Math is the hidden secret to understanding the world | Roger Antonsen

翻译人员: Junyi Sha 校对人员: Hong Li

Hi.

大家好。

I want to talk about understanding, and the nature of understanding,

我想谈谈理解和理解的本质，

and what the essence of understanding is,

理解到底是什么，

because understanding is something we aim for, everyone.

因为我们都在追求理解。

We want to understand things.

我们想理解世间万物。

My claim is that understanding has to do

我认为理解是一种能力，

with the ability to change your perspective.

转变（固有）观点的能力。

If you don't have that, you don't have understanding.

如果我们缺乏它， 就说明我们缺乏理解力。

So that is my claim.

这是我的结论。

And I want to focus on mathematics.

我想重点讲讲数学。

Many of us think of mathematics as addition, subtraction,

很多人认为，数学就是

multiplication, division,

加，减，乘，除，

fractions, percent, geometry, algebra -- all that stuff.

分数，百分数，几何，代数等等。

But actually, I want to talk about the essence of mathematics as well.

但今天，我也想讲讲数学的本质，

And my claim is that mathematics has to do with patterns.

我的观点是，数学跟模式有关。

Behind me, you see a beautiful pattern,

在我身后，是一个美丽的图案，

and this pattern actually emerges just from drawing circles

而这个图案，实际上是通过特定方式

in a very particular way.

不断画圆组成的。

So my day-to-day definition of mathematics that I use every day

所以我对数学有一个的定义

is the following:

非常直白：

First of all, it's about finding patterns.

首先，数学的关键是寻找模式。

And by "pattern," I mean a connection, a structure, some regularity,

这里的模式指的是 某种联系、结构，或者规律、规则，

some rules that govern what we see.

这些东西控制了我们所见的事物。

Second of all,

其次，我认为数学是一种语言， 用来描述各种模式。

I think it is about representing these patterns with a language.

We make up language if we don't have it,

如果没有现成的语言， 就需要创造一种。

and in mathematics, this is essential.

在数学中，这点尤为重要。

It's also about making assumptions

同时，数学也需要进行假设，

and playing around with these assumptions and just seeing what happens.

对假设进行多方验证，看看结果如何。

We're going to do that very soon.

我们一会儿就会这么做。

And finally, it's about doing cool stuff.

最后，数学可以用来做很酷的事情。

Mathematics enables us to do so many things.

能帮我们完成很多事。

So let's have a look at these patterns.

下面我们来看一些模式。

If you want to tie a tie knot,

如果你想系领带，

there are patterns.

会有很多种样式。

Tie knots have names.

每一种都有名字。

And you can also do the mathematics of tie knots.

因此领带结也包含数学。

This is a left-out, right-in, center-out and tie.

这是从左侧绕出，右侧绕入， 中间抽出然后系紧的东方结。

This is a left-in, right-out, left-in, center-out and tie.

这是从左侧绕入，右侧绕出，再左侧绕入， 中间抽出，最后系紧的四手结。

This is a language we made up for the patterns of tie knots,

这就是我们专门 为领带结创造的语言，

and a half-Windsor is all that.

最后还有半温莎结。

This is a mathematics book about tying shoelaces

这是一本关于系鞋带的数学书，

at the university level,

大学级别的，

because there are patterns in shoelaces.

因为系鞋带也有很多种模式。

You can do it in so many different ways.

你可以用成千上万种方式来系鞋带。

We can analyze it.

我们可以进行分析。

We can make up languages for it.

然后为系鞋带也创造一种语言。

And representations are all over mathematics.

这些都可以用数学方法来表达。

This is Leibniz's notation from 1675.

这是莱布尼茨 在1675年使用的符号。

He invented a language for patterns in nature.

他创造了一种语言， 来描述自然界的模式。

When we throw something up in the air,

当我们把物品抛向空中，

it falls down.

它会掉下来。

Why?

为什么？

We're not sure, but we can represent this with mathematics in a pattern.

我们并不确定，但我们可以 用数学把其归结成一种模式。

This is also a pattern.

这也是一种模式。

This is also an invented language.

是一种被发明的语言。

Can you guess for what?

你能猜到这是什么吗？

It is actually a notation system for dancing, for tap dancing.

这是一套表示 舞蹈动作的符号，踢踏舞。

That enables him as a choreographer to do cool stuff, to do new things,

这能让舞蹈编排者， 编一些炫酷的，新的动作，

because he has represented it.

因为他能用符号来描述动作。

I want you to think about how amazing representing something actually is.

请大家想一想， 表达是多么神奇的东西。

Here it says the word "mathematics."

这里写的是“数学”这个词。

But actually, they're just dots, right?

实际上就是一些点，对吧？

So how in the world can these dots represent the word?

一些点怎么能表示单词呢？

Well, they do.

确实可以。

They represent the word "mathematics,"

他们代表了单词“数学”，

and these symbols also represent that word

这些符号也一样，

and this we can listen to.

这次我们可以用听的。

It sounds like this.

听起来就像这样。

(Beeps)

（滴滴声）

Somehow these sounds represent the word and the concept.

可以说，这些声音 也代表了这个词和它的含义。

How does this happen?

这是怎么做到的呢？

There's something amazing going on about representing stuff.

表达是一种很神奇的过程。

So I want to talk about that magic that happens

所以我想跟你们 讨论一下在表达过程中

when we actually represent something.

发生的神奇的事情。

Here you see just lines with different widths.

现在你们看到的 只是不同宽度的线条。

They stand for numbers for a particular book.

这些线条代表了一本书。

And I can actually recommend this book, it's a very nice book.

强烈推荐这本书，非常不错。

(Laughter)

（笑声）

Just trust me.

真的，不骗你们。

OK, so let's just do an experiment,

好吧，让我们来做一个实验。

just to play around with some straight lines.

来玩一下直线。

This is a straight line.

这是一条直线。

Let's make another one.

再画另外一条。

So every time we move, we move one down and one across,

每一次我们都往下、往右移动一格，

and we draw a new straight line, right?

画出一条新的直线。

We do this over and over and over,

如此反复，

and we look for patterns.

从中寻找一种模式。

So this pattern emerges,

我们得到了这个图案，

and it's a rather nice pattern.

是一个非常好看的图案。

It looks like a curve, right?

它看起来就像一道弧，对吧？

Just from drawing simple, straight lines.

我们仅仅画了些简单的直线。

Now I can change my perspective a little bit. I can rotate it.

现在，稍微改变一下角度，旋转一下。

Have a look at the curve.

再看这段弧。

What does it look like?

像什么？

Is it a part of a circle?

是不是像圆的一部分？

It's actually not a part of a circle.

其实它不是圆的一部分。

So I have to continue my investigation and look for the true pattern.

所以我继续探寻，找出真正的模式。

Perhaps if I copy it and make some art?

也许我可以复制它，画一幅画？

Well, no.

好像不行。

Perhaps I should extend the lines like this,

也许我应该延长这些线条，

and look for the pattern there.

再来寻找模式。

Let's make more lines.

再多画一些线条。

We do this.

然后这样。

And then let's zoom out and change our perspective again.

把它缩小，再变换角度。

Then we can actually see that what started out as just straight lines

然后我们就会发现，开始的直线

is actually a curve called a parabola.

变成了抛物线。

This is represented by a simple equation,

这可以用一个简单的等式表达，

and it's a beautiful pattern.

很美的图案。

So this is the stuff that we do.

这就是我们所做的。

We find patterns, and we represent them.

找到某种模式，然后表达出来。

And I think this is a nice day-to-day definition.

这是一种很直白的定义。

But today I want to go a little bit deeper,

但是今天，我想讨论得更深入一些，

and think about what the nature of this is.

思考它们的本质是什么。

What makes it possible?

是什么造就了这一切？

There's one thing that's a little bit deeper,

要看得更深入一些，

and that has to do with the ability to change your perspective.

就要求我们有转换角度的能力。

And I claim that when you change your perspective,

当你换一种角度来看问题，

and if you take another point of view,

当你接受另一种观点，

you learn something new about what you are watching

你就能在所见所闻中，

or looking at or hearing.

学到新的东西。

And I think this is a really important thing that we do all the time.

我认为这一点非常重要。

So let's just look at this simple equation,

让我们看看这个简单的方程，

x + x = 2 • x.

x+x=2x

This is a very nice pattern, and it's true,

这是一个很好的模式，也是正确的。

because 5 + 5 = 2 • 5, etc.

因为5+5=2x5。

We've seen this over and over, and we represent it like this.

这个等式我们司空见惯了。

But think about it: this is an equation.

但是仔细想一想：这是一个等式。

It says that something is equal to something else,

它代表一个事物 与另一个事物相等，

and that's two different perspectives.

这么表述有两种角度。

One perspective is, it's a sum.

一种是总和。

It's something you plus together.

是相加的过程。

On the other hand, it's a multiplication,

另一种是相乘。

and those are two different perspectives.

这是两种不同的角度。

And I would go as far as to say that every equation is like this,

我会进一步说， 每个等式都像这样，

every mathematical equation where you use that equality sign

每一个使用等号连接的数学方程

is actually a metaphor.

实际上都是隐喻。

It's an analogy between two things.

是两种事物间的类比。

You're just viewing something and taking two different points of view,

你观察一件事情，产生两种观点，

and you're expressing that in a language.

然后用一种语言来表达。

Have a look at this equation.

看这个方程。

This is one of the most beautiful equations.

它是最美的等式之一。

It simply says that, well,

简单表明了，

two things, they're both -1.

等式两边都是-1。

This thing on the left-hand side is -1, and the other one is.

左手边的是-1，右边的也是。

And that, I think, is one of the essential parts

我认为这是数学中很重要的部分

of mathematics -- you take different points of view.

——采取不同的观点。

So let's just play around.

我们继续。

Let's take a number.

选一个数字好了。

We know four-thirds. We know what four-thirds is.

我们知道4/3，知道它的含义。

It's 1.333, but we have to have those three dots,

就是1.333……，但是 一定要加上后面的省略号，

otherwise it's not exactly four-thirds.

否则就不是准确的4/3了。

But this is only in base 10.

但只有在使用十进制时才如此。

You know, the number system, we use 10 digits.

我们的数字系统用的是10位计数。

If we change that around and only use two digits,

如果我们改成2位计数，

that's called the binary system.

也就是二进制。

It's written like this.

就变成了这样。

So we're now talking about the number.

我们现在在讨论数字。

The number is four-thirds.

讨论4/3这个数字。

We can write it like this,

我们也可以这样表示，

and we can change the base, change the number of digits,

我们改变进制，改变数位，

and we can write it differently.

就可以用不同的方式书写。

So these are all representations of the same number.

所有这些都代表同一个数。

We can even write it simply, like 1.3 or 1.6.

我们甚至可以把它 简单写作1.3或1.6。

It all depends on how many digits you have.

取决于我们选用哪种进制。

Or perhaps we just simplify and write it like this.

或者我们还可以简单写成这样。

I like this one, because this says four divided by three.

我喜欢这种，因为它表示4被3除。

And this number expresses a relation between two numbers.

表现了两个数字间的关系。

You have four on the one hand and three on the other.

上边是4，下边是3。

And you can visualize this in many ways.

你可以用许多方式 来把这个数字可视化。

What I'm doing now is viewing that number from different perspectives.

从不同的角度来看这个数字。

I'm playing around.

我在不断尝试。

I'm playing around with how we view something,

改变观察事物的角度。

and I'm doing it very deliberately.

我是故意这么做的。

We can take a grid.

让我们画一个网格。

If it's four across and three up, this line equals five, always.

假如为4行3列， 那么这条线就始终代表5。

It has to be like this. This is a beautiful pattern.

肯定如此。这是一个美丽的图案。

Four and three and five.

4和3和5。

And this rectangle, which is 4 x 3,

这个长方形，长宽比为4:3，

you've seen a lot of times.

你们见过很多次的。

This is your average computer screen.

就是你们的屏幕大小的平均值。

800 x 600 or 1,600 x 1,200

800 x 600 或是 1600 x 1200

is a television or a computer screen.

分别是电脑和电视的屏幕。

So these are all nice representations,

这都是很好的表达方式，

but I want to go a little bit further and just play more with this number.

但是我还想再深入一点点， 再玩一下这些数字。

Here you see two circles. I'm going to rotate them like this.

现在，你能看到两个圆。 我要像这样旋转它们。

Observe the upper-left one.

看一下左上角的那个，

It goes a little bit faster, right?

它转得更快一点儿，对吧？

You can see this.

你们都能看到。

It actually goes exactly four-thirds as fast.

准确来说，它的旋转速度 是慢速的4/3倍。

That means that when it goes around four times,

也就是说，它每转4圈，

the other one goes around three times.

另一个圆就会转3圈。

Now let's make two lines, and draw this dot where the lines meet.

现在，画两条线， 并标明相交处的点。

We get this dot dancing around.

我们就能得到一个跳舞的点。

(Laughter)

（笑声）

And this dot comes from that number.

这个点就来源于4/3这个数字。

Right? Now we should trace it.

是吧？现在，让我来看看它的轨迹。

Let's trace it and see what happens.

把轨迹画出来，看看是什么样子。

This is what mathematics is all about.

这就是数学。

It's about seeing what happens.

就是不断探索会发生什么。

And this emerges from four-thirds.

而这来自于4/3这个数字。

I like to say that this is the image of four-thirds.

我觉得，这就是4/3的肖像。

It's much nicer -- (Cheers)

比数字好看多了——（欢呼）

Thank you!

谢谢！

(Applause)

（掌声）

This is not new.

其实这不算新鲜事了。

This has been known for a long time, but --

很早以前就被发现了， 但是——

(Laughter)

（笑声）

But this is four-thirds.

但是这仅仅是4/3。

Let's do another experiment.

让我们再做一个实验。

Let's now take a sound, this sound: (Beep)

让我们选一个声音， 是这样的：（嘟）

This is a perfect A, 440Hz.

这是一个完美的A，440Hz。

Let's multiply it by two.

把它翻倍。

We get this sound. (Beep)

就得到了这个声音。（嘟）

When we play them together, it sounds like this.

同时播放这两种声音， 听起来是这个效果。

This is an octave, right?

这是一个八度音，对吧？

We can do this game. We can play a sound, play the same A.

我们来玩一个游戏。 我们再放一次A。

We can multiply it by three-halves.

然后我们把它翻为1.5倍。

(Beep)

（嘟）

This is what we call a perfect fifth.

我们称之为纯五度音。

(Beep)

（嘟）

They sound really nice together.

把它们一起播放，听起来很不错。

Let's multiply this sound by four-thirds. (Beep)

让我们把这个声音翻4/3倍。

What happens?

会怎么样？

You get this sound. (Beep)

你们会得到这个声音。

This is the perfect fourth.

纯四度音。

If the first one is an A, this is a D.

如果第一个音是A， 那么这就是一个D。

They sound like this together. (Beeps)

一起播放，是这样的声音。

This is the sound of four-thirds.

这就是4/3的声音。

What I'm doing now, I'm changing my perspective.

这就是改变角度。

I'm just viewing a number from another perspective.

我是在从另一个角度看一个数字。

I can even do this with rhythms, right?

也可以用节奏来表示。

I can take a rhythm and play three beats at one time (Drumbeats)

我可以选一个节奏， 在一段时间内敲3下（鼓点声）

in a period of time,

一段固定的时间，

and I can play another sound four times in that same space.

然后在同样的时间内敲4下。

(Clanking sounds)

（铛铛声）

Sounds kind of boring, but listen to them together.

单独听很枯燥， 但如果放在一起。

(Drumbeats and clanking sounds)

（鼓点和铛铛声）

(Laughter)

（笑声）

Hey! So.

嘿！好多了。

(Laughter)

（笑声）

I can even make a little hi-hat.

我还可以加点儿踩镲声。

(Drumbeats and cymbals)

（鼓点和踩镲声）

Can you hear this?

听到了吗？

So, this is the sound of four-thirds.

所以，这就是4/3的声音。

Again, this is as a rhythm.

4/3的节律。

(Drumbeats and cowbell)

（鼓点声和踩镲声）

And I can keep doing this and play games with this number.

我还可以继续玩， 用这个数字做游戏。

Four-thirds is a really great number. I love four-thirds!

4/3是一个超棒的数字。 我爱死4/3了！

(Laughter)

（笑声）

Truly -- it's an undervalued number.

说真的——4/3的价值被低估了。

So if you take a sphere and look at the volume of the sphere,

如果你拿一个球体，看看它的体积，

it's actually four-thirds of some particular cylinder.

会发现其实球体体积就是 某个圆柱体积的4/3倍。

So four-thirds is in the sphere. It's the volume of the sphere.

所以4/3出现在了球体里， 是球的体积。

OK, so why am I doing all this?

好，我为什么玩这些？

Well, I want to talk about what it means to understand something

是想跟你们谈谈 理解一件事物的意义，

and what we mean by understanding something.

谈谈我们所说的理解是什么。

That's my aim here.

这就是我的目的。

And my claim is that you understand something

我认为，只有当我们 从多个角度去审视同一事物时，

if you have the ability to view it from different perspectives.

才能说我们理解了它。

Let's look at this letter. It's a beautiful R, right?

让我们看看这个字母。 这是一个漂亮的R，对吧？

How do you know that?

你们怎么判断这是个R？

Well, as a matter of fact, you've seen a bunch of R's,

因为你们看过各种各样的R，

and you've generalized

然后进行归纳，

and abstracted all of these and found a pattern.

提取它们的共性， 找到了一种模式。

So you know that this is an R.

然后你们确认这是一个R。

So what I'm aiming for here is saying something

所以，我要说的是

about how understanding and changing your perspective

理解事物和变换角度

are linked.

是有关的。

And I'm a teacher and a lecturer,

我是一名教师和演讲者，

and I can actually use this to teach something,

我可以利用这一点去教课，

because when I give someone else another story, a metaphor, an analogy,

因为我用隐喻和类比的方法， 给学生们换一种方式讲故事，

if I tell a story from a different point of view,

从不同的角度去讲述一件事，

I enable understanding.

我就能让他们真正理解。

I make understanding possible,

我让理解变为了可能，

because you have to generalize over everything you see and hear,

因为你们必须要 归纳自己的所见所闻，

and if I give you another perspective, that will become easier for you.

如果我给你们另一个角度， 你们做起来就会更容易。

Let's do a simple example again.

让我们再举一个例子。

This is four and three. This is four triangles.

这是4和3。 这是4个三角形。

So this is also four-thirds, in a way.

这也是某种4/3。

Let's just join them together.

让我们把它们连起来。

Now we're going to play a game; we're going to fold it up

现在我们再玩一个游戏， 把它们折叠起来，

into a three-dimensional structure.

形成一个三维结构。

I love this.

我喜欢这个。

This is a square pyramid.

这是一个金字塔形。

And let's just take two of them and put them together.

让我们再做一个，把它们放在一起。

So this is what is called an octahedron.

就形成了一个八面体。

It's one of the five platonic solids.

这是5种正多面体 （又叫柏拉图立体）之一。

Now we can quite literally change our perspective,

现在我们可以真的来改变角度，

because we can rotate it around all of the axes

绕各种轴旋转它，

and view it from different perspectives.

从其它角度来观察。

And I can change the axis,

我可以改变旋转轴，

and then I can view it from another point of view,

改变观察角度，

but it's the same thing, but it looks a little different.

还是同一个物体， 只是看起来有一些不同。

I can do it even one more time.

我可以再做一次。

Every time I do this, something else appears,

我每调整一次， 就会有新东西出现，

so I'm actually learning more about the object

所以通过改变角度，

when I change my perspective.

我能更加了解这个物体。

I can use this as a tool for creating understanding.

我可以把它作为创造理解的工具。

I can take two of these and put them together like this

我可以把两个正四面体， 像这样穿起来，

and see what happens.

看看会发生什么。

And it looks a little bit like the octahedron.

有点儿像正八面体。

Have a look at it if I spin it around like this.

把它旋转起来再看。

What happens?

发生了什么？

Well, if you take two of these, join them together and spin it around,

如果你把这两个物体 拼在一起，旋转它，

there's your octahedron again,

你就又得到了一个正八面体，

a beautiful structure.

一个漂亮的结构。

If you lay it out flat on the floor,

如果你把它平摊在地上，

this is the octahedron.

这就是一个正八面体。

This is the graph structure of an octahedron.

正八面体的平面结构图。

And I can continue doing this.

我还可以继续玩。

You can draw three great circles around the octahedron,

在正八面体周围画三个大圈，

and you rotate around,

转动看看，

so actually three great circles is related to the octahedron.

三个大圈实际上 是与正八面体相连的。

And if I take a bicycle pump and just pump it up,

如果我拿一个自行车泵， 把它充满气，

you can see that this is also a little bit like the octahedron.

你会发现，它看起来还是 有点儿像正八面体的。

Do you see what I'm doing here?

看出来我在做什么了吗？

I am changing the perspective every time.

我在不停改变角度。

So let's now take a step back --

让我们退后一步

and that's actually a metaphor, stepping back --

——这其实是一个隐喻，退后一步——

and have a look at what we're doing.

看看我们在做的事情。

I'm playing around with metaphors.

我在使用隐喻，

I'm playing around with perspectives and analogies.

在变换角度，进行类比。

I'm telling one story in different ways.

变换不同的角度，

I'm telling stories.

来讲同一个故事。

I'm making a narrative; I'm making several narratives.

我在叙述， 而且做了好几种叙述。

And I think all of these things make understanding possible.

我认为这一切使得理解变成可能。

I think this actually is the essence of understanding something.

我认为这是理解事物的关键。

I truly believe this.

我深信这点。

So this thing about changing your perspective --

所以，关于改变你们的角度——

it's absolutely fundamental for humans.

对人类来说十分重要。

Let's play around with the Earth.

让我们来看看地球。

Let's zoom into the ocean, have a look at the ocean.

让我们放大到海洋，看看海洋。

We can do this with anything.

我们可以放大任何事物。

We can take the ocean and view it up close.

我们以海洋为例，仔细的看看它。

We can look at the waves.

我们能观察海浪。

We can go to the beach.

或是沙滩。

We can view the ocean from another perspective.

我们也可以从另一个角度看海洋。

Every time we do this, we learn a little bit more about the ocean.

每变一次角度，我们就能 对海洋了解得多一些。

If we go to the shore, we can kind of smell it, right?

如果我们走到海边， 就能闻到海水的味道，对吧？

We can hear the sound of the waves.

能听到海浪的声音。

We can feel salt on our tongues.

能尝到风中咸咸的味道。

So all of these are different perspectives.

所有这些，都是不同的角度。

And this is the best one.

而这个（角度）是最棒的。

We can go into the water.

我们进入水中。

We can see the water from the inside.

从内部来观察。

And you know what?

你们知道吗？

This is absolutely essential in mathematics and computer science.

这对数学和计算机科学 来说都绝对重要。

If you're able to view a structure from the inside,

如果你能从一个 结构的内部去进行观察，

then you really learn something about it.

那你就能够真正认识它。

That's somehow the essence of something.

认识到它的本质。

So when we do this, and we've taken this journey

所以，当我们一路前行，

into the ocean,

进入海洋，

we use our imagination.

我们发挥了想象力。

And I think this is one level deeper,

我认为这又更深入了一层，

and it's actually a requirement for changing your perspective.

是改变角度的必然要求。

We can do a little game.

我们可以做个游戏。

You can imagine that you're sitting there.

想象一下你正坐在那儿。

You can imagine that you're up here, and that you're sitting here.

然后你同时又在上面。

You can view yourselves from the outside.

你就可以从外部审视你自己了。

That's really a strange thing.

这听起来很奇怪。

You're changing your perspective.

你在改变你的角度。

You're using your imagination,

你在使用你的想象力，

and you're viewing yourself from the outside.

你在从外部审视你自己。

That requires imagination.

这需要有想象力。

Mathematics and computer science are the most imaginative art forms ever.

数学和计算机科学是 最具想象力的艺术形式。

And this thing about changing perspectives

还有一种改变角度的方式，

should sound a little bit familiar to you,

可能更被你们熟知，

because we do it every day.

因为我们每天都在做，

And then it's called empathy.

叫做共情。

When I view the world from your perspective,

当我从你的角度看世界的时候，

I have empathy with you.

我就与你产生了共情。

If I really, truly understand

如果我能够真正的

what the world looks like from your perspective,

理解你们眼中的世界，

I am empathetic.

那我就与你产生了共情。

That requires imagination.

这需要想象力。

And that is how we obtain understanding.

这就是我们获得理解的方式。

And this is all over mathematics and this is all over computer science,

而这种方式充斥了 数学和计算机科学领域。

and there's a really deep connection between empathy and these sciences.

共情和这些学科间有着深刻的联系。

So my conclusion is the following:

所以，我的结论是：

understanding something really deeply

深入的理解一件事

has to do with the ability to change your perspective.

与转换角度的能力密切相关。

So my advice to you is: try to change your perspective.

所以我的建议是： 尝试转换你的角度。

You can study mathematics.

你可以学习数学。

It's a wonderful way to train your brain.

这是锻炼大脑的好方法。

Changing your perspective makes your mind more flexible.

变换你们的角度， 让思维变得更灵活。

It makes you open to new things,

它能够让你们易于接受新事物，

and it makes you able to understand things.

能够理解事物。

And to use yet another metaphor:

请允许我再使用一次隐喻：

have a mind like water.

让思维像水一样吧。

That's nice.

会很不错的。谢谢大家。

Thank you.

(Applause)