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.