如何透過現象看本質？

最新 09-12

本期TED演講者Roger Antonsen先生，通過最具想像力的藝術形式—數學，揭秘世界的奧秘和內部運轉本質。他向我們解釋，數學是理解萬物之源。

演說者：Roger Antonsen

演說題目：Math is the hidden secret to understanding the world

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。我想談談理解和理解的本質，理解到底是什麼，因為我們都在追求理解，我們想理解世間萬物。我認為理解是一種能力，轉變（固有）觀點的能力。如果我們缺乏它，就說明我們缺乏理解力，這是我的結論。

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.

我想重點講講數學。很多人認為，數學就是加、減、乘、除、分數、百分數、幾何、代數等等。但今天，我也想講講數學的本質，我的觀點是，數學跟模式有關。

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.

在我身後，是一個美麗的圖案，而這個圖案，實際上是通過特定方式不斷畫圓組成的。所以我對數學有一個的定義非常直白：首先，數學的關鍵是尋找模式。這裡的模式指的是某種聯繫、結構，或者規律、規則，這些東西控制了我們所見的事物。

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，我認為數學是一種語言，用來描述各種模式。如果沒有現成的語言，就需要創造一種。在數學中，這點尤為重要。同時，數學也需要進行假設，對假設進行多方驗證，看看結果如何。我們一會兒就會這麼做。最後，數學可以用來做很酷的事情。能幫我們完成很多事。

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.

下面我們來看一些模式。如果你想系領帶，會有很多種樣式。每一種都有名字，因此領帶結也包含數學。這是從左側繞出，右側繞入，中間抽出然後繫緊的東方結。這是從左側繞入，右側繞出，再左側繞入，中間抽出，最後繫緊的四手結。

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.

這就是我們專門為領帶結創造的語言，最後還有半溫莎結。這是一本關於系鞋帶的數學書，大學級別的，因為系鞋帶也有很多種模式，你可以用成千上萬種方式來系鞋帶。我們可以進行分析。然後為系鞋帶也創造一種語言。

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.

這些都可以用數學方法來表達。這是萊布尼茨在1675年使用的符號，他創造了一種語言，來描述自然界的模式。當我們把物品拋向空中，它會掉下來。為什麼？我們並不確定，但我們可以用數學把其歸結成一種模式。

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.

這也是一種模式，是一種被發明的語言。你能猜到這是什麼嗎？這是一套表示舞蹈動作的符號，踢踏舞。這能讓舞蹈編排者，編一些炫酷的，新的動作，因為他能用符號來描述動作。

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.

請大家想一想，表達是多麼神奇的東西。這裡寫的是「數學」這個詞，實際上就是一些點，對吧？一些點怎麼能表示單詞呢？確實可以。他們代表了單詞「數學」，這些符號也一樣，這次我們可以用聽的，聽起來就像這樣。（滴滴聲）

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

可以說，這些聲音也代表了這個詞和它的含義。這是怎麼做到的呢？表達是一種很神奇的過程。

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.

所以我想跟你們討論一下在表達過程中發生的神奇的事情。現在你們看到的只是不同寬度的線條。這些線條代表了一本書，強烈推薦這本書，非常不錯。真的，不騙你們。

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，讓我們來做一個實驗，來玩一下直線。這是一條直線，再畫另外一條，每一次我們都往下、往右移動一格，畫出一條新的直線。如此反覆，從中尋找一種模式。我們得到了這個圖案，是一個非常好看的圖案。它看起來就像一道弧，對吧？我們僅僅畫了些簡單的直線。

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.

現在，稍微改變一下角度，旋轉一下。再看這段弧，像什麼？是不是像圓的一部分？其實它不是圓的一部分。所以我繼續探尋，找出真正的模式。也許我可以複製它，畫一幅畫？好像不行。也許我應該延長這些線條，再來尋找模式，再多畫一些線條，然後這樣。把它縮小，再變換角度。然後我們就會發現，開始的直線變成了拋物線。這可以用一個簡單的等式表達，很美的圖案。

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.

這就是我們所做的。找到某種模式，然後表達出來，這是一種很直白的定義。但是今天，我想討論得更深入一些，思考它們的本質是什麼。是什麼造就了這一切？要看得更深入一些，就要求我們有轉換角度的能力。當你換一種角度來看問題，當你接受另一種觀點，你就能在所見所聞中，學到新的東西。我認為這一點非常重要。

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.

讓我們看看這個簡單的方程，x+x=2x這是一個很好的模式，也是正確的。因為5+5=2x5。這個等式我們司空見慣了。但是仔細想一想：這是一個等式。它代表一個事物與另一個事物相等，這麼表述有兩種角度。一種是總和。是相加的過程。另一種是相乘。這是兩種不同的角度。我會進一步說，每個等式都像這樣，每一個使用等號連接的數學方程實際上都是隱喻。是兩種事物間的類比。你觀察一件事情，產生兩種觀點，然後用一種語言來表達。

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.

看這個方程，它是最美的等式之一。簡單表明了，等式兩邊都是-1。左手邊的是-1，右邊的也是。我認為這是數學中很重要的部分——採取不同的觀點。

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.

我們繼續選一個數字好了。我們知道4/3，知道它的含義。就是1.333……，但是一定要加上後面的省略號，否則就不是準確的4/3了。但只有在使用十進位時才如此，我們的數字系統用的是10位計數。如果我們改成2位計數，也就是二進位，就變成了這樣。我們現在在討論數字，討論4/3這個數字。我們也可以這樣表示，我們改變進位，改變數位，就可以用不同的方式書寫。

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.

所有這些都代表同一個數。我們甚至可以把它簡單寫作1.3或1.6。取決於我們選用哪種進位。或者我們還可以簡單寫成這樣，我喜歡這種，因為它表示4被3除。表現了兩個數字間的關係。上邊是4，下邊是3。你可以用許多方式來把這個數字可視化，從不同的角度來看這個數字。

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.

我在不斷嘗試改變觀察事物的角度，我是故意這麼做的。讓我們畫一個網格。假如為4行3列，那麼這條線就始終代表5，肯定如此，這是一個美麗的圖案。4和3和5。這個長方形，長寬比為4:3，你們見過很多次的。就是你們的屏幕大小的平均值。800 x 600 或是1600 x 1200分別是電腦和電視的屏幕。

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.

這都是很好的表達方式，但是我還想再深入一點點，再玩一下這些數字。現在，你能看到兩個圓，我要像這樣旋轉它們。看一下左上角的那個，它轉得更快一點兒，對吧？你們都能看到。準確來說，它的旋轉速度是慢速的4/3倍。也就是說，它每轉4圈，另一個圓就會轉3圈。現在，畫兩條線，並標明相交處的點，我們就能得到一個跳舞的點。

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)

這個點就來源於4/3這個數字。是吧？現在，讓我來看看它的軌跡。把軌跡畫出來，看看是什麼樣子。這就是數學，就是不斷探索會發生什麼。而這來自於4/3這個數字，我覺得，這就是4/3的肖像。比數字好看多了——（歡呼）

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)

其實這不算新鮮事了。很早以前就被發現了，但是——但是這僅僅是4/3。讓我們再做一個實驗，讓我們選一個聲音，是這樣的：（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)

這是一個完美的A，440Hz。把它翻倍。就得到了這個聲音。（嘟）

同時播放這兩種聲音，聽起來是這個效果。這是一個八度音，對吧？我們來玩一個遊戲。我們再放一次A。然後我們把它翻為1.5倍。（嘟）我們稱之為純五度音。（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)

把它們一起播放，聽起來很不錯。讓我們把這個聲音翻4/3倍。會怎麼樣？你們會得到這個聲音。

純四度音。如果第一個音是A，那麼這就是一個D。一起播放，是這樣的聲音。

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.

這就是4/3的聲音。這就是改變角度。我是在從另一個角度看一個數字。可以用節奏來表示。

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)

我可以選一個節奏，在一段時間內敲3下（鼓點聲）一段固定的時間，然後在同樣的時間內敲4下。（鐺鐺聲）

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

單獨聽很枯燥，但如果放在一起。（鼓點和鐺鐺聲）

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。我還可以加點兒踩鑔聲。（鼓點和踩鑔聲）聽到了嗎？所以，這就是4/3的聲音，4/3的節律（鼓點聲和踩鑔聲）我還可以繼續玩，用這個數字做遊戲。4/3是一個超棒的數字，我愛死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的價值被低估了。如果你拿一個球體，看看它的體積，會發現其實球體體積就是某個圓柱體積的4/3倍。所以4/3出現在了球體里，是球的體積。

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，我為什麼玩這些？是想跟你們談談理解一件事物的意義，談談我們所說的理解是什麼。這就是我的目的。我認為，只有當我們從多個角度去審視同一事物時，才能說我們理解了它。讓我們看看這個字母，這是一個漂亮的R，對吧？你們怎麼判斷這是個R？因為你們看過各種各樣的R，然後進行歸納，提取它們的共性，找到了一種模式。然後你們確認這是一個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，我要說的是理解事物和變換角度是有關的。我是一名教師和演講者，我可以利用這一點去教課，因為我用隱喻和類比的方法，給學生們換一種方式講故事，從不同的角度去講述一件事，我就能讓他們真正理解。我讓理解變為了可能，因為你們必須要歸納自己的所見所聞，如果我給你們另一個角度，你們做起來就會更容易。

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.

讓我們再舉一個例子，這是4和3，這是4個三角形，這也是某種4/3。讓我們把它們連起來。現在我們再玩一個遊戲，把它們摺疊起來，形成一個三維結構，我喜歡這個，這是一個金字塔形。讓我們再做一個，把它們放在一起。就形成了一個八面體。這是5種正多面體（又叫柏拉圖立體）之一。現在我們可以真的來改變角度，繞各種軸旋轉它，從其它角度來觀察。我可以改變旋轉軸，改變觀察角度，還是同一個物體，只是看起來有一些不同。，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.

我每調整一次，就會有新東西出現，所以通過改變角度，我能更加了解這個物體。我可以把它作為創造理解的工具，我可以把兩個正四面體，像這樣穿起來，看看會發生什麼。有點兒像正八面體，把它旋轉起來再看，發生了什麼？如果你把這兩個物體拼在一起，旋轉它，你就又得到了一個正八面體，一個漂亮的結構。如果你把它平攤在地上，這就是一個正八面體，正八面體的平面結構圖。我還可以繼續玩，在正八面體周圍畫三個大圈，轉動看看，三個大圈實際上是與正八面體相連的。如果我拿一個自行車泵，把它充滿氣，你會發現，它看起來還是有點兒像正八面體的。看出來我在做什麼了嗎？我在不停改變角度。

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.

讓我們退後一步——這其實是一個隱喻，退後一步——看看我們在做的事情。我在使用隱喻，在變換角度，進行類比。變換不同的角度，來講同一個故事。我在敘述，而且做了好幾種敘述。我認為這一切使得理解變成可能。我認為這是理解事物的關鍵，我深信這點。

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，關於改變你們的角度——對人類來說十分重要。讓我們來看看地球。讓我們放大到海洋，看看海洋。我們可以放大任何事物。我們以海洋為例，仔細的看看它。我們能觀察海浪或是沙灘。我們也可以從另一個角度看海洋，每變一次角度，我們就能對海洋了解得多一些。如果我們走到海邊，就能聞到海水的味道，對吧？能聽到海浪的聲音。能嘗到風中鹹鹹的味道。所有這些，都是不同的角度。而這個（角度）是最棒的。我們進入水中。從內部來觀察。你們知道嗎？這對數學和計算機科學來說都絕對重要如果你能從一個結構的內部去進行觀察，那你就能夠真正認識它，認識到它的本質。

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，當我們一路前行，進入海洋，我們發揮了想像力。我認為這又更深入了一層，是改變角度的必然要求。我們可以做個遊戲，想像一下你正坐在那兒。然後你同時又在上面。，你就可以從外部審視你自己了。這聽起來很奇怪，你在改變你的角度，你在使用你的想像力，你在從外部審視你自己，這需要有想像力。

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.

數學和計算機科學是最具想像力的藝術形式。還有一種改變角度的方式，可能更被你們熟知，因為我們每天都在做，叫做共情。當我從你的角度看世界的時候，我就與你產生了共情。如果我能夠真正的理解你們眼中的世界，那我就與你產生了共情。這需要想像力，這就是我們獲得理解的方式。而這種方式充斥了數學和計算機科學領域。共情和這些學科間有著深刻的聯繫。

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，我的結論是：深入的理解一件事與轉換角度的能力密切相關。所以我的建議是：嘗試轉換你的角度。你可以學習數學，這是鍛煉大腦的好方法，變換你們的角度，讓思維變得更靈活。它能夠讓你們易於接受新事物，能夠理解事物。請允許我再使用一次隱喻：讓思維像水一樣吧，會很不錯的。

Thank you.(Applause)

謝謝

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