Misplaced Pages

#459540

104-519: In mathematics and physics , many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple yet ambiguous names such as Euler's function , Euler's equation , and Euler's formula . Euler's work touched upon so many fields that he

208-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . 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: The study of types of algebraic structures as mathematical objects

312-486: A demonstration in the form of chains of syllogisms (though they do not always conform strictly to Aristotelian templates). Aristotle's syllogistic logic , together with its exemplification by Euclid's Elements , are recognized as scientific achievements of ancient Greece, and remained as the foundations of mathematics for centuries. During Middle Ages , Euclid's Elements stood as a perfectly solid foundation for mathematics, and philosophy of mathematics concentrated on

416-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

520-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

624-485: A logical way, they were defined in terms of infinitesimals that are hypothetical numbers that are infinitely close to zero. The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopher George Berkeley (1685–1753), who wrote "[Infinitesimals] are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them

728-404: 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 that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

832-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

936-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . 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

1040-681: A non-differential equation, as in these three cases: Selected topics from above, grouped by subject, and additional topics from the fields of music and physical systems Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number 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), and set theory (presently used as

1144-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

SECTION 10

#1732772435460

1248-515: A sentence such that "if x is very small then ..." must be understood as "there is a (sufficiently large) natural number n such that | x | < 1/ n ". In the proofs he used this in a way that predated the modern (ε, δ)-definition of limit . The modern (ε, δ)-definition of limits and continuous functions was first developed by Bolzano in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work. Karl Weierstrass (1815–1897) formalized and popularized

1352-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

1456-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

1560-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

1664-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

1768-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

1872-469: Is flat " and "a field is always a ring ". Foundational crisis of mathematics The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle 's logic and systematically applied in Euclid 's Elements . A mathematical assertion is considered as truth only if it

1976-403: Is "the power of the mind" which allows conceiving of the indefinite repetition of the same act. This applies in particular to the use of the last Peano axiom for showing that the successor function generates all natural numbers. Also, Leopold Kronecker said "God made the integers, all else is the work of man". This may be interpreted as "the integers cannot be mathematically defined". Before

2080-608: Is a theorem that is proved from true premises by means of a sequence of syllogisms ( inference rules ), the premises being either already proved theorems or self-evident assertions called axioms or postulates . These foundations were tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. This new area of mathematics involved new methods of reasoning and new basic concepts ( continuous functions , derivatives , limits ) that were not well founded, but had astonishing consequences, such as

2184-560: Is a set then" or "if φ {\displaystyle \varphi } is a predicate then". So, Peano's axioms induce a quantification on infinite sets, and this means that Peano arithmetic is what is presently called a Second-order logic . This was not well understood at that times, but the fact that infinity occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example, Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it

SECTION 20

#1732772435460

2288-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

2392-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

2496-407: 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, "every free module

2600-402: Is either provable or refutable; that is, its negation is provable), and decidability (there is a decision procedure to test every statement). By near the turn of the century, Bertrand Russell popularized Frege's work and discovered Russel's paradox which implies that the phrase "the set of all sets" is self-contradictory. This paradox seemed to make the whole mathematics inconsistent and

2704-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

2808-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

2912-418: Is no concept of distance in a projective space , and the cross-ratio , which is a number, is a basic concept of synthetic projective geometry. Karl von Staudt developed a purely geometric approach to this problem by introducing "throws" that form what is presently called a field , in which the cross ratio can be expressed. Apparently, the problem of the equivalence between analytic and synthetic approach

3016-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

3120-1192: 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." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , 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

3224-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

List of topics named after Leonhard Euler - Misplaced Pages Continue

3328-406: Is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler. Usually, Euler's equation refers to one of (or a set of) differential equations (DEs). It is customary to classify them into ODEs and PDEs . Otherwise, Euler's equation may refer to

3432-464: Is one of the major causes of the foundational crisis of mathematics. The foundational crisis of mathematics arose at the end of the 19th century and the beginning of the 20th century with the discovery of several paradoxes or counter-intuitive results. The first one was the proof that the parallel postulate cannot be proved. This results from a construction of a Non-Euclidean geometry inside Euclidean geometry , whose inconsistency would imply

3536-435: Is one of the motivation of the development of higher-order logics during the first half of the 20th century. Before the 19th century, there were many failed attempts to derive the parallel postulate from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction, Johann Heinrich Lambert (1728–1777) started to build hyperbolic geometry and introduced the hyperbolic functions and computed

3640-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

3744-474: Is only in the 20th century that a formal definition of infinitesimals has been given, with the proof that the whole infinitesimal can be deduced from them. Despite its lack of firm logical foundations, infinitesimal calculus was quickly adopted by mathematicians, and validated by its numerous applications; in particular the fact that the planet trajectories can be deduced from the Newton's law of gravitation . In

3848-410: Is required for defining and using real numbers involves a quantification on infinite sets. Indeed, this property may be expressed either as for every infinite sequence of real numbers, if it is a Cauchy sequence , it has a limit that is a real number , or as every subset of the real numbers that is bounded has a least upper bound that is a real number . This need of quantification over infinite sets

3952-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

4056-402: Is specified in terms of real numbers called coordinates . Mathematicians did not worry much about the contradiction between these two approaches before the mid-nineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry , the two sides accusing each other of mixing projective and metric concepts". Indeed, there

4160-444: Is still used for guiding mathematical intuition : physical reality is still used by mathematicians to choose axioms, find which theorems are interesting to prove, and obtain indications of possible proofs. Most civilisations developed some mathematics, mainly for practical purposes, such as counting (merchants), surveying (delimitation of fields), prosody , astronomy , and astrology . It seems that ancient Greek philosophers were

4264-454: Is the discovery that there are strictly more real numbers than natural numbers (the cardinal of the continuum of the real numbers is greater than that of the natural numbers). These results were rejected by many mathematicians and philosophers, and led to debates that are a part of the foundational crisis of mathematics . The crisis was amplified with the Russel's paradox that asserts that

List of topics named after Leonhard Euler - Misplaced Pages Continue

4368-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

4472-405: 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 their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

4576-426: Is the starting point of mathematization logic and the basis of propositional calculus Independently, in the 1870's, Charles Sanders Peirce and Gottlob Frege extended propositional calculus by introducing quantifiers , for building predicate logic . Frege pointed out three desired properties of a logical theory: consistency (impossibility of proving contradictory statements), completeness (any statement

4680-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

4784-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

4888-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

4992-511: The Pythagoreans appeared to have considered it a subject in its own right. 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. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

5096-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

5200-446: The controversy 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 ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

5304-595: The foundational crisis of mathematics . The following subsections describe the main such foundational problems revealed during the 19th century. Cauchy (1789–1857) started the project of giving rigorous bases to infinitesimal calculus . In particular, he rejected the heuristic principle that he called the generality of algebra , which consisted to apply properties of algebraic operations to infinite sequences without proper proofs. In his Cours d'Analyse (1821), he considered very small quantities , which could presently be called "sufficiently small quantities"; that is,

SECTION 50

#1732772435460

5408-507: The logic for organizing a field of knowledge by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the foundation of mathematics for centuries. This method resembles the modern axiomatic method but with a big philosophical difference: axioms and postulates were supposed to be true, being either self-evident or resulting from experiments , while no other truth than

5512-428: The ontological status of mathematical concepts; the question was whether they exist independently of perception ( realism ) or within the mind only ( conceptualism ); or even whether they are simply names of collection of individual objects ( nominalism ). In Elements , the only numbers that are considered are natural numbers and ratios of lengths. This geometrical view of non-integer numbers remained dominant until

5616-449: The (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as continuous, nowhere-differentiable functions . Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph. At this point, the program of arithmetization of analysis (reduction of mathematical analysis to arithmetic and algebraic operations) advocated by Weierstrass

5720-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

5824-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

5928-426: The 19th century, mathematics developed quickly in many directions. Several of the problems that were considered led to questions on the foundations of mathematics. Frequently, the proposed solutions led to further questions that were often simultaneously of philosophical and mathematical nature. All these questions led, at the end of the 19th century and the beginning of the 20th century, to debates which have been called

6032-514: The 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry , another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining points as pairs of antipodal points on a sphere (or hypersphere ), and lines as great circles on the sphere. These proofs of unprovability of the parallel postulate lead to several philosophical problems,

6136-668: The 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of axiomatic method and on set theory, specifically ZFC , the Zermelo – Fraenkel set theory with the axiom of choice . It results from this that the basic mathematical concepts, such as numbers , points , lines , and geometrical spaces are not defined as abstractions from reality but from basic properties ( axioms ). Their adequation with their physical origins does not belong to mathematics anymore, although their relation with reality

6240-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: 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, and logic , although used for mathematical proofs, belonged to philosophy and

6344-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

SECTION 60

#1732772435460

6448-447: The area of a hyperbolic triangle (where the sum of angles is less than 180°). Continuing the construction of this new geometry, several mathematicians proved independently that if it is inconsistent , then Euclidean geometry is also inconsistent and thus that the parallel postulate cannot be proved. This was proved by Nikolai Lobachevsky in 1826, János Bolyai (1802–1860) in 1832 and Carl Friedrich Gauss (unpublished). Later in

6552-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

6656-449: The classical foundations of mathematics since all properties of numbers that were used can be deduced from their geometrical definition. In 1637, René Descartes published La Géométrie , in which he showed that geometry can be reduced to algebra by means coordinates , which are numbers determining the position of a point. This gives to the numbers that he called real numbers a more foundational role (before him, numbers were defined as

6760-459: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

6864-410: The correctness of the proof is involved in the axiomatic method. So, for Aristotle, a proved theorem is true, while in the axiomatic methods, the proof says only that the axioms imply the statement of the theorem. Aristotle's logic reached its high point with Euclid 's Elements (300 BC), a treatise on mathematics structured with very high standards of rigor: Euclid justifies each proposition by

6968-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

7072-422: The deduction from Newton's law of gravitation that the orbits of the planets are ellipses . During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the natural and real numbers. This led, near the end of the 19th century, to a series of seemingly paradoxical mathematical results that challenged the general confidence in

7176-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

7280-419: The elements of a Cauchy sequence), and Cantor's set theory was published several years later. The third problem is more subtle: and is related to the foundations of logic: classical logic is a first order logic ; that is, quantifiers apply to variables representing individual elements, not to variables representing (infinite) sets of elements. The basic property of the completeness of the real numbers that

7384-440: The end of Middle Ages, although the rise of algebra led to consider them independently from geometry, which implies implicitly that there are foundational primitives of mathematics. For example, the transformations of equations introduced by Al-Khwarizmi and the cubic and quartic formulas discovered in the 16th century result from algebraic manipulations that have no geometric counterpart. Nevertheless, this did not challenge

7488-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

7592-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

7696-474: The first to study the nature of mathematics and its relation with the real world. Zeno of Elea (490 – c. 430 BC) produced several paradoxes he used to support his thesis that movement does not exist. These paradoxes involve mathematical infinity , a concept that was outside the mathematical foundations of that time and was not well understood before the end of the 19th century. The Pythagorean school of mathematics originally insisted that

7800-431: The geometry is the same as the affine or projective geometry over k . The work of making rigorous real analysis and the definition of real numbers , consisted of reducing everything to rational numbers and thus to natural numbers , since positive rational numbers are fractions of natural numbers. There was therefore a need of a formal definition of natural numbers, which imply as axiomatic theory of arithmetic . This

7904-400: The ghosts of departed quantities?". Also, a lack of rigor has been frequently invoked, because infinitesimals and the associated concepts were not formally defined ( lines and planes were not formally defined either, but people were more accustomed to them). Real numbers, continuous functions, derivatives were not formally defined before the 19th century, as well as Euclidean geometry . It

8008-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

8112-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried 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

8216-405: The main one being that before this discovery, the parallel postulate and all its consequences were considered as true . So, the non-Euclidean geometries challenged the concept of mathematical truth . Since the introduction of analytic geometry by René Descartes in the 17th century, there were two approaches to geometry, the old one called synthetic geometry , and the new one, where everything

8320-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

8424-400: 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 by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

8528-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

8632-556: The only numbers are natural numbers and ratios of natural numbers. The discovery (around 5th century BC) that the ratio of the diagonal of a square to its side is not the ratio of two natural numbers was a shock to them which they only reluctantly accepted. A testimony of this is the modern terminology of irrational number for referring to a number that is not the quotient of two integers, since "irrational" means originally "not reasonable" or "not accessible with reason". The fact that length ratios are not represented by rational numbers

8736-437: The other one by Georg Cantor as equivalence classes of Cauchy sequences . Several problems were left open by these definitions, which contributed to the foundational crisis of mathematics . Firstly both definitions suppose that rational numbers and thus natural numbers are rigorously defined; this was done a few years later with Peano axioms . Secondly, both definitions involve infinite sets (Dedekind cuts and sets of

8840-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

8944-700: The phrase "the set of all sets" is self-contradictory. This condradiction introduced a doubt on the consistency of all mathematics. With the introduction of the Zermelo–Fraenkel set theory ( c.  1925 ) and its adoption by the mathematical community, the doubt about the consistency was essentially removed, although consistency of set theory cannot be proved because of Gödel's incompleteness theorem . In 1847, De Morgan published his laws and George Boole devised an algebra, now called Boolean algebra , that allows expressing Aristotle's logic in terms of formulas and algebraic operations . Boolean algebra

9048-605: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

9152-581: The ratio of two lengths). Descartes' book became famous after 1649 and paved the way to infinitesimal calculus . Isaac Newton (1642–1727) in England and Leibniz (1646–1716) in Germany independently developed the infinitesimal calculus for dealing with mobile points (such as planets in the sky) and variable quantities. This needed the introduction of new concepts such as continuous functions , derivatives and limits . For dealing with these concepts in

9256-414: The reliability and truth of mathematical results. This has been called the foundational crisis of mathematics . The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic that includes set theory , model theory , proof theory , computability and computational complexity theory , and more recently, parts of computer science . Subsequent discoveries in

9360-423: The second half of the 19th century, infinity was a philosophical concept that did not belong to mathematics. However, with the rise of infinitesimal calculus , mathematicians became to be accustomed to infinity, mainly through potential infinity , that is, as the result of an endless process, such as the definition of an infinite sequence , an infinite series or a limit . The possibility of an actual infinity

9464-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

9568-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

9672-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

9776-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

9880-508: 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 term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

9984-514: Was completely solved only with Emil Artin 's book Geometric Algebra published in 1957. It was well known that, given a field k , one may define affine and projective spaces over k in terms of k - vector spaces . In these spaces, the Pappus hexagon theorem holds. Conversely, if the Pappus hexagon theorem is included in the axioms of a plane geometry, then one can define a field k such that

10088-445: Was essentially completed, except for two points. Firstly, a formal definition of real numbers was still lacking. Indeed, beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, including Hermann Hankel , Charles Méray , and Eduard Heine , but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means of Dedekind cuts ;

10192-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

10296-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

10400-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

10504-518: Was resolved by Eudoxus of Cnidus (408–355 BC), a student of Plato , who reduced the comparison of two irrational ratios to comparisons of integer multiples of the magnitudes involved. His method anticipated that of Dedekind cuts in the modern definition of real numbers by Richard Dedekind (1831–1916); see Eudoxus of Cnidus § Eudoxus' proportions . In the Posterior Analytics , Aristotle (384–322 BC) laid down

10608-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

10712-452: Was started with Charles Sanders Peirce in 1881 and Richard Dedekind in 1888, who defined a natural numbers as the cardinality of a finite set . . However, this involves set theory , which was not formalized at this time. Giuseppe Peano provided in 1888 a complete axiomatisation based on the ordinal property of the natural numbers. The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "if S

10816-534: Was the subject of many philosophical disputes. Sets , and more specially infinite sets were not considered as a mathematical concept; in particular, there was no fixed term for them. A dramatic change arose with the work of Georg Cantor who was the first mathematician to systematically study infinite sets. In particular, he introduced cardinal numbers that measure the size of infinite sets, and ordinal numbers that, roughly speaking, allow one to continue to count after having reach infinity. One of his major results

#459540