The map of mathematics | 现代数学启蒙

slug

type

status

tags

password

icon

The mathematics we learned in school doesn’t quite do the Field of mathematics justice.

We only get a glimpse of the one corner of it

but mathematics as a whole is a huge and wonderfully diverse subject.

My aim with video is to show you all of that amazing stuff.

We will 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.

Evidence for human counting goes back to prehistoric times, with checkmarks made in bones ,there were several innovation over the years with the Egyptian having the first equation.

The ancient Greeks making strides in many areas like geometry and numerology.

Negative numbers were invented in China, zero as a number was first used in India.

Then in the golden age of Islam Persian mess Mathematicians made further strides and the first book on algebra was written.

Then mathematics boomed in the renaissance along with the science.

Now there are a lot more to the history of mathematics than what I’ve just said.

But I’m gonna jump to the modern age in the mathematics as we know it now.

Modern mathematics can broadly be broken down into two areas. Pure maths ,the study of mathematics for its own sake and apply maths.

But there’s a lot of crossover. In fact many times in history Someone is 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 is nice and interesting but isn’t really do anything useful, but then say 100 years later someone will be working on some problem at the cutting edge of physics or computer science.

And they will discover that these old theories in pure math is exactly what they need to solve their real world problems which is amazing I think.

This kind of thing has happened so many times over the last centuries it’s interesting how often something so abstract ends up being really useful.

But I should also mention that 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 almost becomes art.

OK enough of this highfalutin~ Let’s get into it.

Pure math is made of several sections. The study of the numbers starts with the natural numbers and what you can do with them with arithmetic operations.

Then it looks at other kinds of numbers, like integers which contains negative numbers ,rational numbers like fractions Real numbers which include numbers like pi, which go off to infinite decimal points, and the complex numbers and a whole bunch of others.

Some numbers have interesting properties, like prime numbers or pi or the exponential, they’re all properties of these number Systems example ,

for example even 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 structure 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 this equations.

Here you will also also find vectors and matrices which is multi dimensional numbers and the rules of how they’re related to each other are captured in linear algebra.

Number theory studies the features of everything in the last section on numbers.

Like the property of prime numbers, combinatorics looks at the property of certain structures, Like trees ,graphs and other things that are made of discrete chunks that you can count.

Group theory looks objects that are related to each other in groups, A familiar example is a Rubik‘s cube, which is an example of a permutation group.

And all the order theory investigates how to arrange objects following certain rules. Like how something is a larger quantity than something else.

The natural numbers are in 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 geometrics which includes Pythagoras and is close to trigonometry which I’m sure we are all familiar with from 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 they always kind of look the same.

Topology looks at different properties of space where you are allowed to continuously deform them but not tear or glue them.

For example Mobius strip has only one surface and one edge, whatever you do to it , and coffee cups and the donuts are the same thing topologically speaking.

Measure Theory is a way to assign values to spaces or sets tying together numbers and spaces.

Finally differential geometry looks at the properties of shapes on curved surfaces

For example triangles have got different angles on a curve surface.

Bring us to the next section

Which is changes

The study of changes contains calculus

Which involves intergrals and differentials

Which looks at the spanned out by functions, or the behavior of gradient of functions.

Vector calculus looks at the same thing but for vectors. Here we also find a bunch of other areas like dynamical systems

Which looks at systems that involve in time from one state to another state, like fluid flows or things with feedback loop 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 probably worth mentioning that everything here is a lot more interrelated then that I have drawn in reality this map should look more like a web, tying together all the different subjects.

But you can only do so much our a two dimensional plane.

So I have laid them out as best as I can.

We will start with physics which uses just about everything on the left hand side to some degree. Mathematical and

Theoretical physics is a very close relationship with Pure math .

Mathematics is also used to in the other natural science for example mathematical chemistry and bio mathematics , which look at loads of stuff from modeling miraculous to evolutionary biology.

Mathematics is also used extensively in the engineering.

Building things has taken a lot of maths since Egyptian and Babylonia times very complex electrical systems like aero planes or the power grid.

Use methods in dynamic systems called control Theory numerical analysis is a mathematical tool commonly used in places where mathematics becomes too complex to solve completely. So instead you use lots of simple approximation and combined them all together, to get a good approximate answers for example if you put a circle inside a square throw darts at it,

And then compare the number of dots in the circle in the square portions, you can approximate the value of pi, but in the real word numerical analysis is done on huge computers,

Game theory looks at what is the best choices are, Given a set of rules and rational players, it’s used in economics when the players can be intelligent, but not always ,in other areas like physiology and biology.

And other areas like psychology and biology .

Probability is the study of random events, like coin tosses or dice or humans

In the statistics is the study of large collection of random processes or the organization and analysis of data, this is obviously related to mathematical finance, where are you want to model financial assistance and get an edge to win all those fat stacks.

Related to this is optimization where you are trying to calculate the best choice. Amongst a set of many different option or constraints.

Which you normally visualize is trying to find the highest or lowest point of a function.

Optimization problems is the second nature to us humans, we do them all the time trying to get the best value of money or trying to maximize a happiness in someway.

Another area that’s very deeply related to pure mathematics is computer science And the rules of computer science were actually derived in pure math. And is another example of some thing that was worked out and is another example of something that was worked out ,

way before Programable computers were built

Machine learning the creation of intelligent computer systems uses many areas in mathematics like linear algebra optimization dynamical systems and probabilities,

finally the Theory of cryptography is very important to computation. And uses a lot of pure maths like Combinatory and the number Theory.

So that covers the main sections of pure and applied mathematics, but I can’t end it within looking at the foundation of mathematics.

This area tries to work out 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 the mathematics comes from? And can we prove that it’s are all consistent with itself?

Mathematical logic ,set Theory and category theory try to answer this in a famous result in mathematical logic,

Are Godels incompleteness theories ,

Which for most people means that mathematics doesn’t have a complete and consistent set of axioms, which means that it’s all kind of made up by as humans,

Which is weird as mathematics explain 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.

We also 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 an isn’t computable,

And how much memory in time you would need, which for most interesting Problems is an insane amount,

so it is the map of mathematics, the thing of learned I’ve loved most about learning maths is that feeling you get where’s some thing that seems so confusing finally clicks in your brain

And everything makes sense like an epiphany moment kind of like seeing through the matrix .

Kind of like seeing through the matrix

In fact some of my most satisfying intellectual moments , have been understanding some part of mathematics, and the same 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.