Wednesday, August 17, 2005

The study of mathematics is extremely rewarding!

Mathematics is the study of quantity, structure, space, and change. Historically, mathematics developed from counting, calculation, measurement, and the study of the shapes and motions of physical objects, through the use of abstraction and deductive reasoning.

Mathematics is also used to refer to the insight gained by mathematicians by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or models and is an indispensable tool in the natural sciences, engineering and economics.
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The word "mathematics" comes from the Greek μάθημα (m�thema) meaning "science, knowledge, or learning" and μαθηματικός (mathematik�s) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in American English.
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The evolution of mathematics can be seen to be an ever increasing series of abstractions. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. From counting, naturally followed arithmetic (e.g. addition, subtraction, multiplication and division).
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However, mathematics undoubtedly could not have developed out of simple counting and arithmetic without writing and a way of writing numbers. Perhaps prehistoric peoples first expressed quantity by drawing lines in the ground or scratching wood. People of the Inca empire, which lacked any other writing system, represented and stored numerical data using a complex system of knots and rope called khipu.
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Historically, the major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the studies of structure, space and change.
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Mathematics arises wherever there are difficult problems that merit careful mental investigation. At first these were found in commerce, land measurement and later astronomy. Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language. Some notable examples of this happening are Newton inventing calculus and Feynman inventing his Feynman path integral, but it also happens with results from string theory.
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The mathematics arising from this immediately has relevance for the subject which inspired it and can be applied to solve problems in that subject. Mathematics which can be so used is called applied mathematics as opposed to pure mathematics. In this way applied mathematics is an indispensable tool. With the increase in our mathematical knowledge, mathematics itself has become a source of inspiration. Mathematics is inspiring to mathematicians because it has some intrinsic aesthetics or inner beauty, which is hard to explain. Mathematicians value especially simplicity and generality and when these seemingly incompatible properties combine in a piece of mathematics, as in a unifying generalization for several subfields, or in a helpful tool for common calculations, often that piece of mathematics is called beautiful.
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Since the result of mathematics inspired by mathematics is often pure mathematics and thus has no applications outside of mathematics yet, the only value it has is in its aesthetics. Surprisingly often, it has happened that pure mathematics, which was considered only of interest to mathematicians, has become applied mathematics because of some new insight, as if it anticipated later needs.
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