The earliest mathematical writer whose name we know was the Egyptian scribe Ahmes, who in c.1650 ¬BC copied an earlier text on handling fractions and solving arithmetical problems. But for at least 1000 years before that, the scribes in the great river civilizations of Egypt and Mesopotamia were developing ways of representing numbers and solving problems which are recognizably precursors of today's mathematical activity.
A significant change was introduced by Greek-speaking people around the E Mediterranean during 500--200 ¬BC: the development of the notion of proving results as a fundamental characteristic of mathematical activity. A research tradition in geometry grew up, whose basic results were codified by Euclid in The Elements (c.300 ¬BC), and culminated in the work of Archimedes and Apollonius.
Later, the idea that the cosmos is intrinsically mathematical, an influential notion found in the work of Plato, was retrospectively attributed to the semi-mythical figure of Pythagoras. The Greek mathematical tradition lasted for several further centuries, notably in Alexandria, and ranged from the astronomical and geographical work, exemplified by Ptolemy, to the arithmetical investigations of Diophantus, who raised the solving of number-problems to a new height. It was in Alexandria, too, that the first well-attested woman mathematician, Hypatia, lived, and where she was murdered in ¬AD 415 by a mob of zealous early Christians.
Meanwhile, mathematical activity had been vigorously pursued in the civilizations of China and India. It is not easy to reconstruct Chinese mathematics before the Emperor Shih Huang Ti's great book-burning of 212 ¬BC. Nonetheless, during the following centuries scholars such as Lui Hui (¬AD c.260) worked to reconstruct and comment on earlier mathematical works, besides developing both geometry and a characteristically Chinese arithmetical-algebraic computational style, which they brought to bear on such areas as computing bounds for p, and solving determinate and indeterminate problems.
India, too, has a long mathematical tradition, initially in a religious context of astronomy and altar-construction, first recorded by Baudhayana c.800--600 ¬BC. The later mathematician-astronomers, Aryabhata (late 5th-c) and Brahmagupta (early 7th-c), wrote important works involving arithmetic, algebra, and trigonometry, whose influence spread to the West in succeeding centuries.
Baghdad in the 9th-c was an important centre for the pivotal Islamic contribution to mathematics. There al-Khwarizmi wrote a number of books, drawing together Babylonian, Greek, and Indian influence. His Arithmetic introduced the Indian decimal-place-value numerals, and his Algebra has given the subject its name (his own name still being remembered in the word algorithm).
By the late 12th-c much mathematical knowledge had been developed and held in the Islamic culture around the S shores of the Mediterranean, and it was beginning to percolate into Christian Europe through trading posts in such places as Sicily and Spain. In particular, the Islamic world was using for its numerals a decimal-place-value system, invented in India some centuries before. An Italian with trading links to Sicily, Leonardo of Pisa, noticed that the Arabs were using much more efficient numerals - which one could calculate with as well as record number values - and wrote a book called Liber abaci (1202, Book of the Abacus) to publicize this.
This period is one of rich mathematical activity in many parts of the world. China at this time was home to mathematicians of the calibre of Yang Hui, who had explored the binomial pattern several centuries before "Pascal's triangle' became known in the West; and Chu Shih Chieh, who took the Chinese arithmetical-algebraic computational style to new heights.
In India, Bhaskara (12th-c) wrote valuable works on arithmetic, algebra, and trigonometry; and Madhava (c.1340--1425) headed a research tradition in Kerala whose work in infinite series and trigonometrical functions, anticipating later European work in mathematical analysis, is only now beginning to be discovered. In Iran, Omar Khayyam worked to develop arithmetic, algebra, and geometry, as well as astronomy and philosophy.
During the late Renaissance, European mathematics had begun to absorb ancient Greek mathematical works, made available by the efforts of such scholars as Regiomontanus, Maurolico (1494--1575), and Commandino (1509--75). An important development was the solution of cubic and quartic equations in Italy, notably in the Ars magna (1545, Great Art) of Gerolamo Cardano (1501--76), and the further development of algebraic analysis by François Viète (1540--1603).
During the 17th-c, Europe saw not only a spectacular flourishing of mathematical creativity - with such mathematicians as John Napier, René Descartes, Pierre Fermat, Christiaan Huygens, Isaac Newton and Gottfried Leibniz - but also the growth of institutions for promoting scientific activity, and journals to communicate and broadcast the results. During the next century, mathematization of many diverse fields of human interest became fashionable in the wake of the enormous success of Newton's Principia (1687).
Mathematicians of the calibre of the Bernoulli family, Leonhard Euler, and Joseph Louis Lagrange consolidated the methods of calculus, applied them to mechanics, and developed new mathematical areas and approaches - notably, an increasing movement from geometry to algebra as the natural language of mathematics.
One consequence of the French Revolution was the promotion of mathematics in education. About this time textbooks were increasingly used, along with tests and examinations, to create mathematics syllabuses and new educational practices. Mathematical activity in France (Gaspard Monge, Pierre Simon Laplace, Augustin Louis Cauchy), and subsequently in Germany (Carl Gauss, Carl Jacobi, Peter Dirichlet, Bernhard Riemann) was strongly developed and professionalized, in both research and teaching directions.
The works of Niels Hendrik and Evariste Galois, two remarkable talents who died young, proved influential on later generations, as did the non-Euclidean geometry of Jçnos Bolyai and Nicolai Lobachevsky. The growing importance of numerical data in society led to the development of statistical thinking, notably in the work of Adolphe Quételet (1796--1874), with applications in both the physical sciences (eg James Clerk Maxwell) and biological and human sciences (eg Karl Pearson). The work of Charles Babbage on computing engines also attests to the recognized need for accurate mathematical tables and efficient handling of numbers.
The foundations of mathematics received growing attention through the 19th-c, from the need to teach and explain the theorems of analysis. Georg Cantor explored the infinite, and founded the theories of sets, while Richard Dedekind, with his definition of real numbers, helped consolidate the process of arithmetizing analysis, as did the work of Karl Weierstrass. One of Weierstrass's pupils who benefited from the slowly opening higher educational opportunities for women was Sonya Kowalevskaya (1850--91).
During the 20th-c much new mathematics has developed, partly through exploring structures common to a range of theories, thus counteracting the tendency to split into more and more distinct specialized areas. Topology has, under the considerable influence of Henri Poincaré, reached new heights of geometrical generality and unifying power, while algebra too has become even more general in its exploration of structural depth, as in the work of Emmy Noether. David Hilbert set the agenda for much 20th-c research through his prescient outline, at the International Mathematical Congress in 1900, of the major problems for mathematics.
In this century, too, mathematics has become applied more diversely than ever before, from the traditional applications in physical science to new applications in economics, biology, and the organization of systems. The development of electronic computers, particularly associated with Alan Turing and John von Neumann, grew out of mathematical, logical, and number-handling activity, and has affected mathematics in a variety of ways. Recent explorations of mathematics which use the computer as a research tool can be seen as restoring mathematics to its roots as an experimental science.
Dr John Fauvel, Open University
Additional information provided by Cambridge Dictionary of Scientists
The History of Mathematics
The earliest mathematical writer whose name we know was the Egyptian scribe Ahmes, who in c.1650 BC copied an earlier text on handling fractions and solving arithmetical problems. But for at least 1000 years before that, scribes in the great river civilizations of Egypt and Mesopotamia were developing ways of representing numbers and solving problems that are recognizably precursors of today's mathematical activity.
A significant change was introduced by Greek-speaking people around the E Mediterranean during 500-200 BC: the development of the notion of proving results as a fundamental characteristic of mathematical activity. A research tradition in geometry grew up, whose basic results were codified by EUCLID in Elements of Geometry (c.300 BC), and culminated in the work of ARCHIMEDES and APOLLONIUS.
Later, the idea that the cosmos is intrinsically mathematical, an influential idea found in the work of Plato, was retrospectively attributed to the semi-mythical figure of PYTHAGORAS. The Greek mathematical tradition lasted for several further centuries, notably in Alexandria, and ranged from the astronomical and geographical work, exemplified by PTOLEMY, to the arithmetical investigations of DIOPHANTUS, who raised the solving of number-problems to a new height. It was in Alexandria, too, that the first well-attested woman mathematician, HYPATIA, lived and where she was murdered in 415, probably by a mob of zealous early Christians.
Meanwhile, mathematical activity had been vigorously pursued in the civilizations of China and India. It is not easy to reconstruct Chinese mathematics before the Emperor Shih Huang Ti's great book-burning of 212 BC. Nonetheless, during the following centuries scholars such as Lui Hui (c.260) worked to reconstruct and comment on earlier mathematical works, besides developing both geometry and a characteristically Chinese arithmetical-algebraic computational style, which they brought to bear on such areas as computing bounds for \fp\n and solving determinate and indeterminate problems.
India, too, has a long mathematical tradition, initially in a religious context of astronomy and altar-construction, first recorded by Baudhayana c.800-600 BC. The later mathematician-astronomer Aryabhata (late 5th-c) and Brahmagupta (early 7th-c) wrote important works involving arithmetic, algebra, and trigonometry, whose influence spread to the West in succeeding centuries.
Baghdad in the 9th-c was an important centre for the pivotal Islamic contributions to mathematics. There AL-KHWARIZMI wrote a number of books, drawing together Babylonian, Greek and Indian influence. His Arithmetic introduced the Indian decimal-place-value numerals, and his Algebra has given the subject its name (his own name still being remembered in the word algorithm).
By the late 12th-c much mathematical knowledge had been developed and held in the Islamic culture around the southern shores of the Mediterranean and it was beginning to percolate into Christian Europe through trading posts in such places as Sicily and Spain. In particular, the Islamic world was using for its numerals a decimal-place-value system. An Italian with trading links to Sicily, FIBONACCI, noticed that the Arabs were using much more efficient numerals-which one could calculate with as well as record number values-and wrote his book Liber abaci (1202, Book of the Abacus) to publicize this.
This period is one of rich mathematical activity in many parts of the world. China at this time was home to mathematicians of the calibre of Yang Hui, who had explored the binomial pattern several centuries before 'Pascal's triangle' became known in the West: and Chu Shih Chieh, who took the Chinese arithmetical-algebraic computational style to new heights.
In India, Bhaskara (12th-c) wrote valuable works on arithmetic, algebra and trigonometry; and Madhava (c. 1340-1425) headed a research tradition in Kerala whose work in infinite series and trigonometrical functions, anticipating later European work in mathematical analysis, is only now beginning to be discovered. In Iran, Omar Khayyam (c. l048- c. ll22) worked to develop arithmetic, algebra and geometry, as well as astronomy and philosophy.
During the late Renaissance, European mathematics had begun to absorb ancient Greek mathematical works, made available by the efforts of such scholars as Regiomontanus (1436-76), Maurolico (1494-1575), and Commandino (1509-75). An important development was the solution of cubic and quartic equations in Italy, notably in the Ars magna (1545, Great Art) of CARDANO, and the further development of algebraic analysis by VIÈTE.
During the 17th-c, Europe saw not only a spectacular flourishing of mathematical creativity-with such mathematicians as NAPIER, DESCARTES, FERMAT, HUYGENS, NEWTON and LEIBNITZ-but also the growth of institutions for promoting scientific activity and journals to communicate and broadcast the results.
During the next century, mathematization of many diverse fields of human interest became fashionable in the wake of the enormous success of Newton's Principia (1687). Mathematicians of the calibre of the BERNOULLI family, EULER and LAGRANGE consolidated the methods of calculus, applied them to mechanics and developed new mathematical areas and approaches-notably, an increasing movement from geometry to algebra as the natural language of mathematics.
One consequence of the French Revolution was the promotion of mathematics in education. About this time text-books were increasingly used, along with tests and examinations, to create mathematics syllabuses and new educational practices. Mathematical activity in France (MONGE, LAPLACE, CAUCHY) and subsequently in Germany (GAUSS, JACOBI, DIRCHLET, RIEMANN) was strongly developed and professionalized, in both research and teaching directions.
The works of Niels Hendrik and GALOIS, two remarkable talents who died young, proved influential on later generations, as did the non-Euclidean geometry of Janos Bolyai (1802-60) and LOBACHEVSKY. The growing importance of the numerical data in society led to the development of statistical thinking, notably in the work of Adolphe Quételet (1796-1874), with applications in both the physical sciences (e.g. MAXWELL) and biological and human sciences (e.g. PEARSON). The work of BABBAGE on computing engines also attests to the recognized need for accurate mathematical tables and efficient handling of numbers.
The foundations of mathematics received growing attention through the 19th-c, from the need to teach and explain the theorems of analysis. Georg Cantor (1845-1918) explored the infinite and founded the theories of sets, while DEDEKIND, with his definition of real numbers, helped consolidate the process of arithmetizing analysis, as did the work of WEIERSTRASS. One of Weierstrass's pupils who benefited from the slowly opening higher educational opportunities for women was SONYA KOVALEVSKY.
During the 20th-c much new mathematics has developed, partly through exploring structures common to a range of theories, thus counteracting the tendency to split into more and more distinct specialized areas. Topology has, under the considerable influence of POINCARÉ, reached new heights of geometrical generality and unifying power, while algebra too has become even more general in its exploration of structural depth, as in the work of EMMY NOETHER.
HILBERT set the agenda for much 20th-c research through his prescient outline, at the International Mathematical Congress in 1900, of the major problems for mathematics. In this century, too, mathematics has become applied more diversely than ever before, from the traditional applications in physical science to new applications in economics, biology and the organization of systems. The development of electronic computers, particularly associated with TURING and VON NEUMANN, grew out of mathematical, logical and number-handling activity and has affected mathematics in a variety of ways. Recent explorations of mathematics which use the computer as a research tool can be seen as restoring mathematics to its roots as an experimental science.
Mather, Cotton1663-1728) clergyman, author; born in Boston, Mass. (son of Increase Mather). He entered Harvard at age twelve and graduated when he was fifteen. He was ordained in 1685 and held office at Boston's Second Church for the rest of his life (as his father's colleague until 1723). He advocated rebellion against the unpopular Sir Edmund Andros with his political writings (1689).
He supported the new Massachusetts charter (1691) and the new royal governor, Sir William Phips, of whom he wrote a biography, Pietas in Patriam (1697). His writings on witchcraft may have increased the mind-set that led to Salem witch trials (1692), but he believed that fasting and prayer were the proper methods for fighting witchcraft. His political popularity declined after 1692, but his religious leadership remained strong and he began to sponsor Yale, rather than Harvard, as the center for Congregational education. He wrote over 450 books during his lifetime.
