Mathematics

Mathematics includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis). There is no general consensus about its exact scope and its epistemological status.

Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of abstract objects. These objects are either abstractions from nature (such as a natural number and a line), or (in modern mathematics) abstract entities that are defined by their basic properties, called axioms. A proof consists of a succession of applications of some deducing rules to already known results, including axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points. The result of a proof is called a theorem. Contrarily to physical laws, the validity of a theorem (its truth) does not rely on any experimentation, but only on the correctness of the reasoning (however, experimentation is often useful for discovering theorems of interest).

Mathematics is widely used in science, for modeling phenomena. This allows extracting quantitative predictions from experimental laws. For example, the movement of planets can be predicted with high accuracy using Newton's law of gravitation combined with mathematical computation. 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. So, when some inaccurate predictions arise, this means that the model must be improved or changed, not that mathematics is wrong. 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. This experimental validation of Einstein's theory showed that Newton's law of gravitation is only an approximation (very accurate in everyday life).

Mathematics is therefore essential in many fields, including natural science, engineering, medicine, finance, computer science and social sciences. Some areas of mathematics, such as statistics and game theory, are developed in a direct relation with their applications, and are often grouped under the name of applied mathematics. Other mathematical areas were developed independently from any application (pure mathematics), but practical applications are often discovered later. An astonishing example is the problem of integer factorization that goes back to Euclid, which had no practical application before its use through the RSA cryptosystem for the security of computer communications.