General Conclusions

What I have sought to explain in the preceding pages is how the scientist should guide himself in choosing among the innumerable facts offered to his curiosity, since indeed the natural limitations of his mind compel him to make a choice, even though a choice be always a sacrifice. I have expounded¹ it first by general considerations, recalling on the one hand the nature of the problem to be solved and on the other hand seeking to better comprehend that of the human mind, which is the principal instrument of the solution. I then have explained it by examples; I have not multiplied them indefinitely; I also have had to make a choice, and I have chosen naturally the questions I had studied most. Others would doubtless have made a different choice; but what difference, because I believe they would have reached the same conclusions.

There is a hierarchy² of facts; some have no reach; they teach us nothing but themselves. The scientist who has ascertained³ them has learned nothing but a fact, and has not become more capable of foreseeing new facts. Such facts, it seems, come once, but are not destined⁴ to reappear.

There are, on the other hand, facts of great yield; each of them teaches us a new law. And since a choice must be made, it is to these that the scientist should devote himself.

Doubtless this classification is relative and depends upon the weakness of our mind. The facts of slight outcome are the complex facts, upon which various circumstances may exercise a sensible influence, circumstances too numerous and too diverse for us to discern them all. But I should rather say that these are the facts we think complex, since the intricacy of these circumstances surpasses the range of our mind. Doubtless a mind vaster and finer than ours would think differently of them. But what matter; we can not use that superior mind, but only our own.

The facts of great outcome are those we think simple; may be they really are so, because they are influenced only by a small number of well-defined circumstances, may be they take on an appearance of simplicity⁵ because the various circumstances upon which they depend obey the laws of chance and so come to mutually compensate⁶. And this is what happens most often. And so we have been obliged to examine somewhat more closely what chance is.

Facts where the laws of chance apply become easy of access to the scientist who would be discouraged before the extraordinary complication of the problems where these laws are not applicable. We have seen that these considerations apply not only to the physical sciences, but to the mathematical sciences. The method of demonstration⁷ is not the same for the physicist⁸ and the mathematician⁹. But the methods of invention are very much alike. In both cases they consist in passing up from the fact to the law, and in finding the facts capable of leading to a law.

To bring out this point, I have shown the mind of the mathematician at work, and under three forms: the mind of the mathematical inventor and creator; that of the unconscious geometer who among our far distant ancestors, or in the misty¹⁰ years of our infancy¹¹, has constructed for us our instinctive¹² notion of space; that of the adolescent to whom the teachers of secondary education unveil the first principles of the science, seeking to give understanding of the fundamental definitions. Everywhere we have seen the r?le of intuition and of the spirit of generalization¹³ without which these three stages of mathematicians¹⁴, if I may so express myself, would be reduced to an equal impotence.

And in the demonstration itself, the logic¹⁵ is not all; the true mathematical reasoning is a veritable induction¹⁶, different in many regards from the induction of physics, but proceeding¹⁷ like it from the particular to the general. All the efforts that have been made to reverse this order and to carry back mathematical induction to the rules of logic have eventuated only in failures, illy concealed¹⁸ by the employment of a language inaccessible¹⁹ to the uninitiated. The examples I have taken from the physical sciences have shown us very different cases of facts of great outcome. An experiment of Kaufmann on radium rays revolutionizes at the same time mechanics, optics and astronomy. Why? Because in proportion as these sciences have developed, we have the better recognized the bonds uniting them, and then we have perceived a species of general design of the chart of universal science. There are facts common to several sciences, which seem the common source of streams diverging²⁰ in all directions and which are comparable to that knoll²¹ of Saint Gothard whence spring waters which fertilize²² four different valleys.

And then we can make choice of facts with more discernment than our predecessors²³ who regarded these valleys as distinct and separated by impassable barriers.

It is always simple facts which must be chosen, but among these simple facts we must prefer those which are situated²⁴ upon these sorts of knolls²⁵ of Saint Gothard of which I have just spoken.

And when sciences have no direct bond, they still mutually throw light upon one another by analogy. When we studied the laws obeyed by gases we knew we had attacked a fact of great outcome; and yet this outcome was still estimated beneath its value, since gases are, from a certain point of view, the image of the milky²⁶ way, and those facts which seemed of interest only for the physicist, ere long opened new vistas²⁷ to astronomy quite unexpected.

And finally when the geodesist sees it is necessary to move his telescope some seconds to see a signal he has set up with great pains, this is a very small fact; but this is a fact of great outcome, not only because this reveals to him the existence of a small protuberance upon the terrestrial globe, that little hump would be by itself of no great interest, but because this protuberance gives him information about the distribution of matter in the interior of the globe, and through that about the past of our planet, about its future, about the laws of its development.

The End

1 expounded   

论述，详细讲解( expound的过去式和过去分词 )

参考例句：

• He expounded his views on the subject to me at great length. 他详细地向我阐述了他在这个问题上的观点。
• He warmed up as he expounded his views. 他在阐明自己的意见时激动起来了。

2 hierarchy   

n.等级制度；统治集团，领导层

参考例句：

• There is a rigid hierarchy of power in that country.那个国家有一套严密的权力等级制度。
• She's high up in the management hierarchy.她在管理阶层中地位很高。

3 ascertained   

v.弄清，确定，查明( ascertain的过去式和过去分词 )

参考例句：

• The previously unidentified objects have now been definitely ascertained as being satellites. 原来所说的不明飞行物现在已证实是卫星。 来自《简明英汉词典》
• I ascertained that she was dead. 我断定她已经死了。 来自《简明英汉词典》

4 destined   

adj.命中注定的；(for)以…为目的地的

参考例句：

• It was destined that they would marry.他们结婚是缘分。
• The shipment is destined for America.这批货物将运往美国。

5 simplicity   

n.简单，简易；朴素；直率，单纯

参考例句：

• She dressed with elegant simplicity.她穿着朴素高雅。
• The beauty of this plan is its simplicity.简明扼要是这个计划的一大特点。

6 compensate   

vt.补偿，赔偿；酬报 vi.弥补；补偿；抵消

参考例句：

• She used her good looks to compensate her lack of intelligence. 她利用她漂亮的外表来弥补智力的不足。
• Nothing can compensate for the loss of one's health. 一个人失去了键康是不可弥补的。

7 demonstration   

n.表明，示范，论证，示威

参考例句：

• His new book is a demonstration of his patriotism.他写的新书是他的爱国精神的证明。
• He gave a demonstration of the new technique then and there.他当场表演了这种新的操作方法。

8 physicist   

n.物理学家，研究物理学的人

参考例句：

• He is a physicist of the first rank.他是一流的物理学家。
• The successful physicist never puts on airs.这位卓有成就的物理学家从不摆架子。

9 mathematician   

n.数学家

参考例句：

• The man with his back to the camera is a mathematician.背对着照相机的人是位数学家。
• The mathematician analyzed his figures again.这位数学家再次分析研究了他的这些数字。

10 misty   

adj.雾蒙蒙的，有雾的

参考例句：

• He crossed over to the window to see if it was still misty.他走到窗户那儿，看看是不是还有雾霭。
• The misty scene had a dreamy quality about it.雾景给人以梦幻般的感觉。

11 infancy   

n.婴儿期；幼年期；初期

参考例句：

• He came to England in his infancy.他幼年时期来到英国。
• Their research is only in its infancy.他们的研究处于初级阶段。

12 instinctive   

adj.(出于)本能的；直觉的；(出于)天性的

参考例句：

• He tried to conceal his instinctive revulsion at the idea.他试图饰盖自己对这一想法本能的厌恶。
• Animals have an instinctive fear of fire.动物本能地怕火。

13 generalization   

n.普遍性，一般性，概括

参考例句：

• This sweeping generalization is the law of conservation of energy.这一透彻的概括就是能量守恒定律。
• The evaluation of conduct involves some amount of generalization.对操行的评价会含有一些泛泛之论。

14 mathematicians   

数学家( mathematician的名词复数 )

参考例句：

• Do you suppose our mathematicians are unequal to that? 你以为我们的数学家做不到这一点吗？ 来自英汉文学
• Mathematicians can solve problems with two variables. 数学家们可以用两个变数来解决问题。 来自哲学部分

15 logic   

n.逻辑(学)；逻辑性

参考例句：

• What sort of logic is that?这是什么逻辑？
• I don't follow the logic of your argument.我不明白你的论点逻辑性何在。

16 induction   

n.感应，感应现象

参考例句：

• His induction as a teacher was a turning point in his life.他就任教师工作是他一生的转折点。
• The magnetic signals are sensed by induction coils.磁信号由感应线圈所检测。

17 proceeding   

n.行动，进行，(pl.)会议录，学报

参考例句：

• This train is now proceeding from Paris to London.这次列车从巴黎开往伦敦。
• The work is proceeding briskly.工作很有生气地进展着。

18 concealed   

a.隐藏的，隐蔽的

参考例句：

• The paintings were concealed beneath a thick layer of plaster. 那些画被隐藏在厚厚的灰泥层下面。
• I think he had a gun concealed about his person. 我认为他当时身上藏有一支枪。

19 inaccessible   

adj.达不到的，难接近的

参考例句：

• This novel seems to me among the most inaccessible.这本书对我来说是最难懂的小说之一。
• The top of Mount Everest is the most inaccessible place in the world.珠穆朗玛峰是世界上最难到达的地方。

20 diverging   

分开( diverge的现在分词 )； 偏离； 分歧； 分道扬镳

参考例句：

• Plants had gradually evolved along diverging and converging pathways. 植物是沿着趋异和趋同两种途径逐渐演化的。
• With member-country bond yields now diverging, 'it's a fragmented set of markets. 但随着成员国债券收益率之差扩大，市场已经分割开来。

21 knoll   

n.小山，小丘

参考例句：

• Silver had terrible hard work getting up the knoll.对于希尔弗来说，爬上那小山丘真不是件容易事。
• He crawled up a small knoll and surveyed the prospect.他慢腾腾地登上一个小丘，看了看周围的地形。

22 fertilize   

v.使受精，施肥于，使肥沃

参考例句：

• Fertilizer is a substance put on land to fertilize it.肥料是施在地里使之肥沃的物质。
• Reading will fertilize his vocabulary.阅读会丰富他的词汇。

23 predecessors   

n.前任( predecessor的名词复数 )；前辈；（被取代的）原有事物；前身

参考例句：

• The new government set about dismantling their predecessors' legislation. 新政府正着手废除其前任所制定的法律。 来自《简明英汉词典》
• Will new plan be any more acceptable than its predecessors? 新计划比原先的计划更能令人满意吗？ 来自《简明英汉词典》

24 situated   

adj.坐落在...的，处于某种境地的

参考例句：

• The village is situated at the margin of a forest.村子位于森林的边缘。
• She is awkwardly situated.她的处境困难。

25 knolls   

n.小圆丘，小土墩( knoll的名词复数 )

参考例句：

• He carefully surveyed the ridges and knolls once more, and also the ravines and gullies. 他又注意地巡视着那些梁和峁，还有沟和壑。 来自互联网

26 milky   

adj.牛奶的，多奶的；乳白色的

参考例句：

• Alexander always has milky coffee at lunchtime.亚历山大总是在午餐时喝掺奶的咖啡。
• I like a hot milky drink at bedtime.我喜欢睡前喝杯热奶饮料。

27 vistas   

长条形景色( vista的名词复数 )； 回顾； 展望； （未来可能发生的）一系列情景

参考例句：

• This new job could open up whole new vistas for her. 这项新工作可能给她开辟全新的前景。
• The picture is small but It'shows broad vistas. 画幅虽然不大，所表现的天地却十分广阔。