直觉与潜意识（二）

在15天的时间里，我努力证明不可能有任何类似于我后来称之为 Fuchsian 函数的函数。那时我很无知；每天我坐在工作桌前，待上一两个小时，尝试各种组合，但都没有结果。一天晚上，与我的平时习惯相反，我喝了黑咖啡，这让我无法入睡。想法成群涌现；我感觉到它们相互碰撞，直到成对地相互连锁，可以说，形成一个稳定的组合。到第二天早上，我已经确定了一类 Fuchsian 函数的存在性，它们来自超几何级数；我只需写出结果，这只花了几个小时。

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.

随后，我想用两个级数的商来表示这些函数；这个想法是完全有意识，且深思熟虑的，与椭圆函数的类比指导着我。我问自己，如果这些级数存在，它们一定具有什么性质，我毫不费力地成功构造出了我称之为 theta-Fuchsian 的级数。

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.

就在这个时候，我离开了我当时居住的卡昂，在矿山学校的赞助下进行一次地质考察。旅行的变化使我忘记了数学工作。到了科孔茨，我们便坐上公共马车，到什么地方去。就在我踏上这一步的时候，我突然想到，我用来定义富克斯函数的变换与那些非欧几里得几何的变换是相同的，尽管在此之前，我似乎没有任何想法来为这个想法铺平道路。我没有证实这个想法，当时我没有时间，因为我一在公共马车上坐下，就继续着已经开始的谈话，但我觉得这是完全肯定的。我回到卡昂后，跟随内心的想法，在闲暇时刻从容不迫地验证了一下结果。

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.

然后，我把注意力转向了一些算术问题，显然没有多大成功，也没有想到这与我之前的研究有任何联系。我对我的失败感到厌恶，去海边呆了几天，想了些别的事情。一天早上，我走在悬崖上，突然有了一个想法，它和非欧几何一样，具有简洁、突然性和立即可确定的特点，它是一个不定三元二次型的算术变换。

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.

回到卡昂后，我对这个结果进行了思考，并推导出了结果。二次型的例子告诉我，除了超几何级数对应的群之外，还存在 Fuchsian 群；我发现我可以将 theta-Fuchsian 级数的理论应用于它们，并且因此，除了来自超几何级数的函数之外，还存在我当时知道的 Fuchsian 函数。很自然地，我构造出了所有的这些函数。我对他们进行了系统的攻破并一个接一个地承担了所有的工作。然而，还存在着一个问题，解决不了它会使所有努力白费。但是，我所有的那些仅在一开始时起作用的努力，使我更清楚地看到了困难，这的确是件了不起的事。所有这些工作都是在完全有意识的情况下进行的。

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.

于是我去了Mont-Valérien，我要在那里服完兵役，所以我的工作完全不同。一天，我在街上走着，突然想到了解决这个使我停下来的难题的办法。我并没有试图立即深入研究，在我服役之后，我才重拾这个问题。我有了所有的元素，只需要把它们排列起来，然后把它们放在一起。于是，我毫不费力地一笔一划地写出了最终的回忆录。

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

我将仅限于讨论这个例子，举再多的例子是无用的......

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...

在这个例子中，最引人注目的是这种突然式启发的出现，这是长期的、潜意识的前置工作的明显迹象。在我看来，这种潜意识的工作在数学创造中的作用是无可争辩的，在其他不那么明显的情况下，也会发现它的痕迹。当一个人努力解决一个难题时，往往一开始就没有什么好的结果。然后或长或短地休息一会儿，再坐下来重新开始工作。在最初的半小时里，就像以前一样什么也找不到，然后突然间，一个决定性的想法出现在脑海里……

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...

关于这种潜意识工作的条件，还有一点值得注意的是：潜意识工作有可能出现在有意识工作之前，也有可能出现在一段有意识的工作之后。对于我们要解决的问题，这样的潜意识工作肯定是富有成效的。如果没有经过几天自发的努力，这些突然的灵感(前面提到的例子证明了这一点)永远不会出现，而之前有意识工作中的努力似乎毫无结果，似乎没有任何好处，所走的路似乎完全错误。但这些努力也并非像人们想象的那样毫无结果；它们启动了这台无意识的机器，没有它们（指有意识的工作），机器就不会移动，也不会产生任何东西……

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?

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