数学家族族谱有多大！

The Map of Mathematics.mp311:6

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The mathematics we learn in school doesn't quite do the field of mathematics justice.

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。

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

我们只在一个角落里窥见一斑，但数学作为一个整体是一个巨大的和精彩多样的科目。

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

|我这部影片的目的是向大家展示数学这一神奇的学科全貌。

We'll start back at the very beginning.

我们从最早的时候开始。

The origin of mathematics lies in counting.

数学的起源在于计数。

In fact counting is not just a human trait, other animals are able to count as well and evidence for human counting goes back to prehistoric times with check marks made in bones.

事实上计数不只是人类的特质，其他动物也会计数，人类计数行为的证据可以追溯到史前时代，那时候做复选标记骨头。

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.

多年来，埃及人在数学领域有多项创新，第一个方程便是在那列出的，古希腊人在数学的许多领域都有过建树像的几何学和数字命理学（占卜算数），负数是在中国发明的。

And zero as a number was first used in India.

零作为一个数字是在印度首先使用的。

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

之后，在伊斯兰教黄金时代波斯数学家引领了数学的进一步发展，第一本书代数书便在这一时期著成。

Then mathematics boomed in the renaissance along with the sciences.

后来数学便向科学技术一样蓬勃发展。

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.

当然，还有很多相关的元超出我刚才所说的，|现在跳到当代，谈下我们现在对数学的认知了解。

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.

现代数学可广义地被分为两个方面，一：纯数学，即研究数学为了让数学更好；二：应用数学即发展数学来处理解决一些现实问题。

But there is a lot of crossover.

但这样分，有很多交叉重叠领域。

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

事实上，在历史上太多次有谁一头扎进数学的汪洋，

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

大多纯粹出于好奇心在美学意识引导下去学习数学。

And then they have created a whole bunch of new mathematics which was nice and interesting but doesn't really do anything useful.

然后，他们便创造一大堆新的数学分支，虽然有的很好很有趣，然并卵!

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!

不过，大约一百年后，一些工作在前沿物理学或计算机的人发现，在解决某些问题上，这些老的旧的纯数学理论正是他们解决实际问题所需要的！

Which is amazing, I think!

这就厉害了，我的数学！

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

而这种事情过去几个世纪已经发生了太多次。

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

如此抽象的东西最终真的是很有用的，这也是有点儿搞笑。

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.

|但我仍要强调的是，纯粹数学研究对数学本身发展仍然是一个非常有价值的东西因为数学它可以是迷人的，唯美而优雅的，几乎变得像艺术一样。

Okay enough of this highfalutin, lets get into it.

好吧，吹够了！现在让我们正式遍历一下数学。

Pure maths is made of several sections.

纯数学是由几个部分组成。

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

数的研究开始于天然数字和你可以用算术运算做什么。

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.

然后它着眼于其他类型的数字喜欢整数，其中包含负数，有理数像馏分，实数其中包括像圆周率的数字哪去了无限小数点，然后复杂数字和一大堆其他人。

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

一些数字有有趣的属性，如素数，或PI或指数。

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.

也有这些号码的属性系统中，例如，即使有无穷大的整数和实数，有数实数比整数多。

So some infinities are bigger than others.

因此，一些无穷比别人更大。

The study of structures is where you start taking numbers and putting them into equations in the form of variables.

结构的研究是你开始以数字和将它们放入方程在变量的形式。

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

代数包含如何你的规则，那么控制这些方程。

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.

在这里，你还会发现向量和矩阵这是多维数字和他们如何彼此相关规则线性代数捕获。

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

|数论研究了上一节中关于数字属性的素数的一切特征。

Combinatorics looks at the properties of certain structures like trees, graphs, and other things that are made of discreet chunks that you can count.

组合数学着眼于特定的属性结构如树，图，和其他的东西这是由离散块，你可以指望。

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

群论着眼于那些相关的对象的联系，在，群中。

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

一个熟悉的例子就是一个魔方其中是置换群的一个例子。

And order theory investigates how to arrange objects following certain rules like, how something is a larger quantity than something else.

和秩序的理论探讨如何安排按照一定的规则一样，对象如何东西比一些更大的量其他。

The natural numbers are an example of an ordered set of objects, but anything with any two way relationship can be ordered.

自然数是有序的例子设置的对象，但任何与任何两个这样的关系可以订购。

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

|纯数学的另一部分看形状和它们如何在空间表现。

The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are all familiar with form school.

起源于几何形状，其中包括毕达哥拉斯，并靠近三角，我们都所有熟悉形式的学校。

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.

|还有一些有趣的事情，就像分形几何这是数学模式这是尺度不变，这意味着你可以放大到无限但外观一直是一样的。

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

拓扑着眼于不同性质空间让你被允许继续它们变形但不破裂或胶水他们。

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

|例如一个莫比乌斯带只有一个表面和一个边缘无论你做什么吧。

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

咖啡杯和甜甜圈是一回事-拓扑抽象的话。

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

测量原理是将值赋给方式空格，把数字和空间联系在一起。

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.

|最后，微分几何看起来弯曲的表面上的形状的性质，对于例如三角形已经得到不同的角度在弯曲的表面上，并且给我们带来了下一个部分，该部分是关于变化。

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.

变化的研究包括微积分这包括积分和微分哪些看起来在跨越区域由出功能或功能梯度的行为。

And vector calculus looks at the same things for vectors.

和矢量微积分看同样的东西为载体。

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.

在这里，我们也发现了一堆像其他领域动力系统看起来在该系统在时间演变从一个状态到另一个状态，像流体流动或事情反馈循环像生态系统。

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

和混沌理论研究其动力系统那些对初始条件非常敏感。

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

最后，复杂的分析着眼于性能与复数函数。

This brings us to applied mathematics.

这给我们带来了应用数学。

At this point it is worth mentioning that everything here is a lot more interrelated than I have drawn.

在这一点上值得一提的是这里的一切是一个很大的关联比我画的这个还要强。

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.

在现实中，这地图看起来应该是更多的网络捆绑在一起的不同科目，但你只能做一个这么大的地图，所以我已经尽量把内容铺开了。

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

好吧，我们将从物理开始，它使用的只是地图上左边的部分到一定程度。

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

|数学和理论物理有用纯数学非常密切的关系。

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.

|数学也是在其它天然使用以数学化学和生物数学科学它们关心一大堆东西，从分子模型一直到进化生物学。

Mathematics is also used extensively in engineering, building things has taken a lot of maths since Egyptian and Babylonian times.

|数学也广泛用于工程，建筑已经采取，很多数学应用追溯到埃及和巴比伦时代。

Very complex electrical systems like aeroplanes or the power grid use methods in dynamical systems called control theory.

非常复杂的电气系统，如飞机或电网使用方法，在动力系统称为控制理论。

Numerical analysis is a mathematical tool commonly used in places where the mathematics becomes too complex to solve completely.

|数值分析是一种数学工具，常用于在某些过于复杂无法求完美解的地方。

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

|因此，你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。

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.

例如，如果你把一个圆放在方形内，向它扔飞镖，然后比较数在圈内飞镖和矩形的部分，可以近似圆周率的值。

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

但在现实世界中的数值分析是巨大的电脑上完成。

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.

博弈论看什么最好的选择给出的一组规则和合理的玩家并且它在经济学中当玩家使用可智能，但并非总是如此，和其他领域，如心理学，生物学。

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.

概率是随机事件的研究像掷硬币，骰子或人，和统计数据是随机大集合的研究流程或组织与分析数据。

This is obviously related to mathematical finance, where you want model financial systems and get an edge to win all those fat stacks.

|这显然与数学金融，要将金融系统进行建模并得到一个优势来赢得所有那些肥肉。

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.

与此相关的优化，你在哪里正试图计算出最佳的选择之中一组的许多不同的选择或限制，你可以想像，通常是试图查找函数的最高或最低点。

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.

优化问题是第二天性我们人类，我们做他们所有的时间：尝试以获取最佳的经济效益，或者试图最大限度地在某些方面我们的幸福。

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.

这是非常深刻关系到另一个领域纯数学是计算机科学和计算机科学的规则，实际上衍生纯数学，是另一个例子的东西，已经筋疲力尽了前路可编程计算机建成。

Machine learning: the creation of intelligent computer systems uses many areas in mathematics like linear algebra, optimisation, dynamical systems and probability.

机器学习：创造智能计算机系统使用的数学许多领域像线性代数，优化，动态系统和概率。

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

|最后加密的理论是要计算非常重要，采用了大量的纯数学像组合数学和数量理论。

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

所以，涵盖纯的主要部分和应用数学，但我不能没有结束看着数学基础。

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

该区域尝试以性能摸出数学本身的，并要求依据什么数学的所有规则。

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

有没有一套完整的基本规则，所谓的公理，可以解释所有数学的来由.

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

|我们可以证明它与自身是一致的吗？

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.

|数理逻辑，集合论和类别理论试图回答这个和著名的结果数理逻辑是哥德尔不完备定理其中，对于大多数人来说，这意味着数学没有一个完整和一致集公理，这意味着它是所有还挺由我们人类组成。

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

这是奇怪的看到，因为数学解释在宇宙中那么多的东西这么好。

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

为什么会由人类组成的事情能要做到这一点？

That is a deep mystery right there.

这是一个深刻的奥秘就在这里。

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.

另外，我们有计算的理论，着眼于计算的不同型号，如何有效，他们可以解决问题，包含复杂理论，着眼于是什么，是不可计算的，又有多少内存和时间你需要，其中，对于最有趣的问题，是一个疯狂的金额。

Ending So that is the map of mathematics.

所以这是数学的地图。

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.

现在我最喜欢关于学习的东西数学是感觉你得到的东西在哪里似乎如此混乱终于在点击你的大脑，一切都有道理：像一种顿悟的时刻，那种像似看穿了矩阵。

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.

事实上，一些我最满意的知识分子时刻已经了解某些部分数学，然后感觉像我在的根本性质一瞥宇宙在其所有对称的奇迹。

It's great, I love it.

牛逼啊，我喜欢。

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.

结尾制作地图的数学是最热门要求我，我真的很高兴因为我喜欢数学和高兴地看到在这么多的兴趣。

So I hope you enjoyed it.

所以，我希望你喜欢它。

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.

显然，只有这么多，我可以得到这个时间表，但希望我有做了主题公正和你发现它有用。

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.

因此会有更多的视频从我来了很快，这里的所有的常规的东西，这是我的荣幸下次再见。