Learning Mathematics » History Of Mathematics

History of Mathematics

Admin — Sat, 24 Apr 2010 04:58:41 +0000

If you’ve taken a first-year college history course – or read through a basic history textbook – you may have noticed a small gap. It’s only a thousand years or so.

For a long time, the history of Western culture was told like this: around the fifth century BCE, math, philosophy and science developed, thanks to the hard work of some very smart Greeks such as Thales, Plato, Archimedes and Aristotle. Then Rome took over Greece, and Rome fell, and things went dark for a thousand years or so. Then the Renaissance came along, and thinkers like Galilei and Johannes Kepler took up where the Greeks had, in effect, left off.

Thanks to new historical research – and broader awareness of non-Western countries and of the very rich intellectual cultures being developed east of the Urals – this picture of the history of math, philosophy and science is changing, slowly. But still, teachers tend all too often to skip over one of the most interesting stories in intellectual history – the way that math and logic, including the best insights of Greek logicians, became the property of Muslim countries during the long twilight period, from Rome’s fall to the Renaissance, when most Europeans could no longer read Greek. Without the work of these great Muslim scholars, math today might be a very different animal. The Islamic Arab Empire, beginning in the eighth century, was a world intellectual capital, and Arabic became a language of learning to rival Latin. Some of the best mathematical reasoning in the world was done here.

We may as well start with Muhammad ibn Musa al-Hwarizmi (9th century), a Persian astronomer deeply learned in the mathematical lore of ancient India. From his name (in its Latin form) we get the word algorithm, and from one of his book titles we derive algebra. It’s appropriate that he should be associated with the history of algebra – after all, his books preserved most of what the ancient world knew about algebra (as well as his own brilliant innovations), and his works helped to spread the use of Arabic numerals (the numbers we know and use today) to the West, thus making algebra a good deal more feasible. (To understand why this is important, imagine trying to do algebra problems while using Roman numerals: XIIa times XXVb equals c? No, thanks.)

Then there’s Al-Karaji, who around 1000AD invented the proof by mathematical induction – one of the most basic logical maneuvers in math. Poet Omar Khayyam, writing in the twelfth century, laid the groundwork for non-Euclidean geometry. During this period, Muslim mathematicians invented spherical trigonometry, figured out how to use decimal points with Arabic numerals (though the decimal itself had long been invented by Hindu mathematicians), and developed cryptography, algebraic calculus, analytic geometry, among other things.

As important as any of these contributions, though, was the rescue of Aristotle’s texts from obscurity by Arab scholars. For long periods during the middle ages, Aristotle was considered by Western intellectuals as one of the world’s great thinkers – but most of them hadn’t read him. The few of his works that had survived the twin falls of Greece and Rome were available in sometimes poor, or rather freehanded and inaccurate, Latin translations, and many of his most important works weren’t available at all. Here and there a Greek manuscript survived, but almost nobody, at this point, could read Greek. (Widespread teaching of Greek had to wait for the Renaissance – even famously learned scholars such as the poet Petrarch struggled over it.)

The same went for such seminal works as Euclid’s Elements, the greatest known treatise on geometry. During this long period, when it was thought that these brilliantly logical works were gone forever, Islamic scholars kept their own copies and translations. When European scholars began traveling to Spain and Sicily (then under Muslim rule) during the 12th century, these works and others were rediscovered in the West, leading to great intellectual ferment, including the theology of Thomas Aquinas – and to an understanding of logic that helped the discipline of mathematics to survive and, slowly, thrive again in the Western countries.

By: Ann R Knapp

History of Mathematics

Admin — Wed, 17 Mar 2010 21:02:16 +0000

HISTORY OF MATHEMATICS I INTRODUCTION Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. The Babylonian system of numeration was quite different from the Egyptian system. Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation. A Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes. B Applied Mathematics in Greece Paralleling the studies described in pure mathematics were studies made in optics, mechanics, and astronomy. Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system for recording fractions and, at about the same time, composed tables of chords in a circle. III MEDIEVAL AND RENAISSANCE MATHEMATICS Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centres of Greek learning. In algebra, al-Karaji completed Muhammad al-Khwarizmi’s algebra of polynomials to include even polynomials with an infinite number of terms. Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. The discovery drew the attention of mathematicians to complex numbers and stimulated a search for solutions to equations of degree higher than four. It was this search, in turn, that led to the first work on group theory at the end of the 18th century, and to the French mathematician Évariste Galois’s theory of equations in the early 19th century. The 16th century also saw the beginnings of modern algebraic and mathematical symbols, as well as the remarkable work on the solution of equations by the French mathematician François Viète. V MATHEMATICS SINCE THE 16TH CENTURY Europeans dominated in the development of mathematics after the Renaissance. A 17th Century During the 17th century, the greatest advances were made in mathematics since the time of Archimedes and Apollonius. The science of number theory, which had lain dormant since the medieval period, illustrates the 17th-century advances built on ancient learning. It was Diophantus’ Arithmetica that stimulated Fermat to advance the theory of numbers greatly. The second development in geometry was the publication by the French engineer Gérard Desargues in 1639 of his discovery of projective geometry. Although the work was much appreciated by Descartes and the French philosopher and scientist Blaise Pascal, its eccentric terminology and the excitement of the earlier publication of analytic geometry delayed the development of its ideas until the early 19th century and the works of the French mathematician Jean Victor Poncelet. Another major step in mathematics in the 17th century was the beginning of probability theory in the correspondence of Pascal and Fermat on a problem in gambling, called the problem of points. Without question, however, the crowning mathematical event of the 17th century was Newton’s discovery, between 1664 and 1666, of differential and integral calculus. B 18th Century The remainder of the 17th century and a good part of the 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas to solving a variety of problems in physics, astronomy, and engineering. For example, Johann and Jakob Bernoulli invented the calculus of variations, and French mathematician Gaspard Monge invented differential geometry. The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas. C 19th Century In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. This solution posed another problem, however, that of a logical definition of “real number”. Although Cauchy’s explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. Gauss was one of the greatest mathematicians who ever lived. A third major step was the development of group theory, from its beginnings in the work of Lagrange. Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by English mathematician George Boole’s Laws of Thought (1854) and Cantor’s set theory, was the foundations of mathematics. Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising, but they left open the question of whether other paradoxes might arise in these restricted theories—that is, whether the theories were consistent. VI CURRENT MATHEMATICS At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert’s address at Göttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-colour problem first proposed in the mid-19th century.

History of Mathematics

Admin — Sat, 23 Jan 2010 05:14:18 +0000