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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Omar Khayyam used the new numbering system to develop a language of calculation that went beyond the efforts of al-Khowârizmî and served as a basis for the more complicated language of algebra. In addition, Omar Khayyam used technical mathematical observations to reform the calendar and to devise a triangular rearrangement of numbers that facilitated the figuring of squares, cubes, and higher powers of mathematics; this triangle formed the basis of concepts developed by the seventeenth-century French mathematician Blaise Pascal, one of the fathers of the theory of choice, chance, and probability.
The impressive achievements of the Arabs suggest once again that an idea can go so far and still stop short of a logical conclusion. Why, given their advanced mathematical ideas, did the Arabs not proceed to probability theory and risk management? The answer, I believe, has to do with their view of life. Who determines our future: the fates, the gods, or ourselves? The idea of risk management emerges only when people believe that they are to some degree free agents. Like the Greeks and the early Christians, the fatalistic Muslims were not yet ready to take the leap.
By the year 1000, the new numbering system was being popularized by Moorish universities in Spain and elsewhere and by the Saracens in Sicily. A Sicilian coin, issued by the Normans and dated “1134 Annoy Domini,” is the first known example of the system in actual use. Still, the new numbers were not widely used until the thirteenth century.
Despite Emperor Frederick’s patronage of Fibonacci’s book and the book’s widespread distribution across Europe, introduction of the Hindu-Arabic numbering system provoked intense and bitter resistance up to the early 1500s. Here, for once, we can explain the delay. Two factors were at work.
Part of the resistance stemmed from the inertial forces that oppose any change in matters hallowed by centuries of use. Learning radically new methods never finds an easy welcome.
The second factor was based on more solid ground: it was easier to commit fraud with the new numbers than with the old. Turning a 0 into a 6 or a 9 was temptingly easy, and a 1 could be readily converted into a 4, 6, 7, or 9 (one reason Europeans write 7 as 7). Although the new numbers had gained their first foothold in Italy, where education levels were high, Florence issued an edict in 1229 that forbade bankers from using the “infidel” symbols. As a result, many people who wanted to learn the new system had to disguise themselves as Moslems in order to do so.40
The invention of printing with movable type in the middle of the fifteenth century was the catalyst that finally overcame opposition to the full use of the new numbers. Now the fraudulent alterations were no longer possible. Now the ridiculous complications of using Roman numerals became clear to everyone. The breakthrough gave a great lift to commercial transactions. Now al-Khowârizmî’s multiplication tables became something that all school children have had to learn forever after. Finally, with the first inklings of the laws of probability, gambling took on a whole new dimension.
The algebraic solution to the epigram about Diophantus is as follows. If x was his age when he died, then:
Diophantus lived to be 84 years old.
1200–1700: A THOUSAND OUTSTANDING FACTS
Chapter 3
The Renaissance Gambler
Piero della Francesca, who painted the picture of the Virgin that appears on the following page (“The Brera Madonna”), lived from about 1420 to 1492, more than two hundred years after Fibonacci. His dates place him at the center of the Italian Renaissance, and his work epitomizes the break between the new spirit of the fifteenth century and the spirit of the Middle Ages.
Delia Francesca’s figures, even that of the Virgin herself, represent human beings. They have no halos, they stand solidly on the ground, they are portraits of individuals, and they occupy their own three-dimensional space. Although they are presumably there to receive the Virgin and the Christ Child, most of them seem to be directing their attention to other matters. The Gothic use of shadows in architectural space to create mystery has disappeared; here the shadows serve to emphasize the weight of the structure and the delineation of space that frames the figures.
The egg seems to be hanging over the Virgin’s head. More careful study of the painting suggests some uncertainty as to exactly where this heavenly symbol of fertility does hang. And why are these earthly, if pious, men and women so unaware of the strange phenomenon that has appeared above them?
Madonna of Duke Federico II di Montefeltro. Pinacoteca di Brera, Milan, Italy.
(Reproduction courtesy of Scala / Art Resource, NY.)
Greek philosophy has been turned upside down. Now the mystery is in the heavens. On earth, men and women are free-standing human beings. These people respect representations of divinity but are by no means subservient to it – a message that appears over and over again in the art of the Renaissance. Donatello’s charming statue of David was among the first male nude sculptures created since the days of classical Greece and Rome; the great poet-hero of the Old Testament stands confidently before us, unashamed of his pre-adolescent body, Goliath’s head at his feet. Brunelleschi’s great dome in Florence and the cathedral, with its clearly defined mass and unadorned interior, proclaims that religion has literally been brought down to earth.
The Renaissance was a time of discovery. Columbus set sail in the year Piero died; not long afterward, Copernicus revolutionized humanity’s view of the heavens themselves. Copernicus’s achievements required a high level of mathematical skill, and during the sixteenth century advances in mathematics were swift and exciting, especially in Italy. Following the introduction of printing from movable type around 1450, many of the classics in mathematics were translated into Italian and published either in Latin or in the vernacular. Mathematicians engaged in spirited public debates over solutions to complex algebraic equations while the crowds cheered on their favorites.
The stimulus for much of this interest dates from 1494, with the publication of a remarkable book written by a Franciscan monk named Luca Paccioli.41 Paccioli was born about 1445, in Piero della Francesca’s hometown of Borgo San Sepulcro. Although Paccioli’s family urged the boy to prepare for a career in business, Piero taught him writing, art, and history and urged him to make use of the famous library at the nearby Court of Urbino. There Paccioli’s studies laid the foundation for his subsequent fame as a mathematician.
At the age of twenty, Paccioli obtained a position in Venice as tutor to the sons of a rich merchant. He attended public lectures in philosophy and theology and studied mathematics with a private tutor. An apt student, he wrote his first published work in mathematics while in Venice. His Uncle Benedetto, a military officer stationed in Venice, taught Paccioli about architecture as well as military affairs.
In 1470, Paccioli moved to Rome to continue his studies and at the age of 27 he became a Franciscan monk. He continued to move about, however. He taught mathematics in Perugia, Rome, Naples, Pisa, and Venice before settling down as professor of mathematics in Milan in 1496. Ten years earlier, he had received the title of magister, equivalent to a doctorate.
Paccioli’s masterwork, Summa de arithmetic, geometria et proportionalità (most serious academic works were still being written in Latin), appeared СКАЧАТЬ
Hogben, 1968, p. 245.
The background material on Paccioli comes primarily from David, 1962, pp. 36–39, and Kemp, 1981, pp. 146–148.