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Mathematics (ancient Greek mathematike techne: the art of learning) is a formal science that arose from the study of geometric figures and arithmetic with numbers. There is no universally accepted definition for mathematics; today it is usually described as a science that uses logic to examine self-created abstract structures for their properties and patterns.

Mathematics is one of the oldest sciences. It first flourished before antiquity in Mesopotamia, India, and China, and later in antiquity in Greece and Hellenism. From there dates the orientation to the task of "pure logical proof" and the first axiomatization, namely Euclidean geometry. In the Middle Ages it survived independently in the early humanism of the universities and in the Arab world.

Characteristic of mathematics continues to be the way in which it progresses by working on problems that are "actually too difficult". Often many centuries have passed between the formulation of a problem and its solution, and a completely new subfield has finally been founded with the solution of the problem: for example, infinitesimal calculus in the 17th century made it possible to solve problems that had been open since antiquity. Even a negative answer, the proof of the unsolvability of a problem, can advance mathematics: thus group theory arose from failed attempts to solve algebraic equations.