Saturday, October 17, 2009

Fjordman: A History of Algebra

Fjordman’s latest essay, “A History of Algebra”, has been posted at Jihad Watch. Some excerpts are below:

Victor J. Katz sums up the state of global mathematics around the year 1300, with a special emphasis on the major Eurasian civilizations, Europe, India, China and the Middle East:

“European algebra of this time period, like its Islamic counterpart, did not consider negative numbers at all. India and China, however, were very fluent in the use of negative quantities in calculation, even if they were still hesitant about using them as answers to mathematical problems. The one mathematical subject present in Europe in this time period which was apparently not considered in the other areas was the complex of ideas surrounding motion. It was apparently only in Europe that mathematicians considered the mathematical question of the meaning of instantaneous velocity and therefore were able to develop the mean speed rule. Thus the seed was planted which ultimately grew into one branch of the subject of calculus nearly three centuries later. It appears that the level of mathematics in these four areas of the world was comparable at the turn of the fourteenth century. Although there were specific techniques available in each culture that were not available in others, there were many mathematical ideas and methods common to two or more.”

If the level of knowledge was comparable across the major regions of Eurasia by the early fourteenth century, why was modern mathematics developed in Europe? In the Islamic world, mathematical sciences and natural philosophy tended to be classified as “foreign sciences” and treated with some suspicion, not integrated into the core curriculum at places of learning. In Europe there was a growing body of universities where natural sciences were viewed more favorably and where students enjoyed much more free inquiry and legal protection. The Islamic world did not develop calculus, analytic geometry or heliocentric astronomy.

In China, the education system was a part of the imperial bureaucracy, which did not encourage studies in science or mathematics but memorization of ancient literary classics. Those who did mathematical work usually did so in isolation, independent of each other and often unknown to each other, and their work was in many cases not followed up. This does not mean that Chinese mathematicians did not make valuable contributions, but like in the Islamic world this often happened more in spite of than because of the education system.

The practical handbook Jiuzhang Suanshu (Nine Chapters on the Mathematical Art) is the longest surviving Chinese mathematical work, and prominent Chinese mathematicians, among them Liu Hui in 263 AD, published commentaries on it. Zu Chongzhi (ca. 429-500 AD) calculated π to seven decimals, the most accurate known estimate in the world until the Persian Jamshid al-Kashi (ca. 1380-1429) surpassed this. The Chinese were proficient in solving many kinds of algebraic problems. One of the most dynamic periods of mathematics in China was the late thirteenth century, with men such as Qin Jiushao (ca. 1202-1261).
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Katz states that “Chinese scholars were primarily interested in solving problems of importance to the Chinese bureaucracy. Although some development of better techniques evidently occurred over the centuries, to a large extent ‘progress’ was stifled by the general Chinese reverence for the past. Hence even incorrect methods from such works as the Jiuzhang suanshu were repeated through the centuries. Although the thirteenth-century mathematicians exploited the counting board to the fullest, its very use imposed limits. Equations remained numerical, so the Chinese were unable to develop a theory of equations comparable to the one developed several centuries later in the West… Finally, in the late sixteenth century, with the arrival of the Jesuit priest Matteo Ricci (1552-1610), Western mathematics entered China and the indigenous tradition began to disappear.”

Madhava of Sangamagramma (ca.1350-ca.1425) was an innovative mathematician and founder of the Kerala school of astronomy and mathematics in South India, which did some interesting work from the fourteenth to sixteenth centuries, but there is today little proof of transfer of knowledge to other regions. According to author John North, “Indian religious tradition was a powerful controlling force, not only of content, but also of form and of the ways of learning by rote. As a result, a typical work of eighteenth-century astronomy can be easily mistaken for one of the previous millennium. We are reminded of the situation in China, markedly different from that in the West.” There was never any strong drive in India to link astronomy with other systems of knowledge, for instance physics, as happened in Europe.

The most important Asian mathematician of the early modern era was arguably Seki Kowa or Seki Takakazu (ca. 1642-1708) in Japan, a rough contemporary of Gottfried Leibniz and Isaac Newton who anticipated some European advances. Born into a samurai family, he was a leading figure in the wasan (“Japanese calculation”) movement. He was the first person to study determinants and independently discovered Bernoulli numbers at the same time as or slightly before the brilliant Jacob Bernoulli (1654-1705) in Switzerland. While you can find a handful of Asian exceptions here and there, Victor J. Katz concludes that “Nevertheless, the locus of the history of mathematics after the fourteenth century was primarily in Europe.” If I concentrate on Europe in the reminder of this story it is because almost all global advances in mathematics between the fourteenth and the mid-twentieth century happened there.

By the fourteenth century a commercial revolution had begun in Western Europe where the “new capitalists” could remain at home and hire others to travel to various ports as their agents. This led to the creation of international trading companies centered in major cities, and these companies needed more sophisticated mathematics than their predecessors did because they had to deal with letters of credit, bills of exchange, promissory notes and interest calculations. Double-entry bookkeeping began as a way of keeping track of these various transactions. A new class of “professional” mathematicians grew up in response to these growing needs, the maestri d’abbaco or abacists. Italian abacists and merchants were instrumental in teaching Europeans the Hindu-Arabic decimal place-value system

Read the rest at Jihad Watch.