China is the source of mathematics in the world: arithmetic, algebra, and geometry were created by ancient Chinese mathematicians中国是世界数学之源：算术、代数、几何都是中国古代数学家创造
China is the source of mathematics in the world: arithmetic, algebra, and geometry were created by ancient Chinese mathematicians
Chinese mathematics has always been in a dominant position in the world, and it was far ahead in many major fields until the end of the Song Dynasty and the beginning of the Ming Dynasty. Today's elementary and middle school mathematics, such as arithmetic, algebra, geometry, etc., are all derived from the creations of ancient Chinese mathematicians, most of which were completed in the "Nine Chapters of Arithmetic" in the Han Dynasty. However, India, Arabia and Europe to the west of China were very late. During the Renaissance, middle schools spread to the west, which inspired Europeans to get rid of the ignorant thinking of the supremacy of theology. For example, calculus is the product of Chinese mathematics over Greek mathematics. It was mainly Chinese mathematics rather than Greek mathematics that determined the development of mathematics at that time. After that, modern mathematics advanced by leaps and bounds, driving the rapid development of western science and technology. In recent years, the most superficial thinking of some peoples on the natural society is to blindly follow the vague and simple language of emotional expression; while rational people analyze specific phenomena, until they use mathematics and other tools as the main scientific thinking. Both scientific experiments and scientific hypotheses require the support of engineering technology. Both theory and technology enrich the body of science, and we must not stop at the representational thinking of language tools.
The essence of scientific theory is that scientists use mathematical tools to make qualitative and quantitative explanations of natural society. This issue of Sciences briefly introduces the sources of the history of mathematics such as arithmetic, geometry, and algebra. Mathematics textbooks for primary and secondary schools around the world today, such as arithmetic, algebra, and geometry, are actually created by ancient Chinese mathematicians. Most of them were prepared in the "Nine Chapters of Arithmetic" in the Han Dynasty. Arabia and Europe are very late. During the Renaissance, middle schools spread to the west, which inspired Europeans to get rid of the ignorance of the theological supremacy. Academician Wu Wenjun pointed out that "the invention of calculus is the product of the Chinese mathematical style defeating the Greek-style mathematics." It can even be said, "The reason why modern mathematics can develop to today is mainly due to Chinese mathematics, not Greek mathematics. The development of mathematics is mainly Chinese mathematics rather than Greek mathematics." Based on this appeal, "The history that has been reversed must be reversed!". The following is Mr. Wen Xing's article.
1. China is the source of mathematics in the world
The Opium War began, the great powers invaded, and the country fell. After the defeat in the Sino-Japanese War of 1894-1895, the Chinese national self-confidence was severely frustrated, and they lost their souls. They began to deny and overthrow Chinese culture, and even shouted "If Chinese characters are not destroyed, China must perish." Even after the founding of the People's Republic of China, this was still the case. In the 1980s, "river cups" flooded, slandering Chinese history as dark and useless. Naturally, mathematics cannot be separated from the torrential "river".
Mr. Wu Wenjun, academician of the Chinese Academy of Sciences and mathematician, pointed out in the article "The Great Contribution of Ancient Chinese Mathematics to World Culture", "Most historians of mathematics in the West, except that they must call Greece, distort history and create a lot of Babylonian mathematics in the East. Myths and Indian myths try to belittle the brilliant achievements of Chinese mathematics, or even turn a blind eye and wipe them out."
For these racist fallacies, domestic scholars do not refute them. Instead, they either follow suit, parroting their tongues, and shouting "the words must be called Greece", or they keep silent. This is mainly due to "ignorance of the ancient mathematics of the motherland" due to the weakness of modern China. Some recent studies have shown that these are only part of the fake history conceived and fabricated by the West in order to meet the needs of constructing the history of capitalist development and Eurocentrism since the Great Geographical Discovery.
Restoring the historical truth
In fact, "Chinese mathematics has always been in a dominant position in the world and was far ahead in many major fields until the end of the Song Dynasty and the beginning of the Ming Dynasty." Judging from the content of mathematics textbooks for primary and secondary schools in various countries today, such as arithmetic, algebra, geometry, etc. The content is all created by ancient Chinese mathematicians. Most of the "Nine Chapters of Arithmetic" was completed in the Han Dynasty, while India, Arabia and Europe to the west of China were very late.
Qian Baocong, a famous Chinese historian of mathematics, discussed in detail the evolution of China's "Nine Chapters of Mathematical Sciences" in India, Arabia, and Europe in the article "Great Achievements in Ancient Chinese Mathematics":
1. "China's (integer) calculation multiplication and division rule evolved into the soil plate rule in India, spread to Arabia, evolved into the pen calculation method, spread to Europe, and gradually improved to the current algorithm. There are reliable historical materials for the evolution process It can be proved clearly.”
2. Fraction notation (the numerator is on the top and the denominator is on the bottom) and "later Indian and Arabic fractional algorithms were also passed down from China."
3. Concepts such as proportion and the golden rule in Europe were also spread from China to India, Arabia, and then to Europe.
4. China's surplus-deficiency technique was passed to Arabia when China had been eliminated by more advanced methods. "The Arabs attached great importance to it, and compiled it into their algebra books. It has the title of 'Khitan Algorithm'. It was passed from Arabia to Europe , was also commonly used in algebra books in the sixteenth and seventeenth centuries, and was renamed 'rule of double false position'."
5. China's "simultaneous linear equation algorithm" was passed to India. The Indians changed the calculations to horizontal writing and added unknown numbers. This is the origin of the current algebraic writing. Indians know negative numbers and say that they have the meaning of debt Probably also learned from Chinese mathematics."
6. The so-called "Horner method" (1819) in textbooks for solving higher-order equations is actually the "increasing multiplication and opening method" of the Song Dynasty in China. Although Qian Baocong did not explicitly say that Europe originated from China, it is obvious that this actually It is plagiarized from China.
When Chinese scholars make a comparison between China and the West, they often only mention "how many years earlier" China is, and dare not directly mention "the gradual development of Chinese learning to the West". How sad that a nation that once felt so inferior that it wanted to eliminate Chinese characters and lost its soul is still so "cautious" today!
Academician Wu Wenjun quoted Kaye, an Indian historian of mathematics, as saying that there are many parallels between Indian and Chinese mathematics, and India owes China a debt.
Middle School Xijian concluded that "Chinese mathematics has a teaching and receiving relationship with India, Arabia, Japan and Western countries". The development of modern mathematics in Europe benefited from Arabic mathematics and Chinese mathematics, and was the product of the "combination of Chinese and Arab".
Wu Wenjun pointed out that "Chinese mathematics can be said to have been "until the 16th century. Our country's mathematics has always been in the most advanced position in many of the most important fields." It is generally believed that these creations are purely the achievements of Western European mathematics. But ancient Chinese mathematics is by no means insignificant (or even decisive). "Academician Wu Wenjun conducted a detailed analysis and demonstration on this.
"The invention of calculus is the product of Chinese mathematics defeating Greek mathematics." It can even be said, "The reason why modern mathematics can develop to this day is mainly due to Chinese mathematics, not Greek mathematics, which determines the development of mathematics. It is mainly Chinese mathematics rather than Greek mathematics." Based on this, Academician Wu Wenjun appealed, "The history that has been reversed must be reversed!"
"Baojian of Computing and Learning" and the Rewriting of Ming History
The above elaboration is the research conclusion of Qian Baocong in 1951 and Wu Wenjun in 1975, that is, before Wang Wensu's "Xinji Token Mathematical Treasure" was really valued and studied. "Baojian of Mathematical Sciences" is "a copy of Lange found in the old bookstore in the Beijing Library during the Republic of China and was collected." It was mentioned in an article. In the 1960s, the "History of Chinese Mathematics" edited by the historian of mathematics Qian Baocong also mentioned this book, but no one had seen or really understood this book at that time. Wang Wensu's "Baojian of Mathematics" was really valued and studied after it was recommended by Professor Zhao Qinghuan from the Physics Department of Beijing Normal University in 1992.
Judging from the current research, "Baojian of Computing Science" represents the highest level of Chinese mathematics in the past dynasties, and it was also the highest level in the world at that time. ""Baojian of Mathematical Sciences" studies the numerical solutions of higher-order equations in one variable. The content is detailed and valuable, which fully shows that the numerical solutions of higher-order equations in one variable, as well as the Tianyuan technique and the four-yuan technique were not completely lost in the Ming Dynasty. The terms used by Wang Wensu in the solutions Terminology and calculus procedures are basically consistent with Song and Yuan mathematics, and have been developed and innovated. Wang Wensu’s mathematical achievements are strong evidence of the continuity of Chinese mathematics history.”
"Wang Wensu's method of solving higher-order equations was nearly 300 years earlier than that of England's Horner (1786-1837) and Italy's Ruffini (1765-1822); in solving algebraic equations, he walked in the 17th century. Newton (I. Newton, 1642-1727) and Raphson (J. Raphson, 1648-1715) took the lead in using derivatives to solve iteratively for more than 140 years, writing a glorious chapter for the history of Chinese mathematics; for the creation of calculus in the 17th century Wang Wensu was the first to discover and use the derivatives that appeared during the 16th century, so it is not enough to explore the origin of derivatives only from the perspective of calculus, so Wang Wensu’s contribution to world mathematics should be further studied.”
These latest research findings have greatly strengthened Academician Wu Wenjun's conclusion that "the development of modern mathematics mainly depends on Chinese mathematics". Some people who depreciate China are still "entangled" with Professor Wu Wenjun's "Chinese-style mathematics" and "Chinese mathematics", but the appearance of "Baojian of Computing" allows the word "style" to be removed. Mr. Wen Xing said in "Missionaries Steal Chinese Civilization and Reverse World History", "Wang Wensu is not 'early' or 'first', but European modern mathematics is completely a product of Chinese mathematics passed by missionaries to the west. "Middle School's Westward Progress', including Newton's and Leibniz's calculus system originated from the 'derivative' of Wang Wensu in the Ming Dynasty, is not a European invention at all." Of course, the effect of "South Orange and North Trifoliate" has occurred——Using Arabic Mathematics The bottle contains the wine of Chinese mathematics.
However, Xiang Guanjie said in his book "Achievements of Ancient Chinese Mathematics" (1988), "Since the birth of Chinese mathematics, it has been developing continuously... After Zhu Shijie, the development of ancient Chinese mathematics Suddenly there was a serious interruption. In the nearly three hundred years from Zhu Shijie to Cheng Dawei in the Ming Dynasty, there has not been a single important mathematician, nor has there been an important mathematical work. Not only is there no new development, Even the precious legacy left by Song and Yuan mathematics has not been preserved."
There are many such arguments, such as: "The mathematics of the recent history, from the early Ming Dynasty to the early Qing Dynasty, was about 1367 AD to 1750 AD, about 400 years... Folk mathematics masters followed one after another, which is called the quiet period of the Chinese calculation. ", "After the middle of the Ming Dynasty, many abacus reading books written by businessmen were published. Compared with the advanced mathematics of the Song and Yuan Dynasties, there was only a lack of it. The ancient Chinese traditional mathematics was almost lost in the Ming Dynasty." "In the 14th century... the ancestors worked hard to create The Tianyuan technique was completely lost. Before the introduction of Western academics, the most important and most widely circulated mathematics book was Cheng Dawei's "Algorithm Tongzong" (1592). This book contains no new information except abacus and song formulas. The creation of it. It is basically a book that sorts out previous works, and omits important parts such as higher-order equations and multivariate higher-order equations", etc.
Xiang Guanjie asked: "Why did the development of ancient Chinese mathematics suddenly stop in the 14th century? This question has always attracted the attention of Chinese and foreign historians." Interestingly, Xiang Guanjie also mentioned in the book, "In the Ming Dynasty At the end of the period, the error between the Datong calendar and the Hijri calendar became larger and larger, and revision of the calendar became an urgent task. However, the huge Ming Dynasty could not find a person who could preside over the revision of the calendar. This shows that after the Ming Dynasty, ancient Chinese astronomy and What a pathetic level the maths has fallen to.”
However, in Mr. Wen Xing's "Missionaries Stealing Chinese Civilization and Reversing World History", Mr. Li Liang of the Institute of Natural Science History of the Chinese Academy of Sciences quoted an article "The "Missed" Communion—Missionaries' Contributions to the Chongzhen Calendar Reform Period." "Selective Deletion of Food Records" mentioned that missionaries deleted and edited Chinese historical documents in large quantities, making us mistakenly believe that the traditional Chinese calendar is getting worse and worse, but the actual situation is that the Western calendar is not as good as the Chinese historical documents currently seen. As shown, it completely outperforms the traditional Chinese calendar. The imperial decree of Emperor Chongzhen clearly stated that "the direction of the first loss of the solar eclipse and the direction of the recirculation are all consistent with the "Da Tong Li". The eclipse time predicted by Wei Wenkui's method is more consistent with the measured results. What's more terrible is that, as Nan Huairen said, the results calculated by the most famous astronomers in Europe will be very different from the measured results, but in China they can be accurate to "a moment". Nan Huairen is very excited about this, and In fact, this is due to the superb mathematics level of the Ming Dynasty.
In fact, Mr. Wen Xing mentioned in "From Qu Yuan Being Kicked Out of History Textbooks by the West" that the Qing Dynasty "in addition to burning books, the Qing Dynasty also systematically destroyed Ming Dynasty archives. , Chongzhen Dynasty military archives, there are also a small number of official documents of Hongwu, Yongle, Xuande, Chenghua, Zhengde, Jiajing, Longqing, Wanli, and Taichang dynasties. The rest are estimated to be no less than 10 million copies of Ming Dynasty archives, all of which have been destroyed. Except for the destroyed In addition to books and archives, the Great Qing also systematically tampered with the remaining books and archives." Only three ten-thousandths (3‱) of the archives of the Ming Dynasty have been handed down, and even the remaining 3‱ has been systematically deleted.
However, Wang Wensu's "Baojian of Mathematical Sciences" survived because it escaped the compilation of "Siku Quanshu" - "There were no collectors and public and private bibliographies for four hundred years, and it was discovered by the Beijing Library in the old book shop during the Republic of China Lange's manuscript was included in the collection", so that we can have a glimpse of the brilliance of mathematics in the Ming Dynasty and restore the upside-down history.
Mr. Wen Xing mentioned in "Missionaries Steal Chinese Civilization and Reverse World History" that Tang Ruowang deleted and tampered with the "Chongzhen Almanac", and deleted "The Origin of Zhili" from twelve volumes to eight volumes. This has been confirmed and is firmly established. . The Korean collection allows us to have a glimpse of the brilliance of astronomy, calendar, and mathematics in the Ming Dynasty, and to restore the history of astronomy that was tampered with by missionaries.
To sum up, combined with Mr. Li Zhaoliang's textual research on "Kunyu Wanguo Quantu", which has been mentioned many times, we can get a glimpse of the glory of the surveying and mapping world in the Ming Dynasty. We can firmly say that there is sufficient evidence that the Manchus and missionaries They conspired to carry out large-scale and systematic deletion and destruction of a large amount of Ming Dynasty materials, belittle and castrate the civilization achievements of the Ming Dynasty, and reverse the "middle school westward transfer" of Ming Dynasty knowledge and technology to the west as "Western learning eastward transfer".
The missionaries came to China at the end of Ming Dynasty and broke into China’s top ranks, followed by the demise of the Ming Dynasty and the establishment of the Qing Dynasty. "Bandit", that is, the patriotic triumphal culture (Voltaire), the Orientalism revealed by Said, must arouse our high vigilance, and must be fully considered in our related research.
We can reason reasonably: the Qing Dynasty also deleted the information about Zheng He's voyages to the West, and then blamed it on Liu Daxia; even the loss of "Yongle Dadian" was not unrelated to the Qing Dynasty. Based on the above, we must be highly vigilant about tacit evidence about the Ming Dynasty. We cannot say that there is no, non-existent, or never happened just because there is no record.
Therefore, we cannot think that only Wang Wensu can reach this level in the entire Ming Dynasty, but we should think that Wang Wensu is a representative of the highest level of mathematics in the Ming Dynasty. Funding cannot be published. The reason why Wang Wensu's "Baojian of Computing and Learning" can be handed down depends on escaping the compilation, deletion, modification and destruction of "Siku Quanshu", and those mathematics works of the same level or higher level were destroyed by the Qing Dynasty and missionaries. Therefore, it is entirely possible for Western missionaries to have access to the highest level of mathematics in the Ming Dynasty, and to spread this knowledge to Europe. European modern mathematics was born under the premise of the introduction of Arabic numerals and Latin Greek letters in Europe. On the contrary, we mistakenly think that the development of mathematics in the Ming Dynasty was interrupted, stagnated, and regressed, and it was "down to a pitiful level". The result of scholars conspiring to falsify history.
Xiang Guanjie also talked about Zhu Shijie's high-order arithmetic sequence summation formula in "Achievements of Ancient Chinese Mathematics",
It is impossible to get this formula without reasoning. It is just that ancient China did not like "empty talk" (just like famous scholars in the Spring and Autumn Period), but emphasized practical results and practical applications. Therefore, the tedious process was not written out. As far as Zhu Shijie’s above-mentioned formula is concerned, Westerners could not reach this level until the 18th century. Obviously, it should not be understood as the Westerners invented it out of thin air, but should be understood as “learning from middle school to the West”. We can imagine that when other nations exposed to Chinese mathematics, they will not only be amazed by the conclusion, but also be interested in the process of proof, and put more emphasis on the process of proof. This is one of the basis for judging originality and circulation. Therefore, when Chinese mathematics was exported to the domestic market, we mistakenly thought that it was a different kind of mathematics.
Mr. Wen Xing pointed out in "From Qu Yuan Being Kicked Out of History Textbooks by the West" that Europe during the Renaissance was far from being secular, rational, scientific, and logical as we now think, and quoted the latest Western research results: "The Renaissance Movement Humanism is not a philosophical trend or system at all... At best, it is something related to literature... Although the humanities school at that time included a philosophical category, morality, it included logic, natural philosophy, metaphysics, and even Other fields of study, including mathematics, astronomy, medicine, law, and theology, were shut out." (Kristler in Classics and Renaissance Thought, p. 7)
"It seems to me that the humanism of the Renaissance tried in every possible way to integrate itself with the philosophy, science, and education of the whole period; and the present facts seem to provide counter-evidence to this effort." (Christler on p. 7 of Classics and Renaissance Thought)
"During the renaissance movement, there was no spiritual or philosophical trend of 'humanism' at all; its matching with various ideas, such as human-centered literary creation and spiritual liberation and freedom, are the worldviews and ideals of later generations It has nothing to do with the spirit of the times." (Nahia Jakovaki in "Europe From Greece" p. 64)
"Actually, in the West until the end of the 17th century, wisdom and divination, philosophy and alchemy were still synonyms. Only in the 18th century did these two pairs of synonyms completely part ways." The cognition that Europe has been brilliant since the Great Geographical Discovery It must be completely reversed, and here I illustrate it with a picture.
Therefore, various signs and evidences show that during the Renaissance, Europe was still very ignorant, far from being as advanced as we imagined, and the progress of modern European civilization was entirely derived from the gift of China in the Ming Dynasty. We can even say that the achievements of modern European development were copied and transplanted by the Ming Dynasty, and modern European civilization is the "South Orange and North Trifoliate" of Chinese civilization. This of course includes mathematics.
China in the Ming Dynasty was slandered, and materials such as "History of the Ming Dynasty" were severely altered and destroyed. In this case, we should not be too limited to historical materials, but should use more logical analysis and common sense to discover the distorted history and restore it. the truth. For example, it is not easy to deny Menzies’ relevant research results ("1421" and "1434") based on the records of Chinese historical documents, and it cannot be denied that some of his research may be wrong. From the current point of view, the direction of his research and the general Most are right, and it's time to take his research seriously (see "Black Athena", "Silver Capital", "The Eastern Origins of Western Civilization", "Europe in China", "A Hundred and Fifty Years of Archeology" wait) .
In my opinion, the Ming Dynasty was extremely brilliant, far beyond our imagination and cognition. Since a large number of documents have been severely tampered with and destroyed, the truth of the Ming Dynasty needs to be rediscovered by us. History of the Ming Dynasty must be rewritten!
The source of world mathematics
As mentioned above, Academician Wu Wenjun agrees with Qian Baocong, the historian of mathematics, on the development of mathematics in Europe.
One of the core directions of Mr. Wen Xing's official account is to expose Western pseudo-history. According to his article "How is Classical Greek Pseudo-History Made?" ", "Ancient Greek Fake Books of Ancient History" and "The Treacherous Greek Empire", the Greek-Gris of the Renaissance actually refers to Eastern Rome, and the Greek actually refers to the text of Eastern Rome. According to Mr. Wen Xing's "Ancient Roman Gold Coins Exposing the Pseudo-History of Western Civilization", the Greek language was actually born during the reform period of Heraclius, and then gradually improved. In the 9th century, all Roman gold coins were written in Greek, which shows that the Greek language has been basically perfected.
Therefore, the "Greece" that Qian Baocong called is actually the Eastern Rome after the 9th century. Therefore, combined with Mr. Wen Xing's public account's views on the pseudo-history of ancient Greece, Mr. Wen Xing based Academician Wu Wenjun on the basis of Mr. Qian Baocong's opinion
2. The great contribution of ancient Chinese mathematics to world culture
In 221 BC, Qin Shihuang destroyed Xingguo and established the first centralized feudal state in Chinese history. The Han inherited the Qin system, and the two hundred years from Qin to the middle of the Western Han Dynasty was a period of consolidation, development and rise of the dictatorship of the emerging landlord class, and the legalist line dominated. Legalists paid more attention to industrial and agricultural production and science and technology, which promoted the rapid development of mathematics, and a group of high-level mathematicians appeared, such as Zhang Cang and Geng Shouchang. "Zhou Bi Suan Jing", "Xu Shang Suan Shu" and "Du Zhong Suan Shu" (the latter two have been lost) all appeared in this period. The most important mathematical work handed down to later generations in my country, "Nine Chapters on Arithmetic", was basically written in the early Western Han Dynasty, and its content laid the foundation for the brilliant achievements of the next thousand years. From the Western Han Dynasty to the Song and Yuan Dynasties, although Confucianism continued to interfere, with the process of the struggle between Confucianism and Legalism, the process of the struggle between idealism and materialism, and with the creative development of my country's social economy and working people, mathematical talents and mathematical creations have continued to evolve from generation to generation. Undoubtedly, it can be said that Chinese mathematics has always been in a dominant position in the world and was far ahead in many major fields until the end of Song Dynasty and the beginning of Ming Dynasty. For other reasons, the development of science and technology has been stifled. In addition to the important development of folk computing technology, mathematics has declined correspondingly. Since Matteo Ricci infiltrated my country's ruling group with bad intentions in the name of introducing Western mathematics at the end of the Ming Dynasty, compared with ancient times, China's mathematics has been far from being innovative. Crawled behind. As Chairman Mao criticized, "Except for saying Greek, I am sorry and forget about my ancestors." Most historians of mathematics in the West distort the history of Eastern mathematics, except for saying Greek. Many Babylonian myths and Indian myths have been created, and the brilliant achievements of Chinese mathematics have been belittled as much as possible, or even turned a blind eye, and wiped out. "Yes, the history of mathematics I have seen is all from "Western historians", and I am very ignorant of the ancient mathematics of the motherland, so I have no way to distinguish some fabrications and distortions of Western mathematics historians. If I follow the words, I must call it Greece, so I have to keep silent.
But history that has been turned upside down must be turned upside down!
As a specific example of ancient Chinese mathematics achievements, we might as well take a look at Chinese and foreign mathematics textbooks for primary and secondary schools until at least the 1950s and 1960s. There is a History of Mathematics in the West ([12], which is basically a better history of mathematics) which says: "In many middle schools, algebra is still taught as a bunch of formulas rather than a deductive science". When referring to the oriental origin of algebra, he said: "Algebra and geometry in schools today still retain the symbols of these different sources."
The so-called Eastern origin or Eastern mathematics here refers to the so-called "Euclidean treatment" based on "a strictly logical deductive method from some definitions, assumptions, and axioms to theorems". Mathematically speaking, the East mentioned in this book originally meant Babylon or India, and we will see that this East should refer to China. It is true that the geometry courses in primary and middle school mathematics textbooks have a special form of expression, which is obviously different from the expression forms of other arithmetic, algebra (even trigonometry and analytic geometry). It is not difficult for us to imagine, if the arithmetic and algebra parts of the oriental color (that is, the Chinese color) are removed, and only the so-called Greek geometry part is kept, what will happen to our primary and middle school mathematics?
Parts of mathematics in primary and middle schools, such as algebra and algebra, from counting to solving simultaneous linear equations and quadratic equations, are essentially inventions and creations of ancient Chinese mathematicians. Bi Suan Jing and other books, according to Qian Baocong's textual research ([3]), Liu Hui's annotation of Jiuzhang Suanshu was written in 263 AD, and the complete book was completed between 50 and 100 AD. But except for a few fragments, the basic content should be completed in 200 BC or earlier (this is the opinion of some western mathematics historians. Some even as early as 1000 BC, such as [9]), the predecessor of Chapter Nine It is the work of Zhang Cang and Geng Shouchang. Zhang Cang had been a palace in Qin Dynasty, and Geng Shouchang was also from the time of Emperor Xuan of the Western Han Dynasty. Some Western claims have a certain basis. But we still might as well say that it was written between 50 and 100 AD. The other Zhoubi Suanjing was written around 100 BC according to Qian Baocong's textual research ([3]).
The following is a Chinese-foreign comparison table about the inventions and creations of arithmetic algebra. It should be pointed out here that although Indian inventions are listed in the table, as Kaye, an expert on the history of mathematics in India, said: Indian and Chinese mathematics have many parallels. Whereas India is in debt to China. (See eg Cajori, [6], pages 97 and 84, and eg Scott [9]). Moreover, what is listed in the table is based on the statements of the two historians of Indian mathematics, Datta and Singh ([7]), who defended Indian mathematics. These statements are highly doubtful even if they do not consider Chinese factors.
In the long-term practice process, the working people in China have created and developed the methods of counting, fractions, decimals, positive and negative numbers, and infinitely approaching any real number, which has essentially achieved the completion of the entire real number system. In particular, there has been a perfect decimal notation system since ancient times. This is a unique creation of China, which is not available in other ancient nations in the world. This creation has made a great contribution to world culture. If it cannot be compared with the invention of fire, it can also be compared with inventions such as gunpowder, compass, and printing.
Algebra is indisputably a Chinese creation. The development clues from Jiuzhang to Qin Jiushao and Zhu Shijie in the Song and Yuan Dynasties are very clear. It can even be said that before the 16th century, except for some Arabic works, algebra was basically a Chinese creation. Single-handedly arranged. But the achievements of ancient Chinese mathematics are by no means limited to arithmetic and algebra. In terms of geometry, the keystone of Greek Euclidean geometry is the Pythagorean theorem (from Bourbaki, [5]) or the Pythagorean theorem ． This theorem has naturally existed in ancient my country. However, when Chinese and foreign historians of mathematics mentioned the Pythagorean theorem in China, they either cited the special case of the three-legged four-five theorem, or they quoted the general theorem but only covered nine chapters at the earliest. Statement of theorem:
"If you want to find the evil solstice, use the sun's setting as the hook, and the sun's high as the stock, take the hook and the stock respectively, and open the square to get rid of them, and you will get the evil solstice."
Not only that, the Pythagorean theorem is also specifically applied to the direct mutual finding of Pythagorean chords, and even to complex problems such as measuring the height of the sun. This is quite different from the situation in which theory is divorced from reality in Euclidean geometry. There are many differences between Chinese geometry and Greek geometry, and their detailed comparison remains to be elucidated.
China is also one of the earliest inventors of trigonometry. Western historians of mathematics generally regard Ptolemy (Ptolemy, around 150 AD), the author of "Almagest", as the founder of trigonometry, and regard Chinese trigonometry as influenced by him, for example, In the History of Western Mathematics [12] it is said:
China "has some trigonometry, chiefly in the Sea Island Suanjing, but since this sutra is attributed to the third century after the Epoch, we cannot ignore Western influences."
It is true that Liu Hui, the author of "Hai Dao Suan Jing" was born in the third century AD, but according to the preface of Liu Hui's Nine Chapters Annotations, "Hai Dao Suan Jing" was originally the tenth volume of Nine Chapters Annotated "Heavy Differences", while Zheng Xuan's "Nine Chapters" at the end of the Eastern Han Dynasty "Zhou Li Zhu" quotes Zheng Zhong's "Nine Numbers" (about 50 AD) in Zhou Li's "Nine Numbers", which says "today there are heavy differences and Pythagorean shares". It can be seen that the predecessor of Liu Hui's "Sea Island" was the heavy difference technique in the Han Dynasty. If we analyze the method of measuring altitude in "Sea Island" in detail, we can see that the method of weight difference has a long history. Zhou Bizhong:
"The Zhoubi is eight feet long, and the sundial on the summer solstice is one foot and six inches. If it is a thousand miles south, it will be one foot and five inches. If it is a thousand miles north, it will be one foot and seven inches. From here on up to the sun, it will be eighty thousand miles."
This is exactly the same as the first question of "Today's hopeful island" in "Sea Island Suanjing". It is true that Zhou Bi’s view of the earth as flat is a mistake, but the triangulation principle it is based on is correct, and because of this, Zhou Bi used it to observe the sky, Liu Hui used it to measure the earth, and established the heavy Difference-based triangulation. The purpose and method of this kind of triangulation have already been explained in Zhou Bi. Zhou Bi quoted Chen Zizhi's words "looking high and starting far" is its purpose, and quoted Shang Gao's words "a straight moment is used to straighten a rope, Looking at the height, measuring the depth by compounding the moment, and knowing the distance by lying down the moment" is its method. Liu Hui changed "to measure the image of the round sky" to "to measure the height of Mount Tai and the breadth of the river and the sea," and by analogy, it became "the one with the highest degree pays more attention to the table, the one who measures the depth accumulates moments, the one who is isolated is destroyed three times, and the one who is far away is destroyed three times. And begging to look around" is nothing more than the development and deduction of Zhou Bi's triangulation technique of setting up two watches to measure the height of the sun.
In Western trigonometry, there is first spherical triangle and then plane triangle. Torremy's "Astronomical Book" mainly refers to the center of the earth, and his trigonometry comes from astrometry, so it is spherical trigonometry. As for plane trigonometry, it was not established until 1250 A.D. by the Persian astronomer Nafaraddin. But in our country, no matter whether it is the observation of the sky by the Zhoubi or the geodesy of the island, it is plane trigonometry from the beginning. The development paths of Eastern and Western trigonometry are exactly opposite, and it is difficult to talk about any mutual influence. If there is influence, then Zhou Bi was several hundred years earlier than "Tian Wen Shu", and it is more reasonable to say that Torremi was influenced by China. The previous quotations of western mathematics historians on Chinese trigonometry obviously reversed the historical facts.
In Western Europe, many pictures describing triangulation appeared in the 16th century, one of which was called "Drum Trigonometry", which was drawn just like the Rigao Diagram (that is, the principle diagram of gravity difference) attached to Zhou Bi and Zhao Shuang's note. A replica, which can also see the advanced level of triangulation in our country. (See Smith for the drawing, [10], p. 355)
Analytic geometry and calculus, which did not appear until the 17th century in Western Europe, are the two main creations leading to the so-called modern mathematics. It is generally believed that these creations are purely the achievements of Western European mathematics. But ancient Chinese mathematics is by no means insignificant (or even decisive).
Let’s talk about analytic geometry first. Smith ([11], page 316,) once believed that the development of analytic geometry has three main steps: (1) the invention of the coordinate system; (2) the understanding of the one-to-one correspondence between geometry and algebra; (3) ) The graphical representation of the function y=f(x), the first step belongs to ancient times, the second step belongs to the middle ages, and the third step belongs to modern times.
In the west, it is believed that Descartes (and Fermat) in the 17th century is the founder of analytic geometry, but in fact, there is neither the concept of coordinates nor coordinate axes in Descartes' major works, let alone the equations of straight lines and curves. The contribution of Descartes is in the second step, that is, the establishment of the relationship between geometry and algebra. In his main hard work, he gave the geometric solution of the quadratic equation, but the earliest algebra in Arabia is the AI-Khowarizmi Works (9th century AD) have already used another method better than Descartes to draw the geometric solution of the quadratic equation. In fact, the unified treatment of geometry and algebra is a traditional feature of ancient Chinese mathematics. It has been like this since Chapter Nine. It must have originated in China. Judging from the style of the work, the latter is not impossible. This period of history is naturally worthy of attention and clarification. It is now known that Huacizimo was sent as an envoy to the Western Turkic Khazar State north of Persia from 842 to 847, which is an important commercial hub between the East and the West. Khasa could speak Chinese, and the imperial court followed Chinese etiquette ([13], Addenda). Details need to be further studied investigation.
As for the first step, Western historians of mathematics agree that the real concept of coordinates appeared in Oresme's work about using "longitude" and "latitude" to represent the position of a point in the middle of the 14th century. Mention Smith pointed out ([11], p. 320 note), Oresme's work may be derived from a work in the 10th century. The 10th century works here are estimated to be Arabic. In our country, there are such words as "the scale is used to rule the meridian" and "the ruler where the instrument is located is the skin poisonous insect" in the Zhoubi. .” China also has the world’s earliest star catalog (Gan and Shi’s Celestial Book, the middle of the Warring States period, 356 BC), Zhang Heng made a star map and armillary sphere in the 2nd century AD, and the world’s earliest stone-carved star map (Song Dynasty , 1247 AD, in Suzhou). It can be seen from this that our country is one of the earliest founders of the coordinate concept of using latitude and longitude to represent the position of stars. Astronomy and mathematics in our country have always been closely integrated. my country is also the inventor of the compass and was once one of the most developed countries in navigation. The concept and method of using longitude and latitude table position must have been developed later, and its evolution and its relationship with Arabia and Western Europe are worth tracing.
Micro-scientific points, this is a major invention that made Western European mathematics leap to the leading position in the world, and it seems that it has no share in our country. But the invention of micro-score has a difficult process from Kepler to Newton. Among the preparatory conditions necessary for the production of micro-sense points, some have already existed in our country, but are beyond the reach of Greek-style mathematics. For example (see Scott, [19], p. 138):
"The concept of the limit, the very basis of differential calculus, was quite a foreigner to the Greek mind".
The theory of irrational numbers, which is considered one of the most brilliant creations in Greek mathematics, is flashy for the limit, but the numbering method of decimals in my country from Liu Hui to the Song Dynasty has nothing to do with the concept of limit. Since the decimal decimal was reinvented in Western Europe as late as the 16th century, it directly led to the invention of logarithms. As one of the pioneers of calculus, Kepler "applied logarithms and decimal fractions extensively, and enthusiastically disseminated knowledge in this area." (Cajori, [6], p. 160), there is some reason. The calculation of area body consciousness is another important problem that led to the invention of calculus. However, the "exhaustion method" used by Euclid in Greece until Archimedes was very ineffective. Kepler used it with little effort, until Galileo's student Cavalieri gave up the rigorous method of equatorial means and switched to the rough method of impossibility. The component method has made a major breakthrough. The so-called Cavalieri principle, which played such an important role in the creation of micro-scientific points and is widely known by western mathematics historians, has actually been seen in the works of Zu Chongzhi and Zu Pao's father and son, that is, the so-called "if the power potential is the same, the accumulation cannot be different " And specifically used in the calculation of the volume of the sphere, more than 1100 years earlier than Cavarieli's discovery.
The invention of calculus has gone through a long and arduous process from Kepler and Galileo to Newton and Leibniz. The above two examples can show that the role of ancient Chinese mathematics in the process of invention is far superior to that of Greek mathematics. In other words, the invention of calculus is the product of Chinese-style mathematics defeating Greek-style mathematics.
We can also point out that the so-called interpolation method (and the binomial coefficient) was valued throughout the 17th century, with the participation of the most famous mathematicians from Kepler to Huygens and Newton. In practical application, it is necessary to fabricate various tables (trigonometric tables, logarithmic tables, nautical tables and astronomical tables), and in theory, it is one of the important ways to obtain micro-knowledge in order to obtain precise approximation , it deserves attention (see Bourbaki, [51]). However, the interpolation method is the method of recruiting in ancient Chinese mathematics. From the linear interpolation method of the Nine Chapters of Arithmetic Surplus and Insufficiency, it has gone through the four times of Liu Hong in the Eastern Han Dynasty, Liu Zhuo in the Sui Dynasty, Tang Seng and his party and Xu Ang, and Guo Shoujing and Zhu Shijie in the Yuan Dynasty. The method of recruiting has actually reached the so-called Newton's general interpolation formula, and the latter appeared in 1676, and Zhu Shijie was in the 13th and 4th centuries, about 300 years earlier than Newton. Taylor's formula is obtained from the interpolation formula through the limit. It turns out that Taylor obtained the so-called Taylor expansion through this method.
Ancient Chinese mathematics has been in the leading position in the world in many aspects at least since it was recorded in the Qin and Han Dynasties. It developed into the Song and Yuan Dynasties, and it has already met many conditions on the eve of the invention of calculus in Western Europe in the 17th century. It may be said that we are close to the gate of micro-sense. Although there have been conflicts between Confucianism and Legalism in the past dynasties, and the obstruction of Confucianism has slowed down the development of mathematics, and even caused many creations to be obliterated or lost forever, it is still possible for us to invent calculus before Europe. However, Cheng-Zhu Neo-Confucianism in the Song Dynasty had caused some outstanding mathematicians (such as Yang Hui) to waste their energy on mathematical games such as vertical and horizontal diagrams, and fell into mysticism, which violated the fine traditions of our country since ancient times. Scholars, Neo Confucianism dominated the thinking of academic circles, and mathematics in our country has since plummeted.
Western historians of mathematics often flaunt the Greek-style rigorous reasoning, and criticize Chinese mathematics for having never reached the form of deduction. However, we have seen the fragility of the Greek form in the invention of the integral and the vitality of the Chinese form in comparison. Some historians of mathematics such as Bourbaki[5] have also pointed out that Euclid's system hindered the development of algebra and paralyzed it. When comparing Cavalieri with Archimedes, Bourbaki also pointed out that Archimedes In order to obtain the "proof" of his special case, Archimedes had to use his The famous so-called Archimedes rigor was left behind. Ancient Chinese mathematics did not develop a formal system of deductive reasoning, but there is another system with more vitality. In the preface to Chapter Nine of Liu Hui, it is said that "the theory is broken down with words, and the disintegration is with pictures". Liu Hui’s Haidao Mathematical Sutra originally had annotations and circles, which were annotated with analysis and intended to be disintegrated, but they have been lost. This is a dialectical thinking method used by ancient mathematics to analyze and resolve contradictions. The working people in ancient China have always paid attention to reality, and are good at discovering and refining problems from reality, and then analyzing and solving problems. On the basis of in-depth and extensive practice, they have established the most advanced Chinese ancient mathematics in the world. Mathematics in China is firmly rooted among the working people and derived from the long-term practical experience of the working people. This is different from the formalism of Greek geometry, which divorced from reality and the masses and went to pure logical reasoning. ． This is the fundamental reason why Chinese mathematics has been in the most advanced position in many of the most important fields until the 16th century, and it is also the fundamental reason why Chinese-style mathematics is far superior to Greek-style mathematics in the invention of calculus. The backwardness of Chinese mathematics since the Ming and Qing Dynasties was caused by the ideological domination of Confucianism which blocked the development of mathematics in the Song and Ming Dynasties. Western historians of mathematics attribute it to the lack of deductive reasoning in Chinese mathematics, which is completely inconsistent with historical facts. Engels once said (see [1]):
"Mathematical calculations are suitable for material proofs and tests, because they are based on material intuition (albeit abstract); pure logical calculations are only suitable for reasoning and proof, so they do not have the empirical reliability of mathematical calculations." sex - and many of them are wrong!"
This is a very good illustration of the great superiority of Chinese mathematics over Greek numbers.
Qian Baocong said in his article "The Great Achievements of Ancient Chinese Mathematics" ([4]):
"After the fifth century, most of Indian mathematics was of Chinese style, and after the ninth century, most of Arabic mathematics was of Greek style. By the middle of the tenth century, the two schools of mathematics merged and spread through the Muslims in northern Africa and Spain. to all parts of Europe, so the Europeans restored the lost Greek mathematics on the one hand, and on the other hand absorbed the vital force E of Chinese mathematics, and modern mathematics began to develop dialectically.”
This period of mathematical development can be summarized as the following diagram (c represents century):
Based on the previous arguments, we think there are reasons to go further: the reason why modern mathematics can develop to today is mainly due to Chinese mathematics rather than Greek mathematics, and it is mainly Chinese mathematics rather than Greek mathematics that determines the historical development of mathematics. math.
The above is a good introduction, and the argument is rough and incomplete, and I hope to add further elucidation. If the argument is extreme and inappropriate, I hope to arouse controversy.
中国是世界数学之源：算术、代数、几何都是中国古代数学家创造
科学理论的本质是科学家用数学工具对自然社会做从出定性定量解释。本期科学Sciences 简述 算术、几何、代数等数学历史的来源。今天世界各国中小学数学教材，如算术、代数、几何这些内容，其实都源自中国古代数学家的创造，大部分在汉朝的 《九章算术》就俱已齐备，而中国以西的印度、阿拉伯和欧洲是非常晚的。 文艺复兴时期 中学西渐，启发欧洲人破除了 神学至尊的愚昧思想。 吴文俊院士指出 “ 微积分的发明乃是中国数学式战胜了希腊式数学的产物。” 甚至可以说，“ 近代数学之所以能够发展到今天，主要是靠中国的数学，而非希腊的数学，决定数学发展进程的主要是中国的数学而非希腊的数学。” 基于此呼吁，“ 被颠倒了的历史必须颠倒回来！” 。下面是 文行先生文章。
1. 中国是世界数学之源
鸦片战争开始，列强入侵、国土沦陷。甲午战败，中国人 民族自信心严重受挫，失魂落魄，开始否定中华文化、打倒中华文化，乃至呐喊 “ 汉字不灭，中国必亡” 。即使建国之后，还是如此，延至80 年代，竟然“ 河觞” 泛滥，污蔑中国历史一片黑暗，一无是处。数学自然也不能自外于滔滔“ 河觞” 。
中科院院士、数学家 吴文俊先生 《中国古代数学对世界文化的伟大贡献》一文指出， “ 西方的大多数数学史家，除了言必称希腊以外，对于东方的数学，则歪曲历史，制造了不少巴比伦神话和印度神话，把中国数学的辉煌成就尽量贬低，甚至视而不见，一笔抹煞” 。
对于这些种族主义的谬论，国内学者不加以批驳，反而要么亦步亦趋、鹦鹉学舌，跟着叫嚷着“ 言必称希腊” ，要么不吱声。这主要是由于中国近代孱弱导致“ 对祖国古代数学一无所知” 。近年来一些最新研究显示，这些只不过是西方为了配合其建构地理大发现以来的资本主义发展史和欧洲中心论的需要而臆想和编造出来的伪史的一部分。
还原历史真相
事实上，“ 中国数学，在世界上可以说一直居于主导地位并在许多主要的领域内遥遥领先直至宋末明初” ，以今天世界各国中小学数学教材内容看，如算术、代数、几何这些内容，都是中国古代数学家的创造，大部分在 汉朝的 《九章算术》就俱已齐备，而中国以西的印度、阿拉伯和欧洲是非常晚的。如下表：
中国著名数学史家 钱宝琮在 《中国古代数学的伟大成就》一文中详细地谈到中国 《九章算术》西传印度、阿拉伯、欧洲的演变情况：
1.“ 中国的( 整数) 筹算 乘除法则到印度演变为 土盘法则，传到阿拉伯，演变为笔算的削去法，传到欧洲，逐渐改进到现在的算法。演变的经过都有可靠的史料可以考证明白。 ”
2. 分数表示法( 分子在上，分母在下) 和“ 后来印度、阿拉伯的 分数算法，亦是从中国传过去的。 ”
3. 欧洲的 比例、黄金法则等概念，也是由中国传到印度、阿拉伯，再传到欧洲。
4. 中国的 盈不足术在中国已经被更先进的方法淘汰的时候，传到了阿拉伯， “ 阿拉伯人十分重视，编入他们的 代数书内，有 ‘ 契丹算法’ 的称谓。由阿拉伯传到欧洲，在十六、七世纪中代数学书亦普遍采用，改称‘rule ofdouble false position’ 。”
5. 中国“ 联立一次方程式算法传到印度，印度人把各项筹算改为横写，并添辅未知数明色，就是现在 代数写法的渊源，印度人认识负数，说它有欠债的意思大约亦是从中国数学学习到的 ” 。
6. 现在教科书解高次方程式所谓的“霍纳方法”(1819)实际上就是中国宋代的“ 增乘开方法” ，虽然这里钱宝琮没有明确说欧洲源自中国，但很明显，这实际上就是抄袭中国的。
中国学者在进行中西方对比的时候，经常只提中国“ 早多少年” ，不敢直接提“ 中学西渐” 。一个曾经自卑到要消灭汉字、失魂落魄的民族，直到今天依然如此“ 谨慎” ，悲乎！
吴文俊院士引用了印度数学史家Kaye说法，即：印度与中国的数学有很多平行之处，而印度是欠了中国的债。
中学西渐总结道，“ 中国算学与印度、阿拉伯、日本及西洋各国算学均有授受关系” 。欧洲近代数学的发展是得益于 阿拉伯数学和 中国数学，是 “ 中阿合璧” 的产物，如下图：
吴文俊指出，“ 中国数学可以说一直“ 直到16 世纪以前我国数学在许多最主要的领域一直居于最先进地位” 。“ 到西欧17 世纪以后才出现的解析几何与微积分，乃是通往近代数学的主要的两大创造，一般认为这些创造纯粹是西欧数学的成就。但是中国古代数学绝不是不起着重大作用的( 甚或还是决定性作用) 。” 吴文俊院士对此进行了详细的分析论证。
“ 微积分的发明乃是中国数学式战胜了希腊式数学的产物。” 甚至可以说，“ 近代数学之所以能够发展到今天，主要是靠中国的数学，而非希腊的数学，决定数学发展进程的主要是中国的数学而非希腊的数学。” 基于此，吴文俊院士呼吁，“ 被颠倒了的历史必须颠倒回来！”
《算学宝鉴》与重写明史
以上阐述是 钱宝琮于 1951 年、 吴文俊在 1975 年的研究结论，即在 王文素的 《新集通证算学宝鉴》真正得到重视和研究之前。 《算学宝鉴》系 “ 民国年间由北京图书馆于旧书肆中发现一兰格抄本而得以入藏” ，虽然“ 抗战前，中算史家 李俨曾看过此书， 《中算史论丛》中有文提及。六十年代，数学史家 钱宝琮主编 《中国数学史》也提到此书 ” ，但是当时谁也没有见过或真正了解此书。王文素的 《算学宝鉴》真正得到重视和研究是 1992 年北京师范大学物理系 赵擎寰教授推介之后。
从目前研究来看， 《算学宝鉴》代表着中国历代数学的最高水准，也是当时世界的最高水平。 “ 《算学宝鉴》研究了一元高次方程的数值解法，内容详实可贵，这充分说明一元高次方程数值解法及天元术、四元术在明朝并未完全失传。王文素在解法中所用名词术语、演算程序，基本上与宋元数学一致，并有所发展和创新。王文素的数学成就是中国数学史连续性的有力证据。”
“ 王文素解高次方程的方法，较英国的 霍纳(Hirner，1786－1837)、意大利的 鲁非尼(Ruffini，l765－1822)早近300 年；在解代数方程上，他走在17 世纪 牛顿(I．Newton，1642—1727)、拉夫森(J．Raphson，1648－1715)的前面140 多年，率先用导数逐步迭代求解，为中国数学史谱写了光辉的篇章；对于17 世纪微积分创立时期出现的导数， 王文素在 16 世纪已率先发现并使用，因而，只从微积分的角度探索导数的起源是不够的，由此看来王文素对世界数学的贡献还应更深入的研究。”
这些最新的研究发现极大地强化了吴文俊院士关于“ 近代数学的发展主要靠中国数学” 的结论。有些贬抑中国的人还在“ 纠缠” 吴文俊教授的“ 中国式数学” 和“ 中国数学” ，而《算学宝鉴》的出现使得“ 式” 字可以拿掉了。 文行先生在 《传教士盗取中华文明、颠倒世界历史》谈到，“ 王文素不是‘ 早’ 、‘ 率先’ ，而是欧洲的近代数学完全系中国数学通过传教士西传的产物，是‘ 中学西渐’ ，包括 牛顿和 莱布尼茨的微积分系源自明朝 王文素的 ‘ 导数’ ，根本不是欧洲的发明。” 当然，发生了“ 南橘北枳” 的效应了—— 用阿拉伯数学的瓶子装中国数学的酒。
然而， 项观捷在其所著的 《中国古代数学成就》(1988)一书中说道，“ 我国数学自产生之日起，就一直持续发展着┅┅而在朱世杰之后，我国古代数学的发展突然发生了严重的中断。从朱世杰到明朝程大位将近三百年光景，没有出现一位重要的数学家，也没有出现一部重要的数学著作。而且不仅仅是没有什么新的发展，就是宋元数学所留下的那份宝贵遗产也没有保住。”
这种论调还有很多，如：“ 近史期算学，自 明初至清初约当公元1367年迄1750年，前后约400年……民间算学大师又继起无人，是称中算沉寂时期” ，“ 明代中叶以后，出版了很多商人所写的珠算读本，对比较高深的宋元数学只能付之阙如，中国古代传统数学到明代几乎失传” ，“ 十四世纪…… 先辈们辛勤创造的 天元术竟完全失传了。在西方学术输入之前，最重要的也是流传最广的数学书是程大位的 《算法统宗》(1592年)，这书除了算盘和歌诀之外，没有新的创造。它基本是整理前人作品的书，并且漏掉高次方程和多元高次方程等重要部分” 等。
项观捷发问道：“ 为什么 我国古代数学发展到十四世纪突然发生了中断？这个问题历来受到中外史家的注意。 ” 有趣的是，项观捷还在书中还讲到，“ 在明朝末期， 大统历和 回回历的 误差越来越大，修改历法已成当务之急，但 偌大的明朝居然找不出一个能主持修改历法的人了。这说明经过一个明朝，我国古代天文学和数学水平已经下降到多么可怜的水平。 ”
但是，在文行先生 《传教士盗取中华文明、颠倒世界历史》所引用的中科院自然科学史研究所 李亮先生的文章 《被“遗漏”的交食——传教士对崇祯改历时期交食记录的选择性删除》中谈到，传教士大量删改中国的历史文献，使得我们误以为中国传统历法越来越差劲，而实际情况是，西洋历法并没有像目前所看到的中国历史文献所显示那样完胜中国传统历法，崇祯皇帝的圣旨明确说“ 日食初亏、复圆时刻方向皆与 《大统历》合，其食甚时刻及分数，魏文魁所推为合 ” ，即大统历和魏文魁法预报的交食时刻与实测结果更吻合。更要命的是，如南怀仁所言，欧洲最著名的天文学家计算出的结果都会与实测结果有巨大差异，但在中国却能精确到“ 刻” ， 南怀仁为此激动万分，而实际上，这要归功于明代高超的数学水平。
事实上，文行先生 《从屈原被西方踢出历史教材说起》谈到，清朝“ 除了焚毁书籍， 大清还系统地对明代档案进行了销毁。明代档案仅三千余件，主要是天启、崇祯朝兵部档案，也有少量洪武、永乐、宣德、成化、正德、嘉靖、隆庆、万历、泰昌朝的官方文书。其余估计不少于 1000 万份明代档案，已经全部被销毁了。除了销毁书籍和档案外，大清还系统的对残存书籍和档案，进行篡改” 。明代档案仅万分之三(3‱) 流传下来，就是仅存的这3‱ 也是经过系统性删改的。
而 王文素的 《算学宝鉴》因逃过了 《四库全书》的编撰而得以幸存 ——“ 四百年间未见各收藏家及公私书目著录，民国年间由北京图书馆于旧书肆中发现一兰格抄本而得以入藏” ，使我们得以一窥明代数学的辉煌，得以还原被颠倒的历史。
文行先生 《传教士盗取中华文明、颠倒世界历史》谈到， 汤若望删除、篡改 《崇祯历书》，把 《治历缘起》从十二卷删除到八卷，这已经是证实了的，铁板钉钉。 韩国藏本使我们得以一窥明代天文、历法、数学的辉煌，得以还原被传教士篡改的天文史。
综上所述，结合屡屡提及的 李兆良先生对 《坤舆万国全图》的考证，使我们得以一窥明代测绘世界的辉煌，我们可以斩钉截铁地说，充分的证据表明， 满清和传教士合谋进行大规模、系统性删改、销毁大量明朝资料，贬抑和阉割明朝的文明成就，把明朝知识、科技西传的“中学西渐”颠倒为“西学东渐”。
传教士明末来华并打入中国高层以及随后的明朝灭亡和清朝新立，满清蛮荒出身和贬抑明朝的政策，传教士的宗教狂热催使的篡改历史的阴暗心理和西方的“ 强盗” 即爱国的 凯旋文化(伏尔泰语)， 萨义德揭示的东方主义，必须引起我们高度警惕，必须在我们的相关研究中得到充分的考虑。
我们可以合理推理： 郑和下西洋的资料也是清朝删掉，然后甩锅给 刘大夏的；甚至 《永乐大典》的失传与满清也不无关系。基于上述，我们要高度警惕关于明朝的默证问题，不能因为没有记载就说没有、不存在、不曾发生过，不能因为现存的中国明代文献记载某方面很差就认为真的很差。
因此，我们不能认为整个明朝只有 王文素一个人能达到这个水平，而应该认为王文素是明朝数学最高水准的一个代表，甚至王文素的 《算学宝鉴》并不是明朝的最高水准，因而即使有人资助也出版不了。而王文素的 《算学宝鉴》之所以能够流传下来，仰赖于逃过《 四库全书》的编撰删改销毁，而那些同样水平或更高水准的的数学著作都被满清和传教士销毁了。因此， 西方传教士完全可能接触到明代最高水平的数学，并把这些知识西传欧洲，在欧洲引入阿拉伯数字和拉丁希腊字母的前提环境下诞生了欧洲 近代数学。 相反，我们误以为明代数学发展出现中断、停滞、退步， “ 下降到可怜的地步” ，是西方传教士介绍先进文明( 如天文学、数学) 给中国—— 事实上不是这样，是满清和传教士合谋篡改历史的结果。
项观捷还在 《中国古代数学成就》讲到关于朱世杰的一个高阶等差序列求和公式，如下图：
能够得到这个公式，不靠推理是不行的，只是中国古代不喜“ 空谈”( 就像春秋名家一样) ，而重实效，重实际应用，所以，没有把繁琐的过程书写出来。就 朱世杰的上述公式而言，西方人直到 18 世纪才能达到这个水平，而显然，不能理解为西方人凭空自己发明出来，而应理解为“ 中学西渐” 。我们可以想象，当其他接触到中国数学的民族，不但对结论会感到神奇，更会对证明过程感兴趣，更重证明的过程。这是判断原创和流传的依据之一。因此，当中国数学出口转内销之后，我们误以为那是别样的数学。
文行先生 《从屈原被西方踢出历史教材说起》指出，欧洲在文艺复兴时期远非我们现在所认为的那么世俗、理性、科学、逻辑，并引用西方最新研究成果：“ 文艺复兴运动的人文主义，根本算不上一个哲学思潮或体系┅┅充其量不过是一些和文学沾边的东西…… 当时的人文学校虽然包含了一个哲学范畴即道德，却将诸如 逻辑学、自然哲学、玄学、甚至包括 数学、天文学、医学、法律学以及 神学在内的其他学科领域统统拒之门外。 ”( 克利斯特勒在 《古典名著和文艺复兴思想》第 7 页)
“ 在我看来，文艺复兴运动的人文主义，千方百计地想要将自己与整个时期的哲学、科学及教育结合在一起；而眼下的确确凿事实，却好像为这一努力提供了反面证据。”( 克利斯特勒在 《古典名著和文艺复兴思想》第 7 页)
“ 在文艺复兴运动期间，根本不存在什么精神或哲学思潮上的' 人文主义' ；它与各种思想观念的撮合，比如说以人为本的文学创作以及精神解放和自由，都是后人世界观及理想主义的发挥，与那个时代精神毫无瓜葛。”( 娜希亚·雅克瓦基在 《欧洲由希腊走来》第 64 页)
“ 其实，西方直到17 世纪末期之前，智慧与预卜、哲学与炼金术还是同义词，只有到了18 世纪，这两对同义词才彻底分道扬镳。” 那种认为欧洲自从地理大发现开始就辉煌无比的认知必须得到彻底的扭转，这里我以一幅画来形象说明。如下图：
因此，种种迹象和证据表明， 文艺复兴期间，欧洲还是很愚昧的，远非我们所想象的那么先进， 欧洲近代的文明进步完全源于明代中国的恩赐。甚至，我们可以说，欧洲近代的发展成就系明朝的复制、移植，欧洲近代文明是中华文明的 “ 南橘北枳” 。这当然包括数学。
明代中国是被污蔑的， 《明史》等资料被严重篡改和销毁了，在这种情况下，我们不能过于囿于史料，要更多地运用逻辑分析和常识，去发现被歪曲的历史，还原真相。例如，不能轻易以中国历史文献的记载去否定 孟席斯的相关研究成果 ( 《1421》和 《1434》) ，不能否认其某些研究可能是错误的，从目前看，他研究的方向以及大多数是对的，是时候必须认真对待他的研究了( 参见 《黑色雅典娜》、《白银资本》、《西方文明的东方起源》、《中国之欧洲》、《考古学一百五十年》等 ) 。
我认为，明朝极其辉煌，远超我们的想象和认知，由于大量文献被严重篡改和销毁，明朝的真相有待于我们去重新发现。 明史要重写！
世界数学之源
上文中提到 吴文俊院士赞成数学史家 钱宝琮关于欧洲数学发展的说法，如下图 ( 吴文俊所作) ：
文行先生公号核心方向之一是揭露西方伪史。根据其文章 《古典希腊伪史是如何炼成的？》、《古史辨之古希腊伪书》和《诡谲的希腊帝国》，文艺复兴时期的 希腊-格里斯实际上指的是 东罗马， 希腊文实际上指的是东罗马的文字。根据文行先生 《古罗马金币揭穿西方文明伪史》，希腊文实际上诞生于 希拉克略改革时期，之后逐步完善，在 9 世纪，罗马金币上全部为希腊文，显示希腊文已经基本完善。
所以， 钱宝琮所谓的那个 “ 希腊” 实际上是9世纪之后的东罗马，因此，结合文行先生公众号阐述的古希腊伪史的观点，文行先生把吴文俊院士根据钱宝琮先生的观点而所作的图修改如下：
( 完) (注1：资料来自文行先生公号等[1-4]。)
附录：数学学报Vol.18,No.I(March,1975) 顾今用文章 《中国古代数学对世界文化的伟大贡献》。
2. 中国古代数学对世界文化的伟大贡献
公元前221 年，秦始皇灭兴国，建立了中国历史上第一个中央集权的封建国家．汉承秦制，自 秦至西汉中期这两百来年间，是新兴地主阶级专政巩固发展与上升的时期，法家路线占着主导地位．法家对工农业生产与科学技术比较重视，由此促进了数学的迅猛发展，出现了一批高水平的数学家，如 张苍、耿寿昌等． 《周髀算经》、《许商算术》与《杜忠算术》( 后二者已失传) 都在这时期出现．我国最主要的一部传于后世的数学著作 《九章算术》也基本上成书于 西汉初年，其内容为以后一千多年的辉煌成就奠定了系础．从西汉以迄宋元，虽然有儒家思想不断干扰，但随着儒法斗争的过程，唯心论与唯物论斗争的过程，随着我国社会经济和劳动人民创造性发展，数学人才与数学创作仍世代不绝，中国的数学，在世界上可以说一直居于主导地位并在许多主要的领域内遥遥领先，直到宋末明初， 宋明理学成为垄断一切的统治思想，明代并以 八股取士，以及其他一些原因，科学技术的发展受到扼杀，除了民间的计算技术还有重要发展外，数学巳相应地大为衰落．从明末 利玛窦怀着不良企图以介绍西方数学为名打入我国统治集团内部以来，我国数学上与古代相比已谈不上什么创造，基本上依靠国外的技术输入，在外国人屁股后面爬行了．正如 毛主席批评的那样 “ 言必称希腊以外，对于自己的祖宗，则对不住，忘记了．” 西方的大多数数学史家，除了言必称希腊以外，对于东方的数学，则歪曲历史，制造了不少巴比伦神话与印度神话，把中国数学的辉煌成就尽量贬低，甚至视而不见，一笔抹煞，对于在已成为半封建半殖民地社会中生活过来的一些旧知识分子，接触的数学都是“ 西方” 的，看到的数学史都是“ 西方史家” 的，对于祖国古代数学又十分无知，因而对于西方数学史家的一些捏造与歪曲无从辨别，不是跟着言必称希腊，就只好不吭声．
但是，被颠倒了的历史必须颠倒过来！
作为中国古代数学成就的具体例子，不妨着一看至少直到本世纪五、六十年代中小学的 中外数学教本．有一本 西方数学史([12] ，这基本上是一本比较好的数学史) 说：“ 在许多中学中，代数学现仍被教成一堆公式而不是一种演绎的科学” ．在谈到 代数学的东方起源时又说： “ 今日学校中的代数学和几何学仍然保持这些 不同来源的标志． ”
这里所谓 东方起源或 东方数学，乃是针对 “ 由一些定义、假定和公理到定理的一种严格逻辑的演绎法” 作为根据的所谓“ 欧几里得的处理” 这种西方( 或希腊) 式数学来说，这本书所说的东方，原意是指 巴比伦或是 印度，而我们将会看到这个东方应指 中国才算确当．诚然，中小学数学教本中的几何课程有一种特殊的表达形式，与其他算术、代数 ( 甚至三角与解析几何) 的表达形式显然有别．我们不难想到，如果抽掉了东方色彩( 也就是中国色彩) 的算术、代数的这些部分，只保留所谓希腊式的几何部分，我们的中小学数学将成什么局面？
中小学数学中的第术、代数这些部分，从记数、以至解联立线性方程与二次方程，实质上都是中国古代数学家的发明创造，早就见之于中国的 九章算术甚至是 周髀算经等书，据 钱宝琮考证 ([3]) ，九章算术 刘徽作注是在公元 263 年，全书完成则在 公元50一100年间．但除个别片段外，基本内容应完成于公元前200 年或更前一些( 这是某些西方数学史家的意见．有的甚至提早到 公元前1000年，例如[9]) ，九章的前身是 张苍、耿寿昌的著作，张苍在秦时就已做宫，耿寿昌也是西汉宣帝时人．西方的某些说法是有一定依据的．但我们仍不妨依钱说，认为是公元 50-100 年间写成的著作．另一部周髀算经则据 钱宝琮考证 ([3]) 成书于 公元前100年前后．
下面是关于算术代数部分发明创造的一张中外对照表，这里应该指出，表中虽列有印度的发明，但诚如一位印度数学史专家Kaye所说的那样：印度与中国的数学有很多平行之处，而印度是欠了中国的债．( 参阅例如Cajori，[6] ，页97 与84 ，又如Scott[9]) ．而且，表中所列是依据两位褐力为印度数学辩护的印度数学史家Datta以及Singh([7]) 自己的说法，这些说法即使不考虑中国的因素，也是大有疑问的．
中国的劳动人民，在长期的实践过程中，创造与发展了从 记数、分数、小数、正负数以及 无限逼近任一实数的方法，实质上达到了 整个实数系统的完成．特别是自古就有了完美的 10进位位值制的记数法．这是中国的独特创造，是世界其他古代民族都没有的．这一创造对世界文化贡献之大，如果不能与火的发明相比，也是可以与火药、指南针、印刷术一类发明相媲美的．
代数学无可争辩地是 中国的创造，从九章以至宋元的 秦九韶与 朱世杰发展的线索甚为分明，甚至可以说在 16 世纪以前，除了 阿拉伯某些著作之外，代数学基本上是中国一手包办了的．但中国古代数学的成就决不止于算术与代数方面，以几何而论，希腊欧几里得几何的 拱心石是毕达哥拉斯定理 ( 语出Bourbaki ，[5]) 或即 勾股定理．这一定理我国古代自然也早已有之．但中外数学史家提到中国的勾股定理时，或则引述髀而只及勾三股四五这一特例，或则虽引述一般定理而最早只及九章，其实在周髀中就已有一般定理的叙述：
“ 若求邪至日者，以日下为勾，日高为股，勾、股各自乘，并而开方除之，得邪至日．”
不仅如此，勾股定理还被具体用于勾股弦的直接互求，甚至应用于测日之高远这一类复杂问题．这与欧几里得几何中理论脱离实际的情况是迥不相同的．中国的几何学与希腊的几何学有许多不同之处，其详细比较有待阐发．
对于三角学中国也是最早发明者之一．西方数学史家一般都把 《天文书》(Almagest)的作者 托雷米(Ptolemy，公元150 年左右) 作为 三角术的创始人，而把中国的三角术视为是受了他的影响，例如，在西方数学史 [12] 中说：
中国“ 有一些三角学，主要是在 《海岛算经》中，但是，由于这算经被归之于纪元后第三世纪，我们就不可以不考虑西方影响了． ”
诚然， 《海岛算经》的作者 刘徽是公元三世纪时人，但据刘徽九章注自序， 《海岛算经》本是九章注第十卷 《重差》，而东汉末郑玄 《周礼注》引郑众注周礼 “ 九数”( 约公元50 年) 语云“ 今有重差、勾股” ．可见刘徽 《海岛》的前身乃是 汉时的重差术．如果把 《海岛》测高远之法具体分析，可见重差之法由来已久，周髀中：
“ 周髀长八尺，夏至之日晷一尺六寸．┅┅正南千里，勾一尺五寸．正北千里，勾一尺七寸．┅┅从此以上至日，则八万里．”
这正与 《海岛算经》中 “ 今有望海岛” 的第一题是一样的．诚然 周髀视地为平地是一种错误，但它所依据的三角测量原理是正确无误的，也正因为如此， 周髀以之观天者， 刘徽以之测地，而建立了以重差为基础的三角测量术．这种三角测量术的目的与方法在周髀中都早已有所说明， 周髀引 陈子之言 “ 望高起远” 是它的目的，引商高之言“ 平矩以正绳，偃矩以望高，复矩以测深，卧矩以知远” 是它的方法．刘徽把“ 度天圆穹之象” 改为度“ 泰山之高与江海之广，” 又触类而长为“ 度高者重表，测深者累矩，孤离者三毁，离而又旁求着四望” ，无非是周髀立两表以测日高这这一三角测量术的发展与推演而已．
西方的三角术是先有球面三角后有平面三角． 托雷米的 《天文书》主 地球中心说，他的三角术由测天而来，因而是 球面三角术．至于 平面三角术则迟至公元 1250 年才由波斯天文学家 纳法刺丁所建立．但在我国则不论是 周髀观天还是 海岛测地，一开始就是平面三角术．东西方三角术的发展途径是刚巧相反的，很难谈到有什么相互影响．如果说有影响，那么周髀早于《天文书》有好几百年，只有说 托雷米受到中国的影响才更合情理．前引西方数学史学家关于中国三角学之说显然是颠倒了历史事实．
在西欧，16 世纪中出现了不少描述三角测量的图画，其中有一张名“ 鼓皮三角法” ，所画正如 周髀赵爽注所附的日高图 ( 也即重差原理图) 的一个翻版，这也可以见到我国三角测量术的先进程度．( 画见Smith ，[10] ，页355)
到西欧17 世纪以后才出现的 解析几何与 微识分，乃是通向所谓近代数学的主要的两大创造，一般认为这些创造纯粹是西欧数学的成就．但是中国的古代数学决不是不起着重大作用 ( 甚或还是决定性的作用) ．
先说解析几何，Smith([11] ，页316 ，) 曾认为解析几何的发展有三个主要步骤：(1) 座标系统的发明；(2) 几何与代数间一一对应的认识；(3) 函数y=f(x) 的图形表示，第一步属于古代，第二步属于中世纪，第三步则是近代的．
西方向来认为17 世纪的Descartes( 以及Fermat) 是解析几何的创始人，但实际上Descartes的有关主要著作中既无坐标也无坐标轴的概念，更无所谓直线与曲线的方程．Descartes的贡献是在第二步即几何与代数建立关系方面，在他的主要辛苦作中，给出了二次方程的几何解法，但阿拉伯最早的代数学即AI-Khowarizmi(花刺子模) 的著作( 公元9 世纪) 也早已用另一种较Descartes 更好的方法绘出了二次方程的几何解．事实上几何与代数的统一处理乃是我国古代数学的一个传统特色，从九章以来就向来如此， 花刺子模的著作据 Cajori[6] 与希腊印度无关，如果不是阿拉伯自己的发明创造，则必然渊源于中国，从著作的风格看来，后者是不无可能的．这一段历史自然是值得重视并予以澄消的．现已知 花刺子模在 842-847 年曾出使 波斯以北当东西方商业要冲的 西突厥可萨国，而可萨通中国语，朝廷依中国礼仪 ([13] ，Addenda) ，详情有待进一步调查．
至于第一步，西方数学史家比较一致地认为真正的坐标概念出现于14 世纪中叶Oresme 关于以“ 经度” 、“ 纬度” 来表示点的位置的一个著作．提Smith 指出([11] ，页320 注) ，Oresme 的著作可能导源于10 世纪时的一个作品．这里10 世纪的作品估计应是阿拉伯的．在我国，则周髀中已有“ 分度以定则正管经络” 以及“ 游仪所至之尺为皮毒虫” 等语，注中并屡言“ 引绳圣经纬之交，以望之．” 中国又有世界上最早的星表( 甘石星经，战国中叶，公元前三百五六十年) ，公元2 世纪 张衡就作星图与浑天仪，又有世界上最早的石刻星图 ( 宋，公元1247 年，在苏州) ．由此可以看到以经纬度表星的位置的这种座标概念我国是最早的创始人之一．我国的天文数学历来紧密结合．我国又是罗盘的发明者并曾经是航海最发达的国家之一．用经纬度表位置的概念与方法在后来必然有所发展，其演变以及与阿拉伯西欧的关系，是值得把它追查清楚的．
微识分，这是使西欧数学一跃而居世界领导地位的重大发明创造，在我国似乎是没有份的．但是微识分的发明从Kepler 到牛顿有一段艰难的过程．在作为产生微识分所必要的准备条件中，有些是在我国早已有之，而为希腊式的数学所力所不及的．例如( 见Scott ，[19] ，页138) ：
“ 极限的概念，作为微分学的真正基础，对于希腊头脑来说完全象是一个外国人” ．
希腊数学中被认为最辉煌的创造之一的 无理数论，对于极限来说是华而不实的，而从 刘徽以至宋代的我国十迸位小数的记数法，却与极限概念一衣带水．十进位小数迟至 16 世纪在西欧重被发明以来，直接导致了对数的发明．作为微积分先驱者之一的Kepler ，“ 广泛应用了对数与十迸位分数，且热情地传播这方面的知识．”(Cajori ，[6] ，页160) ，是有一定的道理的．面积体识的计算乃是导致微积分发明的另一重要问题．然而，原来希腊欧几里得以至阿基米德所使用的“ 穷竭法” 是很不得力的，Kepler 用之劳而少功，直到伽利略学生Cavalieri 放弃了严密的穷均法改用粗糙的不可分量法才取得了重大的突破．在微识分的创造过程中起了如此重大作用为西方数学史家盛称的所谓Cavalieri 原理，事实上早就见之于 祖冲之、祖咆父子的著作，即所谓 “ 幂势既同则积不容异” 并具体用之于球体积的计算，比Cavarieli 的发现要早了1100 多年．
微积分的发明从Kepler与Galileo以至Newton与Leibniz经历过一段艰苦漫长的过程，上面所举两个例子可以说明发明过程中中国古代数学的作用远优于希腊式的数学，我们甚至不无理由可以这么说，微积分的发明乃是中国式数学战胜了希腊式数学的产物．
我们还可以指出，所谓 插值法(以及二项式系数)在整个17 世纪中受到重视，从Kepler 以至Huygens 与Newton 这些最著名的数学家都参加了这一工作．在实际应用上这是编造各种表格( 三角表，对数表，航海用表以及天文表) ，所必需，在理论上又为求得精密逼近而为通向微识分的重要途径之一，它的受到重视是应当的( 参阅Bourbaki ，[51]) ．然而，插值法即我国古代数学中的招差术，从九章算术盈而不足术的直线内插法历经东汉 刘洪，隋刘焯，唐僧一行与 徐昂，到元 郭守敬与 朱世杰的四次招差术，实质上已到达了所谓 Newton 的一般插值公式，而后者出现于1676 年，朱世杰则是元十三、四世纪时，早于Newton 约300 年．从插值公式通过极限即得Taylor公式，原来Taylor即是通过这一方式来得到所谓Taylor 展开的．
中国古代数学至少自秦汉有记载以来，许多方面一直属于世界上的遥遥领先的地位，发展到宋元之世，已经具备了西欧17 世纪发明微积分前夕的许多条件．不妨说我们已经接近了微识分的大门．尽管历代都有儒法斗争，儒家思想的阻扰放慢了数学发展的速度，甚至使许多创造湮没不彰或从此失传，但我们还是有可能先于欧洲发明微积分的．然而，宋朝的 程朱理学已使当时的一些优秀数学家 ( 例如 扬辉) 浪费精力于纵横图之类的数学游戏，陷入神秘主义，违反了我国自古以来的优良传统， 到了明朝八段取士，理学统治了学术界的思想，我国的数学也就从此一落千丈了．
西方数学史家往往以希腊式的严密推理相标榜，并以中国数学从来没有达到演绎学的形式相指责．然而，我们已经看到，在放积分的发明上希腊形式的那种脆弱性以及与之相较中国式数学的生命力．某些数学史家例如Bourbaki[5] 也曾指出欧几里得的那种系统阻确了代数学的发展并使之瘫痪，在将Cavalieri 与亚基米德作比较时，Bourbaki 又指出亚基米德只能得到Cavalieri 原理很特殊的情况，而与Cavalieri 作出他的原理用了不很科学的所谓“ 证明” 相仿，亚基米德为了获得他的特殊情况的“ 证明” ，也不得不把他著名的所谓亚基米德严密性弃之脑后．我国古代数学并没有发展出一套演绎推理的形式系统，但却另有一套更有生命力的系统．刘徽九章注序中说“ 折理以辞，解体用图” ．刘徽海岛算经本来有注有圈，注以析理，图以解体，只是已失传而己，这是古代数学用以分析矛盾解决矛盾的一种辩证思维方法．中国古代的劳动人民向来重视实际，善于从实际中发现问题提炼问题，进而分析问题解决问题，在深入广泛实践的基础上往高里提，建立了世界上最先进的我国古代数学．中国的数学是牢牢扎根于广大劳动人民之中，是导源于劳动人民长期实践经验的基础之上的，这与希腊几何学脱离实际脱离群众走到纯逻辑推理的形式主义道路是有别的．这正是直至16 世纪以前我国数学在许多最主要的领域内一直居于最先进地位的根本原因，也是在微积分的发明上中国式的数学远远优越于希腊式数学的根本原因．明清以来我国数学的落后，乃是宋明理学八股取士堵塞了数学的发展道路，是儒家的思想统治所造成．西方数学史家把它归之于我国数学的缺少演绎推理与历史事实完全不符． 恩格斯曾经说过 ( 见[1]) ：
“数学演算适合于物质的证明，适合于检验，为它们是建立在物质直观(尽管是抽象的)的基础上的；纯逻辑演算只适合于推理证明，因此没有数学演算所具有的实证的可靠性——而且其中许多还是错误的！”
这是我国数学对希腊式数字来说具有极大优越性的一个很好的说明．
钱宝琮在 《中国古代数学的伟大成就》一文 ([4]) 中曾说：
“ 第五世纪以后，大部分印度数学是中国式的，第九世纪以后，大部分阿拉伯数学是希腊式的，到第十世纪中这两派数学合流，通过非洲北部与西班牙的回教徒，传到欧洲各地，于是欧洲人一方面恢复已经失去的希腊数学，一方面吸收有生力量E 的中国数学，近代数学才得开始辩证的发展．”
这段数学发展过程可概括为下面的简图(c 表示世纪) ：
根据前面的论证，我们认为有理由可以进一步说：近代数学之所以能够发展到今天，主要是靠中国的数学，而非希腊的数学，决定数学历史发展进程的主要是中国的数学而非希腊的数学．
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