In the history of mathematics, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks). Other mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later.
Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Mathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences. This experimental validation of Einstein's theory shows that Newton's law of gravitation is only an approximation, though accurate in everyday application. For example, the perihelion precession of Mercury cannot be explained by Newton's law of gravitation but is accurately explained by Einstein's general relativity. Inaccurate predictions imply the need for improving or changing mathematical models, not that mathematics is wrong in the models themselves. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model for describing the reality. For example, the movement of planets can be accurately predicted using Newton's law of gravitation combined with mathematical computation. This enables the extraction of quantitative predictions from experimental laws. Mathematics is widely used in science for modeling phenomena. The result of a proof is called a theorem. A proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration. These objects are either abstractions from nature, such as natural numbers or lines, or - in modern mathematics - entities that are stipulated with certain properties, called axioms. Most mathematical activity involves discovering and proving properties of abstract objects by pure reasoning. Mathematics (from Ancient Greek μάθημα máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers ( arithmetic, number theory), formulas and related structures ( algebra), shapes and the spaces in which they are contained ( geometry), and quantities and their changes ( calculus and analysis). 3rd century BC Greek mathematician Euclid holding calipers, as imagined by Raphael in this detail from The School of Athens (1509–1511)
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