Mathematical Creation

By Henri Poincaré

The genesis of mathematical creation is a problem which should intensely interest the psychologist. It is the activity in which the human mind seems to take least from the outside world, in which it acts or seems to act only of itself and on itself, so that in studying the procedure of geometric thought we may hope to reach what is most essential in man’s mind…

A first fact should surprise us, or rather would surprise us if we were not so used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds; if its evidence is based on principles common to all men, and that none could deny without being mad, how does it come about that so many persons are here refractory?

That not every one can invent is nowise mysterious. That not every one can retain a demonstration once learned may also pass. But that not every one can understand mathematical reasoning when explained appears very surprising when we think of it. And yet those who can follow this reasoning only with difficulty are in the majority; that is undeniable, and will surely not be gainsaid by the experience of secondary-school teachers.

And further: how is error possible in mathematics? A sane mind should not be guilty of a logical fallacy, and yet there are very fine minds who do not trip in brief reasoning such as occurs in the ordinary doings of life, and who are incapable of following or repeating without error the mathematical demonstrations which are longer, but which after all are only an accumulation of brief reasoning wholly analogous to those they make so easily. Need we add that mathematicians themselves are not infallible?…

As for myself, I must confess, I am absolutely incapable even of adding without mistakes… My memory is not bad, but it would be insufficient to make me a good chess-player. Why then does it not fail me in a difficult piece of mathematical reasoning where most chess-players would lose themselves? Evidently because it is guided by the general march of the reasoning. A mathematical demonstration is not a simple juxtaposition of syllogisms, it is syllogisms placed in a certain order, and the order in which these elements are placed is much more important than the elements themselves. If I have the feeling, the intuition, so to speak, of this order, so as to perceive at a glance the reasoning as a whole, I need no longer fear lest I forget one of the elements, for each of them will take its allotted place in the array, and that without any effort of memory on my part.

We know that this feeling, this intuition of mathematical order, that makes us divine hidden harmonies and relations, cannot be possessed by every one. Some will not have either this delicate feeling so difficult to define, or a strength of memory and attention beyond the ordinary, and then they will be absolutely incapable of understanding higher mathematics. Such are the majority. Others will have this feeling only in a slight degree, but they will be gifted with an uncommon memory and a great power of attention. They will learn by heart the details one after another; they can understand mathematics and sometimes make applications, but they cannot create. Others, finally, will possess in a less or greater degree the special intuition referred to, and then not only can they understand mathematics even if their memory is nothing extraordinary, but they may become creators and try to invent with more or less success according as this intuition is more or less developed in them.

In fact, what is mathematical creation? It does not consist in making new combinations with mathematical entities already known. Anyone could do that, but the combinations so made would be infinite in number and most of them absolutely without interest. To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice.

It is time to penetrate deeper and to see what goes on in the very soul of the mathematician. For this, I believe, I can do best by recalling memories of my own. But I shall limit myself to telling how I wrote my first memoir on Fuchsian functions. I beg the reader’s pardon; I am about to use some technical expressions, but they need not frighten him, for he is not obliged to understand them. I shall say, for example, that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances.

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.

Then I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and I succeeded without difficulty in forming the series I have called theta-Fuchsian.

Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure.

Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.

Returned to Caen, I meditated on this result and deduced the consequences. The example of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series; I saw that I could apply to them the theory of theta-Fuchsian series and that consequently there existed Fuchsian functions other than those from the hypergeometric series, the ones I then knew. Naturally I set myself to form all these functions. I made a systematic attack upon them and carried all the outworks, one after another. There was one, however, that still held out, whose fall would involve that of the whole place. But all my efforts only served at first the better to show me the difficulty, which indeed was something. All this work was perfectly conscious.

Thereupon I left for Mont-Valérien, where I was to go through my military service; so I was very differently occupied. One day, going along the street, the solution of the difficulty which had stopped me suddenly appeared to me. I did not try to go deep into it immediately, and only after my service did I again take up the question. I had all the elements and had only to arrange them and put them together. So I wrote out my final memoir at a single stroke and without difficulty.

I shall limit myself to this single example; it is useless to multiply them…

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind…

There is another remark to be made about the conditions of this unconscious work; it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations (and the examples already cited prove this) never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray. These efforts then have not been as sterile as one thinks; they have set agoing the unconscious machine and without them it would not have moved and would have produced nothing…

Such are the realities; now for the thoughts they force upon us. The unconscious, or, as we say, the subliminal self plays an important role in mathematical creation; this follows from what we have said. But usually the subliminal self is considered as purely automatic. Now we have seen that mathematical work is not simply mechanical, that it could not be done by a machine, however perfect. It is not merely a question of applying rules, of making the most combinations possible according to certain fixed laws. The combinations so obtained would be exceedingly numerous, useless and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones or rather to avoid the trouble of making them, and the rules which must guide this choice are extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated. Under these conditions, how imagine a sieve capable of applying them mechanically?

A first hypothesis now presents itself; the subliminal self is in no way inferior to the conscious self; it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine. What do I say? It knows better how to divine than the conscious self, since it succeeds where that has failed. In a word, is not the subliminal self superior to the conscious self? You recognize the full importance of this question…

Is this affirmative answer forced upon us by the facts I have just given? I confess that, for my part, I should hate to accept it. Re-examine the facts then and see if they are not compatible with another explanation.

It is certain that the combinations which present themselves to the mind in a sort of sudden illumination, after an unconscious working somewhat prolonged, are generally useful and fertile combinations, which seem the result of a first impression. Does it follow that the subliminal self, having divined by a delicate intuition that these combinations would be useful, has formed only these, or has it rather formed many others which were lacking in interest and have remained unconscious?

In this second way of looking at it, all the combinations would be formed in consequence of the automatism of the subliminal self, but only the interesting ones would break into the domain of consciousness. And this is still very mysterious. What is the cause that, among the thousand products of our unconscious activity, some are called to pass the threshold, while others remain below? Is it a simple chance which confers this privilege? Evidently not; among all the stimuli of our senses, for example, only the most intense fix our attention, unless it has been drawn to them by other causes. More generally the privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility.

It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility.

Now, what are the mathematic entities to which we attribute this character of beauty and elegance, and which are capable of developing in us a sort of esthetic emotion? They are those whose elements are harmoniously disposed so that the mind without effort can embrace their totality while realizing the details. This harmony is at once a satisfaction of our esthetic needs and an aid to the mind, sustaining and guiding. And at the same time, in putting under our eyes a well-ordered whole, it makes us foresee a mathematical law… Thus it is this special esthetic sensibility which plays the role of the delicate sieve of which I spoke, and that sufficiently explains why the one lacking it will never be a real creator.

Yet all the difficulties have not disappeared. The conscious self is narrowly limited, and as for the subliminal self we know not its limitations, and this is why we are not too reluctant in supposing that it has been able in a short time to make more different combinations than the whole life of a conscious being could encompass. Yet these limitations exist. Is it likely that it is able to form all the possible combinations, whose number would frighten the imagination? Nevertheless that would seem necessary, because if it produces only a small part of these combinations, and if it makes them at random, there would be small chance that the good, the one we should choose, would be found among them.

Perhaps we ought to seek the explanation in that preliminary period of conscious work which always precedes all fruitful unconscious labor. Permit me a rough comparison. Figure the future elements of our combinations as something like the hooked atoms of Epicurus. During the complete repose of the mind, these atoms are motionless, they are, so to speak, hooked to the wall…

On the other hand, during a period of apparent rest and unconscious work, certain of them are detached from the wall and put in motion. They flash in every direction through the space (I was about to say the room) where they are enclosed, as would, for example, a swarm of gnats or, if you prefer a more learned comparison, like the molecules of gas in the kinematic theory of gases. Then their mutual impacts may produce new combinations.

What is the role of the preliminary conscious work? It is evidently to mobilize certain of these atoms, to unhook them from the wall and put them in swing. We think we have done no good, because we have moved these elements a thousand different ways in seeking to assemble them, and have found no satisfactory aggregate. But, after this shaking up imposed upon them by our will, these atoms do not return to their primitive rest. They freely continue their dance.

Now, our will did not choose them at random; it pursued a perfectly determined aim. The mobilized atoms are therefore not any atoms whatsoever; they are those from which we might reasonably expect the desired solution. Then the mobilized atoms undergo impacts which make them enter into combinations among themselves or with other atoms at rest which they struck against in their course. Again I beg pardon, my comparison is very rough, but I scarcely know how otherwise to make my thought understood.

However it may be, the only combinations that have a chance of forming are those where at least one of the elements is one of those atoms freely chosen by our will. Now, it is evidently among these that is found what I called the good combination. Perhaps this is a way of lessening the paradoxical in the original hypothesis…

I shall make a last remark: when above I made certain personal observations, I spoke of a night of excitement when I worked in spite of myself. Such cases are frequent, and it is not necessary that the abnormal cerebral activity be caused by a physical excitant as in that I mentioned. It seems, in such cases, that one is present at his own unconscious work, made partially perceptible to the over-excited consciousness, yet without having changed its nature. Then we vaguely comprehend what distinguishes the two mechanisms or, if you wish, the working methods of the two egos. And the psychologic observations I have been able thus to make seem to me to confirm in their general outlines the views I have given.

Surely they have need of [confirmation], for they are and remain in spite of all very hypothetical: the interest of the questions is so great that I do not repent of having submitted them to the reader.


Clipped on 12-April-2011, 1:21 AM from 复杂理论收集

2011-03-24 20:53:36 来自: Jack

隐藏的逻辑 http://www.douban.com/subject/3766438/

1. 复杂的两个核心问题: 系统对于初始条件的敏感和反馈。





2. 社会原子,人的心理和群体行为是两回事,分析群体行为更应该把人作为分子或者原子。

互动模式,人类并没有传统社会学认为的那样“特殊”,人不过是自然界的一部分,是人类社会的“社会原子”;而人类社会之所以纷繁复杂并不是因为 “人”本身复杂,而是因为人们的“互动模式”的千变万化。“互动模式”,大概就是包括布坎南先生在内的很多社会物理学研究者眼里的人类社会之物理学法则。


同一性,当人们自由自在的时候,他们会相互模仿… 一个赋予个体无限自由的社会,往往会达到令人惶恐的同一性。


3. 模式分析。

3.1 要思考的是模式,不是人,大规模的模式形成和个体的特征无多大关系。

3.2 我们对于辨别模式,适应不断变化的世界,具有超级好的能力。

3.3 要关注重要的细节,忽略不重要的细节,市场往往在可预测性和不可预测性边缘徘徊。



从牛顿达尔文到巴菲特投资的格栅理论 http://www.douban.com/subject/1022739/

5. 复杂适应系统,每一个复杂系统其实都是由许多平行作用且相互影响的独立个体组成的网络。一个系统成为复杂系统的关键因素,是系统中的个体能够在于其他个体的相互作用中积累经验,在适应的环境中变化自己。

7. 经济特性,圣塔菲认为经济的四大特点是:松散的相互作用,不存在称霸世界之王,不断的适应过程和动态不平衡。

8. 反馈,复杂适应系统的基本元素是反馈,系统中的个体首先形成自己的期望或者模型,根据这些模型计算出的预测来行动。

10. 自组织,自组织系统有三大明显特征。(1)复杂的全球性行为是由简单的当地处理者组成的。(2)各种个人意见的贡献构成了解决方法。(3)系统强大的功能远比任何一个独立处理者要大得多。

12. 集体决定,只有系统中的个体都往单一集体选择方向积累信息时才可能产生集体决定。为了达到这个共同决定,所有的个体并不必拥有相同的信息,但他们必须对不 同机会的理解相同。这个共同理解在所有复杂系统的稳定性中起着关键的作用。系统的共识程度越低,其不稳定性越大。



成败就在刹那间 http://www.douban.com/subject/3346153/

1. 认知的进化,认知的进化是建立在装满了“本能”的适应性工具箱的基础之上的。适应性工具箱包括三层:进化能力,利用进化能力构建的积木块,由积木块构成的经验法则。

2. 推测,无论是感知,信念还是欺骗,我们大部分的直觉行为都可以被某种已经适应于我们周围世界的简单机制所描述和形容,也就是推测。我们大脑正是通过对世界的推测来帮助我们,如果没有推测,我们虽然能够看到细节,却毫无结构。

3. 感知系统,一个良好的感觉系统必须深入到所给予的信息表层的背后,它必须“创造”一些东西。你的大脑看到的东西,远比你的眼睛看到的多。

4. 自由选择的两难困境,拥有的选择机会越多,其内心冲突的可能性就越大,对照比较的困难程度也越高。过多的选择/产品和意见,反而会损害商家和顾客双方。

5. 简单,在一个不确定的世界中,简单法则对于复杂现象的预测效果高于复杂法则能达到的程度。

7. 模仿,经验法则让我们可以以一种对环境敏感的方式进行模仿。如果你的周围世界变化很慢,那么就进行模仿,否则就根据自身的经验进行学习。(或者模仿那些比你更聪明,对新环境更适应更快更迅速的人)

8. 简单,简单是对不确定性的适应。在一个不确定的环境当中,良好的直觉必须忽略掉一些信息。

9. 人类理性行为的剪刀,构成两侧刀片的,是任务所在的环境结构和行动者的计算能力(Herbert Simon)

10. 逻辑与直觉的冲突,逻辑准则对我们的文化视而不见,忽略了我们进化的能力和周遭的环境结构,那些从纯逻辑角度通常被看作是推理错误的感知,事后被证明反而是在现实中更高智慧的社会判断。


链接网络新科学 http://www.douban.com/subject/2149971/

1. 引爆点,来自连接者,连接者(带有极大量链接的节点)是引爆流行的真正原因。

2. 中心节点,在大部分复杂网络中,都存在中心节点,他们的广泛存在,成为这个广泛联系的世界的基本特性。中心节点是网络中的基本组成部分,它们起的作用非常重要,保证了网络的高可靠性,也使网络呈现小世界的特点。

3. 钟形分布与幕率分布,感觉这个和黑天鹅那本书很类似,钟形就是平均斯坦,幕率分布就是极端斯坦。

4. 符合钟形分布的是随机网络,比如公路图,符合幕率分布的是无尺度网络,比如飞机航线图。

5. 钟形分布同时意味着无序状态,幕率分布同时意味着有序状态。

幂律分布的最突出特点,不仅是其中有许多小事件,而且是许多小事件,而且是许多小事件伴随着少数极大的事件。这种超乎寻常的大事件是不可能存在于 钟形曲线内的。注意,在这分布图的末端,幂律分布和钟形分布也存在重要的性质差异,钟形曲线末端呈指数递减,递减速度比幂律曲线大。出现这种呈指数级递减 的末端,原因在于钟形曲线上缺乏中心节点。相比之下,幂律曲线递减速度较慢,允许罕见事件如中心节点的存在。

中心结点、80/20、幂率分布,说的都是同一件事情,然而长尾的流行和Niche Market的日益重要告诉我们:要重视中心结点,但不要忽视普通节点

6. 相变,相变意味着从无序到有序。从无序过渡到有序的关键:节点由无规则分布到社会化分布


1965年,leo kadanoff突然意识到:在临界点附近,我们就不能再把原子当成独立的粒子看待,而应该把它们看作是属于一个个社区,共同行动的群体。可以把原子看做是装在一个个盒子里,每个盒子里的原子都有同样的行为方式。

kenneth wilson的重正化理论证明了每当无序变成有序的临界点,即由混沌到有序的临界点的时候都会发现幂律的存在,他给相变理论的金字塔添上了顶端的最后一块 石头,并于1982年获得诺贝尔物理学奖。一旦系统被迫发生相变,一切随之改变,继而出现幂律。相变理论表明了从混沌到有序的过程受到自组织的影响,这是 幂律造成的。幂律的存在,将复杂网络从er模型的随机性的丛林里拯救出来,将其放在色彩斑斓的,充满了丰富理论营养的自组织的舞台的中心。

7. 无尺度网络的两个特征:增长、优先情结。造就了富者愈富;而合理利用适应性,是后来者打破这一法则的关键

8. 适应性概念:后发先至,在呈现适者致富的网络中,竞争会导致无尺度拓扑结构的产生。




非理性市场与蜥蜴式大脑思维 http://www.douban.com/subject/3099576/


情绪,. 一项有趣的测试,第一, 所有的商人—即使是经验最丰富的人—对市场信息都反应出明显的情绪波动。 第二, 经验丰富者的情绪化反应比经验不足的人要弱一些。






协同学 http://www.douban.com/subject/1013152/

8. 在各种领域中结构的发展有着相同的规律:某种有序状态不断增长,直到最后它占据优势并支配一个系统的所有部分,迫使其他部分进入这种有序状态。常常是一种不可预见的涨落时两个等价的有序状态间作出了最终选择。

5. 外部条件的改变将会使系统的某种确定状态变得不稳定,并且能够为一种新的、有时甚至完全不同的状态所代替,系统的不同部分,将被迫进入新的状态,他们将受到序参数的支配。

6. 条件的微小变动也可能导致完全不同的状态。

7. 流行的舆论起着序参数的作用,它支配着个人的意见,形成一种大体上是一致的舆论,借以维持其自身的存在。


沙地上的图案——计算机、复杂性和生命 http://www.douban.com/subject/1838594/


科学的两条规律 一、试图解释一切的理论(万能理论)往往什么都解释不了。二、观察事物的新方法、特别是这些方法能够将表面看起来不同的事物联系起来时,通常会导致对自然界更深刻的见解。

9. 线性系统倾向于消除微小变化,而非线性系统倾向于放大该变化…变化的加强是因为简单个体间存在的无数的相互作用而形成的。

14. 生物系统是高度适应性的,但他们也具有可观的冗余度…..如果我们没有完全理解复杂系统的动力学,我们就无法把事情真正做好。


隐秩序:适应性造就复杂性 http://www.douban.com/subject/1071936/


复杂性、风险与金融市场 http://www.douban.com/subject/1192910/


复杂系统具有局部杂乱无序和整体结构有序的特性。整体的结构维持整体的聚合力,局部的杂乱无序导致创新和活力。在自由市场经济中,竞争是局部杂乱 无序的源泉,而规则是维持整体结构有序的保证。因此,竞争需要高度的不确定性,这与我们在现实中的感受是一致的。企图减少不确定性的做法只能破坏自由市场 经济的特性。


这两大研究领域的结合就可以体现自由市场经济的精髓,特别是,我们将看到不确定性对自由市场的存在是多么必要。事实上,所有既需要变化又需要稳定 性的系统都需要不确定性。自由市场需要稳定,以便使人们对经济保持信心,与此同时,自由市场也需要一定的涨落。保持自由市场基本活力的机制是竞争,而竞争 需要不确定性

任何复杂系统都具有一定的功能或目的。这种功能是一种状态,而不是一个结果。整个生态系统通过稳定的能量转换来维持有机体的生存。自由市场经济通 过想赚钱的人们之间的商品和服务贸易来促进经济的发展。复杂系统追求的目标不是一种静态‘平衡’,相反,它追求的是一个不断变化,不断创新,同时还要保持 相对稳定的动态演化状态。至于它是怎样达到这种状态的,其详细过程并非一成不变。因为复杂系统到达某一稳定状态并不需要特定顺序的事件发生,所以,尽管环 境中存在着一些无法预料的变化,系统也总是能够加以克服,并能够创造性地调整自身以实现其目标。系统能够从不确定性中产生秩序,但这种秩序的‘独特’特征 是不可预知的。复杂系统在将不确定性转化为秩序的同时,也在产生着更多的不确定性

复杂系统是随着时间不断演化的,虽然它们也难以预测,但是他们需要依据过去来理解现在,未来的可能性取决于人们过去的选择,而适合我们的选择在很 大程度上依赖于我们过去所做出的决定。尽管复杂系统具有‘路径依赖’性,但它们并不具有‘前定’性。事实上,复杂系统的一个明显特征就是它能够通过多种手 段来实现同一目标,如果气候、土壤等条件具备,橡树种子总是能够长成橡树。

复杂系统的局部具有不确定性,但其整体却具有确定性。复杂系统不断地变化,并且能够带来一些意想不到的惊奇。它们将不确定性转化为秩序,又将秩序 转化为不确定性。复杂系统随时间不断地演化和变化着,其背后并不存在什么中心策划者。复杂系统随处可见,事实上,现实生活本身就是一个巨大的复杂系统





“当考虑一个多因素的过程,例如气候系统,一个小的影响后来能引起一个大的反应。这类混沌过程在短期内通常能够进行预测,因为短期内的相互关系近 似是线性的。……所以,这是一种非线性的关系,在近处,几乎是线性的;可是,离开当前位置越远,非线性就越明显。因此,一个非线性的动力系统(通常被称为 混沌系统)在短期内是可以预测的。但在长期内不能加以预测。



“另一种典型的非线性相关称做非线性随机相关。假设大多数的‘无序’过程均服从正态分布(或钟形曲线),这是错误的。存在一些随机过程,不服从该 假设,它们的概率分布是完全不同的。尽管很难描述,一个非线性随机过程的概率分布变化是有规律的。但它们不会收敛到一些通常的分布族,例如正态分布。


“复杂过程,我们已经谈到,其特征是整体的结构性和局部的随机性。这些特征能够转变成高的长期可预测性和低的短期可预测性。整体结构性允许我们预 测除了细部之外的典型的特征。……复杂性让我们能够预测长期的结果,但是近期的结果总是未知的和不可能知道的。如果复杂过程在近期内是可以预测的,那么它 将不再是复杂的过程。


http://www.douban.com/note/87408954/ 崩落的沙粒——思考自组织的临界性


By 郭凯












甲 乙 丙

a   b   c

c   a   b

b   c   a

孔多赛悖论就是上面这个例子,假设有甲乙丙三个人,甲认为a比c好,c比b好。乙和丙的偏好如图。他就说,如果在a和b之间投票选一个最好的,那么乙和丙都会选b,所以b赢。也就是b>a.要是在b和c之间投票,甲和丙都会投c,所以c赢,于是c>b.但是要是在a和c之间投票,甲和乙都会投a,于是a>c.所以我们就有了a>c, c>b和b>a.如果a,b,c是三个候选人,那么谁能选出来当总统,就完全取决于投票的顺序,任何人都可能当总统。用阿罗的语言就是,多数投票的结果不满足传递性。


下面再说量化这件事情吧。如果形式逻辑这种东西,未必一定需要数学才能实现。量化这件事,要是没有数学(统计),那根本就是完全不可能了。拿我最近正在看的一篇论文举例吧,这是Bernard,Eaton,Jenson和Kortum的一篇American Economic Review的论文,这篇论文的内容很丰富,但是其中一个问题是:如果世界上所有的贸易壁垒一夜之间全部消失,那么对美国会有什么影响?你可能会想,这样也许会有更多的中国商品进入美国,导致更多的美国人失业。你也可以想,也许会有更多的美国商品进入中国,所以导致更多的美国人就业。但是,究竟是增加就业还是减少就业,是多少?BEJK用他们的模型给出了一个回答。他们用的是三步走的办法,他们先估计模型中的重要参数,第二步是用这个模型去预测现在的贸易数字,在得到了比较精确的预测之后,他们就把模型中贸易壁垒的参数调为零,然后看对美国的影响。他们也许是对的,当然也很有可能是错的。但是,这不是重点。重点是,只有运用了数学,我们才有可能对这样一个问题进行估计,不用数学的估计至多只能是拍脑袋。




  一个英国某大学的数学教授发现自己家的下水道堵了,就请来一个水管工来修。30分钟后,水管疏通了。教授相当满意水管工的表现,但当他看到账单后不禁大叫:“what!就30分钟你收的钱够我一个月收入的1/3了!我去当水管工好了!”。水管工说,“你可以去啊。我们公司正招人呢,还包培训。不过你得说你只是小学毕业。公司不喜欢学历太高的人”。于是教授就去参加培训,当了水管工。他的收入一下翻了三倍。他比以前高兴多了。几年后,公司突然决定把水管工们的文化水平提高到初中毕业,便要求旗下的工人们都去上夜校。夜校的第一堂课是数学。老师想先看一下这些水管工的基础有多好,于是他随便抽了一个人上来写圆面积的公式。这个教授被抽中了,不过干了这么多年水管工,他已经忘了圆面积的公式是PI * R^2。于是他只好从头推导:把圆无限分割后积分。但他得出的结果是负的PI * R^2。尴尬ing,教授从来又来,结果还是负的。他非常尴尬,于是回过头向教室里坐着的几十个水管工同事求助。只见这些同事正在交头接耳,纷纷给他说:把积分上下限交换一下。


  数学家、生物学家和物理学家坐在街头咖啡屋里, 看着人们从街对面的一间房子走进走出.他们先看到两个人进去. 时光流逝. 他们又看到三个人出来.





  工程师、化学家和数学家住在一家老客栈的三个相邻房间里. 当晚先是工程师的咖啡机着了火, 他嗅到烟味醒来, 拔出咖啡机的电插头, 将之扔出窗外,然后接着睡觉.

  过一会儿化学家也嗅到烟味醒来, 他发现原来是烟头燃着了垃圾桶. 他自言自语道:“怎样灭火呢? 应该把燃料温度降低到燃点以下, 把燃烧物与氧气隔离. 浇水可以同时做到这两点.” 于是他把垃圾桶拖进浴室, 打开水龙头浇灭了火, 就回去接着睡觉.

  数学家在窗外看到了这一切, 所以, 当过了一会儿他发现他的烟灰燃着了床单时, 他可一点儿也不担心. 说:“嗨, 解是存在的!”就接着睡觉了.








  物理学家和工程师乘着热气球,在大峡谷中迷失了方向。他们高声呼救:“喂——!我们在哪儿?”过了大约15分钟,他们听到回应在山谷中回荡: “喂——!你们在热气球里!”物理学家道:“那家伙一定是个数学家。”工程师不解道:“为什么?”物理学家道:“因为他用了很长的时间,给出一个完全正确的答案,但答案一点用也没有。”


  常函数和指数函数e的x次方走在街上,远远看到微分算子,常函数吓得慌忙躲藏,说:“被它微分一下,我就什么都没有啦!”指数函数不慌不忙道: “它可不能把我怎么样,我是e的x次方!”指数函数与微分算子相遇。指数函数自我介绍道:“你好,我是e的x次方。”微分算子道:“你好,我是 d/dy!”