Mathematical Creation
数学创造
原题：L’invention mathématique（法语）
亨利·庞加莱 Henri Poincaré（1854–1912）
译者：George Bruce Halsted（1853–1922），1913 年
1908 年在巴黎心理学会的讲演

冷冻图书馆 · 开架书库 · S-006019 · https://coldlibrary.com/s/006019
权利：法文原著 1908 年，庞加莱卒于 1912 年。译者哈尔斯特德卒于 1922 年，译本出版于 1913 年。已按澳大利亚、欧盟、英国、中国大陆和美国的规则核对（截至 2026 年 10 月 11 日）。仅供参考，不是法律意见。
来源：Project Gutenberg，https://www.gutenberg.org/cache/epub/39713/pg39713.txt（取于 2026-10-11）
中文译文：Claude（AI），2026 · CC0 1.0
生成于 2026-10-14，来自 https://coldlibrary.com/open-stacks/poincare-mathematical-creation

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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.

This has long been appreciated, and some time back the journal called *L'enseignement mathématique*, edited by Laisant and Fehr, began an investigation of the mental habits and methods of work of different mathematicians. I had finished the main outlines of this article when the results of that inquiry were published, so I have hardly been able to utilize them and shall confine myself to saying that the majority of witnesses confirm my conclusions; I do not say all, for when the appeal is to universal suffrage unanimity is not to be hoped.

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

The answer seems to me evident. Imagine a long series of syllogisms, and that the conclusions of the first serve as premises of the following: we shall be able to catch each of these syllogisms, and it is not in passing from premises to conclusion that we are in danger of deceiving ourselves. But between the moment in which we first meet a proposition as conclusion of one syllogism, and that in which we reencounter it as premise of another syllogism occasionally some time will elapse, several links of the chain will have unrolled; so it may happen that we have forgotten it, or worse, that we have forgotten its meaning. So it may happen that we replace it by a slightly different proposition, or that, while retaining the same enunciation, we attribute to it a slightly different meaning, and thus it is that we are exposed to error.

Often the mathematician uses a rule. Naturally he begins by demonstrating this rule; and at the time when this proof is fresh in his memory he understands perfectly its meaning and its bearing, and he is in no danger of changing it. But subsequently he trusts his memory and afterward only applies it in a mechanical way; and then if his memory fails him, he may apply it all wrong. Thus it is, to take a simple example, that we sometimes make slips in calculation because we have forgotten our multiplication table.

According to this, the special aptitude for mathematics would be due only to a very sure memory or to a prodigious force of attention. It would be a power like that of the whist-player who remembers the cards played; or, to go up a step, like that of the chess-player who can visualize a great number of combinations and hold them in his memory. Every good mathematician ought to be a good chess-player, and inversely; likewise he should be a good computer. Of course that sometimes happens; thus Gauss was at the same time a geometer of genius and a very precocious and accurate computer.

But there are exceptions; or rather I err; I can not call them exceptions without the exceptions being more than the rule. Gauss it is, on the contrary, who was an exception. As for myself, I must confess, I am absolutely incapable even of adding without mistakes. In the same way I should be but a poor chess-player; I would perceive that by a certain play I should expose myself to a certain danger; I would pass in review several other plays, rejecting them for other reasons, and then finally I should make the move first examined, having meantime forgotten the danger I had foreseen.

In a word, 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.

It seems to me then, in repeating a reasoning learned, that I could have invented it. This is often only an illusion; but even then, even if I am not so gifted as to create it by myself, I myself re-invent it in so far as I repeat it.

We know that this feeling, this intuition of mathematical order, that makes us divine hidden harmonies and relations, can not 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. Any one 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.

How to make this choice I have before explained; the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another.

Among chosen combinations the most fertile will often be those formed of elements drawn from domains which are far apart. Not that I mean as sufficing for invention the bringing together of objects as disparate as possible; most combinations so formed would be entirely sterile. But certain among them, very rare, are the most fruitful of all.

To invent, I have said, is to choose; but the word is perhaps not wholly exact. It makes one think of a purchaser before whom are displayed a large number of samples, and who examines them, one after the other, to make a choice. Here the samples would be so numerous that a whole lifetime would not suffice to examine them. This is not the actual state of things. The sterile combinations do not even present themselves to the mind of the inventor. Never in the field of his consciousness do combinations appear that are not really useful, except some that he rejects but which have to some extent the characteristics of useful combinations. All goes on as if the inventor were an examiner for the second degree who would only have to question the candidates who had passed a previous examination.

But what I have hitherto said is what may be observed or inferred in reading the writings of the geometers, reading reflectively.

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' 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. In regard to my other researches I would have to say analogous things, and the observations of other mathematicians given in *L'enseignement mathématique* would only confirm them.

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The rôle 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. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness. But it is more probable that this rest has been filled out with unconscious work and that the result of this work has afterward revealed itself to the geometer just as in the cases I have cited; only the revelation, instead of coming during a walk or a journey, has happened during a period of conscious work, but independently of this work which plays at most a rôle of excitant, as if it were the goad stimulating the results already reached during rest, but remaining unconscious, to assume the conscious form.

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 sufficiently 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.

The need for the second period of conscious work, after the inspiration, is still easier to understand. It is necessary to put in shape the results of this inspiration, to deduce from them the immediate consequences, to arrange them, to word the demonstrations, but above all is verification necessary. I have spoken of the feeling of absolute certitude accompanying the inspiration; in the cases cited this feeling was no deceiver, nor is it usually. But do not think this a rule without exception; often this feeling deceives us without being any the less vivid, and we only find it out when we seek to put on foot the demonstration. I have especially noticed this fact in regard to ideas coming to me in the morning or evening in bed while in a semi-hypnagogic state.

Such are the realities; now for the thoughts they force upon us. The unconscious, or, as we say, the subliminal self plays an important rôle 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. Boutroux in a recent lecture has shown how it came up on a very different occasion, and what consequences would follow an affirmative answer. (See also, by the same author, *Science et Religion*, pp. 313 ff.)

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. Reexamine 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. Now, as we have said above, the only mathematical facts worthy of fixing our attention and capable of being useful are those which can teach us a mathematical law. So that we reach the following conclusion: The useful combinations are precisely the most beautiful, I mean those best able to charm this special sensibility that all mathematicians know, but of which the profane are so ignorant as often to be tempted to smile at it.

What happens then? Among the great numbers of combinations blindly formed by the subliminal self, almost all are without interest and without utility; but just for that reason they are also without effect upon the esthetic sensibility. Consciousness will never know them; only certain ones are harmonious, and, consequently, at once useful and beautiful. They will be capable of touching this special sensibility of the geometer of which I have just spoken, and which, once aroused, will call our attention to them, and thus give them occasion to become conscious.

This is only a hypothesis, and yet here is an observation which may confirm it: when a sudden illumination seizes upon the mind of the mathematician, it usually happens that it does not deceive him, but it also sometimes happens, as I have said, that it does not stand the test of verification; well, we almost always notice that this false idea, had it been true, would have gratified our natural feeling for mathematical elegance.

Thus it is this special esthetic sensibility which plays the rôle 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; so this complete rest may be indefinitely prolonged without the atoms meeting, and consequently without any combination between them.

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 rôle 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.

Another observation. It never happens that the unconscious work gives us the result of a somewhat long calculation *all made*, where we have only to apply fixed rules. We might think the wholly automatic subliminal self particularly apt for this sort of work, which is in a way exclusively mechanical. It seems that thinking in the evening upon the factors of a multiplication we might hope to find the product ready made upon our awakening, or again that an algebraic calculation, for example a verification, would be made unconsciously. Nothing of the sort, as observation proves. All one may hope from these inspirations, fruits of unconscious work, is a point of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. The rules of these calculations are strict and complicated. They require discipline, attention, will, and therefore consciousness. In the subliminal self, on the contrary, reigns what I should call liberty, if we might give this name to the simple absence of discipline and to the disorder born of chance. Only, this disorder itself permits unexpected combinations.

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 it, 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.

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中文译文 · Claude（AI）译

数学创造的起源，是一个应当引起心理学家浓厚兴趣的问题。在这种活动中，人的心智似乎从外部世界取得的最少，它仅仅依靠自身、作用于自身，或者看上去是这样；因此，研究几何思维的进程，我们就有希望触及人类心智中最本质的东西。

人们早就认识到了这一点。前些时候，由莱桑和费尔主编的《数学教育》杂志，就对不同数学家的思维习惯和工作方法开展了一项调查。那项调查的结果发表时，我已经写完了这篇文章的主要轮廓，所以几乎没能利用它们，只能说一句：大多数受访者证实了我的结论；我不说全部，因为既然诉诸普选，就别指望全体一致。

第一个事实应当让我们吃惊，或者说，若不是我们对它太习以为常，它本会让我们吃惊。怎么会有人不懂数学呢？如果数学所援引的只是一切正常心智都接受的逻辑规则；如果它的明证所依据的原理是一切人所共有的，谁要否认它们就只能是疯了；那么，怎么会有这么多人在这里冥顽不化呢？

并非人人都能发明，这一点毫不神秘。并非人人都能记住一个学过的证明，这也说得过去。但并非人人都能在听人讲解时理解数学推理，细想起来，这就显得非常令人吃惊了。然而，只能吃力地跟上这种推理的人占大多数：这是无可否认的，中学教师的经验也一定不会与此相悖。

还有：数学中怎么可能出错呢？一个健全的心智不应犯逻辑谬误；可是有一些非常优秀的心智，在日常生活中出现的简短推理上从不失足，却不能毫无差错地跟上或复述那些较长的数学证明——而这些证明归根到底不过是一连串简短推理的累积，与他们做起来那么轻松的推理完全类似。还需要补充说，数学家自己也并非不会犯错吗？

在我看来，答案是显然的。设想有一长串三段论，前面那些的结论充当后面那些的前提：我们能够抓住其中每一个三段论，我们有可能弄错的地方，并不在于从前提到结论的过渡。但从我们第一次遇到一个命题作为某个三段论的结论，到我们再次遇到它作为另一个三段论的前提，其间有时会过去一段时间，链条上的好几环已经展开了；于是可能发生这样的事：我们已经把它忘了，或者更糟，忘了它的意思。于是又可能发生这样的事：我们把它换成了一个稍有不同的命题，或者虽然保留了同样的表述，却赋予它稍有不同的意思；我们就是这样陷入错误的。

数学家常常使用一条法则。自然，他先要证明这条法则；在这个证明在他记忆中还很新鲜的时候，他完全明白它的意义和适用范围，不会有改动它的危险。但后来他就信赖自己的记忆，此后只是机械地应用它；这时如果记忆辜负了他，他就可能完全用错。举一个简单的例子：我们有时算错，就是因为忘了乘法表。

照这样说，数学的特殊才能就只是出于非常可靠的记忆，或者惊人的注意力。那就会是一种类似惠斯特牌手记住已出之牌的能力；或者更高一级，类似象棋手能够在脑中看到大量的组合、并把它们记住的能力。每个好的数学家都应当是好的棋手，反之亦然；同样，他也应当是一个好的计算者。当然，这种情形有时确实会出现；比如高斯，他既是一位天才的几何学家，又是一个非常早慧而精确的计算者。

但也有例外；或者不如说我错了；我不能把它们称作例外，否则例外就比规则还多了。恰恰相反，高斯才是一个例外。至于我自己，我必须承认，我连做加法都绝对做不到不出错。同样，我也只会是一个蹩脚的棋手；我会看出走某一步棋会使自己陷入某种危险；我会逐一考虑其他几步棋，又因为别的理由把它们一一否定，最后我却走了最初考虑的那一步，因为在此期间我已经把预见到的危险忘了。

总之，我的记忆力并不坏，但它不足以使我成为一个好棋手。那么，在一段艰深的数学推理中——大多数棋手会在其中迷失——为什么它又不辜负我呢？显然是因为它受到推理的总体进程的引导。一个数学证明并不是三段论的简单并列，而是*按某种次序排列起来的*三段论，而这些要素排列的次序，比要素本身重要得多。如果我对这种次序有一种感觉，可以说是一种直觉，从而能一眼看清整个推理，我就不必再担心忘记其中某个要素，因为每一个要素都会在整个阵列中占据分派给它的位置，而且我在记忆上无需作任何努力。

所以在我看来，当我复述一个学过的推理时，我似乎本来也能把它发明出来。这常常只是一种错觉；但即使如此，即使我没有那样的天分能独自创造它，我在复述它的时候，也是在亲自把它重新发明一遍。

我们知道，这种感觉，这种对数学次序的直觉，使我们能猜出隐藏的和谐与关系，但它并不是人人都具备的。有些人既没有这种难以界定的细腻感觉，也没有超乎寻常的记忆力和注意力，那么他们就完全不能理解高等数学。这样的人占大多数。另一些人只在很小的程度上有这种感觉，但他们天生有不寻常的记忆力和很强的注意力。他们会把细节一个接一个地背下来；他们能理解数学，有时也能加以应用，但他们不能创造。最后，还有一些人或多或少地具有我所说的这种特殊直觉，那么，即使他们的记忆力毫不出众，他们不仅能理解数学，而且可能成为创造者，并尝试去发明，成功的大小则视这种直觉在他们身上发达的程度而定。

事实上，什么是数学创造？它并不在于用已知的数学对象作出新的组合。谁都能做到这一点，但这样作出的组合会多到无穷，而且其中绝大多数毫无意思。创造恰恰在于不去作无用的组合，而去作那些有用的组合，而有用的组合只占很小的一部分。发明就是辨别，就是选择。

怎样作出这种选择，我以前已经说明过：值得研究的数学事实，是那些能够凭借与其他事实的类比，引导我们认识一条数学规律的事实，正如实验事实引导我们认识一条物理规律一样。它们向我们揭示出其他一些事实之间意想不到的亲缘关系，那些事实早已为人所知，却一直被错误地认为彼此毫不相干。

在选出的组合中，最富成果的，往往是那些由取自相距遥远之领域的元素构成的组合。我并不是说，只要把尽可能不相干的东西拼在一起，就足以发明；这样构成的组合，大多数完全是不毛的。但其中有某些——非常稀少——是一切组合中最富成果的。

我说过，发明就是选择；但这个词也许并不完全确切。它让人想到一位买主，面前摆着大量样品，他一件一件地察看，好作出选择。而在这里，样品会多到用一生的时间也看不完。实际情形并非如此。不毛的组合甚至根本不会出现在发明者的脑海里。在他意识的范围里，除了少数他会拒绝、但多少带有有用组合之特征的组合之外，从不出现真正无用的组合。一切都好像发明者是第二轮考试的考官，只需要考问那些已经通过了初试的考生。

不过，我到此为止所说的，是人们在读几何学家的著作时——用心地读——可以观察到或推断出来的。

现在是深入下去、看看数学家心灵深处究竟发生了什么的时候了。为此，我相信最好的办法是回忆我自己的经历。但我只讲我是怎样写出我关于富克斯函数的第一篇论文的。请读者原谅，我要用到一些专门的术语，但读者不必害怕，因为他并不需要懂得它们。比如我会说，我在某某情形下找到了某某定理的证明。这条定理会有一个古怪的名字，许多人都不熟悉，但这并不要紧；对心理学家来说，有意思的不是定理，而是当时的情形。

有十五天，我竭力想证明：不可能存在任何像我后来称为富克斯函数那样的函数。那时我非常无知；每天我都坐到工作台前，待上一两个小时，试了大量的组合，却毫无结果。一天晚上，我一反常态地喝了黑咖啡，睡不着觉。想法成群地涌现；我感到它们相互碰撞，直到其中两两勾连起来，可以说，形成了一个稳定的组合。到第二天早上，我已经确立了一类富克斯函数的存在，即由超几何级数得出的那一类；我只需把结果写出来，这只花了几个小时。

接着我想用两个级数之商来表示这些函数；这个想法是完全有意识的、经过考虑的，与椭圆函数的类比引导着我。我问自己，如果这样的级数存在，它们必须具有什么性质；我没费什么力气就构造出了我后来称为“θ富克斯级数”的那些级数。

恰在这时，我离开了当时居住的卡昂，去参加矿业学校组织的一次地质考察。旅途的变化让我忘记了我的数学工作。到了库唐斯，我们登上一辆公共马车，要去什么地方。就在我把脚踏上踏板的那一刻，一个想法来到了我心里，我先前的思考中似乎没有任何东西为它铺过路：我用来定义富克斯函数的那些变换，与非欧几何的变换是同一的。我没有验证这个想法；我也来不及验证，因为我在马车里坐下之后，又接着进行一场已经开始的谈话；但我感到完全确定。回到卡昂之后，为了问心无愧，我才从从容容地验证了这个结果。

随后，我把注意力转向某些算术问题，显然没有多大成功，也丝毫没有想到它们与我先前的研究有什么联系。我对自己的失败感到厌烦，便到海边去住了几天，想些别的事情。一天早上，我在悬崖上散步，一个想法来到我心里，同样具有简短、突然和当即确定的特征：不定三元二次型的算术变换，与非欧几何的变换是同一的。

回到卡昂之后，我思考了这个结果，推出了它的推论。二次型的例子向我表明，除了与超几何级数相对应的富克斯群之外，还有别的富克斯群；我看出，我可以把θ富克斯级数的理论应用于它们，因而除了由超几何级数得出的、我当时所知的那些富克斯函数之外，还存在别的富克斯函数。我自然着手去构造所有这些函数。我对它们发起了一场系统的进攻，把外围的堡垒一个接一个地攻了下来。但还有一座始终坚守着，它一陷落，整个要塞也就随之陷落。可是起初，我的一切努力只是让我更清楚地看到了困难所在，这本身倒也算是有所收获。这一切工作都是完全有意识的。

此后我去了瓦勒里安山，要在那里服兵役；这样我就忙于完全不同的事了。一天，我走在街上，那个曾经难住我的困难的解答，突然出现在我面前。我没有试图立刻深究它，直到服役期满，才重新拾起这个问题。我已经掌握了所有的要素，只需把它们排列起来、组合在一起。于是我一气呵成、毫不费力地写出了我最后的论文。

我只举这一个例子；再多举也没有用。关于我的其他研究，我也得说类似的话，而《数学教育》中刊载的其他数学家的观察，只会证实这些话。

最初最引人注目的，是这种突然顿悟的样子，这是长时间无意识的先期工作的明显标志。在我看来，这种无意识工作在数学发明中的作用是无可争辩的，在另一些它不那么明显的情形中，也能找到它的痕迹。人们在钻研一个难题时，常常第一次进攻毫无所成。于是休息一下，或长或短，然后重新坐下来工作。头半个小时里，和先前一样，什么也没发现，然后突然之间，那个决定性的想法出现在脑海里。人们可能会说，有意识的工作之所以更有成效，是因为它被打断了，休息使心智恢复了力量和清新。但更可能的是，这段休息被无意识的工作填满了，而这一工作的结果后来向几何学家显露出来，正如我所举的那些情形一样；只不过，这种显露不是在散步或旅行中到来，而是在一段有意识工作的期间发生的，但它与这一工作无关，这一工作至多起到一种刺激的作用，好像是一根刺棒，催促那些在休息期间已经得到、却仍处于无意识状态的结果，采取有意识的形式。

关于这种无意识工作的条件，还有一点要说：只有在它之前有一段有意识的工作、之后又有一段有意识的工作时，它才是可能的，而且肯定也只有这样才有成效。这些突然的灵感（前面所举的例子已足以证明这一点）只会在几天自觉的努力之后出现，那些努力看上去完全没有成果，似乎什么好结果也没有得出，所走的路似乎完全走错了。可见这些努力并不像人们以为的那样徒劳；它们开动了那台无意识的机器，没有它们，这台机器就不会运转，也就什么也产生不出来。

灵感之后需要第二段有意识的工作，这一点更容易理解。必须把灵感的结果整理成形，从中推出直接的推论，把它们排列起来，把证明写出来，但最需要的是验证。我说过伴随灵感而来的那种绝对确定的感觉；在我所举的那些情形中，这种感觉没有骗人，通常它也不骗人。但不要以为这是一条没有例外的规则；这种感觉常常欺骗我们，而它的鲜明程度丝毫不减，我们只有在着手建立证明时才发现这一点。我特别注意到，那些在早晨或晚上躺在床上、处于半睡半醒状态时来到我心中的想法，就有这种情况。

这些就是事实；现在来看它们迫使我们作出的思考。无意识的自我，或者如我们所说的阈下自我，在数学创造中起着重要的作用；这是从我们上面所说的推出来的。但阈下自我通常被认为是纯粹自动的。而我们已经看到，数学工作不只是机械的，它不可能由一台机器来完成，无论那台机器多么完善。问题不仅仅在于应用规则，按照某些固定的法则作出尽可能多的组合。这样得到的组合会多得惊人，既无用又累赘。发明者真正的工作，在于在这些组合中作出选择，以便剔除无用的组合，或者不如说，免去作出这些组合的麻烦；而必须指导这种选择的规则，是极其精微细致的。几乎不可能把它们准确地表述出来；它们是被感觉到的，而不是被表述出来的。在这样的条件下，怎么能设想有一个筛子能机械地应用它们呢？

于是出现了第一个假说：阈下自我丝毫不比有意识的自我低劣；它不是纯粹自动的；它能够辨别；它有分寸，有灵敏；它知道怎样选择，怎样猜测。我在说什么呢？它比有意识的自我更善于猜测，因为它在后者失败的地方成功了。总之，阈下自我难道不比有意识的自我更高明吗？你们会认识到这个问题的全部重要性。布特鲁在最近的一次讲演中曾经表明，这个问题是怎样在一个很不相同的场合提出来的，以及肯定的回答会带来什么后果。（另见同一作者的《科学与宗教》，第 313 页以下。）

我刚才举出的事实，是否迫使我们作出这个肯定的回答呢？我承认，就我个人而言，我很不愿意接受它。那么，让我们重新审视这些事实，看看它们是否与另一种解释相容。

可以肯定，在一段相当长的无意识工作之后，以一种突然顿悟的方式出现在脑海中的那些组合，一般都是有用而富有成果的组合，它们似乎是第一印象的结果。是否由此可以推出：阈下自我以一种细腻的直觉猜到了这些组合会有用，于是只构成了这些组合？还是说，它构成了许多别的组合，那些组合缺乏意义，始终停留在无意识之中？

按照第二种看法，所有的组合都是由于阈下自我的自动作用而构成的，但只有那些有意义的组合才会闯入意识的领域。而这仍然非常神秘。在我们无意识活动的上千种产物之中，为什么有一些被召唤越过门槛，另一些却留在下面？是单纯的偶然赋予了这种特权吗？显然不是；比如说，在我们感官所受的一切刺激中，只有最强烈的那些才能吸引我们的注意，除非注意力是由别的原因引向它们的。更一般地说，享有特权的无意识现象，即那些可能变成有意识的现象，是直接或间接地最深刻地影响我们情感感受力的那些现象。

看到在谈论数学证明时援引情感感受力，人们也许会感到惊讶，因为数学证明似乎只能引起理智的兴趣。但这就忘记了对数学美的感觉，对数与形之和谐的感觉，对几何之优雅的感觉。这是一切真正的数学家都懂得的一种真正的审美感受，它当然属于情感感受力。

那么，我们认为具有这种美与优雅的特性、能够在我们心中激起一种审美情感的数学对象，是哪些呢？是那些元素安排得和谐、使心智能够毫不费力地把握其整体同时又领会其细节的对象。这种和谐既满足了我们的审美需要，又是对心智的一种帮助，支撑它，引导它；同时，它把一个井然有序的整体摆在我们眼前，使我们预见到一条数学规律。而正如我们上面所说的，唯一值得我们注意、能够有用的数学事实，就是那些能教给我们一条数学规律的事实。于是我们得出如下的结论：有用的组合恰恰是最美的组合，我是说，最能打动那种特殊感受力的组合——一切数学家都懂得这种感受力，而外行人对它却如此无知，以致常常忍不住要对它一笑置之。

那么，会发生什么呢？在阈下自我盲目地构成的大量组合中，几乎全都没有意义、没有用处；但正因为如此，它们对审美感受力也毫无影响。意识永远不会知道它们；只有某些组合是和谐的，因而既有用又美。它们能够触动我刚才说过的几何学家的那种特殊感受力，而这种感受力一旦被唤起，就会把我们的注意力引向它们，从而给它们变成有意识的机会。

这只是一个假说，然而这里有一个观察或许可以证实它：当一个突然的顿悟攫住数学家的心智时，它通常不会骗他，但正如我说过的，有时它也经不起验证的考验；而我们几乎总是注意到，这个错误的想法，如果它是对的，本会满足我们对数学优雅的天然感觉。

因此，正是这种特殊的审美感受力，扮演了我所说的那个精细的筛子的角色；这也足以解释，为什么缺乏这种感受力的人永远不会成为真正的创造者。

然而，并不是所有的困难都消失了。有意识的自我受到狭窄的限制，至于阈下自我，我们不知道它的限度，因此我们并不太难于假定，它能够在短时间内作出比一个有意识的人一生所能涉及的还要多的不同组合。然而这些限度是存在的。它是否可能构成一切可能的组合呢？那些组合的数目会让想象力都为之惊恐。然而这似乎又是必要的，因为如果它只产生这些组合中的一小部分，而且是随意产生的，那么*好的*组合——我们应当选择的那一个——在其中的机会就会很小。

也许我们应当到那段先期的有意识工作中去寻找解释，那段工作总是在一切富有成果的无意识劳动之前。请允许我作一个粗糙的比较。把我们组合的未来元素设想成某种像伊壁鸠鲁的带钩原子那样的东西。在心智完全静止的时候，这些原子是不动的，可以说，它们被钩在墙上；所以这种完全的静止可以无限期地延续下去，原子却不会相遇，因而它们之间也不会有任何组合。

另一方面，在一段表面上的休息和无意识工作期间，其中某些原子从墙上脱落下来，开始运动。它们在关住它们的那个空间（我差点要说那个房间）里向四面八方飞窜，就像，比如说，一群蚊蚋，或者，如果你们喜欢一个更有学问的比较，就像气体运动论中的气体分子。于是它们的相互碰撞就可能产生新的组合。

先期的有意识工作起什么作用呢？显然是调动其中某些原子，把它们从墙上摘下来，让它们摆动起来。我们以为自己一无所获，因为我们为了把这些元素拼合起来，把它们挪动了上千种不同的方式，却没有找到令人满意的集合。但是，在我们的意志强加给它们的这番摇动之后，这些原子并不回到它们原初的静止状态。它们自由地继续着它们的舞蹈。

而我们的意志并不是随意选择它们的；它追求的是一个完全确定的目标。因此，被调动起来的原子并不是随便什么原子；它们是那些我们可以合理地期望从中得到所求解答的原子。然后，被调动起来的原子经受碰撞，这些碰撞使它们彼此结合，或者与它们在行进中撞上的其他静止原子结合。我再次请求原谅，我的比较非常粗糙，但我几乎不知道还能用别的什么办法让人理解我的想法。

无论如何，唯一有机会形成的组合，是那些其中至少有一个元素是我们的意志自由选定的原子的组合。而显然，我所说的那个*好的组合*，就在这些组合之中。也许这是一种减轻原先那个假说的悖论色彩的办法。

再说一个观察。无意识的工作从来不会把一个稍长的计算——只需应用固定规则的计算——的结果*现成地*交给我们。我们可能会以为，完全自动的阈下自我特别适合做这种工作，因为这种工作在某种意义上是纯粹机械的。似乎晚上想着一个乘法的各个因数，就可以指望醒来时发现乘积已经现成地算好了；或者一个代数计算，比如一次验算，会在无意识中完成。观察证明，根本不是这样。从这些灵感——无意识工作的果实——中所能指望的，只是这类计算的一个出发点。至于计算本身，必须在第二段有意识的工作中进行，也就是在灵感之后的那段时期，在那段时期里，人们验证灵感的结果，推出它们的推论。这些计算的规则是严格而复杂的。它们需要纪律、注意力、意志，因而需要意识。相反，在阈下自我中，统治着的是我所谓的自由，如果我们可以把这个名字给予单纯的纪律之缺乏和偶然所生的无序的话。只是，这种无序本身使意想不到的组合成为可能。

我要作最后一点评论：上面我讲述某些个人观察时，提到过一个兴奋的夜晚，那一夜我身不由己地工作着。这样的情形是常有的，而且这种不正常的大脑活动未必是像我提到的那样由某种生理上的刺激物引起的。在这样的情形中，人似乎在旁观自己的无意识工作，这种工作部分地被过度兴奋的意识所察觉，却没有因此改变它的性质。这时我们就模模糊糊地领会到，是什么区分了这两种机制，或者，如果你们愿意的话，区分了这两个自我的工作方法。我这样所能作出的心理观察，在我看来，在大体轮廓上证实了我所提出的观点。

它们确实需要证实，因为无论如何，它们现在是、将来也仍然是非常假设性的：但这些问题的意义如此重大，我并不后悔把它们提交给读者。