直觉与潜意识（一）

亨利·庞加莱（Henri Poincaré）是19世纪末，二十世纪初的领袖数学家，法国最伟大的数学家之一。他在数学、物理和哲学等方面有着众多创造性和基础性的贡献，也被称为“最后一位全才数学家”。与此同时，庞加莱还是少有的，对自己思考数学时大脑的工作状态有极大兴趣和深刻理解的数学家。下面这篇文章来自庞加莱的论文Mathematical Creation，向读者描述和展示了直觉与潜意识这两种数学创造中必不可少的元素。

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...

数学创造的源头是一个应该引起心理学家强烈兴趣的问题。 在数学创造活动中，人类的思维似乎对外界的依赖最少，且只对人类自身或只在人类自身上发挥作用，因此，在研究几何思想的过程中，我们希望寻得人类思维中必不可少的东西......

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?

第一个事实会使我们感到惊讶——或者更确切地说——如果我们不太习惯它的话，我们便会感到惊讶：怎么会有人无法理解数学呢？如果数学仅仅调用逻辑的规则，比如一些所有正常人都会接受的规则；如果它的论据是基于一种对所有人来说都很普遍，而且正常人都不会去否认它的原则，那么怎么会有那么多人深陷其中呢？

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.

不是每个人都能进行创造性的工作，这是很正常的。此外，也不是每个人都能记住学习过的例子。与之不同且令人惊讶的是，并不是每个人都能理解数学推导的过程，（从逻辑上看这是很奇怪的，见下文）。事实上，大多数人都很难跟上推导的节奏，这是不可否认的，当然，一个有经验的中学老师肯定也会同意这点。

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?...

让我们更进一步：在学习数学或研究数学时，为什么会出错？理智的头脑不应该犯逻辑谬误，有好头脑的人也不会被困在简短的推导中，因为对他们来说这就像处理日常事务一样简单，然而这些人却不能无误地跟上数学推导和演算的节奏，但这些数学演示毕竟只是简单推理的累积，完全类似于他们能够轻易得出的结论。难道说数学家们也做不到这点吗？

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.

我必须承认，对于我来说，我没法准确无误地做加法运算。我的记忆力也不差，但我并不是一个好的棋手。那么为什么大多数棋手会迷失在数学推导中而我却不会？显然，下棋时的推理是由普遍的、一般的推导来引导的。但数学证明并不是简单地把三段论并列起来，而是把三段论按一定的顺序排列，而这些排列的顺序比思考某一个具体的三段论要重要得多。如果我有这样的感觉，或者说直觉，对于这个三段论的顺序，我看一眼就能对整个推导过程略知一二，那么我根本不用担心会忘记其中一个的一个具体步骤，它们自动就会被分配到一个序列中，这不需要我动用任何记忆。

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，这种对于数学顺序的直觉，使我们能够预知到隐藏在其中的和谐与联系，这种直觉不是每个人都能拥有的。有些人既没有这种难以定义的微妙感觉，也没有超乎寻常的记忆力和注意力，那么他们就绝对无法理解高等数学，大多数人都是这样。另一些人可能只是在很小的程度上有这种感觉，但他们被赋予了罕见的记忆力和巨大的注意力，他们能一个接一个地记住细节，他们能理解数学，有时也能应用，但他们不能创造。最后，另一些人，他们或多或少地拥有这种特殊的直觉，即使他们的记忆力并不超群，他们也可以理解数学，甚至可能成为创造者。他们在数学创造上获得成就的大小与他们的这种直觉被开发了多少相对应。

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？它不在于用已知的数学实体进行新的组合。任何人都可以这样做，但这样得到的组合在数量上将会是无限的，而且其中的大多数都是我们完全不感兴趣的。创造就是不做无用的组合，而只做那些有用的，这往往只是少数的组合。因此，创造是洞察力和选择。

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.

现在是时候深入探究，看看在数学家的灵魂深处究竟发生了什么。为此，我相信，通过回顾我的记忆，我能做到最好。但我将仅限于叙述我是如何写出我的第一本关于 Fuchsian 函数的回忆录的。请读者原谅，我将使用一些专业用语，但不必吓唬自己，因为你们不需要去理解它们。例如，我将会说，我已经找到了这个定理在这种情况下的证明，而这个定理有一个吓人的名字，很多人都不熟悉，但这并不重要；心理学家感兴趣的不是定理，而是环境。

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.

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