TED 如何透过现象看本质 （视频+中英对照翻译+音频）

Math is the hidden secret to understanding the world

TED简介：本期TED演讲者Roger Antonsen先生，通过最具想象力的艺术形式—数学，揭秘世界的奥秘和内部运转本质。他向我们解释，数学是理解万物之源。

演讲者：Roger Antonsen

片长：17：09

理解世界的秘诀：数学 (换个角度看世界)_腾讯视频

视频+音频+中英对照翻译：TED | 如何透过现象看本质

TED与纪录片

中英文对照翻译

Hi. I want to talk about understanding, and the nature of understanding, and what the essence of understanding is, because understanding is something we aim for, everyone. We want to understand things. My claim is that understanding has to do with the ability to change your perspective. If you don't have that, you don't have understanding. So that is my claim.

大家好。我想谈谈理解和理解的本质，理解到底是什么，因为我们都在追求理解，我们想理解世间万物。我认为理解是一种能力，转变（固有）观点的能力。如果我们缺乏它， 就说明我们缺乏理解力，这是我的结论。

And I want to focus on mathematics. Many of us think of mathematics as addition, subtraction,multiplication, division, fractions, percent, geometry, algebra — all that stuff. But actually, I want to talk about the essence of mathematics as well. And my claim is that mathematics has to do with patterns.

我想重点讲讲数学。很多人认为，数学就是 加、减、乘、除、分数、百分数、几何、代数等等。但今天，我也想讲讲数学的本质，我的观点是，数学跟模式有关。

Behind me, you see a beautiful pattern, and this pattern actually emerges just from drawing circles in a very particular way. So my day-to-day definition of mathematics that I use every day is the following: First of all, it's about finding patterns. And by "pattern," I mean a connection, a structure, some regularity,some rules that govern what we see.

在我身后，是一个美丽的图案，而这个图案，实际上是通过特定方式不断画圆组成的。所以我对数学有一个的定义非常直白：首先，数学的关键是寻找模式。这里的模式指的是某种联系、结构，或者规律、规则，这些东西控制了我们所见的事物。

Second of all, I think it is about representing these patterns with a language. We make up language if we don't have it, and in mathematics, this is essential. It's also about making assumptions and playing around with these assumptions and just seeing what happens. We're going to do that very soon. And finally, it's about doing cool stuff. Mathematics enables us to do so many things.

其次，我认为数学是一种语言，用来描述各种模式。如果没有现成的语言，就需要创造一种。在数学中，这点尤为重要。同时，数学也需要进行假设，对假设进行多方验证，看看结果如何。我们一会儿就会这么做。最后，数学可以用来做很酷的事情。能帮我们完成很多事。

So let's have a look at these patterns. If you want to tie a tie knot, there are patterns. Tie knots have names. And you can also do the mathematics of tie knots. This is a left-out, right-in, center-out and tie.This is a left-in, right-out, left-in, center-out and tie.

下面我们来看一些模式。如果你想系领带，会有很多种样式。每一种都有名字，因此领带结也包含数学。这是从左侧绕出，右侧绕入，中间抽出然后系紧的东方结。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, and a half-Windsor is all that. This is a mathematics book about tying shoelaces at the university level, because there are patterns in shoelaces. You can do it in so many different ways. We can analyze it. We can make up languages for it.

这就是我们专门为领带结创造的语言，最后还有半温莎结。这是一本关于系鞋带的数学书，大学级别的，因为系鞋带也有很多种模式，你可以用成千上万种方式来系鞋带。我们可以进行分析。然后为系鞋带也创造一种语言。

And representations are all over mathematics. This is Leibniz's notation from 1675. He invented a language for patterns in nature. When we throw something up in the air, it falls down. Why? We're not sure, but we can represent this with mathematics in a pattern.

这些都可以用数学方法来表达。这是莱布尼茨在1675年使用的符号，他创造了一种语言，来描述自然界的模式。当我们把物品抛向空中，它会掉下来。为什么？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. This is also an invented language. Can you guess for what? It is actually a notation system for dancing, for tap dancing. That enables him as a choreographer to do cool stuff, to do new things, because he has represented it.

这也是一种模式，是一种被发明的语言。你能猜到这是什么吗？这是一套表示舞蹈动作的符号，踢踏舞。这能让舞蹈编排者，编一些炫酷的，新的动作，因为他能用符号来描述动作。

I want you to think about how amazing representing something actually is. Here it says the word "mathematics." But actually, they're just dots, right? So how in the world can these dots represent the word? Well, they do. They represent the word "mathematics," and these symbols also represent that word and this we can listen to. It sounds like this.

请大家想一想，表达是多么神奇的东西。这里写的是“数学”这个词，实际上就是一些点，对吧？一些点怎么能表示单词呢？确实可以。他们代表了单词“数学”，这些符号也一样，这次我们可以用听的，听起来就像这样。（滴滴声）

Somehow these sounds represent the word and the concept. How does this happen? There's something amazing going on about representing stuff.

可以说，这些声音也代表了这个词和它的含义。这是怎么做到的呢？表达是一种很神奇的过程。

So I want to talk about that magic that happens when we actually represent something. Here you see just lines with different widths. They stand for numbers for a particular book. And I can actually recommend this book, it's a very nice book.Just trust me.

所以我想跟你们讨论一下在表达过程中 发生的神奇的事情。现在你们看到的只是不同宽度的线条。这些线条代表了一本书，强烈推荐这本书，非常不错。真的，不骗你们。

OK, so let's just do an experiment, just to play around with some straight lines. This is a straight line.Let's make another one. So every time we move, we move one down and one across, and we draw a new straight line, right? We do this over and over and over, and we look for patterns. So this pattern emerges, and it's a rather nice pattern. It looks like a curve, right? Just from drawing simple, straight lines.

好吧，让我们来做一个实验，来玩一下直线。这是一条直线，再画另外一条，每一次我们都往下、往右移动一格，画出一条新的直线。如此反复，从中寻找一种模式。我们得到了这个图案，是一个非常好看的图案。它看起来就像一道弧，对吧？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. Have a look at the curve. What does it look like? Is it a part of a circle? It's actually not a part of a circle. So I have to continue my investigation and look for the true pattern. Perhaps if I copy it and make some art? Well, no. Perhaps I should extend the lines like this, and look for the pattern there. Let's make more lines. We do this. And then let's zoom out and change our perspective again. Then we can actually see that what started out as just straight lines is actually a curve called a parabola. This is represented by a simple equation, and it's a beautiful pattern.

现在，稍微改变一下角度，旋转一下。再看这段弧，像什么？是不是像圆的一部分？其实它不是圆的一部分。所以我继续探寻，找出真正的模式。也许我可以复制它，画一幅画？好像不行。也许我应该延长这些线条，再来寻找模式，再多画一些线条，然后这样。把它缩小，再变换角度。然后我们就会发现，开始的直线变成了抛物线。这可以用一个简单的等式表达，很美的图案。

So this is the stuff that we do. We find patterns, and we represent them. And I think this is a nice day-to-day definition. But today I want to go a little bit deeper, and think about what the nature of this is. What makes it possible? There's one thing that's a little bit deeper, and that has to do with the ability to change your perspective. And I claim that when you change your perspective, and if you take another point of view, you learn something new about what you are watching or looking at or hearing. And I think this is a really important thing that we do all the time.

这就是我们所做的。找到某种模式，然后表达出来，这是一种很直白的定义。但是今天，我想讨论得更深入一些，思考它们的本质是什么。是什么造就了这一切？要看得更深入一些，就要求我们有转换角度的能力。当你换一种角度来看问题，当你接受另一种观点，你就能在所见所闻中，学到新的东西。我认为这一点非常重要。

So let's just look at this simple equation, x + x = 2 • x. This is a very nice pattern, and it's true, because 5 + 5 = 2 • 5, etc. We've seen this over and over, and we represent it like this. But think about it: this is an equation. It says that something is equal to something else, and that's two different perspectives. One perspective is, it's a sum. It's something you plus together. On the other hand, it's a multiplication, and those are two different perspectives. And I would go as far as to say that every equation is like this, every mathematical equation where you use that equality sign is actually a metaphor. It's an analogy between two things. You're just viewing something and taking two different points of view, and you're expressing that in a language.

让我们看看这个简单的方程，x+x=2x这是一个很好的模式，也是正确的。因为5+5=2x5。这个等式我们司空见惯了。但是仔细想一想：这是一个等式。它代表一个事物与另一个事物相等， 这么表述有两种角度。一种是总和。是相加的过程。另一种是相乘。这是两种不同的角度。我会进一步说，每个等式都像这样，每一个使用等号连接的数学方程实际上都是隐喻。是两种事物间的类比。 你观察一件事情，产生两种观点，然后用一种语言来表达。

Have a look at this equation. This is one of the most beautiful equations. It simply says that, well, two things, they're both -1. This thing on the left-hand side is -1, and the other one is. And that, I think, is one of the essential parts of mathematics — you take different points of view.

看这个方程，它是最美的等式之一。简单表明了，等式两边都是-1。左手边的是-1，右边的也是。我认为这是数学中很重要的部分——采取不同的观点。

So let's just play around. Let's take a number. We know four-thirds. We know what four-thirds is. It's 1.333, but we have to have those three dots, otherwise it's not exactly four-thirds. But this is only in base 10. You know, the number system, we use 10 digits. If we change that around and only use two digits,that's called the binary system. It's written like this. So we're now talking about the number. The number is four-thirds. We can write it like this, and we can change the base, change the number of digits, and we can write it differently.

我们继续选一个数字好了。我们知道4/3，知道它的含义。就是1.333……，但是 一定要加上后面的省略号，否则就不是准确的4/3了。但只有在使用十进制时才如此，我们的数字系统用的是10位计数。如果我们改成2位计数，也就是二进制，就变成了这样。我们现在在讨论数字，讨论4/3这个数字。我们也可以这样表示，我们改变进制，改变数位，就可以用不同的方式书写。

So these are all representations of the same number. We can even write it simply, like 1.3 or 1.6. It all depends on how many digits you have. Or perhaps we just simplify and write it like this. I like this one, because this says four divided by three. And this number expresses a relation between two numbers.You have four on the one hand and three on the other. And you can visualize this in many ways.

所有这些都代表同一个数。我们甚至可以把它简单写作1.3或1.6。取决于我们选用哪种进制。或者我们还可以简单写成这样，我喜欢这种，因为它表示4被3除。表现了两个数字间的关系。 上边是4，下边是3。你可以用许多方式来把这个数字可视化，从不同的角度来看这个数字。

What I'm doing now is viewing that number from different perspectives. I'm playing around. I'm playing around with how we view something, and I'm doing it very deliberately. We can take a grid. If it's four across and three up, this line equals five, always. It has to be like this. This is a beautiful pattern. Four and three and five. And this rectangle, which is 4 x 3, you've seen a lot of times. This is your average computer screen.800 x 600 or 1,600 x 1,200 is a television or a computer screen.

我在不断尝试改变观察事物的角度，我是故意这么做的。让我们画一个网格。假如为4行3列，那么这条线就始终代表5，肯定如此，这是一个美丽的图案。4和3和5。这个长方形，长宽比为4:3，你们见过很多次的。就是你们的屏幕大小的平均值。800 x 600 或是1600 x 1200分别是电脑和电视的屏幕。

So these are all nice representations, but I want to go a little bit further and just play more with this number. Here you see two circles. I'm going to rotate them like this. Observe the upper-left one. It goes a little bit faster, right? You can see this. It actually goes exactly four-thirds as fast. That means that when it goes around four times, the other one goes around three times. Now let's make two lines, and draw this dot where the lines meet. We get this dot dancing around.

这都是很好的表达方式，但是我还想再深入一点点，再玩一下这些数字。现在，你能看到两个圆，我要像这样旋转它们。看一下左上角的那个，它转得更快一点儿，对吧？你们都能看到。准确来说，它的旋转速度是慢速的4/3倍。也就是说，它每转4圈，另一个圆就会转3圈。现在，画两条线，并标明相交处的点，我们就能得到一个跳舞的点。

And this dot comes from that number. Right? Now we should trace it. Let's trace it and see what happens. This is what mathematics is all about. It's about seeing what happens. And this emerges from four-thirds. I like to say that this is the image of four-thirds. It's much nicer — (Cheers)

这个点就来源于4/3这个数字。是吧？现在，让我来看看它的轨迹。把轨迹画出来，看看是什么样子。这就是数学，就是不断探索会发生什么。而这来自于4/3这个数字，我觉得，这就是4/3的肖像。比数字好看多了——（欢呼）

Thank you!(Applause) This is not new. This has been known for a long time, but —But this is four-thirds.Let's do another experiment. Let's now take a sound, this sound: (Beep)

其实这不算新鲜事了。 很早以前就被发现了， 但是——但是这仅仅是4/3。让我们再做一个实验，让我们选一个声音，是这样的：（嘟）

This is a perfect A, 440Hz. Let's multiply it by two. We get this sound. (Beep)

When we play them together, it sounds like this. This is an octave, right? We can do this game. We can play a sound, play the same A. We can multiply it by three-halves.(Beep)

This is what we call a perfect fifth.(Beep)

这是一个完美的A，440Hz。 把它翻倍。 就得到了这个声音。（嘟）

同时播放这两种声音， 听起来是这个效果。 这是一个八度音，对吧？ 我们来玩一个游戏。 我们再放一次A。 然后我们把它翻为1.5倍。（嘟）我们称之为纯五度音。（嘟）

They sound really nice together. Let's multiply this sound by four-thirds. (Beep)What happens? You get this sound. (Beep)

This is the perfect fourth. If the first one is an A, this is a D. They sound like this together. (Beeps)

把它们一起播放，听起来很不错。 让我们把这个声音翻4/3倍。会怎么样？ 你们会得到这个声音。

纯四度音。 如果第一个音是A， 那么这就是一个D。 一起播放，是这样的声音。

This is the sound of four-thirds. What I'm doing now, I'm changing my perspective. I'm just viewing a number from another perspective.

这就是4/3的声音。 这就是改变角度。 我是在从另一个角度看一个数字。可以用节奏来表示。

I can even do this with rhythms, right? I can take a rhythm and play three beats at one time (Drumbeats)in a period of time, and I can play another sound four times in that same space.(Clanking sounds)

我可以选一个节奏， 在一段时间内敲3下（鼓点声）一段固定的时间， 然后在同样的时间内敲4下。（铛铛声）

Sounds kind of boring, but listen to them together.(Drumbeats and clanking sounds)

单独听很枯燥， 但如果放在一起。（鼓点和铛铛声）

Hey! So.I can even make a little hi-hat.(Drumbeats and cymbals)Can you hear this? So, this is the sound of four-thirds. Again, this is as a rhythm(Drumbeats and cowbell)And I can keep doing this and play games with this number. Four-thirds is a really great number. I love four-thirds!

嘿！好多了。我还可以加点儿踩镲声。（鼓点和踩镲声）听到了吗？所以，这就是4/3的声音，4/3的节律（鼓点声和踩镲声）我还可以继续玩，用这个数字做游戏。4/3是一个超棒的数字，我爱死4/3了！

Truly — it's an undervalued number. So if you take a sphere and look at the volume of the sphere, it's actually four-thirds of some particular cylinder. So four-thirds is in the sphere. It's the volume of the sphere.

4/3的价值被低估了。如果你拿一个球体，看看它的体积，会发现其实球体体积就是某个圆柱体积的4/3倍。所以4/3出现在了球体里，是球的体积。

OK, so why am I doing all this? Well, I want to talk about what it means to understand something and what we mean by understanding something. That's my aim here. And my claim is that you understand something if you have the ability to view it from different perspectives. Let's look at this letter. It's a beautiful R, right? How do you know that? Well, as a matter of fact, you've seen a bunch of R's, and you've generalized and abstracted all of these and found a pattern. So you know that this is an R.

好，我为什么玩这些？是想跟你们谈谈理解一件事物的意义，谈谈我们所说的理解是什么。这就是我的目的。我认为，只有当我们从多个角度去审视同一事物时，才能说我们理解了它。让我们看看这个字母，这是一个漂亮的R，对吧？你们怎么判断这是个R？因为你们看过各种各样的R，然后进行归纳，提取它们的共性，找到了一种模式。然后你们确认这是一个R。

So what I'm aiming for here is saying something about how understanding and changing your perspective are linked. And I'm a teacher and a lecturer, and I can actually use this to teach something,because when I give someone else another story, a metaphor, an analogy, if I tell a story from a different point of view, I enable understanding. I make understanding possible, because you have to generalize over everything you see and hear, and if I give you another perspective, that will become easier for you.

所以，我要说的是理解事物和变换角度是有关的。我是一名教师和演讲者，我可以利用这一点去教课，因为我用隐喻和类比的方法，给学生们换一种方式讲故事，从不同的角度去讲述一件事，我就能让他们真正理解。我让理解变为了可能，因为你们必须要归纳自己的所见所闻，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. This is four and three. This is four triangles. So this is also four-thirds, in a way. Let's just join them together. Now we're going to play a game; we're going to fold it up into a three-dimensional structure. I love this. This is a square pyramid. And let's just take two of them and put them together. So this is what is called an octahedron. It's one of the five platonic solids. Now we can quite literally change our perspective, because we can rotate it around all of the axes and view it from different perspectives. And I can change the axis, and then I can view it from another point of view, but it's the same thing, but it looks a little different. I can do it even one more time.

让我们再举一个例子，这是4和3，这是4个三角形，这也是某种4/3。让我们把它们连起来。现在我们再玩一个游戏，把它们折叠起来，形成一个三维结构，我喜欢这个，这是一个金字塔形。让我们再做一个，把它们放在一起。就形成了一个八面体。这是5种正多面体 （又叫柏拉图立体）之一。现在我们可以真的来改变角度，绕各种轴旋转它，从其它角度来观察。我可以改变旋转轴，改变观察角度，还是同一个物体，只是看起来有一些不同。，我可以再做一次。

Every time I do this, something else appears, so I'm actually learning more about the object when I change my perspective. I can use this as a tool for creating understanding. I can take two of these and put them together like this and see what happens. And it looks a little bit like the octahedron. Have a look at it if I spin it around like this. What happens? Well, if you take two of these, join them together and spin it around, there's your octahedron again, a beautiful structure. If you lay it out flat on the floor, this is the octahedron. This is the graph structure of an octahedron. And I can continue doing this. You can draw three great circles around the octahedron, and you rotate around, so actually three great circles is related to the octahedron. And if I take a bicycle pump and just pump it up, you can see that this is also a little bit like the octahedron. Do you see what I'm doing here? I am changing the perspective every time.

我每调整一次，就会有新东西出现，所以通过改变角度，我能更加了解这个物体。我可以把它作为创造理解的工具，我可以把两个正四面体，像这样穿起来，看看会发生什么。有点儿像正八面体，把它旋转起来再看，发生了什么？如果你把这两个物体拼在一起，旋转它，你就又得到了一个正八面体，一个漂亮的结构。如果你把它平摊在地上，这就是一个正八面体，正八面体的平面结构图。我还可以继续玩，在正八面体周围画三个大圈，转动看看，三个大圈实际上是与正八面体相连的。如果我拿一个自行车泵，把它充满气，你会发现，它看起来还是有点儿像正八面体的。看出来我在做什么了吗？我在不停改变角度。

So let's now take a step back — and that's actually a metaphor, stepping back — and have a look at what we're doing. I'm playing around with metaphors. I'm playing around with perspectives and analogies. I'm telling one story in different ways. I'm telling stories. I'm making a narrative; I'm making several narratives. And I think all of these things make understanding possible. I think this actually is the essence of understanding something. I truly believe this.

让我们退后一步——这其实是一个隐喻，退后一步——看看我们在做的事情。我在使用隐喻，在变换角度，进行类比。变换不同的角度，来讲同一个故事。我在叙述，而且做了好几种叙述。我认为这一切使得理解变成可能。我认为这是理解事物的关键，我深信这点。

So this thing about changing your perspective — it's absolutely fundamental for humans. Let's play around with the Earth. Let's zoom into the ocean, have a look at the ocean. We can do this with anything.We can take the ocean and view it up close. We can look at the waves. We can go to the beach. We can view the ocean from another perspective. Every time we do this, we learn a little bit more about the ocean. If we go to the shore, we can kind of smell it, right? We can hear the sound of the waves. We can feel salt on our tongues. So all of these are different perspectives. And this is the best one. We can go into the water. We can see the water from the inside. And you know what? This is absolutely essential in mathematics and computer science. If you're able to view a structure from the inside, then you really learn something about it. That's somehow the essence of something.

所以，关于改变你们的角度——对人类来说十分重要。让我们来看看地球。让我们放大到海洋，看看海洋。我们可以放大任何事物。我们以海洋为例，仔细的看看它。我们能观察海浪或是沙滩。我们也可以从另一个角度看海洋，每变一次角度，我们就能对海洋了解得多一些。如果我们走到海边，就能闻到海水的味道，对吧？能听到海浪的声音。能尝到风中咸咸的味道。所有这些，都是不同的角度。而这个（角度）是最棒的。我们进入水中。从内部来观察。你们知道吗？这对数学和计算机科学来说都绝对重要如果你能从一个结构的内部去进行观察，那你就能够真正认识它，认识到它的本质。

So when we do this, and we've taken this journey into the ocean, we use our imagination. And I think this is one level deeper, and it's actually a requirement for changing your perspective. We can do a little game. You can imagine that you're sitting there. You can imagine that you're up here, and that you're sitting here. You can view yourselves from the outside. That's really a strange thing. You're changing your perspective. You're using your imagination, and you're viewing yourself from the outside. That requires imagination.

所以，当我们一路前行，进入海洋，我们发挥了想象力。我认为这又更深入了一层，是改变角度的必然要求。我们可以做个游戏，想象一下你正坐在那儿。然后你同时又在上面。，你就可以从外部审视你自己了。这听起来很奇怪，你在改变你的角度，你在使用你的想象力，你在从外部审视你自己，这需要有想象力。

Mathematics and computer science are the most imaginative art forms ever. And this thing about changing perspectives should sound a little bit familiar to you, because we do it every day. And then it's called empathy. When I view the world from your perspective, I have empathy with you. If I really, truly understand what the world looks like from your perspective, I am empathetic. That requires imagination.And that is how we obtain understanding. And this is all over mathematics and this is all over computer science, and there's a really deep connection between empathy and these sciences.

数学和计算机科学是最具想象力的艺术形式。还有一种改变角度的方式，可能更被你们熟知，因为我们每天都在做，叫做共情。当我从你的角度看世界的时候，我就与你产生了共情。如果我能够真正的理解你们眼中的世界，那我就与你产生了共情。这需要想象力，这就是我们获得理解的方式。而这种方式充斥了数学和计算机科学领域。共情和这些学科间有着深刻的联系。

So my conclusion is the following: understanding something really deeply has to do with the ability to change your perspective. So my advice to you is: try to change your perspective. You can study mathematics. It's a wonderful way to train your brain. Changing your perspective makes your mind more flexible. It makes you open to new things, and it makes you able to understand things. And to use yet another metaphor: have a mind like water. That's nice.

所以，我的结论是：深入的理解一件事与转换角度的能力密切相关。所以我的建议是：尝试转换你的角度。你可以学习数学，这是锻炼大脑的好方法，变换你们的角度，让思维变得更灵活。它能够让你们易于接受新事物，能够理解事物。请允许我再使用一次隐喻：让思维像水一样吧，会很不错的。

Thank you.(Applause)

谢谢大家。