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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.

All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms.

This article offers a history of mathematics from ancient times to the present.

As a consequence of the exponential growth of science, most mathematics has developed since the 15th century , and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America.

India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years.

A separate article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system.

Although, in general outline, the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic method, the pre-Euclidean theory of ratios, and the discovery of the conic sections, historians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount of conjecture.

Many important treatises from the early period of Islamic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the relationship between early Islamic mathematics and the mathematics of Greece and India.

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.

For these reasons, the bulk of this article is devoted to European developments since 1500.

This does not mean, however, that developments elsewhere have been unimportant.

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.