There was also some pseudo-science, such as numerology and astrology that were not clearly distinguished from mathematics.Īround the Renaissance, two new main areas appeared. The abacus is a simple calculating tool used since ancient times.īefore the Renaissance, mathematics was divided into two main areas, arithmetic, devoted to the manipulation of numbers, and geometry, devoted to the study of shapes. This, in turn, gave rise to a dramatic increase in the number of mathematics areas and their fields of applications a witness of this is the Mathematics Subject Classification, which lists more than sixty first-level areas of mathematics. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. Since then the interaction between mathematical innovations and scientific discoveries have led to a rapid increase in the rate of mathematical discoveries. 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. However, the concept of a "proof" and its associated " mathematical rigour" first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics has been a human activity from as far back as written records exist. A fitting example is the problem of integer factorization which goes back to Euclid but 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 direct correlation with their applications, and are often grouped under the name of 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 (which still is very accurate in everyday life). 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. So when some inaccurate predictions arise, it means that the model must be improved or changed, not that the mathematics is wrong.
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 predicted with high accuracy 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. Contrary to physical laws, the validity of a theorem (its truth) does not rely on any experimentation but on the correctness of its reasoning (though experimentation is often useful for discovering new theorems of interest). 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 "a line"), or (in modern mathematics) abstract entities that are defined by their basic properties, called axioms. Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of abstract objects. There is no general consensus about its exact scope or epistemological status. Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') 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).
Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha.Greek mathematician Euclid (holding calipers), 3rd century BC, as imagined by Raphael in this detail from The School of Athens (1509–1511)
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