《数学是打开科学大门的钥匙，是一切知识中的最高形式》

《数学是打开科学大门的钥匙，是一切知识中的最高形式》

《Mathematics is the key to open the door to science and is the highest form of all knowledge》

在数学的领域中，提出问题的艺术比解答问题的艺术更为重要。

--- 格奥尔格·康托尔，德国数学家，集合论的创始人，生于俄国圣彼得堡。

In mathematics, the art of asking questions is more important than the art of answering them.

--- Georg Cantor，German mathematician, founder of set theory, born in St. Petersburg, Russia.

数学起源于人类早期的生产活动，是古代中国六艺之一（六艺，指中国周朝贵族教育体系中的六种技能，即：礼、乐、射、御、书、数），亦被古希腊学者视为哲学的起点。数学的希腊语μαθηματικός (mathematikós)的含义是“爱好学问”，源于μάθημα（máthema）（“科学，知识，学问”）。数学最早是研究量、结构、变化以及空间模型的学科，在现代，数学又是利用逻辑形式研究现实世界的空间形式和数量关系的学科。

Mathematics originated from early human production activities and is one of the six arts in ancient China (the six arts refer to the six skills in the aristocratic education system of the Zhou Dynasty in China, namely: etiquette, music, archery, carriage driving, calligraphy, and mathematics). Considered by Greek scholars as the starting point of philosophy. The Greek word for mathematics μαθηματικός (mathematikós) means "love of learning" and is derived from μάθημα (máthema) ("science, knowledge, learning"). Mathematics was originally a discipline that studied quantities, structures, changes, and spatial models. In modern times, mathematics is a discipline that uses logical forms to study spatial forms and quantitative relationships in the real world.

数学是知识的工具，亦是其它知识工具的泉源，所有研究顺序和度量的科学均和数学有关，数学是符号加逻辑，数学是各式各样的证明技巧，学习数学要多做习题，边做边思索。先知其然，然后知其所以然，不光会做题还要会提问题，下面是youtube上很有名的一个数学科学的科普视频翻译，希望你们可以初步了解数学学科的概貌，培养你们学习数学的兴趣。

Mathematics is a tool of knowledge and the source of other knowledge tools. All sciences that study order and measurement are related to mathematics. Mathematics is symbols plus logic. Mathematics is a variety of proof techniques. To learn mathematics, you need to do more exercises. Thinking by doing. know what is happening, and then know why it is happening. You can not only solve problems but also ask questions. The following is a translation of a famous popular science video on mathematics on YouTube. I hope you can have a preliminary understanding of the overview of mathematics and cultivate your interest in learning mathematics.

The Video subtitles as followed：（视频字幕如下）

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.

我们在学校里学到的数学并没有很好的解释目前数学涉及哪些领域。我们只是管中窥豹，但数学作为一个整体是一个巨大的和精彩多样的主题。我这段视频的目的是向你们展示数学的精彩之处。

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 evidence for human counting goes back toprehistoric 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.

我们从最早的时候开始。数学的起源在于计数。 事实上计数不只是人类的特质，他动物也会计数，人类计数行为的证据可以追溯到史前时代在骨头上做的标记。有很多发明：埃及人发明了第一个方程，古希腊人在几何和占卜术等许多领域取得了长足进步，负数在中国发明， 零作为数字首先在印度使用。 然后在伊斯兰教的黄金时代，波斯数学家取得了进一步的进步，写出了第一本代数书籍。 随后，数学与科学一起在文艺复兴时期蓬勃发展。 现在关于数学史的内容比我刚才说的还要多，但我要跳到我们现在所知道的现代数学。

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.

现代数学大致可以分为两个领域：纯数学（为了数学本身而研究数学）和应用数学（发展数学来帮助解决一些现实世界的问题， 两者有很多交叉。

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.

事实上，历史上很多时候有人纯粹出于好奇心和某种审美感而进入数学荒野。 然后他们创造了一大堆新的数学，这些数学很好、很有趣，但并没有真正做任何有用的事情。 但是，说一百次之后，有人将研究物理学或计算机科学前沿的某个问题，他们会发现纯数学中的这种古老理论正是他们解决现实世界问题所需要的！ 这太棒了！ 而这样的事情在过去的几个世纪里已经发生过很多次了。 有趣的是，如此抽象的东西往往最终变得非常有用。 但我还应该提到，纯数学本身仍然是一件非常有价值的事情，因为它可以令人着迷，并且本身可以具有真正的美丽和优雅，几乎变得像艺术一样。 好了，废话已经说得够多了，让我们开始吧。

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.

纯数学由几个部分组成。 数字的研究从自然数以及算术运算开始，然后它会研究其他类型的数字，例如整数，其中包含负数，有理数（例如分数），实数（包括像 pi 这样的数字，其小数点无限小），然后是复数和一大堆其他数字。 有些数字具有有趣的属性，例如素数、圆周率或指数。 这些数字系统也有一些性质，例如，尽管整数和实数都有无限数量，但实数比整数多。 所以有些无穷大比其他无穷大。

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.

对结构的研究是从获取数字并将它们以变量的形式放入方程中开始的。 代数包含了如何操作这些方程的规则。 在这里，您还可以找到多维数字的向量和矩阵，以及它们如何相互关联的规则是在线性代数中捕获的。数论研究了上一节中关于数字的所有特征，例如素数的属性。组合学着眼于某些结构的属性，如树、图和其他由可计算的离散块组成的东西。群论着眼于群体中彼此相关的对象。 一个熟悉的例子是魔方，它是排列群的一个例子。秩序理论研究如何按照某些规则排列对象，例如某事物如何比其他事物的数量更大。 自然数是有序对象集的一个示例，但任何具有任意双向关系的事物都可以排序。

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.

纯数学的另一部分着眼于形状以及它们在空间中的行为方式。 起源于几何学，其中包括毕达哥拉斯，并且接近三角学，这是我们在学校中都熟悉的。 还有一些有趣的东西，比如分形几何，它们是比例不变的数学模式，这意味着你可以永远放大它们，而且看起来总是一样的。 拓扑学着眼于空间的不同属性，允许你不断地使它们变形，但不能撕裂或粘合它们。 例如，无论你对它做什么，莫比乌斯带都只有一个表面和一个边缘。 从拓扑上来说，咖啡杯和甜甜圈是同一件事。测度论是一种为空间或将数字和空间联系在一起的集合赋值的方法。 最后，微分几何研究曲面上形状的属性，例如三角形在曲面上具有不同的角度，并将我们带到下一部分，即变化。

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.

对变化的研究包含涉及积分和微分的微积分，它着眼于函数跨越的面积或函数梯度的行为。 向量微积分对于向量来说也是同样的事情。 在这里，我们还发现了许多其他领域，例如动力系统，它研究随时间从一种状态演变到另一种状态的系统，例如流体流动或具有反馈循环的事物（例如生态系统）。混沌理论研究对初始条件非常敏感的动力系统。最后，复变分析着眼于复数函数的属性。

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.

这给我们带来了应用数学。 在这一点上值得一提的是，这里的一切都比我所描绘的更加相互关联。 事实上，这张地图应该看起来更像是一个将所有不同主题联系在一起的网络，但你只能在二维平面上做这么多，所以我已经尽我所能将它们展示出来。

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.

好吧，我们将从物理学开始，它在某种程度上使用了左侧的所有内容。 数学和理论物理与纯数学有着非常密切的关系。 数学还用于其他自然科学，如数学化学和生物数学，研究从分子建模到进化生物学的大量内容。 数学也广泛应用于工程领域，自埃及和巴比伦时代以来，建筑就需要大量数学知识。 飞机或电网等非常复杂的电力系统使用称为控制理论的动力系统方法。

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.

数值分析是一种数学工具，常用于数学过于复杂而无法完全解决的地方。 因此，您可以使用大量简单的近似值并将它们组合在一起以获得良好的近似答案。 例如，如果将一个圆放在一个正方形内，向其扔飞镖，然后比较圆和正方形部分中飞镖的数量，您就可以近似得出 pi 的值。 但在现实世界中，数值分析是在大型计算机上完成的。

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.

博弈论着眼于在给定一组规则和理性参与者的情况下最好的选择，当参与者可能很聪明（但并非总是如此）时，它被用于经济学，以及心理学和生物学等其他领域。

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.

概率论是对抛硬币、骰子或人类等随机事件的研究，而统计学是对大量随机过程或数据的组织和分析的研究。

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 optimization, 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.

这显然与数学金融有关，在数学金融中，您希望对金融系统进行建模并获得赢得所有这些大筹码的优势。 与此相关的是优化，您试图在一组许多不同的选项或约束中计算最佳选择，您通常可以将其想象为试图找到函数的最高点或最低点。 优化问题是我们人类的第二天性，我们一直在做这些问题：试图获得最佳的金钱价值，或者试图以某种方式最大化我们的幸福。

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.

与纯数学密切相关的另一个领域是计算机科学，计算机科学的规则实际上是从纯数学中导出的，并且是在制造可编程计算机之前就已经解决的问题的另一个例子。 机器学习：智能计算机系统的创建使用了数学的许多领域，如线性代数、优化、动力系统和概率。 最后，密码学理论对于计算非常重要，并且使用大量纯数学，例如组合学和数论。

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.

这涵盖了纯粹数学和应用数学的主要部分，但我不能不看数学基础就结束。该领域试图弄清楚数学本身的性质，并询问所有数学规则的基础是什么。 是否存在一套完整的基本规则，称为公理，所有数学都源自这些规则？ 我们能否证明这一切都与其自身一致？ 数理逻辑、集合论和范畴论试图回答这个问题，数理逻辑中的一个著名结果是哥德尔不完备定理，对于大多数人来说，这意味着数学没有一套完整且一致的公理，这意味着它有点 由我们人类组成。 这很奇怪，因为数学很好地解释了宇宙中的很多东西。 为什么人类制造的东西能够做到这一点？ 这是一个很深的谜团。

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.

此外，我们还有计算理论，它着眼于不同的计算模型以及它们解决问题的效率，并包含复杂性理论，该理论着眼于什么是可计算的和不可计算的以及您需要多少内存和时间，对于最有趣的 问题，是一个疯狂的数量。

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.

这就是数学地图。 现在，我在学习数学时最喜欢的事情是，感觉你得到了一些看似令人困惑的东西，最终在你的大脑中点击，一切都有意义：就像顿悟的时刻，有点像看穿矩阵。 事实上，我最满意的一些智力时刻是理解数学的某些部分，然后感觉我瞥见了宇宙所有对称奇迹的基本性质。 太棒了，我喜欢它。

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.

制作数学地图是我收到的最受欢迎的请求，我对此感到非常高兴，因为我热爱数学，很高兴看到人们对数学如此感兴趣。 所以我希望你喜欢它。 显然，在这个时间范围内我能讲的内容就这么多，但希望我已经正确地阐述了这个主题，并且您发现它很有用。 因此，我很快就会发布更多视频，这是所有常规内容，很高兴下次见到您。

英语学习时间：（English Study Time）

mathematical terms：

Origins	counting计数、equation方程式, geometry几何学，and numerology占卜术, negative numbers 负数，algebra 代数
Pure maths	Number System 数字系统	Natural Numbers自然数，Rational Numbers有理数，Integers整数，Real Numbers实数、Complex Numbers复数
	Structures 结构	Linear algebra线性代数，Number Theory 数论，Combinatorics 组合学，Group Theory群论，Graph Theory图论，Order theory秩序论，
	Spaces 空间	Geometry 几何，Trigonometry 三角学，Fractal Geometry 分形，Topology 拓扑学，Differential geometry微分几何，Measure theory 测量理论
	Changes 变化	Calculus 微积分，integrals and differentials（积分和微分），Vector Calculus 向量微积分，Dynamical System动力系统，Chaos theory混沌理论，complex analysis 复变分析
Applied maths	Mathematical physics 数学物理，Mathematical chemistry 数学化学，Biomathematics生物数学，Control theory 控制理论，Numerical Analysis 数值分析，Mathematical Finance金融数学，Statistics 统计，Probability 概率，Optimization 最优化，Computer Science 计算机科学，Cryptography 密码学，Machine learning 机器学习
Foundations 基础	Fundamental rules基本规则，Mathematical logic 数理逻辑，Set theory 集合论，category theory 运筹学，Theory of Computation计算理论，complexity theory复杂性理论

PS：

1、《The Map of Mathematics》Youtube Video，https://www.youtube.com/watch?v=OmJ-4B-mS-Y&list=PLOYRlicwLG3St5aEm02ncj-sPDJwmojIS；

2、《The Map of Mathematics》Bilibili Video，https://www.bilibili.com/video/BV1Sb411k7fn/?spm_id_from=333.337.search-card.all.click&vd_source=8df1212821cd1df387f893ab3bdc4799；

2、The picture of The Map of Mathematics，https://www.flickr.com/photos/95869671@N08/32264483720/in/dateposted-public/；

4、《what is the Mathematics》，http://www.shuxueji.com；