Against the Gods. Bernstein Peter L.
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Название: Against the Gods
Автор: Bernstein Peter L.
Издательство: Автор
Жанр: Зарубежная образовательная литература
isbn: 9780470534533
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But the discovery of a superior numbering system would not occur until about 500 AD, when the Hindus developed the numbering system we use today. Who contrived this miraculous invention, and what circumstances led to its spread throughout the Indian subcontinent, remain mysteries. The Arabs encountered the new numbers for the first time some ninety years after Mohammed established Islam as a proselytizing religion in 622 and his followers, united into a powerful nation, swept into India and beyond.
The new system of numbering had a galvanizing effect on intellectual activity in lands to the west. Baghdad, already a great center of learning, emerged as a hub of mathematical research and activity, and the Caliph retained Jewish scholars to translate the works of such pioneers of mathematics as Ptolemy and Euclid. The major works of mathematics were soon circulating throughout the Arab empire and by the ninth and tenth centuries were in use as far west as Spain.
Actually, one westerner had suggested a numbering system at least two centuries earlier than the Hindus. About 250 AD, an Alexandrian mathematician named Diophantus wrote a treatise setting forth the advantages of a system of true numbers to replace letters substituting for numbers.30
Not much is known about Diophantus, but the little we do know is amusing. According to Herbert Warren Turnbull, a historian of mathematics, a Greek epigram about Diophantus states that “his boyhood lasted l/6th of his life; his beard grew after l/12th more; he married after l/7th more, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.” How old was Diophantus when he died?31 Algebra enthusiasts will find the answer at the end of this chapter.
Diophantus carried the idea of symbolic algebra – the use of symbols to stand for numbers – a long way, but he could not quite make it all the way. He comments on “the impossible solution of the absurd equation 4 = 4x + 20. “32 Impossible? Absurd? The equation requires x to be a negative number: −4. Without the concept of zero, which Diophantus lacked, a negative number is a logical impossibility.
Diophantus’s remarkable innovations seem to have been ignored. Almost a millennium and a half passed before anyone took note of his work. At last his achievements received their due: his treatise played a central role in the flowering of algebra in the seventeenth century. The algebraic equations we are all familiar with today – equations like a + bx = c– are known as Diophantine equations.
The centerpiece of the Hindu-Arabic system was the invention of zero —sunya as the Indians called it, and cifr as it became in Arabic.33 The term has come down to us as “cipher,” which means empty and refers to the empty column in the abacus or counting frame.34
The concept of zero was difficult to grasp for people who had used counting only to keep track of the number of animals killed or the number of days passed or the number of units traveled. Zero had nothing to do with what counting was for in that sense. As the twentieth-century English philosopher Alfred North Whitehead put it,
The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.35
Whitehead’s phrase “cultivated modes of thought” suggests that the concept of zero unleashed something more profound than just an enhanced method of counting and calculating. As Diophantus had sensed, a proper numbering system would enable mathematics to develop into a science of the abstract as well as a technique for measurement. Zero blew out the limits to ideas and to progress.
Zero revolutionized the old numbering system in two ways. First, it meant that people could use only ten digits, from zero to nine, to perform every conceivable calculation and to write any conceivable number. Second, it meant that a sequence of numbers like 1, 10, 100 would indicate that the next number in the sequence would be 1000. Zero makes the whole structure of the numbering system immediately visible and clear. Try that with the Roman numerals I, X, and C, or with V, L, and D – what is the next number in those sequences?
The earliest known work in Arabic arithmetic was written by al-Khowârizmî, a mathematician who lived around 825, some four hundred years before Fibonacci.36 Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al-Khowârizmî” fast. That’s where we get the word “algorithm,” which means rules for computing.37 It was al-Khowârizmî who was the first mathematician to establish rules for adding, subtracting, multiplying, and dividing with the new Hindu numerals. In another treatise, Hisâb al-jabr w’ almuqâbalah, or “Science of transposition and cancellation,” he specifies the process for manipulating algebraic equations. The word al-jabr thus gives us our word algebra, the science of equations.38
One of the most important, surely the most famous, early mathematician was Omar Khayyam, who lived from about 1050 to about 1130 and was the author of the collection of poems known as the Rubaiyat.39 His haunting sequence of 75 four-line poems (the word Rubaiyat defines the poetic form) was translated in Victorian times by the English poet Edward Fitzgerald. This slim volume has more to do with the delights of drinking wine and taking advantage of the transitory nature of life than with science or mathematics. Indeed, in number XXVII, Omar Khayyam writes:
Myself when young did eagerly frequent
Doctor and Saint, and heard great Argument
About it and about; but evermore
Came out by the same Door as in I went.
According to Fitzgerald, Omar Khayyam was educated along with two friends, both as bright as he: Nizam al Mulk and Hasan al Sabbah. One day Hasan proposed that, since at least one of the three would attain wealth and power, they should vow that “to whomsoever this fortune falls, he shall share it equally with the rest, and preserve no preeminence for himself.” They all took the oath, and in time Nizam became vizier to the sultan. His two friends sought him out and claimed their due, which he granted as promised.
Hasan demanded and received a place in the government, but, dissatisfied with his advancement, left to become head of a sect of fanatics who spread terror throughout the Mohammedan world. Many years later, Hasan would end up assassinating his old friend Nizam.
Omar Khayyam asked for neither title nor office. “The greatest boon you can confer on me,” he said to Nizam, “is to let me live in a corner under the shadow of your fortune, to spread wide the advantages of science and pray for your long life and prosperity.” Although the sultan СКАЧАТЬ
The background material on Diophantus is from Turnbull, 1951, p. 113.
See Hogben, 1968, pp. 244–246.
The Arabic term survives even in Russian, where it appears as
From Newman, 1988a, p. 433.
The background material on al-Khowârizmî is primarily from Muir, 1961, and Hogben, 1968.
Hogben, 1968, p. 243.
See Hogben, 1968, Chapter VI, for an extended and stimulating discussion of the development of algebra and the uses of zero.
The background material on Omar Khayyam is from Fitzgerald.