MATHEMATICS

the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences. In many culturesunder the stimulus of the needs of practical pursuits, such as commerce and agriculturemathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians. For full treatment of this aspect of mathematics, see mathematics, history of. All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics, foundations of. The substantive branches of mathematics are treated in several articles. See algebra; analysis; arithmetic; combinatorics; Euclidean geometry; game theory; number theory; numerical analysis; optimization; probability theory; set theory; statistics; trigonometry.