What is mathematics? 1 what is mathematics, co-written by Courant and Robin, is a world-famous mathematics book. It has been 60 years since the first edition. About 20 years ago, two publishing houses published some Chinese versions. Fortunately, Oxford University Press published an updated version at 1996, and its Chinese translation was recently published by Fudan University Press.
As an outstanding mathematician in the 20th century, Courant studied with mathematicians such as Hilbert at the University of G? ttingen, the holy place of mathematics at that time. After the Nazis came to power, he came to the United States and founded the world-famous Courant Institute. Regarding Courant, Rhett has a biography A Tale of Two Cities by a Mathematician translated and published in China, which contains many stories about Courant and contemporary algebra. Just reading the photos in the book, the collective image of outstanding intellectuals at that time jumped into our eyes with a loud name, which was enough to make us schoolmates admire. Interestingly, the awesome mathematicians in G? ttingen have written wonderful books on the popularization of mathematics, such as Hilbert's Intuitive Geometry, Klein's Elementary Mathematics with High Viewpoint, Wei's Symmetry and Courant's What is Mathematics. The common feature of these works is strategic positioning and accumulation.
Abel once said that we should learn from the master, not from his disciples. Because a master can quickly lead you to the right path.
"What is Mathematics" was highly praised by all sides as soon as it was published. Einstein believed that this book was "a thorough and clear exposition of the basic concepts and methods in the whole field of mathematics". Mathematical masters such as Weil and Morse also praised it. The New York Times is also willing to spend a page to introduce.
Judging from the title alone, there are many choices in the content and genre of this book (too wide a choice, sometimes it is both a freedom and a problem). This book, for example, can be written as a book for children or a monograph for great people (similar to the well-known Mathematics by bourbaki). The genre chosen by Courand is roughly the so-called "advanced science popularization" today. The difficulty in the creation of advanced popular science lies not in the depth of knowledge, but in how to maintain the necessary affinity between the author and the reader. It is necessary to fully reflect the author's own ideas and give consideration to non-expert readers. There are many examples of failures and successes in this respect. However, the fact that it has been circulated for decades and has to be updated by the famous mathematician Stuart proves that What is Mathematics is destined to be a successful classic. Maybe there will be a Stewart 2 update in the future! Writing here, the author is thinking that the value of the paper lies in the citation rate, so does the vitality of popular science works lie in its revision or update? Maybe this is a good indicator.
Besides genre, Courand has to face another problem. Mathematics in the 20th century has developed to the point of sighing at the ocean. How to present the face of this extremely developed subject to readers in a book that we can just flip through when we take it out for an outing? Courant's approach is to collect a large number of mathematical "treasures". Every aspect of the story is not bottomless, but it is not superficial. Deep enough to finish when it's time to finish. This method, which is neither a blind man touching an elephant nor dissecting an elephant, can make ordinary readers understand the exquisite structural beauty of mathematics. This probably follows the idea that mathematics is an organic whole advocated by Hilbert.
Courant subtitled the book "Basic Research on Thought and Method". Why talk about the so-called "research"? Stuart revealed this to us. It turns out that between the relatively simple lines, there is such an ideological skeleton, and that is the subject of mathematics. This subject is not the free creation of some people, but abstract and abstract; However, it is not entirely based on physical objects, although mathematics is widely used in real life. Mathematics is like botany or astronomy. The inherent "rhythm" of discipline promotes its development, and it is our duty to fulfill this discipline. For example, if a botanist discovers a new species and an astronomer discovers a new star, it is necessary to record it. Not recording is incompetence. If this new species happens to be of great significance to human beings in fighting cancer, then this botanist may not win the Nobel Prize. If this plant is useless to human beings, botanists can at most mention it briefly in the encyclopedia. Questioning whether this kind of knowledge has practical value from the beginning is a departure from the inherent principles of the subject and a complete ignorance and mistake. What is valuable, what is worthless and what should be eliminated should be decided by history, not man-made. Although Hilbert raised 23 questions cautiously, he also warned that it is often impossible to judge the value of a question in advance. Now it seems that in the history of mathematics development, some of these problems are not as valuable as originally thought. Poincare said, "The correct way to foresee the future of mathematics is to study its history and present situation." What is Mathematics selects some valuable fields, which are mature and attractive.
"What is Mathematics?" has a clear content and distinct levels. The three parts of mathematics-algebra, geometry and analysis-are expounded chapter by chapter. The author also noticed the proper connection of different chapters. The whole book starts with natural numbers, and then extends to the expansion of number theory and number system until the most general objects are set. In the third chapter, we turn to geometric drawing related to number field algebra. In the next two chapters, the author will discuss topology from the perspective of projective geometry and non-Euclidean geometry. The last three chapters focus on calculus and its application.
Major problems in mathematics or related disciplines have always been the source and stimulation of the development of mathematical theory. The importance of the problem lies not in the degree of difficulty, nor in whether it is "advanced" Through the questions interspersed in the book, we can see the living mathematical research process. Take solving algebraic equations as an example. Because the number of times is more, it is associated with geometric drawing, and the final findings are enriched: first, the basic theorems of complex numbers and algebra are put forward; The second is the invention of group theory. On the other hand, increasing the number of elements in the equation leads to the concepts of matrix and linear space, which is ultimately related to the group. So many tricks can be made just by solving equations!
Calculus is a completely different field from algebraic equations, and it is always triggered by some interesting questions. These problems are more from physics, among which the most famous are the steepest descent line, three-body and Prato problem about the minimal surface formed by soap film. There are also pure mathematical problems, such as four-color problems. These seemingly unrelated problems make mathematicians extend calculus to important branches such as differential equations, variational methods, topology and differential dynamic systems. The author also added many famous "elementary extreme problems", such as isoperimetric problems, optical triangles, shortest networks and so on. It not only increases readability, but also emphasizes the indelible contribution of these famous historical questions to the development of mathematics.
The purpose of asking questions is to solve problems and ask new questions. The ultimate goal is not to show off your problem-solving skills, but to strengthen theoretical weapons and achieve a higher realm and a broader vision. So mathematicians are not engineers. The whole history of mathematics is that mathematicians find problems, not solve them for mathematicians. Engineers and doctors always want fewer problems, while mathematicians do the opposite. The book has a meticulous grasp of the new concepts behind the problem, and it often feels "promoted" when reading. For centuries, mathematicians' efforts to find unity at the root of piecemeal problems have undoubtedly set a great milestone in human rationality.
Of course, Courand didn't see some exciting new progress in mathematics, such as Fermat's last theorem, the proof of four-color problem, and the prime number problem, knot, fractal and continuum hypothesis. All these are introduced by Stuart in Chapter 9, "Latest Development".
The references in this book are also quite good, and it must have taken the author a lot of thought to recommend reading bibliography. This is also the characteristic of a good popular science book.
Good works should make readers often read and always be new. For example, The Journey to the West, compared with those Buddhist classics, is really good to read, but behind the funny stories and simple words, his ideological mystery is really not a word, and one person can tell the truth, so he has been commenting constantly; Even ordinary readers, when encountering some social phenomena, will make some analogies with the plot in the novel and have new insights. So can scientific works achieve the same effect? At least, the book "What is Mathematics" is made.
Reflections on "What is Mathematics" 2 As the saying goes, learning without thinking is useless. Once I wandered around a math forum and found that many people were discussing this book, which was highly praised. I thought it had something to do with mathematics, so I bought this book on a whim. Until I really finished reading this book, it had accumulated dust in the drawer for a long time. After reading it, I found that there were really too many gains.
What is Mathematics is written for beginners and experts, students and teachers, philosophers and engineers. It is a world-famous popular mathematics reading. The book brings together many classic mathematical treasures, gives a group of interesting and simple pictures of the mathematical world, and profoundly and vividly expounds the basic concepts and methods in the whole mathematical field.
I Stewart added a new chapter, expounded the latest progress of mathematics from a new angle, and described the proof of four-color theorem and Fermat's last theorem. These problems were not solved when Courant and Robin wrote this book, but they have been solved now.
Einstein commented: "What is mathematics" is a thorough and clear exposition of the basic concepts and methods in the whole field of mathematics. "Reading this book makes us know clearly what mathematics is. Mathematics is the study of ideas and methods. At present, our mathematics teaching sometimes becomes empty problem-solving training. Although this kind of training can improve the ability of formal deduction, it cannot lead to real understanding and in-depth independent thinking. There is a trend of over-specialization and over-emphasis on abstraction in mathematical research, while ignoring the application of mathematics and its connection with other fields. Therefore, we must realize that the ultimate goal of mathematics teaching should be to cultivate thinking ability. Reading "What is Mathematics" will have a constructive transformation for teachers, students and educatees, so that everyone can truly understand that mathematics is an organic whole and the basis of scientific thinking and action.
As a math teacher, we should not only help students learn and master math knowledge, but also pay attention to cultivating students' thinking ability and mastering math ideas and methods. Mathematics is a way of thinking, not problem-solving training. This is what every math teacher should pay attention to. Back to my own teaching, I feel that if students have a general understanding of mathematics, they will no longer feel that mathematics is so boring and terrible. But if we want to be as noble as the author of this book, we must rely on our mathematics background as teachers to judge what is the knowledge of mathematics essence in the problems that students have in class, and skillfully handle the relevant mathematics content. As a math teacher, we should not only help students learn and master math knowledge, but also pay attention to cultivating students' thinking ability and mastering math ideas and methods. Therefore, we must realize that the ultimate goal of mathematics teaching should be to cultivate thinking ability, not problem-solving training. This is what every math teacher should pay attention to, and it is also where I will work hard in the future.
What is mathematics? 3 What is mathematics? Basic research on thoughts and methods is co-authored by R. Courant and H. Robin in the United States.
There are two paragraphs in the preface: first, it is not important what the mathematical object is, but what it does. Mathematics struggles between reality and non-reality, and its significance does not lie in abstract forms or in concrete objects; This may be a problem for philosophers who like mathematical concepts, but this is the great power of mathematics-what we call "unrealistic reality". Mathematics connects the abstract world of mind perception with the real material world without life at all.
Second, meaningful mathematics is like newspapers and magazines telling interesting stories, but unlike some newspapers and magazines, its stories must be true, and the best mathematics should be like literary works. The story comes from your real life, which makes you put your energy and feelings into it.
From these two paragraphs, I think of the "life class" we are studying. We strive to make our classroom communicate with the real world, and make the classroom content combine with the students' existing life experience. This will undoubtedly make our classroom more lively and developmental. If our math class is just a problem-solving class, just an empty calculus and reasoning, there will be no strong vitality. If it is divorced from the real world, such mathematics is just a tool, cold without temperature and lifeless.
How to realize the connection and integration of the two is a problem that all our teachers, especially math teachers, should think about and solve. I hope to find some answers from this book.
On the fifth page of the article, there is a passage: Fortunately, despite some dogmatic philosophical beliefs, creative thinking continues to develop. If thinking succumbs to this belief, it will hinder constructive achievements. No matter for experts or ordinary people, the only thing that can answer this question is not philosophy, but the life experience of mathematics itself.
It can be seen that mathematics comes from life and is higher than life. Mathematics is the abstraction and high generalization of real life, and mathematics is the induction and synthesis of some phenomena and laws in life. Therefore, life is the land, and mathematics is the flower or towering tree nourished by this land. The development of mathematics must be nourished by real life in order to obtain a steady stream of nutrients. Therefore, life is the source of mathematics, and our "life classroom" research must be closely related to the development of life and real society, so that our classroom can truly have vitality and constant vitality. This is also the direction of our future research and efforts.
What is mathematics? 4 What is mathematics? Mathematicians R, Ke, H and Robin wrote a book on popular mathematics to tell you. Whether you are a math major or want to learn math, you can read this book. Especially for high school students, college students and middle school math teachers, it is an excellent reference book. This book profoundly and vividly expounds the basic concepts and methods in the whole field of mathematics. The New York Times commented that this book is a perfect book for beginners, experts, students and teachers, philosophers and engineers.
This reminds me that when I was in junior high school, I was not interested in mathematics. I thought mathematics was too simple, and I could understand it as soon as I learned it. It can be measured that a later math class influenced my life.
That was when Sue was in the third grade. When he was in XX middle school, a teacher Yang who had just returned from studying in Tokyo taught mathematics. In the first class, Mr. Yang didn't talk about math, but told stories. He said: A long time ago, the law of the jungle prevailed in the world. The world powers relied on their ships to build guns and gain profits, and they all wanted to eat and carve up China. The danger of China's national subjugation and extinction is imminent, so we must revitalize science, develop industry and save the nation. Every man is responsible for the rise and fall of the world, and every student here is responsible. He quoted and described the great role of mathematics in the development of modern science and technology. The last sentence of this lesson is: in order to save the country and survive, science must be revitalized. Mathematics is the pioneer of science. In order to develop science, we must learn math well. I don't know how many lessons Sue took in her life, but I will never forget this lesson.
Teacher Yang's class deeply touched me, and he injected new ideas into my mind. Reading is not only to get rid of personal difficulties, but to strengthen the country; Reading is not only to find a way for individuals, but to create for the Chinese nation. That night, I tossed and turned and couldn't sleep all night. Under the influence of Teacher Yang, Su's interest shifted from literature to mathematics, and since then, she has set the motto "Never forget to save the country when reading, and never forget to save the country when reading". I am fascinated by mathematics. No matter in hot winter, frosty morning or snowy night, I only know reading, thinking, solving problems and calculating. I have worked out tens of thousands of math exercises in four years. When I graduated from middle school, my scores in all subjects were above 90.
/kloc-At the age of 0/7, Su went to Japan to study, and won the first place in Tokyo Technical School, where she studied eagerly. The belief of winning glory for our country drove Su to enter the field of mathematics research earlier. At the same time, he has written more than 30 papers, and made great achievements in differential geometry, and obtained the doctor of science degree in 193 1. Before receiving her doctorate, Su was a lecturer in the Department of Mathematics of Imperial University of Japan. Just as a Japanese university was preparing to hire him as an associate professor with a high salary, Su decided to return to China to teach with his ancestors. After the professor of Zhejiang University returned to Suzhou, his life was very hard. In the face of difficulties, Sue's answer is that suffering is nothing, and I am willing, because I have chosen a correct road, which is a patriotic and bright road!
I read it, and I have skills in math. As long as you master the skills, you will succeed.
Only when I opened this book did I know how lacking my professional knowledge of mathematics was, and I felt that my mathematics level was still in the primary school stage, and even what I learned in middle school was almost forgotten. Especially since the implementation of the new curriculum, I often feel that I can't deeply understand the textbook and see through its essence. What is Mathematics is a professional book that studies mathematical thoughts and methods. The basic concepts and methods of the whole mathematics field are expounded profoundly and vividly. Knowledge points are linked one by one, and strict logical reasoning is followed, instead of jumping out of thin air to a conclusion for you to accept. The knowledge points inside should be carefully tasted, chewed and digested, and a bucket of water should be unfolded to truly understand what mathematics is.
As an expression of human thinking, mathematics embodies people's initiative, meticulous reasoning and the will to pursue perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. In this sentence, I seem to understand why some wise teachers always say that the core of mathematics is philosophy. I think it is more important for us as math teachers to guide students to understand what we have learned dialectically. For example, 1/2 is greater than 1/5, and sometimes 1/2 is less than 1/5 when the unit is different.
As a math teacher, we should not only help students learn and master math knowledge, but also pay attention to cultivating students' thinking ability and mastering math ideas and methods. Mathematics is a way of thinking, not problem-solving training. This is what every math teacher should pay attention to.