# 庞加莱 关于数学创造（翻译）

昨天刘未鹏在日志中提到庞加莱关于数学创造的文章，顺手译成了中文，如下：

Mathematical Creation

数学创造

How is mathematics made? What sort of brain is it that can compose the propositions and systems of mathematics? How do the mental processes of the geometer or algebraist compare with those of the musician, the poet, the painter, the chess player? In mathematical creation which are the key elements? Intuition? An exquisite sense of space and time? The precision of a calculating machine? A powerful memory? Formidable skill in following complex logical sequences? A supreme capacity for concentration?

数学是由什么构成的？哪种类型的大脑能够创造数学的定理和系统？几何学家或是代数学家的思维活动和音乐家、是人画家和象棋选手有什么不同？数学创造中，哪些是关键元素？直觉？空间和时间的精确感觉？机器一般的计算准确性？超强的记忆力？复杂逻辑推导的超强技能？极好的注意力？

The essay below, delivered in the first years of this century as a lecture before the Psychological Society in Paris, is the most celebrated of the attempts to describe what goes on in the mathematician's brain. Its author, Henri Poincaré, cousin of Raymond, the politician, was peculiarly fitted to undertake the task. One of the foremost mathematicians of all time, unrivaled as an analyst and mathematical physicist, Poincaré was known also as a brilliantly lucid expositor of the philosophy of science. These writings are of the first importance as professional treatises for scientists and are at the same time accessible, in large part, to the understanding of the thoughtful layman.

下面的文章，是本世纪（20世纪）初期在巴黎心理学会上做的一次报告，是描述关于数学家大脑如何运转的最著名的尝试。该报告的作者，亨利·庞加莱（是政治家雷蒙德的表亲），是承担该任务的极其适当的人选。作为历史上最重要的数学家之一，一位无可匹敌的分析和数学物理学家，庞加莱同样擅长对科学哲学作出清晰准确的阐释。这份报告对于科学家们而言是一极其重要的专业论述，同时在很大程度上，也可以为非专业人士所理解。

Poincaré on Mathematical Creation

庞加莱 关于数学创造

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...

数学创造的起源是一个引起心理学家强烈兴趣的问题。这种创造活动似乎是涉及外部世界最少的、大脑内部发生的活动，仅仅、或者看起来仅仅依赖于和作用于它自身，因此，研究几何思考的过程，我们也许希望能够深入到人类思维的最本质。

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?

一个简单事实可能会使我们惊讶，因为我们对此并不熟悉。为什么有人不能理解数学？如果数学仅仅唤起的是逻辑法则，那么正常的思维应该都能接受它；如果它基于所有人都了解的通常法则，除非神经错乱否则不能否认，为什么这么多人不能掌握数学呢？

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.

不是每个人都能创造数学，这一点毫不神奇。不是每个人学习了某个定理证明以后都能灵活运用它，这也毫不为怪。然而不是每个人都能够领会数学分析思维，即使反复解释也不能——仔细思考这一点就让人感到惊异了。而且能跟上分析思维的人，大多数也感到困难，这是不可否认的；相信每个当过中学教师的人都对此深有体验。

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 reasonings wholly analogous to those they make so easily. Need we add that mathematicians themselves are not infallible?...

此外：数学怎么会产生错误？理智的思维不会犯下逻辑错误，然而在日常生活中不会被简单分析所难倒的人，在重复较长的数学证明时往往难免出错；然而数学证明终归不过是一系列简短分析的集合而已，分开来看的话它们看似如此简易。我们是不是应该加一句，数学家们并不是不会犯错的？

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.

对于我自己，我承认，我甚至连做加法都不会不犯错。我的记忆力不坏，但也不足以让我成为一个好的象棋选手。为什么令大多数象棋选手感到吃力的数学题，却不会难倒我呢？显然是因为它是由普通的分析步骤构成的。数学证明并不是简单的演绎法的排列，它是由演绎法按特定顺序排列而成，而且排列顺序比元素本身更为重要。如果我对这个顺序产生某种感觉，或称直觉，只需要一眼就能感知到推理的整体，那么我不会担心我忘记其中的一个元素，因为每个元素都是特定方式放置在这个阵列中的，而不需要我用记忆去牢记。

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.

我们知道，这种感觉，这种数学顺序的直觉，带给我们隐含的和谐和关系的神圣感，并不能被每个人所掌握。有些人既缺乏这种难以被定义的精妙感觉，又缺乏超乎常人的记忆和专注的程度，以至完全不能把握高等数学。大多数人都是如此。有些人会产生少许这类感觉，但他们天生拥有与众不同的记忆力和注意力。他们会记住一个接一个的细节，他们能够理解数学，并且作出应用，但他们不能创造。其他人，最终或多或少地能够掌握一些这种直觉，这样他们不知能够理解数学，并且即使他们的记忆力并不超凡，也能变成创造者，并伴随着这种直觉的多少，获得相应程度的成功。

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.

事实上，什么是数学创造？它并不意味着对已知的数学事实重新组合。任何人都可以做到重新组合，但这种组合的数量是无限的，并且大多数毫无价值。创造，意味着不制造无用的组合，而仅制造那些少量且有用的。创造即鉴别，即选择。

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.

现在可以挖掘得更深一些，来看看数学家的灵魂深处是什么了。我相信，我可以通过回忆自己的经历来进行充分说明。不过我将仅限于讲述我写第一本关于Fuchsian函数论文集的情况。我请求听众原谅：我将会使用某些技术术语，但你们无需感到害怕，因为你们并不需要理解。比如，我会这么说，基于某种情况，我得到了这个定理的证明。这个定理可能会有一个奇怪的名字，大部分人会感到不习惯，但并不重要；你们心理学家感兴趣的并不是定理，而是情况本身。

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.

我挣扎了15天来证明不可能存在像后来我称之为Fuchsian函数那样的函数。当时我一无所知；每天我坐在书案前，工作一到两个小时，尝试大量的组合却一无所获。一个晚上，和我平日的习惯不同，我喝了黑咖啡，没有睡觉。大量的思绪汹涌，我感到它们相互碰撞直到契合，也就是说，慢慢地稳定下来。第二天早上之前，我已经建立好一类Fuchsian函数的存在性证明，这些函数来自于超几何序列；我只需要把结果写下来即可，前后花了不过几个小时。

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.

后来我想用两个序列的商来表达这些函数；想法很清晰而且经过深思熟虑，因为椭圆方程的相似性指引着我。我问自己，如果这些序列存在，它们会有怎样的属性，并且我毫无困难地成功构建了这些序列，后来我称之为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.

就是这时期，我离开了卡昂，这个我生活的地方，受我所在学院的资助进行一次旅行。旅途上的经历让我暂且忘记了数学工作。到达古特昂司后，我们上了一辆公共马车以去到另外的什么地方。突然，当我刚登上马车阶梯的刹那，一个想法来临，之前没有任何想法为之做准备，即我用来定义Fuchsian函数的变换实际上与非欧几何中的是等价的。我没有去证明这个想法；我几乎没有时间，因为一上马车我就参与了另一场已经开始的谈话，但我对自己的想法确信无疑。回到卡昂以后，我在良心催促下利用空余时间完成了证明。

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.

随后，我把注意力转回到一些算术问题的研究上，这些问题一直没有进展，而且显然和我之前的研究没有丝毫联系。吸取了之前失败的教训，我到海边去了几天，并且考虑了一些其他事情。一天早晨，正在悬崖边散步，我忽然有了主意，想法和上次一样具有同样简短、突然的性质，而且几乎立即就能肯定，算术问题中的三元不定二次型变换等价于非欧几何中的变换。

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.

回到卡昂，我对此结果进行思考，并得到了一些结论。二次型的例子表明存在Fuchsian群而无须使用超几何序列的方法；我意识到可以对此应用theta-Fuchsian序列的理论，并得出Fuchsian方程而无须引入超几何序列。很自然地，我着手构建这些方程。我系统性地研究他们并且得到一个接一个的结论。然而，有一个问题迟迟不能解决，并且结果很可能影响到全局。因此我的努力一开始似乎效果很好，却只是引入了更加困难的问题。这些工作都是在我有意识地状态下完成的。

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...

后来我去了瓦勒里昂山服兵役；因此我忙着完全不相干的事情。有一天，正沿街走着，难题的解突然出现在我面前。我当时并没有立即深入，直到服完兵役，我才重新拾起该问题。我有了所有的元素，只需要重新安置它们即可、因此我一次性就写完了最后的论文集，没有遇到任何困难。

我只用这一个例子，因为多说无益。

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...

一开始，最令人震惊的是福至心灵的瞬间，反映了长时间、无意识的预备工作。数学创造中，无意识工作的重要性对我而言是无可争议的，并且可以在其他例子中，无意识工作不那么明显的情况下找到它的痕迹。通常，一个人思考困难问题时，最初的工作不会带来太好的结果。休息一会儿，或者更长时间，然后重新坐下来工作。在开始半小时，像以前一样，没什么发现，然而突然决定性的想法呈现在思想面前。

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...

关于无意识工作的条件还有另一个说法：它是可能的，并且某种程度上富有成效，仅当之前和之后都进行了长期有意识的工作。这些突然的灵感（之前的例子已经表明了）发生，仅当之前大量的努力失败，并且没有好的结果发生，而采取的方法又是不确定的。这些努力并不像我们认为的毫无作用；它们启动了无意识的机器，否则这架机器永远不会运转起来，也不会有所成就。

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?

这就是现实；现在我们来分析一下数学家们的想法。无意识，或称之为，潜意识的自我在数学创造中占有重要地位；这和我们之前所说是一致的。然而通常潜意识的自我被认为是完全自发的。现在我们看到数学工作并不是简单的机械运算，一架无论如何精密的仪器是不能胜任数学工作的。它也不仅仅是应用规则的问题，不是根据某些固定法则做出最可能的组合。这样得到的组合可能数量中毒，但毫无用处、繁冗不堪。创造者的真实工作包括在这些组合中进行挑选以减少无用的，避免制造无用结论的麻烦，并且借以挑选的法则必须极其精致准确。几乎不可能准确陈述这些法则；我们更多地是感觉而不是规定。在这些条件下，怎能想象机械地来应用这些法则呢？

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...

现在我们可以得到第一个假设；潜意识的自我并不比意识层面上的自我表现要差；潜意识并不是自动的；它能够鉴别；它手法老练，技巧精湛；它知道如何挑选，它发掘。“发掘”的意思是，它知道如何超越意识层面的自我，因为它能完成有意识不能完成的事情。一句话而言，是否潜意识的自我比有意识的自我更好？至少，你现在应该了解这个问题的重要性。

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.

但，是否我刚才举的实例就能确定无误地说明问题呢？我承认，在我的方面，我不喜欢这个结论。让我们重新检查这个例子，看看能不能得到其他解释。

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?

可以确定的是，在潜意识进行一段时间的工作后，就会产生灵光一现的想法，并且通常是相当有用的结论组合，就像是第一印象得出的结论。这是由于潜意识的自我产生的微妙直觉，判断出哪些组合是有用的，因此只采用了这些组合吗？还是同样产生了很多其他组合，而由于其无用性而停留在潜意识中？

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.

用第二种角度来看，潜意识自动将所有组合产生，但只有最令人感兴趣的能突破意识的界限。这仍是非常神秘的。是什么愿意，使得我们的潜意识活动产生的千百种产品，只有少数能突破界限，而大多数停留在意识层面之下？仅仅是因为机会的原因，让它们得到了这些特权？显然不是；例如就我们的感觉而言，受到的所有的刺激，除非是特殊原因，只有那些最强烈的能够吸引我们的注意力。一般而言，那些比较特殊的潜意识活动，那些可能跃而成为意识层面的，是那些直接或间接能深刻影响我们情感认知的活动。

It may be surprising to see emotional sensibility invoked apropos 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.

看到情感认知能唤起适当的数学证明，也许令人惊异；因为我们本认为这仅仅关系到智力。那是因为我们忘了这其中存在对数学之美的感知，对数和形式的协调感，以及几何优雅性的体验。这是所有真正数学家应当了解的美感，并且显然属于情感认知的范畴。

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.

那么，我们所谓的数学美和优雅，并且可以引起我们的美的情感的主体究竟是什么？它们是元素的和谐放置，以让思想无需费力就能理感知整体和理解细节。这种和谐感也能满足我们的审美需求，并且帮助和引领头脑。同时，随着我们眼前出现一个美妙排序的整体，我们可以看到其中隐含的数学原理。这种特殊的审美就像是一个滤网，就像我说的那样，提炼出我们所需要的东西；这也充分解释了为什么缺乏这种审美的人永远不可能成为数学的创造者。

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.

然而困难未曾消失。意识的自我存在限制，但对于潜意识，我们并不知道限制所在，因此我们并不愿意假设潜意识在短期内能够完成的工作，比有意识在长期内能够进行的更多。然而限制是存在的。如果所有的组合都能够形成，其数量比想象的更多，这可能吗？然而这是必须的，因为如果它只生成少数的组合，而且是随机生成的，那么我们所能挑选的比较好的组合的几率就相对太小了。

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...

也许我们应当到早期的有意识的工作中寻求答案，以往内所有的无意识工作都必须由有意识的先行。请允许我做一个粗略的类比。假设我们潜意识作出的数学组合像是伊壁鸠鲁的原子论假说。在意识得到完全休息的阶段，这些原子是静止的，或者说，它们黏在了墙上。

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.

而另一方面，在表面上休息却是潜意识工作的阶段，某些原子从墙上脱离开来并进入运动。它们向空间的各个方向漂移，就像一小群昆虫一样，或者用更学术的比喻，就像气体运动理论中的气体分子那样。它们之间的互相影响可能形成新的组合。

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.

那么之前有意识工作起到怎样的铺垫呢？该工作推动了某些原子，将它们从墙上释放并进入运动。我们认为我们没有得到结果，因为我们把这些元素推向一千个不同的方向，却希望它们能够汇聚，显然得不到满意的结果。然而，在强加在它们之上的意愿结束之后，这些原子并不会停止。它们开始自由自在地运动。

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.

现在，我们的意愿不会随机挑选它们；它们按照自己的意愿行事。运动起来的原子们不再是普通原子；它们是那些我们可以期待成为解的那些原子。这些运动起来的原子产生冲击，让它们彼此之间产生结合，直到停止下来为止。这里我请听众们原谅，因为我的比喻相当粗疏，但我不知道有什么更好的比喻了。

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...

无论如何，最后可能的组合，至少包含了部分由我们的意志释放的原子。在这些原子中，产生了被我们称之为“好”的组合。也许，这是一种使得原始假设不那么荒谬的说法。

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.

最后我还要做出一些说明：上面这些我个人作出的观察中，描述的是我个人工作的感受。这种事例是很常见的，因此不需要考虑我之前提到的大脑产生某种不同寻常的兴奋。似乎在这些事例中，一个人面对他的无意识工作，部分感受到他自己过度兴奋的意识，却没有改变这种意识的本质。我们可以大概理解这两种工作机制的区别，或者说，两种自我不同的工作方法。我所做出的心理学观察似乎证实了之前我所给出的观点。

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.

这种观点还有待确认，因为它们仍然停留在高度假说的程度；但这个问题如此有趣，以至于这里我毫不犹豫，要将它们提供给我的听众们。

Mathematical Creation

数学创造

How is mathematics made? What sort of brain is it that can compose the propositions and systems of mathematics? How do the mental processes of the geometer or algebraist compare with those of the musician, the poet, the painter, the chess player? In mathematical creation which are the key elements? Intuition? An exquisite sense of space and time? The precision of a calculating machine? A powerful memory? Formidable skill in following complex logical sequences? A supreme capacity for concentration?

数学是由什么构成的？哪种类型的大脑能够创造数学的定理和系统？几何学家或是代数学家的思维活动和音乐家、是人画家和象棋选手有什么不同？数学创造中，哪些是关键元素？直觉？空间和时间的精确感觉？机器一般的计算准确性？超强的记忆力？复杂逻辑推导的超强技能？极好的注意力？

The essay below, delivered in the first years of this century as a lecture before the Psychological Society in Paris, is the most celebrated of the attempts to describe what goes on in the mathematician's brain. Its author, Henri Poincaré, cousin of Raymond, the politician, was peculiarly fitted to undertake the task. One of the foremost mathematicians of all time, unrivaled as an analyst and mathematical physicist, Poincaré was known also as a brilliantly lucid expositor of the philosophy of science. These writings are of the first importance as professional treatises for scientists and are at the same time accessible, in large part, to the understanding of the thoughtful layman.

下面的文章，是本世纪（20世纪）初期在巴黎心理学会上做的一次报告，是描述关于数学家大脑如何运转的最著名的尝试。该报告的作者，亨利·庞加莱（是政治家雷蒙德的表亲），是承担该任务的极其适当的人选。作为历史上最重要的数学家之一，一位无可匹敌的分析和数学物理学家，庞加莱同样擅长对科学哲学作出清晰准确的阐释。这份报告对于科学家们而言是一极其重要的专业论述，同时在很大程度上，也可以为非专业人士所理解。

Poincaré on Mathematical Creation

庞加莱 关于数学创造

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...

数学创造的起源是一个引起心理学家强烈兴趣的问题。这种创造活动似乎是涉及外部世界最少的、大脑内部发生的活动，仅仅、或者看起来仅仅依赖于和作用于它自身，因此，研究几何思考的过程，我们也许希望能够深入到人类思维的最本质。

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?

一个简单事实可能会使我们惊讶，因为我们对此并不熟悉。为什么有人不能理解数学？如果数学仅仅唤起的是逻辑法则，那么正常的思维应该都能接受它；如果它基于所有人都了解的通常法则，除非神经错乱否则不能否认，为什么这么多人不能掌握数学呢？

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.

不是每个人都能创造数学，这一点毫不神奇。不是每个人学习了某个定理证明以后都能灵活运用它，这也毫不为怪。然而不是每个人都能够领会数学分析思维，即使反复解释也不能——仔细思考这一点就让人感到惊异了。而且能跟上分析思维的人，大多数也感到困难，这是不可否认的；相信每个当过中学教师的人都对此深有体验。

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 reasonings wholly analogous to those they make so easily. Need we add that mathematicians themselves are not infallible?...

此外：数学怎么会产生错误？理智的思维不会犯下逻辑错误，然而在日常生活中不会被简单分析所难倒的人，在重复较长的数学证明时往往难免出错；然而数学证明终归不过是一系列简短分析的集合而已，分开来看的话它们看似如此简易。我们是不是应该加一句，数学家们并不是不会犯错的？

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.

对于我自己，我承认，我甚至连做加法都不会不犯错。我的记忆力不坏，但也不足以让我成为一个好的象棋选手。为什么令大多数象棋选手感到吃力的数学题，却不会难倒我呢？显然是因为它是由普通的分析步骤构成的。数学证明并不是简单的演绎法的排列，它是由演绎法按特定顺序排列而成，而且排列顺序比元素本身更为重要。如果我对这个顺序产生某种感觉，或称直觉，只需要一眼就能感知到推理的整体，那么我不会担心我忘记其中的一个元素，因为每个元素都是特定方式放置在这个阵列中的，而不需要我用记忆去牢记。

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.

我们知道，这种感觉，这种数学顺序的直觉，带给我们隐含的和谐和关系的神圣感，并不能被每个人所掌握。有些人既缺乏这种难以被定义的精妙感觉，又缺乏超乎常人的记忆和专注的程度，以至完全不能把握高等数学。大多数人都是如此。有些人会产生少许这类感觉，但他们天生拥有与众不同的记忆力和注意力。他们会记住一个接一个的细节，他们能够理解数学，并且作出应用，但他们不能创造。其他人，最终或多或少地能够掌握一些这种直觉，这样他们不知能够理解数学，并且即使他们的记忆力并不超凡，也能变成创造者，并伴随着这种直觉的多少，获得相应程度的成功。

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.

事实上，什么是数学创造？它并不意味着对已知的数学事实重新组合。任何人都可以做到重新组合，但这种组合的数量是无限的，并且大多数毫无价值。创造，意味着不制造无用的组合，而仅制造那些少量且有用的。创造即鉴别，即选择。

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.

现在可以挖掘得更深一些，来看看数学家的灵魂深处是什么了。我相信，我可以通过回忆自己的经历来进行充分说明。不过我将仅限于讲述我写第一本关于Fuchsian函数论文集的情况。我请求听众原谅：我将会使用某些技术术语，但你们无需感到害怕，因为你们并不需要理解。比如，我会这么说，基于某种情况，我得到了这个定理的证明。这个定理可能会有一个奇怪的名字，大部分人会感到不习惯，但并不重要；你们心理学家感兴趣的并不是定理，而是情况本身。

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.

我挣扎了15天来证明不可能存在像后来我称之为Fuchsian函数那样的函数。当时我一无所知；每天我坐在书案前，工作一到两个小时，尝试大量的组合却一无所获。一个晚上，和我平日的习惯不同，我喝了黑咖啡，没有睡觉。大量的思绪汹涌，我感到它们相互碰撞直到契合，也就是说，慢慢地稳定下来。第二天早上之前，我已经建立好一类Fuchsian函数的存在性证明，这些函数来自于超几何序列；我只需要把结果写下来即可，前后花了不过几个小时。

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.

后来我想用两个序列的商来表达这些函数；想法很清晰而且经过深思熟虑，因为椭圆方程的相似性指引着我。我问自己，如果这些序列存在，它们会有怎样的属性，并且我毫无困难地成功构建了这些序列，后来我称之为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.

就是这时期，我离开了卡昂，这个我生活的地方，受我所在学院的资助进行一次旅行。旅途上的经历让我暂且忘记了数学工作。到达古特昂司后，我们上了一辆公共马车以去到另外的什么地方。突然，当我刚登上马车阶梯的刹那，一个想法来临，之前没有任何想法为之做准备，即我用来定义Fuchsian函数的变换实际上与非欧几何中的是等价的。我没有去证明这个想法；我几乎没有时间，因为一上马车我就参与了另一场已经开始的谈话，但我对自己的想法确信无疑。回到卡昂以后，我在良心催促下利用空余时间完成了证明。

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.

随后，我把注意力转回到一些算术问题的研究上，这些问题一直没有进展，而且显然和我之前的研究没有丝毫联系。吸取了之前失败的教训，我到海边去了几天，并且考虑了一些其他事情。一天早晨，正在悬崖边散步，我忽然有了主意，想法和上次一样具有同样简短、突然的性质，而且几乎立即就能肯定，算术问题中的三元不定二次型变换等价于非欧几何中的变换。

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.

回到卡昂，我对此结果进行思考，并得到了一些结论。二次型的例子表明存在Fuchsian群而无须使用超几何序列的方法；我意识到可以对此应用theta-Fuchsian序列的理论，并得出Fuchsian方程而无须引入超几何序列。很自然地，我着手构建这些方程。我系统性地研究他们并且得到一个接一个的结论。然而，有一个问题迟迟不能解决，并且结果很可能影响到全局。因此我的努力一开始似乎效果很好，却只是引入了更加困难的问题。这些工作都是在我有意识地状态下完成的。

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...

后来我去了瓦勒里昂山服兵役；因此我忙着完全不相干的事情。有一天，正沿街走着，难题的解突然出现在我面前。我当时并没有立即深入，直到服完兵役，我才重新拾起该问题。我有了所有的元素，只需要重新安置它们即可、因此我一次性就写完了最后的论文集，没有遇到任何困难。

我只用这一个例子，因为多说无益。

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...

一开始，最令人震惊的是福至心灵的瞬间，反映了长时间、无意识的预备工作。数学创造中，无意识工作的重要性对我而言是无可争议的，并且可以在其他例子中，无意识工作不那么明显的情况下找到它的痕迹。通常，一个人思考困难问题时，最初的工作不会带来太好的结果。休息一会儿，或者更长时间，然后重新坐下来工作。在开始半小时，像以前一样，没什么发现，然而突然决定性的想法呈现在思想面前。

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...

关于无意识工作的条件还有另一个说法：它是可能的，并且某种程度上富有成效，仅当之前和之后都进行了长期有意识的工作。这些突然的灵感（之前的例子已经表明了）发生，仅当之前大量的努力失败，并且没有好的结果发生，而采取的方法又是不确定的。这些努力并不像我们认为的毫无作用；它们启动了无意识的机器，否则这架机器永远不会运转起来，也不会有所成就。

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?

这就是现实；现在我们来分析一下数学家们的想法。无意识，或称之为，潜意识的自我在数学创造中占有重要地位；这和我们之前所说是一致的。然而通常潜意识的自我被认为是完全自发的。现在我们看到数学工作并不是简单的机械运算，一架无论如何精密的仪器是不能胜任数学工作的。它也不仅仅是应用规则的问题，不是根据某些固定法则做出最可能的组合。这样得到的组合可能数量中毒，但毫无用处、繁冗不堪。创造者的真实工作包括在这些组合中进行挑选以减少无用的，避免制造无用结论的麻烦，并且借以挑选的法则必须极其精致准确。几乎不可能准确陈述这些法则；我们更多地是感觉而不是规定。在这些条件下，怎能想象机械地来应用这些法则呢？

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...

现在我们可以得到第一个假设；潜意识的自我并不比意识层面上的自我表现要差；潜意识并不是自动的；它能够鉴别；它手法老练，技巧精湛；它知道如何挑选，它发掘。“发掘”的意思是，它知道如何超越意识层面的自我，因为它能完成有意识不能完成的事情。一句话而言，是否潜意识的自我比有意识的自我更好？至少，你现在应该了解这个问题的重要性。

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.

但，是否我刚才举的实例就能确定无误地说明问题呢？我承认，在我的方面，我不喜欢这个结论。让我们重新检查这个例子，看看能不能得到其他解释。

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?

可以确定的是，在潜意识进行一段时间的工作后，就会产生灵光一现的想法，并且通常是相当有用的结论组合，就像是第一印象得出的结论。这是由于潜意识的自我产生的微妙直觉，判断出哪些组合是有用的，因此只采用了这些组合吗？还是同样产生了很多其他组合，而由于其无用性而停留在潜意识中？

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.

用第二种角度来看，潜意识自动将所有组合产生，但只有最令人感兴趣的能突破意识的界限。这仍是非常神秘的。是什么愿意，使得我们的潜意识活动产生的千百种产品，只有少数能突破界限，而大多数停留在意识层面之下？仅仅是因为机会的原因，让它们得到了这些特权？显然不是；例如就我们的感觉而言，受到的所有的刺激，除非是特殊原因，只有那些最强烈的能够吸引我们的注意力。一般而言，那些比较特殊的潜意识活动，那些可能跃而成为意识层面的，是那些直接或间接能深刻影响我们情感认知的活动。

It may be surprising to see emotional sensibility invoked apropos 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.

看到情感认知能唤起适当的数学证明，也许令人惊异；因为我们本认为这仅仅关系到智力。那是因为我们忘了这其中存在对数学之美的感知，对数和形式的协调感，以及几何优雅性的体验。这是所有真正数学家应当了解的美感，并且显然属于情感认知的范畴。

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.

那么，我们所谓的数学美和优雅，并且可以引起我们的美的情感的主体究竟是什么？它们是元素的和谐放置，以让思想无需费力就能理感知整体和理解细节。这种和谐感也能满足我们的审美需求，并且帮助和引领头脑。同时，随着我们眼前出现一个美妙排序的整体，我们可以看到其中隐含的数学原理。这种特殊的审美就像是一个滤网，就像我说的那样，提炼出我们所需要的东西；这也充分解释了为什么缺乏这种审美的人永远不可能成为数学的创造者。

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.

然而困难未曾消失。意识的自我存在限制，但对于潜意识，我们并不知道限制所在，因此我们并不愿意假设潜意识在短期内能够完成的工作，比有意识在长期内能够进行的更多。然而限制是存在的。如果所有的组合都能够形成，其数量比想象的更多，这可能吗？然而这是必须的，因为如果它只生成少数的组合，而且是随机生成的，那么我们所能挑选的比较好的组合的几率就相对太小了。

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...

也许我们应当到早期的有意识的工作中寻求答案，以往内所有的无意识工作都必须由有意识的先行。请允许我做一个粗略的类比。假设我们潜意识作出的数学组合像是伊壁鸠鲁的原子论假说。在意识得到完全休息的阶段，这些原子是静止的，或者说，它们黏在了墙上。

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.

而另一方面，在表面上休息却是潜意识工作的阶段，某些原子从墙上脱离开来并进入运动。它们向空间的各个方向漂移，就像一小群昆虫一样，或者用更学术的比喻，就像气体运动理论中的气体分子那样。它们之间的互相影响可能形成新的组合。

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.

那么之前有意识工作起到怎样的铺垫呢？该工作推动了某些原子，将它们从墙上释放并进入运动。我们认为我们没有得到结果，因为我们把这些元素推向一千个不同的方向，却希望它们能够汇聚，显然得不到满意的结果。然而，在强加在它们之上的意愿结束之后，这些原子并不会停止。它们开始自由自在地运动。

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.

现在，我们的意愿不会随机挑选它们；它们按照自己的意愿行事。运动起来的原子们不再是普通原子；它们是那些我们可以期待成为解的那些原子。这些运动起来的原子产生冲击，让它们彼此之间产生结合，直到停止下来为止。这里我请听众们原谅，因为我的比喻相当粗疏，但我不知道有什么更好的比喻了。

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...

无论如何，最后可能的组合，至少包含了部分由我们的意志释放的原子。在这些原子中，产生了被我们称之为“好”的组合。也许，这是一种使得原始假设不那么荒谬的说法。

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.

最后我还要做出一些说明：上面这些我个人作出的观察中，描述的是我个人工作的感受。这种事例是很常见的，因此不需要考虑我之前提到的大脑产生某种不同寻常的兴奋。似乎在这些事例中，一个人面对他的无意识工作，部分感受到他自己过度兴奋的意识，却没有改变这种意识的本质。我们可以大概理解这两种工作机制的区别，或者说，两种自我不同的工作方法。我所做出的心理学观察似乎证实了之前我所给出的观点。

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.

这种观点还有待确认，因为它们仍然停留在高度假说的程度；但这个问题如此有趣，以至于这里我毫不犹豫，要将它们提供给我的听众们。

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