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理解与转换视角的能力

引言

你好。我想讨论一下“理解”，以及理解的本质和它的核心意义，因为理解是我们每个人都在追求的目标。我们都想要理解事物。

我的观点是：理解与能够改变你的视角这一能力密切相关。如果你做不到这一点，你就无法真正达到理解。所以，这是我的观点。

接下来我想重点谈一谈数学。很多人认为数学就是加法、减法、乘法、除法、分数、百分比、几何、代数——这些东西。

但实际上，我想聊聊数学的本质。

我认为数学的本质与模式（patterns）相关。

在我身后，你能看到一个漂亮的图案，而这个图案只是通过按照特定方式画圆而形成的。

数学的定义

我在日常生活中使用的数学定义是这样的：

• 首先，它是关于“寻找模式”。这里的“模式”指的是一种联系、结构、一种规律性，以及支配我们所观察到的事物的规则。

• 其次，它是关于用语言来表达这些模式。如果我们暂时没有合适的语言，那就创造一种语言，而在数学中，这一点尤其重要。

• 另外，数学还涉及做出一些假设，然后围绕这些假设进行探索，看看会发生什么。我们很快就会做类似的事情。

• 最后，它也关乎“做一些很酷的事情”。数学能够让我们实现许多有趣且有用的操作。

让我们来看看这些“模式”。如果你想系领带，领带的各种打法中就包含着模式。不同的领结或领带打法都有各自的名字。你也可以研究领带打法的数学。比如，“左出、右进、中出，然后打好”是一种打法，“左进、右出、左进、中出，然后打好”又是另一种打法。我们为领带打法创造出这种描述模式的语言，而半温莎结就是其中的一种打法方式。

这是一本讲述系鞋带的大学级数学书籍，确实是大学水平，因为在系鞋带的过程中也存在各种模式。系鞋带的方法有许多种，我们可以对它进行分析，并为其创造一套合适的语言。

代表及其重要性

代表无处不在，尤其在数学中。

这是莱布尼茨在 1675 年创立的符号体系。他为自然界的各种模式发明了一种语言。

当我们把某个物体抛向空中时，它会落下。为什么会这样？我们并不完全清楚，但我们可以用数学的方式将其表示为一种模式。

这同样是一种模式，这也是一种被创造出来的语言。你能猜到它是用来做什么的吗？其实，这是一个用来记录舞蹈、特别是踢踏舞的符号系统。有了这种符号系统，他作为编舞者便能够做出许多很酷的新尝试，因为他已经将它以符号的形式代表出来。

我希望你能思考一下，能将事物以某种方式代表出来是多么神奇的事情。

这里写着"mathematics"（数学）这个词。但实际上，它们只是一些小点，对吗？那么，这些点究竟是怎样代表这个单词的呢？但它们确实做到了这一点。它们完美地代表了"mathematics"这个词。这些符号同样能代表这个词，而且我们还能通过听觉来感受它——它的声音是这样的（发出“哔哔”声）。不知为何，这些声音也能够代表这个单词以及它所蕴含的概念。这到底是怎么发生的呢？在代表事物的过程中，似乎存在着某种令人惊叹的奥秘。

模式

所以我想谈谈这种神奇的现象，这种当我们表示某物时所发生的奇妙转变。这里你看到的只是一些宽度不同的线条，它们代表了某本特定书籍中的数字。我真心推荐这本书，这是一本非常棒的书。相信我，准没错。

好，让我们做个实验，来玩玩这些直线。这是一条直线。我们再画一条。每次移动时，我们向下一格，向右一格，然后画一条新的直线，对吧？我们不断重复这个过程，寻找其中的模式。慢慢地，这个模式显现出来了，形成了一个相当优美的图案。看起来像一条曲线，对不对？这仅仅是通过画一些简单的直线形成的。

现在我们稍微改变视角，把它旋转一下。仔细看这条曲线。它像什么？是圆的一部分吗？其实它并不是圆的一部分。所以我必须继续探索，寻找真正的规律。也许我可以把它复制下来，做成某种艺术品？嗯，也许不行。也许我应该这样延长这些线条，在那里寻找规律。让我们画更多的线。就这样做。然后我们拉远视角，换个角度看。这时我们就能发现，最初的那些直线实际上构成了一条叫做抛物线的曲线。它可以用一个简单的方程来表示，形成了一个美妙的模式。

这就是我们在做的事情。我们发现模式，并找到表示它们的方法。我认为这可以作为日常生活中对于“模式”一个不错的定义。不过今天我想更深入地探讨，思考一下这背后的本质是什么。是什么让这一切成为可能？有一点非常重要，那就是改变视角的能力。

我认为当你改变视角，当你从另一个角度看问题时，你会对你正在观察的、看到的或听到的事物有全新的认识。我认为这是我们一直在做的一件极其重要的事情。

方程

让我们来看这个简单的方程：x+x=2⋅x。这个方程非常优美，而且它是对的，比如 5+5=2⋅5，等等。我们在不同场合下都见过它，并且我们用这种形式来表示它。不过仔细想想：这是一个等式，意思是有一样东西等于另一样东西，而这是两种不同的视角。一种视角是它是一个加法，我们把两个东西相加；另一种视角是它是一个乘法，而加法与乘法是两种不同的视角。

我甚至会说，每一个等式都是这样的。每一个使用了等号的数学等式实际上都带有隐喻的性质：它是在两个事物之间做类比。你只是用两种不同的观点去看同一个东西，并用一种语言将其表达出来。再看看这个等式。这是最美的等式之一。它基本上说了两件事：两个东西，它们都等于-1。它们都等于 −1。左边的东西是 −1，右边的也是 −1。我认为，这就是数学的一个核心要素——你要以不同视角去看问题。

那我们来玩一玩。取一个数字，我们知道 34​。我们知道 34​ 是 1.333，但必须带上三个点（省略号），否则就不是准确的 34​。不过这只是在十进制的情况下。你知道，就是我们平常用的那个十个数字的系统。如果我们换一种方式，只用两个数字，这就是所谓的二进制系统，写出来会是另一个样子。现在我们讨论的这个数字就是 34​。我们可以这样写它，我们可以改变进制，改变可用的数字数量，这样就能用不同方式表示它。

所以这些都是同一个数的不同表示方式。我们甚至可以把它简单写成 1.3 或者 1.6，这取决于你要保留多少位小数。或者我们干脆写成 34​ 的形式。

我喜欢这一种写法，因为它直接写了 “4÷3”。而这个数字表达的是两个数字之间的关系：一边是 4，一边是 3。你可以用各种方式去把它可视化。我现在在做的就是从不同角度观察这个数。我在探索。我在探索如何用不同方式看待同一个东西，而且我是很有目的地这样做。

我们可以画一个网格。如果横向是 4，纵向是 3，那么这条斜线就一直是 5，这是恒定的。它必须如此。这是一个很美的模式：4、3 和 5。那个 4×3 的矩形你肯定见过很多次，比如常见的电脑显示屏是 800×600 或者 1600×1200，电视屏幕也是如此。

这些都是很好的表现方式，但我想再深入一点，用这个数玩出更多花样。这里你看到两个圆。我会这样转动它们，注意看左上角那个，它转得快一点，对吗？你能看到这个，它实际上正好比另一个快 34​。也就是说，当它转 4 圈的时候，另一个刚好转 3 圈。现在我们画两条线，然后标出它们交汇的那个点，你会看到那个点在“跳舞”（笑声），而这个点的运动正是由那个数字（34​）所决定的。

神奇吧？现在我们来追踪它的轨迹。让我们来追踪看看会发生什么。这就是数学的意义所在：去看看会发生什么。而它从 34​ 中出现。我喜欢把它称作“四分之三的图像”，因为它很漂亮。谢谢！

这其实不是什么新发现。这个很早就被发现了，但是——但这就是 34​。

让我们做另一个实验。现在让我们取一个声音，这个声音是（嗡），这是一个标准的A音，频率是440赫兹。让我们把它的频率乘以 2。就得到这个声音（嗡）。当我们把它们一起播放，就变成这样。这就是一个八度，对吧？我们可以继续玩这个游戏。我们先放一个声音，还是那个A音。然后我们把它的频率乘以 23​。（嗡）这就是我们称为纯五度的音。（嗡）它们听起来非常和谐。

现在让我们把这个声音的频率乘以34​。（嗡）会发生什么？你就会得到这个音（嗡）。这就是纯四度。如果第一个音是 A，那么这个音就是 D。它们合起来就是这样（嗡嗡）。这就是 34​ 的声音。我现在做的其实是在改变看问题的角度。我只是用另一种方式来理解这个数。

我甚至可以用节奏来展示，怎么样？我可以在同一段时间里打出 3 下（鼓声），并在同样的时段里再打 4 下另一种声音（敲击声）。听起来可能有点单调，但你把它们合起来听听。（鼓声和敲击声）嘿！好了。我还可以加入一个镲片的声音。（鼓点和镲片声）听得出来吗？

这就是 34​ 以声音的方式呈现。（鼓点和牛铃声）

我还可以继续玩下去，用这个数字变出各种新花样。34​ 是个非常棒的数字。我爱 34​！说真的——这是一个被严重低估的数字。比如，你拿一个球体来计算它的体积，实际上就是它和某个特定圆柱体体积之间的 34​ 关系。所以 34​ 也存在于球里——那就是球的体积公式中的系数。

改变你的视角

好，那么我为什么要做这些呢？因为我想探讨“理解”某个事物到底意味着什么，我们所谓的“理解”究竟指什么。这就是我今天的目标。

我认为，当你理解某物时，你就具备了从不同角度看待它的能力。

让我们先看这个字母：它是一个漂亮的“R”，对吗？你是怎么知道的？事实上，你见过很多形态各异的 R，你把它们都概括起来（归纳），提取了所有这些特征，抽象出了一个模式，于是你就知道这是一个 R。

所以我在这里想表达的是：理解和改变视角之间有着密切联系。

作为一名教师和演讲者，我实际中可以运用这一点来教学。因为当我给学生讲另一个故事、一个隐喻、一个类比，当我从不同的角度讲述同一个故事时，我就为他们打开了理解的大门——只要他们去把看到和听到的一切进行归纳。如果我再给他们一个视角，这种归纳就会变得更容易。

让我们再举个简单的例子。这是 4 和 3。这里是 4 个三角形，因此某种意义上它也代表了四分之三。让我们把这些三角形连在一起，现在我们来玩个有趣的游戏：我们要把它折叠起来，变成一个立体的结构。我很喜欢这样做。这会形成一个方锥。再把其中两个方锥合在一起，这就是我们所说的八面体（octahedron），它是五种柏拉图立体之一。

现在我们可以非常直观地改变我们的视角，因为可以把它围绕所有轴进行旋转，从不同的角度去看。我可以改变旋转轴，然后从另一个视角来观察它，虽然它还是原来的东西，但看起来略有不同。我还可以再换个角度。每次这么做时，就会看到一些新的细节——所以当我改变视角的时候，其实也在加深对这个物体的认识。我可以把这个方法用作帮助理解的工具。

我还能把两个这样的结构拼在一起，看看会发生什么。看起来会有点像八面体。如果我这样旋转它的时候会怎样。观察一下。你会发现，当我们把两个结构拼在一起再旋转时，八面体的形状又出现了，多么优美的结构啊。如果你把它展开平铺在地上，这就是八面体的“展开图”，也就是它的图形结构。

我还可以继续这么做，比如在八面体上画出三条大圆（great circles），然后再旋转。你会发现这三个大圆圈与八面体有着密切的联系。如果我拿一个打气筒把它稍微充气，你会发现充气后的形状也有点像八面体。

你明白我在做什么吗？我每次都在换一个新的视角。现在，让我们“退后一步”——这本身也是一个隐喻——审视一下所做的事。我在玩各种隐喻、玩各种视角和类比；我在用不同的方式讲述同一个故事。我在构建一个叙事；准确地说，是多个叙事。我相信正是这些方法让理解成为可能。我认为这就是真正理解某事物的本质。我真的深信不疑。所以，说到改变视角——这是人类最根本的能力之一。

让我们拿地球来做个实验。让我们把视角拉近到海洋，仔细观察海洋。其实我们可以对任何事物都这样做。我们可以拉近观察海洋，可以看看海浪，可以到海滩再换一个角度去看它。每一次这样做，都能对海洋了解更多一点。如果我们走到海边，就能闻到海的味道，听到海浪的声音，甚至能尝到唇边的咸味。这些都是认识海洋的不同视角。

而最精彩的是——我们可以潜入水中，从内部去看海。你知道吗？这个道理在数学和计算机科学中极其重要。如果你能从内部去观察一个结构，你就真的能更深入地理解它。那就像是把握了事物的本质。

想象力与共情

当我们这么做，踏上这段探寻海洋的旅程时，我们其实在运用想象力。我认为这触及到了更深层的东西，这实际上是改变视角的必要条件。让我们来玩个小游戏：想象你正坐在那个位置上，或者想象你坐在上面那个地方；你可以想象自己从外部看待自己。

这真的很奇妙。你正在改变你的视角，你在运用想象力，从外部观察自己。这需要想象力。而数学和计算机科学恰恰是有史以来最富想象力的艺术形式。

这种改变视角的能力对你们来说应该不陌生，因为我们每天都在练习。这就是我们所说的同理心。当我尝试从你的视角看世界时，我就对你产生了同理心。如果我能真正地、彻底地理解从你的视角看世界是什么样子，我就拥有了真正的同理心。这需要想象力。这就是我们获得理解的方式。

这种方式贯穿于整个数学和计算机科学，同理心与这些科学之间存在着深刻的联系。

结论

所以我要总结的是：要想真正深入地理解某件事，关键在于改变视角的能力。因此，我给大家的建议是：努力改变你的视角。你可以通过学习数学来做到这一点。这是训练大脑的绝佳方式。改变视角能让你的思维更加灵活，让你对新事物保持开放，让你能够真正理解事物。让我用最后一个隐喻来结束：让你的心智如水般流动。那很美好。谢谢！

# **Understanding and the Ability to Change Your Perspective**

## Introduction

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. 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. Behind me, you see a beautiful pattern, and this pattern actually emerges just from drawing circles in a very particular way.

## Definition of Mathematics

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. 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.

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. 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.

## Representation and Its Significance

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.

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

## Patterns

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. (Laughter) Just trust me.

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.

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.

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.

## Equations

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. 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.

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.

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. 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.

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. (Laughter) 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) Thank you! (Applause)

This is not new. This has been known for a long time, but -- (Laughter) But this is four-thirds. Let's do another experiment. Let's now take a sound, this sound: (Beep) 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) 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) 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. 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) Sounds kind of boring, but listen to them together. (Drumbeats and clanking sounds) (Laughter) Hey! So. (Laughter) 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! (Laughter) 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.

## Changing Your Perspective

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.

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.

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. 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. 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.

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.

## Imagination and Empathy

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. Let's 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. And 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.

## Conclusion

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. Thank you. (Applause)