Chapter 53 Chapter 12 Features and Significance of Ancient Chinese Mathematics

Mathematics is a science that studies the relationship between spatial form and quantity of objective things.It is not limited by any time and space, and strongly demonstrates this essential attribute.However, in different cultural traditions in various periods of ancient times, the expressions of mathematics are often not the same, each showing its own characteristics.For example, Chinese classical mathematics is quite different from other cultural traditions, especially ancient Greek mathematics, in many aspects such as expression form, thinking mode, relationship with social reality, research center, and development process.

The first is its form of expression, which mainly refers to the form of classic works of mathematics.Ancient Greek mathematics often takes the form of abstract axiomatics, while Chinese classical mathematics is in the form of technical examples.Two different forms represent two very different styles.Both forms and styles can also illuminate the foundations of mathematical theory.Some people tend to ignore this point and generally summarize ancient Chinese mathematics works in the form of applied problem sets.As long as we carefully analyze and compare the mathematical works themselves, it is not difficult to find that this conclusion is extremely incorrect.For example, the most important book "Nine Chapters of Arithmetic", in its nine chapters, all of the six chapters on Fangtian, Corn, Shaoguang, Shanggong, Insufficiency, Equation and the three chapters on Decline, Even Losing, and Pythagorean, Either list one or several examples first, and then give very abstract "techniques"; or list very abstract "arts" first, and then give several examples.The "techniques" here are all formulas or abstract calculation procedures; the examples of the former only have questions and answers, while the examples of the latter include questions, answers and "skills".The so-called "art" is to explain various algorithms and specific applications, similar to the fine grass of later generations. Only about one-fifth of the "Nine Chapters of Arithmetic", that is, about 50 topics in the three chapters of Decay, Even Loss, and Pythagorean, can be said to be in the form of a set of applied problems.From this, it is inappropriate to draw the conclusion that "Nine Chapters on Arithmetic" is a collection of applied problems, and the correct formulation should be in the form of commanding examples.The main body of the later "Sun Tzu Suan Jing" should be said to be in the form of a set of applied problems, but some preliminary knowledge is put at the beginning of the volume.Among the works in the upsurge of mathematics in the Song and Yuan Dynasties, Jia Xian's "Nine Chapters of the Yellow Emperor's Sutra" is more abstract than "Nine Chapters of Arithmetic", and other works are sometimes less abstract than "Nine Chapters" due to more complex algorithms. However, valuable efforts have been made. For example, the "Dayan Summarization Technique" and its core "Dayan Seeking One Technique" in "Nine Chapters of Shushu" are the general technique of the congruence solution; "After expounding the method of opening the fourth power in abstract words, he also stated that "the next chapter will follow this", which shows that it is also a common method.Zhu Shijie's two works put a lot of preliminary knowledge and algorithms at the front of the volume, and the front volume of "Siyuan Yujian" also contains examples of solutions for Tianyuan, Binary, Sanyuan, and Quaternary. "Che Yuan Hai Jing" even includes the "Yuancheng Schema" and the definitions and propositions to be used later in the "Recognition Miscellaneous Notes" of Volume 1.Therefore, in general, algorithm (shu) is the key to solving application problems, and "shu" has naturally become the core of ancient Chinese mathematics.Chinese mathematics works are centered on algorithms, which are in the form of examples.

The second is the study of mathematical theory.Ancient Greek mathematics used deductive reasoning to form a rigorous axiomatic system of mathematical knowledge.Many scholars have exaggerated the difference between ancient Chinese mathematics and ancient Greek mathematics, and believe that the achievements of ancient Chinese mathematics are only the accumulation of experience, without reasoning, especially without deductive reasoning.This is a superficial view that lacks a basic understanding of ancient Chinese mathematics.Regrettably, this superficial view was endorsed by some scientific leaders and became popular, and even became the starting point for discussing that modern science did not arise in China.It is true that ancient Chinese mathematics and philosophy were not as closely integrated as in ancient Greece, and most of the ancient Chinese mathematicians were not like the ancient Greek mathematicians who were leading figures in the ideological world or leaders of schools of thought.Generally speaking, Chinese thinkers are far less interested in mathematics than their counterparts in ancient Greece. Among the pre-Qin thinkers, even the Mohists, who were the most educated in mathematics, could hardly match the achievements of ancient Greek thinkers in mathematics.Similarly, Chinese mathematicians, on the whole, pay far less attention to the study of mathematical theory than ancient Greek mathematicians.For example, "Nine Chapters on Arithmetic" and many mathematical works do not define mathematical concepts, and the expressions of many mathematical problems are not rigorous.This requires readers to stand on the author's standpoint and coexist in a harmonious system with the author in order to understand its content, which more or less hinders the development of mathematical theory.It is certainly inappropriate to insist that ancient China and ancient Greece paid equal attention to the study of mathematical theory.Conversely, it is also inconsistent with historical facts to say that ancient Chinese mathematics has no theory and reasoning. "Zhou Bi Suan Jing" records that when Chen Zi, a pre-Qin mathematician, taught Rong Fang, he pointed out that the reason why he could not understand some mathematical principles was that he "did not understand numbers". ", "Speaking about appointments and using blogging" must achieve "being able to classify and combine classes".Chen Zi was probably at the beginning of the compilation process of "Nine Chapters of Sumension".In fact, the compilation of "Nine Chapters" runs through the ideas of "common class" and "class by class". The author of "Nine Chapters of Arithmetic" grouped the problems that can be solved by the same mathematical method into one category, and proposed common and abstract "skills", such as Fangtian technique, Yuantian technique, Jinyou technique, decaying technique, and back-decaying technique. According to their nature or application, these techniques and examples are divided into Fangtian, Corn, Decline, Shaoguang, Shang There are nine types of merit, equal loss, insufficient profit, equation, and Pythagorean shares.Liu Hui further excavated the internal connection of many methods in "Nine Chapters", and attributed the techniques of decay, equalization, and new equations to modern techniques.Liu Hui found out the destination of various methods through "analogous analogy", and found that mathematical knowledge is a big tree with "branches divided but the same root" and "sent from one end", forming his own complete mathematical theory system.Jia Xian summed up the method of prescribing and created the origin of the method of prescribing.Yang Hui summed up the thirteen diagrams of Pythagorean growth and change. Li Ye discussed various relationships between tolerance and circle, and gave more than 600 formulas. theoretical overview.

After summarizing abstract formulas through "combining categories", applying these formulas to solve certain mathematical problems is actually a deductive process from general to special. Here we will specifically talk about whether there is deductive reasoning in ancient Chinese mathematics. .As we all know, mathematical knowledge is obtained through various reasoning approaches such as analogy, induction, and deduction, and to prove the correctness of a mathematical proposition must rely on deductive reasoning.The ancient Chinese mathematics works made extensive use of deductive reasoning.Taking the most developed branch of high-degree equations in ancient China as an example, both Liu Hui and Wang Xiaotong proposed the derivation process of the equations. Jinyuan mathematicians even created Tianyuanshu to set equations of unknown numbers. Li Ye will use Tianyuanshu to formulate equations Most of the required theorems and formulas are given in "Recognition Miscellaneous Notes" in Volume 1.Liu Hui, Wang Xiaotong, Qin Jiushao, Li Ye, Zhu Shijie and others all relied on deductive reasoning to derive higher-order equations, so they were correct.As for Liu Hui's proof of the circle area formula and the cone volume formula by using limit thinking and infinitesimal division; The proof of alchemy; the proof of equation technique, surplus-deficiency technique and many algorithms by using the same principle are all models of deductive reasoning.As long as you are not prejudiced, you will realize that Liu Hui mainly uses induction and analogy when expanding his mathematical knowledge, and when he demonstrates the correctness of the formulas and algorithms in "Nine Chapters of Arithmetic", he criticizes some mistakes in "Nine Chapters of Arithmetic" At the same time, he mainly focuses on deductive reasoning, so as to build his own mathematical knowledge on a reliable theoretical basis.

To say that mathematical research is not closely integrated with the world of thought is to say that as a whole, it does not mean that every mathematician is like this, for example, Liu Hui is an exception.He was deeply influenced by the style of debate in the Wei and Jin Dynasties. He "analyzed with words and disintegrated with diagrams" of "Nine Chapters of Arithmetic". The methods are all in tune with the wind of debate at that time.Of course, even Liu Hui's exploration of many mathematical concepts has not reached the level of ancient Greece.For example, Liu Hui's introduction of infinitesimal division into mathematics proved to be an unprecedented contribution, but he never considered the difference between potential infinitesimals and real infinitesimals.However, this is not necessarily a bad thing.Ancient Greek mathematicians could not satisfactorily solve the problem of potential infinity and real infinity, and had to exclude the concept of infinitesimal from mathematical research. Therefore, they never used limit thinking and infinitesimal division when proving mathematical propositions.Liu Hui is not the case. He believes that the number of sides of a regular polygon inscribed in a circle increases infinitely, and it must finally "fit with the circumference". The barriers composed of turtles can be divided into infinite divisions, which can reach the point of "invisibility".Ancient Greek mathematicians believed that the diagonal of a square has no commensurity with its sides, that is, √2 and 1 have no commensurity, which led to the first crisis in the history of mathematics, turning ancient Greek mathematics to exclude calculation from mathematics, and only Focusing on the study of spatial forms, he is helpless in the face of irrational numbers.But Liu Hui, Zu Chongzhi, etc. are not. They dare to continue to find squares for "inexhaustible" and "undividable" numbers, "find their micronumbers", and use decimal fractions to infinitely approach the approximate value of the irrational root.Without falling into philosophical debates, starting from the reality of mathematical calculations, Chinese mathematicians can bypass the hidden reefs that caused Greek mathematics to change course or stagnate, and reach heights that ancient Greek mathematicians never achieved in mathematical theory and practice.

Being good at calculation and centered on algorithms is a remarkable feature of ancient Chinese mathematics.Ancient Greek mathematics only considered the properties of numbers and shapes, not specific values.For example, they understood very early on that the ratio of the circumference to the diameter of any circle is a constant, but the value of this constant has been ignored for hundreds of years, and it was not until Archimedes found out the range of its value.On the contrary, Chinese classical mathematics seldom studies the properties of graphs without quantitative relationship, but converts practical problems into a kind of mathematical model through practical methods, and then uses a set of programmed or mechanized algorithms to solve them.The "techniques" in the arithmetic are all calculation formulas and calculation programs, or the fine grass of applying these formulas and programs. All questions must be calculated with specific values ​​as the answer. Even for geometry problems, the length, area, and volume.This is the combination of geometric methods and algorithms, or the algorithmization of geometric problems.Liu Hui said: "The method passed down from generation to generation is still the same as the rules, and the measurement can be obtained and shared" ("Nine Chapters on Arithmetic Notes Preface"), which clearly expresses the characteristics of the combination of shape and number in ancient China. "Nine Chapters of Arithmetic" methods of formula extraction, equation, surplus and deficiency, decline and division, and equal loss; Liu Hui's method of calculating pi by cutting a circle and calculating the approximate value of arc field area; Jia Xianqiu and Jia Xian's triangular multiplication Methods, Jia Xian pioneered and Qin Jiushao completed the positive and negative method of finding positive roots of higher-order equations, Qin Jiushao's congruence solution method, Zhu Shijie's quaternary technique, etc., all have quite complicated calculation procedures.The programming of mathematical operations makes it easy to grasp complex calculation problems. Even if you don't understand the mathematical principles, you can also master the procedures, so the auxiliary table "Licheng" for the procedures came into being.These programs above all have the three characteristics of modern algorithms, such as complete determinism, applicability to a whole class of problems, and effectiveness.Many programs can be moved almost verbatim to modern electronic computers.

The advanced counting system and strong positional value system are important factors for the full development of Chinese algorithm theory.China was the first to invent the decimal place value notation method, which is very beneficial to the four arithmetic operations of addition, subtraction, multiplication and division, and the representation of fractions and decimals.In addition, numbers in Chinese are all monosyllabic, which is convenient for formulating formulas, and facilitates the transformation of formulas for calculation, multiplication and division into formulas.And the use of calculation makes the separation coefficient representation a matter of course.The notation of the separation coefficient, the notation of the opening method, the notation of the Tianyuan polynomial, and the notation of the quaternion of the linear equation system are actually a position value system.The power of the unknown number is completely determined by its position in the expression, without having to write the unknown number itself. For example, in the open method, from top to bottom are "quotient", "real" (constant item), "square" (one-time item), "one cheap", "two cheap" (coefficients of the second and third terms) ... corner (the highest sub-term coefficient).The same is true for the Tianyuan formula. It is only because there are positive and negative powers in the calculation that it is necessary to mark a word "太" next to the constant item, or a word "Yuan" next to the primary item. The unknown power is completely determined by the word "太". Or "meta" is determined by the relative position.This notation is particularly convenient for square root or addition, subtraction, multiplication, and division operations, especially multiplication (or division) with the power of Tianyuan, as long as the position of the word "Tai" or "Yuan" is moved up and down.

Mathematical theory is closely connected with practice, which is another notable feature of ancient Chinese mathematics.All the questions in the ancient arithmetic can not be regarded as application questions in daily production and life. Some questions are just examples to illustrate the algorithm. There are such questions in "Nine Chapters of Arithmetic" and "Che Yuan Hai Jing".However, ancient Chinese arithmetic is indeed for the purpose of application, which is one of the significant differences from ancient Greek mathematics.The latter publicly stated that it was not aimed at practical application, but as a spiritual activity of pure ideas. Euclid almost erased all traces of the actual source of "Elementary Geometry".However, Chinese mathematicians have never denied the utilitarian purpose of studying mathematics.From "Han Shu Lu Li Zhi" to Liu Hui and Qin Jiushao, they all summarized the role of mathematics into two aspects: "understanding the gods" and "resembling all things".The meaning of gods here can be understood either in terms of mysticism, or as categories explaining the changing nature of the material world, or both. "Nine Chapters of Arithmetic" Liu Hui did not have any mystical elements in its annotations, nor did it explain the role of gods. Liu Hui did clearly point out the scope of application of each chapter of "Nine Chapters of Arithmetic" in actual production and life: Fangtian is used to control the boundary of the field, corn is used to control the change of quality, the decline is used to control the high and low tax, the Shaoguang is used to control the accumulation of power, and the commercial achievements are used to control the distance and labor costs. The surplus is not enough to control Concealed and mixed are intertwined, equations are used to prevent mistakes from mixing positive and negative, and Pythagorean is used to control high, profound and far-reaching, which is obviously the aspect of "like all things".Qin Jiushao regards "Tongshenming" as a person with a great mathematical function, and his understanding is a combination of mysticism and the nature of world changes, while he regards all things and world affairs as a person with a small mathematical function.Although he stated that he wanted to "introduce mathematics into the Tao", his mathematical research practice made him feel that he was still "not seeing the skin" of the big ones, but focused on the small ones, and realized that "the transmission of mathematics is based on the truth." body", so "set it as a question and answer for use".Except for the first question, his "Nine Chapters of Numerical Books" are mostly application questions about real life, production and various projects. reflect.In short, Chinese mathematics is closely related to practice and has been developed through practical application.Perhaps because of this advantage, Chinese mathematics has basically adhered to the materialist tradition from "Nine Chapters of Arithmetic" to the Song and Yuan climaxes, and has not been affected by number mysticism.There are some mystical things in the works of the Ming Dynasty, and they have the nature of wearing boots and hats, but they still cannot change the general characteristic of being practical.

The ruler's attitude towards mathematics resulted in the different development characteristics of Chinese and Greek mathematics.Ancient Greek rulers attached great importance to mathematics, resulting in strong continuity and inheritance of Greek mathematics.However, most of the rulers in ancient China, except for a few, did not pay much attention to mathematics.Qin Shihuang unified China, and Mohism, which paid more attention to mathematics, was suppressed. After the Han Dynasty, Confucianism was the only one, and Confucianism and Legalism merged.Due to the important role of mathematics in the national economy and the people's livelihood, the ruling class has to admit that "arithmetic is also an important part of the six arts" ("Yan's Family Instructions: Zayi"), but it advocates that "it can be both good and professional" (ibid.).Mathematics has always been regarded as "ninety-nine cheap skills".Liu Hui lamented that "there are few people who are good today", ("Nine Chapters of Arithmetic Notes Preface") Qin Jiushao said that "scholars of later generations will not talk about it", ("Shushu Nine Chapters") Li Ye studied mathematics with great Confucianism, and he claimed that "Those who pity me should be counted as hundreds, and those who laugh at me should be counted as thousands." ("Preface to Surveying Yuanhai Mirror") The Wei and Jin Dynasties where Liu Hui lived, and the Song and Yuan Dynasties where Qin and Li lived were both in the prosperous period of Chinese mathematics.The twenty-four histories, numerous, include numerous emperors, generals, writers, thinkers, and even martyrs, but no biography is written for a mathematician. Zu Chongzhi and Li Ye have biographies, but they are writers and famous ministers. Incoming.The needs of the society, as well as generations of mathematicians who worked hard and relentlessly, from the Han Dynasty to the Yuan Dynasty, have made Chinese mathematics reach one peak after another in the world mathematics world.What is even more strange is that the climax often occurs in the period of war, such as the foundation of the main achievements of "Nine Chapters of Arithmetic" in the Warring States Period, the establishment of mathematical theories in the Wei, Jin, Southern and Northern Dynasties, and the climax of calculation mathematics in the Song, Liao, Jin and Yuan Dynasties; In the unified peace and prosperity, such as the Tang and Ming dynasties, not only were there few mathematical achievements, it was even ridiculous that the great mathematicians could not understand the achievements of the previous generations!Of course, this does not mean that wars and divisions are more conducive to the development of mathematics than stability and unification, but because during wars, the dominance of Confucianism is often impacted, social thoughts are more active, and the mind is more liberated.At the same time, due to the war, the road to studying classics and becoming an official was blocked. Intellectuals were able to use their talents according to their own interests and social needs, and their mathematical talents were fully displayed. As a result, the situation of mathematical research in the cracks was relatively poor. The peaceful and prosperous age of great unity is better.