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 calculations, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics ha 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 cultures under the stimulus of the needs of practical pursuits, such as commerce and agriculture mathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for compensation and the opportunity to build on the achievements of earlier mathematics.
All mathematics systems (for example, Euclidean geometry) are combinations of sets of axioms and of 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.
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. For these reasons, the bulk of this article is devoted to European development since 1500. This does not mean, however, that development, that development elsewhere have been un-important. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments.
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. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea ad Vietnam. The substantive branches of mathematics are treated in several articles.
Ancient mathematical sources
It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians. The tablets indicate that the Mesopotamian had a great deal of remarkable mathematical knowledge , although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamian or about its influence on Greek mathematics.
In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the un-published works of mathematicians. However, the exponential growth of mathematics means that, for the period from 19th century on, historians are able to treat only the major figures in any detail. Mathematics like any other human activity has its own fashions, will look like a wave of the future.For this reason, the present article males no attempt to assess the most recent developments in the subjects.