Among the four civilizations in the ancient world, China's mathematics has the longest lasting prosperity. In14th century BC, China's classical mathematics experienced three development climaxes: Han Dynasty, Wei, Jin, Southern and Northern Dynasties and Song and Yuan Dynasties, and reached its peak in Song and Yuan Dynasties.

Unlike Greek classical mathematics, which centered on proving theorems, China ancient mathematics focused on creating algorithms, especially various algorithms for solving equations. From linear equations to high-order polynomial equations, even indefinite equations, ancient mathematicians in China created a series of advanced algorithms (called "technology" by mathematicians in China), and they used these algorithms to solve corresponding types of algebraic equations, thus solving various scientific and practical problems that led to these equations. In particular, geometric problems are also reduced to algebraic equations, and then solved by programmed algorithms. Therefore, China ancient mathematics has obvious algorithmic and mechanized characteristics. Here are some examples to illustrate this feature of the development of ancient mathematics in China.

1. 1 linear equations and "equation skills"

The "Equation Technique" in the eighth volume of Nine Chapters Arithmetic, the most important mathematical classic in ancient China, is an algorithm for solving linear equations. Taking the title of this volume 1 as an example, expressed by modern symbols, this problem is equivalent to solving a system of linear equations with three variables:

3x+2y+z=39

2x+3y+z=34

x+2y+3z=26

"Nine Chapters" does not use unknown symbols, but uses calculation to calculate X? y？ The coefficients and constant terms of z are arranged in a (long) square matrix:

1 2 3

2 3 2

3 1 1

26 34 39

The key algorithm of "equation technology" is called "multiplication and direct division". In this example, the calculation process is as follows: multiply the numbers of this row and the left row by the coefficient of the right row (x), and then "divide" the right row separately, that is, subtract the corresponding numbers of the right row continuously, so the coefficients of this row and the left row will become 0. This equation can be solved by repeatedly executing this "multiplication and division" algorithm. Obviously, the "multiplication and direct division" algorithm of the equation technology in "Nine Chapters Arithmetic" is essentially the elimination method for solving linear equations that we use today. It used to be called "gauss elimination" in western literature, but in recent years it began to change its title. For example, Professor P.Gabriel, an academician of the French Academy of Sciences and former head of the Department of Mathematics at the University of Zurich, called the elimination method for solving linear equations "Zhang Cang method" in his textbook.

1.2 polynomial equation of higher degree and "positive and negative square root"

There are "Fang" and "Fang" in Volume 4 of Nine Chapters Arithmetic. These algorithms in "Nine Chapters Arithmetic" were gradually extended to high-order cases, and developed into numerical solutions of general high-order polynomial equations in the Song and Yuan Dynasties. Qin is a master in this field. In his book Nine Chapters of Mathematics (1247), he gave a complete algorithm for numerical solution of polynomial equation of higher order, which he called "plus and minus square extraction".

Represented by modern symbols, Qin's idea of "positive and negative square roots" is as follows: for any given equation,

f(x)= a0xn+a 1xn- 1+……+an-2 x2+an- 1x+an = 0( 1)

Where a0≠0, an

f(c+h)= A0(c+h)n+a 1(c+h)n- 1+……+an- 1(c+h)+an = 0

The equation about h can be obtained by combining similar terms according to the power of h:

f(h)= a0hn+a 1hn- 1+……+an- 1h+an = 0(2)

Then the highest number of roots satisfying the new equation (2) can be estimated again. In this way, if the constant term of a new equation is 0, then the root is a rational number; Otherwise, the above process can be continued and the approximate value of the root can be obtained according to the required accuracy.

If the coefficients a0, a6+0, α 1 and α2 of the original equation (1) and the estimated value c are used to find the coefficients A0, a 1, an of the new equation (2), the algorithm needs to be used repeatedly. Qin gave a standardized scheme, which we can call "Qin scheme". He wrote in the ninth chapter of the math book. The maximum number of equations involved is 10. The algorithm of solving these problems in Qin Dynasty is unified and clear, and it is a model of algorithmic and mechanization of ancient mathematics in China.

1.3 multivariate higher order equations and "quaternary technology"

Not all problems can be reduced to a linear equation or a polynomial equation with unknown quantities. In fact, it can be said that if a larger number of practical problems can be solved by algebraic equations, there will be higher-order equations with multiple unknowns.

Even today, it is not easy to solve higher-order multivariate equations. Zhu Shijie, a mathematician of Yuan Dynasty in China, was the first person to systematically deal with multivariate higher-order equations in history. The higher-order equation involved in Zhu Shijie's Four Lessons of Yu Juan (1303) has reached four unknowns. Zhu Shijie used "four elements" to solve these equations. The "four-element method" firstly represents different unknowns with "heaven", "earth", "people" and "things", and at the same time establishes equations, and then solves the equations with the general sequential elimination method. Zhu Shijie met in Siyuan and created various elimination procedures.

Through the concrete examples in Four Elements Jade Mirror, we can clearly understand the characteristics of Zhu Shijie's "Four Elements Skills". It is worth noting that quite a few of these examples are derived from geometric problems. This kind of example of transforming geometric problems into algebraic equations and solving them with a unified algorithm can be found everywhere in the mathematical works of Song and Yuan Dynasties, which fully embodies the algebraic and mechanized tendency of China's ancient geometry.

1.4 linear congruence equations and "China remainder theorem"

Ancient mathematicians in China began to study the shape of calendar for the need of calendar calculation;

X≡Ri (mod ai) i= 1，2，...，n ( 1)

(where ai is a pairwise coprime integer). In the 4th century AD, there was a famous "grandson problem", which was equivalent to solving the following congruence groups:

X≡2 (module 3) ≡3 (module 5) ≡2 (module 7)

The solution given by the author of Sunzi Suanjing guided the general algorithm of solving a congruence group in the Qin Dynasty in Song Dynasty-"Big Extension and Seeking Technique". This general algorithm is usually called "China remainder theorem" in modern literature.

1.5 interpolation and "call difference"

Interpolation algorithm plays an important role in the brewing process of calculus. In China, as early as the Eastern Han Dynasty, scholars used interpolation to calculate the movements of the sun, the moon and the five stars. At first, it was a simple one-time interpolation method, but in the Sui and Tang Dynasties, a second interpolation method appeared (such as a line in Yan Li, 727). Because of the uneven acceleration of celestial motion, quadratic interpolation is still not accurate enough. With the progress of the calendar, in the Song and Yuan Dynasties, three interpolation methods appeared (Guo Shoujing's Chronological Calendar, 1280). On this basis, mathematician Zhu Shijie even created a general high-order interpolation formula, which is what he called "the trick". Zhu Shijie's formula is equivalent to

f(n)=n△+ n(n？ 1)△2+ n(n？ 1)(n？ 2)△3

+ n(n？ 1)(n？ 2)(n？ 3)△4+……

This is a very outstanding achievement.

It is impossible to list all the algorithms of ancient mathematicians in China, but it is not difficult to see from the above introduction that many algorithms created by mathematicians in ancient and medieval China have reached a high level even by modern standards. Some mathematical truths expressed by these algorithms were not regained by modern mathematical tools in Europe until after 18th century (for example, the Qin program for numerical solutions of higher-order algebraic equations mentioned above is basically consistent with Horner's algorithm re-deduced by the British mathematician W. Horner in 18 19; The systematic study of multivariate higher order equations did not appear in the works of E. Bezhu and others until the end of Europe 18. The residue theorem for solving the first-order congruence group was re-obtained by Euler and Gaussian respectively. As for Zhu Shijie's higher-order interpolation formula, it is essentially consistent with Newton-Gregory formula which is widely used now). The structure and complexity of these algorithms are also amazing. For example, the analysis of Qin's "large diffraction method" and "positive and negative root method" shows that the calculation programs of these algorithms contain the basic elements and structures of constructing non-trivial algorithms in modern computer languages. This complex algorithm can hardly be regarded as a simple rule of thumb, but a product of highly generalized thinking ability. It is completely different from the deductive thinking style of Euclid geometry, but it has played a completely incomparable role in the development of mathematics. In fact, the prosperity of China algorithm in ancient times also gave birth to a series of extremely important concepts, which showed the creative significance and dynamic role of algorithmic thinking in mathematical evolution. Here are a few examples.

1.6 introduction of negative numbers

In the elimination program of "equation skills" in "Nine Chapters Arithmetic", when the coefficient of the equation is subtracted, a decimal number will be subtracted by a large number. It is here that the authors of "Nine Chapters Arithmetic" introduced negative numbers and gave the addition and subtraction algorithm of positive and negative numbers, that is, "addition and subtraction".

Understanding negative numbers is an important step in the expansion of human number system. In the 7th century, Indian mathematicians began to use negative numbers, but the understanding of negative numbers in Europe was slow. Even in the16th century, Vedic works avoided negative numbers.

The Discovery of Irrational Numbers of 1.7

Ancient mathematicians in China were exposed to irrational numbers in square root operations. The Nine Chapters of Arithmetic Prescriptions points out that there are infinite situations: "If there are infinite prescriptions, you can't open them." The authors of Nine Chapters Arithmetic have given this endless number a special term-"face". "Face" is an irrational number. Compared with the Pythagorean school in ancient Greece who found that the diagonal of a square was not rational, the ancient mathematicians in China accepted those "endless" irrational numbers relatively naturally, which may be attributed to their long-term use of decimal system, which enabled them to effectively calculate the approximate value of "infinite roots". Liu Hui, a mathematician who annotated Nine Chapters Arithmetic in the Three Kingdoms period, clearly put forward a method of arbitrarily approaching countless roots with decimals, which he called "differential method", and pointed out that in the process of making a square, "one step back, one hundred steps back, one hundred steps back, and the points are all fine, so .. Although there are some abandoned numbers,

Decimal numeration system is an indelible contribution to human civilization. Laplace, a great French mathematician, once praised the invention of decimal system, saying that it "made our arithmetic system first-class in all useful creations". It was on the basis of strictly following the decimal system that ancient mathematicians in China established the Oriental Mathematics Building with algorithmic characteristics.

1.8 jiaxian triangle or yanghui triangle

It can be seen from the introduction of the numerical solution algorithm of higher-order equations (Qin program) that the square method in ancient China is based on? The binomial expansion based on c+h n leads to the discovery of binomial coefficient table. Yang Hui, a mathematician in the Southern Song Dynasty, wrote "Detailed Explanation of the Nine Chapters Algorithm" (126 1), which contains a so-called "root graph of cholesky decomposition", which is actually a binomial coefficient table. This picture is taken from a work by Jia Xian, a mathematician in the Northern Song Dynasty around A.D. 1050. The root map of recipe practice is now called "Jia Xian Triangle" or "Yang Hui Triangle". Binomial coefficient table is called Pascal triangle in the west? 1654.

1.9 trend sign algebra

The mathematical activity of solving equations will inevitably cause people to think about the expression form of equations. In this respect, mathematicians who were good at solving equations in ancient China naturally went ahead. In the mathematical works of Song and Yuan Dynasties, there have been systematic efforts to use specific Chinese characters as symbols of unknowns and then establish equations. This is the "Heavenly Skill" represented by Ye Li and the "Four Great Achievements" represented by Zhu Shijie. The so-called "Tianyuan technique" means "setting Tianyuan as so-and-so", which is equivalent to "setting Tianyuan as so-and-so", and "Tianyuan as one" means unknown, and then arranging "Tianyuan style" on the abacus, that is, one-dimensional equation. This method is extended to many unknown situations, that is, Zhu Shijie's "four-element method" mentioned earlier. Therefore, the method of sorting equations by celestial sphere method and quaternion method is similar to the method of sorting equations in modern algebra.

Symbolization is one of the symbols of modern algebra. Mathematicians in the Song and Yuan Dynasties in China made an important step in this respect. "Tian Shu" and "Si Shu" are the pinnacles of China's ancient mathematics, focusing on creating algorithms, especially solving equations? .

2 the contribution of ancient mathematics in China to the development of mathematics in the world

The development of mathematics includes two main activities: proving theorems and creating algorithms. Theorem proof was initiated by the Greeks, and then formed the pillar of deductive tendency in mathematical development; Algorithm creation flourished in ancient and medieval China and India, forming a strong algorithm tendency in the development of mathematics. Throughout the history of mathematics, we will find that the development of mathematics is not always dominated by deductive tendency. In the history of mathematics, algorithmic tendency and deductive tendency always occupy the dominant position alternately. The original algorithms in ancient Babylon and Egypt were replaced by Greek deductive geometry, but in the Middle Ages, Greek mathematics declined and the algorithms tended to flourish in China, India and other eastern countries. Oriental mathematics spread to Europe through Arabia on the eve of the Renaissance, which had a far-reaching impact on the rise of modern mathematics. In fact, analytic geometry and calculus, as the birth signs of modern mathematics, are not the products of deductive tendency, but the products of algorithmic tendency from the origin of thinking methods.

From the history of calculus, we can know that calculus is the result of finding a universal algorithm to solve a series of practical problems? 6? . These problems include: determining the instantaneous speed of the object, finding the maximum and minimum values, finding the tangent of the curve, finding the center of gravity and the center of gravity of the object, and calculating the area and volume. From the middle of16th century on 100 years, many great mathematicians devoted themselves to obtaining special algorithms to solve these problems. The merit of Newton and Leibniz lies in unifying these special algorithms into two basic operations-differential and integral, and further pointing out their reciprocal relations. No matter Newton's pioneer or Newton himself, the algorithm they used was not rigorous and there was no complete derivation. The logical defects of Newton flow number technology are well known. For the scholars at that time, the first thing is to find an effective algorithm, not to prove it. This trend continued until18th century. /kloc-mathematicians in the 0 th and 8 th centuries often make bold progress regardless of the difficulties in the foundation of calculus. For example, Taylor formula, Euler, Bernoulli, and even the triangle expansion discovered by Fourier at the beginning of 19 century have long lacked strict proof. As von Neumann pointed out: no mathematician would regard the development of this period as heresy; The mathematical achievements produced in this period are recognized as first-rate. On the other hand, if mathematicians at that time had to admit the rationality of the new algorithm after strict deduction, there would be no calculus and the whole analysis building today.

Now let's look at the birth of early analytic geometry. It is generally believed that Descartes' basic idea of inventing analytic geometry is to solve geometric problems by algebraic method. This is quite different from Euclid's deduction. In fact, if we read Descartes' original work, we will find the thorough algorithmic spirit running through it. "Geometry" declared at the beginning: "In order to make myself smarter, I will not hesitate to introduce arithmetic items into geometry". As we all know, Descartes' Geometry is an appendix to his philosophical work Methodology. Descartes strongly criticized the traditional research methods, mainly the Greek method, in another unpublished philosophical work, The Law of Guiding Thinking, and thought that the deductive reasoning of the ancient Greeks could only be used to prove what we already know, "but it could not help us discover the unknown". Therefore, he put forward that "a method of discovering truth is needed" and called it "mathematical universe". Descartes described the blueprint of this common mathematics in the Law. His bold plan, in short, is to transform all scientific problems into mathematical problems for solving algebraic equations:

Any problem → mathematical problem → algebraic problem → equation solution. Descartes' geometry is the concrete realization and demonstration of his above scheme. Analytic geometry plays an important role as a tool in the whole scheme, which turns all geometric problems into algebraic problems and can be solved by a simple, almost automatic or quite mechanical method. This is in line with the problem-solving route of ancient mathematicians in China introduced above.

Therefore, we have every reason to say that in the spring tide from the Renaissance to the rise of modern mathematics in the17th century, the rhythm of oriental mathematics, especially China mathematics, echoed. The whole17-18th century should be regarded as a heroic era for finding the infinitesimal algorithm, although the infinitesimal algorithm in this period has made a qualitative leap compared with the medieval algorithm. However, from the19th century, especially from the 1970s until the middle of the 20th century, the deductive tendency once again dominated on a level far higher than Greek geometry. Therefore, the development of mathematics presents a process of algorithm creation and deductive proof, in which two main streams alternately prosper and spiral upward:

Deductive tradition-theorem proving activity

Algorithm Tradition-Algorithm Creation Activity

Ancient mathematicians in China made great contributions to the formation and development of the algorithm tradition.

We emphasize the algorithmic tradition of ancient mathematics in China, which does not mean that ancient mathematics in China has no deductive tendency. In fact, in the works of some mathematicians in Wei, Jin, Southern and Northern Dynasties, there have been quite profound argumentation ideas. For example, the proof of Zhao Shuang's Pythagorean theorem and Liu Hui's "raising horses"? The proof of the volume of a rectangular cone, and the derivation of the formula for the volume of a sphere by Zu Chongzhi and his son. It can be compared with the corresponding work of ancient Greek mathematicians. The prototype of Zhao Shuang Pythagorean Theorem Proof Diagram "String Diagram" has been adopted as the emblem of the 2002 International Congress of Mathematicians. Confusingly, with the end of the Northern and Southern Dynasties, this tendency to debate can be said to have come to an abrupt end. Limited by the space and the focus of this article, it is impossible to elaborate on this aspect here. Interested readers can refer to it? 3? .

3 Make the past serve the present and innovate and develop

In the 20th century, at least from the mid-term, the emergence of electronic computers has brought a far-reaching impact on the development of mathematics, and a series of remarkable achievements have been born, such as soliton theory, chaotic dynamics, and proof of four-color theorem. With the help of computers and effective algorithms, we can guess and discover new facts, induce and prove new theorems, and even conduct more general automatic reasoning ... all these can be said to be the great prelude to a new era of algorithm prosperity in the history of mathematics. Sharp people of insight in the scientific community have foreseen this trend of mathematical development. In China, as early as 1950s, Professor Hua personally led the establishment of a computer research group, which laid the foundation for the development of computer science and mathematics in China. Since the mid-1970s, Professor Wu Wenjun has resolutely turned from the initial field of topology to the study of theorem machine proof, and started a brand-new field of modern mathematics-mathematical mechanization. The method of mathematical mechanization is known as "Wu method" in the world, which makes China in a leading position in the field of mathematical mechanization. As Professor Wu Wenjun himself said, "The mechanization problem proved by geometric theorem can be found from thinking to method, at least in the Song and Yuan Dynasties", and his work was "mainly inspired by China's ancient mathematics". "Wu Fa" is the development of the algorithmic and mechanized essence of ancient mathematics in China.

Under the influence of computers, the development trend of algorithms naturally aroused some foreign scholars' interest in the algorithm tradition in ancient mathematics in China. As early as the early 1970s, D.E.Knuth, a famous computer scientist, called people's attention to the algorithms of ancient China and India? 5? . Over the years, some progress has been made in this field, but in general, it still needs to be strengthened. As we all know, China's ancient culture, including mathematics, spread to the west through the famous Silk Road, and the Arab region is an important transit point for this cultural spread. Some existing books on mathematics and astronomy in Arabic contain some knowledge of mathematics and astronomy in China. For example, a considerable number of mathematical problems in Al Casey's masterpiece The Key to Arithmetic directly or indirectly show the origin of China. According to Al Cassie, there are many scholars from China in the observatory where he works.

However, for a long time, due to the influence of "western-centrism", especially "Greek-centrism" and the obstacles in language and writing, the relevant materials have not been excavated very far. In order to fully reveal the relationship between oriental mathematics and European mathematical renaissance, Professor Wu Wenjun specially allocated special funds from the highest national science award he won to set up the "Wu Wenjun Silk Road Fund for Mathematics and Astronomy" to encourage and support young scholars to carry out in-depth research in this field, which is of far-reaching significance.

An important significance of studying the history of science is to learn from the development of history and promote realistic scientific research. In layman's terms, it means "making the past serve the present". Wu Wenjun has an incisive exposition on this. He said: "If you understand the historical development of mathematics, the occurrence and development of a field, the rise and fall of a theory, the ins and outs of a concept, the emergence and influence of an important thought and many other historical factors, I think you will know more about mathematics, understand the current situation of mathematics more clearly and deeply, and can also play a guiding role in the future of mathematics, that is. The establishment of mathematical mechanization theory is the result of this principle. The great rejuvenation of science and technology in China calls for more innovations with strong China characteristics and distinctive flavor of the times.