聽力原文如下
<code>1.Thank you very much.
非常感謝。
2.Please excuse me for sitting; I'm very old.
請原諒我坐著講； 我很老了。
3.(Laughter) Well, the topic I'm going to discuss is one which is in a certain sense very peculiar because it's very old.
(笑聲) 我要討論的主題 在某種意義上很古怪， 因為它很古老。
4.Roughness is part of human life forever and forever.
粗糙永永遠遠是 人類生活的一部分。
5.And ancient authors have written about it.
古代的作者描寫過它。
6.It was very much uncontrollable.
它很不受控制。
7.And in a certain sense, it seemed to be the extreme of complexity, just a mess, a mess and a mess.
在某種意義上， 它似乎是極度的複雜， 一片混亂、 亂七八糟。
8.There are many different kinds of mess.
有許多不同類型的混亂。
9.Now, in fact, by a complete fluke, I got involved many years ago in a study of this form of complexity.
那麼，實際上 完全是出於偶然， 我在許多年前 涉足於這種複雜性的研究。
10.And to my utter amazement, I found traces -- very strong traces, I must say -- of order in that roughness.
讓我非常驚訝的是， 我發現了—— 很清晰的蹤跡，我必須說—— 粗糙中秩序的蹤跡
11.And so today, I would like to present to you a few examples of what this represents.
今天，我想向你們展示 幾個 有代表性的例子
12.I prefer the word roughness to the word irregularity because irregularity -- to someone who had Latin in my long-past youth -- means the contrary of regularity.
我喜歡“粗糙”這個詞 而不是“不規則” 因為“不規則”—— 對於象我這樣 年輕時學過拉丁文的人來講—— 是“規則”的反義詞
13.But it is not so.
其實並非如此。
14.Regularity is the contrary of roughness because the basic aspect of the world is very rough.
“規則”是“粗糙”的反義詞 因為世界的基本面 是很粗糙的。
15.So let me show you a few objects.
那麼讓我給你們展示幾個東西。
16.Some of them are artificial.
有些是人造的
17.Others of them are very real, in a certain sense.
另外一些在某種意義上講是非常真實的。
18.Now this is the real. It's a cauliflower.
這個是真實的，這是一個菜花。
19.Now why do I show a cauliflower, a very ordinary and ancient vegetable?
我為什麼展示一個菜花， 一種非常普通和古老的蔬菜？
20.Because old and ancient as it may be, it's very complicated and it's very simple both at the same time.
因為儘管它很古老， 它卻是非常複雜的，同時也是 非常簡單的。
21.If you try to weigh it, of course it's very easy to weigh it.
如果您想稱它的重量，當然稱它是非常容易的。
22.And when you eat it, the weight matters.
當你吃它時，你關心的是重量。
23.But suppose you try to measure its surface.
但是假設您想 測量它的表面積。
24.Well, it's very interesting.
那麼，非常有意思。
25.If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller.
如果您用一把鋒利的刀， 切下其中一朵花， 分別觀察它， 您會看到一棵整菜花，只是小點兒。
26.And then you cut again, again, again, again, again, again, again, again, again.
您然後再切， 再切，再切，再切，….
27.And you still get small cauliflowers.
您得到仍然是小菜花。
28.So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole,
在人類的經驗中 總是有一些形狀 具有奇怪的特性 每個部分就象整體一樣
29.but smaller.
只是更小
30.Now, what did humanity do with that?
現在人類是否對此做了些什麼呢？
31.Very, very little.
非常非常少。
32.(Laughter) So what I did actually is to study this problem, and I found something quite surprising.
(笑聲) 我實際上做的就是 研究這個問題， 我發現了相當驚奇的事情。
33.That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more.
我們可以用數字來度量粗糙度 用一個數字 2.3，1.2. 有時需要多個數字
34.One day, a friend of mine, to bug me, brought a picture, and said, "What is the roughness of this curve?"
一天，我一個朋友 來煩我， 他帶來了一張圖片，說： “這條曲線的粗糙度是多少？”
35.I said, "Well, just short of 1.5."
我說，“好的，小與1.5。”
36.It was 1.48.
是1.48。
37.Now, it didn't take any time.
這一點也不費事。
38.I've been looking at these things for so long.
我觀察這些事物很長時間了。
39.So these numbers are the numbers which denote the roughness of these surfaces.
這些數字表示 這些表面的粗糙度。
40.I hasten to say that these surfaces are completely artificial.
我急切地說這些表面 完全是人造的，
41.They were done on a computer.
是用計算機產生的。
42.And the only input is a number.
唯一的輸入是一個數字.
43.And that number is roughness.
那個數字就是粗糙度.
44.And so on the left, I took the roughness copied from many landscapes.
在左邊 我取的是從許多風景中複製的粗糙度
45.To the right, I took a higher roughness.
在右邊，我採取了更高的粗糙度
46.So the eye, after a while, can distinguish these two very well.
過一會兒 眼睛就可以很好地區分這兩個.
47.Humanity had to learn about measuring roughness.
人類必須瞭解粗糙度的測量.
48.This is very rough, and this is sort of smooth, and this perfectly smooth.
這個非常粗糙，這個有點光滑，這個非常光滑。
49.Very few things are very smooth.
很少東西是很光滑的。
50.So then if you try to ask questions: what's the surface of a cauliflower?
所以你如果要問： 一個菜花的表面積是多少？
51.Well, you measure and measure and measure.
那麼，你反覆地測量。
52.Each time you're closer it gets bigger, down to very, very small distances.
測量得越精確，得到的數值就會越大， 直到非常、 非常小的差距。
53.What's the length of the coastline of these lakes?
這些湖泊的湖岸線 長度是多少？
54.The closer you measure, the longer it is.
你測量得越精確，結果越長。
55.The concept of length of coastline, which seems to be so natural because it's given in many cases, is, in fact, completely fallacy; there's no such thing.
海岸線長度的概念 似乎是那麼自然， 在許多情況下都會用到它， 但實際上，是完全錯誤的。根本沒有這種東西。
56.You must do it differently.
你必須換種方式對待它。
57.What good is that, to know these things?
知道這些事情有什麼好處呢？
58.Well, surprisingly enough, it's good in many ways.
足以讓人吃驚的是， 它的好處是多方面的。
59.To begin with, artificial landscapes, which I invented sort of, are used in cinema all the time.
首先，人工景觀—— 我發明的名詞—— 在電影中經常使用。
60.We see mountains in the distance.
我們看遠處的群山。
61.They may be mountains, but they may be just formulae, just cranked on.
他們可能是山，也可能只是個公式，是手搖出來的。
62.Now it's very easy to do.
現在很容易做。
63.It used to be very time consuming, but now it's nothing.
它曾經是非常耗時的，但現在沒有什麼。
64.Now look at that. That's a real lung.
現在看看這個，這是一個真正的肺。
65.Now a lung is something very strange.
肺是很奇怪的東西。
66.If you take this thing, you know very well it weighs very little.
如果你把它拿在手裡， 你就會知道它的重量很小。
67.The volume of a lung is very small.
肺的體積也很小。
68.But what about the area of the lung?
但肺的面積呢？
69.Anatomists were arguing very much about that.
解剖學家們對此爭論很大。
70.Some say that a normal male's lung has an area of the inside of a basketball [court].
有人說一個正常男性的肺 其面積相當於一個籃球 內部的面積。
71.And the others say, no, five basketball [courts].
有人說，不對，是五個籃球。
72.Enormous disagreements.
分歧很大。
73.Why so? Because, in fact, the area of the lung is something very ill-defined.
為何如此？因為實際上肺的面積的定義 非常含糊不清。
74.The bronchi branch, branch, branch.
支氣管分枝，分枝，分枝。
75.And they stop branching, not because of any matter of principle, but because of physical considerations, the mucus, which is in the lung.
它們停止產生分枝 不是因為規則的緣故， 而是因為物理的考慮—— 肺內的粘液。
76.So what happens is that it's the way you have a much bigger lung, but if it branches and branches, down to distances about the same for a whale, for a man
假如您有一個很大的肺 它的分支產生分支， 那將會怎樣呢？ 對於鯨魚、人和小的齧齒目動物來說
77.and for a little rodent.
沒有兩個距離大致相同。
78.Now, what good is it to have that?
那麼，這有什麼好處呢？
79.Well, surprisingly enough, amazingly enough, the anatomists had a very poor idea of the structure of the lung until very recently.
足以令人吃驚、足以讓人稱奇的是， 解剖學家直到最近才對肺的結構 有了一些正確的認識
80.And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses,
我認為我的數學， 令人吃驚地 為研究肺病 的外科醫生 幫了大忙。 還有腎病.
81.all these branching systems, for which there was no geometry.
這些器官都具有分枝系統， 但沒有幾何結構。
82.So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry.
因此我發現我自己，換句話說, 為這種沒有幾何結構的事物 構造了幾何規則。
83.And a surprising aspect of it is that very often, the rules of this geometry are extremely short.
並且，一個驚奇的方面是， 這幾何規則經常是 極其簡練的。
84.You have formulas that long.
你的公式只有這麼長。
85.And you crank it several times.
你把它迭代多次。
86.Sometimes repeatedly, again, again, again.
有時需要一次一次地重複，
87.The same repetition.
重複同樣的運算。
88.And at the end you get things like that.
最後，你將得到這樣的東西。
89.This cloud is completely, 100 percent artificial.
這朵雲彩是完全地， 100%地人造的。
90.Well, 99.9.
好吧，99.9%。
91.And the only part which is natural is a number, the roughness of the cloud, which is taken from nature.
其中唯一自然的部分 是一個數字，雲的粗糙度， 這是取自於自然的。
92.Something so complicated like a cloud, so unstable, so varying, should have a simple rule behind it.
象雲這種團狀的複雜東西， 如此不穩定，如此易變， 背後應該有一個簡單規則。
93.Now this simple rule is not an explanation of clouds.
這個簡單規則 不是對雲的一個解釋。
94.The seer of clouds had to take account of it.
雲的觀察者必須 把它考慮在內。
95.I don't know how much advanced these pictures are, they're old.
我不知道這些圖片有多先進， 他們是舊的。
96.I was very much involved in it, but then turned my attention to other phenomena.
我曾經很投入地研究它們， 但後來我的注意力轉向了其他現象。
97.Now, here is another thing which is rather interesting.
這是另一件 相當有趣的事情
98.One of the shattering events in the history of mathematics, which is not appreciated by many people, occurred about 130 years ago, 145 years ago.
數學史上的 一次粉碎性事件， 沒有多少人讚賞它， 發生於大約130年前， 145年前。
99.Mathematicians began to create shapes that didn't exist.
數學家開始創造 不存在的形狀
100.Mathematicians got into self-praise to an extent which was absolutely amazing that man can invent things that nature did not know.
數學家們有點沾沾自喜， 甚至在某種程度上喜不自勝， 因為人類能發明出 大自然不知道的事物。
101.In particular, it could invent things like a curve which fills the plane.
具體來說，人類可以發明 填裝飛機的曲線。
102.A curve's a curve, a plane's a plane, and the two won't mix.
曲線是曲線，飛機是飛機， 二者不會混淆
103.Well they do mix.
哦，他們還真混淆了。
104.A man named Peano did define such curves, and it became an object of extraordinary interest.
一個名叫皮諾的人 定義了這種曲線 它成為了非常有意思的對象。
105.It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality on the one hand
它非常重要，但更有趣的是 因為它導致了數學的分裂， 來自現實的數學 和純粹來自人的頭腦的新數學
106.and new mathematics coming from pure man's mind.
之間的分離。
107.Well, I was very sorry to point out that the pure man's mind has, in fact, seen at long last what had been seen for a long time.
那麼，我非常抱歉地指出， 純粹的人腦 實際上 終於看見了 一直是隨處可見的東西
108.And so here I introduce something, the set of rivers of a plane-filling curve.
那麼在這裡我要介紹一下 一套飛機填裝曲線。
109.And well, it's a story unto itself.
那麼， 它本身就是一個故事。
110.So it was in 1875 to 1925, an extraordinary period in which mathematics prepared itself to break out from the world.
那是在1875年至1925年， 一個數學本身 準備在世界上爆發的非凡時期。
111.And the objects which were used as examples, when I was a child and a student, of the break between mathematics and visible reality --
那些在數學與 可見現實分裂時， 那時我還是個孩子和學生， 被用作例子 的事物-
112.those objects, I turned them completely around.
那些對象， 我完全地拿它們另作他用。
113.I used them for describing some of the aspects of the complexity of nature.
我用它們來描述 自然複雜性的某些方面。
114.Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke.
那麼，1919年，一個名叫豪斯多夫的人 介紹了一個數字，這個數字簡直是一個數學笑話。
115.And I found that this number was a good measurement of roughness.
我發現這個數字 是一個很好的測量粗糙度的值。
116.When I first told it to my friends in mathematics they said, "Don't be silly. It's just something [silly]."
當我首先把它告訴我的數學朋友時 他們說： “別傻了。 那只是一個數。”
117.Well actually, I was not silly.
事實上我不傻。
118.The great painter Hokusai knew it very well.
大畫家葛飾北齋很瞭解它。
119.The things on the ground are algae.
地面上長的是海藻。
120.He did not know the mathematics; it didn't yet exist.
他不懂數學;那時還沒有數學。
121.And he was Japanese who had no contact with the West.
他是日本人，沒有接觸過西方文化。
122.But painting for a long time had a fractal side.
但是他的繪畫長期以來就有分數維的一面。
123.I could speak of that for a long time.
我講這個可以將很長時間。
124.The Eiffel Tower has a fractal aspect.
埃佛爾鐵塔也有分數維的方面。
125.And I read the book that Mr. Eiffel wrote about his tower.
我讀了埃菲爾先生寫的關於他這座塔的書。
126.And indeed it was astonishing how much he understood.
他了解的程度的確使我吃驚。
127.This is a mess, mess, mess, Brownian loop.
這是一個亂糟糟的布朗環。
128.One day I decided that halfway through my career, I was held by so many things in my work, I decided to test myself.
一天，我決定 在我職業生涯的半途中， 我被工作中太多的事情所纏繞， 我決定考驗一下自己。
129.Could I just look at something which everybody had been looking at for a long time and find something dramatically new?
我能否在 每個人都很熟悉的事物中 找到一些戲劇性的新發現呢？
130.Well, so I looked at these things called Brownian motion -- just goes around.
於是我觀察這些 被稱作布朗運動的現象——只是來回轉圈.
131.I played with it for a while, and I made it return to the origin.
我玩了一會兒之後， 又把它放回到原處。
132.Then I was telling my assistant, "I don't see anything. Can you paint it?"
然後我對我的助手說: “我沒有看到任何東西。你能畫出它來嗎？”
133.So he painted it, which means he put inside everything. He said: "Well, this thing came out ..." And I said, "Stop! Stop! Stop!
於是他畫將它畫了出來，這意味著 他把一切都裝進心裡了。他說： “那麼，事情是...” 我說：“停！停！停！
134.I see, it's an island."
我看到了，這是一個島。”
135.And amazing.
太神奇了。
136.So Brownian motion, which happens to have a roughness number of two, goes around.
所以布朗運動， 碰巧粗糙度為2，就是轉圈圈。
137.I measured it, 1.33.
我測量了它，1.33
138.Again, again, again.
一次又一次
139.Long measurements, big Brownian motions, 1.33.
長的測量，大型的布朗運動 1.33。
140.Mathematical problem: how to prove it?
數學問題：怎樣證明它？
141.It took my friends 20 years.
這花了我朋友20年的時間。
142.Three of them were having incomplete proofs.
其中三個人得到了一個不完整的證明。
143.They got together, and together they had the proof.
他們不斷地聚在一起研究，得到了這個證明。
144.So they got the big [Fields] medal in mathematics, one of the three medals that people have received for proving things which I've seen
所以他們獲得到了一個數學大獎（菲爾茨獎）， 是三大數學獎項之一， 用來獎勵那些證明了
145.without being able to prove them.
別人看到了但無法證明的事情的人們。
146.Now everybody asks me at one point or another, "How did it all start?
大家經常問我， “這一切是怎麼開始的？
147.What got you in that strange business?"
是什麼讓你做起了這個奇怪的行當？”
148.What got me to be, at the same time, a mechanical engineer, a geographer and a mathematician and so on, a physicist?
是什麼使我 同時成為一名機械工程師、 一名地理學家 和一名數學家，等等，還有物理學家？
149.Well, actually I started, oddly enough, studying stock market prices.
那麼，很奇怪的是，我實際上是從 研究股市價格開始的
150.And so here I had this theory, and I wrote books about it, Financial prices increments.
於是 我提出了這個理論 並且寫了關於它的書， 金融價格增量。
151.To the left you see data over a long period.
在左邊您看到的是長期數據。
152.To the right, on top, you see a theory which is very, very fashionable.
在右上角， 您看到是一個非常非常時髦的理論。
153.It was very easy, and you can write many books very fast about it.
它非常容易，您可以很快地寫出許多關於它的書。
154.(Laughter) There are thousands of books on that.
(笑聲) 有數以千計的寫它的書。
155.Now compare that with real price increments.
現在把它與真實的價格增量比較一下。
156.and where are real price increments?
真實的價格增量在哪裡呢？
157.Well, these other lines include some real price increments and some forgery which I did.
這些曲線包括了 真實的價格增量 和我的偽造。
158.So the idea there was that one must able to -- how do you say? -- model price variation.
這裡的想法是 人必須能 --怎麼說呢? – 模擬價格變化。
159.And it went really well 50 years ago.
50年前這方法運行的很好。
160.For 50 years people were sort of pooh-poohing me because they could do it much, much easier.
50年來，人們有點兒看不起我， 因為他們可以很容易地做到它。
161.But I tell you, at this point, people listened to me.
但是我告訴您，此時此刻，人們聽我的。
162.(Laughter) These two curves are averages.
(笑聲) 這兩條曲線是均線。
163.Standard & Poor, the blue one.
標準普爾，藍色的那個
164.And the red one is Standard & Poor's, from which the five biggest discontinuities are taken out.
而紅色的一個是 去掉不連續性最大的五個股票後的 標準普爾。
165.Now discontinuities are a nuisance.
不連續性是有害的。
166.So in many studies of prices, one puts them aside.
因此所有價格研究， 人們總是把它們放到一邊。
167."Well, acts of God.
“哦，不可抗力
168.And you have the little nonsense which is left.
您就沒有什麼好胡攪蠻纏的了。
169.Acts of God." In this picture five acts of God are as important as everything else.
不可抗力。”在這張圖片中， 五個不可抗力同其它因素是同樣重要的。
170.In other words, it is not acts of God that we should put aside.
換句話說， 不可抗力是不應該被放到一邊的。
171.That is the meat, the problem.
那才是肉，是問題的所在。
172.If you master these, you master price.
如果您掌握了這些，您就掌握了價格。
173.And if you don't master these, you can master the little noise as well as you can, but it's not important.
如果您掌握不了這些， 您可以儘量掌握小噪音。 但是這不重要。
174.Well, here are the curves for it.
那麼，這是它的曲線。
175.Now, I get to the final thing, which is the set of which my name is attached.
現在，我講最後一個事情， 用我名字命名的一個集合。
176.In a way it's the story of my life.
在某種意義上它是我生命的故事。
177.My adolescence was spent during the German occupation of France.
我的青春期是在 德軍佔領下的法國度過的。
178.And since I thought that I might vanish within a day or a week, I had very big dreams.
因為我認為我也許會 在一天或一個星期之內消失 我過有大的夢想。
179.And after the war, I saw an uncle again.
戰爭過後， 我又見到我的叔叔。
180.My uncle was a very prominent mathematician and he told me, "Look, there's a problem which I could not solve 25 years ago, and which nobody can solve.
我的叔叔是一位非常著名數學家，他告訴我， “你看，有一道難題， 我花了25年也沒有解決， 別人也沒有解決。
181.This is a construction of a man named [Gaston] Julia and [Pierre] Fatou.
這是一個名叫（加斯頓）朱麗葉和 一個名叫（皮埃爾）費託的人提出來的。
182.If you could find something new, anything, you will get your career made."
如果你能夠 有任何新發現 你將成就你的事業。”
183.Very simple.
非常簡單。
184.So I looked, and like the thousands of people that had tried before, I found nothing.
於是我就看這道題， 象之前做過嘗試的成千上萬的人一樣， 我什麼也沒有發現。
185.But then the computer came.
然後出現了計算機。
186.And I decided to apply the computer, not to new problems in mathematics -- like this wiggle wiggle, that's a new problem -- but to old problems.
我決定研究計算機， 而不是新的數學問題- 例如這個“擺動”的問題，這是新問題- 而是建立在舊問題上。
187.And I went from what's called real numbers, which are points on a line, to imaginary, complex numbers, which are points on a plane,
我由所謂“實數”開始， 也就是數軸上的點， 到虛的“複數”， 也就是平面上的點，
188.which is what one should do there.
人們應該在平面上研究。
189.And this shape came out.
這個形狀出來了。
190.This shape is of an extraordinary complication.
這個形狀異常複雜。
191.The equation is hidden there, z goes into z squared, plus c.
公式就隱藏在那裡， z等於 z的 平方加c。
192.It's so simple, so dry.
它是那麼簡單，相當簡單。
193.It's so uninteresting.
一點意思也沒有
194.Now you turn the crank once, twice, twice, marvels come out.
現在你把它重複一次、 兩次， 兩次 奇蹟出現了
195.I mean this comes out.
我是說這個出現了
196.I don't want to explain these things.
我不想解釋這些東西。
197.This comes out. This comes out.
這個出來了。這個出來了。
198.Shapes which are of such complication, such harmony and such beauty.
多麼複雜、多麼和諧、 多麼美麗的形狀啊。
199.This comes out repeatedly, again, again, again.
這個出來了， 不斷地，一而再，再而三地出來，
200.And that was one of my major discoveries was to find that these islands were the same as the whole big thing, more or less.
這就是我的一個主要發現 我發現這些小島的形狀 與整體的大形狀相同，或多或少
201.And then you get these extraordinary baroque decorations all over the place.
於是你得到這些 隨處可見的非凡的巴洛克式裝飾。
202.All that from this little formula, which has whatever, five symbols in it.
所有這些來自這個 只有五個符號的小小的公式
203.And then this one.
然後這一個
204.The color was added for two reasons.
加顏色是由於兩個原因
205.First of all, because these shapes are so complicated, that one couldn't make any sense of the numbers.
首先，因為這些形狀 是如此的複雜， 以至於人根本意識不到這些數字。
206.And if you plot them, you must choose some system.
如果你想突出它們，您必須選擇一些系統
207.And so my principle has been to always present the shapes with different colorings, because some colorings emphasize that, and others it is that or that.
所以我的原則是 總是在展示不同的形狀時 塗上不同的顏色 因為有些顏色突出這個， 有些顏色突出那個。
208.It's so complicated.
非常複雜。
209.(Laughter) In 1990, I was in Cambridge, U.K.
（笑聲） 1990 年，我在英國的劍橋大學
210.to receive a prize from the university.
接受了一個獎項。
211.And three days later, a pilot was flying over the landscape and found this thing.
三天後， 一個飛行員在飛行時發現了這個。
212.So where did this come from?
這是從哪裡來的？
213.Obviously, from extraterrestrials.
顯然，從外星人那裡來的。
214.(Laughter) Well, so the newspaper in Cambridge published an article about that "discovery"
(笑聲) 於是劍橋的校報上 發表一篇有關這一“發現”的文章。
215.and received the next day 5,000 letters from people saying, "But that's simply a Mandelbrot set very big."
第二天， 收到了5000封來信，人們說： “那只是一個放得很大的曼德爾布羅特圖形。”
216.Well, let me finish.
好吧，讓我結束演講。
217.This shape here just came out of an exercise in pure mathematics.
這個形狀僅僅出自 純數學的一個練習
218.Bottomless wonders spring from simple rules, which are repeated without end.
無邊的奇蹟源自簡單規則的 無限重複。
219.Thank you very much.
非常感謝。/<code>
結束語
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