability is far from being only some "applied mathematics" or a helper for statisticians. The whole Itô theory extends ideas of integration from deterministic integrators to stochastic processes. Or, for example, I solve deep differential geometrical problems or problems in spectral calculus but using completely different, probabilistic, methods, namely Bismut-type formulae. Also other fields of probability like solving Lévy SDEs or extending existing theory to Feller-type processes is far from being applied mathematics.

## Gee F

Jul 7, 2019, 9:57 pmLearnt more in this video than I did in 3 years doing a degree in pure & applied!

## Vegan Wolf

Jul 7, 2019, 3:27 amWell done!

## Feras Hamdan

Jul 7, 2019, 4:58 amللاسف هذا الرجل لم يذكر فضل العرب ويتكلم بثقه غير مفهومه.

## dawonplace

Jul 7, 2019, 6:51 pmChinese invented negative numbers. Egyptians made first equations. India made 0. Thanks! God bless you.

## tan komi

Jul 7, 2019, 10:37 am…. it's terrifying

## Shanebagel

Jul 7, 2019, 1:06 amHow can I find a job doing one of these things?

## 루이스

Jul 7, 2019, 3:12 amFasinante.

## Dallas Kelley

Jul 7, 2019, 4:21 amIt's not really a mystery as to why Math works. It has to do with how we do science. We make an observation, and based on that observation we make a guess, test that guess, and if it doesn't make sense we tweak it. Math doesn't fit anything in the universe it just fits our perspective of the universe.

## norwin rejuso

Jul 7, 2019, 8:26 amI love math so much but, to be honest, I am not that totally good at it. And now I am taking Bachelor of Secondary Education and I am planning to take math as my major. Love the video.?

## Je Benz

Jul 7, 2019, 8:51 amYou mean the map of hell?

## Victoria Voloshyna

Jul 7, 2019, 4:07 pmThere should be operator algebra and functional analysis. Imo.

## Alain De Vos

Jul 7, 2019, 3:46 amIs it possible to put all the subjects of mathematics in the form of a tree ?

## AVERAGE BiRd

Jul 7, 2019, 10:49 amFinally I got …What i want .. Thank you

## Виталий Кузнецов

Aug 8, 2019, 10:33 amNice video! Thanks!

Make a video about a map of business, please! (economics, management, marketing, accounting, etc.)

It would be so cool!

## Pratik Ahamed

Aug 8, 2019, 9:58 amThanks.

## Blayde

Aug 8, 2019, 7:38 pmDoes anyone have a recommendation for a book that goes over all these topics more in detail?

## Lewis Damasco

Aug 8, 2019, 3:44 amYour video it’s just a quite well explained modelo of mathemathics, you captured the esence of every single part of maths thank you so much

## MRAbhi

Aug 8, 2019, 1:37 pmPersians picked algebra from India.

## YOO HOO

Aug 8, 2019, 10:36 pmThe important part of this whole math is #made up by us humans# Nothing else! And oddly it worked. Otherwise we wouldn't see Voyagers one and two travelling interstellarly!

## Corrado Campisano

Aug 8, 2019, 6:27 amdo you have one for physics?

## Corrado Campisano

Aug 8, 2019, 6:29 am@1:49 take quaternions, now on the rise with 3D graphics…

## Corrado Campisano

Aug 8, 2019, 6:38 am@5:42 oh look, a composite pendulum shown as the "first instance" of chaos theory: so good

but, actually, was it? or was the n-body-problem first? yeah, Galileo came before Newton, but…

you sure?

## Corrado Campisano

Aug 8, 2019, 6:41 am@6:02 well, sort of a web: knowledge is represented on graphs describing ontologies (rdf, owl)

## Aayush Ganesh

Aug 8, 2019, 2:57 pm5:23 not necessarily.

## Arthritic Frodo

Aug 8, 2019, 12:20 am0:06 cos theta = O/A lol what?

## Leoレオ

Aug 8, 2019, 2:23 pmThis guy is not counting the Maya culture in math.

## Imran khan

Aug 8, 2019, 2:36 amGreat

## Jayson Humag

Aug 8, 2019, 12:33 pmHow to learn all of this?

## Jayson Humag

Aug 8, 2019, 12:36 pmSimplex method

## Crawford Rhoderick

Aug 8, 2019, 7:35 pmIn the order of operations which did not come about until around nineteen hundred why is one right and the other wrong. During newton time did not exist when calculus came about.

## Chloroplast ✔

Aug 8, 2019, 5:46 amUniversal rule of math: you use a human brain

## Kami Kuru

Aug 8, 2019, 2:10 amWhy this video was more along the lines of the history of math I'm much more interested in the progression in the way which math should be learned starting with the very basics of numbers moving on up all the way to the algebra level

## MrGood

Aug 8, 2019, 9:29 amYou forgot a thing

Googology

## Auguestine Boakye

Aug 8, 2019, 1:52 pm???????????????

## JiprehBii

Aug 8, 2019, 2:13 amI get curious of math by watching this :3

## Just a Random Channel

Aug 8, 2019, 4:53 amI remember when I was very young and 2x3x3 was something which I was excited to learn about XD.

## Jnlie_judex credo

Aug 8, 2019, 12:14 pmWhy am I here? I dont even do math… smh

## 俞轶伦

Aug 8, 2019, 6:17 pmMath is everthing.

## Jesus Dominguez

Aug 8, 2019, 4:35 amMath is the language of the gods.

## Liam Martin 2: Electric Boogaloo

Aug 8, 2019, 9:44 amVideo ends @ 5:52 . Pure Mathematics Gang Rise Up

## Rohith Krishna

Aug 8, 2019, 2:35 pmI think your mapping skills are very good bro.?????

## Ambuj Kumar

Aug 8, 2019, 7:38 pmThis is wrong… how can china invent negative numbers without conception of zero…

## Jamal Abdulwahid

Aug 8, 2019, 5:43 pmHe started with a hisotirical error when he said the persian mathematicians. They are all arab mathematicians and all the mathematical books are written in arabic language in Baghdad and other great arabic cities.