The word mathematics comes from the greek

Mathematics is an area of knowledge that includes the topics of 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. There is no general consensus among mathematicians about a common definition for their academic discipline.

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 to have 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 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. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. 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 areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications.^[6][7] The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used 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. 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,^[10] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

Etymology

The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning «that which is learnt»,^[11] «what one gets to know», hence also «study» and «science». The word came to have the narrower and more technical meaning of «mathematical study» even in Classical times.^[12] Its adjective is mathēmatikós (μαθηματικός), meaning «related to learning» or «studious», which likewise further came to mean «mathematical».^[13] In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant «the mathematical art».^[11]

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.^[14]

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 change 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.^[15]

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, inherited from Greek.^[16] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.^[17]

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.^[18] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.^[19]

During the Renaissance, two more 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 differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas–arithmetic, geometry, algebra, calculus^[20]–endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics.^[21] The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.^[22]

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.^[23][10] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.^[24] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have «geometry» in their names or are otherwise commonly considered part of 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 or had not previously been considered as mathematics, such as mathematical logic and foundations.^[25]

Number theory

Number theory began with the manipulation of numbers, that is, natural numbers and later expanded to integers and rational numbers Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.^[26] Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.^[27] The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.^[28]

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat’s Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra.^[29] Another example is Goldbach’s conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.^[30]

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).^[25]

Geometry

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.^[31]

A fundamental innovation was the ancient Greeks’ introduction of the concept of proofs, which require 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 are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.^[32][33]

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][31]

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 constituted a major change of paradigm: 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. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.^[34]

Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which 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 Euclidean spaces of higher than three dimensions.^[31]

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate’s truth, this discovery has been viewed as joining Russell’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.^[35][10] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.^[36]

Today’s subareas of geometry include:^[25]

• Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
• Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
• Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
• Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
• Riemannian geometry, the study of distance properties in curved spaces.
• Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
• Topology, the study of properties that are kept under continuous deformations.
  • Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
• Discrete geometry, the study of finite configurations in geometry.
• Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
• Complex geometry, the geometry obtained by replacing real numbers with complex numbers.

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.^[38][39] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one 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’^[40] 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.^[41] Variables allow 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 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.^[42] 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. The scope of algebra thus 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.^[43] (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:^[25]

• group theory;
• field theory;
• vector spaces, whose study is essentially the same as linear algebra;
• ring theory;
• commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
• homological algebra;
• Lie algebra and Lie group theory;
• Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.^[44] 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.^[45]

Calculus and analysis

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.^[46] 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.^[47] 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:^[25]

• Multivariable calculus
• Functional analysis, where variables represent varying functions;
• Integration, measure theory and potential theory, all strongly related with probability theory on a continuum;
• Ordinary differential equations;
• Partial differential equations;
• Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

Discrete mathematics

Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.^[48] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.^[c] Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.^[49]

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.^[50] The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.^[51]

Discrete mathematics includes:^[25]

• Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
• Graph theory and hypergraphs
• Coding theory, including error correcting codes and a part of cryptography
• Matroid theory
• Discrete geometry
• Discrete probability distributions
• Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
• Discrete optimization, including combinatorial optimization, integer programming, constraint programming

Mathematical logic and set theory

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.^[52][53] 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.^[54]

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. Cantor’s work offended many mathematicians not only by considering actually infinite sets^[55] but by showing that this implies different sizes of infinity, per Cantor’s diagonal argument. This led to the controversy over Cantor’s set theory.^[56]

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are «a set is a collection of objects», «natural number is what is used for counting», «a point is a shape with a zero length in every direction», «a curve is a trace left by a moving point», etc.

This became the foundational crisis of mathematics.^[57] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.^[23] For example, in Peano arithmetic, the natural numbers are defined by «zero is a number», «each number has a unique successor», «each number but zero has a unique predecessor», and some rules of reasoning.^[58] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.^[59]

The «nature» of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called «intuition»—to guide their study and proofs. The approach allows considering «logics» (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel’s incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.^[60] This approach to the foundations of 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.^[61][62]

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.^[25] 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.^[63]

Statistics and other decision sciences

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.^[65] The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.^[d]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.^[66] Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.^[67]

Computational mathematics

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.^[68][69] Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors.^[70] Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

History

Ancient

The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,^[71] 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. 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 time—days, seasons, or years.^[72][73]

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[74] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. 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. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. 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.^[75]

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.^[76] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.^[77] His book, Elements, is widely considered the most successful and influential textbook of all time.^[78] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.^[79] 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.^[80] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),^[81] trigonometry (Hipparchus of Nicaea, 2nd century BC),^[82] and the beginnings of algebra (Diophantus, 3rd century AD).^[83]

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics.^[84] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.^[85][86]

Medieval and later

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.^[87] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.^[88] The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.^[89]

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz (1646–1716) in the 17th century. Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems. 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.^[90] In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.^[60]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 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 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.»^[91]

Symbolic notation and terminology

Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas.^[92] More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs,^[93] such as + (plus), × (multiplication), (integral), = (equal), and < (less than).^[94] All these symbols are generally grouped according to specific rules to form expressions and formulas.^[95] Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.

Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary.^[96]

Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism.^[97] Other technical terms are words of the common language that are used in an accurate meaning that may differs slightly from their common meaning. For example, in mathematics, «or» means «one, the other or both», while, in common language, it is either ambiguous or means «one or the other but not both» (in mathematics, the latter is called «exclusive or»). Finally, many mathematical terms are common words that are used with a completely different meaning.^[98] This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, «every free module is flat» and «a field is always a ring».

Relationship with sciences

Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws.^[99] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.^[100] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.^[101] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein’s general relativity, which replaced Newton’s law of gravitation as a better mathematical model.^[102]

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation.^[103] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied «durch planmässiges Tattonieren» (through systematic experimentation).^[104] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.^{[105][106][107][108]}

Pure and applied mathematics

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics.^[109] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.^[110] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece.^[111]

In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics.^[109][112] This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.^[113]

The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere.^[114][115] Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the «pure theory».^[116][117]

An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis.^[118] An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high.^[119] For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.^[120]

In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.^[121][122] The Mathematics Subject Classification has a section for «general applied mathematics» but does not mention «pure mathematics».^[25] However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.

Unreasonable effectiveness

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner.^[7] It is the fact that many mathematical theories, even the «purest» have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.^[123] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.^[124] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.^[125]

In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of the special relativity is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.^[126][127]

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of an unknown particle, and to search these particles. In both cases, these particles were discovered a few years later by specific experiments.^{[128][129][130]}

Specific sciences

Physics

Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,^[131] and is also the motivation of major mathematical developments.^[132] See above for examples of this strong interaction.

Computing

The rise of technology in the 20th century opened the way to a new science: computing.^[e] This field is closely related to mathematics in several ways. Theoretical computer science is essentially mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, graph theory, and so on.^{[citation needed]}

In return, computing has also become essential for obtaining new results. This is a group of techniques known as experimental mathematics, which is the use of experimentation to discover mathematical insights.^[133] The most well-known example is the four-color theorem, which was proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where the rule was that the mathematician should verify each part of the proof. In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer. An international team had since worked on writing a formal proof; it was finished (and verified) in 2015.^[134]

Once written formally, a proof can be verified using a program called a proof assistant.^[135] These programs are useful in situations where one is uncertain about a proof’s correctness.^[135]

A major open problem in theoretical computer science is P versus NP. It is one of the seven Millennium Prize Problems.^[136]

Biology and chemistry

Biology uses probability extensively — for example, in ecology or neurobiology.^[137] Most of the discussion of probability in biology, however, centers on the concept of evolutionary fitness.^[137]

Ecology heavily uses modeling to simulate population dynamics,^[137][138] study ecosystems such as the predator-prey model, measure pollution diffusion,^[139] or to assess climate change.^[140] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.^[141] However, there is the problem of model validation. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.^[142]

Genotype evolution can be modeled with the Hardy-Weinberg principle.^{[citation needed]}

Phylogeography uses probabilistic models.^{[citation needed]}

Medicine uses statistical hypothesis testing, run on data from clinical trials, to determine whether a new treatment works.^{[citation needed]}

Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. It turns out that the form of macromolecules in biology is variable and determines the action. Such modeling uses Euclidean geometry; neighboring atoms form a polyhedron whose distances and angles are fixed by the laws of interaction.^{[citation needed]}

Earth sciences

Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes.^{[citation needed]} Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.^{[citation needed]}

Areas of mathematics used in the social sciences include probability/statistics and differential equations (stochastic or deterministic).^{[citation needed]} These areas used in fields such as sociology, psychology, economics, finance, and linguistics.^{[citation needed]}

The fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. ‘economic man’).^[143] In this model, each individual aims solely to accumulate as much profit as possible,^[143] and always makes optimal choices using perfect information.^{[144][better source needed]} This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be very difficult to discover by a «literary» analysis.^{[citation needed]} For example, explanations of economic cycles are not trivial. Without mathematical modeling, it is hard to go beyond simple statistical observations or unproven speculation.^{[citation needed]}

However, many people have rejected or criticized the concept of Homo economicus.^{[144][better source needed]} Economists note that real people usually have limited information and often make poor choices.^{[144][better source needed]} Also, as shown in laboratory experiments, people care about fairness and sometimes altruism, not just personal gain.^{[144][better source needed]} According to critics, mathematization is a veneer that allows for the material’s scientific valorization.^{[citation needed]}

At the start of the 20th century, there was a movement to express historical movements in formulas.^{[citation needed]} In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis.^[145] Towards the end of the 19th century, Nicolas-Remi Brück [fr] and Charles Henri Lagrange [fr] had extended their analysis into geopolitics. They wanted to establish the historical existence of vast movements that took peoples to their apogee, then to their decline.^{[146][verification needed]} More recently, Peter Turchin has been working on developing cliodynamics since the 1990s.^[147] (In particular, he discovered the Turchin cycle, which predicts that violence spikes in a short cycle of ~50-year intervals, superimposed over a longer cycle of ~200–300 years.^[148])

Even so, mathematization of the social sciences is not without danger. In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences. The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses elementary mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models can be subject to controversy.^{[citation needed]}

Relationship with astrology and esotericism

Mathematics has had a close relationship with astrology for a long time. Biased by astral themes, it had motivated the study of astronomy. Renowned mathematicians have also been considered to be renowned astrologists; for example, Ptolemy, Arab astronomers, Regiomantus, Cardano, Kepler, or John Dee. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, Theodor Zwinger wrote that astrology was a mathematical science that studied the «active movement of bodies as they act on other bodies». He reserved to mathematics the need to «calculate with probability the influences [of stars]» to foresee their «conjunctions and oppositions».^[149]

These disciplines are no longer considered sciences.^[150]

Philosophy

Reality

The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.^[151]

Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.^[128]

Something becomes objective (as opposed to «subjective») as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.^[152] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics …

Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.^[153]

Proposed definitions

There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.^[154][155] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.^[154] There is not even consensus on whether mathematics is an art or a science.^[155] Some just say, «mathematics is what mathematicians do».^[154] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.^[156]

Aristotle defined mathematics as «the science of quantity» and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property «separable in thought» from real instances set mathematics apart.^[157] In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.^[158] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction.^[159] Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.^[160]

Rigor

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules,^[f] without any use of empirical evidence and intuition.^[g][161] Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics’ concision, rigorous proofs can require hundreds of pages to express. The emergence of computer-assisted proofs has allowed proof lengths to further expand,^[h][162] such as the 255-page Feit–Thompson theorem.^[i] The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.^[10]

The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.^[10]

At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell’s paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.^[10] It results that «rigor» is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a «rigorous proof» is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.^[163]

Nevertheless, the concept of «rigor» may remain useful for teaching to beginners what is a mathematical proof.^[164]

Training and practice

Education

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant.^[165]

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.^[166] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE.^[167] The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE in Eygpt.^[168] Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE).^[169] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy.^[170]

Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was L’Enseignement Mathématique, which began publication in 1899.^[171] The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications.^[172] While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.^[173]

During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.^[174] Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.^[175]

Psychology (aesthetic, creativity and intuition)

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.^[176][177] An extreme example is Apery’s theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.^[178]

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles.^[179] This aspect of mathematical activity is emphasized in recreational mathematics.

Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician’s Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.^[180] Paul Erdős expressed this sentiment more ironically by speaking of «The Book», a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. 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.^[181]

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.^[182] 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).^[128] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

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.^[59] 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. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle.^[183]

In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.^[183]

Cultural impact

Artistic expression

Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by .^{[184][185][better source needed]}

This link between frequencies and harmony was discussed in Traité de l’harmonie réduite à ses principes naturels by Jean-Philippe Rameau,^[186] a French baroque composer and music theoretician. It rests on the analysis of harmonics (noted 2 to 15 in the following figure) of a fundamental Do (noted 1); the first harmonics and their octaves sound well together.

The curve in red has a logarithmic shape, which reflects the following two phenomena:

• The pitch of the sound, which in our auditory system is proportional to the logarithm of the sound’s frequency.
• The harmonic frequencies, which are integer multiples of the fundamental frequency.

Humans, as well as some other animals, find symmetric patterns to be more beautiful.^[187] Mathematically, the symmetries of an object form a group known as the symmetry group.^[188]

For example, the group underlying mirror symmetry is the cyclic group of two elements, . A Rorschach test is a figure invariant by this symmetry, as well as a butterfly, and animal bodies more generally (at least on the surface).^{[citation needed]} Waves on the sea surface possess translation symmetry: moving one’s viewpoint by the distance between wave crests does not change one’s view of the sea.^{[citation needed]} Furthermore, fractals possess (usually approximate^{[citation needed]}) self-similarity.^{[189][190][better source needed]}

Popularization

Popular mathematics is the act of presenting mathematics without technical terms.^[191] Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract.^[192] However, popular mathematics writing can overcome this by using applications or cultural links.^[193] Despite this, mathematics is rarely the topic of popularization in printed or televised media.

Awards and prize problems

The most prestigious award in mathematics is the Fields Medal,^[194][195] established in 1936 and awarded every four years (except around World War II) to up to four individuals.^[196][197] It is considered the mathematical equivalent of the Nobel Prize.^[197]

Other prestigious mathematics awards include:^[198]

• The Abel Prize, instituted in 2002^[199] and first awarded in 2003^[200]
• The Chern Medal for lifetime achievement, introduced in 2009^[201] and first awarded in 2010^[202]
• The AMS Leroy P. Steele Prize, awarded since 1970^[203]
• The Wolf Prize in Mathematics, also for lifetime achievement,^[204] instituted in 1978^[205]

A famous list of 23 open problems, called «Hilbert’s problems», was compiled in 1900 by German mathematician David Hilbert.^[206] This list has achieved great celebrity among mathematicians,^[207] and, as of 2022, at least thirteen of the problems (depending how some are interpreted) have been solved.^[208]

A new list of seven important problems, titled the «Millennium Prize Problems», was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert’s problems. A solution to any of these problems carries a 1 million dollar reward.^[209] To date, only one of these problems, the Poincaré conjecture, has been solved.^[210]

See also

Notes

References

Bibliography

Further reading

Etymology

The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means “what one learns”, “what one gets to know”, hence also “study” and “science”, and in modern Greek just “lesson”. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean “to learn”. In Greece, the word for “mathematics” came to have the narrower and more technical meaning “mathematical study”, even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning “related to learning” or “studious”, which likewise further came to mean “mathematical”. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant “the mathematical art”.

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: a particularly notorious one is Saint Augustine’s warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter pluralmathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning 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 the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

Definitions of mathematics

Main article: Definitions of mathematics

Aristotle defined mathematics as “the science of quantity”, and this definition prevailed until the 18th century. Starting 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. Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals. There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some just say, “Mathematics is what mathematicians do.”

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. All have severe problems, none has widespread acceptance, and no reconciliation seems possible.

An early definition of mathematics in terms of logic was Benjamin Peirce’s “the science that draws necessary conclusions” (1870). In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell’s “All Mathematics is Symbolic Logic” (1903).

Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is “Mathematics is the mental activity which consists in carrying out constructs one after the other.”  A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as “the science of formal systems”. A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of “a self-evident truth”. In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

What is Mathematics?

The word Mathematics comes from the Greek word máthēma means knowledge, study, learning.

It includes the study of such topics as:

• Quantity (Number theory)
• Structure (Algebra)
• Space (Geometry)
• Change (Mathematical Analysis)

Simply, The Science or study of numbers, quantities or shapes.

It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter.

Maths is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

A German Mathematician Johann Carl Friedrich Gauss said: Mathematics is the father of all sciences.

Who is the father of mathematics?

Archimedes is known as the Father Of Mathematics.

Applied vs Pure Mathematics

If you want to keep studying maths, you might have to choose between applied and pure mathematics.

But what’s the difference?

The Main Differences

The easiest way to think of it is that pure maths is maths done for its own sake, while applied maths is maths with a practical use. But in fact, it’s not that simple, because even the most abstract maths can have unexpected applications. For example, the branch of mathematics known as “number theory” was once considered one of the most “useless”, but now plays a vital part in computer encryption systems. If you’ve ever bought something online, you can thank number theorists for letting you do it safely.

You could also think about how maths relates to other subjects and to the real world. Applied maths tries to model, predict and explain things in the real world: for example, one area of applied mathematics is fluid mechanics, which analyses how fluids are affected by forces. Other examples of applied maths might be statistics or probability theory.

Pure maths, on the other hand, is separate from the physical world. It solves problems, finds facts and answers questions that don’t depend on the world around us, but on the rules of mathematics itself.

Unfortunately, there is no perfect way to decide what pure maths is and what applied maths is. Even mathematicians can’t agree on it!

Applied Mathematics

It is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.

Pure Mathematics

It is the study of the basic concepts and structures that underlie mathematics. Its purpose is to search for a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry.

Branches of Mathematics

Different Branches of Mathematics

Arithmetic

“Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.”– Bertrand Russell

It is one of the most important branch because its fundamentals are used in daily life.

• Arithmetic deals with numbers and their applications in many ways. Addition, subtraction, multiplication, and division form its basic groundwork as they are used to solve a large number of questions and progress into more complex concepts like exponents, limits, and many other types of calculations.

  Algebra

“The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.”– Rudolf Clausius

A broad field and a fascinating branch of mathematics i.e., ALGEBRA , it involves complicated solutions and formulas to derive answers to the problems.

Algebra deals with solving generic algebraic expressions and manipulating them to arrive at results. Unknown quantities denoted by alphabets that form a part of an equation are solved for and the value of the variable is determined.

Geometry

“The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.”-Isaac Newton

Do you curious about the shapes and sizes of various objects??

Then Geometry is the branch where you must explore.

• Geometry dealing with the shape, sizes, and volumes of figures.

It is a practical branch of mathematics that focuses on the study of polygons, shapes, and geometric objects in both two-dimensions and three-dimensions. Congruence of objects is studied at the same time focussing on their special properties and calculation of their area, volume, and perimeter.

Trigonometry

«A surprise trigonometry quiz that everyone in class fails? Must be in the Lord’s plan to give us challenges.»-Nicholas Sparks

It is derived from Greek words “trigonon” meaning triangle and “metron” meaning “measure”.

Trigonometry focuses on studying angles and sides of triangles to measure the distance and length. Trigonometry is a study of the correlation between the angles and sides of the triangle. It is all about different triangles and their properties!

Calculus

“Calculus is the most powerful weapon of thought yet devised by the wit of man.”– Wallace B. Smith

It is one of the advanced branches of mathematics and studies the rate of change. Earlier maths could only work on static objects but with calculus, mathematical principles began to be applied to objects in motion.

A branch with mind-numbing questions, calculus is an interesting concept introduced to students at a later stage of their study in mathematics.

Probability and Statistics

«Facts are stubborn, but Statics are more pliable» -Mark Twain

«Probability theory is nothing but common sense reduced to calculations»-Pierre Simon Laplace

The abstract branch of mathematics, probability and statistics use mathematical concepts to predict events that are likely to happen and organize, analyze, and interpret a collection of data.

The scope of this branch involves studying the laws and principles governing numerical data and random events. Presenting an interesting study, statistics, and probability is a branch full of surprises.

Number Theory

“Mathematics is the queen of the sciences, and number theory is the queen of mathematics.”– Carl Friedrich Gauss

Number theory is one of the oldest branches of Mathematics which established a relationship between numbers belonging to the set of real numbers.

The basic level of Number Theory includes introduction to properties of integers like addition, subtraction, multiplication, modulus and builds up to complex systems like cryptography, game theory and more.

Topology

“The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.”– Pavel Sergeevich Aleksandrov

Topology is a much recent addition into the branches of Mathematics list.Its application can be observed in differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis.

It is concerned with the deformations in different geometrical shapes under stretching, crumpling, twisting and bedding. Deformations like cutting and tearing are not included in topologies.

Advanced Branches of Mathematics

These branches are studied at an advanced level and involve complex concepts that need strong computational skills. Such advanced branches are listed below:

• Linear Algebra

• Numerical Analysis

• Operation Research

• Game Theory

• Real Analysis

• Complex Analysis

• Cartesian Geometry

• Combinatorics

• Differential Equations

• Integral Equations

Linear Algebra

«Mathematics is not about the numbers, equations, computations, or algorithms; it is about Understand» -William Paul Thurston

It is a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations.

Numerical Analysis

«I don’t like to end my talk with a 700 million dollar loss, even if it shows the importance of Numerical Analysis» -Richard A. Falk

It is the branch of mathematics that deals with the development and use of numerical methods for solving problems.

Operation Research

«So never lose an opportunity of urging a practical beginning, however small, for it’s wonderful how often in such matters the mustard-seed germinates and roots» -Florence Nightingale

A method of mathematically based analysis for providing a quantitive basis for management decisions.

«Operation Research is concerned with optimal decision making in and modelling of deterministic and probabilistic systems that originate from real life» -Hiller and Lieberman

Game Theory

«In terms of the game theory, we might say the Universe is so constituted as to maximize play. The best games are not those in which all goes smoothly and steadily toward a certain conclusion, but those in which the outcome is always in doubt.» -George B. Leonard

It is the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant’s choice of action depends critically on the actions of other participants. Game theory has been applied to contexts in war, business, and biology.

Real Analysis

«The ultimate authority must always rest with the individual’s own reason and critical analysis.» -Dalai Lama

It is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions.

It includes convergence, limits, continuity, smoothness, differentiability and integrability.

Complex Analysis

«The more complex the world situation becomes, the more scientific and rational analysis, you have to have, the less you can do with simple good will and sentiment.» -Reinhold Niebuhr

Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.

Cartesian Geometry

«When you corrdinate your mind and body, you have unlimited access to the wisdom of the universe.» -Koichi Tohei

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions).

Combinatorics

Combinatorics, also called combinatorial mathematics, the field of mathematics concerned with problems of selection, arrangement, and operation within a finite or discrete system. Included is the closely related area of combinatorial geometry.

Differential Equations

«Science is a differential equation. Religion is a boundary condition.» -Alan Turing

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.

In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Integral Equations

«Integral Reality is the world’s transparency, a perceiving of the world as truth; a mutual perceiving and imparting of the truth of the world and of man and of all that transluces both» -Jean Gebser

In mathematics, integral equations are equations in which an unknown function appears under an integral sign.

There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green’s function, Fredholm theory, and Maxwell’s equations.

Mathematics (from Greek μάθημα máthēma «knowledge, study, learning») is the study of quantity, space, structure, and change.^[2][3] Mathematicians seek out patterns^[4][5] and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic. Since the pioneering work of Giuseppe Peano (1858-1932), David Hilbert (1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100^{[citation needed]}, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.^[6]

The mathematician Benjamin Peirce (1809-1880) called mathematics «the science that draws necessary conclusions».^[7] David Hilbert said of mathematics: «We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.»^[8] Albert Einstein (1879-1955) stated that «as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality».^[9]

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.^[10]

Etymology

The word «mathematics» comes from the Greek μάθημα (máthēma), which means in ancient Greek what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson.

The word máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn.

The word «mathematics» in Greek came to have the narrower and more technical meaning «mathematical study», even in Classical times.^[11] Its adjective is μαθηματικός (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art. 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: a particularly notorious one is Saint Augustine’s warning that Christians should beware of «mathematici» meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384-322BC), and meaning 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 the Greek.^[12] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,^[13] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.^[14] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.^{[citation needed]} Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.^{[citation needed]}

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[15] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.^[16]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 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 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs.»^[17]

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today’s string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.^[18] Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the «purest» mathematics often turns out to have practical applications is what Eugene Wigner has called «the unreasonable effectiveness of mathematics».^[19] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.^[20] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid’s proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. 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. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.^[21] Mathematicians often strive to find proofs that are particularly elegant, proofs from «The Book» of God according to Paul Erdős.^[22][23] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century.^[24] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.^[25] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language can be difficult to understand for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and integrable have precise meanings in mathematics. Additionally, shorthand phrases such as «iff» for «if and only if» belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as «rigor».

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken «theorems», based on fallible intuitions, of which many instances have occurred in the history of the subject.^[26] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.^[27]

Axioms in traditional thought were «self-evident truths», but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert’s program to put all of mathematics on a firm axiomatic basis, but according to Gödel’s incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.^[28]

Fields of mathematics

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase «crisis of foundations» describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.^[29] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor’s set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel’s incompleteness theorems which (informally) imply that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science^{[citation needed]}, as well as to Category Theory.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. A famous problem is the «P=NP?» problem, one of the Millennium Prize Problems.^[30] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Pure mathematics

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers («whole numbers») and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat’s Last Theorem. The twin prime conjecture and Goldbach’s conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers («fractions»). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of «infinity». Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Combinatorics	Number theory	Group theory	Graph theory	Order theory

Space

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry was developed to solve problems in number theory and functional analysis but now is pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry	Measure theory

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory	Complex analysis

Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, «applied mathematics» is a mathematical science with specialized knowledge. The term «applied mathematics» also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation, study, and use of mathematical models in science, engineering, and other areas of mathematical practice.

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. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) «create data that makes sense» with random sampling and with randomized experiments;^[31] the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians «make sense of the data» using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.^[32]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, a designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.^[33] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.^[34]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics	Fluid dynamics	Numerical analysis	Optimization	Probability theory	Statistics	Cryptography
Mathematical finance	Game theory	Mathematical biology	Mathematical chemistry	Mathematical economics	Control theory

Mathematics as profession

The best-known award in mathematics is the Fields Medal,^[35][36] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science’s Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field.

A famous list of 23 open problems, called «Hilbert’s problems», was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the «Millennium Prize Problems», was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert’s problems.

Mathematics as science

Carl Friedrich Gauss referred to mathematics as «the Queen of the Sciences».^[38] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a «field of knowledge», and this was the original meaning of «science» in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of «science» to natural science follows the rise of Baconian science, which contrasted «natural science» to scholasticism, the Aristotelean method of inquiring from first principles. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology, biology, or physics. Albert Einstein stated that «as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.«^[9]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.^[39] However, in the 1930s Gödel’s incompleteness theorems convinced many mathematicians^[who?] that mathematics cannot be reduced to logic alone, and Karl Popper concluded that «most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.»^[40] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.^[41] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.^{[citation needed]}

The opinions of mathematicians on this matter are varied. Many mathematicians^[who?] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others^[who?] feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.^{[citation needed]}

See also

Notes

References

Further reading

External links

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from

Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, and predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Other fields of knowledge, such as the natural sciences, engineering, economics, or medicine, make use of many new mathematical discoveries.

The word «mathematics» comes from the Greek μάθημα (máthēma) meaning science, knowledge, or learning, and μαθηματικός (mathēmatikós), meaning fond of learning. It is often abbreviated maths in Commonwealth English and math in North American English.

History

Main article: History of mathematics

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought.
In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Arithmetic (e.g., addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.

From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.

Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 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 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proof.»

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today’s string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the «purest» mathematics often turns out to have practical applications is what Eugene Wigner has called «the unreasonable effectiveness of mathematics.»

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid’s proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. 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.

Notation, language, and rigor

Main article: Mathematical notation

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of computer science, more often now called syntax) and encodes information that would be difficult to write in any other way.

Mathematical language also is hard for beginners. Even common words, such as or and only, have more precise meanings than in everyday speech. Mathematicians, like lawyers, strive to be as unambiguous as possible. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as «homeomorphism» and integrable. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as «rigor».

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken ‘theorems’, based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis). The level of rigor expected in mathematics has varied over time; the Greeks expected detailed arguments, but by the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since errors can be made in a computation, is such a proof sufficiently rigorous?

Axioms in traditional thought were ‘self-evident truths’, but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert’s program to put all of mathematics on a firm axiomatic basis, but according to Gödel’s incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as «the Queen of the Sciences».

If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [1]

In any case, mathematics shares much in common with many fields in the physical sciences, notably
the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).

The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.

The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.

The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. It is used in all the sciences. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically much too large for a human’s capacity; it includes the study of rounding errors or other sources of error in computation.

Major themes in mathematics

An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists.

Quantity

Quantity starts with counting and measurement.

Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

Number – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

Pinning down ideas of size, symmetry, and mathematical structure.

		File:Rubik float.png		File:Lattice of the divisibility of 60.png
Arithmetic	Number theory	Abstract algebra	Group theory	Order theory

Monoids – Rings – Fields – Linear algebra – Algebraic geometry– Universal algebra

Space

A more visual approach to mathematics.

File:Pythagorean.png	File:Taylorsine.png
Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry

Algebraic geometry – Differential topology – Algebraic topology – Linear algebra – Combinatorial geometry – Manifolds

Change

Ways to express and handle change in mathematical functions, and changes between numbers.

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory

Analysis – Real analysis – Complex analysis – Functional analysis – Special functions – Measure theory – Fourier analysis – Calculus of variations

Foundations and methods

Approaches to understanding the nature of mathematics.

Foundations of mathematics – Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Proof theory – Model theory – Reverse mathematics

Discrete mathematics

Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.

Computability theory – Computational complexity theory – Information theory

Applied mathematics

Applied mathematics uses the full knowledge of mathematics to solve real-world problems.

Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography

Important theorems

These theorems have interested mathematicians and non-mathematicians alike.

See list of theorems for more

Pythagorean theorem – Fermat’s last theorem – Gödel’s incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor’s diagonal argument – Four color theorem – Zorn’s lemma – Euler’s identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more

These are some of the major unsolved problems in mathematics.

Goldbach’s conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics

History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Mathematical tools

Old:

• Abacus
• Napier’s bones, slide rule
• Ruler and compass
• Mental calculation

New:

• Calculators and computers
• Programming languages
• Computer algebra systems (listing)
• Internet shorthand notation
• statistical analysis software
  • SPSS
  • SAS programming language
  • R programming language

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:

• misunderstanding of the implications of mathematical rigour;
• attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
• lack of familiarity with, and therefore underestimation of, the existing literature.

The case of Kurt Heegner‘s work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing ‘amateur’ work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.

Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called «points» and «lines» that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also

• Mathematical game
• Mathematical problem
• Mathematical puzzle
• Numerical cognition
• Puzzle
• Where Mathematics Comes From

References

• Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
• Courant, R. and H. Robbins, What Is Mathematics? (1941);
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
• Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
• Gullberg, Jan, Mathematics—From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
• Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
• Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
• Pappas, Theoni, The Joy Of Mathematics (1989).

External links

• Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
• Some mathematics applets, at MIT
• Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics.
• Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
• Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
• Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
• Planet Math. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
• Mathforge. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
• Young Mathematicians Network (YMN). A math-blog «Serving the Community of Young Mathematicians». Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc…
• Metamath. A site and a language, that formalize math from its foundations.
• Math in the Movies. A guide to major motion pictures with scenes of real mathematics
• Mathematics in fiction. Links to works of fiction that refer to mathematics or mathematicians.
• Math Help Forum. A forum, for math help, math discussion and debate.
• S.O.S. Mathematics Cyberboard a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
• Mathematician Bibliography. Extensive history and quotes from all famous mathematicians.
• Maths Is Fun Fun and informative maths resources for all ages . Also includes a Maths Is Fun Forum, where you can ask for help, discuss various aspects of maths and learn about new topics.
• Maths News. Mathematics news, articles and books.