对于我来说，音乐是更难的，数学反倒很简单 | Matthias Kreck

音乐、数学与绘画，绝不仅仅是英文首字母M相同，数学家可以演奏大提琴乐曲，还能用绘画的形式讲数学原理小故事。在SELF讲坛上，世界著名几何学家、业余大提琴演奏家Matthias Kreck，用动画演绎、大提琴演奏，向我们证明了数学家不仅能解决实际问题，还会讲笑话、拉大提琴。

https://v.qq.com/txp/iframe/player.html?vid=u07020pwetx&width=500&height=375&auto=0

Matthias Kreck

世界著名几何学家 业余大提琴演奏家

以下内容为Matthias Kreck演讲实录：

你好！这绝对不是我唯一会的中文，但这是我在今天的表演中唯一会用到的中文单词。

接下来我想用一分钟的时间向你们介绍我的演出 M³，在数学当中M³等于M乘以M再乘以M。第一个M指Music，音乐，我知道你们一定都喜欢它。第二个M，我确定你们不喜欢它，它代表数学—Mathematics， 但是在我演讲结束后你会喜欢上它。第三个M是一个德语单词，Malerei，翻译成英语就是绘画的意思。绘画用中文怎么说？ Huihua，这是我学会的第五个中文单词了。

如果你期待数学能解释音乐或者绘画，那你就错了，这三种艺术只能代表它们本身。所以当你听音乐的时候，就全神贯注地听，忘掉数学，这很简单。思考数学时，要忘掉音乐，好像并不容易。绘画是和数学结合起来的，所以当我们讨论绘画时，你可以思考数学。你们马上就会看到，稍后我展示的绘画都是卡通的形式。

所以听音乐时专注于音乐，思考数学的时候专注于数学，当绘画也会加入进来时，绘画和数学双管齐下。

我先演奏一首意大利作曲家安东尼奥·维瓦尔第的曲子，他最著名的曲子是《四季》，但我不拉这首曲子。我演奏（大提琴）奏鸣曲第3号的两个乐章。（编者注：视频中02:20）

对于我来说音乐的部分是更难的，数学的部分反倒很简单。对于你们来说也许数学更难，但我会让它变得简单。一提到德国人，你们也许就会想到，一群德国人坐在啤酒花园里，一天二十四小时喝着啤酒。我们确实特别喜欢这样，但不是二十四小时。

在动画中，你看到我就在这样一个啤酒花园里，点了一杯啤酒，然后它上来了，我很开心。然而这个桌子在晃动，啤酒洒了出来，我十分生气。

我找啤酒花园的主人抱怨，我掀起桌子生气地说：“看看你们这该死的桌子！”他说：“不不不桌子没问题，是地面的问题。地面不平所以桌子才晃。”好吧，我放过他，的确不是桌子的问题，是啤酒花园的地面不平。

当然地面不可能是完全平整的，我们都经历过这样的事情，而且都知道接下来要做什么。

所以我接下来该怎么做呢？下一个动画中，一个小男孩走过来对我说“非常简单啊，拿一张纸把它垫在桌腿底下就好啦。”但这样只能管用一会儿，因为这张纸过一会就被压塌了，桌子又会开始晃。

数学家讨厌不稳定，他们不接受（不稳定），你们也应该这样。当你的桌子摇晃的时候，照我的方法做，以后你就不会遇到摇晃的桌子了，我的方法非常简单而且实用。

看，我们将桌子旋转90°，顺时针和逆时针都转一下。你会看到在一个神奇的时刻，桌子完全站稳了。你可以在家里或者其他地方做这个实验， 如果你的桌子晃了，只要旋转至多90°，它就稳了。

这并不是偶然，这不是啤酒花园的话题，这是一个数学演讲，并且可以通过数学来证明这个办法有效。首先，我们需要转化一下，将现实里遇到的问题转化成为数学问题，也就是建立数学模型。

我们将桌子标上序号1234，然后将234号固定在地面上，这时1号是悬空的。当然，只有四条腿都挨在地上的时候桌子才不会晃动，桌子晃动就意味着1号桌子腿没有挨到地面。

现在重要的一步来了，纸上的标出来的点，就是四条腿的位置， 其中234号是固定在地面上的，1号是悬空的。如果我把1号桌腿压在地上，那么3号桌腿就会悬空，所以桌子就晃动了。

接下来我们测量一下1号桌腿距离地面的高度，会得到一个具体的数字，比如说一厘米，这个数字是正的，因为桌子腿是悬空在地面之上的一厘米。

然后我们继续旋转桌子，同时保持234号桌腿一直挨在地面上。这个过程中1号桌腿发生了什么呢？有一段时间它是悬空的，但是当我们到达这个位置，1号就会到地面下。

桌子转了90°之后，1号桌腿的位置，是在地面之下的。一开始时1号桌腿在地面之上，我们旋转桌子，保持另外三条腿一直挨着地。现在是这三条桌腿挨着地面，1号桌腿在地面之下。这是一条非常重要的讯息，有了这些信息，我们就可以开始建立数学模型了。

在这条动画中，你会看到1号桌腿是怎么跑到地面之下的，这一步很关键。现在我们将它转换成了数学问题，在数学中我们会画一些曲线然后来研究它们。

现在我们画出这样一条曲线，每转动一下桌子我们就测量一次1号桌腿距离地面的高度，我们把时间坐标轴的起点，也就是旋转之前，标为0，旋转完毕标为1。在0时刻，1号桌腿是悬空的。当时间为1时，1号桌腿是在地面之下的。我们把这个过程中，1号桌腿距地面的高度都测量出来，就会得到一条曲线。

https://v.qq.com/txp/iframe/player.html?vid=b1342hb0uzl&width=500&height=375&auto=0

曲线会上升、下降，在最后的时候高度降到0以下，现在我们非常接近一个有趣的数学定理了。

我们看到234号桌腿一直都挨着地面，对应到坐标轴上是0，但1号桌腿一会儿在地面之上一会儿在地面之下。它的数学定理很重要，叫做中间值定理，大学一年级的学生会详细地学习并且证明它，这一般会花很长时间，但是跟着我你们只需要五分钟，你们比那些学生们要优秀多了。

我们看到这样的连续函数，我们有一条(函数的）曲线 ，f (0)比0大 ，f (1)比0小，中间有某个时刻的值是0，对应到桌子的情况，这个点就是桌子稳定的时刻，因为其他桌子腿都挨着地，现在4号桌腿的高度也是0，所以桌子不再摇晃。

这个定律非常重要，现在我们来证明它，我们通过“狩猎”的方式来证明，想象你有一片土地，上面有一些动物，你想要狩猎，你会怎么做？我们不在美国，所以我们不用枪，枪是绝对禁止的。我们将地划成两半，我们去找动物在哪一边。我们先不管另外一半的地，我们再把这一块地划成两半，我们再看看动物在哪，对半分，建篱笆，然后继续，到最后我们捉到了猎物，这就是我们狩猎的过程。

同样的道理，在这里我们把时间对半分，看看在中间时刻桌腿的高度，会有（正负）两种情况。我们看到，在0时刻它是正数，在1时刻它是负数，在1/2时刻它是负数。然后我们把（1/2到1）这一半扔掉不管，它们的正负性是一样的，都是负的。我们想捕捉的零点在左边这一半，这就像我们的想捕捉的猎物，我们已经把右半边的曲线扔掉了，现在，零点就在0和1/2之间了。

我们再来一次，再对半分，这次零点是在右边，我们不要左边只看右边。如果你一直重复做下去，中间的间隔会越来越小。当然我们可能要做无限多次这样的操作，这是我们在数学中要学习的，“无限” 这个微妙的概念。我们在这里见到了“无限”，我们越来越接近，直到最后我们找到了零点。

这个操作能够有效是有原因的，在定理中这条曲线是连续的，连续的意思就是你可以用笔画出这条曲线而没有间断。如果中间有间断，就意味着你可以从正到负，而不用经过零点。所以粗略地解释，连续性就是没有没有间断。数学上的微妙定义，正是我们现在做的事情，保证我们可以捕捉到零点。

我很自豪，我解决了桌子摇晃的问题，你看人们都对我前呼后拥，但我一点也不喜欢，我只想先喝酒，喝完酒我就溜了。

我希望你们能学到两件事情，首先，数学家是会讲笑话的，第二，数学家可以解决问题。比如这个摇晃的桌子，我们还是很有用的。当然，我们大多时候是在做理论方面的工作，将实际问题转化成数学问题，对于我们来说是思考的起点。

问题可能是来自于物理、化学和其他任何学科，或者是来自数学内部。更多时候我是一个理论数学家，我坐在桌子边，有人进来，他会说我偷懒不工作。因为我就呆呆地坐着，人们以为我在睡觉，但是我在思考。我在思考的明证就是，我会不时地在著名的刊物上发表文章，这就是我们数学家的工作。

请你们记住数学家也会讲笑话，他们能解决实际问题，就像这摇晃的桌子，他们也能给出证明。今天你们见到了一个证明，感谢你们。今天认真的聆听中国观众给我留下了很深的印象，谢谢。

最后的环节仍然是音乐，我希望你们可以和我一起表演。下面我将演奏两首歌曲，希望你们都听过，并且能跟着我一起唱。（编者注：视频中18：35）

刚刚是一首英文歌，下一首是德文歌曲。我很抱歉，但是德国一些作曲家非常不错，我表演其中一首，你们和我一起唱。

https://v.qq.com/txp/iframe/player.html?vid=y0701wepyh2&width=500&height=375&auto=0

谢谢，我的演讲时间还剩1分41秒，我要感谢在座的观众，这是我演奏过的最棒的音乐会，中国观众是如此的有教养，并且熟悉我的同胞贝多芬，这些给我留下了非常深刻的印象。

你们知道这首歌的德语歌词吗？

Freude, schöner Götterfunken, 

Tochter aus Elysium.

欢乐女神圣洁美丽 灿烂光芒照大地

听起来很奇怪吧，这是欢乐与自由之歌，我想在这个世界上我们需要快乐和自由，还有和平。再次感谢。

以下为演讲英文实录：

Ni Hao! This is not my only Chinese word.But it is the only one I will use in this performance.

So I would like to explain you in one minute the idea of my program M³. So in mathematics "cubed" means M times M times M，and the first M is music. You all like that, I'm sure. The second M, I'm sure you don't like it. It's mathematics, but you will like it after the show.And the third M is a German word. It is called Malerei, and in English Painting.What is the Chinese word for painting? Huihua! It's my fifth Chinese word.  

So if you expect that mathematics will explain the music or painting, you are wrong. The idea is that these three arts stand for themselves. So when you listen to the music, only concentrate on the music, forget the mathematics. That might be easy. If you listen to mathematics, forget the music. That's probably not so easy. And then there will be paintings, the paintings are combined with the mathematics. So there you are allowed to think at the mathematics. The paintings will actually be cartoons; you will see them.

So concentrate on the music, concentrate on the mathematics, and then the paintings come,concentrate on both paintings and mathematics.

I promised to say one minute, I still have ten seconds. I have here a wonderful watch, but I don't use this 10secondsI play instead more music. I play a Sonata by AntonioVivaldi, the Italian composer.He's famous for one piece Four Seasons. I don't play this. I play a Sonata for Cello, uh, the Sonata number three, two movements.

So for me, that was the hard part, the mathematics part is easy for me.For you, the mathematics part might be a little bit harder, but I make it easy for you. If you think at Germans, you probably think thatwe sit in a beer garden, twenty-four hours a day and drink beer. We like that very much. Not twenty-four hours.

And on this painting, you see me in such a beer garden. And I ordered a beer, the beer comes.I’m very glad that I have the beer.And now the table is wobbling and I’m extremely angry. Then all beer is poured out. I hate that. 

So I complain to the owner of the place.I throw the table and then I say, “What table do you have?”And he says, “No, no, no. The table is completely alright. The problem is the ground on which the table stand. The ground is not flat, and that's why the table wobbles.”Ok, I excuse him. It's not the table, it's the ground and the beer garden.

Of course, the ground cannot be completely flat. So we all have experienced such a situation and we know what to do. So I wonder what to do. And next video, a little boy comes and tells me,“Very simple. Namely, take a sheet of paper, put it under the leg and the table is okay.”But you know that all only okay for a while. Because of the paper,after a while, it's a little bit compressed and unstablity again.

Mathematicians hate unstability. They don't accept that, and you should also hate it. And I show you something in the future. When you have a wobbling table,you will always apply and you will have never a wobbling table again.You will think this is not possible. And the answer is very simple and can all the time applied.

Namely, we do a quarter turn of the table. We turn it up to a quarter, we turn it forward and backwards.And you see, now we watch at some moment.Now it's completely stable. Try that out at home or wherever you are. If you have a wobbling table, turn it up to a quarter and it will be completely fixed.

And this is not by chance. It's a mathematics talk. It's not a beer garden talk although It's hard to define what is a beer garden talk. It's a mathematics talk. And there is a mathematical proof that this works. And this proof we want to work out together.And for this, we first have to translate.And that's an important part. Also in mathematical research.The problem of reality into a mathematical problem, we have to make a so called mathematical model.

And that we do now together.And we look at the next video We enumerate the four legs, one, two, three, four,and we fix leg two, three, and four on the ground.So that one leg is above the ground.

Yeah, if it would not wobble, then all four would be on the ground.So that wobble means you see leg one.is above the ground.

And that's why it wobbles, and now comes an important step.

We should see the next slide.So you see now the points on the ground where the four legs are, leg two, three, and four are fixed on the ground.And leg one is above the ground.If I push back leg one down, then leg three is above the ground.So it wobbles.

And now please next slide.So we keep the height of the leg one, we measure that.And that's a certain number, say, one centimeter.And this number is positive because of it’s one centimeter above the ground.

And now we turn the table.And while we turn it we keep leg two, three, and four on the floor.two, three, and four on the floor, and we turn it and keep these three on the floor.What will happen with leg one? For a while, it will be above the floor.But if we are there now, we fix these three legs, before this was above the ground.If we make this below the ground, then this will be under the ground.

And you'll see that in the next slide.Now, the position of leg one after the quarter turn.That's. I admit this is something very difficult.And if you don't understand it, ask me after the lecture again.So this is now under the ground.So we turn in the beginning above the ground, we keep these three on the ground. Now these three are on the ground.This means this is under the ground.So that's very important information.And with this in mind, we can now make our mathematical model.

So see the next video here.We see that's a very nice thing.You see how leg one goes under the ground.Can you see can we show this video again? That would be nice to see this important step again.So now we translate into mathematics, you know, in mathematics, you know, from school,we draw certain curves and study them.

And we draw now such a curve.Namely, we measure the height of leg one at every time while we turn,and we turn in time. At the beginning of the time is zero, at the end the time is one.So at time zero, it's above.And at time one, it is below.And now we measure the height at any moment.And we get a curve. Here.

You see now I draw the curve,this measure, measure measure measure, at this time measure measure measure.It goes up again, goes down, and at the end of the time, it's below.I hope you'll see what I did.I take the table, measure the height of this while I turn.This gives such a curve.And now, we are very close to an interesting mathematical theorem.

So we look at the next slide.First we do it again.It's good.We see leg two, three, four, always zero on the ground, but leg one goes up and down.Ok, now comes the next slide with a mathematical theorem.In mathematics, we have theorems.And it's a very important theorem called Intermediate Value Theorem.And students learn it in the first year with all details and proofs.Students take very long to learn it.You learn it in five minutes from me.You are better than the students.

So we have such a continuous function.This is such a curve.You have a curve with f at times zero, greater than zero, f at time one smaller than zero.Then there is some time in the middle where it is zero.And that's in the case of the table, the point where the table is stable because of all others have height zero.And now also leg four has height zero, nothing wobles.

So this important theorem, you could say, I believe it.Now we prove it.And now we come to the proof, and we do the proof by hunting.So imagine you have a field and there is some animal in the field.You want to hunt it.What do you do?We are not in the U.S, we don’t shoot it.No, no, no. Shooting is not allowed.We half the field in two halves.And we look where the animal is.We forget the other half.We half that again.We look where the animal is.Half it, make a fence and so on.And the end, we have the animal.This is how we hunt.

And we hunt here by dividing our time into half.Look where the table height is at half.And then there are two possibilities.You see at time zero, it was positive.At time one it was negative Here at time one half, it is negative.Then we throw this half away, where it has the same sign there.They are both are negative.That should be thrown away because of we hunt, the zero on the left side, this zero we want to hunt.So next slide, we show again the same curve.Now you see the curve on the right.I have thrown away.So this is like our animal we want to hunt.Now, the zero is between zero and one half.

Ok.We do the same again, we half again, this time, the zero is on the right side, we throw the left side away and remain with this part and next slide, we do it again.And you see if you do it again and again and again,the interval will be smaller and smaller.And of course, we perhaps have to it infinitely many times.That's something you learn in mathematics.What infinity is, that's tricky,but for you, no problem.We will see the infinity, we get closer and closer.And in the end, we hunt the zero.

And of course, that this hunting works,in the theorem it is said that the curve has to be continuous.And continuous means the curve can be drawn by a pen without a jump.If you have jumps, then of course, you can go from plus to minus without hitting the zero.So continuous is roughly draw without jumps.Mathematically, the tricky definition is precisely that.What we do here, that you can hunt the zero.

So I'm very proud. I'm very proud because of I managed to solve the problem.You'll see the peoples carry me away, but I hate that because of I want to drink the beer first.So I drink the beer and I disappearand.

I hope you have learned two things.First of all, mathematicians can make jokes.Very important, very important thing.Secondly, mathematicians can solve real problems, like the wobbling table.Yeah, we are sometimes useful,certainly most of the time we work theoretical.The translation into mathematical problem is for us the beginning of thinking,

if the problem comes from physics, chemistry,whatever,or inner mathematics.Most time, I'm a theoretical mathematician.I sit at the table, and when somebody comes in, the person says he's not working at all.Because I just sit and people think I sleep.But I think, and to prove that I think is that,from time to time, I publish a paper in the famous journal.So that's the way mathematicians work.

And you should take at home, mathematicians make jokes.They can solve the real problem, like the wobbling table.And they can really make proofs, which is something normal people don't know any more.You have seen a proof.And so I thank you extremely much for your audience after so many hours of listening. I am deeply impressed by the Chinese audience.

Xie Xie.

So at the end there is, of course, music again.And I count on you.I want to perform some music together with you.I will play now two songs, which I hope most of you know, and you sing with me, please.

So this was an English song.The next song is a German song.I'm sorry, but there exists some not so bad German composers, as you know, and I play one of them, you sing with me.

Thank you.I have one minute and forty-one seconds left, and I use them to thank you for the most wonderful concert I played in my life, to play with such an educated Chinese crowd, who knows my German colleague Beethoven. This is very impressive for me.

Do you know the text of this? 

Freude, schöner Götterfunken, 

Tochter aus Elysium.

very strange, but it is the song of joy and the song of freedomand I think in this world we need both, and peace. Thank you again.

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