Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is widely used to model and solve problems in science, engineering, technology, economics, and everyday life.
There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study of shapes and spaces that contain them), analysis (the quantitative study of approximation and convergence), and set theory (presently used as a foundation for all mathematics).
Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure to prove the properties of objects through , which consist of a succession of applications of to already established results. These results, called , 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.
Mathematics is essential in science and engineering; it is widely used in every field of knowledge and in everyday life, whenever data is manipulated, shapes, patterns, and structures are studied, and logical reasoning is needed. Although mathematics is widely used to model empirical phenomena, its results are established by deductive proof rather than by experiment. The relationship between mathematical truth, logic, and reality is a subject of philosophical debate. Some areas of mathematics, such as 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 but often find practical applications later.45
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Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms.
theorems
These results, called theorems, 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.
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).
This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether, and popularized by Van der Waerden's book Moderne Algebra.
Traditionally, 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.
Terms the article defines in bold, quoted in context.
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Areas of mathematics
Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the study and manipulation of numbers, and geometry, regarding the study of shapes.11 Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.12
Beginning with the Renaissance, two more areas became predominant. New mathematical notation led to modern algebra which, roughly speaking, begins with the study and manipulation of algebraic expressions. Calculus, consisting of the two subfields differential calculus and integral calculus, originated with geometry but evolved into 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, and calculus13—endured until the end of the 19th century. Other areas that were previously studied by mathematicians, such as celestial mechanics and solid mechanics, are now considered to belong to physics.14 The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the 17th century.15
At the end of the 19th century, the foundational crisis in mathematics and the systematic use of the axiomatic method led to an explosion of new areas of mathematics.168 The 2020 Mathematics Subject Classification contains 63 first-level areas.10 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 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.9
This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.
On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales, the sum of the angles of a triangle is not equal to 180°.
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 into many other subfields.22
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.2324
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 systematically.25
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 the view that the truth of the chosen axioms is not a mathematical problem.268 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.27
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 of the concept of length.
Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.2930 Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.31 Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation to the other side.32 The term algebra is derived from the Arabic word al-jabr, meaning 'the reunion of broken parts', which he used for naming one of these methods in the title of his main treatise.3334
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.35 Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.36
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.37 The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. 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,38 and popularized by Van der Waerden's book Moderne Algebra.
Some types of algebraic structures have useful and often fundamental properties in many areas of mathematics. Their study became autonomous parts of algebra, and it includes the following:9
A Cauchy sequence consists of elements such that all terms following a given point become arbitrarily close to each other as the sequence progresses (from left to right).
Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.41 It is fundamentally the study of the relationship between variables that depend continuously on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.42 Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.43
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:9
A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probabilities of flipping the state.
Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.44 Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.cAlgorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.45
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.
The Venn diagram is a commonly used method to illustrate the relations between sets.
The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.4849 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.50
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 sets51 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.52 In the same period, various areas of mathematics concluded that the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigor.53
This became the foundational crisis of mathematics.54 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.16 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.55 This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.56
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 consistentformal 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.57 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).5859
The word mathematics comes from the Ancient Greek word máthēma (μάθημα), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the Late Middle English period through French and Latin.64
Traditionally, 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.65
In Latin and 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.66
An explanation of the sigma (Σ) summation notation
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.104 More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally or letters, and often include . Operations and relations are generally represented by specific symbols or , such as + (), × (), (), = (), and < (). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of and formulas play the role of .
Relationship with sciences
Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws.111 The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.112 Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.113 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.114
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 to science, theories and results (theorems) are often obtained from experimentation. 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). However, some authors emphasize that mathematics differs from the modern notion of science by not on empirical evidence.
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.175
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.176 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 of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality).
Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.194 Comparable evidence has been unearthed for scribal mathematics training in the and then for the starting around 300 BCE. The oldest known mathematics textbook is the , dating from c. 1650 BCE in Egypt. Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized since the (c. 1500 – c. 500 BCE). In during the (618–907 CE), a mathematics curriculum was adopted for the to join the state bureaucracy.
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 .211212
Fractal with a scaling symmetry and a central symmetry
Humans, as well as some other animals, find symmetric patterns to be more beautiful.213 Mathematically, the symmetries of an object form a group known as the symmetry group.214 For example, the group underlying mirror symmetry is the cyclic group of two elements, . A Rorschach test is a figure invariant by this symmetry,215 as are butterfly and animal bodies more generally (at least on the surface).216 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. possess .
cHowever, some advanced methods of analysis are sometimes used; for example, methods of complex analysis applied to generating series.
dFor example, logic belongs to philosophy since Aristotle. Circa the end of the 19th century, the foundational crisis of mathematics implied developments of logic that are specific to mathematics. This eventually allowed the proof of theorems such as Gödel's theorems. Since then, mathematical logic is commonly considered an area of mathematics.
eThis does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible without computers and proof assistants. Even with this modern technology, it may take years of human work to write down a completely detailed proof.
fThis does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
3Hipólito, Inês Viegas (Aug 9–15, 2015). "Abstract Cognition and the Nature of Mathematical Proof". In Kanzian, Christian; Mitterer, Josef; Neges, Katharina (eds.). Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums [Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium] (PDF) (in German and English). Vol. 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. ISSN1022-3398. OCLC236026294. Archived (PDF) from the original on 2022-11-07. Retrieved 2024-01-17. ( November 5, 2022, at the )
Hazewinkel, Michiel, ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers. – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and .
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.67 In English, the noun mathematics takes a singular verb. It is often shortened to maths68 or, in North America, math.69
Ancient
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.7071
Image of Problem 14 from the Egyptian Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.72 Also, fractal geometry designs that are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.73The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.74 Megalithic structures located in Nabta Playa, Upper Egypt featured astronomy, calendar arrangements in alignment with the heliacal rising of Sirius, and supported calibration of the yearly calendar for the annual Nile flood.75 Ancient Nubians established a system of geometric rules which served as the basis for initial sunclocks. Nubians also exercised a trigonometric methodology comparable to their Egyptian counterparts.76777879
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC
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.80 The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC.81 Many early texts mention Pythagorean triples; 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 appears 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.82
By the 5th 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.83 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.84 His book, Elements, covers geometry and number theory and is widely considered the most successful and influential textbook of all time.85 Another notable mathematician of antiquity is Archimedes of Syracuse (c. 287 – c. 212 BC).86 He developed methods for calculating the surface area and volume of solids of revolution, including using the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner reminiscent of modern calculus.87 Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),88trigonometry (Hipparchus of Nicaea, 2nd century BC),89 and the beginnings of pre-modern algebra (Diophantus, 3rd century AD).90
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD
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.91 Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.9293
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.94 Many notable mathematicians from this period were Persian, such as al-Khwarizmi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.95 The Greek and Arabic mathematical texts were in turn translated into Latin during the Middle Ages and made available in Europe.96
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 attributed to François Viète (1540–1603),97 the introduction of logarithms by John Napier in 1614,98 which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of Cartesian coordinates by René Descartes (1596–1650) for reducing geometry to algebra,99 and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716).100Leonhard Euler (1707–1783) unified and built upon these innovations with a standardized terminology, and complemented them with the discovery and proofs of numerous theorems.101
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.102 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.57
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,103
The number of papers and books included in the Mathematical Reviews (MR) 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.
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.108
Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism.109 Other technical terms are words of the common language that are used in an accurate meaning that may differ 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.110 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".
Until the 19th century, there was no clear distinction between pure and applied mathematics as understood today.121 The distinction between developing mathematics for its own sake or for its applications was rather fluid: natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture, and astronomy, but both subjects quickly stood on their own. Later, Isaac Newton used infinitesimal calculus in part to help explain the movement of the planets and his law of gravitation. Moreover, since antiquity, most mathematicians were also scientists, and many scientists were also mathematicians.122 Nonetheless, the Western tradition of pure mathematics traces its roots back to ancient Greece.123 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.124
In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics.121125 This led to mathematics splitting into pure mathematics and applied mathematics, the latter often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.126
The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere.127128 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".129130
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.134135 The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".9 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.5 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.136 Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
In the 19th century, the internal development of geometry (pure mathematics) led to the definition and study of non-Euclidean geometries, spaces of dimension higher than three, and manifolds. At this time, these concepts seemed totally disconnected from physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses these concepts fundamentally. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four.139140
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 of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.141142143
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,144 and is also considered to be the motivation of major mathematical developments.145
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 such as, and especially, probability theory. Statisticians generate data with random sampling or randomized experiments.151
Biology uses probability extensively in fields such as ecology or neurobiology.154 Most discussion of probability centers on the concept of evolutionary fitness.154 Ecology heavily uses modeling to simulate population dynamics,154155 study ecosystems such as the predator-prey model, measure pollution diffusion,156 or to assess climate change.157 The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.158
Statistical hypothesis testing is run on data from clinical trials to determine whether a new treatment works.159 Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.160
Supply and demand curves, like this one, are a staple of mathematical economics.
Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows the creation of structured frameworks for testing hypotheses and analyzing complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.167
Often, one starts from a model of an individual actor – Homo economicus (lit. 'economic man'),168 and one uses mathematics to deduce the global behavior of the society or economy from the interactions between the individual actors. However, this approach is limited by the fact that some important facts cannot be taken into account by the model, such as unpredictable technological breakthroughs and the use of the model by some individuals for maximizing their profit.
At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis.169 Towards the end of the 19th century, mathematicians extended their analysis into geopolitics.170Peter Turchin developed cliodynamics in the 1990s.171
Mathematization of the social sciences is not without risk. 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.172 The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.173174
There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do".178179 A common approach is to define mathematics by its object of study.180181182183
Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. However, Aristotle also noted that 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.184 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.185 With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult.186 For example, in lieu of a definition, Saunders Mac Lane in Mathematics, Form and Function summarizes the basics of several areas of mathematics, emphasizing their interconnectedness, and observes:187
the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.
Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely logical deduction.d188
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,e without any use of empirical evidence and intuition.f189 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, such as the 255-page Feit–Thompson theorem.g The emergence of computer-assisted proofs has allowed proof lengths to further expand.h190 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.8
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.8
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 reintroduction of the axiomatic method pioneered by the ancient Greeks.8 It results that "rigor" is no longer 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 reliable.191
Nevertheless, the concept of "rigor" may remain useful for teaching beginners what a mathematical proof is.192
Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. The oldest journal addressing instruction in mathematics was L'Enseignement Mathématique, which began publication in 1899.199 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.200 While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.201
During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback and motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.202 Some students studying mathematics may develop an apprehension or fear about their performance in the subject. This is known as mathematical anxiety, and it is considered the most prominent of the disorders impacting academic performance. Mathematical 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, interactions with parents and teachers, and tailored treatments for the individual.203
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 of solving them may be a fundamental method of the solving process.204205 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.206
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.207 This aspect of mathematical activity is emphasized in recreational mathematics.
Mathematicians can find an aesthetic value in 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 aesthetics.208Paul 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 include Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.209
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.210 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).141 The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Popular mathematics is the act of presenting mathematics without technical terms.220 Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract.221 However, popular mathematics writing can overcome this by using applications or cultural links.222 Despite this, mathematics is rarely the topic of popularization in printed or televised media.
The most prestigious award in mathematics is the Fields Medal,223224 established by Canadian John Charles Fields in 1936 and awarded every four years (except around World War II) to up to four individuals.225226 It is considered the mathematical equivalent of the Nobel Prize.226
A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.235 This list has achieved great celebrity among mathematicians,236 and at least 13 of the problems (depending on how some are interpreted) have been solved.235
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.237 To date, only one of these problems, the Poincaré conjecture, has been solved.238
g
This is the length of the original paper that does not contain the proofs of some previously published auxiliary results. The book devoted to the complete proof has more than 1,000 pages.
hFor considering a large computation occurring in a proof as reliable, one generally requires two computations using independent software
11Bell, E. T. (1992) [1945]. "General Prospectus". The Development of Mathematics (2nd ed.). Dover Publications. p. 3. ISBN 978-0-486-27239-9. LCCN45010599. OCLC523284. “... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry.”
12Tiwari, Sarju (1992). "A Mirror of Civilization". Mathematics in History, Culture, Philosophy, and Science (1st ed.). New Delhi, India: Mittal Publications. p. 27. ISBN 978-81-7099-404-6. LCCN92909575. OCLC28115124. Retrieved 2025-11-13. “It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself.”
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30Kleiner 2007, "History of Classical Algebra", pp. 3–5.
31Shane, David (2022). "Figurate Numbers: A Historical Survey of an Ancient Mathematics" (PDF). Methodist University. p. 20. Archived (PDF) from the original on 2024-06-05. Retrieved 2024-06-13. “In his work, Diophantus focused on deducing the arithmetic properties of figurate numbers, such as deducing the number of sides, the different ways a number can be expressed as a figurate number, and the formulation of the arithmetic progressions.”
39Riche, Jacques (2007). "From Universal Algebra to Universal Logic". In Beziau, J. Y.; Costa-Leite, Alexandre (eds.). Perspectives on Universal Logic. Milano, Italy: Polimetrica International Scientific Publisher. pp. 3–39. ISBN 978-88-7699-077-9. OCLC647049731. Retrieved 2024-02-08.
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61Marchuk, Gurii Ivanovich (Apr 2020). "G I Marchuk's plenary: ICM 1970". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 2022-11-13. Retrieved 2022-11-13.
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70See, for example, Wilder, Raymond L.Evolution of Mathematical Concepts; an Elementary Study. passim.
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77Depuydt, Leo (Jan 1, 1998). "Gnomons at Meroë and Early Trigonometry". The Journal of Egyptian Archaeology. 84: 171–180. doi:10.2307/3822211. JSTOR3822211.
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81Joseph, George Gheverghese (2011). "The Beginnings of Written Mathematics: Mesopotamia". The Crest of the Peacock. pp. 125–176. doi:10.1515/9781400836369-006. ISBN 978-1-4008-3636-9.
93Kolachana, A.; Mahesh, K.; Ramasubramanian, K. (2019). "Use of series in India". Studies in Indian Mathematics and Astronomy. Sources and Studies in the History of Mathematics and Physical Sciences. Singapore: Springer. pp. 438–461. doi:10.1007/978-981-13-7326-8_20. ISBN 978-981-13-7325-1. S2CID190176726.
94Saliba, George (1994). A history of Arabic astronomy: planetary theories during the golden age of Islam. New York University Press. ISBN 978-0-8147-7962-0. OCLC28723059.
108Rossi, Richard J. (2006). Theorems, Corollaries, Lemmas, and Methods of Proof. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. John Wiley & Sons. pp. 1–14, 47–48. ISBN 978-0-470-04295-3. LCCN2006041609. OCLC64085024.
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119Nickles, Thomas (2013). "The Problem of Demarcation". Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. Chicago: The University of Chicago Press. p. 104. ISBN 978-0-226-05182-6.
122Kuhn, Thomas S. (1976). "Mathematical vs. Experimental Traditions in the Development of Physical Science". The Journal of Interdisciplinary History. 7 (1). The MIT Press: 1–31. doi:10.2307/202372. JSTOR202372.
123Asper, Markus (2009). "The two cultures of mathematics in ancient Greece". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. Oxford Handbooks in Mathematics. OUP Oxford. pp. 107–132. ISBN 978-0-19-921312-2. Retrieved 2022-11-18.
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137Wagstaff, Samuel S. Jr. (2021). "History of Integer Factoring" (PDF). In Bos, Joppe W.; Stam, Martijn (eds.). Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. Archived (PDF) from the original on 2022-11-20. Retrieved 2022-11-20.
138"Curves: Ellipse". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 2022-10-14. Retrieved 2022-11-20.
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155See for example Anne Laurent, Roland Gamet, Jérôme Pantel, Tendances nouvelles en modélisation pour l'environnement, actes du congrès «Programme environnement, vie et sociétés» 15–17 janvier 1996, CNRS
159Salsburg, David (Aug 17, 1992). "Commentary" (PDF). The Use of Statistical Methods in the Analysis of Clinical Studies. 46: 17. Archived (PDF) from the original on 2024-06-01. Retrieved 2024-06-01.
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176See White, L. (1947). "The locus of mathematical reality: An anthropological footnote". Philosophy of Science. 14 (4): 289–303. doi:10.1086/286957. S2CID119887253. 189303; also in Newman, J. R. (1956). The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.
177Dorato, Mauro (2005). "Why are laws mathematical?" (PDF). The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature. Ashgate. pp. 31–66. ISBN 978-0-7546-3994-7. Archived (PDF) from the original on 2023-08-17. Retrieved 2022-12-05.
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179Tobies, Renate; Neunzert, Helmut (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN 978-3-0348-0229-1. Retrieved 2015-06-20. “[I]t is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.”
180Ziegler, Günter M.; Loos, Andreas (Nov 2, 2017). Kaiser, G. (ed.). "What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it. Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer. pp. 63–77. doi:10.1007/978-3-319-62597-3_5. ISBN 978-3-319-62596-6. (Sections "What is Mathematics?" and "What is Mathematics, Really?")
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190Peterson 1988, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Appel proof of the Four Color Theorem)
191Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs". Philosophy of Mathematics. 42 (167 (4)). Revue Internationale de Philosophie: 500–508.
192Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwell, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor". International Journal of Educational Research. 93: 91–100. doi:10.1016/j.ijer.2018.10.002. S2CID149753721.
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199Jones, Phillip S. (1967). "The History of Mathematical Education". The American Mathematical Monthly. 74 (1). Taylor & Francis, Ltd.: 38–55. doi:10.2307/2314867. JSTOR2314867.
210See, for example Bertrand Russell's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his History of Western Philosophy. 1919. p. 60.
212Budden, F. J. (Oct 1967). "Modern mathematics and music". The Mathematical Gazette. 51 (377). Cambridge University Press ({CUP}): 204–215. doi:10.2307/3613237. JSTOR3613237. S2CID126119711.
215Bender, Sara (Sep 2020). "The Rorschach Test". In Carducci, Bernardo J.; Nave, Christopher S.; Mio, Jeffrey S.; Riggio, Ronald E. (eds.). The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment. Wiley. pp. 367–376. doi:10.1002/9781119547167.ch131. ISBN 978-1-119-05751-2.
216Weyl, Hermann (2015). Symmetry. Princeton Science Library. Vol. 47. Princeton University Press. p. 4. ISBN 978-1-4008-7434-7.
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Engineering is the practice of systematically applying natural science and mathematics to design and improve systems, devices, or processes that solve problems under constraints.
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems.
Mathematical term; concerning axioms used to derive theorems
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science.
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as the logical structure of valid arguments.
Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during classical and late antiquity, mostly from the 5th century BC to the 6th century AD.
Foundations of mathematics are the logical and mathematical frameworks that allow the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular.
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
Pseudoscientific divination based on the movements of the stars
Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions of celestial objects.
