RogerAntonsen_2015X[罗杰安东森][理解世界的秘诀数学].pdf

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1、www.XiYuS锡育软件I want to talk about understanding,and the nature ofunderstanding,and what the essence of understanding is,because understanding is something we aim for,everyone.（鼓点和踩镲声）理解到底是什么,因为我们都在追求理解。00:14We want to understand things.我们想理解世间万物。00:24My claim is that understanding has to do with the

2、 ability tochange your perspective.我认为理解是一种能力,转变（固有）观点的能力。00:27If you dont have that,you dont have understanding.如果我们缺乏它,就说明我们缺乏理解力。00:32So that is my claim.这是我的结论。00:36And I want to focus on mathematics.我想重点讲讲数学。00:37Many of us think of mathematics as addition,subtraction,multiplication,division,fr

3、actions,percent,geometry,algebra all that stuff.很多人认为,数学就是 加,减,乘,除,分数,百分数,几何,代数等等。00:40But actually,I want to talk about the essence of mathematicsas well.但今天,我也想讲讲数学的本质,00:50And my claim is that mathematics has to do with patterns.我的观点是,数学跟模式有关。00:53Behind me,you see a beautiful pattern,and this pa

4、tternactually emerges just from drawing circles in a veryparticular way.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。00:57aim for:瞄准;以为目标 subtraction:n.数减法;减少;差集 multiplication:n.数乘法;增加 fractions:n.数分数;小部分,片段(fraction的复数)geometry:n.几何学/几何结构 algebra:n.代数学 emerges:vi.浮现;摆脱;暴露So my day-to-day definition of m

5、athematics that I use everyday is the following:First of all,its about finding patterns.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。01:05And by pattern,I mean a connection,a structure,someregularity,some rules that govern what we see.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。01:16Second of all,I think it is about

6、representing these patternswith a language.其次,我认为数学是一种语言,用来描述各种模式。01:24We make up language if we dont have it,and inmathematics,this is essential.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。01:29Its also about making assumptions and playing around withthese assumptions and just seeing what happens.同时,数学也需要进行假设,对假

7、设进行多方验证,看看结果如何。01:35Were going to do that very soon.我们一会儿就会这么做。01:40And finally,its about doing cool stuff.最后,数学可以用来做很酷的事情。01:42Mathematics enables us to do so many things.能帮我们完成很多事。01:46So lets have a look at these patterns.下面我们来看一些模式。01:50If you want to tie a tie knot,there are patterns.如果你想系领带,会有

8、很多种样式。01:52day-to-day:adj.日常的;逐日的 First of all:adv.首先 regularity:n.规则性;整齐;正规;匀称 representing:v.代表;表示,表现(represent的ing形式)have a look at:看一看,看一眼 knot:n.(绳等的)结；节瘤，疙瘩；海里/小时(航速单位)/vt.打结/vi.打结/Tie knots have names.每一种都有名字。01:56And you can also do the mathematics of tie knots.因此领带结也包含数学。01:58This is a left

9、-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。02:00This 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。02:04This is a language we made up for the patterns of tie knots,and a half-Windsor is all that.这就是我们专门为领带结创造的语言,最后还有半温莎结。02:08This is a mathematics

10、 book about tying shoelaces at theuniversity level,because there are patterns in shoelaces.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。02:15You can do it in so many different ways.你可以用成千上万种方式来系鞋带。02:21We can analyze it.我们可以进行分析。02:23We can make up languages for it.然后为系鞋带也创造一种语言。02:25And representations are all

11、 over mathematics.这些都可以用数学方法来表达。02:28TED演讲者：Roger Antonsen|罗杰?安东森演讲标题：Math is the hidden secret to understanding the world|理解世界的秘诀：数学内容概要：Unlock the mysteries and inner workings of the world through one of the mostimaginative art forms ever mathematics with Roger Antonsen,as he explains how a slight

12、change in perspective can reveal patterns,numbers and formulas as the gateways to empathyand understanding.跟着罗杰?安东森一起，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，细微的角度变化能帮我们理解模式、数字和公式，并指引我们通向与人共鸣和理解万物的大门。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。02:35When we throw something up in the

13、 air,it falls down.当我们把物品抛向空中,它会掉下来。02:39Why?为什么？02:42Were not sure,but we can represent this with mathematicsin 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。02:43tying:n.结子/v.系(tie的ing形式)shoelaces:鞋带(shoelace的复数)representations:代表/表现(representation的复数)up in the air:悬而未决This is also a pattern.这也是一种模式。02:48Thi

14、s is also an invented language.是一种被发明的语言。02:49Can you guess for what?你能猜到这是什么吗？02:52It 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。02:55That enables him as a choreographer to do cool stuff,to donew things,because he has represented it.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

15、。02:59I want you to think about how amazing representingsomething actually is.请大家想一想,表达是多么神奇的东西。03:07Here it says the word mathematics.这里写的是“数学”这个词。03:12But actually,theyre just dots,right?实际上就是一些点,对吧？03:15So how in the world can these dots represent the word?一些点怎么能表示单词呢？03:18Well,they do.确实可以。03:21

16、They represent the word mathematics,他们代表了单词“数学”,03:23and these symbols also represent that word and this we canlisten to.这些符号也一样,这次我们可以用听的。03:25It sounds like this.听起来就像这样。03:29(Beeps)Somehow these sounds represent the word and theconcept.（滴滴声）可以说,这些声音也代表了这个词和它的含义。03:30How does this happen?这是怎么做到的呢？

17、03:36Theres something amazing going on about representingstuff.表达是一种很神奇的过程。03:37notation:n.符号;乐谱;注释;记号法 choreographer:n.编舞者,舞蹈指导So I want to talk about that magic that happens when weactually represent something.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。03:41www.XiYuS锡育软件Here you see just lines with different widt

18、hs.现在你们看到的只是不同宽度的线条。03:49They stand for numbers for a particular book.这些线条代表了一本书。03:52And I can actually recommend this book,its a very nicebook.强烈推荐这本书,非常不错。03:55(Laughter)Just trust me.（笑声）真的,不骗你们。03:58OK,so lets just do an experiment,just to play around withsome straight lines.好吧,让我们来做一个实验。来玩一下直线

19、。04:01This is a straight line.这是一条直线。04:06Lets make another one.再画另外一条。04:07So every time we move,we move one down and one across,and we draw a new straight line,right?每一次我们都往下、往右移动一格,画出一条新的直线。04:08We do this over and over and over,and we look for patterns.如此反复,从中寻找一种模式。04:13So this pattern emerges,

20、and its a rather nice pattern.我们得到了这个图案,是一个非常好看的图案。04:17It looks like a curve,right?它看起来就像一道弧,对吧？04:22Just from drawing simple,straight lines.我们仅仅画了些简单的直线。04:24Now 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。04:27widths:数宽度 stand for:代表;支持;象征;担任的候选人 play around:玩耍;胡搞;轻率

21、对待 over and over:反复;再三Have a look at the curve.再看这段弧。04:30What does it look like?像什么？04:32Is it a part of a circle?是不是像圆的一部分？04:33Its actually not a part of a circle.其实它不是圆的一部分。04:35So I have to continue my investigation and look for the truepattern.所以我继续探寻,找出真正的模式。04:37Perhaps if I copy it and make

22、 some art?也许我可以复制它,画一幅画？04:41Well,no.好像不行。04:45Perhaps I should extend the lines like this,and look for thepattern there.也许我应该延长这些线条,再来寻找模式。04:46Lets make more lines.再多画一些线条。04:50We do this.然后这样。04:52And then lets zoom out and change our perspective again.把它缩小,再变换角度。04:53Then we can actually see tha

23、t what started out as juststraight lines is actually a curve called a parabola.然后我们就会发现,开始的直线 变成了抛物线。04:57This is represented by a simple equation,and its a beautifulpattern.这可以用一个简单的等式表达,很美的图案。05:03So this is the stuff that we do.这就是我们所做的。05:09We find patterns,and we represent them.找到某种模式,然后表达出来。05

24、:11And I think this is a nice day-to-day definition.这是一种很直白的定义。05:13But today I want to go a little bit deeper,and think aboutwhat the nature of this is.但是今天,我想讨论得更深入一些,思考它们的本质是什么。05:16parabola:n.抛物线What makes it possible?是什么造就了这一切？05:22Theres one thing thats a little bit deeper,and that has to dowi

25、th the ability to change your perspective.要看得更深入一些,就要求我们有转换角度的能力。05:24And I claim that when you change your perspective,and ifyou take another point of view,you learn something newabout what you are watching or looking at or hearing.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。05:30And I think this is a r

26、eally important thing that we do all thetime.我认为这一点非常重要。05:41So lets just look at this simple equation,x+x=2?x.让我们看看这个简单的方程,05:45This is a very nice pattern,and its true,because 5+5=2?5,etc.这是一个很好的模式,也是正确的。05:52Weve seen this over and over,and we represent it like this.这个等式我们司空见惯了。05:57But think abo

27、ut it:this is an equation.但是仔细想一想:这是一个等式。06:00It says that something is equal to something else,and thatstwo different perspectives.它代表一个事物与另一个事物相等,这么表述有两种角度。06:03One perspective is,its a sum.一种是总和。06:07Its something you plus together.是相加的过程。06:09On the other hand,its a multiplication,and those are

28、twodifferent perspectives.另一种是相乘。这是两种不同的角度。06:11point of view:观点;见地;立场 equal to:等于;胜任 perspectives:n.数透视,远景,看法;构面;观点展示(perspective的复数形式)On the other hand:另一方面And I would go as far as to say that every equation is like this,every mathematical equation where you use that equalitysign is actually a met

29、aphor.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。06:17Its an analogy between two things.是两种事物间的类比。06:26Youre just viewing something and taking two differentpoints of view,and youre expressing that in a language.你观察一件事情,产生两种观点,然后用一种语言来表达。06:28Have a look at this equation.看这个方程。06:34This is one of the mos

30、t beautiful equations.它是最美的等式之一。06:36It simply says that,well,two things,theyre both-1.简单表明了,等式两边都是-1。06:38This thing on the left-hand side is-1,and the other one is.左手边的是-1,右边的也是。06:44And that,I think,is one of the essential parts of mathematics you take different points of view.我认为这是数学中很重要的部分 采取不同

31、的观点。06:47So lets just play around.我们继续。06:52Lets take a number.选一个数字好了。06:53We know four-thirds.We know what four-thirds is.我们知道4/3,知道它的含义。06:55metaphor:n.暗喻,隐喻;比喻说法 analogy:n.类比；类推；类似 viewing:adj.可见的/v.观察;查看(view的ing形式)expressing:v.表达;表达观点(express的ing形式)equations:n.方程式;等式;均等;均势(equation的复数形式)left-h

32、and:adj.左手的;左侧的Its 1.333,but we have to have those three dots,otherwise itsnot exactly four-thirds.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。06:58But this is only in base 10.但只有在使用十进制时才如此。07:03You know,the number system,we use 10 digits.我们的数字系统用的是10位计数。07:05If we change that around and only use two digits,

33、thatscalled the binary system.如果我们改成2位计数,也就是二进制。07:08Its written like this.就变成了这样。07:12So were now talking about the number.我们现在在讨论数字。07:13The number is four-thirds.讨论4/3这个数字。07:15We can write it like this,and we can change the base,changethe number of digits,and we can write it differently.我们也可以这样表

34、示,我们改变进制,改变数位,就可以用不同的方式书写。07:17So these are all representations of the same number.所有这些都代表同一个数。07:24We 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。07:28It all depends on how many digits you have.取决于我们选用哪种进制。07:31Or perhaps we just simplify and write it like this.或者我们还可以简单写成这样。07:34I

35、 like this one,because this says four divided by three.我喜欢这种,因为它表示4被3除。07:37And this number expresses a relation between two numbers.表现了两个数字间的关系。07:41binary:adj.数二进制的;二元的,二态的 expresses:表达(express的动词单数第三人称形式)/交快车/快递(express的名词复数)You have four on the one hand and three on the other.上边是4,下边是3。07:44And

36、you can visualize this in many ways.你可以用许多方式来把这个数字可视化。07:47What Im doing now is viewing that number from differentperspectives.从不同的角度来看这个数字。07:49Im playing around.我在不断尝试。07:53Im playing around with how we view something,and Imdoing it very deliberately.改变观察事物的角度。我是故意这么做的。07:54We can take a grid.让我们画

37、一个网格。07:58If its 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。08:00It has to be like this.This is a beautiful pattern.肯定如此。这是一个美丽的图案。08:04Four and three and five.4和3和5。08:07And this rectangle,which is 4 x 3,youve seen a lot of times.这个长方形,长宽比为4:3,你们见过很多次的。08:09This is yo

38、ur average computer screen.就是你们的屏幕大小的平均值。08:13800 x 600 or 1,600 x 1,200 is a television or a computerscreen.分别是电脑和电视的屏幕。08:15So these are all nice representations,but I want to go a littlebit further and just play more with this number.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。08:21on the one hand:一方面 visual

39、ize:vt.形象,形象化;想像,设想/vi.显现 deliberately:adv.故意地;谨慎地;慎重地rectangle:n.矩形;长方形Here you see two circles.Im going to rotate them like this.现在,你能看到两个圆。我要像这样旋转它们。08:27Observe the upper-left one.看一下左上角的那个,08:31It goes a little bit faster,right?它转得更快一点儿,对吧？08:32You can see this.你们都能看到。08:35It actually goes exac

40、tly four-thirds as fast.准确来说,它的旋转速度是慢速的4/3倍。08:36That means that when it goes around four times,the otherone goes around three times.也就是说,它每转4圈,另一个圆就会转3圈。08:39Now lets make two lines,and draw this dot where the linesmeet.现在,画两条线,并标明相交处的点。08:44We get this dot dancing around.我们就能得到一个跳舞的点。08:47(Laughte

41、r)And this dot comes from that number.（笑声）这个点就来源于4/3这个数字。08:49Right?Now we should trace it.是吧？现在,让我来看看它的轨迹。08:52Lets trace it and see what happens.把轨迹画出来,看看是什么样子。08:55This is what mathematics is all about.这就是数学。08:57Its about seeing what happens.就是不断探索会发生什么。08:59www.XiYuS锡育软件And this emerges from fo

42、ur-thirds.而这来自于4/3这个数字。09:01I like to say that this is the image of four-thirds.我觉得,这就是4/3的肖像。09:04Its much nicer (Cheers)Thank you!比数字好看多了（欢呼）谢谢！09:07(Applause)This is not new.（掌声）其实这不算新鲜事了。09:09upper-left:左上角This has been known for a long time,but 很早以前就被发现了,但是09:17(Laughter)But this is four-thirds

43、.（笑声）但是这仅仅是4/3。09:19Lets do another experiment.让我们再做一个实验。09:23Lets now take a sound,this sound:(Beep)This is a perfect A,440Hz.让我们选一个声音,是这样的:（嘟）这是一个完美的A,440Hz。09:24Lets multiply it by two.把它翻倍。09:31We get this sound.(Beep)When we play them together,itsounds like this.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。0

44、9:33This is an octave,right?这是一个八度音,对吧？09:37We 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。09:38We can multiply it by three-halves.然后我们把它翻为1.5倍。09:41(Beep)This is what we call a perfect fifth.（嘟）我们称之为纯五度音。09:42(Beep)They sound really nice together.（嘟）把它们一起播放,听起来很不错。09:46Let

45、s multiply this sound by four-thirds.(Beep)Whathappens?让我们把这个声音翻4/3倍。会怎么样？09:49You get this sound.(Beep)This is the perfect fourth.你们会得到这个声音。纯四度音。09:55If 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。09:58They sound like this together.(Beeps)This is the sound offour-thirds.一起播放,是这样的声音。这就是4/3的声

46、音。10:00What Im doing now,Im changing my perspective.这就是改变角度。10:05octave:n.八度音阶;八行诗;十四行诗的前八行;八个一组的事物/adj.八个一组的;高八度音的Im just viewing a number from another perspective.我是在从另一个角度看一个数字。10:07I can even do this with rhythms,right?也可以用节奏来表示。10:10I can take a rhythm and play three beats at one time(Drumbeats

47、)in a period of time,and I can play anothersound four times in that same space.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。10:12(Clanking sounds)Sounds kind of boring,but listen to themtogether.（铛铛声）单独听很枯燥,但如果放在一起。10:22(Drumbeats and clanking sounds)（鼓点和铛铛声）10:25(Laughter)Hey!So.（笑声）嘿！好多了。10:28(Laug

48、hter)I can even make a little hi-hat.（笑声）我还可以加点儿踩镲声。10:31(Drumbeats and cymbals)Can you hear this?（鼓点和踩镲声）听到了吗？10:35So,this is the sound of four-thirds.所以,这就是4/3的声音。10:38Again,this is as a rhythm.4/3的节律。10:40(Drumbeats and cowbell)And I can keep doing this and playgames with this number.（鼓点声和踩镲声）我还可

49、以继续玩,用这个数字做游戏。10:42Four-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！10:47(Laughter)Truly its an undervalued number.（笑声）说真的4/3的价值被低估了。10:50rhythms:韵律 at one time:曾经,一度;同时 clanking:n.发出丁当声 undervalued:adj.贬值的;估值偏低;估价过低/v.低估的价值;轻视(undervalue的过去分词)So if you take a sphere and l

50、ook at the volume of the sphere,its actually four-thirds of some particular cylinder.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。10:53So four-thirds is in the sphere.Its the volume of the sphere.所以4/3出现在了球体里,是球的体积。10:59OK,so why am I doing all this?好,我为什么玩这些？11:03Well,I want to talk about what it means t