Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. 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. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). According to Mikhail B. Sevryuk, in the January2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9million, and more than 75thousand items are added to the database each year. Inaccurate predictions imply the need for improving or changing mathematical models, not that mathematics is wrong in the models themselves. [8] Since its beginning, mathematics were essentially divided into geometry, and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[b] and infinitesimal calculus were introduced as new areas. Qualities like simplicity, symmetry, completeness, and generality are particularly valued in proofs and techniques.

Analytic geometry allows the study of curves that are not related to circles and lines. Physicist Eugene Wigner has named this phenomenon the "unreasonable effectiveness of mathematics". [55] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. [22] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[23] trigonometry (Hipparchus of Nicaea, 2nd century BC),[24] and the beginnings of algebra (Diophantus, 3rd century AD).[25]. The result of a proof is called a theorem. Mathematics has a remarkable ability to cross cultural boundaries and time periods. N ( 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. 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. On the contrary, most mathematical work beyond rote calculations requires clever problem-solving and exploring novel perspectives intuitively. The history of mathematics is an ever-growing series of abstractions. This experimental validation of Einstein's theory shows that Newton's law of gravitation is only an approximation, though accurate in everyday application. [17] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800BC. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. The 1998 book Proofs from THE BOOK, inspired by Erds, is a collection of particularly succinct and revelatory mathematical arguments. 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. [36] To this day, philosophers continue to tackle questions in philosophy of mathematics, such as the nature of mathematical proof.[37].

His book, Elements, is widely considered the most successful and influential textbook of all time. [9] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. Geometry is one of the oldest branches of mathematics. 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. [f][38] On the other hand, proof assistants allow for the verification of details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the 255-page FeitThompson theorem.[g]. ( [19] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. [21] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Euclidean geometry was developed without change of methods or scope until the 17th century, when Ren Descartes introduced what is now called Cartesian coordinates. 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. A solution to any of these problems carries a 1 million dollar reward. Formerly, number theory was called arithmetic, but nowadays this term is mostly used for numerical calculations. The most notable achievement of Islamic mathematics was the development of algebra. ) Therefore, Euclid's depiction in works of art depends on the artist's imagination (see, However, some advanced methods of analysis are sometimes used; for example, methods of, For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software. [31] In English, the noun mathematics takes a singular verb. [54] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science). Analysis is further subdivided into real analysis, where variables represent real numbers and complex analysis where variables represent complex numbers. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. and later expanded to integers [34] Some just say, "Mathematics is what mathematicians do. A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. Mathematical notation led to algebra, which, roughly speaking, consists of the study and the manipulation of formulas. This approach of the foundations of the mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle.

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathmatikoi ()which at the time meant "learners" rather than "mathematicians" in the modern sense. 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). The validity of a mathematical proof is fundamentally a matter of rigor, and misunderstanding rigor is a notable cause for some common misconceptions about mathematics. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. ( This was part of a wider philosophical program known as logicism, which sees mathematics as primarily an extension of logic. The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series. [examples needed], Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially probability theory. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. {\displaystyle (\mathbb {Q} ).} [32], There is no general consensus about the exact definition or epistemological status of mathematics.

Several areas of applied mathematics have even merged with practical fields to become disciplines in their own right, such as statistics, operations research, and computer science. These symbols also contribute to rigor, both by simplifying the expression of mathematical ideas and allowing routine operations that follow consistent rules. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians. 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. This division into four main areas arithmetic, geometry, algebra, calculus[verification needed] endured until the end of the 19th century. Problems inherent in Newton's approach were solved only in the second half of the 19th century, with the formal definitions of real numbers, limits and integrals. It is often shortened to maths or, in North America, math. In the early 20th century, Kurt Gdel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic systemif powerful enough to describe arithmeticwill contain true propositions that cannot be proved. The book containing the complete proof has more than 1,000 pages. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. {\displaystyle (\mathbb {Z} )} This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas. For example, number theory occupies a central place in modern cryptography, and in physics, derivations from Maxwell's equations preempted experimental evidence of radio waves and the constancy of the speed of light. [35], In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Most of the mathematical notation in use today was invented after the 15th century, with many contributions by Leonhard Euler (17071783) in particular. Mathematicians refer to this precision of language and logic as "rigor". "[27], The word mathematics comes from Ancient Greek mthma (), meaning "that which is learnt,"[28] "what one gets to know," hence also "study" and "science". Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,[14] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. In practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences, notably deductive reasoning from assumptions. From this point-of-view, even axioms are just privileged formulas in an axiomatic system, given without being derived procedurally from other elements in the system. Z Such curves can be defined as graph of functions (whose study led to differential geometry). 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). Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented). Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories. This has resulted in several mistranslations. [further explanation needed]. [33] There is not even consensus on whether mathematics is an art or a science. [56] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. Despite mathematics' concision, many proofs require hundreds of pages to express. Mathematics (from Ancient Greek ; mthma:'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic, number theory),[1] 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).[3][4][5]. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretisation with special focus on rounding errors. 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. Number theory began with the manipulation of numbers, that is, natural numbers [20] The greatest mathematician of antiquity is often held to be Archimedes (c. 287212 BC) of Syracuse. Nonetheless, formalist concepts continue to influence mathematics greatly, to the point statements are expected by default to be expressible in set-theoretic formulas. [7], The uncanny connection between abstract mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. This technical vocabulary is both precise and compact, making it possible to mentally process complex ideas. As of 2010, the latest Mathematics Subject Classification of the American Mathematical Society recognizes hundreds of subfields, with the full classification reaching 46 pages. Most mathematical activity involves discovering and proving properties of abstract objects by pure reasoning. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Other words such as open and field are given new meanings for specific mathematical concepts. For example, the physicist Richard Feynman combined mathematical reasoning and physical insight to invent the path integral formulation of quantum mechanics. Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Other mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later. Other areas of computational mathematics include computer algebra and symbolic 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. Computational mathematics is the study of mathematical problems that are typically too large for human numerical capacity.

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. A distinction is often made between applied mathematics and mathematics oriented entirely towards abstract questions and concepts, known as pure mathematics.

The latter applies to every mathematical structure (not only algebraic ones). To allow deductive reasoning, some basic assumptions need to be admitted explicitly as axioms. [42] Experimental mathematics and computational methods like simulation also continue to grow in importance within mathematics. An example of this is the Mathematics Subject Classification, which lists more than sixty first-level areas of mathematics. [definition needed] Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;[11] the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). During the Renaissance, two more areas appeared. One prominent example is Fermat's last theorem. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.[57]. While most mathematicians don't typically concern themselves with the questions raised by Platonism, some more philosophically-minded ones do identify as Platonists, even in contemporary times.[51].

Mathematical discoveries continue to be made to this very day. Analytic geometry also makes it possible to consider spaces of higher than three dimensions. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like timedays, seasons, or years. Algebra became an area in its own right only with Franois Vite (15401603), who introduced the use of letters (variables) for representing unknown or unspecified numbers. {\displaystyle (\mathbb {N} ),} Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects is the object of universal algebra and category theory. 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). Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. During the Golden Age of Islam, especially during the 9th and 10thcenturies, mathematics saw many important innovations building on Greek mathematics. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). 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. Assisted proofs may be erroneous if the proving software has flaws and if they are lengthy, difficult to check. [57], In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. For other uses, see, Examples of shapes encountered in geometry, No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number as a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. For example, Gdel's incompleteness theorems assert, roughly speaking that, in every theory that contains the natural numbers, there are theorems that are true (that is provable in a larger theory), but not provable inside the theory. 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). The emergence of computer-assisted proofs has allowed proof lengths to further expand. Paul Erds expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The two subjects of mathematical logic and set theory have both belonged to mathematics since the end of the 19th century. , (However, several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry.) In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Calculus was expanded in the 18th century by Euler, with the introduction of the concept of a function, and many other results. Mathematicians develop mathematical hypotheses, known as conjectures, using trial and error with intuition too, similarly to scientists. Mathematicians strive to develop their results with systematic reasoning in order to avoid mistaken "theorems". This enables the extraction of quantitative predictions from experimental laws. This led to the controversy over Cantor's set theory. Kurt Gdel proved this goal was fundamentally impossible with his incompleteness theorems, which showed any formal system rich enough to describe even simple arithmetic could not guarantee its own completeness or consistency. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. [citation needed], In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. As with other divisions of mathematics though, the boundary is fluid. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry.

Today, all sciences pose problems studied by mathematicians, and conversely, results from mathematics often lead to new questions and realizations in the sciences. The term algebra is derived from the Arabic word that he used for naming one of these methods in the title of his main treatise. Sometimes, mathematicians even coin entirely new words (e.g. Thus, "applied mathematics" is a mathematical science with specialized knowledge.