We only get a glimpse at one corner of it, but the mathematics as a whole is a huge and

我們只瞭解了它的一個小部分，但事實上數學是一個巨大

wonderfully diverse subject.

且多樣化的一個學科。

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

這個影片的目的是要讓你看看這些令人驚訝的東西

We'll start back at the very beginning.

我們將從頭說起

The origin of mathematics lies in counting.

數學源自於數數

In fact counting is not just a human trait, other animals are able to count as well and

事實上，不是只有人類會數數，其他動物也能數數

e vidence for human counting goes back to prehistoric times with check marks made in

人類計數的證據能追溯到史前時代，當時人類在骨頭上做記號

bones.

有了埃及人的第一個等式在多年中有多次創新

There were several innovations over the years with the Egyptians having the first equation,

古希臘人在像幾何學和數字占卜而負數是在中國發明的，

the ancient Greeks made strides in many areas like geometry and numerology, and negative

零作爲一個數字在印度率先開始使用。

numbers were invented in China.

那樣許多地方走出一些步子。

And zero as a number was first used in India.

隨後在伊斯蘭的黃金時代中波斯數學家又走了幾步，

Then in the Golden Age of Islam Persian mathematicians made further strides and the first book on

並且寫下了第一本講代數的書。

algebra was written.

接著數學跟著科學一起在文藝復興時期發展壯大。

Then mathematics boomed in the renaissance along with the sciences.

除了我剛已說過的，數學史上現在還有更多的。

Now there is a lot more to the history of mathematics then what I have just said, but

但我將跳到現代和現在我們所知道的數學。

I'm gonna jump to the modern age and mathematics as we know it now.

現代數學大約可以分成兩個領域：純數學，為數學本身的緣故而研究；

Modern mathematics can be broadly be broken down into two areas, pure maths: the study

以及應用數學，為了解決真實世界的問題而發展的數學。

of mathematics for its own sake, and applied maths: when you develop mathematics to help

但其中有很多交叉的。

solve some real world problem.

事實上，數學史上多次發生某人純出於

But there is a lot of crossover.

好奇心和由對一種美感的引導而進入數學之荒野。

In fact, many times in history someone's gone off into the mathematical wilderness

而之後他們就創造了一大套新的優美和有趣的數學。

motivated purely by curiosity and kind of guided by a sense of aesthetics.

但實際上做不了任何有用的事。

And then they have created a whole bunch of new mathematics which was nice and interesting

但是後來，比方說一百年之後，有人在研究某個前沿物理

but doesn't really do anything useful.

或計算機科學是的問題時，他們將發現這個在純數學裡老的理論

But then, say a hundred hears later, someone will be working on some problem at the cutting

正是來解決他們在真實世界的問題說需要的

edge of physics or computer science and they'll discover that this old theory in pure maths

我覺得這實在是太奇妙了！

is exactly what they need to solve their real world problems!

而這類事情在過去幾世紀來已經發生過許多次了。

Which is amazing, I think!

這是很有趣的多麽經常一個如此抽象的東西最終是真正有用的。

And this kind of thing has happened so many times over the last few centuries.

但我也必須提到，純數學本身仍是非常有價值來做的事情

It is interesting how often something so abstract ends up being really useful.

因為它非常引人入勝并且就它本身而言可以有一種美麗和優雅，那

But I should also mention, pure mathematics on its own is still a very valuable thing

簡直就是像藝術一樣。

to do because it can be fascinating and on its own can have a real beauty and elegance

好了，這大話夠了，讓我們到這裏來吧。

that almost becomes like art.

純數學由幾個部分組成。

Okay enough of this highfalutin, lets get into it.

對數字的研究從自然數開始，以及

Pure maths is made of several sections.

你用它們可以做些什麽樣的算術運算。

The study of numbers starts with the natural numbers and what you can do with them with

然後它想著是其它類型的數字，這包括包含負數的整數，

arithmetic operations.

像分數那樣的有理數，實數，包含像π，小數點後面有無窮多位的，

And then it looks at other kinds of numbers like integers, which contain negative numbers,

然後是複數和一大堆其他的。

rational numbers like fractions, real numbers which include numbers like pi which go off

有些數字有些有趣的性質，像質數，或者π，或者指數。

to infinite decimal points, and then complex numbers and a whole bunch of others.

這些數的系統本身也有些性質，比如說

Some numbers have interesting properties like Prime Numbers, or pi or the exponential.

雖然整數和實數都有無窮多個的，但實數的更多

There are also properties of these number systems, for example, even though there is

因此有些無窮大比別的更大。

an infinite amount of both integers and real numbers, there are more real numbers than

對結構的探討始於把數字並把它們代入一些等式。

integers.

代數含括你之後如何來變化這些等式的一些規則。

So some infinities are bigger than others.

在這裡你也將發現向量和矩陣，他們是多維度的數

The study of structures is where you start taking numbers and putting them into equations

他們之間的關係的規則都包括在線性代數裏。

in the form of variables.

數論探討所有前面章節論關數字的性質

Algebra contains the rules of how you then manipulate these equations.

像上面提到過的質數的性質。

Here you will also find vectors and matrices which are multi-dimensional numbers, and the

組合數學關心的事某些結構的性質，像是樹、圖論，以及其它的那些

rules of how they relate to each other are captured in linear algebra.

你可以來數的由一堆離散的數字的一些東西。

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

群論考慮一些互相關聯著物體，嗯，就是一些郡。

of prime numbers.

一個熟悉的例子就是魔術方塊它就是一個交換群的例子

Combinatorics looks at the properties of certain structures like trees, graphs, and other things

而序論探討如何將事物根據某些規則來安排物體

that are made of discreet chunks that you can count.

像某個東西的數量大於其它的。

Group theory looks at objects that are related to each other in, well, groups.

自然數是一個有序物體集的例子，但是

A familiar example is a Rubik's cube which is an example of a permutation group.

任何有雙向關係的也都可以是有序的。

And order theory investigates how to arrange objects following certain rules like, how

純數學的另一個部分關注的是形狀和它們在一些空間中怎樣行為。

something is a larger quantity than something else.

起源是在包括畢達哥拉斯的幾何學，是和三角有關的，

The natural numbers are an example of an ordered set of objects, but anything with any two

這我們在學校裡都熟悉的。

way relationship can be ordered.

還有一些有趣的像碎形幾何，它們是一些不受尺寸大小影響的

Another part of pure mathematics looks at shapes and how they behave in spaces.

這意味著你可以把它們的細節無限地

The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are

放大而它們看起來還像是一樣的。

all familiar with form school.

拓墣學著眼於空間的一些不同的性質，

Also there are fun things like fractal geometry which are mathematical patterns which are

在空間裏你可以連續地對它們變形只要你不撕裂或者黏合它們。

scale invariant, which means you can zoom into them forever and the always look kind

比如說一條莫比烏斯帶（Möbius strip）就只有一個表面和一條邊界而不管你對它怎麼變形。

of the same.

而一些咖啡杯和甜甜圈是同樣的東西從拓墣學上來說。

Topology looks at different properties of spaces where you are allowed to continuously

測度理論是對一些空間或者集設定一些數值一種方法

deform them but not tear or glue them.

把數字和空間聯在一起。

For example a Möbius strip has only one surface and one edge whatever you do to it.

而最後，微分幾何尋找在曲面上的形狀的性質例如

And coffee cups and donuts are the same thing - topologically speaking.

現在相似三角形在不同的曲面上會有不同的角度。

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

下一節：那就是變化

spaces.

變化的研究包括著微積分它涉及到積分和微分它們尋找

And finally, differential geometry looks the properties of shapes on curved surfaces, for

由函數覆蓋下的面積或者函數的梯度行爲。

example triangles have got different angles on a curved surface, and brings us to the

而矢量微積分為矢量尋找同樣的東西。

next section, which is changes.

這裏我們又發現一大批其它的領域，像動態系統

The study of changes contains calculus which involves integrals and differentials which

它尋找隨時間從一種狀態變到另一種的系統，像液體流動

looks at area spanned out by functions or the behaviour of gradients of functions.

或是有回饋圈的一下東西，像生態系統。

And vector calculus looks at the same things for vectors.

而混沌理論研究對初始條件非常敏感的動態系統。

Here we also find a bunch of other areas like dynamical systems which looks at systems that

最後複變量分析探討有複數的函數性質。

evolve in time from one state to another, like fluid flows or things with feedback loops

而這就把我們帶到應用數學了。

like ecosystems.

在這一點值得提明一下，這裏所有的東西比

And chaos theory which studies dynamical systems that are very sensitive to initial conditions.

我已經畫的有著更多的互相聯係。

Finally complex analysis looks at the properties of functions with complex numbers.

實際上這張地圖應該看上去更像把所有不同的主題連結的一張網

This brings us to applied mathematics.

但在二維平面上你就只能做那麽多

At this point it is worth mentioning that everything here is a lot more interrelated

因此盡我最好的把它們列出來了。

than I have drawn.

那麽我們將從物理開始，它幾乎在

In reality this map should look like more of a web tying together all the different

某種程度用上了所有在左邊的東西。

subjects but you can only do so much on a two dimensional plane so I have laid them

數學物理和理論物理與純數有一種非常密切的關係。

out as best I can.

數學也以數學化學和生物數學用於其他自然科學，

Okay we'll start with physics, which uses just about everything on the left hand side

它們尋找從分子模擬到進化生物學等許多東西。

to some degree.

從古埃及和巴比倫時代開始，數學也被大量的使用在工程上，

Mathematical and theoretical physics has a very close relationship with pure maths.

建造東西要用很多數學。

Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics

非常複雜的一些電力系統，像飛機或是電力網路使用一些在

which look at loads of stuff from modelling molecules to evolutionary biology.

動態系統中的理論叫做控制論。

Mathematics is also used extensively in engineering, building things has taken a lot of maths since

數值分析是一種數學工具通常被用在數

Egyptian and Babylonian times.

學變得太複雜來完全解決的一些地方。

Very complex electrical systems like aeroplanes or the power grid use methods in dynamical

因此取代你用很多的簡單的近似的方法，

systems called control theory.

而是把它們放在一起來得到一個好的近似值答案。

Numerical analysis is a mathematical tool commonly used in places where the mathematics

比如說如果你在在一個方形裡畫一個圓，然後對著它投飛鏢，

becomes too complex to solve completely.

然後比較飛鏢在圓圈和方塊的的數量就能得到π的近似值。

So instead you use lots of simple approximations and combine them all together to get good

然而在現實世界中數值分析是在大型電腦裡做的。

approximate answers.

博弈理論研究對在給定一套規則和理性的玩家條件下，尋找什麽是最好的選擇。

For example if you put a circle inside a square, throw darts at it, and then compare the number

而這用在經濟學中，如果玩家可以很聰明，但不總是如此，以及其它的地方

of darts in the circle and square portions, you can approximate the value of pi.

像在心理學，和生物學上。

But in the real world numerical analysis is done on huge computers.

機率是研究隨機的事件，像是丟硬幣、骰子或是人們，

Game theory looks at what the best choices are given a set of rules and rational players

而統計是研究隨機過程或是組織的

and it's used in economics when the players can be intelligent, but not always, and other

大量數據的收集和分析。

areas like psychology, and biology.

這顯然跟數學金融有關，你想一點模擬金融系統並

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

找到優勢來贏得所有那些優厚的下注。

is the study of large collections of random processes or the organisation and analysis

與此有關的是最佳化，你在那裏試圖來計算出在一組許多不同的選項或者限制

of data.

中找到最佳的選擇，而後者你通常可以試圖來找到

This is obviously related to mathematical finance, where you want model financial systems

一個函數的中找最高點或最低點時看得到。

and get an edge to win all those fat stacks.

最佳化問題對我們人類是第二天性，我們總是在做的：

Related to this is optimisation, where you are trying to calculate the best choice amongst

用某種方法來為金錢得到最好的價值，或是嘗試來把我們的快樂最優化。

a set of many different options or constraints, which you can normally visualise as trying

還有一個跟純數學深深相關的是計算機科學，而

to find the highest or lowest point of a function.

計算機科學的規則其實就是從純數中引導出來的而這是另一個例子某件東西

Optimisation problems are second nature to us humans, we do them all the time: trying

是遠在可程式化的電腦出現之前就解決了。

to get the best value for money, or trying to maximise our happiness in some way.

機器學習試圖創造出聰明的電腦系統，這用上了許多數學包含

Another area that is very deeply related to pure mathematics is computer science, and

線性代數、最佳化、動態系統、概率。

the rules of computer science were actually derived in pure maths and is another example

而最後密碼學對電腦是非常重要的並用

of something that was worked out way before programmable computers were built.

使用先排列組合和數論那樣很多的純數學。

Machine learning: the creation of intelligent computer systems uses many areas in mathematics

這樣那就涵蓋了純數學和應用數學的主要部分，

like linear algebra, optimisation, dynamical systems and probability.

但是沒有考慮一下數學的基礎我是無法結束的。

And finally the theory of cryptography is very important to computation and uses a lot

這個領域想要得出數學本身的一些性質，并且問

of pure maths like combinatorics and number theory.

什麽是所有數學規則的事實或思想。

So that covers the main sections of pure and applied mathematics, but I can't end without

有沒有完整的一套叫做公理的，所有的

looking at the foundations of mathematics.

數學都來自於的基本規則？

This area tries to work out at the properties of mathematics itself, and asks what the basis

而我們能證明它和它的本身全部都一致的嗎？

of all the rules of mathematics is.

數學邏輯、集論、範疇論試圖解來回答這個而在數學邏輯中一個著明的結果

Is there a complete set of fundamental rules, called axioms, which all of mathematics comes

是哥德爾不完備定理，對大多數的人來說，意思是

from?

數學沒有完整的和前後一致的一套公理，

And can we prove that it is all consistent with itself?

這意味著它像是由我們人類造出來的。

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

看到數學把宇宙中這麼多東西解釋得這麼好，這是難以理解的。[作者要求忽略這個結論]

in mathematical logic are Gödel's incompleteness theorems which, for most people, means that

為什麼人自己編造的東西會可以來做到那樣的？

Mathematics does not have a complete and consistent set of axioms, which mean that it is all kinda

就在那裏是個深奧的神秘。

made up by us humans.

我們也有計算理論，它研究不同的計算模型

Which is weird seeing as mathematics explains so much stuff in the Universe so well.

以及以怎樣的效率它們可以解答問題和包括著複雜性理論

Why would a thing made up by humans be able to do that?

它考慮什麽是和不是是可計算的以及會需要多少記憶存儲和時間

That is a deep mystery right there.

它對大多數有趣的問題，是發瘋那樣的數量。

Also we have the theory of computation which looks at different models of computing and

這樣數學的版圖就結束了。

how efficiently they can solve problems and contains complexity theory which looks at

我對學數學最喜歡的一件事是感覺到

what is and isn't computable and how much memory and time you would need, which, for

看來是很混亂的什麽東西終於在你的腦袋裏豁然開朗而且什麽都理解了，

most interesting problems, is an insane amount.

一種顯靈的時刻，像透過矩陣看見似的。

Ending So that is the map of mathematics.

事實說我的一些在智力上最得到滿足的時刻一直是在理解數學裏的某個部分

Now the thing I have loved most about learning maths is that feeling you get where something

然後感覺到像是我宇宙法則的根本性質

that seemed so confusing finally clicks in your brain and everything makes sense: like

以它對稱的奇跡之美有了一瞥。

an epiphany moment, kind of like seeing through the matrix.

這太棒了，我愛這種感覺。

In fact some of my most satisfying intellectual moments have been understanding some part

做完這個數學版圖是我收到要求最多的

of mathematics and then feeling like I had a glimpse at the fundamental nature of the

對此我真的很開心因為我愛好數學，

Universe in all of its symmetrical wonder.

而我也很開心知道有這麼多人對它有興趣。

It's great, I love it.

因此我希望你喜歡它。

Ending Making a map of mathematics was the most popular

顯然在這段時間內我只能講到這麼多

request I got, which I was really happy about because I love maths and its great to see

但是希望我對得起這個主題以及你發現這是有用的。

so much interest in it.

這樣不久將有更多影片從我這裏出來，全都是普通的東西

So I hope you enjoyed it.

而且這是我的快樂，下次再見。

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.