People seem compelled to organize. They also have a practical need to count certain things: cattle, cornstalks, and so on. There is the need to deal with simple geometrical situations in providing shelter and dealing with land. Once some form of writing is added into the mix, mathematics cannot be far behind. It might even be said that the symbolic approach precedes and leads to the invention of writing.
The Maya, the Chinese, the Civilization of the Indus Valley, the Egyptians, and the region of Mesopotamia between the Tigris and Euphrates rivers -- all had developed impressive bodies of mathematical knowledge by the dawn of their written histories. In each case, what we know of their mathematics comes from a combination of archaeology, the references of later writers, and their own written record.
Mathematical documents from Ancient Egypt date back to 1900 B.C. The practical need to redraw field boundaries after the annual flooding of the Nile, and the fact that there was a small leisure class with time to think, helped to create a problem oriented, practical mathematics. A base-ten numeration system was able to handle positive whole numbers and some fractions. Algebra was developed only far enough to solve linear equations and, of course, calculate the volume of a pyramid. It is thought that only special cases of The Pythagorean Theorem were known; ropes knotted in the ratio 3:4:5 may have been used to construct right angles.
In India mathematics was also mainly practical. Methods of solving equations were largely centered around problems in astronomy. Negative and irrational numbers were used. Of course, India is noted for developing the concept of zero, that was passed into Western mathematics via the Arabic tradition, and is so important as a place holder in our modern decimal number system.
The Classic Maya civilization (250 BC to 900 AD) also developed the zero and used it as a place holder in a base-twenty numeration system. Again, astronomy played a central role in their religion and motivated them to develop mathematics. It is noteworthy that the Maya calendar was more accurate than the European at the time the Spanish landed in The Yukatan Peninsula.
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Ancient Greece
The axiomatic method came into full force in Ancient Greek times; it has characterized mathematics ever since. Geometry was center stage in ancient times. Mathematical models, or idealizations of the real world, were built around points, lines, and planes. Numbers were represented as lengths of line segments. Modern mathematics still relies on the axiomatic method, but tends to be more algebraically based.
Key to the axiomatic method are abstraction and proof. For example, the idea of a point as a pure location with no extension is an abstraction since a point cannot physically exist. A dot differs from a point in that a dot has extension, and represents only an approximate location. Nevertheless, since they can be seen, we use dots to represent points which cannot be seen. Lines, planes and circles are also abstract ideas. That is, they represent idealizations, rather than concrete objects which actually exist. After all, a plane has no thickness, and cannot be anything except a boundary between two regions in space.
One of the best ways to learn more about the history of mathematics is by looking into the lives and work of mathematicians. What follows is a brief list of mathematicians. You can read about each by clicking on the hyperlinks. Use the back button on your browser to get back here.
The Middle Ages
In 476 A.D. The Roman Empire came to an end in the West; the last author of mathematical textbooks, Boethius, was executed in 524; the Eastern Roman Emperor, Justinian, closed the academies in Athens in 529 -- The Middle Ages had been born -- Mathematics, along with the rest of scholarly life, would fall into a decline which would last 1000 years.
Fortunately, during this period Chinese mathematics, the mathematics of India and The Arabic World would continue to flourish. Our modern base-ten number system featuring zero as a place holder was developed in the Eighth Century in India. The basis for algebra was developed in The Arabic World in the Eighth and Ninth Centuries. In fact, the word algebra comes from the Arabic al-jabr which refers to transposing a quantity from one side of an equation to the other.
One of the few bright spots in European mathematics during this period was the work of Fibonacci (1175-1250 A.D). He was the son of an Italian merchant who traveled widely and studied under a Muslim teacher. He helped to open Europe to the Arabic mathematical methods, including the use of 'Arabic Numerals,' which actually were invented in India, as we have seen. Many cegep students will have studied the Fibonacci sequence which has broad use in far-flung areas of mathematics.
By about 1500 A.D. the intellectual climate of Europe was changing. The Middle Ages were coming to an end and the Modern World was being born. Each century from that time until the present day would see the creation of powerful, new, mathematics.
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The Seventeenth Century
The 1600's were an especially high point in scientific and mathematical history. This is the century of Kepler, Galileo, Descartes, Newton, and Leibniz . But, one of the most exciting advancements of the time was the introduction of Logarithms in 1614 by John Napier. Logarithms greatly reduced the labour involved in calculations, and was welcomed by a wider public than most mathematical ideas.
However, the two big new ideas are Descartes' founding of analytic geometry, or geometry based on algebra, and the simultaneous but independent invention of calculus by Newton and Leibniz. This is also the time when Fermat proposed his famous "Last Theorem" which has only just been proved by Andrew Wiles in the 1990's. As with the quest to find a general solution for the fifth degree (quintic) polynomial equation, the challenge presented by Fermat occupied great minds over a period of centuries and produced enormously rich benefits, but no solution until recently. Fermat's main contribution to mathematics was, however, the founding of number theory -- that branch of mathematics which deals with the arithmetic properties of the natural numbers.
Blaise Pascal worked closely with Fermat on number theory and also founded probability as we now know it. His name is commemorated in Pascal's Triangle as well as the Pascal programming language. In England, John Wallis, developed the analytic approach to the conic sections, an area dear to the hearts of many students even today. The Binomial Theorem, another favourite which dates to this period, was introduced by Newton himself. The Seventeenth Century, along with the works of Ancient Greece, establish the roots of the mathematical tradition which lives to the present day.
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The Twentieth Century
By the beginning of The Twentieth Century mathematics had grown wide and deep, so vast that it is impossible to summarize the subject here. Let us just mention one thread.
The invention of mathematical logic lead to a deep analysis of the fundamentals underlying mathematics. It seemed that in the background there was a desire to mechanize intelligent thought itself. All of this was, of course, closely related to the impending introduction of computers. Then, in 1931, Kurt Gödel proved that statements can be formed that are neither provable nor disprovable in any complete and consistent axiom set. It follows that within a given mathematical system it is not possible to prove or disprove all of the statements that can be formed. Essentially, what Gödel did was to confirm that the human mind, and its spark of insight, can never be replaced by mechanical processes.
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