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理解世界的秘诀:数学 – 译学馆
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理解世界的秘诀:数学

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

大家好
Hi.
我想谈谈理解和理解的本质
I want to talk about understanding, and the nature of understanding,
理解到底是什么
and what the essence of understanding is,
因为我们都在追求理解
because understanding is something we aim for, everyone.
我们想理解世间万物
We want to understand things.
我认为理解是一种能力
My claim is that understanding has to do
转变(固有)观点的能力
with the ability to change your perspective.
如果我们缺乏它 就说明我们缺乏理解力
If you don’t have that, you don’t have understanding.
这是我的结论
So that is my claim.
我想重点讲讲数学
And I want to focus on mathematics.
很多人认为 数学就是
Many of us think of mathematics as addition, subtraction,
加 减 乘 除
multiplication, division,
分数 百分数 几何 代数等等
fractions, percent, geometry, algebra — all that stuff.
但今天 我也想讲讲数学的本质
But actually, I want to talk about the essence of mathematics as well.
我的观点是数学跟模式有关
And my claim is that mathematics has to do with patterns.
在我身后 是一个美丽的图案
Behind me, you see a beautiful pattern,
而这个图案 实际上是通过特定方式
and this pattern actually emerges just from drawing circles
不断画圆组成的
in a very particular way.
所以我对数学有一个的定义
So my day-to-day definition of mathematics that I use every day
非常直白
is the following:
首先 数学的关键是寻找模式
First of all, it’s about finding patterns.
这里的模式指的是 某种联系 结构 或者规律 规则
And by “pattern,” I mean a connection, a structure, some regularity,
一些我们可以看见掌控的规则
some rules that govern what we see.
第二点
Second of all,
我认为数学是用一种语言来描述这些模式
I think it is about representing these patterns with a language.
如果没有现成的语言  就需要创造一种
We make up language if we don’t have it,
在数学中 这点尤为重要
and in mathematics, this is essential.
同时 数学也需要进行假设
It’s also about making assumptions
对假设进行多方验证 看看结果如何
and playing around with these assumptions and just seeing what happens.
我们一会儿就会这么做
We’re going to do that very soon.
最后 数学可以用来做很酷的事情
And finally, it’s about doing cool stuff.
能帮我们完成很多事
Mathematics enables us to do so many things.
下面我们来看一些模式
So let’s have a look at these patterns.
如果你想系领带
If you want to tie a tie knot,
会有很多种样式
there are patterns.
每一种都有名字
Tie knots have names.
因此领带结也包含数学
And you can also do the mathematics of tie knots.
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-out, right-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 left-in, right-out, left-in, center-out and tie.
这就是我们专门为领带结创造的语言
This is a language we made up for the patterns of tie knots,
最后还有半温莎结
and a half-Windsor is all that.
这是一本关于系鞋带的数学书
This is a mathematics book about tying shoelaces
大学级别的
at the university level,
因为系鞋带也有很多种模式
because there are patterns in shoelaces.
你可以用成千上万种方式来系鞋带
You can do it in so many different ways.
我们可以进行分析
We can analyze it.
然后为系鞋带也创造一种语言
We can make up languages for it.
这些都可以用数学方法来表达
And representations are all over mathematics.
这是莱布尼茨在1675年使用的符号
This is Leibniz’s notation from 1675.
他创造了一种语言 来描述自然界的模式
He invented a language for patterns in nature.
当我们把物品抛向空中
When we throw something up in the air,
它会掉下来
it falls down.
为什么
Why?
我们并不确定 但我们可以用数学把其归结成一种模式
We’re not sure, but we can represent this with mathematics in a pattern.
这也是一种模式
This is also a pattern.
是一种被发明的语言
This is also an invented language.
你能猜到这是什么吗
Can you guess for what?
它是一种用于舞蹈 用于踢踏舞的标记方法
It is actually a notation system for dancing, for tap dancing.
这能让舞蹈编排者 编一些炫酷的 新的动作
That enables him as a choreographer to do cool stuff, to do new things,
因为他能用符号来描述动作
because he has represented it.
请大家想一想 表达是多么神奇的东西
I want you to think about how amazing representing something actually is.
这里写的是“数学”这个词
Here it says the word “mathematics.”
实际上就是一些点 对吧
But actually, they’re just dots, right?
一些点怎么能表示单词呢
So how in the world can these dots represent the word?
确实可以
Well, they do.
他们代表了单词“数学”
They represent the word “mathematics,”
这些符号也一样
and these symbols also represent that word
这次我们可以用听的
and this we can listen to.
听起来就像这样
It sounds like this.
(滴滴声)
(Beeps)
可以说这些声音也代表了这个词和它的含义
Somehow these sounds represent the word and the concept.
这是怎么做到的呢
How does this happen?
表达是一种很神奇的过程
There’s something amazing going on about representing stuff.
所以我想跟你们讨论一下在表达过程中
So I want to talk about that magic that happens
发生的神奇的事情
when we actually represent something.
现在你们看到的只是不同宽度的线条
Here you see just lines with different widths.
这些线条代表了一本书
They stand for numbers for a particular book.
强烈推荐这本书 非常不错
And I can actually recommend this book, it’s a very nice book.
(笑声)
(Laughter)
真的 不骗你们
Just trust me.
好吧 让我们来做一个实验
OK, so let’s just do an experiment,
来玩一下直线
just to play around with some straight lines.
这是一条直线
This is a straight line.
再画另外一条
Let’s make another one.
每一次我们都往下 往右移动一格
So every time we move, we move one down and one across,
画出一条新的直线
and we draw a new straight line, right?
如此反复
We do this over and over and over,
从中寻找一种模式
and we look for patterns.
我们得到了这个图案
So this pattern emerges,
是一个非常好看的图案
and it’s a rather nice pattern.
它看起来就像一道弧 对吧
It looks like a curve, right?
我们仅仅画了些简单的直线
Just from drawing simple, straight lines.
现在 稍微改变一下角度 旋转一下
Now I can change my perspective a little bit. I can rotate it.
再看这段弧
Have a look at the curve.
像什么
What does it look like?
是不是像圆的一部分
Is it a part of a circle?
其实它不是圆的一部分
It’s actually not a part of a circle.
所以我继续探寻 找出真正的模式
So I have to continue my investigation and look for the true pattern.
也许我可以复制它 画一幅画
Perhaps if I copy it and make some art?
好像不行
Well, no.
也许我应该延长这些线条
Perhaps I should extend the lines like this,
再来寻找模式
and look for the pattern there.
再多画一些线条
Let’s make more lines.
然后这样
We do this.
把它缩小 再变换角度
And then let’s zoom out and change our perspective again.
然后我们就会发现 开始的直线
Then we can actually see that what started out as just straight lines
变成了抛物线
is actually a curve called a parabola.
这可以用一个简单的等式表达
This is represented by a simple equation,
很美的图案
and it’s a beautiful pattern.
这就是我们所做的
So this is the stuff that we do.
找到某种模式 然后表达出来
We find patterns, and we represent them.
这是一种很直白的定义
And I think this is a nice day-to-day definition.
但是今天 我想讨论得更深入一些
But today I want to go a little bit deeper,
思考它们的本质是什么
and think about what the nature of this is.
是什么造就了这一切
What makes it possible?
要看得更深入一些
There’s one thing that’s a little bit deeper,
就要求我们有转换角度的能力
and that has to do with the ability to change your perspective.
当你换一种角度来看问题
And I claim that when you change your perspective,
当你接受另一种观点
and if you take another point of view,
你就能在所见所闻中
you learn something new about what you are watching
学到新的东西
or looking at or hearing.
我认为这一点非常重要
And I think this is a really important thing that we do all the time.
让我们看看这个简单的方程
So let’s just look at this simple equation,
x+x=2x
x + x = 2 • x.
这是一个很好的模式 也是正确的
This is a very nice pattern, and it’s true,
因为5+5=2×5
because 5 + 5 = 2 • 5, etc.
这个等式我们司空见惯了
We’ve seen this over and over, and we represent it like this.
但是仔细想一想 这是一个等式
But think about it: this is an equation.
它代表一个事物与另一个事物相等
It says that something is equal to something else,
这么表述有两种角度
and that’s two different perspectives.
一种是总和
One perspective is, it’s a sum.
是相加的过程
It’s something you plus together.
另一种是相乘
On the other hand, it’s a multiplication,
这是两种不同的角度
and those are two different perspectives.
我会进一步说每个等式都像这样
And I would go as far as to say that every equation is like this,
每一个使用等号连接的数学方程
every mathematical equation where you use that equality sign
实际上都是隐喻
is actually a metaphor.
是两种事物间的类比
It’s an analogy between two things.
你观察一件事情 产生两种观点
You’re just viewing something and taking two different points of view,
然后用一种语言来表达
and you’re expressing that in a language.
看这个方程
Have a look at this equation.
它是最美的等式之一
This is one of the most beautiful equations.
简单表明了
It simply says that, well,
等式两边都是-1
two things, they’re both -1.
左手边的是-1 右边的也是
This thing on the left-hand side is -1, and the other one is.
我认为这是数学中很重要的部分
And that, I think, is one of the essential parts
——采取不同的观点
of mathematics — you take different points of view.
我们继续
So let’s just play around.
选一个数字好了
Let’s take a number.
我们知道4/3 知道它的含义
We know four-thirds. We know what four-thirds is.
就是1.333…… 但是一定要加上后面的省略号
It’s 1.333, but we have to have those three dots,
否则就不是准确的4/3了
otherwise it’s not exactly four-thirds.
但只有在使用十进制时才如此
But this is only in base 10.
我们的数字系统用的是10位计数
You know, the number system, we use 10 digits.
如果我们改成2位计数
If we change that around and only use two digits,
也就是二进制
that’s called the binary system.
就变成了这样
It’s written like this.
我们现在在讨论数字
So we’re now talking about the number.
讨论4/3这个数字
The number is four-thirds.
我们也可以这样表示
We can write it like this,
我们改变进制 改变数位
and we can change the base, change the number of digits,
就可以用不同的方式书写
and we can write it differently.
所有这些都代表同一个数
So these are all representations of the same number.
我们甚至可以把它简单写作1.3或1.6
We can even write it simply, like 1.3 or 1.6.
取决于我们选用哪种进制
It all depends on how many digits you have.
或者我们还可以简单写成这样
Or perhaps we just simplify and write it like this.
我喜欢这种 因为它表示4被3除
I like this one, because this says four divided by three.
表现了两个数字间的关系
And this number expresses a relation between two numbers.
上边是4 下边是3
You have four on the one hand and three on the other.
你可以用许多方式来把这个数字可视化
And you can visualize this in many ways.
从不同的角度来看这个数字
What I’m doing now is viewing that number from different perspectives.
我在不断尝试
I’m playing around.
改变观察事物的角度
I’m playing around with how we view something,
我是故意这么做的
and I’m doing it very deliberately.
让我们画一个网格
We can take a grid.
假如为4行3列 那么这条线就始终代表5
If it’s four across and three up, this line equals five, always.
肯定如此 这是一个美丽的图案
It has to be like this. This is a beautiful pattern.
4和3和5
Four and three and five.
这个长方形 长宽比为4:3
And this rectangle, which is 4 x 3,
你们见过很多次的
you’ve seen a lot of times.
就是你们的屏幕大小的平均值
This is your average computer screen.
800 x 600 或是 1600 x 1200
800 x 600 or 1,600 x 1,200
分别是电脑和电视的屏幕
is a television or a computer screen.
这都是很好的表达方式
So these are all nice representations,
但是我还想再深入一点点 再玩一下这些数字
but I want to go a little bit further and just play more with this number.
现在 你能看到两个圆 我要像这样旋转它们
Here you see two circles. I’m going to rotate them like this.
看一下左上角的那个
Observe the upper-left one.
它转得更快一点儿 对吧
It goes a little bit faster, right?
你们都能看到
You can see this.
准确来说 它的旋转速度是慢速的4/3倍
It actually goes exactly four-thirds as fast.
也就是说 它每转4圈
That means that when it goes around four times,
另一个圆就会转3圈
the other one goes around three times.
现在 画两条线 并标明相交处的点
Now let’s make two lines, and draw this dot where the lines meet.
我们就能得到一个跳舞的点
We get this dot dancing around.
(笑声)
(Laughter)
这个点就来源于4/3这个数字
And this dot comes from that number.
是吧 现在 让我来看看它的轨迹
Right? Now we should trace it.
把轨迹画出来看看是什么样子
Let’s trace it and see what happens.
这就是数学
This is what mathematics is all about.
就是不断探索会发生什么
It’s about seeing what happens.
而这来自于4/3这个数字
And this emerges from four-thirds.
我觉得 这就是4/3的肖像
I like to say that this is the image of four-thirds.
比数字好看多了——(欢呼)
It’s much nicer — (Cheers)
谢谢
Thank you!
(掌声)
(Applause)
其实这不算新鲜事了
This is not new.
很早以前就被发现了 但是——
This has been known for a long time, but —
(笑声)
(Laughter)
但是这仅仅是4/3
But this is four-thirds.
让我们再做一个实验
Let’s do another experiment.
让我们选一个声音 是这样的(嘟)
Let’s now take a sound, this sound: (Beep)
这是一个完美的A 440Hz
This is a perfect A, 440Hz.
把它翻倍
Let’s multiply it by two.
就得到了这个声音 (嘟)
We get this sound. (Beep)
同时播放这两种声音 听起来是这个效果
When we play them together, it sounds like this.
这是一个八度音 对吧
This is an octave, right?
我们来玩一个游戏 我们再放一次A
We can do this game. We can play a sound, play the same A.
然后我们把它翻为1.5倍
We can multiply it by three-halves.
(嘟)
(Beep)
我们称之为纯五度音
This is what we call a perfect fifth.
(嘟)
(Beep)
把它们一起播放 听起来很不错
They sound really nice together.
让我们把这个声音翻4/3倍
Let’s multiply this sound by four-thirds. (Beep)
会怎么样
What happens?
你们会得到这个声音
You get this sound. (Beep)
纯四度音
This is the perfect fourth.
如果第一个音是A 那么这就是一个D
If the first one is an A, this is a D.
一起播放 是这样的声音
They sound like this together. (Beeps)
这就是4/3的声音
This is the sound of four-thirds.
这就是改变角度
What I’m doing now, I’m changing my perspective.
我是在从另一个角度看一个数字
I’m just viewing a number from another perspective.
也可以用节奏来表示
I can even do this with rhythms, right?
我可以选一个节奏 在一段时间内敲3下(鼓点声)
I can take a rhythm and play three beats at one time (Drumbeats)
一段固定的时间
in a period of time,
然后在同样的时间内敲4下
and I can play another sound four times in that same space.
(铛铛声)
(Clanking sounds)
单独听很枯燥 但如果放在一起
Sounds kind of boring, but listen to them together.
(鼓点和铛铛声)
(Drumbeats and clanking sounds)
(笑声)
(Laughter)
嘿 好多了
Hey! So.
(笑声)
(Laughter)
我还可以加点儿踩镲声
I can even make a little hi-hat.
(鼓点和踩镲声)
(Drumbeats and cymbals)
听到了吗
Can you hear this?
所以 这就是4/3的声音
So, this is the sound of four-thirds.
4/3的节律
Again, this is as a rhythm.
(鼓点声和踩镲声)
(Drumbeats and cowbell)
我还可以继续玩 用这个数字做游戏
And I can keep doing this and play games with this number.
4/3是一个超棒的数字 我爱死4/3了
Four-thirds is a really great number. I love four-thirds!
(笑声)
(Laughter)
说真的——4/3的价值被低估了
Truly — it’s an undervalued number.
如果你拿一个球体 看看它的体积
So if you take a sphere and look at the volume of the sphere,
会发现其实球体体积就是某个圆柱体积的4/3倍
it’s actually four-thirds of some particular cylinder.
所以4/3出现在了球体里 是球的体积
So four-thirds is in the sphere. It’s the volume of the sphere.
好 我为什么玩这些
OK, so why am I doing all this?
是想跟你们谈谈 理解一件事物的意义
Well, I want to talk about what it means to understand something
谈谈我们所说的理解是什么
and what we mean by understanding something.
这就是我的目的
That’s my aim here.
我认为要弄懂一些事
And my claim is that you understand something
如果你能够从不同的角度去看待一件事情.
if you have the ability to view it from different perspectives.
让我们看看这个字母 这是一个漂亮的R 对吧
Let’s look at this letter. It’s a beautiful R, right?
你们怎么判断这是个R
How do you know that?
因为你们看过各种各样的R
Well, as a matter of fact, you’ve seen a bunch of R’s,
然后进行归纳
and you’ve generalized
提取它们的共性 找到了一种模式
and abstracted all of these and found a pattern.
然后你们确认这是一个R
So you know that this is an R.
所以 我要说的是
So what I’m aiming for here is saying something
理解事物和变换角度
about how understanding and changing your perspective
是有关的
are linked.
我是一名教师和演讲者
And I’m a teacher and a lecturer,
我可以利用这一点去教课
and I can actually use this to teach something,
因为我用隐喻和类比的方法 给学生们换一种方式讲故事
because when I give someone else another story, a metaphor, an analogy,
从不同的角度去讲述一件事
if I tell a story from a different point of view,
我就能让他们真正理解
I enable understanding.
我让理解变为了可能
I make understanding possible,
因为你们必须要 归纳自己的所见所闻
because you have to generalize over everything you see and hear,
如果我给你们另一个角度你们做起来就会更容易
and if I give you another perspective, that will become easier for you.
让我们再举一个例子
Let’s do a simple example again.
这是4和3 这是4个三角形
This is four and three. This is four triangles.
这也是某种4/3
So this is also four-thirds, in a way.
让我们把它们连起来
Let’s just join them together.
现在我们再玩一个游戏 把它们折叠起来
Now we’re going to play a game; we’re going to fold it up
形成一个三维结构
into a three-dimensional structure.
我喜欢这个
I love this.
这是一个金字塔型
This is a square pyramid.
让我们再做一个 把它们放在一起
And let’s just take two of them and put them together.
就形成了一个八面体
So this is what is called an octahedron.
这是5种正多面体(又叫柏拉图立体)之一
It’s one of the five platonic solids.
现在我们可以真的来改变角度
Now we can quite literally change our perspective,
绕各种轴旋转它
because we can rotate it around all of the axes
从其它角度来观察
and view it from different perspectives.
我可以改变旋转轴
And I can change the axis,
改变观察角度
and then I can view it from another point of view,
还是同一个物体 只是看起来有一些不同
but it’s the same thing, but it looks a little different.
我可以再做一次
I can do it even one more time.
我每调整一次 就会有新东西出现
Every time I do this, something else appears,
所以通过改变角度
so I’m actually learning more about the object
我能更加了解这个物体
when I change my perspective.
我可以把它作为创造理解的工具
I can use this as a tool for creating understanding.
我可以把两个正四面体 像这样穿起来
I can take two of these and put them together like this
看看会发生什么
and see what happens.
有点儿像正八面体
And it looks a little bit like the octahedron.
把它旋转起来再看
Have a look at it if I spin it around like this.
发生了什么
What happens?
如果你把这两个物体 拼在一起 旋转它
Well, if you take two of these, join them together and spin it around,
你就又得到了一个正八面体
there’s your octahedron again,
一个漂亮的结构
a beautiful structure.
如果你把它平摊在地上
If you lay it out flat on the floor,
这就是一个正八面体
this is the octahedron.
正八面体的平面结构图
This is the graph structure of an octahedron.
我还可以继续玩
And I can continue doing this.
在正八面体周围画三个大圈
You can draw three great circles around the octahedron,
转动看看
and you rotate around,
三个大圈实际上是与正八面体相连的
so actually three great circles is related to the octahedron.
如果我拿一个自行车泵 把它充满气
And if I take a bicycle pump and just pump it up,
你会发现 它看起来还是有点儿像正八面体的
you can see that this is also a little bit like the octahedron.
看出来我在做什么了吗
Do you see what I’m doing here?
我在不停改变角度
I am changing the perspective every time.
所以,现在让我们退一步——
So let’s now take a step back —
——这其实是一个隐喻 退后一步——
and that’s actually a metaphor, stepping back —
看看我们在做的事情
and have a look at what we’re doing.
我在使用隐喻
I’m playing around with metaphors.
在变换角度 进行类比
I’m playing around with perspectives and analogies.
变换不同的角度
I’m telling one story in different ways.
来讲同一个故事
I’m telling stories.
我在叙述 而且做了好几种叙述
I’m making a narrative; I’m making several narratives.
我认为这一切使得理解变成可能
And I think all of these things make understanding possible.
我认为这是理解事物的关键
I think this actually is the essence of understanding something.
我深信这点
I truly believe this.
所以 关于改变你们的角度——
So this thing about changing your perspective —
对人类来说十分重要
it’s absolutely fundamental for humans.
让我们来看看地球
Let’s play around with the Earth.
让我们放大到海洋 看看海洋
Let’s zoom into the ocean, have a look at the ocean.
我们可以放大任何事物
We can do this with anything.
我们以海洋为例 仔细的看看它
We can take the ocean and view it up close.
我们能观察海浪
We can look at the waves.
或是沙滩
We can go to the beach.
我们也可以从另一个角度看海洋
We can view the ocean from another perspective.
每变一次角度 我们就能对海洋了解得多一些
Every time we do this, we learn a little bit more about the ocean.
如果我们走到海边 就能闻到海水的味道 对吧
If we go to the shore, we can kind of smell it, right?
能听到海浪的声音
We can hear the sound of the waves.
能尝到风中咸咸的味道
We can feel salt on our tongues.
所有这些 都是不同的角度
So all of these are different perspectives.
而这个(角度)是最棒的
And this is the best one.
我们进入水中
We can go into the water.
从内部来观察
We can see the water from the inside.
你们知道吗
And you know what?
这对数学和计算机科学来说都绝对重要
This is absolutely essential in mathematics and computer science.
如果你能从一个结构的内部去进行观察
If you’re able to view a structure from the inside,
那你就能够真正认识它
then you really learn something about it.
认识到它的本质
That’s somehow the essence of something.
所以 当我们一路前行
So when we do this, and we’ve taken this journey
进入海洋
into the ocean,
我们发挥了想象力
we use our imagination.
我认为这又更深入了一层
And I think this is one level deeper,
是改变角度的必然要求
and it’s actually a requirement for changing your perspective.
我们可以做个游戏
We can do a little game.
想象一下你正坐在那儿
You can imagine that you’re sitting there.
然后你同时又在上面
You can imagine that you’re up here, and that you’re sitting here.
你就可以从外部审视你自己了
You can view yourselves from the outside.
这听起来很奇怪
That’s really a strange thing.
你在改变你的角度
You’re changing your perspective.
你在使用你的想象力
You’re using your imagination,
你在从外部审视你自己
and you’re viewing yourself from the outside.
这需要有想象力
That requires imagination.
数学和计算机科学是最具想象力的艺术形式
Mathematics and computer science are the most imaginative art forms ever.
还有一种改变角度的方式
And this thing about changing perspectives
可能更被你们熟知
should sound a little bit familiar to you,
因为我们每天都在做
because we do it every day.
叫做共鸣
And then it’s called empathy.
当我从你的角度看世界的时候
When I view the world from your perspective,
我就与你产生了共鸣
I have empathy with you.
如果我能够真正的
If I really, truly understand
理解你们眼中的世界
what the world looks like from your perspective,
那我就与你产生了共鸣
I am empathetic.
这需要想象力
That requires imagination.
这就是我们获得理解的方式
And that is how we obtain understanding.
而这种方式充斥了数学和计算机科学领域
And this is all over mathematics and this is all over computer science,
共情和这些学科间有着深刻的联系
and there’s a really deep connection between empathy and these sciences.
所以 我的结论是
So my conclusion is the following:
深入的理解一件事
understanding something really deeply
与转换角度的能力密切相关
has to do with the ability to change your perspective.
所以我的建议是 尝试转换你的角度
So my advice to you is: try to change your perspective.
你可以学习数学
You can study mathematics.
这是锻炼大脑的好方法
It’s a wonderful way to train your brain.
变换你们的角度 让思维变得更灵活
Changing your perspective makes your mind more flexible.
它能够让你们易于接受新事物
It makes you open to new things,
能够理解事物
and it makes you able to understand things.
请允许我再使用一次隐喻
And to use yet another metaphor:
让思维像水一样吧
have a mind like water.
那会很美妙
That’s nice.
谢谢
Thank you.
(掌声)
(Applause)

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视频概述

一个数学老师讲人文真的挺辛苦,一个简单的道理复杂化,说的是从不同角度看问题理解世界的本质,中间用的数学例子还是比较有趣的

听录译者

收集自网络

翻译译者

血蔷薇

审核员

与光同尘

视频来源

https://www.youtube.com/watch?v=ZQElzjCsl9o

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