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Author: Sun Mathematics

Link:/question/21081945/answer/171412.

Source: Zhihu.

First, the Soviet Union and Eastern Europe's research institute system

Because the Soviet Union and Eastern Europe adopted the graduate school system, the career after graduation will probably enter similar institutions for further study. Therefore, it is necessary to deepen the difficulty and expand the scope of undergraduate teaching materials in the Soviet Union and Eastern Europe. The fundamental reason is to intensify the academic crackdown so that students will not start from scratch without any tools after graduation. In fact, we don't expect undergraduates to do research right away, at least for now, so there is nothing wrong with appropriately reducing the difficulty of textbooks. Wrong is to reduce the breadth of teaching materials and class hours-for example, the courses of analytic geometry and affine geometry are almost completely removed (unfortunately, this trend was also started by the Soviets, such as the famous well-written Kostrikin &; Manin linear algebra and geometry) leads to the fragmentation of students' knowledge structure.

For example, Kostrikin's "Introduction to China Algebra" is an all-encompassing survey reader in our view, and there are quite few topics in it. From linear space to Lie algebra, there are both applications and quite rigorous proofs. Therefore, we tend to think that Soviet textbooks and problem sets are two different components of a course, and the textbooks can be returned for reference later, so the textbooks should be expanded in scope and depth as much as possible. For example, Nathanson's theory of real variable function points out the A set constructed by Suslin for no reason, which I think can only be explained by this viewpoint of separating teaching materials from exercises. Problem sets are tools for students interested in this direction to hone their skills. For example, Gregory Lunts's problem set of complex variable function theory contains quite a few baby steps to study the level of problems. These steps are very simple in themselves, but you will encounter some similar situations when doing research.

Second, the general education and postgraduate system in Europe and America

Students in Europe and America usually haven't decided to devote their careers to mathematics at the undergraduate level, so the courses in Europe and America tend to be simplified. The most prominent example is that levinson wrote Ode, and his partner also wrote the introduction of Ode. The comparison between these two textbooks reflects that Europe and America believe that there is an overall vague understanding before training skills. But this has led to the situation that the input and output are not proportional, and only a few geniuses have done it. So far, Russians are far behind Americans on average, and among the top mathematicians, there are many more Russians than Americans. Technically speaking, I think American students should do relatively simple topics and have less skills, but they may be more flexible in application.

When European and American students enter the research, they usually stop reading intro's textbooks and choose another one as their own basis. Now GTM is one of the options. At this time, the difficulty and depth of their postgraduate textbooks are much more difficult than those of the Soviet Union. However, the research mechanism of the Soviet Union basically made up for this small flaw in the postgraduate stage.

However, in recent years, more and more scholars have begun to realize the importance of coherence. For example, Singer tried to combine geometry and topology, and Adin tried to show the original posture of abstract algebra-transformation and structure. These are great attempts made by quite excellent scholars. In this sense, I think the Soviet Union's separation model of textbook problem sets is more suitable for technical researchers, while the gradual approach in Europe and America is more suitable for cultivating bird researchers (broad vision and good integrity).

I think there is no difference between these two mathematicians. Both mathematicians are the same in the final top stage, but they have taken different paths, and the efforts they need are no different from those of geniuses.

Third, a small message

I don't want to criticize domestic textbooks. In fact, there are many excellent textbooks by excellent authors in China (such as homology theory by Jiang Boju, Concise Analysis of Complex Variables, Real Variable Function and Functional Analysis by Hu, etc.). It is important that we take each textbook seriously, have a thorough understanding of the whole subject through other textbooks, and learn other branches at the knowledge level.

I only feel after reading, and my speech is not good. Please forgive me.