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Mathematics Source: en.wikipedia.org/wiki/Mathematics

Mathematics is an area of knowledge that includes topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3][4] respectively.

Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered as true starting points of the theory under consideration.[5]

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. The fundamental truths of mathematics are independent from any scientific experimentation, although mathematics is extensively used for modeling phenomena. 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 mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later.[6][7] A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks).

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[8] Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new areas of the subject. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both.[9] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method.[10] This gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. This can be seen, for example, in the contemporary Mathematics Subject Classification, which lists more than 60 first-level areas of mathematics.

Etymology[edit]

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

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.[14]

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[15]

The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[16] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[17]

Areas of mathematics[edit]

Before the Renaissance, mathematics was divided into two main areas: arithmetic—regarding the manipulation of numbers, and geometry—regarding the study of shapes.[18] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[19]

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

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.[23][10] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[24] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.[25]

Number theory[edit]

This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

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

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

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

Geometry[edit]

On the surface of a sphere, Euclidian geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.[31]

A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[32][33]

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.[b][31]

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This was a major change of paradigm, since 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). This allows one to use algebra (and later, calculus) to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.[34]

Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose study 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 spaces of higher than three dimensions.[31]

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning the truth of that postulate, this discovery has been viewed as joining Russel's paradox in revealing the foundational crisis of mathematics.This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.[35][10] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space.[36]

In the present day, the subareas of geometry include:[25]

Algebra[edit]

The quadratic formula, which concisely expresses the solutions of all quadratic equations
The Rubik's Cube group is a concrete application of group theory[37]

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.[38][39] The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term algebra is derived from the Arabic word al-jabr meaning "the reunion of broken parts"[40] that he used for naming one of these methods in the title of his main treatise.

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.[41] This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.[42] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.[43] (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:[25]

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.[44] The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.[45]

Calculus and analysis[edit]

A Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.[46] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.[47] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:[25]

Discrete mathematics[edit]

A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.

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

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

Discrete mathematics includes:[25]

Mathematical logic and set theory[edit]

The Venn diagram is a commonly used method to illustrate the relations between sets.

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.[52][53] Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.[54]

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets[55] but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.[56]

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

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

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.[60] This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.[61][62]

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory.[25] Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.[63]

Statistics and other decision sciences[edit]

Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.[64]

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

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

Computational mathematics[edit]

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

History[edit]

Ancient[edit]

The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,[71] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.[72][73]

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[74] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.[75]

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.[76] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.[77] His book, Elements, is widely considered the most successful and influential textbook of all time.[78] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.[79] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[80] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[81] trigonometry (Hipparchus of Nicaea, 2nd century BC),[82] and the beginnings of algebra (Diophantus, 3rd century AD).[83]

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

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

Medieval and later[edit]

A page from al-Khwārizmī's Algebra

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

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz (1646–1716) in the 17th century. Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.[89] In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.[60]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[90]

Symbolic notation and terminology[edit]

An explanation of the sigma (Σ) summation notation

Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas.[91] 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,[92] such as + (plus), × (multiplication), (integral), = (equal), and < (less than).[93] All these symbols are generally grouped according to specific rules to form expressions and formulas.[94] Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.

Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corallary.[95]

Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism.[96] Other technical terms are words of the common language that are used in an accurate meaning that may differs slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either amiguous 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.[97] This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring".

Relationship with science[edit]

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws.[98] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.[99] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.[100] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.[101]

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation.[102] 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 (one of the greatest mathematicians of the 19th century) once replied "durch planmässiges Tattonieren" (through systematic experimentation).[103] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.[104][105][106][107]

Pure and applied mathematics[edit]

Isaac Newton
Gottfried Wilhelm von Leibniz
Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics.[108] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.[109] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece.[110]

In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics.[108][111] This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.[112]

The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere.[113][114] During the second half of the 20th century, 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 the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis.[115] An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high.[116] For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.[117]

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.[118][119] The Mathematics Subject Classification does not mention "pure mathematics" nor "applied mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.

Unreasonable effectiveness[edit]

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner.[7] It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.[120] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.[121] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.[122]

In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of the special relativity is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.[123][124]

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of a unknown particle, and to search these particles. In both cases, these particles were discovered a few years later by specific experiments.[125][126][127]

Philosophy[edit]

Reality[edit]

The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.[128]

Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.[125]

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.[129] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...

Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.[130]

Proposed definitions[edit]

There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.[131][132] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[131] There is not even consensus on whether mathematics is an art or a science.[132] Some just say, "mathematics is what mathematicians do".[131] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.[133]

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.[134] 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.[135] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems —assertions whose validity relies on a proof, that is, a purely-logical deduction.[136] Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.[137]

Rigor[edit]

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules,[e] without any use of empirical evidence and intuition.[f][138] Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision, rigorous proofs can require hundreds of pages to express. The emergence of computer-assisted proofs has allowed proof lengths to further expand,[g][139] such as the 255-page Feit–Thompson theorem.[h] The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.[10]

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

At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russel's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.[10] It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.[140]

Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.[141]

Psychology (aesthetic, creativity and intuition)[edit]

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.[142][143] An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.[144]

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles.[145] This aspect of mathematical activity is emphasized in recreational mathematics.

Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.[146] Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.[147]

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.[148] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science).[125] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind.[59] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle.[149]

In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.[149]

Education[edit]

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. While the content of courses varies, many countries in the world teach mathematics to students for significant amounts of time.[150]

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.[151] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE.[152] The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE in Eygpt.[153] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy.[154]

Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was the L'Enseignement Mathématique, which began publication in 1899.[155]

Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.[156]

Awards and prize problems[edit]

The front side of the Fields Medal with an illustration of the Greek polymath Archimedes

The most prestigious award in mathematics is the Fields Medal,[157][158] established in 1936 and awarded every four years (except around World War II) to up to four individuals.[159][160] It is considered the mathematical equivalent of the Nobel Prize.[160]

Other prestigious mathematics awards include:[161]

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[169] This list has achieved great celebrity among mathematicians,[170][unreliable source?] and, as of 2022, at least thirteen of the problems (depending how some are interpreted) have been solved.[171]

A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[172] To date, only one of these problems, the Poincaré conjecture, has been solved.[173]

See also[edit]

Notes[edit]

  1. ^ Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating formulas.
  2. ^ This includes conic sections, which are intersections of circular cylinders and planes.
  3. ^ However, some advanced methods of analysis are sometimes used; for example, methods of complex analysis applied to generating series.
  4. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.
  5. ^ This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without computers and proof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.
  6. ^ This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
  7. ^ For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software
  8. ^ The book containing the complete proof has more than 1,000 pages.

References[edit]

  1. ^ a b "mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Archived from the original on November 16, 2019. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
  2. ^ Kneebone, G. T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Archived from the original on January 7, 2017. Retrieved June 20, 2015. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
  3. ^ LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Archived from the original on January 7, 2017. Retrieved June 20, 2015. Calculus is the study of change—how things change, and how quickly they change.
  4. ^ Ramana, B. V. (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 978-0-07-066753-2. Archived from the original on July 12, 2022. Retrieved July 30, 2022. The mathematical study of change, motion, growth or decay is calculus.
  5. ^ Hipólito, Inês Viegas (2015). "Abstract Cognition and the Nature of Mathematical Proof" (PDF). In Kanzian, Christian; Mitterer, Josef; Neges, Katharina (eds.). Realism – Relativism – Constructivism. 38th International Wittgenstein Symposium, August 9–15, 2015. Kirchberg am Wechsel, Austria. Austrian Ludwig Wittgenstein Society. pp. 132–134. Retrieved November 5, 2022. (at ResearchGate)
  6. ^ Peterson 2001, p. 12.
  7. ^ a b Wigner, Eugene (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. Archived from the original on February 28, 2011.
  8. ^ Wise, David. "Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion". jwilson.coe.uga.edu. Archived from the original on June 1, 2019. Retrieved October 26, 2019.
  9. ^ Alexander, Amir (2011). "The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?". Isis. 102 (3): 475–480. doi:10.1086/661620. PMID 22073771. S2CID 21629993.
  10. ^ a b c d e f Kleiner, Israel (December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine. Taylor & Francis, Ltd. 64 (5): 291–314. doi:10.1080/0025570X.1991.11977625. JSTOR 2690647.
  11. ^ a b Harper, Douglas. "mathematic (n.)". Online Etymology Dictionary. Retrieved November 26, 2022.{{cite encyclopedia}}: CS1 maint: url-status (link)
  12. ^ Both meanings can be found in Plato, the narrower in Republic. 510c. Archived February 24, 2021, at the Wayback Machine, but Plato did not use a math- word; Aristotle did, commenting on it. μαθηματική. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project. OED Online. Mathematics.
  13. ^ Harper, Douglas. "mathematical (adj.)". Online Etymology Dictionary. Retrieved November 26, 2022.
  14. ^ Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms". Pi Mu Epsilon Journal. 4 (2): 62–66. JSTOR 24338341.
  15. ^ Boas, Ralph (1995) [1991]. "What Augustine Didn't Say About Mathematicians". Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas, Jr. Cambridge University Press. p. 257. ISBN 978-0-88385-323-8. Archived from the original on May 20, 2020. Retrieved January 17, 2018.
  16. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics".
  17. ^ "maths, n." and "math, n.3" Archived April 4, 2020, at the Wayback Machine. Oxford English Dictionary, on-line version (2012).
  18. ^ Bell, E. T. (2012). The Development of Mathematics. Dover Books on Mathematics (reprint, revised ed.). Courier Corporation. p. 3. ISBN 9780486152288.
  19. ^ Tiwari, Sarju (1992). Mathematics in History, Culture, Philosophy, and Science. Mittal Publications. p. 27. ISBN 9788170994046.
  20. ^ Restivo, S. (December 2013). Mathematics in Society and History. Springer Netherlands. pp. 14–15. ISBN 9789401129442.
  21. ^ Musielak, Dora (2022). Leonhard Euler and the Foundations of Celestial Mechanics. Springer International Publishing. pp. 1–183. ISBN 9783031123221.
  22. ^ Biggs, N. L. (May 1979). "The roots of combinatorics". Historia Mathematica. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0.
  23. ^ a b Warner, Evan (2013). "Splash Talk: The Foundational Crisis of Mathematics" (PDF). Columbia University. pp. 1–17. Retrieved November 4, 2022.
  24. ^ Dunne, Edward; Hulek, Klaus (March 2020). "Mathematics Subject Classification 2020" (PDF). Notices of the American Mathematical Society. 67 (3). Retrieved November 4, 2022.
  25. ^ a b c d e f g "MSC2020-Mathematics Subject Classification System" (PDF). zbMath. Associate Editors of Mathematical Reviews and zbMATH. Retrieved November 26, 2022.
  26. ^ LeVeque, William J. (January 5, 2014). Fundamentals of Number Theory. Dover Publications. pp. 1–30. ISBN 9780486141503.
  27. ^ Goldman, Jay (1997). The Queen of Mathematics: A Historically Motivated Guide to Number Theory. CRC Press. pp. 1–3. ISBN 9781439864623.
  28. ^ Weil, André (2007). Number Theory, An Approach Through History From Hammurapi to Legendre. Birkhäuser Boston. pp. 1–3. ISBN 9780817645717.
  29. ^ Kleiner, Israel (February 2000). "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem". Elemente der Mathematik. 55: 19–37. doi:10.1007/PL00000079. S2CID 53319514.
  30. ^ Wang, Yuan (2002). The Goldbach Conjecture. Series in pure mathematics. Vol. 4 (revised ed.). World Scientific. pp. 1–18. ISBN 9789812776600.
  31. ^ a b c Straume, Eldar (September 2014). "A Survey of the Development of Geometry up to 1870". ePrint. arXiv:1409.1140. Bibcode:2014arXiv1409.1140S.
  32. ^ Hilbert, David (1962). The Foundations of Geometry. Open Court Publishing Company. p. 1.
  33. ^ Hartshorne, Robin (November 11, 2013). Geometry: Euclid and Beyond. Springer New York. pp. 9–13. ISBN 9780387226767.
  34. ^ Boyer, Carl B. (June 28, 2012). History of Analytic Geometry. Dover Publications. pp. 74–102. ISBN 9780486154510.
  35. ^ Stump, David J. (1997). "Reconstructing the Unity of Mathematics circa 1900". Perspectives on Science. 5 (3): 383. doi:10.1162/posc_a_00532. S2CID 117709681.
  36. ^ O'Connor, J. J.; Robertson, E. F. (February 1996). "Non-Euclidean geometry". MacTuror. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 6, 2022.
  37. ^ Joyner, D. (2008). Adventures in group theory: Rubik's Cube, Merlin's machine, and other mathematical toys (2nd ed.). Johns Hopkins University Press. pp. 219–232. ISBN 9780801890130.
  38. ^ Christianidis, Jean; Oaks, Jeffrey (May 2013). "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria". Historia Mathematica. 40 (2): 127–163. doi:10.1016/j.hm.2012.09.001.
  39. ^ Kleiner 2007, "History of Classical Algebra" pp. 3–5.
  40. ^ Lim, Lisa (December 21, 2018). "Where algebra got its x from, and Xmas its X". South China Morning Post. Retrieved August 9, 2022.
  41. ^ Oaks, J. A. (2018). "François Viète's revolution in algebra" (PDF). Archive for History of Exact Sciences. 72 (3): 245–302. doi:10.1007/s00407-018-0208-0. S2CID 125704699. Retrieved November 8, 2022.
  42. ^ Kleiner 2007, "History of Linear Algebra" pp. 79–101.
  43. ^ Corry, Leo (December 6, 2012). Modern Algebra and the Rise of Mathematical Structures. Birkhäuser Basel. pp. 247–252. ISBN 9783034879170.
  44. ^ Riche, Jacques (2007). Beziau, J. Y.; Costa-Leite, Alexandre (eds.). From Universal Algebra to Universal Logic. Perspective on Universal Logic. Italy: Polimetrica International Scientific. pp. 3–39. ISBN 9788876990779.
  45. ^ Krömer, Ralph (2007). Tool and Object: A History and Philosophy of Category Theory. Science Networks. Historical Studies. Vol. 32. Springer Science & Business Media. pp. xxi–xxv, 1–91. ISBN 9783764375249.
  46. ^ Guicciardini, Niccolo (2017). "The Newton–Leibniz Calculus Controversy, 1708–1730" (PDF). In Schliesser, Eric; Smeenk, Chris (eds.). The Oxford Handbook of Newton. Oxford handbooks. Oxford University Press. doi:10.1093/oxfordhb/9780199930418.013.9. ISBN 9780199930418. Retrieved November 9, 2022.
  47. ^ O'Connor, J. J.; Robertson, E. F. (September 1998). "Leonhard Euler". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 9, 2022.
  48. ^ Franklin, James (2017). "Discrete and continuous: a fundamental dichotomy in mathematics". Journal of Humanistic Mathematics. 7 (2): 355–378. doi:10.5642/jhummath.201702.18. S2CID 6945363. Retrieved November 10, 2022.
  49. ^ Maurer, Stephen B. (1997). Rosenstein, Joseph G.; Franzblau, Deborah S.; Roberts, Fred S. (eds.). What is Discrete Mathematics? The Many Answers. Discrete Mathematics in the Schools. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. Vol. 36. American Mathematical Society. pp. 121–124. ISBN 9780821885789.
  50. ^ Downey, Rod (2014). Turing's Legacy: Developments from Turing's Ideas in Logic. Issue 42 of Lecture Notes in Logic. Cambridge University Press. pp. 260–261. ISBN 9781107043480.
  51. ^ Sipser, Michael (July 1992). The history and status of the P versus NP question. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp. 603–618. doi:10.1145/129712.129771.
  52. ^ Ewald, William (November 17, 2018). "The Emergence of First-Order Logic". Stanford Encyclopedia of Philosophy. Retrieved November 2, 2022.
  53. ^ Ferreirós, José (June 18, 2020). "The Early Development of Set Theory". Stanford Encyclopedia of Philosophy. Retrieved November 2, 2022.
  54. ^ Ferreirós, José (2001). "The Road to Modern Logic—An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676. Retrieved November 11, 2022.
  55. ^ Wolchover, Natalie (December 3, 2013). "Dispute over Infinity Divides Mathematicians". Scientific American. Retrieved November 1, 2022.
  56. ^ Zhuang, C. "Wittgenstein's analysis on Cantor's diagonal argument". PhilArchive. Retrieved November 18, 2022.
  57. ^ Avigad, Jeremy; Reck, Erich H. (December 11, 2001). ""Clarifying the nature of the infinite": the development of metamathematics and proof theory" (PDF). Carnegie Mellon Technical Report CMU-PHIL-120. Retrieved November 12, 2022.
  58. ^ Hamilton, Alan G. (1982). Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press. pp. 3–4. ISBN 9780521287616.
  59. ^ a b Snapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". Mathematics Magazine. 52 (4): 207–16. doi:10.2307/2689412. JSTOR 2689412.
  60. ^ a b Raatikainen, Panu (October 2005). "On the Philosophical Relevance of Gödel's Incompleteness Theorems". Revue Internationale de Philosophie. 59 (4): 513–534. doi:10.3917/rip.234.0513. JSTOR 23955909. S2CID 52083793. Retrieved November 12, 2022.
  61. ^ Moschovakis, Joan (September 4, 2018). "Intuitionistic Logic". Stanford Encyclopedia of Philosophy. Retrieved November 12, 2022.
  62. ^ McCarty, Charles (2006). "At the Heart of Analysis: Intuitionism and Philosophy". Philosophia Scientiæ, Cahier spécial 6: 81–94. doi:10.4000/philosophiascientiae.411.
  63. ^ Halpern, Joseph; Harper, Robert; Immerman, Neil; Kolaitis, Phokion; Vardi, Moshe; Vianu, Victor (2001). "On the Unusual Effectiveness of Logic in Computer Science" (PDF). Archived (PDF) from the original on March 3, 2021. Retrieved January 15, 2021.
  64. ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 10. Archived (PDF) from the original on October 9, 2022.
  65. ^ Rao, C.R. (1997). Statistics and Truth: Putting Chance to Work. World Scientific. ISBN 978-981-02-3111-8.
  66. ^ Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah (eds.). Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. ISBN 978-0-471-08073-2. MR 0607328.
  67. ^ Whittle 1994, pp. 10–11, 14–18.
  68. ^ Marchuk, Gurii Ivanovich (April 2020). "G I Marchuk's plenary: ICM 1970". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 13, 2022.
  69. ^ Johnson, Gary M.; Cavallini, John S. (September 1991). Phua, Kang Hoh; Loe, Kia Fock (eds.). Grand Challenges, High Performance Computing, and Computational Science. Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage. World Scientific. p. 28.
  70. ^ Trefethen, Lloyd N. (2008). Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). Numerical Analysis (PDF). The Princeton Companion to Mathematics. Princeton University Press. pp. 604–615. ISBN 9780691118802. Retrieved November 13, 2022.
  71. ^ Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (August 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neurosciences. 21 (8): 355–61. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604. S2CID 17414557.
  72. ^ See, for example, Wilder, Raymond L. Evolution of Mathematical Concepts; an Elementary Study. passim.
  73. ^ Zaslavsky, Claudia (1999). Africa Counts: Number and Pattern in African Culture. Chicago Review Press. ISBN 978-1-61374-115-3. OCLC 843204342.
  74. ^ Kline 1990, Chapter 1.
  75. ^ Boyer 1991, "Mesopotamia" pp. 24–27.
  76. ^ Heath, Thomas Little (1981) [1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. p. 1. ISBN 978-0-486-24073-2.
  77. ^ Mueller, I. (1969). "Euclid's Elements and the Axiomatic Method". The British Journal for the Philosophy of Science. 20 (4): 289–309. doi:10.1093/bjps/20.4.289. ISSN 0007-0882. JSTOR 686258.
  78. ^ Boyer 1991, "Euclid of Alexandria" p. 119.
  79. ^ Boyer 1991, "Archimedes of Syracuse" p. 120.
  80. ^ Boyer 1991, "Archimedes of Syracuse" p. 130.
  81. ^ Boyer 1991, "Apollonius of Perga" p. 145.
  82. ^ Boyer 1991, "Greek Trigonometry and Mensuration" p. 162.
  83. ^ Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.
  84. ^ Ore, Øystein (1988). Number Theory and Its History. Courier Corporation. pp. 19–24. ISBN 9780486656205.
  85. ^ Singh, A. N. (January 1936). "On the Use of Series in Hindu Mathematics". Osiris. 1: 606–628. doi:10.1086/368443. JSTOR 301627. S2CID 144760421.
  86. ^ Kolachana, A.; Mahesh, K.; Ramasubramanian, K. (2019). Use of series in India. Studies in Indian Mathematics and Astronomy. Sources and Studies in the History of Mathematics and Physical Sciences. Singapore: Springer. doi:10.1007/978-981-13-7326-8_20. ISBN 978-981-13-7325-1. S2CID 190176726.
  87. ^ Saliba, George (1994). A history of Arabic astronomy: planetary theories during the golden age of Islam. New York University Press. ISBN 978-0-8147-7962-0. OCLC 28723059.
  88. ^ Faruqi, Yasmeen M. (2006). "Contributions of Islamic scholars to the scientific enterprise". International Education Journal. Shannon Research Press. 7 (4): 391–399. Retrieved November 14, 2022.
  89. ^ Archibald, Raymond Clare (January 1949). "History of Mathematics After the Sixteenth Century". The American Mathematical Monthly. Part 2: Outline of the History of Mathematics. 56 (1): 35–56. doi:10.2307/2304570. JSTOR 2304570.
  90. ^ Sevryuk 2006, pp. 101–09.
  91. ^ Wolfram, Stephan (October 2000). Mathematical notation: Past and future. MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA. Retrieved November 16, 2022.
  92. ^ Douglas, Heather; Headley, Marcia Gail; Hadden, Stephanie; LeFevre, Jo-Anne (December 3, 2020). "Knowledge of Mathematical Symbols Goes Beyond Numbers". Journal of Numerical Cognition. 6 (3): 322–354. doi:10.5964/jnc.v6i3.293. S2CID 228085700.
  93. ^ Letourneau, Mary; Wright Sharp, Jennifer (2017). "AMS style guide" (PDF). American Mathematical Society. p. 75. Retrieved November 16, 2022.
  94. ^ Jansen, A. R.; Marriott, K.; Yelland, G. W. (2000). Constituent structure in mathematical expressions (PDF). Proceedings of the 22th Annual Meeting of the Cognitive Science Society. Vol. 22. p. 238. Retrieved November 16, 2022.
  95. ^ Rossi, Richard J. (2011). Theorems, Corollaries, Lemmas, and Methods of Proof. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. John Wiley & Sons. pp. 1–14, 47–48. ISBN 9781118030578.
  96. ^ "Earliest Uses of Some Words of Mathematics". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 19, 2022.
  97. ^ Silver, Daniel S. (2017). "The New Language of Mathematics". The American Scientist. 105 (6): 364–371. doi:10.1511/2017.105.6.364. Retrieved November 19, 2022.
  98. ^ Bellomo, Nicola; Preziosi, Luigi (December 22, 1994). Modelling Mathematical Methods and Scientific Computation. Mathematical Modeling. Vol. 1. CRC Press. p. 1. ISBN 9780849383311.
  99. ^ Hennig, Christian (2010). "Mathematical Models and Reality: A Constructivist Perspective". Foundations of Science. 15: 29–48. doi:10.1007/s10699-009-9167-x. S2CID 6229200. Retrieved November 17, 2022.
  100. ^ Frigg, Roman; Hartmann, Stephan (February 4, 2020). "Models in Science". Stanford Encyclopedia of Philosophy. Retrieved November 17, 2022.
  101. ^ Stewart, Ian (2018). "Mathematics, Maps, and Models". In Wuppuluri, Shyam; Doria, Francisco Antonio (eds.). The Map and the Territory: Exploring the Foundations of Science, Thought and Reality. The Frontiers Collection. Springer. pp. 345–356. doi:10.1007/978-3-319-72478-2_18. ISBN 9783319724782.
  102. ^ "The science checklist applied: Mathematics". undsci.berkeley.edu. Archived from the original on October 27, 2019. Retrieved October 27, 2019.
  103. ^ Mackay, A. L. (1991). Dictionary of Scientific Quotations. London: Taylor & Francis. p. 100. ISBN 9780750301060.
  104. ^ Bishop, Alan (1991). "Environmental activities and mathematical culture". Mathematical Enculturation: A Cultural Perspective on Mathematics Education. Norwell, Massachusetts: Kluwer Academic Publishers. pp. 20–59. ISBN 978-0-7923-1270-3. Archived from the original on December 25, 2020. Retrieved April 5, 2020.
  105. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. ISBN 9780387982694.
  106. ^ Nickles, Thomas (2013). "The Problem of Demarcation". Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. Chicago: The University of Chicago Press. p. 104. ISBN 9780226051826.
  107. ^ Pigliucci, Massimo (2014). "Are There 'Other' Ways of Knowing?". Philosophy Now. Archived from the original on May 13, 2020. Retrieved April 6, 2020.
  108. ^ a b Ferreirós, J. (2007). "Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss". In Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae. Springer Science & Business Media. pp. 235–268. ISBN 9783540347200.
  109. ^ Kuhn, Thomas S. (1976). "Mathematical vs. Experimental Traditions in the Development of Physical Science". The Journal of Interdisciplinary History. The MIT Press. 7 (1): 1–31. doi:10.2307/202372. JSTOR 202372.
  110. ^ Asper, Markus (2009). "The two cultures of mathematics in ancient Greece". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. Oxford Handbooks in Mathematics. OUP Oxford. pp. 107–132. ISBN 9780199213122.
  111. ^ Maddy, P. (2008). "How applied mathematics became pure" (PDF). The Review of Symbolic Logic. 1 (1): 16–41. doi:10.1017/S1755020308080027. S2CID 18122406. Retrieved November 19, 2022.
  112. ^ Silver, Daniel S. (2017). "In Defense of Pure Mathematics". In Pitici, Mircea (ed.). The Best Writing on Mathematics, 2016. Princeton University Press. pp. 17–26. ISBN 9780691175294.
  113. ^ Parshall, Karen Hunger (2022). "The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account". Bulletin of the American Mathematical Society. 59 (3): 405–427. doi:10.1090/bull/1754. S2CID 249561106. Retrieved November 20, 2022.
  114. ^ Stolz, Michael (2002). "The History Of Applied Mathematics And The History Of Society". Synthese. 133: 43–57. doi:10.1023/A:1020823608217. S2CID 34271623. Retrieved November 20, 2022.
  115. ^ Lützen, J. (2011). "Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century". In Schlote, K. H.; Schneider, M. (eds.). Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century. Frankfurt am Main: Verlag Harri Deutsch. Retrieved November 19, 2022.
  116. ^ Marker, Dave (July 1996). "Model theory and exponentiation". Notices of the American Mathematical Society. 43 (7): 753–759. Retrieved November 19, 2022.
  117. ^ Chen, Changbo; Maza, Marc Moreno (August 2014). Cylindrical Algebraic Decomposition in the RegularChains Library. International Congress on Mathematical Software 2014. Lecture Notes in Computer Science. Vol. 8592. Berlin: Springer. doi:10.1007/978-3-662-44199-2_65. Retrieved November 19, 2022.
  118. ^ Pérez-Escobar, José Antonio; Sarikaya, Deniz (2021). "Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy". European Journal for Philosophy of Science. 12 (1): 1–22. doi:10.1007/s13194-021-00435-9. S2CID 245465895.
  119. ^ Takase, M. (2014). "Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity". A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry. Vol. 5. Tokyo: Springer. pp. 393–399. doi:10.1007/978-4-431-55060-0_29. ISBN 978-4-431-55059-4.
  120. ^ Sarukkai, Sundar (February 10, 2005). "Revisiting the 'unreasonable effectiveness' of mathematics". Current Science. 88 (3): 415–423. JSTOR 24110208.
  121. ^ Wagstaff, Jr., Samuel S. (2021). "History of Integer Factoring" (PDF). In Bos, Joppe W.; Stam, Martijn (eds.). Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. Retrieved November 20, 2022.
  122. ^ "Curves: Ellipse". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 20, 2022.
  123. ^ Mukunth, Vasudevan (September 10, 2015). "Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry". The Wire. Retrieved November 20, 2022.
  124. ^ Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics". Proceedings of the American Academy of Arts and Sciences. 48 (11): 389–507. doi:10.2307/20022840. JSTOR 20022840.
  125. ^ a b c Borel, Armand (1983). "Mathematics: Art and Science". The Mathematical Intelligencer. Springer. 5 (4): 9–17. doi:10.4171/news/103/8. ISSN 1027-488X.
  126. ^ Hanson, Norwood Russell (November 1961). "Discovering the Positron (I)". The British Journal for the Philosophy of Science. The University of Chicago Press. 12 (47): 194–214. doi:10.1093/bjps/xiii.49.54. JSTOR 685207.
  127. ^ Ginammi, Michele (February 2016). "Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω particle". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 53: 20–27. Bibcode:2016SHPMP..53...20G. doi:10.1016/j.shpsb.2015.12.001.
  128. ^ Balaguer, Mark (2016). "Platonism in Metaphysics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Spring 2016 ed.). Metaphysics Research Lab, Stanford University. Archived from the original on January 30, 2022. Retrieved April 2, 2022.
  129. ^ See L. White (1947). "The locus of mathematical reality: An anthropological footnote". Philosophy of Science. 14 (4): 289–303. doi:10.1086/286957. S2CID 119887253. 189303; also in J.R. Newman (1956). The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.
  130. ^ Voskoglou, Michael Gr. (2017). "Studying the Winger's "Enigma" about the Unreasonable Effectiveness of Mathematics in the Natural Sciences". American Journal of Applied Mathematics and Statistics. 5 (3): 95–100. doi:10.12691/ajams-5-3-2 (inactive November 25, 2022).{{cite journal}}: CS1 maint: DOI inactive as of November 2022 (link)
  131. ^ a b c Mura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–85. doi:10.1007/BF01273907. JSTOR 3482762. S2CID 122351146.
  132. ^ a b Tobies, Renate; Neunzert, Helmut (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN 978-3-0348-0229-1. [I]t is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
  133. ^ Ziegler, Günter M.; Loos, Andreas (November 2, 2017). Kaiser, G. (ed.). "What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it. Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer. pp. 63–77. doi:10.1007/978-3-319-62597-3_5. ISBN 978-3-319-62596-6.
  134. ^ Franklin, James (July 8, 2009). Philosophy of Mathematics. pp. 104–106. ISBN 978-0-08-093058-9.
  135. ^ Cajori, Florian (1893). A History of Mathematics. American Mathematical Society (1991 reprint). pp. 285–286. ISBN 978-0-8218-2102-2.
  136. ^ Brown, Ronald; Porter, Timothy (January 2000). "The Methodology of Mathematics". The Mathematical Gazette. 79 (485): 321–334. doi:10.2307/3618304. JSTOR 3618304. S2CID 178923299.
  137. ^ Strauss, Danie (2011). "Defining mathematics". Acta Academica. 43 (4): 1–28. Retrieved November 25, 2022.
  138. ^ Hamami, Yacin (June 2022). "Mathematical Rigor and Proof" (PDF). The Review of Symbolic Logic. 15 (2): 409–449. doi:10.1017/S1755020319000443. S2CID 209980693. Retrieved November 21, 2022.
  139. ^ Peterson, Ivars (1988). The Mathematical Tourist. Freeman. p. 4. ISBN 978-0-7167-1953-3. A few complain that the computer program can't be verified properly, (in reference to the Haken–Apple proof of the Four Color Theorem).
  140. ^ Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs". Philosophy of Mathematics. Revue Internationale de Philosophie. 42 (167 (4)): 500–508.
  141. ^ Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwelld, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor". International Journal of Educational Research. 93: 91–100. doi:10.1016/j.ijer.2018.10.002. S2CID 149753721.
  142. ^ Yaftian, Narges (June 2, 2015). "The Outlook of the Mathematicians' Creative Processes". Procedia - Social and Behavioral Sciences. 191: 2519–2525. doi:10.1016/j.sbspro.2015.04.617.
  143. ^ Nadjafikhah, Mehdi; Yaftian, Narges (October 10, 2013). "The Frontage of Creativity and Mathematical Creativity". Procedia - Social and Behavioral Sciences. 90: 344–350. doi:10.1016/j.sbspro.2013.07.101.
  144. ^ van der Poorten, A. (1979). "A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)" (PDF). The Mathematical Intelligencer. 1 (4): 195–203. doi:10.1007/BF03028234. S2CID 121589323. Retrieved November 22, 2022.
  145. ^ Petkovi, Miodrag (September 2, 2009). Famous Puzzles of Great Mathematicians. American Mathematical Society. pp. xiii–xiv. ISBN 9780821848142.
  146. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 978-0-521-42706-7. Retrieved November 22, 2022. See also A Mathematician's Apology.
  147. ^ Alon, Noga; Goldston, Dan; Sárközy, András; Szabados, József; Tenenbaum, Gérald; Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015). Alladi, Krishnaswami; Krantz, Steven G. (eds.). "Reflections on Paul Erdős on His Birth Centenary, Part II". Notices of the American Mathematical Society. 62 (3): 226–247. doi:10.1090/noti1223.
  148. ^ See, for example Bertrand Russell's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his History of Western Philosophy. 1919. p. 60.
  149. ^ a b Iemhoff, Rosalie (2020). "Intuitionism in the Philosophy of Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Fall 2020 ed.). Metaphysics Research Lab, Stanford University. Archived from the original on April 21, 2022. Retrieved April 2, 2022.
  150. ^ Mullins, Ina V.S.; Martin, Micheal O.; Foy, Pierre; Kelly, Dana L.; Fishbein, Bethany (2020). TIMSS 2019 International Results in Mathematics and Science. TIMSS & PIRLS International Study Center, Lynch School of Education and Human Development, Boston College and International Association for the Evaluation of Educational Achievement. pp. 448–451. ISBN 978-1-889938-54-7.
  151. ^ Robson, Eleanor (2009). "Mathematics education in an Old Babylonian scribal school". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. OUP Oxford. ISBN 9780199213122.
  152. ^ Bernard, Alain; Proust, Christine; Ross, Micah (2014). "Mathematics Education in Antiquity". In Karp, A.; Schubring, G. (eds.). Handbook on the History of Mathematics Education. New York: Springer. pp. 27–53. doi:10.1007/978-1-4614-9155-2_3. ISBN 978-1-4614-9154-5.
  153. ^ Dudley, Underwood (April 2002). "The World's First Mathematics Textbook". Math Horizons. Taylor & Francis, Ltd. 9 (4): 8–11. JSTOR 25678363.
  154. ^ Siu, Man Keung (2004). "Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?". How Chinese Learn Mathematics (PDF). Series on Mathematics Education. Vol. 1. pp. 157–185. doi:10.1142/9789812562241_0006. ISBN 978-981-256-014-8. Retrieved November 26, 2022.
  155. ^ Jones, Phillip S. (1967). "The History of Mathematical Education". The American Mathematical Monthly. Taylor & Francis, Ltd. 74 (1): 38–55. doi:10.2307/2314867. JSTOR 2314867.
  156. ^ Luttenberger, Silke; Wimmer, Sigrid; Paechter, Manuela (2018). "Spotlight on math anxiety". Psychology Research Behavior Management. 11: 311–322. doi:10.2147/PRBM.S141421. PMC 6087017. PMID 30123014.
  157. ^ Monastyrsky 2001, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics."
  158. ^ Riehm 2002, pp. 778–82.
  159. ^ "Fields Medal | International Mathematical Union (IMU)". www.mathunion.org. Archived from the original on December 26, 2018. Retrieved February 21, 2022.
  160. ^ a b "Fields Medal". Maths History. Archived from the original on March 22, 2019. Retrieved February 21, 2022.
  161. ^ "Mathematics Prizes". Wolfram MathWorld. Retrieved November 17, 2022.
  162. ^ "About the Abel Prize". The Abel Prize. Archived from the original on April 14, 2022. Retrieved January 23, 2022.
  163. ^ "Abel Prize | mathematics award". Encyclopedia Britannica. Archived from the original on January 26, 2020. Retrieved January 23, 2022.
  164. ^ "CHERN MEDAL AWARD" (PDF). www.mathunion.org. June 1, 2009. Archived (PDF) from the original on June 17, 2009. Retrieved February 21, 2022.
  165. ^ "Chern Medal Award". International Mathematical Union (IMU). Archived from the original on August 25, 2010. Retrieved January 23, 2022.
  166. ^ "The Leroy P Steele Prize of the AMS". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved November 17, 2022.
  167. ^ Chern, S. S.; Hirzebruch, F. (September 2000). Wolf Prize in Mathematics. doi:10.1142/4149. ISBN 978-981-02-3945-9. Archived from the original on February 21, 2022. Retrieved February 21, 2022.
  168. ^ "The Wolf Prize". Wolf Foundation. Archived from the original on January 12, 2020. Retrieved January 23, 2022.
  169. ^ "Hilbert's Problems: 23 and Math". Simons Foundation. May 6, 2020. Archived from the original on January 23, 2022. Retrieved January 23, 2022.
  170. ^ Newton, Tommy (2007). "A New Approach to Hilbert's Third Problem" (PDF). Western Kentucky University. Archived (PDF) from the original on January 22, 2013. Retrieved February 21, 2022.
  171. ^ "Hilbert's Problems: 23 and Math". Simons Foundation. May 6, 2020. Retrieved January 23, 2022.
  172. ^ "The Millennium Prize Problems". Clay Mathematics Institute. Archived from the original on July 3, 2015. Retrieved January 23, 2022.
  173. ^ "Millennium Problems". Clay Mathematics Institute. Archived from the original on December 20, 2018. Retrieved January 23, 2022.

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