Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences.
For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. These results include 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. A proof consists of a succession of applications of deductive rules to already established results. These objects consist of either abstractions from nature or-in modern mathematics-entities that are stipulated with certain properties, called axioms. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. Mathematics (from Ancient Greek μάθημα máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.
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