The Thought of Ancient and Modern Mathematics discusses the historical development of most branches from ancient times to the first few decades of the 20th century, including the emergence of Mesopotamian mathematics, Egyptian mathematics and classical Greek mathematics. This paper expounds the sources of some important mathematical ideas and the relationship between mathematics and other natural sciences, especially mechanics and physics.
I'm afraid no one is more familiar with the ins and outs of mathematics than M Klein. The author makes the history of western mathematics clear and attractive.
After reading "Ancient and Modern Mathematical Thoughts 1", I feel quite touched: it seems that reading anything in any subject requires reading its development history.
We often exaggerate the rational spirit of mathematics. But in fact, the development of this subject has never been separated from experience, otherwise negative numbers, irrational numbers, infinity and infinitesimal will not be unacceptable for thousands of years. Only astronomy can have triangle and spherical geometry, and only painting can have projective geometry. Chapter 1 1, the last section of the Renaissance, "the rise of empiricism", has a wonderful view. With experienced materials, mathematics can make great strides.
Of course, this is also in line with my point of view. I have always believed that there is no pure reason divorced from experience.
But it is also undeniable that reason plays a guiding role in experience.
Without calculus, there would be no modern mathematics. As we all know, from the Greek world to the Middle Ages, it was difficult to produce changing ideas when geometry was always advocated and algebra was despised. There must be a proper transformation from geometry to algebra. Influenced by the Arab world, westerners finally began to emancipate their minds. In chapter 13, 16,17th century algebra, Newton, Leibniz and Fermat began to appear, and algebra was finally separated from geometry.
In the last chapter, projective geometry, on the basis of empirical materials and people's practical application needs, mathematics (geometry) finally began to step down from the altar, and new branches and theories finally began to appear. Since then, the vision of mathematics has been continuously broadened.
In fact, the projective geometry of universities is only the achievement of Dasaga. It turns out that Pascal's most important contribution is projective geometry.
The last part is wonderful. The idea of continuous change began. The ideological basis of calculus gradually infiltrated and pressurized until the second volume caused an explosion.
As far as the whole first book is concerned, there is a feeling that the author is too addicted to the Greek world and then scoffs at the Roman world. This may be a prejudice of the author.
After reading Ancient and Modern Mathematical Thoughts 1, I realized:
In learning mathematics, it is important to understand, not to memorize like other subjects.
Mathematics has a characteristic, that is, "knowing one and knowing ten". If you do a problem, you can sum up the methods and principles contained in this problem, and then use the principle of summary to solve this kind of problem.
It is also very important to learn mathematics, that is, start with what is known and basic, practice steadily, don't aim too high, and don't do all the problems.
In the process of doing the problem, the most taboo is carelessness. It is not worth doing a problem clearly but making a mistake because of carelessness.
Therefore, when you take the math test, you must never rush into it, you must calculate clearly and think clearly; This speed may be a little slower, but it will keep you from losing points. In contrast, I will accept a slightly slower calculation method, think more and think more, and try to be as good as possible.
I think learning is a lifelong thing. Don't worry too much. Step by step, you will certainly achieve unexpected results.
The above is my feeling after reading 1, an ancient and modern mathematical thought.
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