Mathematics is a field of study that discovers and organizes methods,theories andtheorems that are developed andproved for the needs ofempirical sciences and mathematics itself. There are many areas of mathematics, which includenumber theory (the study of numbers),algebra (the study of formulas and related structures),geometry (the study of shapes and spaces that contain them),analysis (the study of continuous changes), andset theory (presently used as a foundation for all mathematics).
Mathematics involves the description and manipulation ofabstract objects that consist of eitherabstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, calledaxioms. Mathematics uses purereason toprove properties of objects, aproof consisting of a succession of applications ofdeductive rules to already established results. These results include previously provedtheorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.[1]
Mathematics is essential in thenatural sciences,engineering,medicine,finance,computer science, and thesocial sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such asstatistics andgame theory, are developed in close correlation with their applications and are often grouped underapplied mathematics. Other areas are developed independently from any application (and are therefore calledpure mathematics) but often later find practical applications.[2][3]
Before theRenaissance, mathematics was divided into two main areas:arithmetic, regarding the manipulation of numbers, andgeometry, regarding the study of shapes.[7] Some types ofpseudoscience, such asnumerology andastrology, were not then clearly distinguished from mathematics.[8]
During the Renaissance, two more areas appeared.Mathematical notation led toalgebra which, roughly speaking, consists of the study and the manipulation offormulas.Calculus, consisting of the two subfieldsdifferential calculus andintegral calculus, is the study ofcontinuous functions, which model the typicallynonlinear relationships between varying quantities, as represented byvariables. This division into four main areas—arithmetic, geometry, algebra, and calculus[9]—endured until the end of the 19th century. Areas such ascelestial mechanics andsolid mechanics were then studied by mathematicians, but now are considered as belonging to physics.[10] The subject ofcombinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.[11]
At the end of the 19th century, thefoundational crisis in mathematics and the resulting systematization of theaxiomatic method led to an explosion of new areas of mathematics.[12][6] The 2020Mathematics Subject Classification contains no less thansixty-three first-level areas.[13] Some of these areas correspond to the older division, as is true regardingnumber theory (the modern name forhigher 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 asmathematical logic andfoundations.[14]
This is theUlam spiral, which illustrates the distribution ofprime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximateindependence between being prime and being a value of a quadratic polynomial, a conjecture now known asHardy 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 aslines,angles andcircles, which were developed mainly for the needs ofsurveying andarchitecture, but has since blossomed out into many other subfields.[20]
A fundamental innovation was the ancient Greeks' introduction of the concept ofproofs, which require that every assertion must beproved. For example, it is not sufficient to verify bymeasurement 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 byEuclid around 300 BC in his bookElements.[21][22]
Euclidean geometry was developed without change of methods or scope until the 17th century, whenRené Descartes introduced what is now calledCartesian coordinates. This constituted a majorchange of paradigm: Instead of definingreal numbers as lengths ofline segments (seenumber line), it allowed the representation of points using theircoordinates, 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, andanalytic geometry, which uses coordinates systemically.[23]
In the 19th century, mathematicians discoverednon-Euclidean geometries, which do not follow theparallel postulate. By questioning that postulate's truth, this discovery has been viewed as joiningRussell's paradox in revealing thefoundational 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.[24][6] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties thatdo not change under specific transformations of thespace.[25]
Projective geometry, introduced in the 16th century byGirard Desargues, extends Euclidean geometry by addingpoints at infinity at whichparallel 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 toparallelism and independent from the concept of length.
Algebra is the art of manipulatingequations and formulas.Diophantus (3rd century) andal-Khwarizmi (9th century) were the two main precursors of algebra.[27][28] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.[29] Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side.[30] The termalgebra is derived from theArabic wordal-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title ofhis main treatise.[31][32]
Algebra became an area in its own right only withFrançois Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.[33] Variables allow mathematicians to describe the operations that have to be done on the numbers represented usingmathematical formulas.[34]
Until the 19th century, algebra consisted mainly of the study oflinear equations (presentlylinear algebra), and polynomial equations in a singleunknown, which were calledalgebraic 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 asmatrices,modular integers, andgeometric transformations), on which generalizations of arithmetic operations are often valid.[35] The concept ofalgebraic structure addresses this, consisting of aset 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 calledmodern algebra orabstract algebra, as established by the influence and works ofEmmy Noether.[36]
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:[14]
ACauchy sequence consists of elements such that all subsequent terms of a term 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 mathematiciansNewton andLeibniz.[39] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century byEuler with the introduction of the concept of afunction and many other results.[40] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.[41]
Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications
A diagram representing a two-stateMarkov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.
Discrete mathematics, broadly speaking, is the study of individual,countable mathematical objects. An example is the set of all integers.[42] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.[c]Algorithms—especially theirimplementation andcomputational complexity—play a major role in discrete mathematics.[43]
Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements orsubsets of a givenset; 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 ofgeometric shapes.
TheVenn 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.[46][47] Before this period, sets were not considered to be mathematical objects, andlogic, although used for mathematical proofs, belonged tophilosophy and was not specifically studied by mathematicians.[48]
BeforeCantor's study ofinfinite sets, mathematicians were reluctant to consideractually infinite collections, and consideredinfinity to be the result of endlessenumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets[49] but by showing that this implies different sizes of infinity, perCantor's diagonal argument. This led to thecontroversy over Cantor's set theory.[50] In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuringmathematical rigour.[51]
This became the foundational crisis of mathematics.[52] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside aformalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.[12] For example, inPeano 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.[53] Thismathematical abstraction from reality is embodied in the modern philosophy offormalism, as founded byDavid Hilbert around 1910.[54]
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 everyconsistentformal 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.[55] This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led byBrouwer, who promotedintuitionistic logic, which explicitly lacks thelaw of excluded middle.[56][57]
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 especiallyprobability theory. Statisticians generate data withrandom sampling or randomizedexperiments.[60]
The wordmathematics comes from theAncient Greek wordmáthēma (μάθημα), meaning'something learned, knowledge, mathematics', and the derived expressionmathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning'mathematical science'. It entered the English language during theLate Middle English period through French and Latin.[66]
Similarly, one of the two main schools of thought inPythagoreanism was known as themathēmatikoi (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 ofarithmetic and geometry. By the time ofAristotle (384–322 BC) this meaning was fully established.[67]
In Latin and English, until around 1700, the termmathematics 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 ofmathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.[68]
The apparentplural form in English goes back to the Latinneuter pluralmathematica (Cicero), based on the Greek pluralta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjectivemathematic(al) and formed the nounmathematics anew, after the pattern ofphysics andmetaphysics, inherited from Greek.[69] In English, the nounmathematics takes a singular verb. It is often shortened tomaths[70] or, in North America,math.[71]
Ancient
The Babylonian mathematical tabletPlimpton 322, dated to 1800 BC
In addition to recognizing how tocount 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 theBabylonians 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 fromMesopotamia andEgypt are from 2000 to 1800 BC.[75] Many early texts mentionPythagorean triples and so, by inference, thePythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics thatelementary arithmetic (addition,subtraction,multiplication, anddivision) first appear in the archaeological record. The Babylonians also possessed a place-value system and used asexagesimal numeral system which is still in use today for measuring angles and time.[76]
In the 6th century BC,Greek mathematics began to emerge as a distinct discipline and someAncient Greeks such as thePythagoreans appeared to have considered it a subject in its own right.[77] 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.[78] His book,Elements, is widely considered the most successful and influential textbook of all time.[79] The greatest mathematician of antiquity is often held to beArchimedes (c. 287 – c. 212 BC) ofSyracuse.[80] He developed formulas for calculating the surface area and volume ofsolids of revolution and used themethod of exhaustion to calculate thearea under the arc of aparabola with thesummation of an infinite series, in a manner not too dissimilar from modern calculus.[81] Other notable achievements of Greek mathematics areconic sections (Apollonius of Perga, 3rd century BC),[82]trigonometry (Hipparchus of Nicaea, 2nd century BC),[83] and the beginnings of algebra (Diophantus, 3rd century AD).[84]
The numerals used in theBakhshali manuscript, dated between the 2nd century BC and the 2nd century AD
During theGolden 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 ofalgebra. Other achievements of the Islamic period include advances inspherical trigonometry and the addition of thedecimal point to the Arabic numeral system.[88] Many notable mathematicians from this period were Persian, such asAl-Khwarizmi,Omar Khayyam andSharaf al-Dīn al-Ṭūsī.[89] The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.[90]
During theearly modern period, mathematics began to develop at an accelerating pace inWestern Europe, with innovations that revolutionized mathematics, such as the introduction of variables andsymbolic notation byFrançois Viète (1540–1603), the introduction oflogarithms byJohn Napier in 1614, which greatly simplified numerical calculations, especially forastronomy andmarine navigation, the introduction of coordinates byRené Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus byIsaac Newton (1643–1727) andGottfried Leibniz (1646–1716).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.[91]
Perhaps the foremost mathematician of the 19th century was the German mathematicianCarl Gauss, who made numerous contributions to fields such as algebra, analysis,differential geometry,matrix theory, number theory, andstatistics.[92] In the early 20th century,Kurt Gödel transformed mathematics by publishinghis 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.[55]
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics andscience, 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 theBulletin of the American Mathematical Society, "The number of papers and books included in theMathematical 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."[93]
Mathematical notation is widely used in science andengineering for representing complexconcepts andproperties in a concise, unambiguous, and accurate way. This notation consists ofsymbols used for representingoperations, unspecified numbers,relations and any other mathematical objects, and then assembling them intoexpressions and formulas.[94] More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generallyLatin orGreek letters, and often includesubscripts. Operation and relations are generally represented by specificsymbols orglyphs,[95] such as+ (plus),× (multiplication), (integral),= (equal), and< (less than).[96] All these symbols are generally grouped according to specific rules to form expressions and formulas.[97] Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role ofnoun phrases and formulas play the role ofclauses.
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 rigorousdefinitions that provide a standard foundation for communication. An axiom orpostulate 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 aconjecture. Through a series of rigorous arguments employingdeductive reasoning, a statement that isproven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called alemma. A proven instance that forms part of a more general finding is termed acorollary.[98]
Numerous technical terms used in mathematics areneologisms, such aspolynomial andhomeomorphism.[99] 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.[100] 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, "everyfree module isflat" and "afield is always aring".
Relationship with sciences
Mathematics is used in mostsciences formodeling phenomena, which then allows predictions to be made from experimental laws.[101] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.[102] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.[103] For example, theperihelion precession of Mercury could only be explained after the emergence ofEinstein'sgeneral relativity, which replacedNewton's law of gravitation as a better mathematical model.[104]
There is still aphilosophical 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 isfalsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing acounterexample. Similarly as in science,theories and results (theorems) are often obtained fromexperimentation.[105] 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).[106] However, some authors emphasize that mathematics differs from the modern notion of science by notrelying on empirical evidence.[107][108][109][110]
Until the 19th century, the development of mathematics in the West was mainly motivated by the needs oftechnology and science, and there was no clear distinction between pure and applied mathematics.[111] 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 theplanets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.[112] However, a notable exception occurred with the tradition ofpure mathematics in Ancient Greece.[113] The problem ofinteger factorization, for example, which goes back toEuclid in 300 BC, had no practical application before its use in theRSA cryptosystem, now widely used for the security ofcomputer networks.[114]
In the 19th century, mathematicians such asKarl Weierstrass andRichard Dedekind increasingly focused their research on internal problems, that is,pure mathematics.[111][115] This led to split mathematics intopure mathematics andapplied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.[116]
The aftermath ofWorld War II led to a surge in the development of applied mathematics in the US and elsewhere.[117][118] 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".[119][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.[124][125] The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".[14] However, these terms are still used in names of someuniversity departments, such as at theFaculty of Mathematics at theUniversity of Cambridge.
Unreasonable effectiveness
Theunreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicistEugene Wigner.[3] 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.[126] 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 definition and study of non-Euclidean geometries, spaces of dimension higher than three andmanifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century,Albert Einstein developed thetheory of relativity that uses fundamentally these concepts. In particular,spacetime ofspecial relativity is a non-Euclidean space of dimension four, and spacetime ofgeneral relativity is a (curved) manifold of dimension four.[129][130]
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of thepositron and thebaryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknownparticle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.[131][132][133]
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,[134] and is also considered to be the motivation of major mathematical developments.[135]
Biology uses probability extensively in fields such as ecology orneurobiology.[140] Most discussion of probability centers on the concept ofevolutionary fitness.[140] Ecology heavily uses modeling to simulatepopulation dynamics,[140][141] study ecosystems such as the predator-prey model, measure pollution diffusion,[142] or to assess climate change.[143] The dynamics of a population can be modeled by coupled differential equations, such as theLotka–Volterra equations.[144]
Statistical hypothesis testing, is run on data fromclinical trials to determine whether a new treatment works.[145] Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.[146]
Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics,economics,sociology,[151] andpsychology.[152]
Supply and demand curves, like this one, are a staple of mathematical economics.
Often the fundamental postulate of mathematical economics is that of the rational individual actor –Homo economicus (lit.'economic man').[153] In this model, the individual seeks to maximize theirself-interest,[153] and always makes optimal choices usingperfect information.[154] This atomistic view of economics allows it to relatively easily mathematize its thinking, because individualcalculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticise the concept ofHomo economicus. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.[155]
Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.[156]
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-longKondratiev cycle, which explains phases of economic growth or crisis.[157] Towards the end of the 19th century, mathematicians extended their analysis intogeopolitics.[158]Peter Turchin developedcliodynamics since the 1990s.[159]
Mathematization of the social sciences is not without risk. In the controversial bookFashionable Nonsense (1997),Sokal andBricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.[160] The study ofcomplex 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.[161][162]
The connection between mathematics and material reality has led to philosophical debates since at least the time ofPythagoras. The ancient philosopherPlato 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 asPlatonism. 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.[163]
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.[164] 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 afeeling of an objective existence, of a reality of mathematics ...
Nevertheless, Platonism and the concurrent views on abstraction do not explain theunreasonable effectiveness of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality).[165]
There is no general consensus about the definition of mathematics or itsepistemological 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".[166][167] A common approach is to define mathematics by its object of study.[168][169][170][171]
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.[172] 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.[173] 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.[174] For example, in lieu of a definition,Saunders Mac Lane inMathematics, form and function summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:[175]
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.[d][176][failed verification]
Mathematical reasoning requiresrigor. This means that the definitions must be absolutely unambiguous and theproofs must be reducible to a succession of applications ofinference rules,[e] without any use of empirical evidence andintuition.[f][177] 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-pageFeit–Thompson theorem.[g] The emergence ofcomputer-assisted proofs has allowed proof lengths to further expand.[h][178] The result of this trend is a philosophy of thequasi-empiricist proof that can not be considered infallible, but has a probability attached to it.[6]
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.[6]
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 andWeierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with theapodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.[6] 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 apleonasm. 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.[179]
Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.[180]
Following theDark Ages, mathematics education in Europe was provided by religious schools as part of theQuadrivium. Formal instruction inpedagogy began withJesuit schools in the 16th and 17th century. Most mathematical curricula 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 wasL'Enseignement Mathématique, which began publication in 1899.[187] 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.[188] While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.[189]
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.[190] Some students studying mathematics may develop an apprehension or fear about their performance in the subject. This is known asmathematical anxiety, and 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, by interactions with parents and teachers, and by tailored treatments for the individual.[191]
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 acomputer 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.[192][193] An extreme example isApery'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.[194]
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 solvingpuzzles.[195] This aspect of mathematical activity is emphasized inrecreational mathematics.
Mathematicians can find anaesthetic value to mathematics. Likebeauty, it is hard to define, it is commonly related toelegance, which involves qualities likesimplicity,symmetry, completeness, and generality. G. H. Hardy inA 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.[196]Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 bookProofs 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 thefast Fourier transform forharmonic analysis.[197]
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditionalliberal arts.[198] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results arecreated (as in art) ordiscovered (as in science).[131] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notes that sound well together to a Western ear are sounds whose fundamentalfrequencies of vibration are in simple ratios. For example, an octave doubles the frequency and aperfect fifth multiplies it by.[199][200]
Fractal with a scaling symmetry and a central symmetry
Humans, as well as some other animals, find symmetric patterns to be more beautiful.[201] Mathematically, the symmetries of an object form a group known as thesymmetry group.[202] For example, the group underlying mirror symmetry is thecyclic group of two elements,. ARorschach test is a figure invariant by this symmetry,[203] as arebutterfly and animal bodies more generally (at least on the surface).[204] 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.[205]Fractals possessself-similarity.[206][207]
Popular mathematics is the act of presenting mathematics without technical terms.[208] Presenting mathematics may be hard since the general public suffers frommathematical anxiety and mathematical objects are highly abstract.[209] However, popular mathematics writing can overcome this by using applications or cultural links.[210] Despite this, mathematics is rarely the topic of popularization in printed or televised media.
The most prestigious award in mathematics is theFields Medal,[211][212] established in 1936 and awarded every four years (except aroundWorld War II) to up to four individuals.[213][214] It is considered the mathematical equivalent of theNobel Prize.[214]
Other prestigious mathematics awards include:[215]
A famous list of 23open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[223] This list has achieved great celebrity among mathematicians,[224] and at least thirteen of the problems (depending how some are interpreted) have been solved.[223]
A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, theRiemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[225] To date, only one of these problems, thePoincaré conjecture, has been solved by the Russian mathematicianGrigori Perelman.[226]
^For example, logic belongs to philosophy sinceAristotle. Circa the end of the 19th century, thefoundational crisis of mathematics implied developments of logic that are specific to mathematics. This allowed eventually the proof of theorems such asGödel's theorems. Since then,mathematical logic is commonly considered as an area of mathematics.
^This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, withoutcomputers andproof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.
^This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
^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.
^For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software
^Bell, E. T. (1945) [1940]. "General Prospectus".The Development of Mathematics (2nd ed.). Dover Publications. p. 3.ISBN978-0-486-27239-9.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.{{cite book}}:ISBN / Date incompatibility (help)
^Tiwari, Sarju (1992). "A Mirror of Civilization".Mathematics in History, Culture, Philosophy, and Science (1st ed.). New Delhi, India: Mittal Publications. p. 27.ISBN978-81-7099-404-6.LCCN92909575.OCLC28115124.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.
^Kleiner 2007, "History of Classical Algebra" pp. 3–5.
^Shane, David (2022)."Figurate Numbers: A Historical Survey of an Ancient Mathematics"(PDF).Methodist University. p. 20.Archived(PDF) from the original on June 5, 2024. RetrievedJune 13, 2024.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.
^Riche, 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.ISBN978-88-7699-077-9.OCLC647049731. RetrievedFebruary 8, 2024.
^Sipser, Michael (July 1992).The History and Status of the P versus NP Question. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp. 603–618.doi:10.1145/129712.129771.S2CID11678884.
^Rouaud, Mathieu (April 2017) [First published July 2013].Probability, Statistics and Estimation(PDF). p. 10.Archived(PDF) from the original on October 9, 2022. RetrievedFebruary 13, 2024.
^Marchuk, Gurii Ivanovich (April 2020)."G I Marchuk's plenary: ICM 1970".MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland.Archived from the original on November 13, 2022. RetrievedNovember 13, 2022.
^Kolachana, 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.ISBN978-981-13-7325-1.S2CID190176726.
^Archibald, Raymond Clare (January 1949). "History of Mathematics After the Sixteenth Century".The American Mathematical Monthly. Part 2: Outline of the History of Mathematics.56 (1):35–56.doi:10.2307/2304570.JSTOR2304570.
^Nickles, Thomas (2013). "The Problem of Demarcation".Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. Chicago: The University of Chicago Press. p. 104.ISBN978-0-226-05182-6.
^Kuhn, 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.
^Gozwami, Pinkimani; Singh, Madan Mohan (2019). "Integer Factorization Problem". In Ahmad, Khaleel; Doja, M. N.; Udzir, Nur Izura; Singh, Manu Pratap (eds.).Emerging Security Algorithms and Techniques. CRC Press. pp. 59–60.ISBN978-0-8153-6145-9.LCCN2019010556.OCLC1082226900.
^Peressini, Anthony (September 1999).Applying Pure Mathematics(PDF). Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers. Vol. 66. pp. S1 –S13.JSTOR188757.Archived(PDF) from the original on January 2, 2024. RetrievedNovember 30, 2022.
^Marker, Dave (July 1996)."Model theory and exponentiation".Notices of the American Mathematical Society.43 (7):753–759.Archived from the original on March 13, 2014. RetrievedNovember 19, 2022.
^Sarukkai, Sundar (February 10, 2005). "Revisiting the 'unreasonable effectiveness' of mathematics".Current Science.88 (3):415–423.JSTOR24110208.
^Wagstaff, 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 November 20, 2022. RetrievedNovember 20, 2022.
^"Curves: Ellipse".MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland.Archived from the original on October 14, 2022. RetrievedNovember 20, 2022.
^Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics".Proceedings of the American Academy of Arts and Sciences.48 (11):389–507.doi:10.2307/20022840.JSTOR20022840.
^Ginammi, Michele (February 2016). "Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω– particle".Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics.53:20–27.Bibcode:2016SHPMP..53...20G.doi:10.1016/j.shpsb.2015.12.001.
^See 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
^Salsburg, David (August 17, 1992)."Commentary"(PDF).The Use of Statistical Methods in the Analysis of Clinical Studies.46: 17.Archived(PDF) from the original on June 1, 2024. RetrievedJune 1, 2024.
^Batchelder, William H. (January 1, 2015)."Mathematical Psychology: History". In Wright, James D. (ed.).International Encyclopedia of the Social & Behavioral Sciences (Second Edition). Oxford: Elsevier. pp. 808–815.ISBN978-0-08-097087-5.Archived from the original on February 17, 2023. RetrievedSeptember 30, 2023.
^Balaguer, Mark (2016)."Platonism in Metaphysics". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Spring 2016 ed.). Metaphysics Research Lab, Stanford University.Archived from the original on January 30, 2022. RetrievedApril 2, 2022.
^SeeWhite, L. (1947). "The locus of mathematical reality: An anthropological footnote".Philosophy of Science.14 (4):289–303.doi:10.1086/286957.S2CID119887253. 189303; also inNewman, J. R. (1956).The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.
^Dorato, 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.ISBN978-0-7546-3994-7.Archived(PDF) from the original on August 17, 2023. RetrievedDecember 5, 2022.
^Mura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences".Educational Studies in Mathematics.25 (4):375–85.doi:10.1007/BF01273907.JSTOR3482762.S2CID122351146.
^Tobies, Renate; Neunzert, Helmut (2012).Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9.ISBN978-3-0348-0229-1. RetrievedJune 20, 2015.[I]t is first necessary to ask what is meant bymathematics 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.
^Ziegler, Günter M.; Loos, Andreas (November 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.ISBN978-3-319-62596-6. (Sections "What is Mathematics?" and "What is Mathematics, Really?")
^Peterson 1988, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Appel proof of theFour Color Theorem)
^Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs".Philosophy of Mathematics.42 (167 (4)). Revue Internationale de Philosophie:500–508.
^Davis, 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.
^Jones, Phillip S. (1967). "The History of Mathematical Education".The American Mathematical Monthly.74 (1). Taylor & Francis, Ltd.:38–55.doi:10.2307/2314867.JSTOR2314867.
^Alon, Noga; Goldston, Dan; Sárközy, András; Szabados, József; Tenenbaum, Gérald; Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015). Alladi, Krishnaswami; Krantz, Steven G. (eds.)."Reflections on Paul Erdős on His Birth Centenary, Part II".Notices of the American Mathematical Society.62 (3):226–247.doi:10.1090/noti1223.
^See, for exampleBertrand Russell's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in hisHistory of Western Philosophy. 1919. p. 60.
^Budden, F. J. (October 1967). "Modern mathematics and music".The Mathematical Gazette.51 (377). Cambridge University Press ({CUP}):204–215.doi:10.2307/3613237.JSTOR3613237.S2CID126119711.
^Bender, Sara (September 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.ISBN978-1-119-05751-2.
^Kissane, Barry (July 2009).Popular mathematics. 22nd Biennial Conference of The Australian Association of Mathematics Teachers. Fremantle, Western Australia: Australian Association of Mathematics Teachers. pp. 125–126.Archived from the original on March 7, 2023. RetrievedDecember 29, 2022.
^"Chern Medal Award"(PDF).mathunion.org. June 1, 2009.Archived(PDF) from the original on June 17, 2009. RetrievedFebruary 21, 2022.
^"Chern Medal Award". International Mathematical Union (IMU).Archived from the original on August 25, 2010. RetrievedJanuary 23, 2022.
^"The Leroy P Steele Prize of the AMS". School of Mathematics and Statistics, University of St Andrews, Scotland.Archived from the original on November 17, 2022. RetrievedNovember 17, 2022.
Peirce, Benjamin (1881).Peirce, Charles Sanders (ed.)."Linear associative algebra".American Journal of Mathematics.4 (1–4) (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.):97–229.doi:10.2307/2369153.hdl:2027/hvd.32044030622997.JSTOR2369153. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. GoogleEprint and as an extract, D. Van Nostrand, 1882, GoogleEprint. RetrievedNovember 17, 2020..
Whittle, Peter (1994)."Almost home". InKelly, F.P. (ed.).Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28.ISBN978-0-471-94829-2.Archived from the original on December 19, 2013.
.jpg)
.jpg)
