Before the time of recorded history, prehistoric people learned to count such things as the animals in their herds and flocks. They probably first used their fingers or pebbles to help keep track of small numbers. They learned to use the length of their hands and arms and other standards of measure. And they learned to use regular shapes when they molded pottery and chipped stone arrowheads. Since the prehistoric times, mathematics has come a long way. In this essay, I will try to sum up the development of mathematics, from the very first systems of mathematics, to mathematics being used to model problems in business, industry, and science.
Around 3000 B.C., the people of ancient Babylonia, China, and Egypt had developed a practical system of mathematics, which used written symbols to stand for numbers. From this, they derived simple arithmetic operations. They also developed a practical geometry helpful in agriculture and engineering. For example, the ancient Egyptians knew how to survey their fields and to make the intricate measurements necessary to build huge pyramids. The Babylonians and Egyptians had even explored some of the fundamental ideas of algebra, but this early mathematics was only used to solve practical problems.
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Between 600 and 300 B.C., the Greeks took the next great leap in mathematics. They inherited a large part of their mathematical knowledge from the Babylonians and Egyptians, but they became the first people to separate mathematics from practical problems. For example, they separated geometry from practical applications and made it into an abstract exploration of space. They based this study of points, lines, and figures, such as triangles and circles, on logical reasoning rather than on facts found in nature. Thales of Miletus, a philosopher, helped begin this new viewpoint of geometry. The philosopher Pythagoras and his followers explored the nature of numbers. In geometry, the Pythagoreans developed the famous theorem that bears their name. Thales, Pythagoras, and many other Greek mathematicians built up a large body of geometrical knowledge. Euclid, one of the foremost Greek mathematicians, organized geometry as a single logical system. His book, The Elements, remains on of the basic works in studying mathematics.
The Greeks also advanced other branches of mathematics. As early as 450 B.C., Greek mathematicians recognized irrational numbers such as the square root of 2. About 370 B.C., Eudoxus of Cnidus, a Greek astronomer and mathematician, formulated a surprisingly masterful definition of proportions. Archimedes, the leading mathematician of ancient times, devised processes that foreshadowed those of integral calculus. The Greek astronomer Ptolemy helped develop trigonometry. Diophantus worked on numbers in equations. He earned the title of the father of algebra.
After the fall of Rome in 476 A.D., Europe saw no new developments in mathematics for hundreds of years, But the Arabs preserved the mathematical tradition of the Greeks and Romans. The Arabs also began to use the zero and the decimal numeral system, both of which had been developed by mathematicians in India. The Arabs also made important contributions of their own. Al-Khowarizmi organized and expanded algebra.
After 1100, Europeans began to borrow the mathematics of the Arab world. For example, European merchants started to use the decimal numbers system. In addition, European scholars began to study Arab works on algebra and geometry. Leonardo Fibonacci made contributions to algebra, arithmetic, and geometry.
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The Renaissance, from the 1400’s to the 1600’s, produced many great advancements in mathematics. The exploration of new lands and continents called for better mathematics for navigation. The growth of business demanded better mathematics for banking and finance. The invention of printing brought the appearance of hundreds of popular arithmetic textbooks. Many of the computation methods that are used today date from this period, such as the procedure for doing a long multiplication or division.
Interest also grew in pure mathematics. Jerome Cardan was one of the pioneers in algebra. One of his colleagues, Francois Viete, introduced the method of using letters to represent unknown numbers. Cardan and Viete also helped develop trigonometry. Nicolaus Copernicus, the astronomer who defended the theory that the universe had the sun as its center, contributed to mathematics through his work in astronomy.
The 1600’s brought many brilliant contributions to mathematics. John Napier, a Scottish mathematician, invented logarithms. The astronomers Galileo and Johannes Kepler expanded mathematical knowledge through their studies of the stars and planets. Rene Descartes invented analytic geometry, and he helped in the development of many other branches of mathematics. Pierre de Fermat founded the modern number theory. Blaise Pascal and Fermat invented the mathematical theory of probability. Then, toward the end of this period, Sir Isaac Newton and Baron von Leibniz invented calculus. The invention of calculus marked the beginning of modern mathematics.
The 1700’s saw wide applications of the new calculus, but one of the greatest contributors to calculus was Leonhard Euler, a Swiss mathematician. Euler worked in almost every branch of mathematics. His contributions to calculus reached into so many fields that many mathematicians call him the founder of modern mathematical analysis.
The 1800’s brought further application of calculus throughout mathematics. Jean Baptiste Fourier used calculus for the study of heat in physics. But the early work in calculus often rested on shaky theoretical foundations. As a result, many disturbing paradoxes appeared. Most of the great achievements included the rebuilding the theoretical foundations of calculus and mathematical analysis. Karl Friedrich Gauss helped carry out this important work. Another outstanding advance of the 1800’s was the invention of non-Euclidean geometry.
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The invention of algebra and geometry and the revision of the theoretical foundations of calculus had far-reaching effects on mathematics. In the 1900’s, mathematicians began to explore the foundations of mathematics itself. Many philosophies of mathematics appeared, as well as attempts to give mathematics a basis in logic. Luitzen Brouwer, made important studies of the foundations of mathematics. The work of Albert Einstein opened a whole new area for mathematical research.
New developments in science required a tremendous expansion of applied mathematics. Such fields as electronics, nuclear physics, and the exploration of space have used new inventions from pure mathematics to solve problems. For example, electronic computers use systems of mathematics designed by mathematicians. Also, mathematical models have been formulated to study many kinds of systems, including underground petroleum reserves and worldwide weather patterns. The models consist of mathematical equations that describe the relations between the parts or processes of a system. Computers are used to solve these equations.
In conclusion, we truly have come a long way from the days when we needed to use math to keep track of our sheep. Today, we use math everyday, from the grocery store, to people using math to investigate crimes and push technology even further. Without math, we would literally be living in the dark.
