The Nine Chapters on the History of ...

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//-->The Nine Chapters on theHistory of Chinese MathematicsRik Brandenburg∗Keimpe Nevenzeel†15 July 2007AbstractThis article explores Chinese mathematics from the first archeologicalevidence of numbers on oracle bones (14thcentury BC) to the time Chi-nese mathematics became a part universal mathematics (halfway the 19thcentury AD).First a concise overview of Chinese history and in philosophy is given.The ethical oriented Confucianism was the dominant philosophy and con-sequently little attention was given to the natural world, hindering thedevelopment of natural sciences and mathematics. Due to historical andphilosophical reasons, Chinese mathematics took quite a different paththan its Western counterpart: Chinese mathematics was focused on alge-bra and practical applications instead of geometry and theoretical reason-ing.TheNine Chapters on the Mathematical Art(ca. 1stcentury AD) isby far the most influential work: it would shape Chinese mathematicsfor centuries to come. Between the 3rdand the 11thcentury AD, Bud-dhist and Indian ideas got a firm grip on China, yet curiously Chinesemathematics is barely influenced. In the ‘Chinese Renaissance’ and thesubsequent Mongol occupation between the 12thand 14thcentury Chinesemathematics will reach its zenith.In the 15thand 16thcenturies mathematical development waned andimportant achievements were forgotten. Only after the arrival of Eu-ropean missionary-scientists at the end of the 16thand during the 17thcentury mathematics made progress again. The Opium Wars of the 19thcentury mark the end of the classical China and the indigenous Chinesemathematics would be assimilated by universal mathematics.Two conclusions are reached: (i) war seems to be good for mathe-matical progress and (ii) from the 15thcentury on Chinese mathematicspassed away in particular due ot the lack of a general algebraic structure,which prevented structural insight.Keywords: Chinese mathematics, history of mathematics, Nine Chapterson the Mathematical Art.IntroductionChina is one of the oldest civilizations, comparable only to Egypt and Baby-lonia, which were well versed in mathematics (for a discourse on Babylonian∗IWI,†IWIUniversity of Groningen- Kapteyn Institute, University of Groningen1and Egyptian mathematics, seeKlassieke algebra door de tijden heenby Vander Elsken, N. and Van der Pol, E.; for an extensive treatment of Egypt, seeWiskunde voor het leven op papyrus beschrevenby Jaarsma, K. and Van Oost,K.). But unlike these nations, China developed upon her own until moderntimes.Since it is impossible to Rule an empire as large as or larger than Rome withoutarithmetic and given the millennia that Chinese culture developed relativelyunimpeded, her mathematics achievements are intriguing.In this article the authors aim to share their discoveries about those achieve-ments from the very start of mathematical development until its assimilationby Western mathematics.To do so, we have divided the history of Chinese mathematics into six periods,treating them from chapter 3 to chapter 8:•Chapter 3, Prelude (< ca. -200): As will be explained later, not much isknown about the mathematics before the 2ndcentury BC in China. Thischapter will focus on the things wedoknow.•Chapter 4, Foundations (ca. -200, ca. 600): Central theme of this chapterwill be the most influential book in Chinese mathematical history. Alsosome wonderful achievements during the Chinese Dark Ages will be dis-cussed. This book together with the mathematical developments duringthe Dark Ages would form the foundations of Chinese mathematics.•Chapter 5, Buddhism (ca. 600 - ca. 1200): After the Dark Ages a new dy-nasty took control of China, who reformed the educational system. Mean-while Buddhistic and Indian influences were at their best.•Chapter 6, Zenith (ca. 1200 - ca. 1400): A new war was unleashed, afterwhich China was conquered by the Mongols. In this period of war andbloodshed, ironically, Chinese mathematics flourished more than ever; itsachievements would be the zenith of Chinese mathematics, unparalleledin Chinese history both before and after. When compared to the West,the Chinese where centuries ahead.•Chapter 7, Decline (ca. 1400 - ca. 1850): China freed herself from herMongolian invaders, but her mathematics would dropNpoints. Shewould, however, slowly recover with the aid of Europeans.•Chapter 8, Westernization (ca. 1850 - ca. 1920): The end of Chinesemathematics. At gunpoint, Chinese mathematics was assimilated by itsWestern counterpart.2But before we immerse ourselves in the development of China’s mathematics,we should keep in mind the following quote:Our initial view of the science as a gradual accumulation of iso-lated discoveries, connected only by their common end is in the lastanalysis teleological. It can pointAas a step on the way toB,butit cannot lead to an understanding of howBevolved out of the in-ner necessity, historical experience, and social consequences ofA....Adid not appear at a certain time because of some final causeordained its time had come, but rather because it was coherent withother ideas, attitudes, and prejudices of its time.In other words:Science has shaped human history. The advent of agriculture, the mastery ofmetallurgy, the development of a script, all this proto-scientific developmentschanged the course of events. Especially after the Scientific Revolution at theend of the European Renainssance, scientific advancements have played a crucialrole in determining the balance of power; it is above all the high level of scienceand technology that gives the Western world her current dominant position inthe world.But the converse is true as well: the development of science – just as anyother human enterprise – is inextricably linked with the social context in whichthose who develop the science live. The value a society attaches to science andthe manner in which scientists are regarded is of profound influence on scientificdevelopment.Therefore the first two chapters will discuss the political history (chapter 1)and the Chinese philosophy (chapter 2), since these may explain why Chinesemathematics developed the way she did.Our summary and conclusions can be found in chapter 9.31The History of ChinaContrary to Western history and philosophy, we expect that most of the readerswill only have rudimentary knowledge of their Chinese counterparts. Therefore,our exploration of the history of Chinese mathematics will start with an overviewof the history and philosophy of China itself.1.1Linear vs. Cyclic historiography: historical patternsWhen history is viewed from the western side of Eurasia, the naturalpattern seems to be for civilizations and empires to rise and wane.By 2000 BC Egypt had eclipsed Sumeria; by 1000 BC Egypt was onthe decline but Babylonia was an impressive power. In time, how-ever, it would be surpassed by the Persians, then the Greeks, then theRomans.(...)Underlying this view of history is an unspoken analogybetween civilizations and human lives or perhaps the lives of compet-itive individual warrior-heroes. Civilizations have an early, creative,aggressive stage followed by the strong, mature age, but over timelose vigour and become less flexible until eventually they are defeatedin battle or succumb of old age....When history is viewed from the eastern edge of Eurasia, a very dif-ferent pattern emerges as natural and normal. There is no sense thatyounger civilizations supplant older ones, but that civilization pro-gresses through a series of yin-yang-like reversals of direction fromexcessive disorder to excessive order and back again.(Ebrey, pg.33)Few historians today accept the cyclic view of history, but nevertheless thehistory which will unfold itself in the coming paragraphs shows some strikingpatterns. A large part of Chinese history can indeed be described as a succes-sion of dynasties. In general, a capable military leader unites a divided Chinaunder a strongly centralized civil government, with a scholar-official bureau-cracy. Often, China itself is enlarged, because the momentum of the of internalunification victories leads the armies beyond the Chinese borders. “Their suc-cessors, however, would not all be supermen able to prevent power struggles atcourt, keep the cost of defense and local administration low, and preserve orenhance sources of revenue, all the while inspiring loyalty through their bearingand virtue. In this view of history, men of ability and integrity – both emperorsand their counselors – could arrest decline or even temporarily reverse it, butinevitably the dynasty would weaken and eventually fall.” (Ebrey, pg. 135.)Consequently, barbarian tribes who are eager to invade the rich China cannotbe hold back any longer. These tribes conquer China and found own dynasty.But because (i) their lack of experience in governing a large state; (ii) theirmuch smaller population size and (iii) their attempts to accommodate the Chi-nese, Chinese scholars are appointed to organize this new state. With the statebased on Confucian values and the immersion of the foreign elite in the Chineseculture, the barbarian tribes are sinified within a few generations, effectivelymaking the alien dynasty a Chinese one. This dynasty would be strong at first,4though slowly decline and eventually there would be civil war, till a capablemilitary leader again unites China, etc.So ironically, with the occupation of China the conquerers would not end Chi-nese civilization, but their own. As a result, Chinese cultural tradition, withits first traces in 8 000 BC, has survived almost ten millennia, although Chinahas been occupied by foreign peoples many times. In the coming paragraphs wewill outline the ten millennia of Chinese civilization, starting by the inventionof the family and ending in the 19thcentury, when the last dynasty falls.1.2In the beginning...not some Divine Creator, but brilliant human beings where responsible for thesteps that transformed the Chinese from primitive people to a civilization, orso the Chinese creation legends say (Ebrey, pg. 10). The Ox-farmer Fu Xiinvented the family and the domestication of animals, Shen Nong, (whose namemeans literally Divine Farmer) invented agriculture and commerce and the Yel-low Emperor expelled barbarian tribes from the Yellow River plane, giving hispeople a place to live. Furthermore, the Yellow Emperor ordered 7 of his sub-jects to respectively observe the sun, observe the moon, observe the stars, fixthe musical scales, construct a sexagimal system1, create arithmetic and to usethe previous 6 to develop a calendar (Mikami,pg. 2).2The view that the cosmos came into being on its own, without a Creator toset everything in motion, would become a fundamental difference between theChinese and Western philosophical schools, having major repercussions on thedevelopment of Chinese and Western science, as we will see in chapter 2.But despite a fundamental different cosmological view, it is safe to say thatthe Chinese are among the most ancient civilizations, comparable only to theBabylonians and Egyptians (Mikami, pg. 1).Archeological excavations reveal that around 8000 BC a transition from a hunter-gatherer to an agricultural society occurred (Gernet, pg. 39), around two thou-sand years later the agricultural civilization was well entrenched. Stone, boneand (later) ceramic tools were used. The first bronze alloys are dated at 1700– 1600 BC (Ebrey, pg. 25) and gradually the Chinese mastered the techniquesto work with bronze to an unparalleled degree of sophistication (Gernet, pg. 40).The first proof of a written language stems from ‘oracle bones’ used for reli-gious purposes. Most inscriptions are inquiries from the royalty to spirits, as toinform when was a proper time to start a war, the worshipping of ancestors anddivinities, agricultural campaigns, etcetera. The writing was rather complex:on the about 100 000 oracle bones found so far, with in total 5 000 differentcharacters, of which 1 500 have been deciphered. On these bones also the firstnumbers are found, as will be described in further detail in chapter 3. (Gernet,pg. 47 and Schirokauer, pg. 9.)numerical system with 60 numbers in its base.there are Chinese creation myths which stage a Creator who sets everything inmotion. However, these myths are younger than the legends regarding the Yellow Emperordescribed here (Schirokauer, pg. 23).2Actually,1A5 [ Pobierz całość w formacie PDF ]

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