Introduction

I bring together here different studies relating more or less directly to questions of scientific methodology. The scientific method consists in observing and experimenting; if the scientist had at his disposal infinite time, it would only be necessary to say to him: ‘Look and notice well’; but, as there is not time to see everything, and as it is better not to see than to see wrongly, it is necessary for him to make choice. The first question, therefore, is how he should make this choice. This question presents itself as well to the physicist¹ as to the historian; it presents itself equally to the mathematician², and the principles which should guide each are not without analogy. The scientist conforms to them instinctively³, and one can, reflecting on these principles, foretell⁴ the future of mathematics.

We shall understand them better yet if we observe the scientist at work, and first of all it is necessary to know the psychologic mechanism⁶ of invention and, in particular, that of mathematical creation. Observation of the processes of the work of the mathematician is particularly instructive for the psychologist.

In all the sciences of observation account must be taken of the errors due to the imperfections of our senses and our instruments. Luckily, we may assume that, under certain conditions, these errors are in part self-compensating, so as to disappear in the average; this compensation is due to chance. But what is chance? This idea is difficult to justify⁷ or even to define; and yet what I have just said about the errors of observation, shows that the scientist can not neglect it. It therefore is necessary to give a definition as precise as possible of this concept, so indispensable yet so illusive⁸.

These are generalities applicable in sum to all the sciences; and for example the mechanism of mathematical invention does not differ sensibly from the mechanism of invention in general. Later I attack questions relating more particularly to certain special sciences and first to pure mathematics.

In the chapters devoted⁹ to these, I have to treat subjects a little more abstract. I have first to speak of the notion of space; every one knows space is relative, or rather every one says so, but many think still as if they believed it absolute; it suffices to reflect a little however to perceive to what contradictions they are exposed.

The questions of teaching have their importance, first in themselves, then because reflecting on the best way to make new ideas penetrate¹⁰ virgin¹¹ minds is at the same time reflecting on how these notions were acquired by our ancestors, and consequently on their true origin, that is to say, in reality on their true nature. Why do children usually understand nothing of the definitions which satisfy scientists? Why is it necessary to give them others? This is the question I set myself in the succeeding chapter and whose solution should, I think, suggest useful reflections to the philosophers occupied with the logic⁵ of the sciences.

On the other hand, many geometers believe we can reduce mathematics to the rules of formal logic. Unheard-of efforts have been made to do this; to accomplish it, some have not hesitated, for example, to reverse the historic order of the genesis of our conceptions and to try to explain the finite by the infinite. I believe I have succeeded in showing, for all those who attack the problem unprejudiced, that here there is a fallacious illusion. I hope the reader will understand the importance of the question and pardon me the aridity¹² of the pages devoted to it.

The concluding chapters relative to mechanics and astronomy will be easier to read.

Mechanics seems on the point of undergoing a complete revolution. Ideas which appeared best established are assailed¹³ by bold innovators. Certainly it would be premature¹⁴ to decide in their favor at once simply because they are innovators.

But it is of interest to make known their doctrines¹⁵, and this is what I have tried to do. As far as possible I have followed the historic order; for the new ideas would seem too astonishing unless we saw how they arose.

Astronomy offers us majestic¹⁶ spectacles and raises gigantic problems. We can not dream of applying to them directly the experimental method; our laboratories are too small. But analogy with phenomena¹⁷ these laboratories permit us to attain¹⁸ may nevertheless guide the astronomer¹⁹. The Milky²⁰ Way, for example, is an assemblage of suns whose movements seem at first capricious. But may not this assemblage be compared to that of the molecules²¹ of a gas, whose properties the kinetic²² theory of gases has made known to us? It is thus by a roundabout way that the method of the physicist may come to the aid of the astronomer.

Finally I have endeavored to give in a few lines the history of the development of French geodesy; I have shown through what persevering²³ efforts, and often what dangers, the geodesists have procured²⁴ for us the knowledge we have of the figure of the earth. Is this then a question of method? Yes, without doubt, this history teaches us in fact by what precautions it is necessary to surround a serious scientific operation and how much time and pains it costs to conquer one new decimal.

1 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。

2 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。

3 instinctively   

adv.本能地

参考例句：

• As he leaned towards her she instinctively recoiled. 他向她靠近，她本能地往后缩。 来自《简明英汉词典》
• He knew instinctively where he would find her. 他本能地知道在哪儿能找到她。 来自《简明英汉词典》

4 foretell   

v.预言，预告，预示

参考例句：

• Willow trees breaking out into buds foretell the coming of spring.柳枝绽青报春来。
• The outcome of the war is hard to foretell.战争胜负难以预卜。

5 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。

6 mechanism   

n.机械装置；机构，结构

参考例句：

• The bones and muscles are parts of the mechanism of the body.骨骼和肌肉是人体的组成部件。
• The mechanism of the machine is very complicated.这台机器的结构是非常复杂的。

7 justify   

vt.证明…正当(或有理)，为…辩护

参考例句：

• He tried to justify his absence with lame excuses.他想用站不住脚的借口为自己的缺席辩解。
• Can you justify your rude behavior to me?你能向我证明你的粗野行为是有道理的吗？

8 illusive   

adj.迷惑人的，错觉的

参考例句：

• I don't wanna hear too much illusive words.我不想听太多虚假的承诺。
• We refuse to partake in the production of illusive advertisements.本公司拒绝承做虚假广告。

9 devoted   

adj.忠诚的，忠实的，热心的，献身于...的

参考例句：

• He devoted his life to the educational cause of the motherland.他为祖国的教育事业贡献了一生。
• We devoted a lengthy and full discussion to this topic.我们对这个题目进行了长时间的充分讨论。

10 penetrate   

v.透(渗)入；刺入，刺穿；洞察，了解

参考例句：

• Western ideas penetrate slowly through the East.西方观念逐渐传入东方。
• The sunshine could not penetrate where the trees were thickest.阳光不能透入树木最浓密的地方。

11 virgin   

n.处女，未婚女子；adj.未经使用的；未经开发的

参考例句：

• Have you ever been to a virgin forest?你去过原始森林吗？
• There are vast expanses of virgin land in the remote regions.在边远地区有大片大片未开垦的土地。

12 aridity   

n.干旱，乏味；干燥性；荒芜

参考例句：

• The name Sahara conjures up images of a desert of aridity. "撒哈拉"这个名字使人想起干旱的沙漠情景。 来自辞典例句
• The name conjures up images of a desert of aridity. “撒哈拉”这个名字使人想起“干旱”的沙漠情景。 来自互联网

13 assailed   

v.攻击( assail的过去式和过去分词 )；困扰；质问；毅然应对

参考例句：

• He was assailed with fierce blows to the head. 他的头遭到猛烈殴打。
• He has been assailed by bad breaks all these years. 这些年来他接二连三地倒霉。 来自《用法词典》

14 premature   

adj.比预期时间早的；不成熟的，仓促的

参考例句：

• It is yet premature to predict the possible outcome of the dialogue.预言这次对话可能有什么结果为时尚早。
• The premature baby is doing well.那个早产的婴儿很健康。

15 doctrines   

n.教条( doctrine的名词复数 )；教义；学说；（政府政策的）正式声明

参考例句：

• To modern eyes, such doctrines appear harsh, even cruel. 从现代的角度看，这样的教义显得苛刻，甚至残酷。 来自《简明英汉词典》
• His doctrines have seduced many into error. 他的学说把许多人诱入歧途。 来自《现代汉英综合大词典》

16 majestic   

adj.雄伟的，壮丽的，庄严的，威严的，崇高的

参考例句：

• In the distance rose the majestic Alps.远处耸立着雄伟的阿尔卑斯山。
• He looks majestic in uniform.他穿上军装显得很威风。

17 phenomena   

n.现象

参考例句：

• Ade couldn't relate the phenomena with any theory he knew.艾德无法用他所知道的任何理论来解释这种现象。
• The object of these experiments was to find the connection,if any,between the two phenomena.这些实验的目的就是探索这两种现象之间的联系，如果存在着任何联系的话。

18 attain   

vt.达到，获得，完成

参考例句：

• I used the scientific method to attain this end. 我用科学的方法来达到这一目的。
• His painstaking to attain his goal in life is praiseworthy. 他为实现人生目标所下的苦功是值得称赞的。

19 astronomer   

n.天文学家

参考例句：

• A new star attracted the notice of the astronomer.新发现的一颗星引起了那位天文学家的注意。
• He is reputed to have been a good astronomer.他以一个优秀的天文学者闻名于世。

20 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。

21 molecules   

分子( molecule的名词复数 )

参考例句：

• The structure of molecules can be seen under an electron microscope. 分子的结构可在电子显微镜下观察到。
• Inside the reactor the large molecules are cracked into smaller molecules. 在反应堆里，大分子裂变为小分子。

22 kinetic   

adj.运动的；动力学的

参考例句：

• There exist many sources of energy both potential and kinetic.存在着许多势能和动能的能源。
• The kinetic theory of gases is the best known example.气体动力学理论就是最有名的例子。

23 persevering   

a.坚忍不拔的

参考例句：

• They will only triumph by persevering in their struggle against natural calamities. 他们只有坚持与自然灾害搏斗，才能取得胜利。
• Success belongs to the persevering. 胜利属于不屈不挠的人。

24 procured   

v.（努力）取得， （设法）获得( procure的过去式和过去分词 )；拉皮条

参考例句：

• These cars are to be procured through open tender. 这些汽车要用公开招标的办法购买。 来自《现代汉英综合大词典》
• A friend procured a position in the bank for my big brother. 一位朋友为我哥哥谋得了一个银行的职位。 来自《用法词典》