Archaeologists, anthropologists, linguists and others studying early societies have found that number ideas evolve slowly. There will typically be a different word or symbol for two people, two birds, or two stones. Only slowly does the idea of 'two' become independent from the things that there are two of. Similarly, of course, for other numbers. In fact, specific numbers beyond three are unknown in some lesser developed languages. A bit of this usage hangs on in our modern English when we speak, for example, of a flock of geese, but a school of fish.
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.
What we know of the mathematics of Mesopotamia comes from cuneiform writing on clay tablets which date back as far as 2100 B.C. Sixty was the number system base -- a system that we have inherited and preserve to this day in our measurement of time and angles. Among the clay tablets are found multiplication tables, tables of reciprocals, squares and square roots. A general method for solving quadratic equations was available, and a few equations of higher degree could be handled. From what we can see today, both the Egyptians and the Mesopotamians (or Babylonians) stuck to specific practical problems; the idea of stating and proving general theorems did not seem to arise in either civilization.Chinese mathematics -- a vast and powerful body of knowledge --, although mainly practical and problem oriented, did contain general statements and proofs. A method similar to Gaussian Reduction with back-substitution for solving systems of linear equations was known two thousand years earlier in China than in the West. The value ofp was known to seven decimal places by 500 A.D., far in advance of the West.
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.
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.
An interest in investigating the properties of abstract objects characterizes Greek mathematics. Precise definitions; a small number of commonly accepted assumptions called axioms or postulates are made; then general results (lemmas, theorems, and corollaries) are proved using logic.
The Middle Ages
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.
The Sixteenth Century
The 1500's saw the emergence of what we recognize as mathematics in the modern world as opposed to the geometrical discussions of ancient times. Negative numbers were slowly gaining acceptance; the + and - signs made their debut; in accounting, Arabic numerals and double-entry book keeping came into use. Many people contributed to the more symbolic, algebraically based, mathematics that was coming into being.
Girolamo Cardano published his Ars Magne in 1545. In this work he presented for the first time a general solution of the cubic equation, and some special cases of the quartic, or fourth-degree polynomial, equation. This sparked a great deal of enthusiasm and the impetus lasted for centuries as mathematicians tried to solve fifth, and higher, degree equations in a general way. Only in the 1820's did Galois and Abel show that a general solution of fifth and higher degree equations was not possible. In the mean time a great deal of fresh mathematics was spun off as a by-product of the quest. Cardano, himself, lead an outrageous life full enough for several good biographies.
François Viète developed the first system of symbolic algebra. He introduced the use of braces and parentheses, and used the + and - signs along with a number of abbreviations for other operations. He was the first to make the crucial distinction between variable quantities and constant unknowns. In his work is found most of the methods of ordinary algebra as we know it today. He even foresaw the invention of logarithms in the next century by using the trig identity
sin a + sin b = 2 sin(a+b)/2 cos(a-b)/2
The Seventeenth Century
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.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.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.
The Eighteenth Century
Joseph Louis Lagrange (1736-1813) may have been France's greatest mathematician of the century. Like several of the top mathematicians of the era he was appointed to the Berlin Academy as Court Mathematician to Frederic the Great. A shy and quiet man, he extended greatly our understanding of solutions to algebraic equations, and of planetary orbits. Another great French mathematician was Pierre Simon de Laplace (1749-1827.) A more outgoing and practical person than Lagrange, his greatest contribution may have been in the area of probability theory. Laplace treated mathematics as a tool, as a means to an end; whereas Lagrange considered mathematics as a thing of great beauty -- like poetry -- and created mathematics as an end in itself. Another important figure was D'Alembert (1717-1783), who did significant work in differential equations, sequences and series, mechanics, and astronomy; he was a member of the French Academie des Sciences.
At some point mention should be made of the Bernoulli's. This Swiss family produced at least thirteen mathematicians over a period of two centuries. In the Eighteenth Century two brothers, Jacob and Johann, played a major role. Another Bernoulli, Jean, tutored Leonard Euler (1707-1783) who was to become, without doubt, the century's greatest mathematician. Euler divided his career between the court of Frederic the Great, at Berlin, and the Russian Academy, at Saint Petersberg. His personal life was quiet and apparently happy despite much hardship, including blindness in old age. Amazingly, he was able to continue doing original mathematics even after the loss of his eyesight by dictating his work to others. Euler's output was truly amazing; his collected works fill 74 volumes. And, he frequently held back his results so that others could claim some credit. Besides developing the use of complex numbers and founding what we now know as topology, he introduced many of today's familiar notations, including p, S, e, log x, sin x, cos x, f(x) for functions, and others.
The Nineteenth Century
Karl Frederic Gauss (1777-1854) was the greatest mathematician of the nineteenth century, and one of the greatest of all times. Despite having been born into a working-class family with a father who did not value education, he got his doctorate proving in his thesis The Fundamental Theorem of Algebra which states that an nth-degree polynomial has n roots in complex numbers.
Apart from extending calculus to the complex numbers and developing more abstract algebras, mathematics branched out in various ways. Non-Euclidean geometry is the study of geometries which result from modifications of Euclid's axioms. Along with Reimann, the names of Lobachevsky and Bolyai are impotortant here. George Boole andGeorg Cantor were key in the foundations of set theory and mathematical logic. Karl Pearson (1857-1936) founded statistics as we know it today.
The Twentieth Century
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.
A few of the key figures in the subject this century are: