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RogerAntonsen_2015X-_理解世界的秘诀:数学_
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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. |
(鼓点和踩镲声) 理解到底是什么, 因为我们都在追求理解。 |
essence:n.本质;实质;精髓;香精;
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We want to understand things. |
我们想理解世间万物。 |
My claim is that understanding has to do with the ability to change your perspective . |
我认为理解是一种能力, 转变(固有)观点的能力。 |
claim:v.要求;声称;需要;认领;n.要求;声称;索赔;断言;值得; perspective:n.观点;远景;透视图;adj.透视的;
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If you don't have that, you don't have understanding. |
如果我们缺乏它,就说明我们缺乏理解力。 |
So that is my claim. |
这是我的结论。 |
And I want to focus on mathematics . |
我想重点讲讲数学。 |
mathematics:n.数学;数学运算;
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Many of us think of mathematics as addition , subtraction , multiplication , division , fractions , 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。 |
addition:n.添加;[数]加法;增加物; subtraction:n.[数]减法;减少;差集; multiplication:n.[数]乘法;增加; division:n.师;分配;分开;分歧; 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; stuff:n.东西:物品:基本特征:v.填满:装满:标本:
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But actually, I want to talk about the essence of mathematics as well. |
但今天,我也想讲讲数学的本质, |
And my claim is that mathematics has to do with patterns. |
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. |
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。 |
emerges:vi.浮现;摆脱;暴露;
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So my day-to-day definition of mathematics that I use every day is the following: |
所以我对数学有一个的定义 非常直白: |
day-to-day:adj.日常的;逐日的; definition:n.定义;清晰度;(尤指词典里的词或短语的)释义;解释;
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First of all , it's about finding patterns. |
首先,数学的关键是寻找模式。 |
First of all:adv.首先;
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And by "pattern," I mean a connection, a structure , some regularity , 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。 |
structure:n.结构;构造;建筑物;vt.组织;构成;建造; regularity:n.规则性;整齐;正规;匀称;
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Second of all,I think it is about representing these patterns with a language. |
其次,我认为数学是一种语言,用来描述各种模式。 |
representing:v.代表;维护…的利益;等于;相当于;(represent的现在分词)
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We make up language if we don't have it, and in mathematics, 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。 |
essential:n.要点;要素;实质;必需品;adj.完全必要的;必不可少的;极其重要的;本质的;
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It's also about making assumptions and playing around with these assumptions and just seeing what happens. |
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。 |
assumptions:n.假定;假设;承担;获得;(assumption的复数)
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We're going to do that very soon. |
我们一会儿就会这么做。 |
And finally , it's about doing cool stuff. |
最后,数学可以用来做很酷的事情。 |
finally:adv.终于;最终;(用于列举)最后;彻底地;
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Mathematics enables us to do so many things. |
能帮我们完成很多事。 |
enables:v.使得; (enable的第三人称单数)
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So let's have a look at these patterns. |
下面我们来看一些模式。 |
have a look at:看一看,看一眼;
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If you want to tie a tie knot , there are patterns. |
如果你想系领带, 会有很多种样式。 |
knot:n.(绳等的)结;节瘤,疙瘩;海里;小时(航速单位);v.打结;
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Tie knots have names. |
每一种都有名字。 |
knots:n.发髻:节子:节疤:v.把…打成结(knot的第三人称单数和复数)
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And you can also do the mathematics of tie knots. |
因此领带结也包含数学。 |
This is a left-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。 |
This 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。 |
This 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。 |
This is a mathematics book about tying shoelaces at the university 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。 |
shoelaces:n.鞋带;(shoelace的复数)
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You can do it in so many different ways. |
你可以用成千上万种方式来系鞋带。 |
We can analyze it. |
我们可以进行分析。 |
analyze:v.对…进行分析,分解(等于analyse);
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We can make up languages for it. |
然后为系鞋带也创造一种语言。 |
And representations are all over mathematics. |
这些都可以用数学方法来表达。 |
representations:代表;表现(representation的复数);
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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。 |
When we throw something up in the air, it falls down. |
当我们把物品抛向空中, 它会掉下来。 |
Why? |
为什么? |
We're not sure, but we can represent this with mathematics in 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。 |
This is also a pattern. |
这也是一种模式。 |
This is also an invented language. |
是一种被发明的语言。 |
Can you guess for what? |
你能猜到这是什么吗? |
It 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。 |
notation:n.符号;乐谱;注释;记号法;
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That enables him as a choreographer to do cool stuff, to do new 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。 |
choreographer:n.编舞者,舞蹈指导; represented:v.代表; (represent的过去式和过去分词)
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I want you to think about how amazing representing something actually is. |
请大家想一想,表达是多么神奇的东西。 |
Here it says the word "mathematics." |
这里写的是“数学”这个词。 |
But actually, they're just dots, right? |
实际上就是一些点,对吧? |
So how in the world can these dots represent the word? |
一些点怎么能表示单词呢? |
Well, they do. |
确实可以。 |
They represent the word "mathematics," |
他们代表了单词“数学”, |
and these symbols also represent that word and this we can listen to. |
这些符号也一样, 这次我们可以用听的。 |
symbols:n.符号;象征;标志;符号表(symbol的复数);
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It sounds like this. |
听起来就像这样。 |
(Beeps) |
(滴滴声) |
Somehow these sounds represent the word and the concept. |
可以说,这些声音也代表了这个词和它的含义。 |
Somehow:adv.以某种方法;莫名其妙地;
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How does this happen? |
这是怎么做到的呢? |
There's something amazing going on about representing stuff. |
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. |
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。 |
Here you see just lines with different widths . |
现在你们看到的只是不同宽度的线条。 |
widths:[数]宽度;
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They stand for numbers for a particular book. |
这些线条代表了一本书。 |
And I can actually recommend this book, it's a very nice book. |
强烈推荐这本书,非常不错。 |
recommend:v.推荐;介绍;劝告;建议;使受欢迎;
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(Laughter) |
(笑声) |
Just trust me. |
真的,不骗你们。 |
OK, so let's just do an experiment, just to play around with some straight lines. |
好吧,让我们来做一个实验。 来玩一下直线。 |
play around:玩耍;胡搞;轻率对待;
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This is a straight line. |
这是一条直线。 |
Let's make another one. |
再画另外一条。 |
So 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。 |
We do this over and over and over, and we look for patterns. |
如此反复, 从中寻找一种模式。 |
over and over:反复;再三;
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So this pattern emerges, and it's a rather nice pattern. |
我们得到了这个图案, 是一个非常好看的图案。 |
It looks like a curve , right? |
它看起来就像一道弧,对吧? |
curve:n.曲线;弯曲;曲面;弧线;v.(使)沿曲线移动;呈曲线形;
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Just from drawing simple, straight lines. |
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. |
现在,稍微改变一下角度,旋转一下。 |
rotate:v.旋转;转动;轮换;使…轮流;
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Have a look at the curve. |
再看这段弧。 |
What does it look like? |
像什么? |
Is it a part of a circle? |
是不是像圆的一部分? |
It's actually not a part of a circle. |
其实它不是圆的一部分。 |
So I have to continue my investigation and look for the true pattern. |
所以我继续探寻,找出真正的模式。 |
Perhaps if I copy it and make some art? |
也许我可以复制它,画一幅画? |
Well, no. |
好像不行。 |
Perhaps I should extend the lines like this, and look for the pattern there. |
也许我应该延长这些线条, 再来寻找模式。 |
extend:vt.延伸; vi.延伸;
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Let's make more lines. |
再多画一些线条。 |
We do this. |
然后这样。 |
And then let's zoom out and change our perspective again. |
把它缩小,再变换角度。 |
Then we can actually see that what started out as just straight 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。 |
parabola:n.抛物线;
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This is represented by a simple equation , and it's a beautiful pattern. |
这可以用一个简单的等式表达, 很美的图案。 |
equation:n.方程式,等式;相等;[化学]反应式;
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So this is the stuff that we do. |
这就是我们所做的。 |
We find patterns, and we represent them. |
找到某种模式,然后表达出来。 |
And I think this is a nice day-to-day definition. |
这是一种很直白的定义。 |
But today I want to go a little bit deeper, and think about what 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。 |
What makes it possible? |
是什么造就了这一切? |
There's one thing that's a little bit deeper, and that has to do with 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。 |
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. |
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。 |
point of view:观点;见地;立场;
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And I think this is a really important thing that we do all the time. |
我认为这一点非常重要。 |
So let's just look at this simple equation, x + x = 2 ? x. |
让我们看看这个简单的方程, |
This is a very nice pattern, and it's true, because 5 + 5 = 2 ? 5, etc. |
这是一个很好的模式,也是正确的。 |
We've seen this over and over, and we represent it like this. |
这个等式我们司空见惯了。 |
But think about it: this is an equation. |
但是仔细想一想:这是一个等式。 |
It says that something is equal to something else, and that's two 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。 |
perspectives:n.[数]透视,远景,看法;构面;观点展示(perspective的复数形式);
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One perspective is, it's a sum. |
一种是总和。 |
It's something you plus together. |
是相加的过程。 |
On the other hand , it's a multiplication, and those are two different perspectives. |
另一种是相乘。 这是两种不同的角度。 |
On the other hand:另一方面;
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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 . |
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。 |
as far as:至于…; mathematical:adj.数学的,数学上的;精确的; metaphor:n.暗喻,隐喻;比喻说法;
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It's an analogy between two things. |
是两种事物间的类比。 |
analogy:n.类比;类推;类似;
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You're just viewing something and taking two different points of view, and you're 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。 |
expressing:v.表示;表达;表露;显而易见;(express的现在分词)
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Have a look at this equation. |
看这个方程。 |
This is one of the most beautiful equations . |
它是最美的等式之一。 |
equations:n.方程式;等式;均等;均势(equation的复数形式);
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It simply says that, well, two things, they're both -1. |
简单表明了, 等式两边都是-1。 |
This thing on the left-hand side is -1, and the other one is. |
左手边的是-1,右边的也是。 |
left-hand:adj.左手的;左侧的;
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And 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 ——采取不同的观点。 |
So let's just play around. |
我们继续。 |
Let's take a number. |
选一个数字好了。 |
We know four-thirds. We know what four-thirds is. |
我们知道4/3,知道它的含义。 |
It's 1.333, but we have to have those three dots, otherwise it's not 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。 |
But this is only in base 10. |
但只有在使用十进制时才如此。 |
You know, the number system, we use 10 digits. |
我们的数字系统用的是10位计数。 |
If we change that around and only use two digits, that's called the binary system. |
如果我们改成2位计数, 也就是二进制。 |
binary:adj.[数]二进制的;二元的,二态的;
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It's written like this. |
就变成了这样。 |
So we're now talking about the number. |
我们现在在讨论数字。 |
The number is four-thirds. |
讨论4/3这个数字。 |
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. |
所有这些都代表同一个数。 |
We 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。 |
It all depends on how many digits you have. |
取决于我们选用哪种进制。 |
Or perhaps we just simplify and write it like this. |
或者我们还可以简单写成这样。 |
simplify:vt.简化;使单纯;使简易;
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I like this one, because this says four divided by three. |
我喜欢这种,因为它表示4被3除。 |
And this number expresses a relation between two numbers. |
表现了两个数字间的关系。 |
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);
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You have four on the one hand and three on the other. |
上边是4,下边是3。 |
on the one hand:一方面;
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And you can visualize this in many ways. |
你可以用许多方式来把这个数字可视化。 |
visualize:vt.形象,形象化;想像,设想;vi.显现;
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What I'm doing now is viewing that number from different perspectives. |
从不同的角度来看这个数字。 |
I'm playing around. |
我在不断尝试。 |
I'm playing around with how we view something, and I'm doing it very deliberately . |
改变观察事物的角度。 我是故意这么做的。 |
deliberately:adv.故意地;谨慎地;慎重地;
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We can take a grid . |
让我们画一个网格。 |
grid:n.网格;格子,栅格;输电网;
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If it's 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。 |
It has to be like this. This is a beautiful pattern. |
肯定如此。这是一个美丽的图案。 |
Four and three and five. |
4和3和5。 |
And this rectangle , which is 4 x 3, you've seen a lot of times. |
这个长方形,长宽比为4:3, 你们见过很多次的。 |
rectangle:n.矩形;长方形;
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This is your average computer screen. |
就是你们的屏幕大小的平均值。 |
800 x 600 or 1,600 x 1,200 is a television or a computer screen. |
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. |
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。 |
Here you see two circles. I'm going to rotate them like this. |
现在,你能看到两个圆。我要像这样旋转它们。 |
Observe the upper-left one. |
看一下左上角的那个, |
Observe:v.观察;看到;庆祝;监视; upper-left:左上角;
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It goes a little bit faster, right? |
它转得更快一点儿,对吧? |
You can see this. |
你们都能看到。 |
It actually goes exactly four-thirds as fast. |
准确来说,它的旋转速度是慢速的4/3倍。 |
That means that when it goes around four times, the other one goes around three times. |
也就是说,它每转4圈, 另一个圆就会转3圈。 |
Now let's make two lines, and draw this dot where the lines meet. |
现在,画两条线,并标明相交处的点。 |
We get this dot dancing around. |
我们就能得到一个跳舞的点。 |
(Laughter) |
(笑声) |
And this dot comes from that number. |
这个点就来源于4/3这个数字。 |
Right? Now we should trace it. |
是吧?现在,让我来看看它的轨迹。 |
trace:追溯,追踪
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Let's trace it and see what happens. |
把轨迹画出来,看看是什么样子。 |
This is what mathematics is all about. |
这就是数学。 |
It's about seeing what happens. |
就是不断探索会发生什么。 |
And this emerges from four-thirds. |
而这来自于4/3这个数字。 |
I like to say that this is the image of four-thirds. |
我觉得,这就是4/3的肖像。 |
It's much nicer — (Cheers) |
比数字好看多了——(欢呼) |
Thank you! |
谢谢! |
(Applause) |
(掌声) |
This is not new. |
其实这不算新鲜事了。 |
This has been known for a long time, but — |
很早以前就被发现了,但是—— |
(Laughter) |
(笑声) |
But this is four-thirds. |
但是这仅仅是4/3。 |
Let's do another experiment. |
让我们再做一个实验。 |
Let's now take a sound, this sound: (Beep) |
让我们选一个声音,是这样的:(嘟) |
This is a perfect A, 440Hz. |
这是一个完美的A,440Hz。 |
Let's multiply it by two. |
把它翻倍。 |
multiply:vt.乘; vi.乘; v.多样地; adj.多层的;
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We get this sound. (Beep) |
就得到了这个声音。(嘟) |
When we play them together, it sounds like this. |
同时播放这两种声音,听起来是这个效果。 |
This is an octave , right? |
这是一个八度音,对吧? |
octave:n.八度音阶; adj.八个一组的;
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We 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。 |
We can multiply it by three-halves. |
然后我们把它翻为1.5倍。 |
(Beep) |
(嘟) |
This is what we call a perfect fifth. |
我们称之为纯五度音。 |
(Beep) |
(嘟) |
They sound really nice together. |
把它们一起播放,听起来很不错。 |
Let's multiply this sound by four-thirds. (Beep) |
让我们把这个声音翻4/3倍。 |
What happens? |
会怎么样? |
You get this sound. (Beep) |
你们会得到这个声音。 |
This is the perfect fourth. |
纯四度音。 |
If 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。 |
They sound like this together. (Beeps) |
一起播放,是这样的声音。 |
This is the sound of four-thirds. |
这就是4/3的声音。 |
What I'm doing now, I'm changing my perspective. |
这就是改变角度。 |
I'm just viewing a number from another perspective. |
我是在从另一个角度看一个数字。 |
I can even do this with rhythms , right? |
也可以用节奏来表示。 |
rhythms:韵律;
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I can take a rhythm and play three beats at one time (Drumbeats) |
我可以选一个节奏,在一段时间内敲3下(鼓点声) |
at one time:曾经,一度;同时;
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in a period of time, and I can play another sound four times in that same space. |
一段固定的时间, 然后在同样的时间内敲4下。 |
(Clanking sounds) |
(铛铛声) |
Sounds kind of boring , but listen to them together. |
单独听很枯燥,但如果放在一起。 |
boring:adj.无聊的;令人厌烦的;n.钻孔;v.使厌烦;钻孔;(bore的现在分词)
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(Drumbeats and clanking sounds) |
(鼓点和铛铛声) |
clanking:v.(使)发出叮当声,发出当啷声;(clank的现在分词)
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(Laughter) |
(笑声) |
Hey! So. |
嘿!好多了。 |
(Laughter) |
(笑声) |
I can even make a little hi-hat . |
我还可以加点儿踩镲声。 |
hi-hat:n.踩镲;脚踏钹;高帽钹;双触发镲片(等于highhat);
|
(Drumbeats and cymbals) |
(鼓点和踩镲声) |
Can you hear this? |
听到了吗? |
So, this is the sound of four-thirds. |
所以,这就是4/3的声音。 |
Again, this is as a rhythm. |
4/3的节律。 |
(Drumbeats and cowbell) |
(鼓点声和踩镲声) |
And I can keep doing this and play games with this number. |
我还可以继续玩,用这个数字做游戏。 |
Four-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! |
(Laughter) |
(笑声) |
Truly — it's an undervalued number. |
说真的——4/3的价值被低估了。 |
undervalued:adj.贬值的; v.低估…的价值; (undervalue的过去式和过去分词)
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So if you take a sphere and look at the volume of the sphere, it's 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。 |
sphere:n.球;领域;球体;球形;v.使成球形;包围;把…放在球内;使处于天体之间; volume:n.体积;容积;音量;响度;一册;合订本 cylinder:n.圆筒;汽缸;[数]柱面;圆柱状物;
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So four-thirds is in the sphere. It's the volume of the sphere. |
所以4/3出现在了球体里,是球的体积。 |
OK, so why am I doing all this? |
好,我为什么玩这些? |
Well, I want to talk about what it means to understand something and what we mean by understanding something. |
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。 |
That's my aim here. |
这就是我的目的。 |
And my claim is that you understand something if you have the ability to view it from different perspectives. |
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。 |
Let's look at this letter. It's a beautiful R, right? |
让我们看看这个字母。这是一个漂亮的R,对吧? |
How do you know that? |
你们怎么判断这是个R? |
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. |
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。 |
as a matter of fact:事实上; a bunch of:一群;一束;一堆; generalized:adj.广义的,普遍的; v.推广(generalize的过去分词); abstracted:adj.心不在焉的;出神的;分离出来的;抽出的;
|
So you know that this is an R. |
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. |
所以,我要说的是 理解事物和变换角度 是有关的。 |
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, |
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, |
I enable understanding. |
我就能让他们真正理解。 |
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. |
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. |
让我们再举一个例子。 |
This is four and three. This is four triangles . |
这是4和3。这是4个三角形。 |
triangles:n.[数]三角形,三角型态(triangle的复数形式);
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So this is also four-thirds, in a way. |
这也是某种4/3。 |
Let's just join them together. |
让我们把它们连起来。 |
Now we're going to play a game; we're going to fold it up into a three-dimensional structure. |
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。 |
fold:v.折叠;包;可叠平(以便贮存或携带);裹;n.褶;褶层;折叠部分;褶痕; three-dimensional:adj.三维的;立体的;真实的;
|
I love this. |
我喜欢这个。 |
This is a square pyramid. |
这是一个金字塔形。 |
And let's just take two of them and put them together. |
让我们再做一个,把它们放在一起。 |
So this is what is called an octahedron . |
就形成了一个八面体。 |
octahedron:n.八面体;
|
It's one of the five platonic solids. |
这是5种正多面体(又叫柏拉图立体)之一。 |
platonic:adj.理想的;柏拉图哲学的;不切实际的;纯精神友谊的;
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Now we can quite literally change our perspective, because we can rotate it around all of the axes and view it from different perspectives. |
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。 |
literally:adv.按字面:字面上:确实地: axes:n.轴线;轴心;坐标轴;斧头(axe的复数);
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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. |
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。 |
axis:n.轴;轴线;轴心国;
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I can do it even one more time. |
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. |
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。 |
I can use this as a tool for creating understanding. |
我可以把它作为创造理解的工具。 |
I can take two of these and put them together like this and see what happens. |
我可以把两个正四面体,像这样穿起来, 看看会发生什么。 |
And it looks a little bit like the octahedron. |
有点儿像正八面体。 |
Have a look at it if I spin it around like this. |
把它旋转起来再看。 |
spin:v.旋转;纺纱;吐丝;纺线;n.头晕;(快速)旋转;常用于英式英语;晕头转向;
|
What happens? |
发生了什么? |
Well, if you take two of these, join them together and spin it around, there's your octahedron again, a beautiful structure. |
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。 |
If you lay it out flat on the floor, this is the octahedron. |
如果你把它平摊在地上, 这就是一个正八面体。 |
This is the graph structure of an octahedron. |
正八面体的平面结构图。 |
And I can continue doing this. |
我还可以继续玩。 |
You can draw three great circles around the octahedron, and you rotate around, so actually three great circles is related to the octahedron. |
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。 |
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. |
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。 |
pump:n.抽水机;泵;打气筒;v.用泵输送;涌出;奔流;快速摇动;
|
Do you see what I'm doing here? |
看出来我在做什么了吗? |
I am changing the perspective every time. |
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. |
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。 |
I'm playing around with metaphors . |
我在使用隐喻, |
metaphors:n.隐喻(metaphor的复数形式);
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I'm playing around with perspectives and analogies . |
在变换角度,进行类比。 |
analogies:n.类比;类推法;
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I'm telling one story in different ways. |
变换不同的角度, |
I'm telling stories. |
来讲同一个故事。 |
I'm making a narrative; I'm making several narratives . |
我在叙述,而且做了好几种叙述。 |
narratives:n.叙述,故事;叙述的手法(narrative复数);
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And I think all of these things make understanding possible. |
我认为这一切使得理解变成可能。 |
I think this actually is the essence of understanding something. |
我认为这是理解事物的关键。 |
I truly believe this. |
我深信这点。 |
So this thing about changing your perspective — it's absolutely fundamental for humans. |
所以,关于改变你们的角度—— 对人类来说十分重要。 |
absolutely:adv.绝对地;完全地; fundamental:n.基础; adj.十分重大的;
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Let's play around with the Earth. |
让我们来看看地球。 |
Let's zoom into the ocean, have a look at the ocean. |
让我们放大到海洋,看看海洋。 |
We can do this with anything. |
我们可以放大任何事物。 |
We can take the ocean and view it up close. |
我们以海洋为例,仔细的看看它。 |
We can look at the waves. |
我们能观察海浪。 |
We can go to the beach. |
或是沙滩。 |
We can view the ocean from another perspective. |
我们也可以从另一个角度看海洋。 |
Every time we do this, we learn a little bit more about the ocean. |
每变一次角度,我们就能对海洋了解得多一些。 |
If we go to the shore, we can kind of smell it, right? |
如果我们走到海边,就能闻到海水的味道,对吧? |
We can hear the sound of the waves. |
能听到海浪的声音。 |
We can feel salt on our tongues. |
能尝到风中咸咸的味道。 |
So all of these are different perspectives. |
所有这些,都是不同的角度。 |
And this is the best one. |
而这个(角度)是最棒的。 |
We can go into the water. |
我们进入水中。 |
We can see the water from the inside. |
从内部来观察。 |
And you know what? |
你们知道吗? |
This is absolutely essential in mathematics and computer science . |
这对数学和计算机科学来说都绝对重要。 |
computer science:n.计算机科学;
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If you're able to view a structure from the inside, then you really learn something about it. |
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。 |
That's somehow the essence of something. |
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 . |
所以,当我们一路前行, 进入海洋, 我们发挥了想象力。 |
journey:n.旅行;行程;vi.旅行; imagination:n.想象;想象力;创造力;想象的事物;
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And I think this is one level deeper, and it's actually a requirement for changing your perspective. |
我认为这又更深入了一层, 是改变角度的必然要求。 |
We can do a little game. |
我们可以做个游戏。 |
You can imagine that you're sitting there. |
想象一下你正坐在那儿。 |
You can imagine that you're up here, and that you're sitting here. |
然后你同时又在上面。 |
You can view yourselves from the outside. |
你就可以从外部审视你自己了。 |
That's really a strange thing. |
这听起来很奇怪。 |
You're changing your perspective. |
你在改变你的角度。 |
You're using your imagination, and you're viewing yourself from the outside. |
你在使用你的想象力, 你在从外部审视你自己。 |
That requires imagination. |
这需要有想象力。 |
Mathematics and computer science are the most imaginative art forms ever. |
数学和计算机科学是最具想象力的艺术形式。 |
imaginative:adj.虚构的;富于想象的;有创造力的;
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And this thing about changing perspectives should sound a little bit familiar to you, because we do it every day. |
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, |
familiar:adj.熟悉的;常见的;亲近的;n.常客;密友;
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And then it's called empathy . |
叫做共情。 |
empathy:n.神入;移情作用;执着;
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When I view the world from your perspective, |
当我从你的角度看世界的时候, |
I have empathy with you. |
我就与你产生了共情。 |
If I really, truly understand what the world looks like from your perspective, |
如果我能够真正的 理解你们眼中的世界, |
I am empathetic . |
那我就与你产生了共情。 |
empathetic:adj.移情作用的;同感的(等于empathic);
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That requires imagination. |
这需要想象力。 |
And that is how we obtain understanding. |
这就是我们获得理解的方式。 |
obtain:vi.获得;流行;vt.获得;
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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. |
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. |
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。 |
conclusion:n.结论;结局;推论;
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So my advice to you is: try to change your perspective. |
所以我的建议是:尝试转换你的角度。 |
You can study mathematics. |
你可以学习数学。 |
It's a wonderful way to train your brain. |
这是锻炼大脑的好方法。 |
Changing your perspective makes your mind more flexible . |
变换你们的角度,让思维变得更灵活。 |
flexible:adj.灵活的;柔韧的;易弯曲的;
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It makes you open to new things, and it makes you able to understand things. |
它能够让你们易于接受新事物, 能够理解事物。 |
And to use yet another metaphor: have a mind like water. |
请允许我再使用一次隐喻: 让思维像水一样吧。 |
That's nice. |
会很不错的。谢谢大家。 |
Thank you. |
|
(Applause) |
(掌声) |