Mathematics

Mathematics (from Ancient Greek ; máthma: 'knowledge, study, learning') is a field of study that includes topics such 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. The term "mathematics" originates from the Ancient Greek word máthma, which means "knowledge, study, learning" (calculus and analysis). The majority of activities in mathematics involve the application of pure reason to find or verify the attributes of abstract objects. These abstract objects might be abstractions from nature or, in contemporary mathematics, entities that are prescribed with particular properties in the form of axioms. A mathematical proof is made up of a series of applications of certain deductive principles to previously proven theorems, axioms, and (in the case of abstraction from nature) some fundamental qualities that are believed to be the genuine starting points of the theory that is now being investigated.

In the scientific community, mathematics is used for the modelling of phenomena, which subsequently enables the formulation of predictions based on empirical laws. Due to the fact that mathematical truth is not dependent on any particular experiment, it follows that the accuracy of such predictions is solely dependent on the appropriateness of the model. Instead of being the result of mathematical errors, inaccurate forecasts suggest that the current mathematical model being used should be modified. The perihelion precession of Mercury, for instance, was only able to be described after the development of Einstein's general relativity, which superseded Newton's law of gravity as a more accurate mathematical model.

The natural sciences, engineering, health, the financial sector, computer science, and even the social sciences all rely heavily on mathematics. Applied mathematics is a broad category that encompasses several subfields of mathematics that are developed in close association with their respective applications. Two examples of these subfields are statistics and game theory. Other branches of mathematics are created separately from any applications (and are thus referred to as pure mathematics), although it is common for practical applications to be found afterwards. The issue of integer factorization, which dates back to Euclid but had no practical use until its usage in the RSA cryptosystem, is a good illustration of this point (for the security of computer networks).

In the annals of mathematical history, the notion of a proof and the mathematical rigour that went along with it made its debut in Greek mathematics, most notably in Euclid's Elements. Since the beginning of mathematics, the discipline has been fundamentally subdivided into geometry and arithmetic (the manipulation of natural numbers and fractions). It wasn't until the 16th and 17th centuries that algebra and infinitesimal calculus were introduced as new subfields of the subject. Since that time, a significant amount of progress has been made in the field of mathematics as a direct result of the interplay between new mathematical developments and scientific discoveries. The systematisation of the axiomatic approach came about as a result of the fundamental crisis that occurred in mathematics during the end of the 19th century. This resulted in a significant growth in the number of subfields within mathematics as well as the application areas for those subfields. One example of this is the Mathematics Subject Classification, which describes more than sixty first-level subfields of mathematics and lists them.