Mathematics

"Math" and "Maths" redirect here. For other uses, seeMathematics (disambiguation)andMath (disambiguation).

Mathematicsis a field of study that discovers and organizes methods,theoriesandtheoremsthat are developed andprovedfor the needs ofempirical sciencesand 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 objectsthat consist of eitherabstractionsfrom nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, calledaxioms.Mathematics uses purereasontoproveproperties of objects, aproofconsisting of a succession of applications ofdeductive rulesto 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.

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 asstatisticsandgame 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.

Historically, the concept of a proof and its associatedmathematical rigourfirst appeared inGreek mathematics,most notably inEuclid'sElements.Since its beginning, mathematics was primarily divided into geometry andarithmetic(the manipulation ofnatural numbersandfractions), until the 16th and 17th centuries, when algebraandinfinitesimal calculuswere introduced as new fields. Since then, the interaction between mathematical innovations andscientific discoverieshas led to a correlated increase in the development of both.At the end of the 19th century, thefoundational crisis of mathematicsled to the systematization of theaxiomatic method,which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporaryMathematics Subject Classificationlists more than sixty first-level areas of mathematics.

Areas of mathematics

Before theRenaissance,mathematics was divided into two main areas:arithmetic,regarding the manipulation of numbers, andgeometry,regarding the study of shapes.Some types ofpseudoscience,such asnumerologyandastrology,were not then clearly distinguished from mathematics.

During the Renaissance, two more areas appeared.Mathematical notationled toalgebrawhich, roughly speaking, consists of the study and the manipulation offormulas.Calculus,consisting of the two subfieldsdifferential calculusandintegral calculus,is the study ofcontinuous functions,which model the typicallynonlinear relationshipsbetween varying quantities, as represented byvariables.This division into four main areas—arithmetic, geometry, algebra, calculus—endured until the end of the 19th century. Areas such ascelestial mechanicsandsolid mechanicswere then studied by mathematicians, but now are considered as belonging to physics.The subject ofcombinatoricshas been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.

At the end of the 19th century, thefoundational crisis in mathematicsand the resulting systematization of theaxiomatic methodled to an explosion of new areas of mathematics.The 2020Mathematics Subject Classificationcontains no less thansixty-threefirst-level areas.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 logicandfoundations.

Number theory

Main article:Number theory

This is theUlam spiral,which illustrates the distribution ofprime numbers.The dark diagonal lines in the spiral hint at the hypothesized approximateindependencebetween being prime and being a value of a quadratic polynomial, a conjecture now known asHardy and Littlewood's Conjecture F.

Number theory began with the manipulation ofnumbers,that is,natural numbers$(\mathbb {N} ),$and later expanded tointegers$(\mathbb {Z} )$andrational numbers$(\mathbb {Q} ).$Number theory was once called arithmetic, but nowadays this term is mostly used fornumerical calculations.Number theory dates back to ancientBabylonand probablyChina.Two prominent early number theorists wereEuclidof ancient Greece andDiophantusof Alexandria.The modern study of number theory in its abstract form is largely attributed toPierre de FermatandLeonhard Euler.The field came to full fruition with the contributions ofAdrien-Marie LegendreandCarl Friedrich Gauss.

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example isFermat's Last Theorem.This conjecture was stated in 1637 by Pierre de Fermat, but itwas provedonly in 1994 byAndrew Wiles,who used tools includingscheme theoryfromalgebraic geometry,category theory,andhomological algebra.Another example isGoldbach's conjecture,which asserts that every even integer greater than 2 is the sum of twoprime numbers.Stated in 1742 byChristian Goldbach,it remains unproven despite considerable effort.

Number theory includes several subareas, includinganalytic number theory,algebraic number theory,geometry of numbers(method oriented),diophantine equations,andtranscendence theory(problem oriented).

Geometry

Main article:Geometry

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,anglesandcircles,which were developed mainly for the needs ofsurveyingandarchitecture,but has since blossomed out into many other subfields.

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 bymeasurementthat, 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 byEuclidaround 300 BC in his bookElements.

The resultingEuclidean geometryis the study of shapes and their arrangementsconstructedfrom lines,planesand circles in theEuclidean plane(plane geometry) and the three-dimensionalEuclidean space.

Euclidean geometry was developed without change of methods or scope until the 17th century, whenRené Descartesintroduced what is now calledCartesian coordinates.This constituted a majorchange of paradigm:Instead of definingreal numbersas 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.

Analytic geometry allows the study ofcurvesunrelated to circles and lines. Such curves can be defined as thegraph of functions,the study of which led todifferential geometry.They can also be defined asimplicit equations,oftenpolynomial equations(which spawnedalgebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.

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 paradoxin 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.In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties thatdo not changeunder specific transformations of thespace.

Today's subareas of geometry include:

• Projective geometry,introduced in the 16th century byGirard Desargues,extends Euclidean geometry by addingpoints at infinityat whichparallel linesintersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
• Affine geometry,the study of properties relative toparallelismand independent from the concept of length.
• Differential geometry,the study of curves, surfaces, and their generalizations, which are defined usingdifferentiable functions.
• Manifold theory,the study of shapes that are not necessarily embedded in a larger space.
• Riemannian geometry,the study of distance properties in curved spaces.
• Algebraic geometry,the study of curves, surfaces, and their generalizations, which are defined usingpolynomials.
• Topology,the study of properties that are kept undercontinuous deformations.
  • Algebraic topology,the use in topology of algebraic methods, mainlyhomological algebra.
• Discrete geometry,the study of finite configurations in geometry.
• Convex geometry,the study ofconvex sets,which takes its importance from its applications inoptimization.
• Complex geometry,the geometry obtained by replacing real numbers withcomplex numbers.

Algebra

Main article:Algebra

Thequadratic formula,which concisely expresses the solutions of allquadratic equations

TheRubik's Cube groupis a concrete application ofgroup theory.

Algebra is the art of manipulatingequationsand formulas.Diophantus(3rd century) andal-Khwarizmi(9th century) were the two main precursors of algebra.Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side.The termalgebrais derived from theArabicwordal-jabrmeaning 'the reunion of broken parts' that he used for naming one of these methods in the title ofhis main treatise.

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.Variables allow mathematicians to describe the operations that have to be done on the numbers represented usingmathematical formulas.

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.The concept ofalgebraic structureaddresses this, consisting of asetwhose 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 algebraorabstract algebra,as established by the influence and works ofEmmy Noether.

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:

• group theory
• field theory
• vector spaces,whose study is essentially the same aslinear algebra
• ring theory
• commutative algebra,which is the study ofcommutative rings,includes the study ofpolynomials,and is a foundational part ofalgebraic geometry
• homological algebra
• Lie algebraandLie grouptheory
• Boolean algebra,which is widely used for the study of the logical structure ofcomputers

The study of types of algebraic structures asmathematical objectsis the purpose ofuniversal algebraandcategory theory.The latter applies to everymathematical structure(not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such astopological spaces;this particular area of application is calledalgebraic topology.

Calculus and analysis

Main articles:CalculusandMathematical analysis

ACauchy sequenceconsists 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 mathematiciansNewtonandLeibniz.It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century byEulerwith the introduction of the concept of afunctionand many other results.Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

Analysis is further subdivided intoreal analysis,where variables representreal numbers,andcomplex analysis,where variables representcomplex numbers.Analysis includes many subareas shared by other areas of mathematics which include:

• Multivariable calculus
• Functional analysis,where variables represent varying functions
• Integration,measure theoryandpotential theory,all strongly related withprobability theoryon acontinuum
• Ordinary differential equations
• Partial differential equations
• Numerical analysis,mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications

Discrete mathematics

Main article:Discrete mathematics

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,countablemathematical objects. An example is the set of all integers.Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.Algorithms—especially theirimplementationandcomputational complexity—play a major role in discrete mathematics.

Thefour color theoremandoptimal sphere packingwere two major problems of discrete mathematics solved in the second half of the 20th century.TheP versus NP problem,which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number ofcomputationally difficultproblems.

Discrete mathematics includes:

• Combinatorics,the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements orsubsetsof 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.
• Graph theoryandhypergraphs
• Coding theory,includingerror correcting codesand a part ofcryptography
• Matroidtheory
• Discrete geometry
• Discrete probability distributions
• Game theory(althoughcontinuous gamesare also studied, most common games, such aschessandpokerare discrete)
• Discrete optimization,includingcombinatorial optimization,integer programming,constraint programming

Mathematical logic and set theory

Main articles:Mathematical logicandSet theory

TheVenn diagramis 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.Before this period, sets were not considered to be mathematical objects, andlogic,although used for mathematical proofs, belonged tophilosophyand was not specifically studied by mathematicians.

BeforeCantor's study ofinfinite sets,mathematicians were reluctant to consideractually infinitecollections, and consideredinfinityto be the result of endlessenumeration.Cantor's work offended many mathematicians not only by considering actually infinite setsbut by showing that this implies different sizes of infinity, perCantor's diagonal argument.This led to thecontroversy over Cantor's set theory.In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuringmathematical rigour.

This became the foundational crisis of mathematics.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.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.Thismathematical abstractionfrom reality is embodied in the modern philosophy offormalism,as founded byDavid Hilbertaround 1910.

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 theoremsassert, roughly speaking that, in everyconsistentformal systemthat contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.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.

These problems and debates led to a wide expansion of mathematical logic, with subareas such asmodel theory(modeling some logical theories inside other theories),proof theory,type theory,computability theoryandcomputational complexity theory.Although these aspects of mathematical logic were introduced before the rise ofcomputers,their use incompilerdesign,formal verification,program analysis,proof assistantsand other aspects ofcomputer science,contributed in turn to the expansion of these logical theories.

Statistics and other decision sciences

Main articles:StatisticsandProbability theory

Whatever the form of a random populationdistribution(μ), the samplingmean(x̄) tends to aGaussiandistribution and itsvariance(σ) is given by thecentral limit theoremof probability theory.

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 samplingor randomizedexperiments.

Statistical theorystudiesdecision problemssuch as minimizing therisk(expected loss) of a statistical action, such as using aprocedurein, for example,parameter estimation,hypothesis testing,andselecting the best.In these traditional areas ofmathematical statistics,a statistical-decision problem is formulated by minimizing anobjective function,like expected loss orcost,under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.Because of its use ofoptimization,the mathematical theory of statistics overlaps with otherdecision sciences,such asoperations research,control theory,andmathematical economics.

Computational mathematics

Main article:Computational mathematics

Computational mathematics is the study ofmathematical problemsthat are typically too large for human, numerical capacity.Numerical analysisstudies methods for problems inanalysisusingfunctional analysisandapproximation theory;numerical analysis broadly includes the study ofapproximationanddiscretizationwith special focus onrounding errors.Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory.Other areas of computational mathematics includecomputer algebraandsymbolic computation.

History

Main article:History of mathematics

Etymology

The wordmathematicscomes from theAncient Greekwordmáthēma(μάθημα), meaning'something learned, knowledge, mathematics',and the derived expressionmathēmatikḗ tékhnē(μαθηματικὴ τέχνη), meaning'mathematical science'.It entered the English language during theLate Middle Englishperiod through French and Latin.

Similarly, one of the two main schools of thought inPythagoreanismwas 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 ofarithmeticand geometry. By the time ofAristotle(384–322 BC) this meaning was fully established.

In Latin and English, until around 1700, the termmathematicsmore 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.

The apparentpluralform in English goes back to the Latinneuterpluralmathematica(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 nounmathematicsanew, after the pattern ofphysicsandmetaphysics,inherited from Greek.In English, the nounmathematicstakes a singular verb. It is often shortened tomathsor, in North America,math.

Ancient

The Babylonian mathematical tabletPlimpton 322,dated to 1800 BC

In addition to recognizing how tocountphysical objects,prehistoricpeoples may have also known how to count abstract quantities, like time—days, seasons, or years.Evidence for more complex mathematics does not appear until around 3000BC,when theBabyloniansand Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.The oldest mathematical texts fromMesopotamiaandEgyptare from 2000 to 1800 BC.Many early texts mentionPythagorean triplesand so, by inference, thePythagorean theoremseems 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 asexagesimalnumeral system which is still in use today for measuring angles and time.

In the 6th century BC,Greek mathematicsbegan to emerge as a distinct discipline and someAncient Greekssuch as thePythagoreansappeared to have considered it a subject in its own right.Around 300 BC,Euclidorganized 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.His book,Elements,is widely considered the most successful and influential textbook of all time.The greatest mathematician of antiquity is often held to beArchimedes(c. 287– c. 212 BC) ofSyracuse.He developed formulas for calculating the surface area and volume ofsolids of revolutionand used themethod of exhaustionto calculate theareaunder the arc of aparabolawith thesummation of an infinite series,in a manner not too dissimilar from modern calculus.Other notable achievements of Greek mathematics areconic sections(Apollonius of Perga,3rd century BC),trigonometry(Hipparchus of Nicaea,2nd century BC),and the beginnings of algebra (Diophantus, 3rd century AD).

The numerals used in theBakhshali manuscript,dated between the 2nd century BC and the 2nd century AD

TheHindu–Arabic numeral systemand the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD inIndiaand were transmitted to theWestern worldviaIslamic mathematics.Other notable developments of Indian mathematics include the modern definition and approximation ofsineandcosine,and an early form ofinfinite series.

Medieval and later

A page fromal-Khwarizmi'sAl-Jabr

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 trigonometryand the addition of thedecimal pointto the Arabic numeral system.Many notable mathematicians from this period were Persian, such asAl-Khwarizmi,Omar KhayyamandSharaf al-Dīn al-Ṭūsī.The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.

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 notationbyFrançois Viète(1540–1603), the introduction oflogarithmsbyJohn Napierin 1614, which greatly simplified numerical calculations, especially forastronomyandmarine 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.

Carl Friedrich Gauss

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.In the early 20th century,Kurt Gödeltransformed 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.

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. "

Symbolic notation and terminology

Main articles:Mathematical notation,Language of mathematics,andGlossary of mathematics

An explanation of the sigma (Σ)summationnotation

Mathematical notation is widely used in science andengineeringfor representing complexconceptsandpropertiesin a concise, unambiguous, and accurate way. This notation consists ofsymbolsused for representingoperations,unspecified numbers,relationsand any other mathematical objects, and then assembling them intoexpressionsand formulas.More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generallyLatinorGreekletters, and often includesubscripts.Operation and relations are generally represented by specificsymbolsorglyphs,such as+(plus),×(multiplication),${\textstyle \int }$(integral),=(equal), and<(less than).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 ofnoun phrasesand 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 rigorousdefinitionsthat provide a standard foundation for communication. An axiom orpostulateis 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 isprovento 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.

Numerous technical terms used in mathematics areneologisms,such aspolynomialandhomeomorphism.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.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 moduleisflat"and" afieldis always aring".

Relationship with sciences

Mathematics is used in mostsciencesformodelingphenomena, which then allows predictions to be made from experimental laws.The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.For example, theperihelion precession of Mercurycould only be explained after the emergence ofEinstein'sgeneral relativity,which replacedNewton's law of gravitationas a better mathematical model.

There is still aphilosophicaldebate 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,theoriesand results (theorems) are often obtained fromexperimentation.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 notrelyingon empirical evidence.

Pure and applied mathematics

Main articles:Applied mathematicsandPure mathematics

Isaac Newton (left) andGottfried Wilhelm Leibnizdeveloped infinitesimal calculus.

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs oftechnologyand science, and there was no clear distinction between pure and applied mathematics.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 Newtonintroduced infinitesimal calculus for explaining the movement of theplanetswith his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.However, a notable exception occurred with the tradition ofpure mathematics in Ancient Greece.The problem ofinteger factorization,for example, which goes back toEuclidin 300 BC, had no practical application before its use in theRSA cryptosystem,now widely used for the security ofcomputer networks.

In the 19th century, mathematicians such asKarl WeierstrassandRichard Dedekindincreasingly focused their research on internal problems, that is,pure mathematics.This led to split mathematics intopure mathematicsandapplied mathematics,the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.

The aftermath ofWorld War IIled to a surge in the development of applied mathematics in the US and elsewhere.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".

An example of the first case is thetheory of distributions,introduced byLaurent Schwartzfor validating computations done inquantum mechanics,which became immediately an important tool of (pure) mathematical analysis.An example of the second case is thedecidability of the first-order theory of the real numbers,a problem of pure mathematics that was proved true byAlfred Tarski,with an algorithm that is impossible toimplementbecause of a computational complexity that is much too high.For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities,George Collinsintroduced thecylindrical algebraic decompositionthat became a fundamental tool inreal algebraic geometry.

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.The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".However, these terms are still used in names of someuniversitydepartments, such as at theFaculty of Mathematicsat theUniversity of Cambridge.

Unreasonable effectiveness

Theunreasonable effectiveness of mathematicsis a phenomenon that was named and first made explicit by physicistEugene Wigner.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.Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is theprime factorizationof natural numbers that was discovered more than 2,000 years before its common use for secureinternetcommunications through theRSA cryptosystem.A second historical example is the theory ofellipses.They were studied by theancient Greek mathematiciansasconic sections(that is, intersections ofconeswith planes). It was almost 2,000 years later thatJohannes Keplerdiscovered that thetrajectoriesof the planets are ellipses.

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 Einsteindeveloped thetheory of relativitythat uses fundamentally these concepts. In particular,spacetimeofspecial relativityis a non-Euclidean space of dimension four, and spacetime ofgeneral relativityis a (curved) manifold of dimension four.

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of thepositronand thebaryon$\Omega ^{-}.$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.

Specific sciences

Physics

Main article:Relationship between mathematics and physics

Diagram of a pendulum

Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,and is also considered to be the motivation of major mathematical developments.

Computing

Further information:Theoretical computer scienceandComputational mathematics

Computing is closely related to mathematics in several ways.Theoretical computer scienceis considered to be mathematical in nature.Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, incryptographyandcoding theory.Discrete mathematicsis useful in many areas of computer science, such ascomplexity theory,information theory,andgraph theory.In 1998, theKepler conjectureonsphere packingseemed to also be partially proven by computer.

Biology and chemistry

Main articles:Mathematical and theoretical biologyandMathematical chemistry

The skin of thisgiant pufferfishexhibits aTuring pattern,which can be modeled byreaction–diffusion systems.

Biologyuses probability extensively in fields such as ecology orneurobiology.Most discussion of probability centers on the concept ofevolutionary fitness.Ecology heavily uses modeling to simulatepopulation dynamics,study ecosystems such as the predator-prey model, measure pollution diffusion,or to assess climate change.The dynamics of a population can be modeled by coupled differential equations, such as theLotka–Volterra equations.

Statistical hypothesis testing,is run on data fromclinical trialsto determine whether a new treatment works.Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.

Earth sciences

Main article:Geomathematics

Structural geologyand climatology use probabilistic models to predict the risk of natural catastrophes.Similarly,meteorology,oceanography,andplanetologyalso use mathematics due to their heavy use of models.

Social sciences

Further information:Mathematical economicsandHistorical dynamics

Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics,economics,sociology,andpsychology.

Supply and demandcurves, 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').In this model, the individual seeks to maximize theirself-interest,and always makes optimal choices usingperfect information.This atomistic view of economics allows it to relatively easily mathematize its thinking, because individualcalculationsare 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.

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.

At the start of the 20th century, there was a development to express historical movements in formulas. In 1922,Nikolai Kondratievdiscerned the ~50-year-longKondratiev cycle,which explains phases of economic growth or crisis.Towards the end of the 19th century, mathematicians extended their analysis intogeopolitics.Peter Turchindevelopedcliodynamicssince the 1990s.

Mathematization of the social sciences is not without risk. In the controversial bookFashionable Nonsense(1997),SokalandBricmontdenounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.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.

Philosophy

Main article:Philosophy of mathematics

Reality

The connection between mathematics and material reality has led to philosophical debates since at least the time ofPythagoras.The ancient philosopherPlatoargued 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.

Armand Borelsummarized this view of mathematics reality as follows, and provided quotations ofG. H. Hardy,Charles Hermite,Henri Poincaréand Albert Einstein that support his views.

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.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 afeelingof an objective existence, of a reality of mathematics...

Nevertheless, Platonism and the concurrent views on abstraction do not explain theunreasonable effectivenessof mathematics.

Proposed definitions

Main article:Definitions of mathematics

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".A common approach is to define mathematics by its object of study.

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.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.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.For example, in lieu of a definition,Saunders Mac LaneinMathematics, form and functionsummarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:

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.

Rigor

See also:Logic

Mathematical reasoning requiresrigor.This means that the definitions must be absolutely unambiguous and theproofsmust be reducible to a succession of applications ofinference rules,without any use of empirical evidence andintuition.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.The emergence ofcomputer-assisted proofshas allowed proof lengths to further expand.The result of this trend is a philosophy of thequasi-empiricistproof that can not be considered infallible, but has a probability attached to it.

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.

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 theapodicticinference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.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.

Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.

Training and practice

Education

Main article:Mathematics education

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

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.Comparable evidence has been unearthed for scribal mathematics training in theancient Near Eastand then for theGreco-Roman worldstarting around 300 BCE.The oldest known mathematics textbook is theRhind papyrus,dated fromc. 1650 BCEin Egypt.Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorizedoral traditionsince theVedic period(c. 1500– c. 500 BCE).InImperial Chinaduring theTang dynasty(618–907 CE), a mathematics curriculum was adopted for thecivil service examto join the state bureaucracy.

Following theDark Ages,mathematics education in Europe was provided by religious schools as part of theQuadrivium.Formal instruction inpedagogybegan withJesuitschools 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.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.While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.

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.Some students studying math may develop an apprehension or fear about their performance in the subject. This is known asmath anxietyor math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.

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.An extreme example isApery's theorem:Roger Aperyprovided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.

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.This aspect of mathematical activity is emphasized inrecreational mathematics.

Mathematicians can find anaestheticvalue 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 Apologyexpressed 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.Paul Erdősexpressed 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 transformforharmonic analysis.

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

Cultural impact

Artistic expression

Main article:Mathematics and art

Notes that sound well together to a Western ear are sounds whose fundamentalfrequenciesof vibration are in simple ratios. For example, an octave doubles the frequency and aperfect fifthmultiplies it by${\frac {3}{2}}$.

Fractalwith a scaling symmetry and a central symmetry

Humans, as well as some other animals, find symmetric patterns to be more beautiful.Mathematically, the symmetries of an object form a group known as thesymmetry group.For example, the group underlying mirror symmetry is thecyclic groupof two elements,$\mathbb {Z} /2\mathbb {Z}$.ARorschach testis a figure invariant by this symmetry,as arebutterflyand animal bodies more generally (at least on the surface).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.Fractalspossessself-similarity.

Popularization

Main article:Popular mathematics

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

Awards and prize problems

Main category:Mathematics awards

The front side of theFields Medalwith an illustration of the GreekpolymathArchimedes

The most prestigious award in mathematics is theFields Medal,established in 1936 and awarded every four years (except aroundWorld War II) to up to four individuals.It is considered the mathematical equivalent of theNobel Prize.

Other prestigious mathematics awards include:

• TheAbel Prize,instituted in 2002and first awarded in 2003
• TheChern Medalfor lifetime achievement, introduced in 2009and first awarded in 2010
• TheAMSLeroy P. Steele Prize,awarded since 1970
• TheWolf Prize in Mathematics,also for lifetime achievement,instituted in 1978

A famous list of 23open problems,called "Hilbert's problems",was compiled in 1900 by German mathematician David Hilbert.This list has achieved great celebrity among mathematicians,and at least thirteen of the problems (depending how some are interpreted) have been solved.

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.To date, only one of these problems, thePoincaré conjecture,has been solved by the Russian mathematicianGrigori Perelman.

See also

• Law (mathematics)
• List of mathematical jargon
• Lists of mathematicians
• Lists of mathematics topics
• Mathematical constant
• Mathematical sciences
• Mathematics and art
• Mathematics education
• Philosophy of mathematics
• Relationship between mathematics and physics
• Science, technology, engineering, and mathematics

References

Sources

• Bouleau, Nicolas(1999).Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur.L'Harmattan.ISBN978-2-7384-8125-2.
• Boyer, Carl Benjamin(1991).A History of Mathematics(2nd ed.). New York:Wiley.ISBN978-0-471-54397-8.
• Cresswell, Julia (2021).Oxford Dictionary of Word Origins(3 ed.). Oxford University Press.ISBN978-0-19-886875-0.
• Devlin, Keith (2018).Sets, Functions, and Logic: An Introduction to Abstract Mathematics(3 ed.). CRC Press.ISBN978-1-4822-8602-1.
• Eves, Howard(1990).An Introduction to the History of Mathematics(6th ed.). Saunders.ISBN978-0-03-029558-4.
• Kleiner, Israel(2007). Kleiner, Israel (ed.).A History of Abstract Algebra.Springer Science & Business Media.doi:10.1007/978-0-8176-4685-1.ISBN978-0-8176-4684-4.LCCN2007932362.OCLC76935733.S2CID117392219.RetrievedFebruary 8,2024.
• Kline, Morris(1990).Mathematical Thought from Ancient to Modern Times.New York: Oxford University Press.ISBN978-0-19-506135-2.
• Monastyrsky, Michael (2001)."Some Trends in Modern Mathematics and the Fields Medal"(PDF).CMS – Notes – de la SMC.33(2–3). Canadian Mathematical Society.Archived(PDF)from the original on August 13, 2006.RetrievedJuly 28,2006.
• Oakley, Barbara(2014).A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra).New York: Penguin Random House.ISBN978-0-399-16524-5.A Mind for Numbers.
• 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. GoogleEprintand as an extract, D. Van Nostrand, 1882, GoogleEprint.RetrievedNovember 17,2020..
• Peterson, Ivars (1988).The Mathematical Tourist: Snapshots of Modern Mathematics.W. H. Freeman and Company.ISBN0-7167-1953-3.LCCN87033078.OCLC17202382.
• Popper, Karl R.(1995). "On knowledge".In Search of a Better World: Lectures and Essays from Thirty Years.New York: Routledge.Bibcode:1992sbwl.book.....P.ISBN978-0-415-13548-1.
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• Sevryuk, Mikhail B. (January 2006)."Book Reviews"(PDF).Bulletin of the American Mathematical Society.43(1): 101–109.doi:10.1090/S0273-0979-05-01069-4.Archived(PDF)from the original on July 23, 2006.RetrievedJune 24,2006.
• 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.Archivedfrom the original on December 19, 2013.

Further reading

• Benson, Donald C. (1999).The Moment of Proof: Mathematical Epiphanies.Oxford University Press.ISBN978-0-19-513919-8.
• Davis, Philip J.;Hersh, Reuben(1999).The Mathematical Experience(Reprint ed.). Boston; New York: Mariner Books.ISBN978-0-395-92968-1.Availableonline(registration required).
• Courant, Richard;Robbins, Herbert(1996).What Is Mathematics?: An Elementary Approach to Ideas and Methods(2nd ed.). New York: Oxford University Press.ISBN978-0-19-510519-3.
• Gullberg, Jan(1997).Mathematics: From the Birth of Numbers.W.W. Norton & Company.ISBN978-0-393-04002-9.
• 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, andonline.ArchivedDecember 20, 2012, atarchive.today.
• Hodgkin, Luke Howard (2005).A History of Mathematics: From Mesopotamia to Modernity.Oxford University Press.ISBN978-0-19-152383-0.
• Jourdain, Philip E. B.(2003). "The Nature of Mathematics". In James R. Newman (ed.).The World of Mathematics.Dover Publications.ISBN978-0-486-43268-7.
• Pappas, Theoni(1986).The Joy Of Mathematics.San Carlos, California: Wide World Publishing.ISBN978-0-933174-65-8.
• Waltershausen, Wolfgang Sartorius von(1965) [1856].Gauss zum Gedächtniss.Sändig Reprint Verlag H. R. Wohlwend.ISBN978-3-253-01702-5.