Repeat the preface of the book "The Preliminary Study of Li's geometric structure"

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Note:

This is the preface to Mr. Wu hongxi's book the preliminary study of the LI Shi ry. Although it is about mathematics, I think our study
It is very helpful, so I will spend some time posting it and share it with you.

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To readers

"Your Career should grow like a tree. It should be natural, uninterrupted, and comprehensive. I hope that your roots will be as deep as possible under the fertile land of this school to make your trunk grow thick and strong. In this way, no matter how thick the leaves are, there will never be problems with water supply. From time to time, there will be storms and lightning in the sky. So do not grow too fast or too high ."

The above is part of the speech given to the students at the Opening Ceremony of the old Vic theatre Institute on 1947. These words have special significance for you. This book is a very preliminary book. If you are interested in reading this book, you should at least find out what you can get from it. At present, in the minds of graduate students, the most urgent question seems to be: Is there a small question that can write an article? Therefore, I need to declare that this book does not discuss small topics of this category. I wrote this book in the hope that "your roots should be as deep as possible ". In the future, whether or not to develop branches or leaves is dependent on your own efforts and talents. The topics discussed in the book are both general and fundamental, and are also familiar to any physicist. If you can master several basic concepts and have a comprehensive understanding of ry in the future, you will naturally be able to pick up some meaningful questions in the future. This is not the attitude of a matcher. This book should be one of your stepping stones. I hope that you will soon be out of the scope of this book.

The author of each book has one thing that is the same as a magician, that is, to expect the audience or the reader to see everything that he wants his readers or the audience to see. In my mind, what do you see in this book?

First, you will understand that the definitions and theorems in the book are both artificial and reasonable. Maybe you think that a book is profound and unpredictable to show the profound knowledge of the author, but I hope that you will feel everything in the book is not taken for granted, in addition, it is easy to make it by yourself as long as you are willing to spend a little effort. To achieve this, in addition to the general "definition-> Theorem-> proof" basic form, I tried to add more words to clarify the ins and outs of each major definition and main theorem and the intuitive meaning. On the other hand, I also want to point out that the concepts and results in the book are considered basic, not because the authority of XX said so, but because after time passes the test, this is exactly the case. That is to say, from the summary of experience, we now know that these concepts and theorems are useful and necessary. Therefore, a beginner should explore why they are useful and necessary. Otherwise, he cannot have a full understanding of what he has learned. This kind of academic attitude is not only applicable to mathematics, but also to all fields of learning, including social science.

Secondly, I hope that you can grasp the key points of the book, as well as each theorem, each proof, and each concept. A good math book should be different from a dictionary. Each word in the latter occupies the same position. However, if there are countless definitions in this book, the theorem and proof are equally important, it is no stranger. For example, the quadratic variational formula of the arc length is just the result of a general technique. The main point is to figure out how to apply it to specific situations, instead of exploring the depth of the formula itself or studying its derivation. Therefore, we should not simply calculate this formula, nor apply it to it. We should not regard this formula as one of the main theorems. For example, the proof of the Synge theorem seems quite cumbersome. However, starting from a very intuitive fact, it is "a shortest closed curve with a non-single-connection condition must exist on any non-single-connected and tightly-connected column ", then everything else is justified.
So I want you to develop a habit and always ask: what are the points of this book? What are the highlights of this chapter? What are the points of this proof? Only by finding the answers to all these questions can we truly understand them.

Finally, I hope that you can fully understand the content of this book from an intuitive perspective. All mathematics books are full of technical terms, because to be clear, the author has no choice. However, a matcher's thinking is mostly driven forward by an intuitive (or even overly simplified) idea. This is especially important in geometry. Therefore, this kind of intuitive discussion in the book is more than that in other mathematics classes. Maybe you are also superstitious about the so-called "strict mathematics", thinking that the most important thing in mathematics is the correctness of every deduction. This argument is equivalent to the benefit of Lu Xun's article, mainly because every sentence is well written. I hope you will not make this mistake.

Of course, the above three points are my personal fantasies. There may be a lot of distance between reality and fantasy. But after knowing my intention, I hope that you can use your imagination to fill in the shortcomings of this book. When I spoke about this lesson, it was just the same as the Olympics. Because of the great harvest of the Olympic Games in the motherland, it naturally caused the question of "why Chinese mathematicians cannot win the gold medal in the field of mathematics. As a result, the "Gold Medal" slogan went viral, and everyone in the summer Center put the same question mark on the face: "When can China win the first gold medal in the field of mathematics ?" This issue was even raised in magazines and newspapers. This idea is very irritating. If you can really think of a serious learning as a sports competition, you will be able to play in the future. For example, the first page of the People's Daily may have the following title: "Poincare and Gauss are fighting in the topology field. Poincare is a great winner, with a ratio of five to zero ." Or: "group discussion finals, Abel fought Galois, unfortunately two to three defeated" and so on. But I guess the main purpose of advocating "Gold Medal" in mathematics is just encouragement. This is a good intention, but it is unfortunately misunderstood. I think that the ultimate goal of mathematics is nothing more than a prize. There is a lot of overlap with the ancient feudal ideology of "getting famous in one fell swoop. You must be aware that, in your own generation, this kind of utilitarian idea is growing and you cannot use the "Gold Medal" as encouragement. What I think is worth doing is to encourage you to cultivate a scholar's demeanor of "seeking truth from facts and doing what you have learned with a long history. However, it is not just a few words that can be clearly explained, and I am afraid there are some preaching tastes, so I will come back and discuss mathematics with you.

Another purpose of "winning the gold medal" is to give a goal and hope that everyone will go in this direction. From the standpoint of a math, this practice does not seem thorough enough. If you really want to stick to this point, simply give the greatest mathematician an example for young people. Gu yuyun: "The method is adopted, and the method is moderate ." According to my views, the numbers that have reached the peak in the history of mathematics are also Gaussian, riman, Abel, and poinare of the 19th century. Hermann Weyl said in his speech in the 1944 s that his academic achievements are not as good as those of gales and riman. But it is also Weyl, without any ambiguity: "in our generation (Weyl myself), there is no mathematician comparable to Hilbert ." Weyl is recognized as one of the Second mathematicians of the century, and is also the last full power in the history of mathematics. However, from his review, we can understand why, if we want to climb the peak of mathematics, we have to take these nineteenth century masters as examples. To understand their achievements, you must read their complete set.

If you only talk about "winning gold medals", but not about this obvious fact, It will virtually turn to encouraging young people to "Learn from the law", and the result will naturally "get the right ". This is out of line with the original intention of advocating this slogan. You must think that "learning from masters" is just a pleasant and impractical sentence. This is understandable. After all, young people love fashion and reading articles will always get better and better. So the articles 10 or 20 years ago are outdated, let alone the articles of the 19th century? However, I do not need to defend this term, because someone who learns more than me will replace me. When I was a graduate student, I once went to the Andre Weil lecture. At the beginning, he said that young people must read the full set of first-stream mathematicians, such as Gauss and Euler. In this regard, Weil is a person with consistent words and deeds. In 1947, he suffered from low mood for a while, but he was inspired by reading Gaussian's collection and made a series of guesses. This is the "weil conjecture" that dominated the development of algebra and ry over the past thirty years ". In fact, there are too many similar examples. We recommend the following articles for your own experience:

An article about Gaussian's creation of modern Curved Surface ry: disquisitiones Generales circa superficies curves. This article has recently published new English translations and comments. Please refer to P. dombroski, 150 years after Gauss 'disquisitiones..., asterisque, vol.62, Soc. Math. france.1979

A short article about the creation of the "riman ry": ueber die Hypothesen, welche der geometrie zu grunde liegen. the English translation and detailed explanation of this article can be found in the reference [S8, ii] of this book.

Three Poincare:
Analysis situs, J. Ecole polytechnique (2) 1 (1895), 1-121;
1R complement, rend. CIRC. Mat. Palermo 13 (1899), 285-343;
2d complement, Proc. London. Math. Soc. 32 (1900), 277-308;
3E complement, Bull. Soc. Math. France 30 (1902), 49-70;
4e complement, J. Math. Pure. Appl. (5) 8 (1902), 169-214;
5E complement, rend. CIRC. Mat. Palermo 18 (1904), 45-110.

These articles are basically understandable to you. At the same time, I can assure you that they will inspire you infinitely.

Finally, let's go back to the "Gold Medal" issue. Most people think that the only purpose of participating in the Olympic Games is to win the gold medal. Last year, Li Ning won three gold medals, and the whole country said they would celebrate the event, while Tong was not able to win the event. Therefore, the separation of success and failure is clear at a glance based on the "Gold Medal" scale. But Li Ning, the "gold medal winner", has his personal thoughts? You can go to the library to review his article published in newspapers at the end of 1984, with the title of "Children are really heroes". Then you can see another point of view. In fact, participating in sports competitions or studying mathematics is just a part of life. Exploring the significance of life is a university question that we can't finish every day after death. The two paragraphs recorded below may be enough to provide some distinctive ideas for your reference. The first section is said by Pierre de Coubertin, founder of the modern Olympics:

The purpose of the sport is not to win but to compete
The meaning of life is not overcome but struggling

The other paragraph is one of the sayings of the ancient Greek Olympics:

Do not claim victory. You only need the courage to go forward. Because from the indomitable struggle, you will receive honors for yourself. But more importantly, you will be honored for all mankind.

Wu hongxi

June 1985 at Peking University

Repeat the preface of the book "The Preliminary Study of Li's geometric structure"