How ancient Chinese math differs from Western math

What is the difference between ancient Chinese math and Western math?

Author: e^iπ+1=0

Ancient Chinese understanding of the world was a cyclic closed system, with some kind of connection behind a thousand different phenomena, which were interdependent, while the Western understanding of the world was based on a straight chain of one-way causation, with particular phenomena being explained from general abstracted concepts and resulting derivations. These two kinds of thinking lead to a fundamental difference, that is, the ancient Chinese focus on the understanding of things, using one phenomenon to explain another phenomenon, to discover the inner connection, while the West is more focused on logic, to establish a general theory to unify all the phenomena under the theory. In turn, we can understand why the West can give birth to a modern axiomatic, highly abstract mathematical system, while the Chinese mathematics is not systematic, to the primitive form presented to the mathematicians.

Based on the above understanding, it is easy to understand that although the origins of Chinese and Western mathematics are very similar, and both are based on the generalization and rational treatment of problems encountered in the practice of life, the development of Chinese mathematics has continued the tradition of the research of the predecessors, i.e., based on the intuitive phenomena or examples, and apply them.

It should be noted that the development of modern mathematics in the West (from the 16th century onwards), and the tradition of the development of modern science in the West, is not a direct inheritance of the axiomatic method of research from the ancient Greek period, which was originally laid down by geometry. In fact, when we look at the development of both modern mathematics and physics, it was based on the reliance on empirical facts and bold speculation and imagination. From this point of view, the difference between China and the West lies in the fact that the West was the first to use a general, abstract approach to explaining particular problems, with the conviction that all the phenomena of the world could be unified in numbers. Not only that, they were good at summarizing "beautiful properties" from complex phenomena, and believed that beautiful, simple theories were the ultimate explanation of the world. Therefore, in the early 16th century, the flourishing of mathematics and science revealed the implementation of this simple philosophical view. For example, Kepler, an early explorer of astronomy and mathematics, pointed out in his important work The Harmony of the Worlds that it was possible to combine astronomy and music in a perfect union, and that this was seen as the harmony of the world. And this simple epistemology was the beginning of modern science in the West.

The second important issue is the creation and derivation of mathematical systems. One thing that must be recognized is that in the establishment and derivation of systems, Chinese and Western mathematics parted ways early on. Nine chapters of arithmetic, for example, from the content, the core of the ancient Chinese mathematical problem lies in the interpretation and reuse of the actual problem, so the volume classification to "square field", "corn", "decline points Therefore, the volume is categorized by "square field", "corn", "bad points", "less wide", "business work" and other real-life scenarios. However, in terms of mathematical content, the Nine Chapters not only dealt with a large number of complex problems, but also contained important philosophical ideas (e.g. limits, division, combination, etc.). The most popular example is "Zu Yi's Principle", that is, to judge the volume of two objects to be the same, we can use the principle of "if the powers are the same, the products cannot be different" to judge, and use this principle to find out the volume of the "Mouhe Square Cover" (the so-called "Mouhe Square Cover"). "Volume (the so-called" Mouhekangai "refers to the same two cylinders orthogonal enclosure of the three-dimensional shape) and the volume of this three-dimensional shape can not be solved by elementary mathematics, strictly speaking, should be used in order to fully solve the calculus. And from its exposition, we can see the simple integral idea, but also shows the outstanding mathematical intuition of ancient mathematicians. At the same time, there is great flexibility in the field of study, from elementary algebra, elementary number theory to elementary geometry (based on the viewpoint of modern mathematics) in the various problems have been involved, and gives an understanding of the important thinking to solve the problem. For example, the opening problem in Volume VIII on equations is solved using the idea of systems of equations, which, from a Western mathematical point of view, utilizes Gaussian elimination. Then there is the widely publicized Chinese Remainder Theorem, and the Hook Theorem, which involves a large number of important core propositions in elementary mathematics. Derivatively, however, the narrative explanations we have given predominate, rather than derivations and calculations. In fact, in the Nine Chapters on Arithmetic, calculations are performed only when practical examples are encountered and on a few formulas, while the principled content appears as understanding. In this case, the development of mathematics relies solely on the unproven insight of a very few mathematicians to give progress, which is fatal to the development of the system itself.

In the West, on the other hand, the development of mathematics was also bold and imaginative in its early stages, though they did not stop at understanding, but expressed it in mathematical language organized in relatively elaborate chains of logic. The vitality of mathematics was first resurrected in the hands of artists, both in painting (the influence of perspective drawing on projective geometry), and in the development of music theory, which stimulated people's thinking.The work of mathematicians in the 16th,17th centuries was from time to time undisciplined, and not even guaranteed by any axiomatic foundations of mathematics, such as Euler's treatment of many infinite series, which was based on some plain understanding, and which, when inspired by the form, was not used with a The use of proofs. At this stage of mathematics, the driving force of thought was in fact the same as that of the ancient Chinese mathematicians, who relied on the intuition of the mathematicians to carry out their research. However, there are two reasons why Western mathematics exploded after a similar period. One, the use of abstract symbols to describe mathematics freed it from practical problems, allowing it to be freely combined, to portray complex things in a simple way, to use the imagination, and to no longer be constrained by figuration. Secondly, compared with ancient Chinese mathematics, Western mathematicians pay more attention to the establishment of logical chains, so the process from cause to effect is more detailed, laying a solid foundation for subsequent research. The axiomatization and abstraction of mathematics that we are so fond of were not done in the early days, but began to be taken seriously by more and more mathematicians in the 18th century. The birth of analytics was in fact a search by mathematicians for a fine chain of logic that would provide a solid theoretical foundation for the calculus and reveal the origin of the correctness of a large number of obvious propositions, leading to a deeper understanding of the calculus system. At the same time, the vigorous development of physics promoted the development of a large number of computational techniques, and the application of calculus to practical research became a kind of **** knowledge. In the 19th century, on the one hand, calculus swept through almost all the original branches of elementary mathematics, and on the other hand, the development of modern algebra provided abstract tools, such as group theory, which was used to explain the theory born from the solution of equations, so that the next branches of mathematics developed became group theory, complex function theory, and geometry also took on a new light. In the 20th century, both axiomatization and abstraction reached their peak, and mathematicians realized that there was a close connection between the various branches of mathematics, and the development of topology, set theory, and abstract algebra encompassed the fragmented studies and branches of mathematics in an interconnected, unified structure that truly became a system. From this point of view, it is impossible for the ancient Chinese mathematical tradition to evolve such a system, not only because of epistemological differences, but because of a deeper problem.

I fully support the argument made in "The Re-Creation of the World" that Chinese science is doomed to be inadequate because there has been no real cultural transplantation from Chinese culture. One, the exchange and collision of very different cultures would have brought new insights to both civilizations. Secondly, the Western writing system is more suitable for abstract thinking, while the Chinese language, due to its strong combinatorial ability and good intuition, did not produce a new symbol system to describe mathematics, and therefore it is difficult to carry out complex and abstract calculations and derivations. However, in the author's opinion, the key question is why there is no collision between ancient Chinese mathematics and Western mathematical systems. In terms of historical period, there is a big time difference between the development of Chinese mathematics and the development of Western mathematics. Chinese mathematical research originated early, there have been outstanding mathematical achievements in the third century AD (the earliest book of nine chapters of arithmetic was also written at this time, compiled by Liu Hui). And although the ancient Greek mathematics also has outstanding achievements, but obviously the influence of the scope of coverage is far less than East Asia, up to the two river basin, and then into the Indian territory, and that has been up to the eighth century AD. In the Tang and Song dynasties, mathematics was highly developed, and the Nine Chapters of Arithmetic gradually evolved into the most important mathematics textbook in East Asia. At the same time, Europe was going through the Middle Ages, dominated by the Church's monopoly on the interpretation of the world. Until the thirteenth and fourteenth centuries, through India, the Arabian region will be the origin of mathematics back to Western European societies, Western mathematics began to develop, however, at this time China was ruled by the Mongols period, the development of mathematics is obviously hindered. After entering the fifteenth century, mathematics began to revive in Europe and entered a period of vigorous development, but Chinese mathematics was still lukewarm and increasingly biased, which determined the gap between Chinese and Western mathematics during this period. Throughout, the break in the development of Chinese and Western mathematics was a strong obstacle to the exchange between the two sides, failing to form exchanges at the same period standing at roughly similar heights. Politically speaking, the existence of mathematics in ancient China is actually for political service, so the research focuses on practicality, bias, and is very good at solving practical problems, but there is not much enthusiasm for establishing systematic theories. Compared with the Western understanding of the metaphysics of mathematics, Chinese mathematics "for the time and use", is "goods with the emperor's family" talent, if there is no political support, then there is no soil for the development of mathematics. For this reason, there were few mathematicians in China, and the spread of mathematical culture was not arbitrary. Importantly, advanced computational skills were unlikely to trickle down to the people, and naturally they were unlikely to spawn the development of mathematics in China as a whole. At the same time, even though some important texts such as the Analects of Confucius, Laozi, and Zhouyi were introduced to the Western world, the mathematical wisdom of the East was not spread to Europe. In terms of research methods and tools, Chinese mathematics emphasized computation and practical results. For example, the achievements of the calendar were based on very high computational skills. The symbolic system on which these skills were based was advanced compared to any ancient mathematical civilization. Because of their simplicity and combinatorial nature, and through generalization and shorthand (such as the creation of the concepts of hundred, thousand, ten thousand, and billion, and the creation of combinations such as million, ten million, and billions), we can easily and intuitively represent a very large number of numbers, which is very helpful in the study of computational skills. So even though Western symbol systems and numerical systems were introduced to China, their computational superiority inevitably led to the fact that they could not replace the millennia-old written language.

Today, there are many comparisons between Chinese and Western scientific development, and many of them are aimed at giving the ancient Chinese science and technology a proper name and boosting the nation's self-confidence, which is not to be denied, but we should explore them in an objective and fair manner. If the core point of view is always based on the differences between the two cultural systems, and then find an equal balance point, I think it is not necessary. Whether it is the previous proposal "if given time China can develop the same degree of science as the West", or the current proposal "Chinese science in the broad sense is the study of gezhi, the museum of life", in fact, are in fact to avoid the issue of the problem. And regardless of whether the Western powers to open the door to China by force is the cause of the abortion of China's indigenous science, even if the two sides do not interfere in the premise of the development of mathematics, the basic disciplines of science, the speed of development is not on a level. China's mathematical development is cumulative, linear, is a steady development, but the development of Western mathematics is explosive, like an exponential function, will only develop faster and faster, this is a huge gap. Then there is the point of view from a broad scientific perspective on the cut, basically rising to the philosophical level of understanding, can not just stay in the exploration of different thoughts and ideas on the long breath, that found equality can be some peace of mind. For Chinese people who are now studying science and researching science, it is very important to learn how to draw on ancient wisdom. This is by no means to abandon the methodology of science, but to understand the world from a perspective different from the mechanical idealism of the West. A case in point is the "Wu Method" developed by the mathematician Wu Wenjun. Starting from the ancient idea of algorithms, Prof. Wu Wenjun achieved remarkable results by constructing programs to prove a large number of elementary and projective geometries. In modern times, when there are so many different branches of science, the large volume of scientific systems has actually become an obstacle that limits people's further exploration, so how to understand science from the holistic view of ancient China is a topic that is likely to be successful and extremely important. In a way, we should be grateful that Chinese philosophical thought alongside the West will bring the world perhaps the most important revelation.