Mathematics

Mathematics
Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra),^[2] shapes and the spaces in which they are contained (geometry),^[1] and quantities and their changes (calculus and analysis). There is no general consensus about its exact scope or epistemological status.
Most mathematical activity involves discovering and proving, by pure reasoning, properties of abstract objects. 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. 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. The result of a proof is called a theorem.
Mathematics is widely used in science for modeling phenomena. This enables the extraction of quantitative predictions from experimental laws. For example, the movement of planets can be accurately predicted 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. Inaccurate predictions imply the need for improving or changing mathematical models, not that mathematics is wrong in the models themselves. 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 shows that Newton's law of gravitation is only an approximation yet nonetheless accurate in everyday application.
Mathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences. 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. Other mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later.^[8][9] 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).
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.^[10] Mathematics developed at a relatively slow pace until the Renaissance, when algebra and infinitesimal calculus were added to arithmetic and geometry as main areas of mathematics. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid increase in the development of mathematics. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This, in turn, gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. An example of this is the Mathematics Subject Classification, which lists more than sixty first-level areas of mathematics.
Areas of mathematics
Before the Renaissance, mathematics was divided into two main areas: arithmetic — regarding the manipulation of numbers, and geometry — regarding the study of shapes. Some pseudosciences, such as numerology and astrology, were not clearly distinguished from mathematics.
During the Renaissance, two main areas appeared. Mathematical notation led to algebra, which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields infinitesimal calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities (variables). This division into four main areas — arithmetic, geometry, algebra, calculus^{[verification needed]} — endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were often then considered as part of mathematics, but now are considered as belonging to physics. Some subjects developed during this period predate mathematics and are divided into such areas as probability theory and combinatorics, which only later became regarded as autonomous areas.
At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. Today, the Mathematics Subject Classification contains more than 60 first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. However, several other first-level areas have "geometry" in their names or are commonly considered to belong to geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century (for example category theory; homological algebra, and computer science) or had not previously been considered as mathematics, such as Mathematical logic and foundations (including model theory, computability theory, set theory, proof theory, and algebraic logic).