开篇

The beginning

1. 数学家其实很着迷。没有迷恋，就没有数学。诺瓦利斯

1. Mathematicians are actually very fascinated. Without fascination, there would be no mathematics. Novalis

2. 没有大胆的猜测，就不可能有伟大的发现。牛顿

2. Without daring guesses, there can be no great discoveries. - Newton

3. 数字统治宇宙。毕达哥拉斯

3. Numbers rule the universe. Pythagoras.

4. 数学，科学女王；数论，数学女王。高斯

4. Mathematics, the queen of sciences; number theory, the queen of mathematics. Gauss

5. 上帝创造了整数，剩下的都是人造的。克隆脖子

5. God created integers, and the rest are man-made. Clone neck.

6. 上帝是数学家——雅各比

6. God is a mathematician - Jacobi

7. 不是诗人的数学家永远不可能是一个完整的数学家。-威尔斯特斯

7. A mathematician who is not a poet can never be a complete mathematician. - Vilsters

8. 在其现代发展阶段，纯数学可以说是人类精神最原始的创造。-白石海

8. In its modern development stage, pure mathematics can be said to be the most primitive creation of the human spirit. - Shibai Hai

9. 无限！没有其他问题如此深刻地触动了人类的心灵。-希尔伯特

9. Infinity! No other question has touched the human heart so deeply. - Hilbert

10. 发现每一个新的组在形式上都是数学的，因为我们不能有其他的指导。-达尔文

10. We find that every new group is formal in mathematics, because we cannot have any other guidance. - Darwin

11. 给我五个系数，我就画一头大象。给我六个系数，大象就会摇尾巴。-柯西

11. Give me five coefficients, and I'll draw an elephant. Give me six coefficients, and the elephant will wag its tail. - Cauchy

12. 纯数学是魔术师真正的魔杖。诺瓦利斯

12. Pure mathematics is the true wand of the magician. Novalis

13. 谁要是不知道正方形的对角线边是一个不可公度的量，那他就不配这个称号。-柏拉图

13. Whosoever does not know that the diagonal of a square is a quantity that cannot be measured by any number, is not worthy of that title. - Plato

14. 几个世纪以来，整数的简单组合一直是数学新生的源泉。-伯克霍夫

14. For several centuries, the simple combinations of integers have been the source of new life in mathematics. - Bernhard Riemann

15. 数学无与伦比的永恒性和万能性，以及它在时间和文化背景上的独立性，是其本质的直接后果。-艾伯

15. The unparalleled eternity and versatility of mathematics, as well as its independence in terms of time and cultural context, are direct consequences of its essence. - Eber

16. 人生只有两件事，发展数学和教授数学。

16. There are only two things in life: developing mathematics and teaching mathematics.

17. 我比笛卡尔看得更远，因为我站在巨人的肩膀上。牛顿

17. I have seen further than Descartes because I have stood on the shoulders of giants. - Newton

18. 如果我继承了可观的财产，我可能在数学上没什么价值。拉格朗日

18. If I inherited considerable wealth, I might not have much value mathematically. Lagrange

19. 我把数学当成一份有趣的工作，而不是自己的丰碑。可以肯定的是，比起自己的作品，我更喜欢别人的作品。我总是对我的工作不满意。拉格朗日

19. I consider mathematics as an interesting occupation, rather than a monument to myself. It is certain that I prefer others' works over my own. I am always dissatisfied with my work. Lagrange

20. 一个人的贡献与他的自负严格成反比，这似乎是一条行为公理。拉格朗日

20. A person's contribution is strictly inversely proportional to his arrogance, which seems to be a behavioral axiom. Lagrange

21. 看在上帝的份上，不要放弃你的工作！这是你最好的药。-达朗贝尔

21. For God's sake, don't give up your job! This is your best medicine. - d'Alembert

22. 有时候，你一开始得不到最简单最奇妙的证明，但正是这种证明，才能深入到高等算术真理的奇妙联系中去。这是我们继续研究的动力，最能让我们有所发现。高斯

22. Sometimes, at first, you may not obtain the simplest and most wonderful proof, but it is precisely this kind of proof that allows you to delve into the wonderful connections of higher arithmetic truths. This is the driving force behind our continued research, and it is the most conducive to our making discoveries. Gauss

23. 如果别人也像我一样深入而执着地思考数学的真理，他们也会发现我的发现。高斯

23. If others also think deeply and persistently about the truths of mathematics as I do, they too will discover my discoveries. Gauss

24. 精致的论证往往不是一蹴而就的，而是长期学习和积累的结果。我是慢慢学会的，我会继续学下去。-亚伯

24. A refined argument is often not achieved overnight but is the result of long-term learning and accumulation. I have learned slowly, and I will continue to learn. - Abo

25. 异常抽象的问题一定要讨论清楚。笛卡尔

用英语翻译为： 25. Issues that are exceptionally abstract must be discussed clearly. Descartes

26. 数学是人类知识活动留下的最强大的知识工具，是一些现象的根源。数学是不可改变的，是客观存在的，上帝会按照数学的规律来建造宇宙。笛卡尔

26. Mathematics is the most powerful knowledge tool left behind by human intellectual activity and is the source of some phenomena. Mathematics is immutable and objectively exists; God will create the universe according to the laws of mathematics. Descartes

27. 直接向大师学习，而不是向他们的学生学习。-亚伯

27. Learn directly from the masters, not from their students. - Abar

28. 选择明确的研究对象，持之以恒。你可能永远也不会到达终点，但是你一定可以在途中发现一些有趣的事情。克莱恩

28. Choose a clear research object and persist in it. You may never reach the end, but you are sure to find some interesting things along the way. Kline.

29. 我从不把工作当成个人的事，也不追求名气和赞誉。我只是为真理的进步尽我所能。不管是我还是别人，对我来说都无所谓。重要的是更接近事实。-威尔斯特斯

29. I never consider work as a personal matter, nor do I seek fame and praise. I merely strive to the best of my ability for the advancement of truth. Whether it is me or someone else, it doesn't matter to me. What is important is getting closer to the truth. - Wilsteds

30. 思维的运动形式通常是这样的:意识研究——潜意识活动——意识研究。-庞加莱

30. The form of movement of thought is usually this: consciousness study - subconscious activity - consciousness study. - Poincaré

31. 生活是一场不断的斗争。如果说我们偶尔享受和平，那是我们的祖先在顽强地战斗。如果我们的精神和警惕性一时放松，我们就会失去祖先为我们赢得的成果。-庞加莱

31. Life is a continuous struggle. If we occasionally enjoy peace, it is because our ancestors have fought valiantly. If our spirit and vigilance relax for a moment, we will lose the achievements won for us by our ancestors. - Poincaré

32. 一个人的贡献和他的自负严格地成反比，这似乎是品行上的一个公理。----拉格朗日

32. A person's contribution is strictly inversely proportional to his egoism, which seems to be an axiom of character. ---- Lagrange

33. 时间是个常数，但对勤奋者来说，是个变数。用分来计算时间的人比用小时来计算时间的人时间多59倍。 俄国历史学家雷巴柯夫

33. Time is a constant, but to the diligent, it is a variable. A person who calculates time in minutes has 59 times more time than a person who calculates time in hours. - Russian historian Rybakov

34. 自然这一巨举是用数学符号写成的。--伽里略

34. Nature's grand design is written in mathematical symbols. -- Galileo

35. 整数的简单构成，若干世纪以来一直是使数学获得新生的源泉。--G D 伯克霍夫

35. The simple composition of integers has been a source of renewal for mathematics for several centuries. -- G D Birkhoff

36. 我国著名数学家华罗庚在谈到学习与探索时指出：在学习中要敢于做减法，就是减去前人已经解决的部分，看看还有那些问题没有解决，需要我们去探索解决。

36. The famous Chinese mathematician Hua Loo-keng pointed out when talking about learning and exploration: In learning, we should dare to do subtraction, that is, subtract the parts that have already been solved by predecessors, and see what other problems have not been solved and need us to explore and solve.

37. 日累月积见功勋，山穷水尽惜寸阴。 --华罗庚

37. Accumulated efforts over time yield meritorious achievements; at the end of the mountain and the end of the water, one regrets even a moment's waste. --Hua Luogeng

38. 锦城虽乐，不如回故乡;乐园虽好 ，非久留之地。归去来兮。--华罗庚

38. Though Jincheng is pleasant, it is not as good as returning to my hometown; though the Paradise is beautiful, it is not a place to stay for long. Return to your home, come back. -- Hua Luogeng

39. 时间是个常数，花掉一天等于浪费24小时。--陈景润

39. Time is a constant; spending a day is equivalent to wasting 24 hours. -- Chen Jingrun

40. 这是一个可靠的规律，当数学或哲学着作的作者以模糊深奥的话写作时，他是在胡说八道。--A N 怀德海

40. This is a reliable rule: when an author of mathematical or philosophical works writes in vague and abstruse language, he is talking nonsense. -- A.N. Whitehead

41. 一种奇特的美统治着数学王国，这种美不像艺术之美与自然之美那么相类似，但她深深地感染着人们的心灵，激起人们对她的欣赏，与艺术之美是十分相象的。--库默

41. A peculiar beauty reigns over the kingdom of mathematics, a beauty that is not so similar to the beauty of art and nature, but it deeply touches people's hearts, arousing their appreciation for it, which is very similar to the beauty of art. -- Kummer

42. 一个没有几分诗人气的数学家永远成不了一个完全的数学家。--维尔斯特拉斯

42. A mathematician without a bit of poet's spirit can never be a complete mathematician. -- Weierstrass

43. 数学的本质在於它的自由。－－－康扥尔（Cantor）

43. The essence of mathematics lies in its freedom. —— Cantor

44. 在数学的领域中，提出问题的艺术比解答问题的艺术更为重要。康扥尔（Cantor）

44. In the field of mathematics, the art of posing questions is more important than the art of solving them. Cantor

45. 没有任何问题可以向无穷那样深深的触动人的情感，很少有别的观念能像无穷那样激励理智产生富有成果的思想，然而也没有任何其他的概念能向无穷那样需要加以阐明。希尔伯特（Hilbert）

45. There is no question that can stir human emotions as deeply as infinity does, and few other ideas can inspire the intellect to generate fruitful thoughts as much as infinity does. However, there is no other concept that requires as much clarification as infinity. - Hilbert

46. 只要一门科学分支能提出大量的问题，它就充满着生命力，而问题缺乏则预示着独立发展的终止或衰亡。－－Hilbert

46. As long as a branch of science can propose a large number of questions, it is full of vitality, and the lack of questions predicts the termination or decline of independent development. -- Hilbert

47. 无限！再也没有其他问题如此深刻地打动过人类的心灵。--D 希尔伯特

47. Infinity! No other question has ever so deeply touched the human heart. -- David Hilbert

48. 我们能够期待，随着教育与娱乐的发展，将有更多的人欣赏音乐与绘画。但是，能够真正欣赏数学的人数是很少的。--贝尔斯

48. We can expect that with the development of education and entertainment, more people will appreciate music and painting. However, the number of people who truly appreciate mathematics is very few. --Bells

49. 要想建立空间观念，必须有动手操作的过程，动手操作的过程，不仅是一个实践的过程，更是尝试、想象、推理和反思的过程。

49. To establish a spatial concept, there must be a process of hands-on operation. The process of hands-on operation is not only a practical process but also a process of trying, imagining, reasoning, and reflecting.

50. 数学教学活动必须建立在学生认知发展水平和已有的生活经验基础上，让学生亲身经历体验、探索，将实际问题抽象成数学模型，并进行解释与应用的过程。

50. Mathematics teaching activities must be based on the students' cognitive development level and their existing life experiences, allowing students to personally experience and explore, abstract practical problems into mathematical models, and explain and apply them.

51. 如果谁不知道正方形的对角线同边是不可通约的量，那他就不值得人的称号。柏拉图

51. If anyone does not know that the diagonal of a square is incommensurable with its side, he does not deserve to be called a human being. — Plato

52. 浮躁的人容易问：我到底该怎么学；别问，学就对了。

52. Impatient people tend to ask: "How should I actually learn?" Just learn, and don't ask.

53. 无限！再也没有其他问题如此深刻地打动过人类的心灵。D希尔伯特

53. Infinity! No other question has so deeply touched the hearts of humanity. David Hilbert

54. 天才是不足恃的，聪明是不可靠的，要想顺手拣来的伟大科学发明是不可想象的。--华罗庚

54. Talent is not enough to rely on, intelligence is not reliable, and it is unimaginable to pick up great scientific inventions easily. -- Hua Luogeng