A brief history of mathematics
Mathematics first arose from the practical need to measure time and to count. The earliest evidence of primitive forms of counting occurs in scored pieces of wood and stone. Early uses of geometry are revealed in patterns found on ancient cave walls and pottery. As civilizations arose in Asia and the Near East, sophisticated number systems and basic knowledge of arithmetic, geometry, and algebra began to develop.
Early Civilizations
The ancient Egyptians were able to solve many different kinds of practical mathematical problems, including the intricate calculations necessary to build the pyramids. Egyptian arithmetic, based on counting in groups of ten, was relatively simple. This Base-10 system probably arose from biological reasons, as we have 8 fingers and 2 thumbs. Numbers are sometimes called digits from the Latin word for finger.
The most remarkable feature of Babylonian arithmetic was its use of a sexagesimal (base 60) place-valued system in addition to a decimal system. Babylonian mathematics is still used to tell time – an hour consists of 60 minutes, and each minute is divided into 60 seconds – and circles are measured in divisions of 360 degrees.
Greek and Hellenistic mathematics
Greek mathematics was more sophisticated than the mathematics that had been developed by earlier cultures. Thales used geometry to solve problems
such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras is credited with the first proof of the Pythagorean Theorem. The Pythagoreans proved the existence of irrational numbers.
Eudoxus developed the method of exhaustion, a precursor of modern integration. Euclid is the earliest example of the format still used in mathematics today: definition, axiom, theorem and proof. He also studied cones. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi. He also studied the spiral bearing his name, formulas for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.
The Middle Ages
Indian mathematicians were especially skilled in arithmetic, methods of calculation, algebra, and trigonometry. It was their decimal place-valued number system, including zero, which was especially suited for easy calculation.
When the Greek civilization declined, Greek mathematics (and the rest of Greek science) was kept alive by the Arabs. The Arabs also learned of the considerable scientific achievements of the Indians, including the invention of a system of numerals (now called ‘»Arabic» numerals) which could be used to write down calculations instead of having to resort to an abacus. One of the greatest scientific minds of Islam was al-Khwarizmi, who introduced the name (al-jabr) that became known as algebra.
From about the 11th century first Abelard of Bath and then Fibonacci brought Islamic mathematics and its knowledge of Greek mathematics back into Europe.
The Renaissance
Major progress in mathematics in Europe turned out to have started at the beginning of the 16th century with the algebraic solution of cubic and quadratic equations. Copernicus and Galileo revolutionized the applications of mathematics to the study of the Universe. The Progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra spread from Italy to Belgium and France.
The Seventeenth and Eighteenth Centuries
In the 17th century Napier, Briggs and others greatly extended the power of mathematics as a calculatory science with the discovery of logarithms.
Cavalieri made progress towards the calculus with his infinitesimal
Department of Philology
Advanced English for Mathematicians 29
methods and Descartes added the power of algebraic methods to geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus is considered to be the topic of most significance in the 17th century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new astronomy.
Newton’s theory of gravitation and his theory of light take us into the 18th century. However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th century when the calculus grew in power and variety of application.
It is Euler who is considered to be the most important mathematician of the 18th century. In addition to work in a wide range of mathematical areas, he invented two new branches, namely calculus of variations and differential geometry. Euler was also important in pushing forward research in number theory started so effectively by Fermat. Toward the end of the 18th century, Lagrange began a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace’s great work of celestial mechanics.
The Nineteenth Century
Rapid progress was made in the 19th century. Non-Euclidian geometry developed by Lobachevsky and Bolyai led to characterization of geometry by Riemann. Gauss, who is thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruencies. His work in differential geometry was to revolutionize the topic. He also contributed in a major way to astronomy and magnetism. The 19th century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Cauchy, basing on the work of Lagrange on functions, began rigorous analysis of the theory of functions of a complex variable. This work was continued by Weierstrass and Riemann. At the end of the 19th century Cantor invented set theory almost single-handedly. Analysis was driven by the requirements of mathematical physics and astronomy. Maxwell is known to have revolutionized the application of analysis to mathematical physics, and Galois’ introduction of the group concept heralded a new direction for mathematical research which has continued through the 20th century.
The Twentieth Century
In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th century mathematics. Today 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not. Famous historical conjectures were finally proved. In 1976, Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem. Andrew Wiles, building on the work of others, proved Fermat’s Last Theorem in 1995. Paul Cohen and Kurt Godel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory.Entirely new areas of mathematics such as mathematical logic, topology, complexity theory, and game theory changed the kinds of questions that could be answered by mathematical methods.
At the same time, deep insights were made about the limitations to mathematics. A consequence of Godel’s two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof;
there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic and David Hilbert’s dream of making all of mathematics complete and consistent failed.
Exercise 5. Find in the text equivalents of the following words and