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?
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.
Poincaré on Mathematical Creation
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 reasonings 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.
康德对这一问题进行了极其重要的探索。他明确把主体意识摆在认识过程中的核心地位。一方面是外在的世界本身，一方面是主观的王国，知识从何而来？康德认为存在某些超逻辑的东西成为知识的先决条件，就像维柯的“具有想象力的普遍本质”（imaginative universal）那样，把经验和概念综合为知识。他指出，人们先天具有的空间直观和时间直观发挥着模型的功能，塑造着我们的经验，人们据此提出描述这些经验的概念框架，如数学就是这样的概念框架。 但困难接着出现了。按照康德的说法，我们的大脑先天具有某种能力，但是我们如何确定事情就是如此？这个问题意味着我自己成为我的认识的对象，因此我不得不面对自己。因此，我们需要比内省更多的东西来支持对这些内容的确认。康德的解决办法求助于共同体，把自我的认识交于他人来裁决。但显然康德本人对这一办法并不十分满意，因为他认为如果个体毫无批判性地接受共同体的信念就是“不成熟”或“未开化”的。 早期浪漫主义思想家很快注意到这一困难。费希特认识到，没有非我，就没有我（without the not-I，there can be no I）。他认为，我是由认知的我（knowing-I）和某种其他的东西构成，这种其他的东西是一种创造性的力量，尽管我永远不能知道它，但我可以把它设想为某种连续的活动。也就是说，“在我之中还有比我更多的东西”。其基本意思是指，在我之中，有超越于我的客观知识的内容。不论它是什么，它先于所有的知识，甚至超越经验。 这样，浪漫主义提出了一个基本原则：自我的这种不可还原的“活动成分”是知识不能达到的。也就是说，它是可感觉、可猜想的对象，但不是科学能够研究的对象。如果科学、逻辑与数学能捕捉到这种力量，就不会有自由来进行创造性的活动，结果将是一切都进入某种严格决定论的框架之中。 值得强调的是，正是在这个问题上，塔西奇敏锐地把握住文化冲突的最深层根源。哲学家们强调这些不能被理性征服的创造力、想象、美学成分的意义，而科学家们却致力消除它们，并试图用理性的途径来代替。这种冲突以不同的形式贯穿了整个19世纪的浪漫主义思想，并一直持续到今天。 在这个冲突中，数学处于重要的地位。尽管康德提出了他自己的数学哲学，却受到很多数学家和科学家的讥笑。 但是，如果康德的目的是指出一般意义上的想象是“心灵不可缺少的一项功能，没有它知识就是不可能的”，那么康德就算得上是试图调和这种冲突的第一人。
[1][2] Roger Jones：Realism About What？Philosophy of Science 58（1991），P185、197 [3][5][6][7][8][9][10][12] Van Fraassen：The Scientific Image，Oxford 1980，P156、7271、73、202、5、2、90； [4]Van Fraasen：Belief and the Will，The Journal of Philosophy1984，P236； [11] Van Fraassen：On the Extension of Beth's Semantics of Physical Theories，Philosophy of Sciehce37，P327； [13][14]Van Fraassen：After Foundationalism：Between Vicious Circle and Infinite Regress，in Proceedings of Conference on the philosophy of Hilary Putnam，Taxco，Mexico，Aug，1992； [15] A。Fine：The shaky Game，Chicago 1986，P142； [16] Van Fraassen：Interpretation in Science and in the Arts。Forthcoming in G。 Levine （ed。）：Realism and Representation，Uniu。ofWisconsin Press； [17][18] P。Churchland and C。A。Hooker（eds。）：Images of Science，Chicago 1985，P258、246； [19] R。Schlagel：Fine's"Shaky Game"（And why NOA is NO Ark for Science），Philosophy of Science 58。P322。P322。
【彭加勒科学方法论的特色ZT】 李醒民
（中国科学院自然辩证法通讯杂志社， 北京 100039）
朱尔•昂利•彭加勒(Jules Henri Poincaré，1854~1912)是法国著名的科学家。他不仅在数学、物理学、天文学的众多分支有开创性的贡献，而且在科学哲学上也有重要建树。在本世纪初，彭加勒先后出版了几本科学哲学著作：《科学与假设》(1902年)、《科学的价值》(1905年)、《科学与方法》(1908年) 和《最后的沉思》(1912年) ，广泛地探讨了有关科学哲学问题，其中也大量涉及到科学方法论问题。本文拟评介一下彭加勒科学方法论中几个有特色的方面——假设、直觉、科学美和事实的选择。
值得注意的是，彭加勒对假设进行了分类研究([1]，pp.135~136、28)。在他看来，假设共有三种：第一种是“极其自然的假设”。这是一种很普遍的假设，人们几乎不可能回避它们，我们用它来做相关判断(judgement of relevance)。例如，我们不得不假定遥远物体的影响完全可以忽略，小位移遵守线性定律，结果是原因的连续函数等等。这类假设只是表面看来是假设，其实只不过是一种隐蔽的约定或定义而已，这类假设在数学及与数学相关的学科中常常遇到。第二种是“中性的假设”。在大多数问题中，解析家在计算之初，或者假定物质是连续的，或者相反，假定物质是由原子构成的。无论他做那一种假定，其结果是一样的，只是求得结果的难易程度不同。当假设是可选择的，而它又不能用实验区分时，它们就是这类假设。这类假设只要它们的特征未被误解，就不会使我们误入歧途。它们对于计算或通过具体图像帮助我们理解是有用的，即有助于整理和坚定我们的思想，从而不存在排除它们的机会。第三类假设是“真正的推广”。它们是实验必须证实或推翻的假设。它们是观察和实验的直接推广，无限期地敞开着通向进一步检验的大门。不管实验宣布其合理或不适用，就它们的启发作用而言，它们总是有价值的。玛丽•妮厄(Mary Jo Nye)在彭加勒论述的启示下，根据假设在科学知识体系中所起的作用，把假设分为“说明性假设”(保留在物理理论中，是一种协调手段，有益于提示不同现象之间的相互关系)， “启发性假设”(不具有存在的意义，仅有建议作进一步实验、观察或探索纲领的联接方式中的信息的启发意义)、“实在论假设”(具有实在论的或存在的意义) 并能由实验直接地或间接地加以验证) 。这两种分类方法在形式上尽管不同，但却具有大致相同的内容。
科学美在于自然的理性美，而这种理性美必须由人的理智来把握，因此科学美必然带有浓厚的主观色彩。彭加勒看到了这一点，他说：“美的事物是其本身最适合于我们理智的事物，因此它们同时是这种理智最了解如何使用的工具。”([1]，p.367)“一言以蔽之，数学的美感只不过是由于解与我们思想需要的任意一致的满足，正因为这种真正的一致，这个解在我们看来才能够成为工具。”([1]，pp.372~373)而且，科学美像艺术美一样，并非每一个人都能领略和体会到的，“只有少数有特殊能力的人才能充分地享受它” ([1]，p.280)。 与科学有关的理性美，即科学美的含义和内容是什么呢?彭加勒对此虽未做出十分肯定的回答，但是他却把“雅致”(elegance)、“和谐”(harmony)、“对称”(symmetry)、“平衡” (balance)、“秩序”(order)、“统—”(unity)、“方法的简单性”(simplicity of the means)、“思维经济”(economy of thought)等赋予科学美。彭加勒说： “在解中、在证明中，给我们以美感的东西是什么呢?是各部分的和谐，是它们的对称、它们的巧妙平衡。总而言之，就是引入秩序，给出统一，容许我们清楚地观察和理解整体与细节的东西。”这种科学美也表现为“方法的简单性和问题集合的复杂性的对立”，而且“与思维经济密切相关” ([1]，p.373)他还说：“被我们赋予优美和雅致特征的、能够在我们身上激起美的情感的数学实体是什么呢?它们是这样的实体：其元素和谐地配置，以致当思想认识到细节时，能够毫不费力地包容整体。这种和谐立即满足了我们的审美需要，有助于它所证实和指导的思想。与此同时，一个秩序井然的整体处于我们的双目之下，使得我们能预见数学定律。”([1]，pp.391~392)但是，在这形形色色的含义中，彭加勒最为强调的是“和谐”，他甚至把其它含义也包容于“和谐”的概念之中，有时干脆认为，“普遍和谐是众美之源”([1]，p.209)，“内部和谐是唯一的美”([1]，p.285)。
参考文献 这三本书的英译本是H. Poincare，The Foundations of Science，Authorized Translation by G.B. Halsted，The Science Press，New York and Garrison，N.Y., 1913. 该书是彭加勒逝世后由其他人集其遗著编辑而成的，英译本是H. Poincare，Mathematics and Science：Last Essays，Translated by John W.Bolduc，Dover Publications，Inc., New York，1963. 恩格斯：《自然辩证法》，北京：人民出版社，第216页。 M.J.妮厄：《十九世纪关于原子的争论与一种“中性假设”的二难推论》，北京：《自然科学哲学问题丛刊》，1980年第4期。 ポアンカレ(H.Poineare)著：《科学者と詩人》，平林初之輔訳，岩波書店，1927年，p.139． 《马克思恩格斯全集》第42卷，北京：人民出版社，第97页。 《爱因斯坦文集》第1卷，许良英等编译，北京：商务印书馆1976年版，第298页。
