Math is the hidden secret to understanding the world | Roger Antonsen

譯者: Regina Chu 審譯者: Kuan-Yi Li

嗨！

Hi.

I want to talk about understanding, and the nature of understanding,

我想跟大家談談理解、理解的本質，

and what the essence of understanding is,

及什麼是理解的精髓，

because understanding is something we aim for, everyone.

因為理解是我們每個人的目標。

We want to understand things.

我們想要理解萬物。

My claim is that understanding has to do

我的主張是理解

與你能不能改變你的觀點大有關係。

with the ability to change your perspective.

If you don't have that, you don't have understanding.

如果你不能，你就不能理解。

所以那是我的主張。

So that is my claim.

And I want to focus on mathematics.

我想把焦點集中在數學。

許多人認為數學就是

Many of us think of mathematics as addition, subtraction,

加、減、乘、除、

multiplication, division,

fractions, percent, geometry, algebra -- all that stuff.

分數、百分比、幾何、 代數這些東西。

可是其實我也想談數學的本質。

But actually, I want to talk about the essence of mathematics as well.

And my claim is that mathematics has to do with patterns.

我的主張是數學跟模式有關。 （又譯胚騰）

在我後面大家可以看到 一個很漂亮的模式，

Behind me, you see a beautiful pattern,

and this pattern actually emerges just from drawing circles

這個模式其實只要

用一種特別的方式畫圓就會出現。

in a very particular way.

So my day-to-day definition of mathematics that I use every day

所以我對數學的簡單定義， 我每天都在用的，

is the following:

就是：

第一，數學就是找出模式。

First of all, it's about finding patterns.

我說的模式是指一種關聯、 一種結構，有某種規律性，

And by "pattern," I mean a connection, a structure, some regularity,

some rules that govern what we see.

某些規則，掌控我們所見。

第二，

Second of all,

我認為它以一種語言表達這些模式。

I think it is about representing these patterns with a language.

We make up language if we don't have it,

如果找不到我們就自己編新語言。

and in mathematics, this is essential.

而在數學，這非常必要。

這也跟你做了假設，

It's also about making assumptions

and playing around with these assumptions and just seeing what happens.

然後玩一下假設 來看看會發生什麼事有關。

We're going to do that very soon.

我們等一下就會這麼做。

And finally, it's about doing cool stuff.

最後，這還跟做出很酷的東西有關。

Mathematics enables us to do so many things.

數學能讓我們做很多事。

So let's have a look at these patterns.

所以現在就來看一下這些模式。

If you want to tie a tie knot,

如果你要打領帶，

這裡有幾種打法。

there are patterns.

Tie knots have names.

領帶打出的結都有名字。

And you can also do the mathematics of tie knots.

你也可以用數學來分析領結。

This is a left-out, right-in, center-out and tie.

有左外、右內、 中間翻出來的打法。

還有左內、右外、左內、 中間翻出來的打法。

This is a left-in, right-out, left-in, center-out and tie.

這就是我們為領結模式 創造出的語言，

This is a language we made up for the patterns of tie knots,

and a half-Windsor is all that.

半溫莎結也能這樣表示。

This is a mathematics book about tying shoelaces

這是本綁鞋帶的數學書，

at the university level,

──大學程度，

because there are patterns in shoelaces.

因為這些是鞋帶的模式。

You can do it in so many different ways.

你可以用很多不同的方式綁。

We can analyze it.

你可以分析綁法。

我們能為之創造語言。

We can make up languages for it.

And representations are all over mathematics.

數學裡充滿了各種表式式。

This is Leibniz's notation from 1675.

這是 1675 年的萊布尼茲表示法。

He invented a language for patterns in nature.

他為自然界的模式創造了一種語言。

When we throw something up in the air,

我們往上丟東西到空中，

it falls down.

它會掉下來。

Why?

為什麼？

We're not sure, but we can represent this with mathematics in a pattern.

我們不太確定，但是我們能用 數學模式表達這種現象。

This is also a pattern.

這也是一種模式。

This is also an invented language.

這也是一種創造出的語言。

Can you guess for what?

你能猜到這是什麼嗎？

It is actually a notation system for dancing, for tap dancing.

這其實是一種舞譜，給踢踏舞用的。

That enables him as a choreographer to do cool stuff, to do new things,

這讓他身為編舞者 能做很酷的動作，做新的動作，

because he has represented it.

因為他能表達出來。

I want you to think about how amazing representing something actually is.

我要大家想一下能把 某種事物表達出來有多厲害！

Here it says the word "mathematics."

這裡寫著「數學」這個詞。

But actually, they're just dots, right?

但是其實只有幾個點，對吧？

So how in the world can these dots represent the word?

所以這幾個點到底怎麼表達這個詞？

Well, they do.

的確可以。

They represent the word "mathematics,"

它們表達「數學」這個詞，

and these symbols also represent that word

這些符號也表達那個詞，

and this we can listen to.

我們可以用聽的。

It sounds like this.

它聽起來像這樣。

(Beeps)

（嗶嗶聲）

Somehow these sounds represent the word and the concept.

這些聲音表達出這個詞和概念。

How does this happen?

怎麼會這樣呢？

There's something amazing going on about representing stuff.

表達是一件很厲害的事。

So I want to talk about that magic that happens

我想談談實際表達某樣東西時，

when we actually represent something.

那種不可思議的魔力。

Here you see just lines with different widths.

這裡你們看到的 只是不同粗細的線條。

They stand for numbers for a particular book.

這是某一本書的條碼。

And I can actually recommend this book, it's a very nice book.

我很推薦這本書，很不錯的一本書。

(Laughter)

（笑聲）

Just trust me.

相信我！

OK, so let's just do an experiment,

好，我們來做個實驗，

just to play around with some straight lines.

就來玩一下幾條直線。

這是一條直線。

This is a straight line.

Let's make another one.

來畫另外一條。

So every time we move, we move one down and one across,

我們每移動一次， 向下一點，衡過去一點，

and we draw a new straight line, right?

我們就畫出一條新的直線，對嗎？

We do this over and over and over,

我們不斷重複這個過程，

and we look for patterns.

我們尋找模式。

So this pattern emerges,

所以這種模式出現，

and it's a rather nice pattern.

還蠻不錯的模式。

It looks like a curve, right?

看起來像弧線，對嗎？

只是畫簡單的直線就得到這個。

Just from drawing simple, straight lines.

Now I can change my perspective a little bit. I can rotate it.

現在我可以改變一下視角。 我可以把它轉一下。

Have a look at the curve.

看一下這條弧線。

What does it look like?

看起來像什麼？

Is it a part of a circle?

像圓的一部分嗎？

It's actually not a part of a circle.

這其實不是圓的一部分。

So I have to continue my investigation and look for the true pattern.

所以我得繼續研究並尋找真的模式。

Perhaps if I copy it and make some art?

或許我能複製一下做成某種藝術品？

Well, no.

嗯，不好。

Perhaps I should extend the lines like this,

或許我應該把線像這樣延長，

然後在那裡找模式。

and look for the pattern there.

Let's make more lines.

再多畫幾條線。

我們這樣做。

We do this.

And then let's zoom out and change our perspective again.

然後我們拉遠再改變一下視角。

Then we can actually see that what started out as just straight lines

我們就會看到原來只是幾條直線，

is actually a curve called a parabola.

變成稱為拋物線的弧線。

This is represented by a simple equation,

這可以用一條簡單的方程式來代表，

而且這是很漂亮的模式。

and it's a beautiful pattern.

So this is the stuff that we do.

所以這就是我們做的事。

We find patterns, and we represent them.

我們找模式，表達出來。

And I think this is a nice day-to-day definition.

我認為這是很好的簡單版定義。

But today I want to go a little bit deeper,

但是今天我想深入一點點，

and think about what the nature of this is.

並思考一下這個的本質。

What makes it possible?

是什麼讓它成為可能？

There's one thing that's a little bit deeper,

我有個稍微深入一點的東西，

and that has to do with the ability to change your perspective.

還跟你能不能改變觀點有關。

我主張當你改變觀點，

And I claim that when you change your perspective,

and if you take another point of view,

如果你能用另一個視角來看，

you learn something new about what you are watching

你會從你正在看、正在觀察、

or looking at or hearing.

正在聽的東西裡學到新東西。

And I think this is a really important thing that we do all the time.

我認為這很重要， 我們應該時時這麼做。

So let's just look at this simple equation,

來看一下這個簡單的等式，

x + x = 2 • x.

x + x = 2 ⋅ x

This is a very nice pattern, and it's true,

這是很棒的模式，而且是準確的，

because 5 + 5 = 2 • 5, etc.

因為 5 + 5 = 2 ⋅ 5，依此類推。

We've seen this over and over, and we represent it like this.

我們不斷看到這個， 所以我們像這樣來表達。

But think about it: this is an equation.

但是想一下：這是條等式。

它說某樣東西等於另一種東西，

It says that something is equal to something else,

and that's two different perspectives.

這是兩種不同的觀點。

One perspective is, it's a sum.

一方面來說這是總和，

It's something you plus together.

是你把東西加起來。

On the other hand, it's a multiplication,

另一方面，這又是乘法，

這是兩個不同的觀點。

and those are two different perspectives.

我甚至要大膽的說 每一條像這樣的等式，

And I would go as far as to say that every equation is like this,

every mathematical equation where you use that equality sign

每一條你用了等號的數學式

都是一種隱喻。

is actually a metaphor.

It's an analogy between two things.

這是兩種事物之間的類比。

You're just viewing something and taking two different points of view,

你用兩種不同的視角看著某樣東西，

and you're expressing that in a language.

而且你用一種語言表示出來。

Have a look at this equation.

看一下這個方程式。

This is one of the most beautiful equations.

這是一條非常漂亮的方程式。

It simply says that, well,

它就是說，嗯，

two things, they're both -1.

兩種東西都是 -1。

This thing on the left-hand side is -1, and the other one is.

左邊這個東西是 -1，右邊也是。

And that, I think, is one of the essential parts

我認為那就是

數學的本質之一： 你用不同的視角觀察。

of mathematics -- you take different points of view.

So let's just play around.

我們就來玩一下。

Let's take a number.

隨便拿一個數字。

We know four-thirds. We know what four-thirds is.

我們知道 4/3。 我們都知道 4/3 是什麼。

它是 1.333，後面還要加上這三個點，

It's 1.333, but we have to have those three dots,

otherwise it's not exactly four-thirds.

要不然就不是精確的 4/3。

But this is only in base 10.

但這只是以 10 為基數的說法。

You know, the number system, we use 10 digits.

你知道，數字系統，我們用十進位。

如果我們變一下只用 2 為基數，

If we change that around and only use two digits,

that's called the binary system.

就是所謂的二進位系統，

It's written like this.

寫起來就像這樣。

So we're now talking about the number.

所以我們現在來談談這個數字。

The number is four-thirds.

數字為 4/3。

We can write it like this,

我們可以像這樣寫，

and we can change the base, change the number of digits,

我們只要改變基數，

and we can write it differently.

就會寫成不同的樣子。

這些是同一個數字的不同表示法。

So these are all representations of the same number.

We can even write it simply, like 1.3 or 1.6.

我們甚至可以簡化它 成為 1.3 或 1.6。

It all depends on how many digits you have.

就看你有幾位數。

Or perhaps we just simplify and write it like this.

或者我們只要簡寫成這樣。

I like this one, because this says four divided by three.

我喜歡這個， 因為這是說把 4 除以 3。

這個數字表示出 這兩個數字間的關係。

And this number expresses a relation between two numbers.

You have four on the one hand and three on the other.

你有 4 在一邊，3 在另一邊。

And you can visualize this in many ways.

你還可以用很多方法把它視覺化。

What I'm doing now is viewing that number from different perspectives.

我現在就是用不同的 視角看這個數字。

I'm playing around.

我在嘗試。

I'm playing around with how we view something,

我在試我們怎麼看某件事物，

而且我還很刻意去做。

and I'm doing it very deliberately.

We can take a grid.

我們可以拿格子。

如果有 4 個橫的 3 個直的， 這條線永遠都等於 5。

If it's four across and three up, this line equals five, always.

It has to be like this. This is a beautiful pattern.

必然如此。 這是一個很漂亮的模式。

Four and three and five.

4 與 3 與 5。

這個長方形，就是 4 × 3，

And this rectangle, which is 4 x 3,

you've seen a lot of times.

你看過很多次。

This is your average computer screen.

這是一般的電腦螢幕。

800 x 600 or 1,600 x 1,200

800 × 600 或 1,600 × 1,200

is a television or a computer screen.

是電視或電腦螢幕。

So these are all nice representations,

所以這些都是很好的表示法，

but I want to go a little bit further and just play more with this number.

但我還想進一步 再多玩一下這個數字。

Here you see two circles. I'm going to rotate them like this.

你在這裡看到兩個圓。 我把它們像這樣轉一下。

觀察一下左上那個。

Observe the upper-left one.

It goes a little bit faster, right?

它轉的比較快，對嗎？

你可以觀察到。

You can see this.

It actually goes exactly four-thirds as fast.

它就是快 4/3。

That means that when it goes around four times,

那代表它每轉四次，

另外一個就轉三次。

the other one goes around three times.

現在畫兩條線，在交會處畫一個點。

Now let's make two lines, and draw this dot where the lines meet.

We get this dot dancing around.

我們就得到轉圈圈的點。

(Laughter)

（笑聲）

And this dot comes from that number.

這個點從那個數字來。

Right? Now we should trace it.

對嗎？我們應該畫它的軌跡。

Let's trace it and see what happens.

我們來畫一下軌跡看看會得到什麼。

This is what mathematics is all about.

這就是數學。

It's about seeing what happens.

想看看會得到什麼。

這得自 4/3。

And this emerges from four-thirds.

我想說這是 4/3 的圖像。

I like to say that this is the image of four-thirds.

It's much nicer -- (Cheers)

這個棒多了。（歡呼聲）

Thank you!

謝謝！

(Applause)

（掌聲）

This is not new.

這不是新發現。

This has been known for a long time, but --

這是早就知道的東西，但是

(Laughter)

（笑聲）

But this is four-thirds.

但是這是 4/3。

再來做另外一個實驗。

Let's do another experiment.

Let's now take a sound, this sound: (Beep)

來弄點聲音，這個聲音：（嗶）

This is a perfect A, 440Hz.

這是標準音高 A440。

Let's multiply it by two.

把頻率乘以 2。

We get this sound. (Beep)

我們就得到這個音（嗶）。

When we play them together, it sounds like this.

兩個音一起彈，聽起來像這樣。

This is an octave, right?

這是個八度音，對吧？

We can do this game. We can play a sound, play the same A.

我們可以玩個遊戲。 我們可以彈一個音，同樣的 A。

We can multiply it by three-halves.

然後乘上 3/2。

(Beep)

（嗶）

This is what we call a perfect fifth.

這是我們說的完全五度。

(Beep)

（嗶）

They sound really nice together.

這幾個音加起來真的很好聽。

Let's multiply this sound by four-thirds. (Beep)

把這個乘上 4/3。（嗶）

What happens?

會得到什麼？

You get this sound. (Beep)

這個音。（嗶）

This is the perfect fourth.

這是完全四度。

If the first one is an A, this is a D.

如果第一個音是 A，這就是 D。

They sound like this together. (Beeps)

合奏聽起來像這樣。（嗶嗶）

This is the sound of four-thirds.

這就是 4/3 的聲音。

What I'm doing now, I'm changing my perspective.

我現在做的就是改變我的觀點。

I'm just viewing a number from another perspective.

我只是用另一個觀點來看一個數字。

I can even do this with rhythms, right?

我甚至可以用節拍來做，對吧？

I can take a rhythm and play three beats at one time (Drumbeats)

我可以加上節拍， 一段時間內打三拍。

in a period of time,

（鼓聲）

我也可以用另一種聲音 在同樣一段時間內敲四次。

and I can play another sound four times in that same space.

(Clanking sounds)

（金屬敲擊聲）

Sounds kind of boring, but listen to them together.

聽起來有點無聊， 但是把它們加起來聽。

(Drumbeats and clanking sounds)

（鼓和金屬聲）

(Laughter)

（笑聲）

嘿！好聽吧？

Hey! So.

(Laughter)

（笑聲）

I can even make a little hi-hat.

我甚至可以加點腳踏鈸。

(Drumbeats and cymbals)

（鼓及鈸聲）

Can you hear this?

聽到了嗎？

So, this is the sound of four-thirds.

所以，這就是 4/3 的聲音。

Again, this is as a rhythm.

同樣的，這也以節拍表達。

(Drumbeats and cowbell)

（鼓聲及牛鈴）

And I can keep doing this and play games with this number.

我可以一直做下去， 繼續玩這個數字，

Four-thirds is a really great number. I love four-thirds!

4/3 真的是很棒的數字， 我愛 4/3！

（笑聲）

(Laughter)

Truly -- it's an undervalued number.

真的，這是個被看扁的數字。

So if you take a sphere and look at the volume of the sphere,

所以如果你拿一個球看它的體積，

it's actually four-thirds of some particular cylinder.

它就是某圓柱的 4/3。

So four-thirds is in the sphere. It's the volume of the sphere.

所以球裡也有 4/3， 就是這個球的體積。

OK, so why am I doing all this?

好，我為什麼要說這些？

Well, I want to talk about what it means to understand something

嗯，我想談一下 理解事物是什麼意思。

以及我們說理解又是什麼意思。

and what we mean by understanding something.

That's my aim here.

那是我今天的目的。

我的主張是你要理解某項事物，

And my claim is that you understand something

if you have the ability to view it from different perspectives.

就要有能力從不同的角度來看它。

Let's look at this letter. It's a beautiful R, right?

來看這個字母， 很漂亮的 R，對吧？

How do you know that?

你怎麼知道？

Well, as a matter of fact, you've seen a bunch of R's,

嗯，其實，你已經看過一堆的 R，

and you've generalized

你歸納

and abstracted all of these and found a pattern.

並抽象化這些 R，並找出模式。

So you know that this is an R.

所以你知道這是 R。

So what I'm aiming for here is saying something

所以我今天的目的是要告訴大家

about how understanding and changing your perspective

理解及改變你的觀點

are linked.

如何連在一起。

And I'm a teacher and a lecturer,

我是老師及講師，

and I can actually use this to teach something,

我其實可以用這個來教某樣東西，

because when I give someone else another story, a metaphor, an analogy,

因為每當我說一個故事、 一段隱喻、一段類比，

if I tell a story from a different point of view,

如果我用不同的角度來說一個故事，

我就在讓你理解。

I enable understanding.

I make understanding possible,

我讓理解成為可能，

because you have to generalize over everything you see and hear,

因為你必須歸納每一樣 你見到及聽到的事物，

and if I give you another perspective, that will become easier for you.

如果我給你另外的觀點， 你就更容易歸納。

Let's do a simple example again.

再來一個簡單的例子。

這是 4 和 3 。 這是四個三角形。

This is four and three. This is four triangles.

So this is also four-thirds, in a way.

所以在某種意義上這也是 4/3。

Let's just join them together.

把它們連在一起。

現在我們來玩一個遊戲； 我們要把它折成

Now we're going to play a game; we're going to fold it up

into a three-dimensional structure.

一個三度空間的結構。

I love this.

我很愛這個。

This is a square pyramid.

這是一個正方錐。

And let's just take two of them and put them together.

就拿兩個出來放在一起。

So this is what is called an octahedron.

這個就叫做八面體。

It's one of the five platonic solids.

這是五種柏拉圖立體之一。

現在我們真的可以改變視角，

Now we can quite literally change our perspective,

because we can rotate it around all of the axes

因為我們可以讓它繞著各軸旋轉，

and view it from different perspectives.

從各種不同的角度來看。

And I can change the axis,

我可以改變軸，

and then I can view it from another point of view,

然後從另一個視點來看它，

but it's the same thing, but it looks a little different.

這是同一個東西， 只是看起來有點不一樣。

I can do it even one more time.

我可以再做一次。

Every time I do this, something else appears,

每次做都有不同的東西出現，

so I'm actually learning more about the object

所以我真的從這個物體中 學到更多東西，

when I change my perspective.

只要我改變角度。

I can use this as a tool for creating understanding.

我能用這個當工具引發理解。

I can take two of these and put them together like this

我能把這兩個像這樣放在一起

然後看會發生什麼。

and see what happens.

And it looks a little bit like the octahedron.

這看起來有點像八面體。

Have a look at it if I spin it around like this.

我把它像這樣轉一轉，看一下。

What happens?

什麼出現了？

嗯，如果你把兩個這樣的東西 放在一起，然後轉一轉，

Well, if you take two of these, join them together and spin it around,

there's your octahedron again,

你的八面體又出現了，

a beautiful structure.

很漂亮的結構。

If you lay it out flat on the floor,

如果你把它平放在地板上，

this is the octahedron.

這是八面體。

This is the graph structure of an octahedron.

這是八面體的圖結構。

And I can continue doing this.

我可以繼續做下去。

You can draw three great circles around the octahedron,

你可以在這個八面體旁畫三個大圓，

and you rotate around,

然後轉一轉，

這三個大圓其實跟這個八面體相關。

so actually three great circles is related to the octahedron.

And if I take a bicycle pump and just pump it up,

如果我拿一個打氣筒來打氣，

you can see that this is also a little bit like the octahedron.

你會看到這也有點像八面體。

Do you see what I'm doing here?

你看懂我在做什麼了嗎？

我一直在改變觀點。

I am changing the perspective every time.

So let's now take a step back --

我們現在退一步，

and that's actually a metaphor, stepping back --

這其實也是一種隱喻，退一步，

and have a look at what we're doing.

來看看我們在做什麼。

I'm playing around with metaphors.

我正在嘗試隱喻。

I'm playing around with perspectives and analogies.

我在嘗試各種觀點及類比。

我用不同的方法說同一個故事。

I'm telling one story in different ways.

I'm telling stories.

我說很多故事。

I'm making a narrative; I'm making several narratives.

我做一種敘述；我做很多種敘述。

And I think all of these things make understanding possible.

我認為這些都在幫助我們理解。

I think this actually is the essence of understanding something.

我認為這其實就是理解事物的精髓。

I truly believe this.

我真的相信這點。

So this thing about changing your perspective --

所以改變你的觀點這件事

it's absolutely fundamental for humans.

對人類是非常重要的。

Let's play around with the Earth.

就來玩一下地球。

Let's zoom into the ocean, have a look at the ocean.

我們來放大海洋，看一下海洋。

什麼東西都可以拿來這樣玩。

We can do this with anything.

We can take the ocean and view it up close.

我們可以近看海洋。

We can look at the waves.

我們可以看海浪。

We can go to the beach.

我們可以去海灘。

We can view the ocean from another perspective.

我們能從不同的觀點看海洋。

Every time we do this, we learn a little bit more about the ocean.

我們每次這麼做， 就對海洋多一點認識。

如果我們到海邊， 好像就可以聞到海，對吧？

If we go to the shore, we can kind of smell it, right?

We can hear the sound of the waves.

我們能聽海浪的聲音。

We can feel salt on our tongues.

我們能用舌頭嘗到鹹味。

So all of these are different perspectives.

所以這些都是不同的觀點。

And this is the best one.

這個最棒。

We can go into the water.

我們可以進到水中。

We can see the water from the inside.

我們可以從裡面看水。

And you know what?

你知道嗎？

This is absolutely essential in mathematics and computer science.

這在數學及電腦科學中不可或缺。

If you're able to view a structure from the inside,

如果你能從裡面看結構，

then you really learn something about it.

那你真的能從中學到東西。

That's somehow the essence of something.

那可以說是事物的精髓。

So when we do this, and we've taken this journey

就在剛剛那段旅程之中，

into the ocean,

我們進入了海洋，

we use our imagination.

──以我們的想像力。

And I think this is one level deeper,

我想這更深一層了，

and it's actually a requirement for changing your perspective.

這實際上是改變觀點的必要條件。

We can do a little game.

我們可以玩一個小遊戲。

你想像一下你坐在那裡。

You can imagine that you're sitting there.

你想像一下你在台上這裡， 你坐在這裡。

You can imagine that you're up here, and that you're sitting here.

You can view yourselves from the outside.

你可以從外面看自己。

That's really a strange thing.

那真的很怪。

You're changing your perspective.

你正在改變你的觀點。

You're using your imagination,

你運用你的想像力，

and you're viewing yourself from the outside.

你從外面看自己。

那需要想像力。

That requires imagination.

數學及電腦科學是最富 想像力的藝術形式。

Mathematics and computer science are the most imaginative art forms ever.

And this thing about changing perspectives

改變觀點這件事

聽起來應該有點熟悉，

should sound a little bit familiar to you,

because we do it every day.

因為我們每天都在做。

And then it's called empathy.

那就叫做同理心。

When I view the world from your perspective,

當我從你的觀點看世界，

I have empathy with you.

我就是站在你的立場設身處地。

If I really, truly understand

如果我真的能理解

what the world looks like from your perspective,

從你的觀點所看到的世界，

I am empathetic.

我就是將心比心。

那需要想像力。

That requires imagination.

And that is how we obtain understanding.

那就是我們獲得理解的方法。

And this is all over mathematics and this is all over computer science,

這在數學界很普遍， 在電腦科學界也很普遍，

and there's a really deep connection between empathy and these sciences.

這兩種科學與同理心 也有很深的淵源。

So my conclusion is the following:

所以我的結論就是：

understanding something really deeply

要非常深入的理解事物，

跟你是否能改變觀點大有關係。

has to do with the ability to change your perspective.

So my advice to you is: try to change your perspective.

所以我給大家的建議是： 試著改變你的觀點。

You can study mathematics.

你能研究數學。

這是非常棒的方法來訓練大腦。

It's a wonderful way to train your brain.

Changing your perspective makes your mind more flexible.

改變你的觀點讓你的心智更靈活。

It makes you open to new things,

它能讓你接受新東西，

and it makes you able to understand things.

讓你能理解事物。

And to use yet another metaphor:

再用一個隱喻來形容：

have a mind like water.

讓你的心像水一樣。

That's nice.

真好。

Thank you.

謝謝。

（掌聲）

(Applause)