How art can be good [zz]

December 2006

I grew up believing that taste is just a matter of personal preference. Each person has things they like, but no one’s preferences are any better than anyone else’s. There is no such thing as good taste.

Like a lot of things I grew up believing, this turns out to be false, and I’m going to try to explain why.

One problem with saying there’s no such thing as good taste is that it also means there’s no such thing as good art. If there were good art, then people who liked it would have better taste than people who didn’t. So if you discard taste, you also have to discard the idea of art being good, and artists being good at making it.

It was pulling on that thread that unravelled my childhood faith in relativism. When you’re trying to make things, taste becomes a practical matter. You have to decide what to do next. Would it make the painting better if I changed that part? If there’s no such thing as better, it doesn’t matter what you do. In fact, it doesn’t matter if you paint at all. You could just go out and buy a ready-made blank canvas. If there’s no such thing as good, that would be just as great an achievement as the ceiling of the Sistine Chapel. Less laborious, certainly, but if you can achieve the same level of performance with less effort, surely that’s more impressive, not less.

Yet that doesn’t seem quite right, does it?


I think the key to this puzzle is to remember that art has an audience. Art has a purpose, which is to interest its audience. Good art (like good anything) is art that achieves its purpose particularly well. The meaning of “interest” can vary. Some works of art are meant to shock, and others to please; some are meant to jump out at you, and others to sit quietly in the background. But all art has to work on an audience, and—here’s the critical point—members of the audience share things in common.

For example, nearly all humans find human faces engaging. It seems to be wired into us. Babies can recognize faces practically from birth. In fact, faces seem to have co-evolved with our interest in them; the face is the body’s billboard. So all other things being equal, a painting with faces in it will interest people more than one without. [1]

One reason it’s easy to believe that taste is merely personal preference is that, if it isn’t, how do you pick out the people with better taste? There are billions of people, each with their own opinion; on what grounds can you prefer one to another? [2]

But if audiences have a lot in common, you’re not in a position of having to choose one out of a random set of individual biases, because the set isn’t random. All humans find faces engaging—practically by definition: face recognition is in our DNA. And so having a notion of good art, in the sense of art that does its job well, doesn’t require you to pick out a few individuals and label their opinions as correct. No matter who you pick, they’ll find faces engaging.

Of course, space aliens probably wouldn’t find human faces engaging. But there might be other things they shared in common with us. The most likely source of examples is math. I expect space aliens would agree with us most of the time about which of two proofs was better. Erdos thought so. He called a maximally elegant proof one out of God’s book, and presumably God’s book is universal. [3]

Once you start talking about audiences, you don’t have to argue simply that there are or aren’t standards of taste. Instead tastes are a series of concentric rings, like ripples in a pond. There are some things that will appeal to you and your friends, others that will appeal to most people your age, others that will appeal to most humans, and perhaps others that would appeal to most sentient beings (whatever that means).

The picture is slightly more complicated than that, because in the middle of the pond there are overlapping sets of ripples. For example, there might be things that appealed particularly to men, or to people from a certain culture.

If good art is art that interests its audience, then when you talk about art being good, you also have to say for what audience. So is it meaningless to talk about art simply being good or bad? No, because one audience is the set of all possible humans. I think that’s the audience people are implicitly talking about when they say a work of art is good: they mean it would engage any human.[4]

And that is a meaningful test, because although, like any everyday concept, “human” is fuzzy around the edges, there are a lot of things practically all humans have in common. In addition to our interest in faces, there’s something special about primary colors for nearly all of us, because it’s an artifact of the way our eyes work. Most humans will also find images of 3D objects engaging, because that also seems to be built into our visual perception. [5]And beneath that there’s edge-finding, which makes images with definite shapes more engaging than mere blur.

Humans have a lot more in common than this, of course. My goal is not to compile a complete list, just to show that there’s some solid ground here. People’s preferences aren’t random. So an artist working on a painting and trying to decide whether to change some part of it doesn’t have to think “Why bother? I might as well flip a coin.” Instead he can ask “What would make the painting more interesting to people?” And the reason you can’t equal Michelangelo by going out and buying a blank canvas is that the ceiling of the Sistine Chapel is more interesting to people.

A lot of philosophers have had a hard time believing it was possible for there to be objective standards for art. It seemed obvious that beauty, for example, was something that happened in the head of the observer, not something that was a property of objects. It was thus “subjective” rather than “objective.” But in fact if you narrow the definition of beauty to something that works a certain way on humans, and you observe how much humans have in common, it turns out to be a property of objects after all. You don’t have to choose between something being a property of the subject or the object if subjects all react similarly. Being good art is thus a property of objects as much as, say, being toxic to humans is: it’s good art if it consistently affects humans in a certain way.


So could we figure out what the best art is by taking a vote? After all, if appealing to humans is the test, we should be able to just ask them, right?

Well, not quite. For products of nature that might work. I’d be willing to eat the apple the world’s population had voted most delicious, and I’d probably be willing to visit the beach they voted most beautiful, but having to look at the painting they voted the best would be a crapshoot.

Man-made stuff is different. For one thing, artists, unlike apple trees, often deliberately try to trick us. Some tricks are quite subtle. For example, any work of art sets expectations by its level of finish. You don’t expect photographic accuracy in something that looks like a quick sketch. So one widely used trick, especially among illustrators, is to intentionally make a painting or drawing look like it was done faster than it was. The average person looks at it and thinks: how amazingly skillful. It’s like saying something clever in a conversation as if you’d thought of it on the spur of the moment, when in fact you’d worked it out the day before.

Another much less subtle influence is brand. If you go to see the Mona Lisa, you’ll probably be disappointed, because it’s hidden behind a thick glass wall and surrounded by a frenzied crowd taking pictures of themselves in front of it. At best you can see it the way you see a friend across the room at a crowded party. The Louvre might as well replace it with copy; no one would be able to tell. And yet the Mona Lisa is a small, dark painting. If you found people who’d never seen an image of it and sent them to a museum in which it was hanging among other paintings with a tag labelling it as a portrait by an unknown fifteenth century artist, most would walk by without giving it a second look.

For the average person, brand dominates all other factors in the judgement of art. Seeing a painting they recognize from reproductions is so overwhelming that their response to it as a painting is drowned out.

And then of course there are the tricks people play on themselves. Most adults looking at art worry that if they don’t like what they’re supposed to, they’ll be thought uncultured. This doesn’t just affect what they claim to like; they actually make themselves like things they’re supposed to.

That’s why you can’t just take a vote. Though appeal to people is a meaningful test, in practice you can’t measure it, just as you can’t find north using a compass with a magnet sitting next to it. There are sources of error so powerful that if you take a vote, all you’re measuring is the error.

We can, however, approach our goal from another direction, by using ourselves as guinea pigs. You’re human. If you want to know what the basic human reaction to a piece of art would be, you can at least approach that by getting rid of the sources of error in your own judgements.

For example, while anyone’s reaction to a famous painting will be warped at first by its fame, there are ways to decrease its effects. One is to come back to the painting over and over. After a few days the fame wears off, and you can start to see it as a painting. Another is to stand close. A painting familiar from reproductions looks more familiar from ten feet away; close in you see details that get lost in reproductions, and which you’re therefore seeing for the first time.

There are two main kinds of error that get in the way of seeing a work of art: biases you bring from your own circumstances, and tricks played by the artist. Tricks are straightforward to correct for. Merely being aware of them usually prevents them from working. For example, when I was ten I used to be very impressed by airbrushed lettering that looked like shiny metal. But once you study how it’s done, you see that it’s a pretty cheesy trick—one of the sort that relies on pushing a few visual buttons really hard to temporarily overwhelm the viewer. It’s like trying to convince someone by shouting at them.

The way not to be vulnerable to tricks is to explicitly seek out and catalog them. When you notice a whiff of dishonesty coming from some kind of art, stop and figure out what’s going on. When someone is obviously pandering to an audience that’s easily fooled, whether it’s someone making shiny stuff to impress ten year olds, or someone making conspicuously avant-garde stuff to impress would-be intellectuals, learn how they do it. Once you’ve seen enough examples of specific types of tricks, you start to become a connoisseur of trickery in general, just as professional magicians are.

What counts as a trick? Roughly, it’s something done with contempt for the audience. For example, the guys designing Ferraris in the 1950s were probably designing cars that they themselves admired. Whereas I suspect over at General Motors the marketing people are telling the designers, “Most people who buy SUVs do it to seem manly, not to drive off-road. So don’t worry about the suspension; just make that sucker as big and tough-looking as you can.” [6]

I think with some effort you can make yourself nearly immune to tricks. It’s harder to escape the influence of your own circumstances, but you can at least move in that direction. The way to do it is to travel widely, in both time and space. If you go and see all the different kinds of things people like in other cultures, and learn about all the different things people have liked in the past, you’ll probably find it changes what you like. I doubt you could ever make yourself into a completely universal person, if only because you can only travel in one direction in time. But if you find a work of art that would appeal equally to your friends, to people in Nepal, and to the ancient Greeks, you’re probably onto something.

My main point here is not how to have good taste, but that there can even be such a thing. And I think I’ve shown that. There is such a thing as good art. It’s art that interests its human audience, and since humans have a lot in common, what interests them is not random. Since there’s such a thing as good art, there’s also such a thing as good taste, which is the ability to recognize it.

If we were talking about the taste of apples, I’d agree that taste is just personal preference. Some people like certain kinds of apples and others like other kinds, but how can you say that one is right and the other wrong? [7]

The thing is, art isn’t apples. Art is man-made. It comes with a lot of cultural baggage, and in addition the people who make it often try to trick us. Most people’s judgement of art is dominated by these extraneous factors; they’re like someone trying to judge the taste of apples in a dish made of equal parts apples and jalapeno peppers. All they’re tasting is the peppers. So it turns out you can pick out some people and say that they have better taste than others: they’re the ones who actually taste art like apples.

Or to put it more prosaically, they’re the people who (a) are hard to trick, and (b) don’t just like whatever they grew up with. If you could find people who’d eliminated all such influences on their judgement, you’d probably still see variation in what they liked. But because humans have so much in common, you’d also find they agreed on a lot. They’d nearly all prefer the ceiling of the Sistine Chapel to a blank canvas.

Making It

I wrote this essay because I was tired of hearing “taste is subjective” and wanted to kill it once and for all. Anyone who makes things knows intuitively that’s not true. When you’re trying to make art, the temptation to be lazy is as great as in any other kind of work. Of course it matters to do a good job. And yet you can see how great a hold “taste is subjective” has even in the art world by how nervous it makes people to talk about art being good or bad. Those whose jobs require them to judge art, like curators, mostly resort to euphemisms like “significant” or “important” or (getting dangerously close) “realized.” [8]

I don’t have any illusions that being able to talk about art being good or bad will cause the people who talk about it to have anything more useful to say. Indeed, one of the reasons “taste is subjective” found such a receptive audience is that, historically, the things people have said about good taste have generally been such nonsense.

It’s not for the people who talk about art that I want to free the idea of good art, but for those who make it. Right now, ambitious kids going to art school run smack into a brick wall. They arrive hoping one day to be as good as the famous artists they’ve seen in books, and the first thing they learn is that the concept of good has been retired. Instead everyone is just supposed to explore their own personal vision. [9]

When I was in art school, we were looking one day at a slide of some great fifteenth century painting, and one of the students asked “Why don’t artists paint like that now?” The room suddenly got quiet. Though rarely asked out loud, this question lurks uncomfortably in the back of every art student’s mind. It was as if someone had brought up the topic of lung cancer in a meeting within Philip Morris.

“Well,” the professor replied, “we’re interested in different questions now.” He was a pretty nice guy, but at the time I couldn’t help wishing I could send him back to fifteenth century Florence to explain in person to Leonardo & Co. how we had moved beyond their early, limited concept of art. Just imagine that conversation.

In fact, one of the reasons artists in fifteenth century Florence made such great things was that they believed you could make great things[10] They were intensely competitive and were always trying to outdo one another, like mathematicians or physicists today—maybe like anyone who has ever done anything really well.

The idea that you could make great things was not just a useful illusion. They were actually right. So the most important consequence of realizing there can be good art is that it frees artists to try to make it. To the ambitious kids arriving at art school this year hoping one day to make great things, I say: don’t believe it when they tell you this is a naive and outdated ambition. There is such a thing as good art, and if you try to make it, there are people who will notice.


[1] This is not to say, of course, that good paintings must have faces in them, just that everyone’s visual piano has that key on it. There are situations in which you want to avoid faces, precisely because they attract so much attention. But you can see how universally faces work by their prevalence in advertising.

[2] The other reason it’s easy to believe is that it makes people feel good. To a kid, this idea is crack. In every other respect they’re constantly being told that they have a lot to learn. But in this they’re perfect. Their opinion carries the same weight as any adult’s. You should probably question anything you believed as a kid that you’d want to believe this much.

[3] It’s conceivable that the elegance of proofs is quantifiable, in the sense that there may be some formal measure that turns out to coincide with mathematicians’ judgements. Perhaps it would be worth trying to make a formal language for proofs in which those considered more elegant consistently came out shorter (perhaps after being macroexpanded or compiled).

[4] Maybe it would be possible to make art that would appeal to space aliens, but I’m not going to get into that because (a) it’s too hard to answer, and (b) I’m satisfied if I can establish that good art is a meaningful idea for human audiences.

[5] If early abstract paintings seem more interesting than later ones, it may be because the first abstract painters were trained to paint from life, and their hands thus tended to make the kind of gestures you use in representing physical things. In effect they were saying “scaramara” instead of “uebfgbsb.”

[6] It’s a bit more complicated, because sometimes artists unconsciously use tricks by imitating art that does.

[7] I phrased this in terms of the taste of apples because if people can see the apples, they can be fooled. When I was a kid most apples were a variety called Red Delicious that had been bred to look appealing in stores, but which didn’t taste very good.

[8] To be fair, curators are in a difficult position. If they’re dealing with recent art, they have to include things in shows that they think are bad. That’s because the test for what gets included in shows is basically the market price, and for recent art that is largely determined by successful businessmen and their wives. So it’s not always intellectual dishonesty that makes curators and dealers use neutral-sounding language.

[9] What happens in practice is that everyone gets really good attalking about art. As the art itself gets more random, the effort that would have gone into the work goes instead into the intellectual sounding theory behind it. “My work represents an exploration of gender and sexuality in an urban context,” etc. Different people win at that game.

[10] There were several other reasons, including that Florence was then the richest and most sophisticated city in the world, and that they lived in a time before photography had (a) killed portraiture as a source of income and (b) made brand the dominant factor in the sale of art.

Incidentally, I’m not saying that good art = fifteenth century European art. I’m not saying we should make what they made, but that we should work like they worked. There are fields now in which many people work with the same energy and honesty that fifteenth century artists did, but art is not one of them.

Thanks to Trevor Blackwell, Jessica Livingston, and Robert Morris for reading drafts of this, and to Paul Watson for permission to use the image at the top.

【zz】Birds & Frogs

17世纪初,两位伟大的哲学家,英国的弗兰西斯•培根(Francis Bacon)和法国的勒奈•笛卡尔(Rene Descartes),正式宣告了现代科学的诞生。笛卡尔是一只鸟,培根是一只青蛙。两人分别描述了对未来的远景,但观点大相径庭。培根说:“一切均基于眼睛所见自然之确凿事实。”笛卡尔说:“我思,故我在。”





我是一个培根学派的信徒。对我而言,布尔巴基纲领的一个主要不足是错失了一种惊喜元素。布尔巴基纲领努力让数学更有逻辑。当我回顾数学的历史时,我看见不断有非逻辑的跳跃、难以置信的巧合和自然的玩笑。大自然所开的最深刻玩笑之一是负1的平方根,1926年,物理学家埃尔文•薛定谔(Erwin Schrodinger)在发明波动力学时,将这个数放入他的波动方程。


结果,薛定谔方程准确描述了我们今天所知原子的每一种行为。这是整个化学和绝大部分物理学的基础。负1的平方根意味着大自然是以复数而不是实数的方式运行。这一发现让薛定谔和其他所有人耳目一新。薛定谔记得,当时,他14岁大的“女朋友”伊萨•荣格尔(Itha Junger)曾对他说:“嗨,开始时,你从来没想过会出现这么多有意义的结果吧?”


大自然所开的第二个玩笑是量子力学的精确线性。事实上,物理对象的各种可能状态构成了一个线性空间。在量子力学被发明之前,经典物理总是非线性的,线性模式只是近似有效。在量子力学之后,大自然本身突然变成了线性。这对数学产生了深刻的影响。19世纪,索菲斯•李(Sophus Lie)发展了他关于连续群的精致理论(elaborate theory),以期弄清楚经典力学系统的行为。当时的数学家和物理学家对李群几乎没有任何兴趣。李群的非线性理论对数学家来说过于复杂,对物理学家来说又过于晦涩。索菲斯•李在失望中离开了人世。50年后,人们发现大自然本身就是线性的,李代数的线性表示竟然是粒子物理的自然语言。作为20世纪数学的中心主题之一,李群和李代数获得了新生。

大自然的第三个玩笑是拟晶体(Quasi-crystals)的存在。19世纪,对晶体的研究导致了对欧几里德空间中可能存在的离散对称群种类的完整列举。人们已经证明:在三维欧几里德空间中,所有离散对称群仅包含3级、4级或6级的旋转。之后,1984年,拟晶体被发现了,从液体金属阵列中长出的真正固体物显示了包含5重旋转的二十面体的对称性。与此同时,数学家罗杰•彭罗斯(Roger Penrose)发现了平面“彭罗斯拼砖法”。拟晶阵列是二维彭罗斯拼砖法的三维模拟。在这些发现之后,数学家不得不扩大晶体群理论,将合金拟晶体包含其中。这是还在发展中的一个重要研究项目。

大自然开的第四个玩笑是拟晶和黎曼ζ函數零点(zeros of the Riemann Zeta function)在行为的相似性。黎曼ζ函數零点令数学家们着迷,因为所有的零点都落在一条直线上,没有人知道这是为什么。著名的黎曼猜想是指:除了平凡的例外,黎曼ζ函数零点都在一条直线上。100多年来,证明黎曼猜想一直是年轻数学家们的梦想。我现在大胆提议:也许可以用拟晶体来证明黎曼猜想。你们中的部分数学家也许认为这个建议无关紧要。那些不是数学家的人可能对这个建议不感兴趣。然而,我将这个问题放到你们面前,希望你们严肃思考。年轻时的物理学家里奥•齐拉特(Leo Szilard)不满意摩西的十条诫命,写了新十诫来替换它们。齐拉特的第二条诫律说:“行动起来,向有价值的目标前进,不问这些目标是否能达到:行动是模范和例子,而不是终结。” 齐拉特践行了他的理论。他是第一个想象出核武器的物理学家,也是第一个积极以行动反对核武器使用的物理学家。他的第二条诫律也适用于这里。黎明猜想的证明是一个值得为之的目标,我们不应该问这个目标是否能实现。我将给你们一些这个目标可以实现的暗示。我将给数学家们一些建议,这是我在50年前成为一名物理学家之前获得的忠告。我先谈黎明猜想,再谈拟晶体。

直到最近,纯数学领域还有两个未解决的超级问题:费马大定理的证明和黎曼猜想的证明。12年前,我在普林斯顿的同事安德鲁•怀尔斯(Andrew Wiles)证明了费马大定理,如今,只剩下黎曼猜想有待证明。怀尔斯对费马大定理的证明不只是一个技术绝技,它的证明还需要发现和探索数学思想的新领域,这比费马大定理本身更辽阔更重要。正因如此,对黎曼猜想的证明也将导致对数学甚至物理学诸多不同领域的深刻认识。黎曼ζ函數和其他ζ函數也类似,它们在数论、动力系统、几何学、函数论和物理学中普遍存在。ζ函數仿佛是通向各方路径的交叉结合点。对黎曼猜想的证明将阐明所有这些关联。就像每一位纯数学领域里严肃的学生一样,我年轻时的梦想是证明黎曼猜想。我有一些模糊不清的想法,认为可以引导自己证明这个猜想。最近几年,在拟晶体被发现后,我的想法不再模糊。我在这里把它们呈现给有雄心壮志赢得菲尔茨奖的年轻数学家们。


将普通晶体排除在外,三维中的拟晶体只有极为有限的变形,它们均与二十面体有关。二维拟晶体数目众多,粗略地讲,一个独特的类型与平面上每个正多边形都相关联。含五边 形对称的二维拟晶体是著名的平面彭罗斯拼砖。最后,一维拟晶体有更为丰富的结构,因为它们不受制于任何旋转对称。就我所知,目前还没有对一维拟晶体存在情况的全数调查。现已知,一种独特拟晶体的存在与每个皮索特-维贡伊拉卡文数(pisot Vijayaraghavan number)或PV数对应。一个PV数是一个真正的代数整数,是有整数系数(integer coefficients)多项式方程的根,其他所有根的绝对值都有小于1的绝对值。全部PV数的集合是无限的,并有非凡的拓扑结构。所有一维拟晶体的集合都有一种结构,其丰富程度可与所有的PV数集合相比,甚至更丰富。我们并不确切地知道,一个由与PV数没有关联的一维拟晶体构成的大世界正等待探索。

现在谈一维准晶体与黎曼猜想的联系。如果黎曼猜想是正确的,那么根据定义,ζ函數零点就会形成一个一维拟晶体。它们在一条直线上构成了点质量(point masses)的一个分布,它们的傅利叶变化同样也是一个点质量分布,前者的点质量位于每个素数的对数处,其傅里叶变换点质量位于每个素数的幂的对数处。我的朋友安德鲁•奥德泽科(Andrew Odlyzko)发表了一个漂亮的ζ函數零点的傅利叶变换的计算机运算。这个运算精确地显示了傅利叶变换的预期结构,在每一个素数或素数的幂的对数上有明显的间断性。



1941年,我作为一名学生来到英国剑桥大学,极其幸运地受教于俄罗斯数学家艾伯拉姆•萨莫罗维奇•伯西柯维奇(Abram Samoilovich Besicovitch)。时值第二次世界大战,剑桥只有很少的学生,几乎没有研究生。尽管当时我只有17岁,而伯西柯维奇已是一位著名教授,但是,他给了我相当多的时间和关注,我们成为终身朋友。在我开始从事和思考数学时,他塑造了我的性格。他在测量理论和积分方面上了许多精彩的课程,在我们因他大胆地滥用英语而哈哈大笑时,他只是亲切地笑笑。我记得仅有一次,他被我们之间的玩笑惹怒。在沉默了一会后,他说:“先生们,有5000万英国人讲你们所讲的英文。有1.5亿俄罗斯人讲我所讲的英文。”

伯西柯维奇是一只青蛙,年轻时,因解决一个名为挂谷问题(Kakeya Problem)的初等本平面几何问题而出名。挂谷问题是这样描述的:让一条长度为1的线段按360度的角度在一个平面上自由转动,这条线扫过的最小面积是多少?日本数学家挂谷宗一(Soichi Kakeya)在1917年提出这个问题,并成为之后十年内未解决的著名问题。当时,美国数学界领袖乔治•伯克霍夫(George Birkhoff)公开声称,挂谷问题和四色问题是最著名的未解决问题。数学家们普遍相信,最小的面积应该是π/8,即棒在三尖点内摆线的面积(three-cusped hypocycloid)。三尖点内摆线是一条优美的三尖点曲线,它是一个半径为四分之一的小圆圈在一个半径为四分之三的定圆内滑动时,动圆圆周上的一个点所绘制的轨迹。长度为1的线段在旋转时始终与内摆线相切,它的两端也在内摆线上。一条线段在旋转时与内摆线的三个点相切,这是一幅多么优美的画,绝大多数人相信它一定给出了最小面积。然后,伯西柯维奇给了大家一个惊喜:他证明,对任何正∈(positive ∈)来说,这一线段在旋转时所扫过的面积小于∈。

实际上,在挂谷问题成为著名问题之前,伯西柯维奇已经在1920年解决了这个问题,但在当时,伯西柯维奇本人甚至不知道挂谷提出了这个问题。1920年,他将解决方案用俄文发表在《彼尔姆物理和数学学会期刊》(Journal of the Perm Physics and Mathematics Society)上,这是一份不被广泛阅读的期刊。彼尔姆大学位于距离莫斯科东面1100公里的彼尔姆城,在俄罗斯革命之后,这个城市成为许多著名数学家的短暂避难所。他们出版了两期《彼尔姆物理和数学学会期刊》,之后,期刊便在革命和内战的混乱中停刊了。在俄罗斯之外,这份期刊不仅不为人知,而且不可获取。1925年,伯西柯维奇离开俄罗斯,来到哥本哈根,并在这里获知到他已经在5年前解决的著名挂谷问题。他将解决方案重新出版,这一次,论文用英文发表在德国著名的《数学期刊》(Mathematische Zeitschrift)上。正如伯西柯维奇所说,挂谷问题是一个典型的青蛙问题,一个与数学的其它方面没有太多联系的具体问题。伯西柯维奇给出了一个优雅、深刻的解决方案,揭示出它与平面中点集结构的一般定理之间的联系。

伯西柯维奇的风格体现在他的三篇最好的经典文章中,这些文章的标题是:“平面点集之线性可测量的基本几何性质”(On the fundamental geometric properties),它们分别发表在1928年、1938年和1939年的《数学年鉴》(Mathematische Annalen)上。在这些论文中,他证明:平面上的每个线性可测量集可被分解为有规则和无规则的分支,规则分支在每个地方几乎都有一个切线,而无规律分支都有一个零测量投射向几乎所有方向。简而言之,规则分支看起来像连续曲线,而无规则分支看起来不像连续曲线。无规则分支的存在和性质与挂谷问题的伯西柯维奇解有联系。他给我的工作之一是,在高维空间中将可测量集分为规则分支组件和无规则分支。虽然我在这个问题上一事无成,却永远被烙上了伯西柯维奇风格。伯西柯维奇风格是建筑学风格。他用简单元素建造出精美、复杂的建筑结构,通常情况下有层次计划;当大厦建成时,通过简单的论证就可从完整结构中推导出意外的结论。伯西柯维奇的每项工作都是一件艺术品,像巴赫的赋格曲一样精心构成。
在跟随伯西柯维奇做了几年的学生后,我来到美国普林斯顿,认识了赫尔曼•外尔(Hermann Weyl)。外尔是一只典型的鸟,正如伯西柯维奇是一只典型的青蛙。幸运的是,在外尔退休回到位于苏黎世的老家之前,我在普林斯顿高等研究所与他有一年的相处时间。他喜欢我,因为在这一年间,我在《数学年鉴》(Annals of Mathematics)上发表了有关数论的论文,在《物理评论》(Physics Review)上发表了量子辐射理论的论文。他是当时活在世上的少数几位同时精通这两领域的专家之一。他欢迎我到普林斯顿研究所,希望我像他一样成为一只鸟。他失望了,我始终是一只固执的青蛙。尽管我总是在各种各样的泥洞附近闲逛,我一次只能关注一个问题,没有寻找问题之间的联系。对我而言,数论和量子理论是拥有各自美丽的两个世界。我不像外尔一样去发现构建大设计的线索。




外尔逝世后的五十年是实验物理和观察天文学的黄金时代,也培根学派旅行者收集事实、青蛙们在我们生存的小片沼泽地上探索的黄金时代。在这50年中,青蛙们积累了大量的有关宇宙结构、众多粒子和其间相互作用的详尽知识。在持续探索新领域的同时,宇宙变得越来越复杂。不再是展现外尔数学简洁和美丽的大设计 ,探索者发现了夸克和伽玛射线爆等奇异事件,以及超对称和多重宇宙等新奇概念。与此同时,在持续探索混沌和许多被电子计算机打开的新领域时,数学在变得越来越复杂。数学家发现了可计算性的中心谜团,这个猜想表示为P不等于NP。这个猜想声称:存在这样的数学问题,它的个案可以被很快解决,但没有适用于所有情形的快速算法可解决所有问题。这个问题中最著名的例子是旅行销售员问题,即在知道每两个城市之间距离的前提下,寻找这位销售员在这一系列城市间旅行的最短路径。所有的专家都相信这是猜想是正确的,旅行销售员的问题是P不等于NP的实际问题。但没有人知道证明这一问题的一点线索。在赫尔曼•外尔19世纪的数学世界中,这个谜团甚至还没有形成。

对鸟们来说,最近五十年是艰难时光。然而,即使在艰难时代,也有事情等着鸟们去做,他们勇敢地去解决这些事情。在赫尔曼•外尔离开普林斯顿后不久,杨振宁(Frank Yang)从芝加哥来到普林斯顿,搬进了外尔的旧居,在我这一代的物理学家中,他接替外尔的位置成为一只领头鸟。在外尔还活着时,杨振宁和他的学生罗伯特•米尔斯(Robert Mills)发现了非阿贝尔规范场(non-Abelian gauge fields)的杨—米尔斯理论,这是外尔规范场思想的一个漂亮外推。外尔的规范场是一个经典数量,满足了乘法交换定律。杨-米尔斯理论有一个不交换的三重规范场(triplet of gauge fields)。它们满足量子力学自旋三分量的交换法则,这是最简单的非阿贝尔躺代数A2(non-abelian lie algebra A2)的生成子。这个理论后来如此普遍,以至规范场论成为任何有限元李代数的生成子。有了这种普遍性,杨—米尔斯规范场理论为所有已知粒子和其相互作用提供了一个模型框架,这个模型就是今天粒子物理学的标准模型。通过证明爱因斯坦的重力场论适合于同样的框架,以克里斯托夫三指标符号规取代范场的作用,杨振宁为这个理论上写下点睛之笔。


我深深敬重的另一只鸟是俄罗斯数学家尤里•曼宁(Yuri Manin),他最近出版了一本名为《数学如隐喻》(Mathematics as Metaphor)的随笔。这本书以俄文在莫斯科出版,美国数学协会将之译为英文出版。我为英文版书作序。在这里,我简单引用我的序言:“对鸟们来说,《数学如隐喻》是一个好口号。它意味着数学中最深刻的概念是将一个世界的思想与另一个世界的思想联系起来。在17世纪,笛卡尔用他的坐标概念将彼此不相干的代数学和几何学联系起来;牛顿用他的流数(fluxions)概念将几何学和力学的世界联系起,今天,我们将这种方法称为微积分学。19世纪,布尔(Boole)用他的符号逻辑(symbolic logic)概念将逻辑与代数联系起来;黎曼用他的黎曼曲面概念将几何和分析的世界联系起来。坐标、流数、符号逻辑和黎曼曲面,都是隐喻,将词的意义从熟悉的语境拓展到陌生的语境。曼宁将数学的未来看成是对可见但仍不可知的隐喻的一个探索。最深刻的一个隐喻是数论和物理学之间在结构上的相似性。在这两个领域中,他看到并行概念诱人的一暼,对称性将连续与离散联结起来。他期待一种名为数学量化(quantization of mathematics)的统一。”



约翰•冯•诺伊曼(John von Neumann)是20世纪数学中另一位重要人物。冯•诺伊曼是一只青蛙,他用自己惊人的技术技能解决了数学和物理学众多分支领域中的问题。从创立数学的基础开始,他发现了集合论的第一个令人满意的公理集,避免了康托(Cantor)在试图解决无穷集和无穷数时遇到的逻辑悖论。几年后,冯•诺伊曼的鸟类朋友库特•哥德尔(Kurt Godel)用他的公理集证明了数学中的不可判定性命题。


冯•诺伊曼从数学基础的奠定迈向了量子力学基础的奠定。为了给量子力学一个坚实的数学基础,他创立了一个宏大的算子环理论(theory of rings of operator)。每个可观察量都可以由一个线性算子来代表,量子行为的特殊性可由算术代数忠实地代表。正如牛顿发明了描述经典力学的微积分,冯•诺伊曼发明了描述量子力学的算子环理论。

冯•诺伊曼在几个领域做出了奠基性贡献,特别是从博弈论到数字计算机的设计。在他生命的最后十年里,他深深了陷到计算机里。他对计算机的兴趣如此强烈,以至决定不仅要研究它们的设计,而且还要用真正的硬件和软件构建一台可做科学研究的计算机。我对冯•诺伊曼在普林斯顿高等研究所的早期计算机有生动清晰的记忆。那时,他有两个主要的科学兴趣:氢弹和气象学。夜晚,他用计算机做氢弹问题,白天,则做气象学问题。白天,游荡在计算机大楼里的许多人都是气象学家,他们的领导是朱尔•查耐(Jule Charney)。查耐是一位真正的气象学家,妥善谦卑地讨论天气变幻莫测的神秘,怀疑计算机解决这个神秘的能力。我听过冯•诺伊曼以这个问题为主题的一次演讲。如往常一样,他充满自信地说:“计算机将使我们能够在任何时刻将大气划分为稳定域和不稳定域。我们可以预测稳定域,我们能够控制不稳定域。”





究竟发生了什么事?我知道所发生的事情,因为1954年9月2日,星期四,下午3:00,我正坐在阿姆斯特丹音乐厅的听众席上。大厅里挤满了数学家,所有人都期望在这样一个历史时刻聆听一个精彩绝伦的演讲。演讲结果却是令人非常失望。冯•诺伊曼可能在几年前就接受邀请做这样一个演讲,然后将之忘到九宵云外。诸事缠身,他忽略了准备演讲之事。然后,在最一刻,他想起来他将旅行到阿姆斯特丹,谈一些有关数学的事;他拉开一个抽屉,从中抽出一份20世纪30年代的老演讲稿,弹掉上面灰尘。 这是一个有关算子环的演讲,在30年代是一个全新、时髦的话题。没有谈任何未解决的问题,没有谈任何未来的问题。没有谈任何计算机,我们知道这是冯•诺伊曼心中最亲爱的话题,他至少应该谈一些有关计算机的新的、激动人心的事。音乐厅里的听众开始变得焦躁不安。有人用全音乐厅里的人都能听见的声音大声说:“Aufgewarmte suppe”,这是一句德国,意思是“先将汤加热(warmed-up soup)”。1954年,绝大多数数学家都懂德语,他们明白这句玩笑的意思。冯•诺伊曼陷入深深的尴尬,匆匆结束演讲,没有等待任何提问就离开了音乐厅。



在长期积分(long-term integration)做出来之前,人们从未想象过太阳系中的混沌行为,因为这种混沌是弱的。弱混沌意味着相邻轨道呈指数级离散,却不会离散得太远。这种离散开始时以指数级速度增长,但随后就维持在边界处。因为行星运动的离散是弱的,所以太阳系能在40亿多年的时光里得以生存。尽管这种运动是混沌的,但行星从来不会在远离它们所熟悉的地区漫游,因此,太阳系作为一个整体从来不曾分崩离析。尽管混沌无处不在,但拉普拉斯将太阳系当作像时钟运动一样完美的观点离事实并不遥远。


混沌的特征已被众多的数据和无止境的美丽图片所勾勒,但却缺少严格理论。严谨理论赋予一个课题以智力的深度和精确。在你能证明一个严格理论之前,你不可能全面理解你所关注的概念的意义。在混沌领域,我知道只有一个严格理论在1975年被李天岩(Tien-Yien Li)和吉姆• 约克(Jim Yorke)所证明,这篇短论文的题目是:《周期三蕴含混沌》(Period Three Implies Chaos)。李-约克论文是数学文献中不朽的珍宝。他们的理论将非线性地图的区间扩展至它本身。当被当作是一个经典粒子的轨道时,点位置的连续性就能重复。如果一个点在N次映像之后又回到它原始的位置,那么这个轨道就有N个周期。由此而论,如果一个轨道从所有的周期轨道中离散,那么这个轨道就被定义为混沌。这个理论表明,如果单个轨道拥有三个存在周期,那么混沌轨道就是存在的。这个证明简洁、短小。在我的印象里,这个理论和它的证明投向混沌基本特征的光芒胜过几千张美丽图片。它解释了混沌为什么在这个世界里普遍存在,但没有解释混沌为什么总是这样弱,这是留给未来的一个任务。我相信,在证明有关弱混沌的严谨定理之前,我们是不会从根本上理解弱混沌。


我想在弦理论上讲几句。只讲几句,是因为我对弦理论知之甚少。我从来没有劳心费神地学习这个理论,或自己花功夫去研究它。但是,当我在普林斯顿研究所有一个家时,我周围环绕着弦理论专家,我有时能听到他们之间的谈话。偶尔,我也能明白一点点他们谈话的内容。有三件事情是显而易见:第一,他们正在做第一流的数学,从而让迈克尔•阿蒂亚(Michael Atiyah)、伊萨多•辛格(Isadore Singer)这样的领袖级纯数学家也爱上弦理论,它开启了一个有新想法和新问题的全新数学分枝,最不寻常的是,它赋予数学一种解决老问题的新方法,这些老问题以前是不能解决的;第二,这些弦理论学家认为自己是物理学家而非数学家。他们相信自己的理论描述了物质世界的一些真实东西;第三,还没有任何证明显示这个理论与物理学相关。这个理论至今尚未被实验所证明。这个理论还在它自己的世界里,远离物理学。弦理论学家们付出艰苦努力,试图演绎这个可能在真实世界里被检验的理论的结果,但至今尚未成功。

我的同事爱德华•威腾(Ed Witten)、胡安•马尔达西那(Juan Maldacena)和其他创建弦理论的人,都是鸟,他们飞翔在高高的天空,俯览远隔千里的众山全貌。在世界各地的大学里,几千名在弦理论上埋头苦干的谦卑实践者是青蛙,他们探索那些鸟们在地平线上第一次看到的数学结构的细节。我对弦理论的忧虑是从社会学角度而不是科学角度。成为发现新联系和探求新方法的第一批几千名弦理论学家之一,这是一个光荣的事;但成为第二批或万名弦理论学家之一,则不是一件光荣的事。今天,世界各地分布着上万名弦理论学家。对第1万名或第2000名科学家来说,情形是危险的。不可预测事情可能会发生,比如形势变化,弦理论不再时髦。这样的事情也可能发生:9000名弦理论学家可能会失业。他们在一个狭窄的领域接受训练,在其它科学领域可能无法被聘用。


最后,我想谈谈我对弦理论未来的推测。我的推测可能是错的。我从来没有幻想过我能预测未来。我告诉你们我的推测,只是想给你们一些思考的问题。我认为,弦理论不可能完全成功或完全无用。所谓完全成功,我的意思是它是一种完全(完整?)的物理理论,解释了粒子和其间相互作用的所有细节。所谓完全的无用,我的意思是它保留了一种纯数学的美丽。我的推测是,弦理论将在完全成功与完全失败之间的某一处终结。我认为它应该类似于李群,这是索菲斯•李(Sophus Lie)在19世纪为经典物理创建的一个数学框架。所以,只要物理学保持其经典性,李群就是一个失败。它们是一个寻找问题的解决方案。但另一方面,五十年后,量子革命改变了物理学,李代数找到用武之地:成为认识量子世界对称性中心作用的关键。我期望今后五十年或一百年中,物理学的另一场革命会引入我们今天一无所知的新概念,这些新概念将赋予弦理论一种全新的意义。在此之后,弦理论会突然发现自己在宇宙中应有的位置,提出对真实世界可经测试的陈述。我警告你们:这个有关未来的猜测可能是错的,它本身具有证伪性的美德,(科学哲学大师)卡尔 波普尔(karl Popper)说,这正是科学命题的特点。 明天,它可能会被来自大型强子对撞机的新发现所推翻。




三十多年前,歌手莫尼克 莫瑞利(Monique Morelli)录制了一盘皮埃尔 迈克奥兰(Pierre Macorlan)作词的唱片。其中一首歌是《死城》(La ville Morte),萦绕于心的旋律切合着莫瑞利深沉的低音,随着歌声的对位,一个具有强烈冲击力的死城形象生动地出现了。歌声并没有特殊之处:






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