Mathematics, in the very broadest sense, is the systematic study of relationships in the physical world and relationships between symbols which need not pertain to the real world. In relation to the world, mathematics is the language of science. It operates within the laws and constraints of science as it examines physical phenomena. Unlike science, however, mathematics has no constraints. So in relation to symbols, mathematics can be considered a pure mental activity which is capable of generating new concepts within the mind unrelated to anything that presently exists.
Mathematics has many utilitarian uses and was developed for these purposes originally. Agriculture and farming required knowledge of geometry for making things. Astronomy and navigation required knowledge of trigonometry, while most everyday activities required knowledge of number and measurement for keeping account of transactions. Pythagorus (who preceded Euclid) considered number to be everything since it expresses the relationship between a multiplicity of natural phenomena from sounds in music to patterns in flowers, and relationships between man-made objects, from architecture to games.
It is often erroneously considered by students that arithmetic is an inferior part of mathematics concerned with computation and calculation. Mathematics is assumed to be the superior activity involved with reasoning and abstract ideas. In fact, arithmetic is said by mathematicians to be the Queen of Mathematics since number theory is one of the most abstract parts of mathematics. Number theory can be studied for its own sake rather than for its usefulness in science and technology.
It was mentioned earlier that we need numbers for keeping account of transactions. Numerical statements of fact in any area of inquiry are known as statistics. Statistical methods of mathematical processes are used to summarize numerical data and help in their interpretation. For example, instead of listing everyone's test scores on an examination and comparing them to last year's scores, it is more expedient to calculate average scores as a measure of class progress.
Although mathematics is famous for the certainty of its results, statistical methods lead us into areas of uncertainty. For example, in the previous paragraph the "average" we calculate is subject to a degree of error. We recognize this and have ways of calculating the probability that the true score lies within a certain range of values. Probability is that part of mathematics that enables mathematicians to calculate the likelihood of an event happening in the future. Probability is the mathematical engine that drives statistics. It enables us to infer the behavior of a whole population from a small sample.
All branches of mathematics are interrelated, as may be seen from the school curriculum. Mathematics is the study of quantitative relationships. When such relationships are expressed in terms of number, that branch of mathematics is called arithmetic. When relationships are expressed in letters and numbers, with similar rules to arithmetic, the subject is known as algebra. Trigonometry studies relationships between angles. Geometry is concerned with size, shape, area, and volume of objects and position in space.
Calculus deals with the relationship between changing quantities. In differential calculus, the problem is to find the rate at which a known but varying quantity changes. The problem in integral calculus is the reverse of this: to find a quantity when the rate at which it is changing is known. Mathematics is the name for the broad area which is comprised of all these subject areas, and many others not included in the school curriculum, e.g., non-Euclidean geometry.
Understanding of quantitative relationships develops in the pre-school period as children learn concepts such as "greater than," "less than," and "equal to." Understanding concepts in mathematics is more important than memorizing rules. "Coming to grips" with time for example, means more than telling time on a watch or clock. It means having some idea of how long it takes to complete tasks, how to budget time, and so forth. Quantitative reasoning is a part of everyday life, yet mathematics tends to be seen as unrelated to daily living. The extent to which mathematics does pervade all aspects of life is astonishing. All the major advances in electricity and magnetism, thermodynamics, and so forth, were dependent on mathematics. Exploration of space and most of the technological discoveries of the twentieth century have been made through the application of mathematics.
Logicians and philosophers are concerned with mathematics for its own sake. They are interested in pure thought and mathematics as a system of reasoning, unrelated to the physical structure of the world. The correspondence between language and mathematics pursued by this branch of study led to information theory and its outgrowths, cybernetics, and operations research.
Ball, W.W. Rouse. A Short Account of the History of Mathematics. London: Sterling Publications, 2002.
Burton, David M. The History of Mathematics. 5th Ed. New York: McGraw Hill College Division, 2002.
Motz, Lloyd, and Jefferson Hane Weaver. The Story of Mathematics. New York: Plenum Press, 1993.
Nye, Mary Jo. The Cambridge History of Science: Vol. 5, The Modern Physical and Mathematical Sciences. Cambridge: Cambridge University Press, 2002.
Slavin, Steven. All the Math You'll Ever Need. New York: Wiley, 1989.
Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. by New York: CRC Press, 1998.
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