Mathematics is an area of knowledge that includes topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3][4] respectively.
Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. 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.[5]
Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. The fundamental truths of mathematics are independent from any scientific experimentation, although mathematics is extensively used for modeling phenomena. 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.[6][7] 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).
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[8] Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new areas of the subject. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both.[9] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. This can be seen, for example, in the contemporary Mathematics Subject Classification, which lists more than 60 first-level areas of mathematics.
The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt,"[10] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[11] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art."
Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.[12]
In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[13]
The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[14] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[15]
Main article: Geometry
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Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.[27]
A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[28][29]
The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.[b][27]
Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This was a major change of paradigm, since instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates (which are numbers). This allows one to use algebra (and later, calculus) to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.[30]
Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose study led to differential geometry). They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider spaces of higher than three dimensions.[27]
In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning the truth of that postulate, this discovery has been viewed as joining Russel's paradox in revealing the foundational crisis of mathematics.This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.[31] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space.[32]
In the present day, the subareas of geometry include:
Main article: Algebra
Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.[33][34] The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term algebra is derived from the Arabic word al-jabr meaning "the reunion of broken parts"[35] that he used for naming one of these methods in the title of his main treatise.
Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.[36] This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.
Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.[37] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.[38] (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)
Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:
The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.[39] The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.[40]
Main articles: Calculus and Mathematical analysis
Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.[41] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.[42] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.
Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:
