Mathematics - Knowledge

2367: 12044: 2829: 11178: 9687: 4412: 14647: 14583: 4142: 2466: 209: 2820: 971: 14671: 1807: 1533: 3304: 3931: 1978: 2539: 3526:—that is, its place among other human activities. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". 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. 14813: 3530:

in thought" from real instances set mathematics apart. In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task. For example, in lieu of a definition,

1287: 1302: 14683: 14595: 2792:. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not 14659: 2319: 14711: 14779: 14796: 14762: 4035: 728: 14830: 14745: 1647: 5907: 14619: 14728: 14607: 2982:. It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. 2165: 3600:

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.

3770:, which began publication in 1899. The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications. While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time. 705:) 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 3193: 1329:(9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term 3791:. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. An extreme example is 1147:. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that 3549:

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. Others take the perspective that mathematics is an investigation of

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phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical

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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

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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

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wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences " to foresee their "conjunctions and oppositions". As of 2023, astrology is no longer considered

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led to a surge in the development of applied mathematics in the US and elsewhere. Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the

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Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining

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Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of

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In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure

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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"

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Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Tat Dat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor; Mclaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, Thi Hoai An; Tran, Nam Trung;

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During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics. Some students studying math may

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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

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the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is

3112: 2720:"). 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 1899:

that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by

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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."

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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.

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inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a

3616:. 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. 2851:

and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later,

2654:). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of 3763:

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

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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

3400:(evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy. 10493: 4019:

and mathematical objects are highly abstract. However, popular mathematics writing can overcome this by using applications or cultural links. Despite this, mathematics is rarely the topic of popularization in printed or televised media.

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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

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In his work, Diophantus focused on deducing the arithmetic properties of figurate numbers, such as deducing the number of sides, the different ways a number can be expressed as a figurate number, and the formulation of the arithmetic

3850:, 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 10373: 2495:

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.

2462:(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. 4111:", was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and at least thirteen of the problems (depending how some are interpreted) have been solved. 11286:

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

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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,

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and animal bodies more generally (at least on the surface). Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.

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von Davier, Matthias; Foy, Pierre; Martin, Michael O.; Mullis, Ina V.S. (2020). "Examining eTIMSS Country Differences Between eTIMSS Data and Bridge Data: A Look at Country-Level Mode of Administration Effects".

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Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. Comparable evidence has been unearthed for scribal mathematics training in the

1352:(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 1711:; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of 3119:

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

2387:, 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 3227:, study ecosystems such as the predator-prey model, measure pollution diffusion, or to assess climate change. The dynamics of a population can be modeled by coupled differential equations, such as the 2776:

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

10485: 3473:. 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. 3142:

is considered to be mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in

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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

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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

497:(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 350:. 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 2665:

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

6350: 450:), meaning "that which is learnt", "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in 2860:

with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred with the tradition of

3982: 2063:, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of 12323:

Alon, Noga; Goldston, Dan; Sárközy, András; Szabados, József; Tenenbaum, Gérald; Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015). Alladi, Krishnaswami; Krantz, Steven G. (eds.).

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This the length of the original paper that does not contain the proofs of some previously published auxiliary results. The book devoted to the complete proof has more than 1,000 pages.

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mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.

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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

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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

12704:. 22nd Biennial Conference of The Australian Association of Mathematics Teachers. Fremantle, Western Australia: Australian Association of Mathematics Teachers. pp. 125–126. 3802:

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

1872:, 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 11550: 10327: 1342: 3024:

In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and

2712:. 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, " 2938:

because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities,

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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

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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:

13512:(1–4) (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.): 97–229. 3086: 520:") 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, 10893: 12619: 9678:. Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers. Vol. 66. pp. S1–S13. 2927: 12979: 5132:

It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself.

2716:" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " 1377:(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 3051:

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

13742: 13538:. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. 10894:"Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. | The Online Books Page" 11057: 2632: 1506:(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 974:

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

13697: 2503:, 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 11838: 1857:. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring 5088:... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. 1849:. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per 10919: 9395:

Gozwami, Pinkimani; Singh, Madan Mohan (2019). "Integer Factorization Problem". In Ahmad, Khaleel; Doja, M. N.; Udzir, Nur Izura; Singh, Manu Pratap (eds.).

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Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics".

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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

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Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwell, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor".

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Popular mathematics is the act of presenting mathematics without technical terms. Presenting mathematics may be hard since the general public suffers from

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Humans, as well as some other animals, find symmetric patterns to be more beautiful. Mathematically, the symmetries of an object form a group known as the

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are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticise the concept of

1868:. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in 4389: 13144: 4122:, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the 1560:

and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

11050:"L'astrologie à l'épreuve : ça ne marche pas, ça n'a jamais marché ! / Afis Science – Association française pour l'information scientifique" 6493: 6121: 3241:

to determine whether a new treatment works. Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.

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This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a

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consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses (from left to right).

4710: 3580:. Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' 13214: 4515: 2903:, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred. 323:, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration. 10863: 9471: 9263: 6339: 11977: 8697: 3396:

denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences. The study of

2959: 13731:(previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. 12554:

Bender, Sara (September 2020). "The Rorschach Test". In Carducci, Bernardo J.; Nave, Christopher S.; Mio, Jeffrey S.; Riggio, Ronald E. (eds.).

14043: 13445: 12762: 4457: 12860: 10845: 10423: 1093:, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: 13678: 13501: 12728: 4406: 3844:

expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book

2505: 2260:, is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be 1631:, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications. 236: 11088: 8495: 4733: 2129:. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- 1689:, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of 893:

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is

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

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seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that

898: 4490: 3352:. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain. 1888:(that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, 3873:(as in science). The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. 1552:. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by 13081: 8758: 5435: 5337: 4334:

For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software

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Ginammi, Michele (February 2016). "Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω particle".

4962: 3367:, which explains phases of economic growth or crisis. Towards the end of the 19th century, mathematicians extended their analysis into 12220: 11519: 7703: 7621: 6788: 2578:

and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally

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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

666:. This division into four main areas–arithmetic, geometry, algebra, calculus–endured until the end of the 19th century. Areas such as 13916: – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and 10541: 2975: 9812:"Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy" 8840: 13008: 12646: 9967: 3787:

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a

11577:, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Apple proof of the 9990: 678:

has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.

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This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.

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Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians

8009: 7314: 3092:, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments. 3028:. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, 2411:. The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe. 12827: 11812: 3088:

In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown

13650: 12021: 9934: 9027: 4372:(in German and English). Vol. 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. 4299:

This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without

1703:, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or 14011: 13989: 13963: 13941: 13895: 13869: 13834: 13806: 13736: 13624: 13480: 13419: 13362: 13330: 13308: 13278: 13187: 13075: 12772: 12738: 12603: 12571: 12293: 12256: 11995: 11989: 11859: 11837:

Siu, Man Keung (2004). "Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?".

11743: 11699: 11656: 11386: 11349: 11319: 11268: 11172: 11026: 10959: 10826: 10782: 10745: 10638: 10563: 10188: 9872: 9784:. International Congress on Mathematical Software 2014. Lecture Notes in Computer Science. Vol. 8592. Berlin: Springer. 9509: 9404: 9372: 9305: 9233: 9208: 9173: 9137: 9104: 9073: 8942: 8796: 8534: 8404: 8368: 8285: 8140: 8064: 7979: 7831: 7478: 7401: 7028: 6736: 6666: 6620: 6436: 6156: 5228: 5109: 5065: 4930: 4769: 4354:

Hipólito, Inês Viegas (August 9–15, 2015). "Abstract Cognition and the Nature of Mathematical Proof". In Kanzian, Christian;

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provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.

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intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.

1043: 12890: 12676: 9721: 1889: 14406: 13999: 12968: 12946: 9127: 6750: 6658: 3183: 415:, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary 172: 11306:. Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer. pp. 63–77. 4839: 14699: 13240: 9853:"Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity" 7167: 7115: 5451:

The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications.

4922: 2943: 1854: 12916: 12022:"Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career" 7214: 14551: 14036: 13720: 7676: 3865:. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are 2861: 1144: 690: 682: 416: 408: 404: 13670: 12514: 11049: 14623: 13574: 13111: 10449:

Tendances nouvelles en modélisation pour l'environnement, actes du congrès «Programme environnement, vie et sociétés»

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Riche, Jacques (2007). "From Universal Algebra to Universal Logic". In Beziau, J. Y.; Costa-Leite, Alexandre (eds.).

6570: 5993: 5938: 5784: 5647: 5587: 5530: 5522: 5204: 5164: 4540:(September 2011). "The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?". 3687: 2935: 1750: 740: 299:

from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called

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The connection between mathematics and material reality has led to philosophical debates since at least the time of

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Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as

14087: 5489: 2926:, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the 2754: 2591: 2563: 468:), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, 10911: 8355:. Sources and Studies in the History of Mathematics and Physical Sciences. Singapore: Springer. pp. 438–461. 3861:

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional

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Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century

9536: 8738: 8604: 7551: 4307:. Even with this modern technology, it may take years of human work for writing down a completely detailed proof. 3593: 3155: 2331: 1932: 1865: 744: 559:) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective 14717: 10299: 9756: 7920:. Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage. World Scientific. p. 28. 7738: 3952: 3713:

in Egypt. Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized

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At the start of the 20th century, there was a development to express historical movements in formulas. In 1922,

1025:). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by 1013:

that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (

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

3536: 3344:. This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual 3234: 2766: 2499:

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and

1616: 17: 14819: 12595: 10983: 9537:"The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account" 8441: 5031: 4917:(1995). "What Augustine Didn't Say About Mathematicians". In Alexanderson, Gerald L.; Mugler, Dale H. (eds.). 3658:, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the 443: 14785: 14029: 13978: 13716: 13136: 12797:, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics." 11304:"What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it 11208:

Mura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences".

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to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a

1850: 1622: 14802: 8252: 7082:. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp. 603–618. 3540:

summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:

2230:) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a 14865: 14611: 13909: 12309: 11425: 11335: 6837: 6428: 6109: 3835: 3662:

academic disciplines. Prominent careers for professional mathematicians include math teacher or professor,

3228: 3139: 3129: 3002: 2299:, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are 1017:) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( 222: 31: 13921: 10224: 7884: 1128:). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. 14526: 14082: 13824: 13724: 10733: 10694: 9049: 4834: 4806: 4773: 4705: 4171: 4161: 4029: 3432:. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, 2024: 65: 13884: 13858: 13842: 13210: 11919:"Introduction: the history of mathematics teaching. Indicators for modernization processes in societies" 11018:

Comprendre et maîtriser la nature au Moyen Age: mélanges d'histoire des sciences offerts à Guy Beaujouan

10144:

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

9714:"Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century" 6274: 4505: 2847:

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of

2828: 14637: 14097: 13781: 9440: 9253: 8099: 7971: 7445: 7377: 7012: 5156: 4765: 4057: 3651: 3517: 2780:, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a 1777: 710: 13795: 13293: 10663: 8240: 3779:

approaches, by interactions with parents and teachers, and by tailored treatments for the individual.

2346:. Other notable developments of Indian mathematics include the modern definition and approximation of 14734: 14675: 14511: 14483: 14120: 10998:"Modern Macroeconomic Models as Tools for Economic Policy | Federal Reserve Bank of Minneapolis" 10825:

Kremer, Michael; Rao, Gautam; Schilbach, Frank (2019). "Chapter 5 Behavioral development economics".

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Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium

4361:

Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums

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axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.

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Often the fundamental postulate of mathematical economics is that of the rational individual actor –

3133: 2838: 2773: 2092: 1690: 1594: 826: 761: 296: 187: 177: 167: 13430: 12359:'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his 9933:. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. 9895:

Sarukkai, Sundar (February 10, 2005). "Revisiting the 'unreasonable effectiveness' of mathematics".

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were then studied by mathematicians, but now are considered as belonging to physics. The subject of

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concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a

2575: 2130: 1982: 1873: 1685:

were two major problems of discrete mathematics solved in the second half of the 20th century. The

1671: 1654:. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state. 1386: 1341:

meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of

1132: 922: 894: 3061: 14441: 14431: 14401: 14335: 14070: 10274: 9670: 9640: 8550: 8464: 7110: 6720: 6034: 5027: 4996: 4452: 4166: 3589: 3258:

and climatology use probabilistic models to predict the risk of natural catastrophes. Similarly,

2963: 2947: 2915: 2567: 2533: 2529: 2399:

to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as

2052: 1194: 1148: 1065:

Euclidean geometry was developed without change of methods or scope until the 17th century, when

941: 407:

has led to a correlated increase in the development of both. At the end of the 19th century, the

13030: 11080: 10249: 8129: 6705: 1915:

These problems and debates led to a wide expansion of mathematical logic, with subareas such as

403:

were introduced as new fields. Since then, the interaction between mathematical innovations and

14751: 14539: 14436: 14416: 14411: 14340: 14065: 13497: 11726:; Ross, Micah (2014). "Mathematics Education in Antiquity". In Karp, A.; Schubring, G. (eds.). 10104: 7826:. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. 6996: 6315: 3851: 3470: 3356:

theoretical concepts into quantifiable predictions that can be tested against real-world data.

3337: 3279: 3187: 2571: 2392: 2227: 2154: 2106: 2080: 1909: 1785: 1610: 1583: 1557: 1503: 1226: 937: 874:

of Alexandria. The modern study of number theory in its abstract form is largely attributed to

859: 663: 400: 182: 157: 60: 13542: 11683: 11376: 11339: 11016: 10772: 9852: 9360: 9289: 9053: 7438:""Clarifying the nature of the infinite": the development of metamathematics and proof theory" 7000: 6913: 6560: 4123: 3607:) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the 3592:

has allowed proof lengths to further expand, The result of this trend is a philosophy of the

2989:

of natural numbers that was discovered more than 2,000 years before its common use for secure

2422:, with innovations that revolutionized mathematics, such as the introduction of variables and 697:

first-level areas. Some of these areas correspond to the older division, as is true regarding

14566: 14496: 14373: 14297: 14236: 14221: 14216: 14193: 14075: 13057: 12852: 12246: 11165:

The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature

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Johnson, Gary M.; Cavallini, John S. (September 1991). Phua, Kang Hoh; Loe, Kia Fock (eds.).

7437: 6652: 6310: 5378: 5348: 4919:

Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories

4863: 4191: 4181: 4176: 4108: 4089: 3643: 3633: 2865: 2559: 2525: 2480: 2431: 2211: 2014: 1998: 1773: 1730: 1527: 1353: 1306: 1190: 1113: 1070: 883: 855: 644: 627: 281: 192: 106: 35: 12699: 12556:

The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment

11646: 11258: 10419: 10178: 7504:(September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". 7468: 6610: 6338: 5924: 14855: 14546: 14426: 14421: 14345: 14246: 13612: 12537: 12474: 12187: 10151: 9324:(1976). "Mathematical vs. Experimental Traditions in the Development of Physical Science". 9196: 8788: 8609:. MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA. 7997: 7849: 7793: 7696: 7613: 6776: 6728: 6282: 4666: 4569: 4466: 4016: 3846: 3741: 3604: 3384: 3220: 3151: 3033: 2666: 2511: 2469: 2323: 2312: 2296: 2284: 2280: 2256: 2142: 2114: 2098: 2040: 1928: 1905: 1641: 1378: 1140: 1031: 953: 949: 887: 743:

between being prime and being a value of a quadratic polynomial, a conjecture now known as

655: 383: 152: 111: 80: 13170: 11299: 8216: 8204: 5829:

Straume, Eldar (September 4, 2014). "A Survey of the Development of Geometry up to 1870".

8: 14561: 14471: 14393: 14292: 14226: 14183: 14173: 14153: 13288: 13004: 12620:"Lecture 8: Translation Symmetry | Physics III: Vibrations and Waves | Physics" 10710: 10545: 10223:. Proceedings of the International Congress of Mathematicians. p. 31. Archived from 9293: 8828: 6924:. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. Vol. 36. 5344: 4186: 4010: 3887: 3774:

develop an apprehension or fear about their performance in the subject. This is known as

3655: 3638:

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a

3569: 3341: 3283: 3224: 2986: 2805: 2785: 2729: 2713: 2686: 2678: 2611: 2427: 2415: 2384: 2343: 2304: 2207: 2110: 2072: 2048: 1944: 1858: 1781: 1686: 1588: 1491: 1463: 1445: 1423: 1390: 1349: 1258: 1236: 1206: 1184: 1162: 1152: 1121: 1109: 945: 918: 667: 528:, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians. 370: 362:. Other areas are developed independently from any application (and are therefore called 359: 358:, are developed in close correlation with their applications and are often grouped under 308: 292: 253: 13616: 12638: 12478: 10155: 9959: 8192: 4470: 14651: 14587: 14506: 14446: 14378: 14368: 14307: 14282: 14158: 14115: 14110: 13905: 13531: 13384: 12506: 12443: 12435: 12400: 12392: 12212: 12097: 12070: 11940: 11899: 11784: 11776: 11627: 11578: 11542: 11457: 11449: 11233: 11225: 11130: 10365: 10124: 10037: 9904: 9833: 9679: 9609: 9558: 9463: 9341: 8986: 8913: 8742: 8667: 8659: 8570: 8487: 8374: 8333: 8325: 8228: 8181: 7946: 7579: 7571: 7529: 7389: 7285: 7261: 7237: 7091: 7050: 6971: 6887: 6485: 6453: 6243: 6211: 6190:"Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria" 6145:

Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys

6083: 6051: 5985: 5888: 5830: 5806: 5745: 5713: 5669: 5609: 5552: 5514: 5403: 5291: 5250: 5216: 4880: 4678: 4646: 4630: 4581: 4506:"Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion" 4482: 4282:

However, some advanced methods of analysis are sometimes used; for example, methods of

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of vibration are in simple ratios. For example, an octave doubles the frequency and a

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Mathematization of the social sciences is not without risk. In the controversial book

2465: 1579:. Analysis includes many subareas shared by other areas of mathematics which include: 14663: 14646: 14582: 14302: 14287: 14231: 14178: 14007: 13985: 13959: 13937: 13891: 13865: 13830: 13802: 13732: 13620: 13605: 13588: 13580: 13570: 13476: 13415: 13408: 13388: 13376: 13368: 13358: 13326: 13304: 13300: 13274: 13183: 13071: 12768: 12734: 12599: 12567: 12498: 12490: 12447: 12404: 12289: 12252: 12216: 12102: 12040: 11985: 11944: 11855: 11788: 11739: 11695: 11652: 11631: 11546: 11488: 11461: 11382: 11345: 11315: 11264: 11237: 11168: 11157: 11134: 11022: 10965: 10955: 10949: 10778: 10741: 10714: 10569: 10559: 10400: 10369: 10357: 10184: 10086: 9868: 9837: 9779: 9654: 9635: 9589: 9562: 9505: 9418: 9410: 9400: 9368: 9301: 9249: 9229: 9204: 9169: 9133: 9069: 8966: 8938: 8917: 8905: 8897: 8889: 8810: 8802: 8792: 8750: 8671: 8530: 8491: 8410: 8400: 8378: 8364: 8337: 8281: 8173: 8136: 8070: 8060: 8052: 8035: 8001: 7985: 7975: 7962: 7954: 7921: 7853: 7837: 7827: 7797: 7781: 7771: 7657: 7644: 7521: 7474: 7415: 7407: 7397: 7310: 7277: 7269: 7253: 7188: 7180: 7172: 7136: 7128: 7120: 7042: 7034: 7024: 6963: 6955: 6947: 6937: 6879: 6871: 6863: 6742: 6732: 6680: 6672: 6662: 6626: 6616: 6584: 6576: 6566: 6489: 6477: 6469: 6461: 6402: 6394: 6386: 6247: 6235: 6227: 6219: 6170: 6162: 6152: 6087: 6075: 6067: 6059: 6007: 5999: 5989: 5952: 5944: 5934: 5930: 5892: 5880: 5872: 5798: 5790: 5780: 5737: 5729: 5721: 5673: 5661: 5653: 5643: 5613: 5601: 5593: 5583: 5556: 5544: 5536: 5526: 5427: 5419: 5411: 5315: 5307: 5299: 5286: 5269: 5254: 5242: 5234: 5224: 5178: 5170: 5160: 5123: 5115: 5105: 5079: 5071: 5061: 4944: 4936: 4926: 4914: 4896: 4888: 4872: 4670: 4654: 4638: 4573: 4557: 4381: 4373: 4287: 4270: 4156: 4141: 3855: 3699: 3695: 3671: 3531: 3307: 3167: 2939: 2923: 2455: 2443: 2238: 1838: 1745: 1708: 1682: 1507: 1495: 1394: 1389:), on which generalizations of arithmetic operations are often valid. The concept of 1370: 1291: 1244: 1117: 1098: 702: 650: 374: 208: 13635: 12819: 11803: 11111:

White, L. (1947). "The locus of mathematical reality: An anthropological footnote".

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The Map and the Territory: Exploring the Foundations of Science, Thought and Reality

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Perhaps the foremost mathematician of the 19th century was the German mathematician

2447: 1215:, the study of curves, surfaces, and their generalizations, which are defined using 1193:, the study of curves, surfaces, and their generalizations, which are defined using 1066: 854:

Number theory was once called arithmetic, but nowadays this term is mostly used for

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International Encyclopedia of the Social & Behavioral Sciences (Second Edition)

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proof that can not be considered infallible, but has a probability attached to it.

3504:

Nevertheless, Platonism and the concurrent views on abstraction do not explain the

3479:

summarized this view of mathematics reality as follows, and provided quotations of

3421: 3364: 3197: 2994: 2919: 2888: 2884: 2877: 2873: 2809: 2138: 2068: 1956: 1948: 1901: 1869: 1602: 1572: 1449: 1403: 1264: 1166: 1139:. By questioning that postulate's truth, this discovery has been viewed as joining 1047: 875: 686: 659: 536: 412: 363: 343: 13917: 13776: 13692: 13549: 13471:

A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

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is taken in its modern sense, which is, roughly speaking, the art of manipulating

2819: 1935:. Although these aspects of mathematical logic were introduced before the rise of 14456: 14383: 14312: 14105: 13951: 13853: 13849: 13816: 13493: 13469: 13344: 13266: 13177: 12882: 12171: 12154: 12138: 12121: 9931:

Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL

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Kolachana, A.; Mahesh, K.; Ramasubramanian, K. (2019). "Use of series in India".

7993: 7845: 7789: 7319: 7020: 6917: 5854: 4779: 4662: 4565: 4420: 4304: 4060:) to up to four individuals. It is considered the mathematical equivalent of the 3998: 3584:, rigorous proofs can require hundreds of pages to express, such as the 255-page 3484: 3433: 3397: 3316: 3029: 3014: 2789: 2753:

concepts, imply the need to change the mathematical model used. For example, the

2450:(1596–1650) for reducing geometry to algebra, and the development of calculus by 2355: 2276: 2010: 1952: 1759: 1712: 1537: 1499: 1479: 1360: 1250: 1200: 1180: 1059: 1051: 979: 970: 914: 821: 671: 521: 451: 347: 327: 134: 12938: 12563: 11735: 11311: 10547:

Beyond the Molecular Frontier: Challenges for Chemistry and Chemical Engineering

10163: 8787:. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. 8397:

A history of Arabic astronomy: planetary theories during the golden age of Islam

6709: 6615:. Milano, Italy: Polimetrica International Scientific Publisher. pp. 3–39. 1001:

A fundamental innovation was the ancient Greeks' introduction of the concept of

14687: 14534: 14461: 14168: 13464: 13271:

Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur

12594:. Princeton Science Library. Vol. 47. Princeton University Press. p. 12388: 12283: 11851: 11687: 11679: 11623: 11372: 10211: 9828: 9811: 9436: 9258: 8465:"Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages" 7950: 7672: 7075: 6858: 6372: 5868: 4829: 4537: 4355: 3985: 3942: 3733: 3714: 3639: 3573: 3250: 3238: 3201: 3171: 2781: 2651: 2583: 2579: 2459: 2419: 2315:, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD). 2300: 2242: 2223: 2126: 2122: 2076: 1667: 1598: 1576: 1532: 1433: 1365: 1268: 1174: 1074: 1055: 879: 756: 490: 479: 392: 316: 162: 14768: 13354: 13236: 11935: 11918: 11538: 10120: 9605: 9459: 9422: 8982: 8483: 8271: 7419: 6933: 6449: 5639: 5246: 5212: 4801: 14849: 14322: 14254: 14206: 13973: 13925: 12912: 12494: 12375:

Cazden, Norman (October 1959). "Musical intervals and simple number ratios".

11254: 10945: 10718: 10361: 10090: 9201:

Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists

8893: 8746: 8663: 8392: 8177: 8074: 8005: 7819: 7680: 7525: 7501: 7433: 7411: 7393: 7257: 7241: 7176: 7124: 7046: 6951: 6883: 6867: 6746: 6630: 6465: 6457: 6390: 6223: 6215: 6206: 6189: 6174: 6063: 6055: 5977: 5850: 5725: 5717: 5415: 5407: 5303: 5295: 5238: 5220: 5196: 4948: 4876: 4761: 4700: 4642: 4634: 4613:(December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". 4561: 4448: 4385: 4377: 4266: 3897: 3892:

Notes that sound well together to a Western ear are sounds whose fundamental

3703: 3647: 3438: 3393: 3372: 3147: 3006: 2979: 2931: 2708: 2635: 2484: 2396: 2339: 2199: 2118: 2032: 1977: 1896: 1881: 1726: 1700: 1564: 1082: 906: 863: 739:. The dark diagonal lines in the spiral hint at the hypothesized approximate 736: 722: 698: 675: 612: 432: 269: 87: 13592: 13380: 13033:. School of Mathematics and Statistics, University of St Andrews, Scotland. 10969: 9966:. School of Mathematics and Statistics, University of St Andrews, Scotland. 9859:. Mathematics for Industry. Vol. 5. Tokyo: Springer. pp. 393–399. 8814: 8754: 8414: 8169: 7891:. School of Mathematics and Statistics, University of St Andrews, Scotland. 7801: 7281: 7192: 7140: 6967: 6684: 6588: 6406: 6079: 6026: 6011: 5956: 5802: 5627: 5605: 5182: 5127: 3841: 3303: 2492: 14264: 14259: 14163: 13879: 13820: 13403: 12587: 12462: 12106: 10612: 10573: 10081: 10064: 10060: 9290:"Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss" 9166:

Mathematical Enculturation: A Cultural Perspective on Mathematics Education

8909: 8884: 8863: 7857: 7692: 6481: 6239: 5741: 5665: 5548: 5431: 5319: 4900: 4861:

Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms".

4674: 4577: 4478: 4104: 4053: 4039: 3946: 3862: 3775: 3737: 3718: 3663: 3523: 3476: 3409: 3376: 3345: 3263: 3216: 3163: 3143: 3056: 3018: 2907: 2853: 2717: 2682: 2538: 2451: 2404: 2400: 2371: 2308: 2246: 2170: 2134: 2064: 1920: 1916: 1834: 1830: 1811: 1734: 1717: 1651: 1545: 1429: 1417: 1407: 1326: 1310: 995: 926: 902: 387:. Since its beginning, mathematics was primarily divided into geometry and 268:

and mathematics itself. There are many areas of mathematics, which include

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Dudley, Underwood (April 2002). "The World's First Mathematics Textbook".

11260:

Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry

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Singh, A. N. (January 1936). "On the Use of Series in Hindu Mathematics".

7385: 7087: 5884: 5579: 5083: 2689:. A proven instance that forms part of a more general finding is termed a 2249:

appeared to have considered it a subject in its own right. Around 300 BC,

2234:

numeral system which is still in use today for measuring angles and time.

1203:, the study of shapes that are not necessarily embedded in a larger space. 14466: 14130: 13600: 13318: 12279: 12188:"A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)" 12087: 9321: 9011: 8654: 8637: 7958: 7688: 7567: 7350: 7342: 6047: 5572:

The Queen of Mathematics: A Historically Motivated Guide to Number Theory

5374: 5148: 5144: 5017: 4986: 4079: 4061: 3930: 3893: 3796: 3568:. This means that the definitions must be absolutely unambiguous and the 3480: 3417: 3368: 3267: 3259: 2777: 2725: 2721: 2655: 2643: 2551: 2438:

in 1614, which greatly simplified numerical calculations, especially for

2435: 2231: 2219: 2195: 2183: 1924: 1846: 1755: 1568: 1469: 1439: 1086: 1078: 1010: 925:, which asserts that every even integer greater than 2 is the sum of two 732: 600: 575: 355: 331: 125: 12020:

Rowan-Kenyon, Heather T.; Swan, Amy K.; Creager, Marie F. (March 2012).

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Wagh, Sanjay Moreshwar; Deshpande, Dilip Abasaheb (September 27, 2012).

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The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae

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study of these applications may give new insights on the "pure theory".

319:

to already established results. These results include previously proved

14451: 14330: 14125: 13535: 12439: 12396: 12208: 11982:

International Association for the Evaluation of Educational Achievement

11903: 11453: 11229: 11221: 11087:(Spring 2016 ed.). Metaphysics Research Lab, Stanford University. 10941: 10352: 10343: 9991:"Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry" 8574: 7917:

Grand Challenges, High Performance Computing, and Computational Science

7533: 7265: 7249: 7004: 5708: 5691: 4650: 4453:"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" 4072: 4046: 3826: 3752: 3679: 3522:

There is no general consensus about a definition of mathematics or its

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in a concise, unambiguous, and accurate way. This notation consists of

2488: 2479:, who made numerous contributions to fields such as algebra, analysis, 2476: 2318: 2279:. He developed formulas for calculating the surface area and volume of 2261: 1968: 1893: 1823: 1801: 1721: 1453: 1322: 1254: 1216: 871: 634:

which, roughly speaking, consists of the study and the manipulation of

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Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs".

10128: 9857:

A Mathematical Approach to Research Problems of Science and Technology

9744: 9683: 9345: 9168:. Norwell, Massachusetts: Kluwer Academic Publishers. pp. 20–59. 8553:(January 1949). "History of Mathematics After the Sixteenth Century". 8329: 8185: 7729: 4082:

for lifetime achievement, introduced in 2009 and first awarded in 2010

3702:

starting around 300 BCE. The oldest known mathematics textbook is the

3408:

Some renowned mathematicians have also been considered to be renowned

12486: 12341: 12324: 10033: 7763: 6815: 6565:(2nd revised ed.). Germany: Birkhäuser Basel. pp. 247–252. 6556: 5632:

Number Theory: An Approach Through History From Hammurapi to Legendre

5398: 3989: 3814: 3675: 3608: 3581: 3577: 3293: 3289: 3037: 2856:

introduced infinitesimal calculus for explaining the movement of the

2697: 2690: 2670: 2603: 2587: 2543: 2439: 2191: 1806: 1663: 1659: 1473: 1018: 991: 620: 517: 513: 502: 455: 396: 13729:

Probability, statistics and optimisation: A Tribute to Peter Whittle

13517: 12431: 11917:

Schubring, Gert; Furinghetti, Fulvia; Siu, Man Keung (August 2012).

11895: 11811:. History and Pedagogy of Mathematics conference, July 16–20, 2012. 11445: 11429: 11054:

Afis Science – Association française pour l’information scientifique

8566: 7517: 7233: 6810: 2978:

is a phenomenon that was named and first made explicit by physicist

1658:

Discrete mathematics, broadly speaking, is the study of individual,

1286: 14355: 14274: 14201: 11882:

Jones, Phillip S. (1967). "The History of Mathematical Education".

11651:. Practical Career Guides. Rowman & Littlefield. pp. 1–3. 11126: 10555: 9337: 8875: 8429: 8321: 7815: 7552:"On the Philosophical Relevance of Gödel's Incompleteness Theorems" 5104:(1st ed.). New Delhi, India: Mittal Publications. p. 27. 4553: 4300: 4043: 4034: 3994: 3830: 3782: 3756: 3748: 3613: 3429: 3089: 3052: 3025: 2990: 2954:

mathematics". However, these terms are still used in names of some

2891:

increasingly focused their research on internal problems, that is,

2758: 2292: 2215: 2179: 1994: 1990: 1940: 1936: 1842: 1814:

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

1523: 1483: 1318: 1222: 965: 727: 639: 608: 335: 277: 102: 92: 13607:

In Search of a Better World: Lectures and Essays from Thirty Years

13372: 11846:. Series on Mathematics Education. Vol. 1. pp. 157–185. 11381:. American Mathematical Society (1991 reprint). pp. 285–286. 10524:

The Use of Statistical Methods in the Analysis of Clinical Studies

9414: 9226:

Philosophy of Pseudoscience: Reconsidering the Demarcation Problem

8806: 8731:

Proceedings of the Annual Meeting of the Cognitive Science Society

8156:

Mueller, I. (1969). "Euclid's Elements and the Axiomatic Method".

7989: 7184: 7132: 7038: 6875: 6676: 6580: 6398: 6166: 6003: 5835: 5794: 5423: 4056:, established in 1936 and awarded every four years (except around 2669:

that provide a standard foundation for communication. An axiom or

2574:

and any other mathematical objects, and then assembling them into

1959:, contributed in turn to the expansion of these logical theories. 1108:

unrelated to circles and lines. Such curves can be defined as the

14140: 13860:

What Is Mathematics?: An Elementary Approach to Ideas and Methods

13067: 10664:"Using Mathematical Models to Investigate Planetary Habitability" 10326:

Trieu, Thi Diep; Urban, Josef; Vu, Ky; Zumkeller, Roland (2017).

9367:. Oxford Handbooks in Mathematics. OUP Oxford. pp. 107–132. 6842:"Discrete and Continuous: A Fundamental Dichotomy in Mathematics" 6275:"Figurate Numbers: A Historical Survey of an Ancient Mathematics" 5776: 4236: 3934: 3667: 3413: 3212: 3111: 2998: 2745: 2555: 2500: 2388: 2376: 1739: 1549: 1482:, which is widely used for the study of the logical structure of 1359:

Until the 19th century, algebra consisted mainly of the study of

1301: 1281: 1014: 897:. This conjecture was stated in 1637 by Pierre de Fermat, but it 790: 635: 631: 569: 339: 320: 273: 257: 97: 14836: 13584: 11145:. Vol. 4. New York: Simon and Schuster. pp. 2348–2364. 9781:

Cylindrical Algebraic Decomposition in the RegularChains Library

8901: 8430:"Contributions of Islamic scholars to the scientific enterprise" 7925: 7841: 7785: 7273: 6959: 6657:. Science Networks – Historical Studies. Vol. 32. Germany: 6473: 6231: 6071: 5948: 5876: 5733: 5657: 5597: 5540: 5311: 5174: 5119: 5075: 4940: 4892: 4658: 3642:, the practice of mathematics has a social side, which includes 2590:. Operation and relations are generally represented by specific 2055:, a statistical-decision problem is formulated by minimizing an 1646: 13954:(2003). "The Nature of Mathematics". In James R. Newman (ed.). 12069:

Luttenberger, Silke; Wimmer, Sigrid; Paechter, Manuela (2018).

9590:"The History Of Applied Mathematics And The History Of Society" 8967:"Mathematical Models and Reality: A Constructivist Perspective" 7864: 5692:"From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem" 3818: 3803: 3760: 3425: 2869: 2857: 2659: 2391:. Other achievements of the Islamic period include advances in 2351: 2250: 2164: 1704: 1334: 1026: 987: 867: 752: 546: 532: 378: 304: 12418:

Budden, F. J. (October 1967). "Modern mathematics and music".

10828:

Handbook of Behavioral Economics: Applications and Foundations

8350: 3403: 3310:

curves, like this one, are a staple of mathematical economics.

2872:

in 300 BC, had no practical application before its use in the

14693: 13171:"Deciding the undecidable: Wrestling with Hilbert's problems" 11973: 11969: 7671: 6811:"Calculus (Differential and Integral Calculus with Examples)" 3559: 3465: 3138:

Computing is closely related to mathematics in several ways.

3044:

is a non-Euclidean space of dimension four, and spacetime of

2595: 1819: 1767: 1763: 1553: 1105: 1022: 983: 473: 463: 300: 8937:. Mathematical Modeling. Vol. 1. CRC Press. p. 1. 8722:

Jansen, Anthony R.; Marriott, Kim; Yelland, Greg W. (2000).

7001:"Turing's Legacy: Developments from Turing's Ideas in Logic" 6654:

Tool and Object: A History and Philosophy of Category Theory

6340:"Where the x we use in algebra came from, and the X in Xmas" 3219:. Most discussion of probability centers on the concept of 2101:

that are typically too large for human, numerical capacity.

2067:, the mathematical theory of statistics overlaps with other 882:. The field came to full fruition with the contributions of 12325:"Reflections on Paul Erdős on His Birth Centenary, Part II" 12322: 11962:

TIMSS 2019 International Results in Mathematics and Science

11684:"Mathematics education in an Old Babylonian scribal school" 10984:"Biden's Misleading Unemployment Statistic – FactCheck.org" 10671: 10454: 10447:

See for example Anne Laurent, Roland Gamet, Jérôme Pantel,

9300:. Springer Science & Business Media. pp. 235–268. 8632:

Douglas, Heather; Headley, Marcia Gail; Hadden, Stephanie;

8631: 8581: 7697:"On the Unusual Effectiveness of Logic in Computer Science" 3945:. For example, the group underlying mirror symmetry is the 3740:(618–907 CE), a mathematics curriculum was adopted for the 3659: 3010: 2347: 2288: 2060: 2028: 1986: 12068: 11805:

Indian pedagogy and problem solving in ancient Thamizhakam

11648:

Mathematicians and Statisticians: A Practical Career Guide

8081: 5767:. Series in Pure Mathematics. Vol. 4 (2nd ed.). 3500:

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

3215:

uses probability extensively in fields such as ecology or

2928:

decidability of the first-order theory of the real numbers

2418:, mathematics began to develop at an accelerating pace in 1021:), or are part of the definition of the subject of study ( 689:

led to an explosion of new areas of mathematics. The 2020

252:

is a field of study that discovers and organizes methods,

13567:

The Mathematical Tourist: Snapshots of Modern Mathematics

12800: 12788: 11957: 11609: 10774:

Moral Markets: The Critical Role of Values in the Economy

10540: 9228:. Chicago: The University of Chicago Press. p. 104. 8934:

Modelling Mathematical Methods and Scientific Computation

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5201:

Leonhard Euler and the Foundations of Celestial Mechanics

3336:). In this model, the individual seeks to maximize their 2326:, dated between the 2nd century BC and the 2nd century AD 1962: 998:, but has since blossomed out into many other subfields. 291:

Mathematics involves the description and manipulation of

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Nadjafikhah, Mehdi; Yaftian, Narges (October 10, 2013).

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Proceedings of the American Academy of Arts and Sciences

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are from 2000 to 1800 BC. Many early texts mention

1919:(modeling some logical theories inside other theories), 623:, were not then clearly distinguished from mathematics. 419:

lists more than sixty first-level areas of mathematics.

369:

Historically, the concept of a proof and its associated

13934:

A History of Mathematics: From Mesopotamia to Modernity

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Evolution of Mathematical Concepts; an Elementary Study

7645:"At the Heart of Analysis: Intuitionism and Philosophy" 7376:. Cambridge Elements in the Philosophy of Mathematics. 7343:"Wittgenstein's analysis on Cantor's diagonal argument" 3270:

also use mathematics due to their heavy use of models.

3036:

that uses fundamentally these concepts. In particular,

1267:, the geometry obtained by replacing real numbers with 1089:), it allowed the representation of points using their 399:), until the 16th and 17th centuries, when algebra and 11302:; Loos, Andreas (November 2, 2017). Kaiser, G. (ed.). 9224:

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Modern Algebra and the Rise of Mathematical Structures

2620: 2423: 1257:, which takes its importance from its applications in 505:(384–322 BC) this meaning was fully established. 288:(presently used as a foundation for all mathematics). 30:"Math" and "Maths" redirect here. For other uses, see 14635: 13829:(Reprint ed.). Boston; New York: Mariner Books. 9810:

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A History of Greek Mathematics: From Thales to Euclid

6774: 6107: 5338:"Splash Talk: The Foundational Crisis of Mathematics" 4426: 3955: 3906: 3572:

must be reducible to a succession of applications of

3154:

is useful in many areas of computer science, such as

3064: 1876:

from reality is embodied in the modern philosophy of

1822:, although used for mathematical proofs, belonged to 829: 798: 764: 489:

Similarly, one of the two main schools of thought in

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Africa Counts: Number and Pattern in African Culture

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6710:"The Newton–Leibniz Calculus Controversy, 1708–1730" 6027:"Reconstructing the Unity of Mathematics circa 1900" 4252:, Book 6 Section 510c. However, Plato did not use a 4137: 4114:

A new list of seven important problems, titled the "

2550:

Mathematical notation is widely used in science and

2519: 1826:

and was not specifically studied by mathematicians.

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"A Mirror of Civilization". 3976: 3919: 3821:, it is hard to define, it is commonly related to 3545:guided by a variety of general insights and ideas. 3080: 3013:with planes). It is almost 2,000 years later that 2626: 2509:, "The number of papers and books included in the 2241:began to emerge as a distinct discipline and some 1348:Algebra became an area in its own right only with 846: 812: 781: 508:In Latin and English, until around 1700, the term 13984:. San Carlos, California: Wide World Publishing. 13410:Mathematical Thought from Ancient to Modern Times 12842: 11721: 11692:The Oxford Handbook of the History of Mathematics 10805:Levin, Jonathan; Milgrom, Paul (September 2004). 10412:The Oxford Handbook of Probability and Philosophy 10410:. In Hájek, Alan; Hitchcock, Christopher (eds.). 10109:The British Journal for the Philosophy of Science 10107:(November 1961). "Discovering the Positron (I)". 9365:The Oxford Handbook of the History of Mathematics 8557:. Part 2: Outline of the History of Mathematics. 8158:The British Journal for the Philosophy of Science 6309:Overbay, Shawn; Schorer, Jimmy; Conger, Heather. 6187: 4207:Science, technology, engineering, and mathematics 4052:The most prestigious award in mathematics is the 2696:Numerous technical terms used in mathematics are 1294:, which concisely expresses the solutions of all 1247:, the study of finite configurations in geometry. 1009:. For example, it is not sufficient to verify by 626:During the Renaissance, two more areas appeared. 14847: 12461:Enquist, Magnus; Arak, Anthony (November 1994). 11728:Handbook on the History of Mathematics Education 11408: 10300:"Real-Life Applications of Discrete Mathematics" 10115:(47). The University of Chicago Press: 194–214. 9778:Chen, Changbo; Maza, Marc Moreno (August 2014). 7913: 7540: 6914:"What is Discrete Mathematics? The Many Answers" 6308: 5570:Goldman, Jay R. (1998). "The Founding Fathers". 4118:", was published in 2000. Only one of them, the 3783:Psychology (aesthetic, creativity and intuition) 3174:seemed to also be partially proven by computer. 1510:; this particular area of application is called 583:takes a singular verb. It is often shortened to 276:(the study of formulas and related structures), 13545:and as an extract, D. Van Nostrand, 1882, 13056:Chern, S. S.; Hirzebruch, F. (September 2000). 13055: 12185: 11978:Lynch School of Education and Human Development 9278: 8930: 6559:(2004). "Emmy Noether: Ideals and Structures". 6188:Christianidis, Jean; Oaks, Jeffrey (May 2013). 5824: 5822: 5820: 5060:(2nd ed.). Dover Publications. p. 3. 4126:, has been solved by the Russian mathematician 3988:is a figure invariant by this symmetry, as are 3833:, completeness, and generality. G. H. Hardy in 3505: 2997:. A second historical example is the theory of 2799: 2788:and results (theorems) are often obtained from 2757:could only be explained after the emergence of 1406:, as established by the influence and works of 1131:In the 19th century, mathematicians discovered 990:, which were developed mainly for the needs of 611:, regarding the study of shapes. Some types of 603:, mathematics was divided into two main areas: 366:) but often later find practical applications. 11253: 11072: 10250:"Course 18C Mathematics with Computer Science" 9504:. Princeton University Press. pp. 17–26. 9242: 8249:, "Greek Trigonometry and Mensuration" p. 162. 7315:"Dispute over Infinity Divides Mathematicians" 5331: 5329: 4605: 4603: 4601: 4599: 4597: 4595: 4458:Communications on Pure and Applied Mathematics 4075:, instituted in 2002 and first awarded in 2003 4067:Other prestigious mathematics awards include: 3937:with a scaling symmetry and a central symmetry 1591:, where variables represent varying functions; 1490:The study of types of algebraic structures as 1225:, the study of properties that are kept under 1042:is the study of shapes and their arrangements 662:between varying quantities, as represented by 315:consisting of a succession of applications of 14037: 13679:Bulletin of the American Mathematical Society 13323:An Introduction to the History of Mathematics 11612:International Journal of Educational Research 11282:t is first necessary to ask what is meant by 10486:"1.4: The Lotka-Volterra Predator-Prey Model" 10405:"Probability in Biology: The Case of Fitness" 10176: 9541:Bulletin of the American Mathematical Society 9132:. London: Taylor & Francis. p. 100. 9010: 6715:. In Schliesser, Eric; Smeenk, Chris (eds.). 5147:(1992). "Mathematics from the Ground Up". In 2969: 2739: 2506:Bulletin of the American Mathematical Society 1762:are also studied, most common games, such as 1097:, which uses purely geometrical methods, and 1005:, which require that every assertion must be 751:Number theory began with the manipulation of 607:, regarding the manipulation of numbers, and 579:, inherited from Greek. In English, the noun 230: 13797:The Moment of Proof: Mathematical Epiphanies 12329:Notices of the American Mathematical Society 10804: 9749:Notices of the American Mathematical Society 9394: 9217: 9195: 9097:"The science checklist applied: Mathematics" 8862:Silver, Daniel S. (November–December 2017). 8829:"Earliest Uses of Some Words of Mathematics" 8478:(1–2). Cambridge University Press: 313–331. 7768:Statistics and Truth: Putting Chance to Work 7473:. Cambridge University Press. pp. 3–4. 7215:"The Road to Modern Logic—An Interpretation" 6704: 5817: 5386:Notices of the American Mathematical Society 4760: 4443: 4441: 4202:Relationship between mathematics and physics 4023: 3576:, without any use of empirical evidence and 3491:and Albert Einstein that support his views. 3223:. Ecology heavily uses modeling to simulate 3107:Relationship between mathematics and physics 2883:In the 19th century, mathematicians such as 2086: 1837:, mathematicians were reluctant to consider 1674:—play a major role in discrete mathematics. 866:. Two prominent early number theorists were 524:'s warning that Christians should beware of 14051: 13428: 12794: 12460: 12075:Psychology Research and Behavior Management 11424: 11298: 10777:. Princeton University Press. p. 158. 10019: 9720:. Frankfurt am Main: Verlag Harri Deutsch. 9716:. In Schlote, K. H.; Schneider, M. (eds.). 9397:Emerging Security Algorithms and Techniques 9153: 8353:Studies in Indian Mathematics and Astronomy 8135:. New York: Dover Publications. p. 1. 7608: 7549: 5574:. Wellesley, MA: A K Peters. pp. 2–3. 5326: 4976:"mathematics", "mathematic", "mathematics". 4592: 3404:Relationship with astrology and esotericism 3321: 2035:) of a statistical action, such as using a 1187:and independent from the concept of length. 554: 540: 14044: 14030: 13007:. International Mathematical Union (IMU). 11021:(in French). Librairie Droz. p. 130. 10732:Batchelder, William H. (January 1, 2015). 10731: 9929:. In Bos, Joppe W.; Stam, Martijn (eds.). 9816:European Journal for Philosophy of Science 8529:. Vol. 2. Bibliotex Digital Library. 7432: 5919: 5372: 4963:The Oxford Dictionary of English Etymology 4621:(5). Taylor & Francis, Ltd.: 291–314. 3977:{\displaystyle \mathbb {Z} /2\mathbb {Z} } 3048:is a (curved) manifold of dimension four. 2934:, with an algorithm that is impossible to 2097:Computational mathematics is the study of 1556:with the introduction of the concept of a 237: 223: 13691: 13525: 13349:. Springer Science & Business Media. 12340: 12170: 12159:Procedia – Social and Behavioral Sciences 12137: 12126:Procedia – Social and Behavioral Sciences 12096: 12086: 11934: 11328: 11249: 11247: 11014: 10639:"HOW MATH IS USED IN WEATHER FORECASTING" 10399: 10351: 10328:"A Formal Proof of the Kepler Conjecture" 10080: 10055: 10053: 10051: 9989:Mukunth, Vasudevan (September 10, 2015). 9827: 9668: 9653: 9552: 9287: 9248: 8883: 8653: 8549: 8051: 7945: 7656: 7309: 7212: 7160: 6857: 6546:, "History of Linear Algebra" pp. 79–101. 6263:, "History of Classical Algebra" pp. 3–5. 6205: 5834: 5707: 5397: 5379:"Mathematics Subject Classification 2020" 5285: 4536: 4438: 3970: 3957: 3340:, and always makes optimal choices using 2976:unreasonable effectiveness of mathematics 1981:Whatever the form of a random population 1432:, whose study is essentially the same as 834: 803: 769: 685:and the resulting systematization of the 14006:. Sändig Reprint Verlag H. R. 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Oxford: Elsevier. pp. 808–815. 10692: 10496:from the original on December 29, 2022 10395: 10393: 10391: 10210: 10103: 10048: 10001:from the original on November 20, 2022 9940:from the original on November 20, 2022 9850: 9742: 9711: 9569:from the original on November 20, 2022 9495: 9159: 9125: 9103:. University of California, Berkeley. 9030:from the original on November 17, 2022 8964: 8861: 8764:from the original on November 16, 2022 8613:from the original on November 16, 2022 8606:Mathematical Notation: Past and Future 8501:from the original on December 17, 2022 8444:from the original on November 14, 2022 8440:(4). Shannon Research Press: 391–399. 8427: 8391: 7964:The Princeton Companion to Mathematics 7895:from the original on November 13, 2022 7824:Mathematical programming in statistics 7808: 7731:Probability, Statistics and Estimation 7727: 7649:Philosophia Scientiæ, Cahier spécial 6 7624:from the original on December 16, 2022 7590:from the original on November 12, 2022 7436:; Reck, Erich H. 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New York: Oxford University Press. 13402: 13287: 13147:from the original on January 23, 2022 13114:from the original on January 12, 2020 13031:"The Leroy P Steele Prize of the AMS" 12949:from the original on January 26, 2020 12806: 12726: 12278: 11881: 11556:from the original on December 5, 2022 11196: 11110: 11091:from the original on January 30, 2022 11060:from the original on January 29, 2023 10420:10.1093/oxfordhb/9780199607617.013.27 10376:from the original on December 4, 2020 10059: 9970:from the original on October 14, 2022 9669:Peressini, Anthony (September 1999). 9587: 9502:The Best Writing on Mathematics, 2016 9441:"How applied mathematics became pure" 9435: 9358: 9107:from the original on October 27, 2019 8782: 8703:from the original on December 8, 2022 8462: 8307: 8258: 8246: 8234: 8222: 8210: 8198: 8123: 8111: 8087: 7163:"The Early Development of Set Theory" 7108: 6995: 6791:from the original on November 9, 2022 6756:from the original on November 9, 2022 6659:Springer Science & Business Media 6608: 6555: 6499:from the original on November 8, 2022 6437:Archive for History of Exact Sciences 6272: 6124:from the original on November 6, 2022 6024: 5976: 5267: 4828:Harper, Douglas (December 22, 2018). 4395:from the original on November 7, 2022 4248:This meaning can be found in Plato's 2895:. This led to split mathematics into 2446:, the introduction of coordinates by 2361: 2187: 14700: 14606: 13669:Sevryuk, Mikhail B. (January 2006). 13451:from the original on August 13, 2006 13011:from the original on August 25, 2010 12764:The Best Writing on Mathematics 2016 12586: 11207: 11184:from the original on August 17, 2023 10711:10.1146/annurev.soc.28.110601.140942 10636: 10183:. PHI Learning Pvt. Ltd. p. 3. 9693:from the original on January 2, 2024 9636:"On the role of applied mathematics" 9477:from the original on August 12, 2017 9320: 9254:"Are There 'Other' Ways of Knowing?" 8678: 8519: 7744:from the original on October 9, 2022 7111:"The Emergence of First-Order Logic" 7109:Ewald, William (November 17, 2018). 6729:10.1093/oxfordhb/9780199930418.013.9 6426: 6371: 5762: 5626: 5495:from the original on January 2, 2024 5052: 4913: 4907: 4689: 3184:Mathematical and theoretical biology 3095: 2922:for validating computations done in 2914:An example of the first case is the 1855:controversy over Cantor's set theory 1563:Analysis is further subdivided into 1165:, introduced in the 16th century by 681:At the end of the 19th century, the 14618: 13126: 12919:from the original on April 14, 2022 12863:from the original on March 22, 2019 11836: 11468:from the original on March 23, 2023 10770: 10761: 10637:Hill, Mullica (September 7, 2022). 10515:Salsburg, David (August 17, 1992). 10388: 9759:from the original on March 13, 2014 9724:from the original on March 23, 2023 9633: 9129:Dictionary of Scientific Quotations 9024:Stanford Encyclopedia of Philosophy 8270: 7814: 7762: 7618:Stanford Encyclopedia of Philosophy 7556:Revue Internationale de Philosophie 7168:Stanford Encyclopedia of Philosophy 7116:Stanford Encyclopedia of Philosophy 6922:Discrete Mathematics in the Schools 6336: 5634:. Birkhäuser Boston. pp. 2–3. 5456: 5441:from the original on August 3, 2021 4923:Mathematical Association of America 2944:cylindrical algebraic decomposition 2168:The Babylonian mathematical tablet 2009:. Statisticians generate data with 1650:A diagram representing a two-state 1317:Algebra is the art of manipulating 745:Hardy and Littlewood's Conjecture F 24: 13751: 13703:from the original on July 23, 2006 13475:. New York: Penguin Random House. 12985:from the original on June 17, 2009 12708:from the original on March 7, 2023 12679:from the original on March 2, 2023 12649:from the original on March 7, 2023 11428:; Porter, Timothy (January 2000). 11210:Educational Studies in Mathematics 10734:"Mathematical Psychology: History" 10429:from the original on March 7, 2023 8225:, "Archimedes of Syracuse" p. 130. 8213:, "Archimedes of Syracuse" p. 120. 8034: 8015:from the original on March 7, 2023 7709:from the original on March 3, 2021 7550:Raatikainen, Panu (October 2005). 7340: 7161:Ferreirós, José (June 18, 2020) . 5335: 5153:Mathematics in Society and History 5034:from the original on April 4, 2020 4800:Harper, Douglas (April 20, 2022). 4713:from the original on March 7, 2013 4699:Harper, Douglas (March 28, 2019). 4042:with an illustration of the Greek 3876: 3273: 3066: 2946:that became a fundamental tool in 2868:, for example, which goes back to 2862:pure mathematics in Ancient Greece 2178:In addition to recognizing how to 1751:Discrete probability distributions 1145:foundational crisis of mathematics 691:Mathematics Subject Classification 683:foundational crisis in mathematics 642:, consisting of the two subfields 417:Mathematics Subject Classification 411:led to the systematization of the 409:foundational crisis of mathematics 25: 14877: 13217:from the original on July 3, 2015 13137:"Hilbert's Problems: 23 and Math" 12639:"Fractals – Chaos & Fractals" 11884:The American Mathematical Monthly 10874:from the original on July 1, 2016 9745:"Model theory and exponentiation" 9266:from the original on May 13, 2020 8864:"The New Language of Mathematics" 8555:The American Mathematical Monthly 7949:(2008). "Numerical Analysis". In 7885:"G I Marchuk's plenary: ICM 1970" 7372:Tanswell, Fenner Stanley (2024). 7213:Ferreirós, José (December 2001). 6846:Journal of Humanistic Mathematics 5980:(2004) . "Fermat and Descartes". 5523:Addison-Wesley Publishing Company 5205:Springer International Publishing 4731: 4518:from the original on June 1, 2019 4004: 3244: 2841:developed infinitesimal calculus. 2520:Symbolic notation and terminology 2430:(1540–1603), the introduction of 1792:Mathematical logic and set theory 14828: 14811: 14794: 14777: 14760: 14743: 14726: 14709: 14681: 14669: 14657: 14645: 14617: 14605: 14594: 14593: 14581: 13229: 13203: 13159: 13096: 13049: 13023: 12997: 12961: 12939:"Abel Prize | mathematics award" 12931: 12905: 12875: 12812: 12754: 12720: 12691: 12661: 12630: 12612: 12580: 12547: 12529: 12463:"Symmetry, beauty and evolution" 12454: 12411: 12368: 12349: 12316: 12272: 12238: 12179: 12146: 12120:Yaftian, Narges (June 2, 2015). 12113: 12062: 12041:10.1002/j.2161-0045.2012.00001.x 12029:The Career Development Quarterly 12013: 11951: 11910: 11875: 11830: 11795: 11752: 11715: 11672: 11638: 11603: 11584: 11568: 11511: 11480: 11430:"The Methodology of Mathematics" 11418: 11402: 11292: 11149: 11103: 11042: 11008: 10990: 10976: 10934: 10904: 10886: 10856: 10838: 10818: 10798: 10725: 10686: 10656: 10630: 10605: 10580: 10534: 10508: 10478: 10441: 10318: 10292: 10267: 10242: 10204: 10170: 10135: 10097: 10013: 9982: 9952: 9922:Wagstaff, Samuel S. Jr. (2021). 9915: 9888: 9844: 9803: 9771: 9736: 9705: 9662: 9627: 9581: 9525: 9498:"In Defense of Pure Mathematics" 9489: 9429: 9388: 9352: 9314: 9189: 9119: 9089: 9042: 9004: 8958: 8924: 8855: 8821: 8776: 8715: 8625: 8593: 8543: 8201:, "Euclid of Alexandria" p. 119. 7658:10.4000/philosophiascientiae.411 5905: 4503: 4410: 4328: 4319: 4310: 4293: 4140: 3825:, which involves qualities like 3564:Mathematical reasoning requires 2827: 2818: 2769:as a better mathematical model. 2755:perihelion precession of Mercury 2542:An explanation of the sigma (Σ) 2105:studies methods for problems in 2051:. In these traditional areas of 1456:, and is a foundational part of 1393:addresses this, consisting of a 716: 486:) meant "the mathematical art". 326:Mathematics is essential in the 207: 14502:Computational complexity theory 13506:American Journal of Mathematics 13343:(2007). Kleiner, Israel (ed.). 13211:"The Millennium Prize Problems" 10588:"Catastrophe Models (Property)" 9054:"Mathematics, Maps, and Models" 8739:University of California Merced 8513: 8456: 8434:International Education Journal 8421: 8385: 8344: 8301: 8264: 8237:, "Apollonius of Perga" p. 145. 8149: 8117: 8093: 8045: 8027: 7939: 7907: 7876: 7756: 7728:Rouaud, Mathieu (April 2017) . 7721: 7665: 7636: 7602: 7460: 7426: 7365: 7334: 7303: 7206: 7154: 7102: 7068: 6989: 6905: 6830: 6803: 6768: 6698: 6644: 6612:Perspectives on Universal Logic 6602: 6549: 6511: 6420: 6365: 6337:Lim, Lisa (December 21, 2018). 6330: 6302: 6266: 6181: 6136: 6101: 6018: 5970: 5913: 5843: 5756: 5680: 5620: 5563: 5507: 5366: 5261: 5189: 5137: 5093: 5056:(1945) . "General Prospectus". 5046: 5010: 4979: 4955: 4854: 4821: 4793: 4754: 4276: 4259: 4242: 3744:to join the state bureaucracy. 2297:summation of an infinite series 1933:computational complexity theory 1890:Gödel's incompleteness theorems 1696:Discrete mathematics includes: 1617:Ordinary differential equations 847:{\displaystyle (\mathbb {Q} ).} 782:{\displaystyle (\mathbb {N} ),} 13239:. Clay Mathematics Institute. 13213:. Clay Mathematics Institute. 12767:. Princeton University Press. 12288:. Cambridge University Press. 12196:The Mathematical Intelligencer 11773:10.1080/10724117.2002.11975154 11520:"Mathematical Rigor and Proof" 11411:Mathematics, form and function 11344:. Elsevier. pp. 104–106. 10693:Edling, Christofer R. (2002). 10275:"Theoretical Computer Science" 10069:The Mathematical Intelligencer 10065:"Mathematics: Art and Science" 9924:"History of Integer Factoring" 8642:Journal of Numerical Cognition 7222:The Bulletin of Symbolic Logic 6149:Johns Hopkins University Press 5058:The Development of Mathematics 4735:Republic, Book 6, Section 510c 4725: 4627:10.1080/0025570X.1991.11977625 4530: 4497: 4347: 4225: 3920:{\displaystyle {\frac {3}{2}}} 3537:Mathematics, form and function 3235:Statistical hypothesis testing 2658:and formulas play the role of 1623:Partial differential equations 1502:. The latter applies to every 1116:. They can also be defined as 1069:introduced what is now called 838: 830: 813:{\displaystyle (\mathbb {Z} )} 807: 799: 773: 765: 13: 1: 13932:Hodgkin, Luke Howard (2005). 13914:. Kluwer Academic Publishers. 13890:. W.W. Norton & Company. 13693:10.1090/S0273-0979-05-01069-4 13569:. W. H. Freeman and Company. 13429:Monastyrsky, Michael (2001). 13346:A History of Abstract Algebra 12362:History of Western Philosophy 11840:How Chinese Learn Mathematics 11083:. In Zalta, Edward N. (ed.). 10898:onlinebooks.library.upenn.edu 10808:Introduction to Choice Theory 10736:. In Wright, James D. (ed.). 9399:. CRC Press. pp. 59–60. 8694:American Mathematical Society 8399:. New York University Press. 8277:Number Theory and Its History 8102:pg 10. Retrieved June 1, 2024 6926:American Mathematical Society 6717:The Oxford Handbook of Newton 6382:Encyclopaedia of Islam Online 5923:(2000). "Euclid's Geometry". 5861:Open Court Publishing Company 5519:Fundamentals of Number Theory 4269:, which are intersections of 4213: 3722: 3707: 3444: 3021:of the planets are ellipses. 2491:. In the early 20th century, 2265: 1908:, which explicitly lacks the 1309:is a concrete application of 549:), based on the Greek plural 501:and geometry. By the time of 13911:Encyclopaedia of Mathematics 13782:Resources in other libraries 13502:"Linear associative algebra" 13498:Peirce, Charles Sanders 12698:Kissane, Barry (July 2009). 12186:van der Poorten, A. (1979). 12172:10.1016/j.sbspro.2013.07.101 12139:10.1016/j.sbspro.2015.04.617 11976:International Study Center, 11527:The Review of Symbolic Logic 11158:"Why are laws mathematical?" 10217:On the Work of Edward Witten 9865:10.1007/978-4-431-55060-0_29 9790:10.1007/978-3-662-44199-2_65 9655:10.1016/0001-8708(76)90024-4 9448:The Review of Symbolic Logic 9066:10.1007/978-3-319-72478-2_18 8463:Lorch, Richard (June 2001). 8361:10.1007/978-981-13-7326-8_20 7021:10.1017/CBO9781107338579.001 5982:History of Analytic Geometry 5287:10.1016/0315-0860(79)90074-0 5270:"The roots of combinatorics" 4341: 3627: 3363:discerned the ~50-year-long 3140:Theoretical computer science 3130:Theoretical computer science 3123: 3081:{\displaystyle \Omega ^{-}.} 3003:ancient Greek mathematicians 2958:departments, such as at the 2800:Pure and applied mathematics 2744:Mathematics is used in most 2338:and were transmitted to the 1845:to be the result of endless 1841:collections, and considered 1605:, all strongly related with 1575:, where variables represent 1567:, where variables represent 1112:, the study of which led to 1058:) and the three-dimensional 658:, which model the typically 422: 32:Mathematics (disambiguation) 7: 13936:. Oxford University Press. 13826:The Mathematical Experience 13801:. Oxford University Press. 13634:Riehm, Carl (August 2002). 12564:10.1002/9781119547167.ch131 12558:. Wiley. pp. 367–376. 12071:"Spotlight on math anxiety" 11736:10.1007/978-1-4614-9155-2_3 11518:Hamami, Yacin (June 2022). 11312:10.1007/978-3-319-62597-3_5 11257:; Neunzert, Helmut (2012). 11167:. Ashgate. pp. 31–66. 10550:. NAP.edu. pp. 71–73. 10164:10.1016/j.shpsb.2015.12.001 9500:. In Pitici, Mircea (ed.). 9199:; Lazere, Cathy A. (1998). 8428:Faruqi, Yasmeen M. (2006). 7313:, ed. (November 26, 2013). 6912:Maurer, Stephen B. (1997). 5926:Geometry: Euclid and Beyond 5856:The Foundations of Geometry 4835:Online Etymology Dictionary 4807:Online Etymology Dictionary 4706:Online Etymology Dictionary 4358:; Neges, Katharina (eds.). 4172:Lists of mathematics topics 4162:List of mathematical jargon 4133: 3817:value to mathematics. Like 3813:Mathematicians can find an 3767:L'Enseignement Mathématique 3009:(that is, intersections of 3001:. They were studied by the 2993:communications through the 2784:. Similarly as in science, 2767:Newton's law of gravitation 2586:letters, and often include 2332:Hindu–Arabic numeral system 1073:. This constituted a major 959: 905:, who used tools including 567:anew, after the pattern of 10: 14882: 14552:Films about mathematicians 13793:Benson, Donald C. (1999). 13325:(6th ed.). Saunders. 13299:(2nd ed.). New York: 13258: 12389:10.1177/002242945900700205 11852:10.1142/9789812562241_0006 11624:10.1016/j.ijer.2018.10.002 11081:"Platonism in Metaphysics" 10699:Annual Review of Sociology 10695:"Mathematics in Sociology" 9829:10.1007/s13194-021-00435-9 9743:Marker, Dave (July 1996). 9634:Lin, C. C . (March 1976). 9496:Silver, Daniel S. (2017). 9332:(1). The MIT Press: 1–31. 8965:Hennig, Christian (2010). 8783:Rossi, Richard J. (2006). 8114:, "Mesopotamia" pp. 24–27. 7972:Princeton University Press 7467:Hamilton, Alan G. (1982). 7446:Carnegie Mellon University 7378:Cambridge University Press 7013:Cambridge University Press 6859:10.5642/jhummath.201702.18 6661:. pp. xxi–xxv, 1–91. 5869:10.1126/science.16.399.307 5157:Kluwer Academic Publishers 5155:. Episteme. Vol. 20. 4027: 4008: 3885: 3631: 3557: 3518:Definitions of mathematics 3515: 3506:unreasonable effectiveness 3464:. The ancient philosopher 3455: 3448: 3277: 3248: 3206:reaction–diffusion systems 3204:, which can be modeled by 3181: 3127: 3104: 3100: 2970:Unreasonable effectiveness 2803: 2740:Relationship with sciences 2523: 2206:and so, by inference, the 2159: 2152: 2148: 2090: 1966: 1851:Cantor's diagonal argument 1795: 1778:combinatorial optimization 1639: 1521: 1279: 1275: 1135:, which do not follow the 1029:around 300 BC in his book 963: 720: 474: 464: 441: 147:Relationship with sciences 29: 14575: 14525: 14482: 14392: 14354: 14321: 14273: 14245: 14192: 14139: 14121:Philosophy of mathematics 14096: 14061: 13777:Resources in your library 13355:10.1007/978-0-8176-4685-1 13059:Wolf Prize in Mathematics 12310:A Mathematician's Apology 12285:A Mathematician's Apology 11936:10.1007/s11858-012-0445-7 11923:ZDM Mathematics Education 11593:Philosophy of Mathematics 11539:10.1017/S1755020319000443 11409:Saunders Maclane (1986). 11341:Philosophy of Mathematics 10542:National Research Council 9672:Applying Pure Mathematics 9460:10.1017/S1755020308080027 9203:. Springer. p. 228. 8983:10.1007/s10699-009-9167-x 8837:University of St. Andrews 8484:10.1017/S0269889701000114 7643:McCarty, Charles (2006). 6450:10.1007/s00407-018-0208-0 6427:Oaks, Jeffery A. (2018). 6118:University of St. Andrews 5640:10.1007/978-0-8176-4571-7 5268:Biggs, N. L. (May 1979). 5213:10.1007/978-3-031-12322-1 5023:Oxford English Dictionary 4992:Oxford English Dictionary 4969:Oxford English Dictionary 4511:The University of Georgia 4419:November 5, 2022, at the 4197:Philosophy of mathematics 4116:Millennium Prize Problems 4097:Wolf Prize in Mathematics 4024:Awards and prize problems 3836:A Mathematician's Apology 3451:Philosophy of mathematics 3134:Computational mathematics 2985:A notable example is the 2839:Gottfried Wilhelm Leibniz 2554:for representing complex 2322:The numerals used in the 2093:Computational mathematics 2087:Computational mathematics 1691:computationally difficult 1387:geometric transformations 555: 442: 311:properties of objects, a 303:. Mathematics uses pure 14557:Recreational mathematics 13956:The World of Mathematics 13845:(registration required). 13603:(1995). "On knowledge". 13565:Peterson, Ivars (1988). 13295:A History of Mathematics 12536:Hestenes, David (1999). 12420:The Mathematical Gazette 11434:The Mathematical Gazette 11378:A History of Mathematics 11143:The World of Mathematics 10451:15–17 janvier 1996, CNRS 10332:Forum of Mathematics, Pi 8791:. pp. 1–14, 47–48. 8551:Archibald, Raymond Clare 8059:. Chicago Review Press. 7873:, pp. 10–11, 14–18. 6785:University of St Andrews 6346:South China Morning Post 6207:10.1016/j.hm.2012.09.001 6110:"Non-Euclidean geometry" 6025:Stump, David J. (1997). 5517:(1977). "Introduction". 4218: 3808:recreational mathematics 3755:. Formal instruction in 3590:computer-assisted proofs 3553: 3229:Lotka–Volterra equations 2837:Isaac Newton (left) and 2395:and the addition of the 2059:, like expected loss or 1874:mathematical abstraction 1672:computational complexity 1452:, includes the study of 1448:, which is the study of 1195:differentiable functions 1133:non-Euclidean geometries 272:(the study of numbers), 14442:Mathematical statistics 14432:Mathematical psychology 14402:Engineering mathematics 14336:Algebraic number theory 13998: 13611:. New York: Routledge. 13527:2027/hvd.32044030622997 13438:CMS – Notes – de la SMC 12943:Encyclopedia Britannica 12761:Pitici, Mircea (2017). 12637:Bradley, Larry (2010). 11645:Endsley, Kezia (2021). 11487:Strauss, Danie (2011). 11263:. Springer. p. 9. 11079:Balaguer, Mark (2016). 11015:Beaujouan, Guy (1994). 10121:10.1093/bjps/xiii.49.54 10105:Hanson, Norwood Russell 9641:Advances in Mathematics 9606:10.1023/A:1020823608217 9588:Stolz, Michael (2002). 8520:Kent, Benjamin (2022). 6721:Oxford University Press 6035:Perspectives on Science 5765:The Goldbach Conjecture 5696:Elemente der Mathematik 5028:Oxford University Press 4997:Oxford University Press 4775:A Greek–English Lexicon 4167:Lists of mathematicians 2964:University of Cambridge 2948:real algebraic geometry 2916:theory of distributions 2796:on empirical evidence. 2570:, unspecified numbers, 2534:Glossary of mathematics 2530:Language of mathematics 2354:, and an early form of 2237:In the 6th century BC, 2174:, dated to 1800 BC 2053:mathematical statistics 2027:such as minimizing the 1227:continuous deformations 942:algebraic number theory 660:nonlinear relationships 295:that consist of either 260:that are developed and 14588:Mathematics portal 14437:Mathematical sociology 14417:Mathematical economics 14412:Mathematical chemistry 14341:Analytic number theory 14222:Differential equations 13980:The Joy Of Mathematics 13958:. Dover Publications. 13952:Jourdain, Philip E. B. 13924:December 20, 2012, at 12913:"About the Abel Prize" 12883:"Honours/Prizes Index" 11686:. In Robson, Eleanor; 11489:"Defining mathematics" 11156:Dorato, Mauro (2005). 11141:Newman, J. R. (1956). 11002:www.minneapolisfed.org 10490:Mathematics LibreTexts 9533:Parshall, Karen Hunger 9359:Asper, Markus (2009). 9288:Ferreirós, J. (2007). 9126:Mackay, A. L. (1991). 8971:Foundations of Science 8885:10.1511/2017.105.6.364 8868:The American Scientist 7614:"Intuitionistic Logic" 6651:Krömer, Ralph (2007). 6316:University of Kentucky 5203:. History of Physics. 4479:10.1002/cpa.3160130102 4049: 4038:The front side of the 3978: 3938: 3921: 3852:fast Fourier transform 3547: 3524:epistemological status 3502: 3311: 3280:Mathematical economics 3237:, is run on data from 3209: 3188:Mathematical chemistry 3116: 3082: 2960:Faculty of Mathematics 2681:, a statement that is 2628: 2566:used for representing 2547: 2472: 2393:spherical trigonometry 2380: 2327: 2175: 2155:History of mathematics 2125:with special focus on 2081:mathematical economics 2002: 2001:of probability theory. 1910:law of excluded middle 1815: 1786:constraint programming 1731:error correcting codes 1683:optimal sphere packing 1655: 1584:Multivariable calculus 1541: 1504:mathematical structure 1314: 1298: 1077:: Instead of defining 975: 938:analytic number theory 870:of ancient Greece and 856:numerical calculations 848: 814: 789:and later expanded to 783: 748: 693:contains no less than 587:or, in North America, 541: 483: 405:scientific discoveries 401:infinitesimal calculus 14567:Mathematics education 14497:Theory of computation 14217:Hypercomplex analysis 14004:Gauss zum Gedächtniss 13237:"Millennium Problems" 13179:In the Light of Logic 12727:Steen, L. A. (2012). 11114:Philosophy of Science 10954:. New York: Picador. 10771:Zak, Paul J. (2010). 10403:(September 8, 2016). 10230:on September 28, 2013 10180:Essentials of Physics 10075:(4). Springer: 9–17. 9197:Shasha, Dennis Elliot 9160:Bishop, Alan (1991). 9101:Understanding Science 8789:John Wiley & Sons 8170:10.1093/bjps/20.4.289 7612:(September 4, 2018). 7386:10.1017/9781009325110 7088:10.1145/129712.129771 6934:10.1090/dimacs/036/13 6706:Guicciardini, Niccolo 6273:Shane, David (2022). 5580:10.1201/9781439864623 4864:Pi Mu Epsilon Journal 4830:"Mathematical (adj.)" 4762:Liddell, Henry George 4192:Mathematics education 4182:Mathematical sciences 4177:Mathematical constant 4090:Leroy P. Steele Prize 4037: 3979: 3933: 3922: 3634:Mathematics education 3623:Training and practice 3586:Feit–Thompson theorem 3542: 3493: 3306: 3278:Further information: 3195: 3178:Biology and chemistry 3128:Further information: 3115:Diagram of a pendulum 3114: 3083: 2866:integer factorization 2629: 2541: 2526:Mathematical notation 2487:, number theory, and 2481:differential geometry 2468: 2409:Sharaf al-Dīn al-Ṭūsī 2369: 2321: 2212:elementary arithmetic 2167: 2099:mathematical problems 1999:central limit theorem 1993:distribution and its 1980: 1955:and other aspects of 1866:formalized set theory 1809: 1774:Discrete optimization 1649: 1535: 1528:Mathematical analysis 1518:Calculus and analysis 1354:mathematical formulas 1304: 1289: 1191:Differential geometry 1114:differential geometry 1071:Cartesian coordinates 973: 950:diophantine equations 923:Goldbach's conjecture 921:. Another example is 895:Fermat's Last Theorem 884:Adrien-Marie Legendre 849: 815: 784: 730: 701:(the modern name for 645:differential calculus 628:Mathematical notation 512:more commonly meant " 391:(the manipulation of 36:Math (disambiguation) 14547:Informal mathematics 14427:Mathematical physics 14422:Mathematical finance 14407:Mathematical biology 14346:Diophantine geometry 13289:Boyer, Carl Benjamin 12088:10.2147/PRBM.S141421 10951:Fashionable Nonsense 10868:www.encyclopedia.com 10414:. pp. 601–622. 9294:Goldstein, Catherine 9018:(February 4, 2020). 8655:10.5964/jnc.v6i3.293 8636:(December 3, 2020). 8125:Heath, Thomas Little 7974:. pp. 604–615. 7816:Rao, C. Radhakrishna 7764:Rao, C. Radhakrishna 7568:10.3917/rip.234.0513 7506:Mathematics Magazine 7015:. pp. 260–261. 6928:. pp. 121–124. 6719:. Oxford Handbooks. 6283:Methodist University 6194:Historia Mathematica 6151:. pp. 219–232. 6048:10.1162/posc_a_00532 5274:Historia Mathematica 4615:Mathematics Magazine 4103:A famous list of 23 4092:, awarded since 1970 4017:mathematical anxiety 3953: 3904: 3847:Proofs from THE BOOK 3605:Weierstrass function 3512:Proposed definitions 3416:, Arab astronomers, 3385:Fashionable Nonsense 3221:evolutionary fitness 3152:Discrete mathematics 3062: 3034:theory of relativity 3017:discovered that the 2618: 2512:Mathematical Reviews 2470:Carl Friedrich Gauss 2324:Bakhshali manuscript 2313:Hipparchus of Nicaea 2285:method of exhaustion 2281:solids of revolution 2143:symbolic computation 2115:approximation theory 2041:parameter estimation 1997:(σ) is given by the 1929:computability theory 1906:intuitionistic logic 1642:Discrete mathematics 1636:Discrete mathematics 1492:mathematical objects 1373:, which were called 1333:is derived from the 1122:polynomial equations 956:(problem oriented). 954:transcendence theory 929:. Stated in 1742 by 888:Carl Friedrich Gauss 827: 796: 762: 656:continuous functions 595:Areas of mathematics 563:and formed the noun 112:Discrete mathematics 14866:Main topic articles 14562:Mathematics and art 14472:Operations research 14227:Functional analysis 13906:Hazewinkel, Michiel 13617:1992sbwl.book.....P 13487:A Mind for Numbers. 13005:"Chern Medal Award" 12969:"Chern Medal Award" 12809:, pp. 778–782. 12701:Popular mathematics 12479:1994Natur.372..169E 12364:. 1919. p. 60. 11688:Stedall, Jacqueline 10492:. January 5, 2022. 10463:, pp. 282–283. 10156:2016SHPMP..53...20G 9851:Takase, M. (2014). 9712:Lützen, J. (2011). 9020:"Models in Science" 8590:, pp. 101–109. 7947:Trefethen, Lloyd N. 6519:"Variable in Maths" 5988:. pp. 74–102. 5763:Wang, Yuan (2002). 5515:LeVeque, William J. 5345:Columbia University 4471:1960CPAM...13....1W 4187:Mathematics and art 4124:Poincaré conjecture 4011:Popular mathematics 3888:Mathematics and art 3882:Artistic expression 3688:computer consultant 3588:. The emergence of 3342:perfect information 3284:Historical dynamics 3225:population dynamics 2987:prime factorization 2901:applied mathematics 2806:Applied mathematics 2679:deductive reasoning 2416:early modern period 2385:Golden Age of Islam 2344:Islamic mathematics 2307:, 3rd century BC), 2305:Apollonius of Perga 2291:under the arc of a 2208:Pythagorean theorem 2204:Pythagorean triples 2111:functional analysis 2073:operations research 1945:formal verification 1859:mathematical rigour 1782:integer programming 1687:P versus NP problem 1589:Functional analysis 1464:homological algebra 1446:commutative algebra 1391:algebraic structure 1375:algebraic equations 1296:quadratic equations 1237:homological algebra 1207:Riemannian geometry 1163:Projective geometry 1050:and circles in the 948:(method oriented), 946:geometry of numbers 919:homological algebra 668:celestial mechanics 371:mathematical rigour 360:applied mathematics 46:Part of a series on 14676:History of science 14507:Numerical analysis 14116:Mathematical logic 14111:Information theory 13643:Notices of the AMS 12915:. The Abel Prize. 12624:MIT OpenCourseWare 12209:10.1007/BF03028234 11579:Four Color Theorem 11300:Ziegler, Günter M. 11222:10.1007/BF01273907 10643:www.mathnasium.com 10401:Millstein, Roberta 10344:10.1017/fmp.2017.1 10082:10.4171/news/103/8 9250:Pigliucci, Massimo 8523:History of Science 8472:Science in Context 8053:Zaslavsky, Claudia 8036:Wilder, Raymond L. 8033:See, for example, 7955:Barrow-Green, June 7311:Wolchover, Natalie 5986:Dover Publications 5709:10.1007/PL00000079 4802:"Mathematics (n.)" 4271:circular cylinders 4148:Mathematics portal 4120:Riemann hypothesis 4109:Hilbert's problems 4050: 4030:Mathematics awards 3974: 3939: 3917: 3742:civil service exam 3361:Nikolai Kondratiev 3312: 3256:Structural geology 3210: 3160:information theory 3117: 3078: 3046:general relativity 3042:special relativity 2763:general relativity 2627:{\textstyle \int } 2624: 2548: 2473: 2381: 2362:Medieval and later 2328: 2182:physical objects, 2176: 2103:Numerical analysis 2057:objective function 2049:selecting the best 2045:hypothesis testing 2021:Statistical theory 2007:probability theory 2003: 1985:(μ), the sampling 1973:Probability theory 1853:. This led to the 1816: 1798:Mathematical logic 1679:four color theorem 1666:—especially their 1656: 1629:Numerical analysis 1607:probability theory 1542: 1512:algebraic topology 1508:topological spaces 1494:is the purpose of 1458:algebraic geometry 1325:(3rd century) and 1315: 1307:Rubik's Cube group 1299: 1233:Algebraic topology 1213:Algebraic geometry 1171:points at infinity 1137:parallel postulate 1126:algebraic geometry 1118:implicit equations 1110:graph of functions 1095:synthetic geometry 1075:change of paradigm 1040:Euclidean geometry 976: 931:Christian Goldbach 911:algebraic geometry 844: 810: 779: 749: 707:mathematical logic 654:, is the study of 470:mathēmatikḗ tékhnē 377:, most notably in 373:first appeared in 266:empirical sciences 213:Mathematics Portal 14633: 14632: 14232:Harmonic analysis 14013:978-3-253-01702-5 13991:978-0-933174-65-8 13965:978-0-486-43268-7 13943:978-0-19-152383-0 13897:978-0-393-04002-9 13871:978-0-19-510519-3 13836:978-0-395-92968-1 13808:978-0-19-513919-8 13758:Library resources 13738:978-0-471-94829-2 13626:978-0-415-13548-1 13482:978-0-399-16524-5 13421:978-0-19-506135-2 13364:978-0-8176-4684-4 13332:978-0-03-029558-4 13310:978-0-471-54397-8 13280:978-2-7384-8125-2 13189:978-0-19-508030-8 13167:Feferman, Solomon 13141:Simons Foundation 13077:978-981-02-3945-9 12976:www.mathunion.org 12824:www.mathunion.org 12774:978-1-4008-8560-2 12740:978-1-4613-9435-8 12669:"Self-similarity" 12605:978-1-4008-7434-7 12573:978-1-119-05751-2 12538:"Symmetry Groups" 12473:(6502): 169–172. 12355:See, for example 12295:978-0-521-42706-7 12258:978-0-8218-4814-2 11991:978-1-889938-54-7 11861:978-981-256-014-8 11745:978-1-4614-9154-5 11724:Proust, Christine 11701:978-0-19-921312-2 11658:978-1-5381-4517-3 11388:978-0-8218-2102-2 11351:978-0-08-093058-9 11321:978-3-319-62596-6 11270:978-3-0348-0229-1 11174:978-0-7546-3994-7 11028:978-2-600-00040-6 10961:978-0-312-19545-8 10784:978-1-4008-3736-6 10747:978-0-08-097087-5 10565:978-0-309-16839-7 10190:978-81-203-4642-0 9960:"Curves: Ellipse" 9874:978-4-431-55059-4 9554:10.1090/bull/1754 9511:978-0-691-17529-4 9406:978-0-8153-6145-9 9374:978-0-19-921312-2 9307:978-3-540-34720-0 9235:978-0-226-05182-6 9210:978-0-387-98269-4 9175:978-0-7923-1270-3 9139:978-0-7503-0106-0 9075:978-3-319-72478-2 9016:Hartmann, Stephan 8944:978-0-8493-8331-1 8798:978-0-470-04295-3 8687:"AMS Style Guide" 8536:978-1-984668-67-7 8406:978-0-8147-7962-0 8370:978-981-13-7325-1 8287:978-0-486-65620-5 8142:978-0-486-24073-2 8066:978-1-61374-115-3 7981:978-0-691-11880-2 7833:978-0-471-08073-2 7685:Kolaitis, Phokion 7610:Moschovakis, Joan 7480:978-0-521-28761-6 7403:978-1-00-949438-0 7341:Zhuang, Chaohui. 7030:978-1-107-04348-0 6738:978-0-19-993041-8 6668:978-3-7643-7523-2 6622:978-88-7699-077-9 6158:978-0-8018-9012-3 5933:. pp. 9–13. 5931:Springer New York 5921:Hartshorne, Robin 5771:. pp. 1–18. 5525:. pp. 1–30. 5354:on March 22, 2023 5230:978-3-031-12321-4 5111:978-81-7099-404-6 5067:978-0-486-27239-9 4932:978-0-88385-323-8 4701:"Mathematic (n.)" 4288:generating series 4157:Law (mathematics) 3949:of two elements, 3915: 3900:multiplies it by 3856:harmonic analysis 3700:Greco-Roman world 3698:and then for the 3696:ancient Near East 3672:financial analyst 3532:Saunders Mac Lane 3379:since the 1990s. 3308:Supply and demand 3196:The skin of this 3168:Kepler conjecture 3156:complexity theory 3096:Specific sciences 2924:quantum mechanics 2906:The aftermath of 2878:computer networks 2864:. The problem of 2772:There is still a 2765:, which replaced 2456:Gottfried Leibniz 2444:marine navigation 2424:symbolic notation 2287:to calculate the 2239:Greek mathematics 2069:decision sciences 2039:in, for example, 2025:decision problems 1839:actually infinite 1746:Discrete geometry 1496:universal algebra 1450:commutative rings 1343:his main treatise 1292:quadratic formula 1245:Discrete geometry 1143:in revealing the 1141:Russell's paradox 1099:analytic geometry 703:higher arithmetic 651:integral calculus 516:" (or sometimes " 493:was known as the 375:Greek mathematics 264:for the needs of 247: 246: 202: 201: 27:Area of knowledge 16:(Redirected from 14873: 14841: 14833: 14832: 14831: 14824: 14823:from Wikiversity 14816: 14815: 14814: 14807: 14799: 14798: 14797: 14790: 14782: 14781: 14780: 14773: 14765: 14764: 14763: 14756: 14748: 14747: 14746: 14739: 14731: 14730: 14729: 14722: 14714: 14713: 14712: 14702: 14686: 14685: 14674: 14673: 14672: 14662: 14661: 14660: 14650: 14649: 14641: 14621: 14620: 14609: 14608: 14597: 14596: 14586: 14585: 14517:Computer algebra 14492:Computer science 14212:Complex analysis 14046: 14039: 14032: 14023: 14022: 14017: 13995: 13983: 13969: 13947: 13915: 13901: 13889: 13875: 13863: 13854:Robbins, Herbert 13850:Courant, Richard 13840: 13817:Davis, Philip J. 13812: 13800: 13746: 13712: 13710: 13708: 13702: 13695: 13675: 13665: 13663: 13661: 13655: 13640: 13630: 13610: 13596: 13560: 13558: 13556: 13529: 13494:Peirce, Benjamin 13489: 13474: 13460: 13458: 13456: 13450: 13435: 13425: 13413: 13399: 13397: 13395: 13336: 13314: 13298: 13284: 13267:Bouleau, Nicolas 13253: 13252: 13250: 13248: 13233: 13227: 13226: 13224: 13222: 13207: 13201: 13200: 13198: 13196: 13175: 13163: 13157: 13156: 13154: 13152: 13133: 13124: 13123: 13121: 13119: 13104:"The Wolf Prize" 13100: 13094: 13093: 13091: 13089: 13053: 13047: 13046: 13044: 13042: 13027: 13021: 13020: 13018: 13016: 13001: 12995: 12994: 12992: 12990: 12984: 12978:. 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Scotland, UK: 8825: 8819: 8818: 8780: 8774: 8773: 8771: 8769: 8763: 8728: 8719: 8713: 8712: 8710: 8708: 8702: 8691: 8682: 8676: 8675: 8657: 8634:LeFevre, Jo-Anne 8629: 8623: 8622: 8620: 8618: 8603:(October 2000). 8601:Wolfram, Stephan 8597: 8591: 8585: 8579: 8578: 8547: 8541: 8540: 8528: 8517: 8511: 8510: 8508: 8506: 8500: 8469: 8460: 8454: 8453: 8451: 8449: 8425: 8419: 8418: 8389: 8383: 8382: 8348: 8342: 8341: 8305: 8299: 8298: 8296: 8294: 8268: 8262: 8256: 8250: 8244: 8238: 8232: 8226: 8220: 8214: 8208: 8202: 8196: 8190: 8189: 8153: 8147: 8146: 8134: 8121: 8115: 8109: 8103: 8097: 8091: 8085: 8079: 8078: 8049: 8043: 8042: 8031: 8025: 8024: 8022: 8020: 8014: 7969: 7943: 7937: 7936: 7934: 7932: 7911: 7905: 7904: 7902: 7900: 7880: 7874: 7868: 7862: 7861: 7812: 7806: 7805: 7760: 7754: 7753: 7751: 7749: 7743: 7736: 7725: 7719: 7718: 7716: 7714: 7708: 7701: 7669: 7663: 7662: 7660: 7640: 7634: 7633: 7631: 7629: 7606: 7600: 7599: 7597: 7595: 7547: 7538: 7537: 7498: 7492: 7491: 7489: 7487: 7464: 7458: 7457: 7455: 7453: 7442: 7430: 7424: 7423: 7369: 7363: 7362: 7360: 7358: 7347: 7338: 7332: 7331: 7329: 7327: 7307: 7301: 7300: 7298: 7296: 7219: 7210: 7204: 7203: 7201: 7199: 7158: 7152: 7151: 7149: 7147: 7106: 7100: 7099: 7072: 7066: 7065: 7063: 7061: 6997:Hales, Thomas C. 6993: 6987: 6986: 6984: 6982: 6918:Roberts, Fred S. 6909: 6903: 6902: 6900: 6898: 6861: 6834: 6828: 6827: 6825: 6823: 6807: 6801: 6800: 6798: 6796: 6783:. Scotland, UK: 6777:"Leonhard Euler" 6772: 6766: 6765: 6763: 6761: 6755: 6714: 6702: 6696: 6695: 6693: 6691: 6648: 6642: 6641: 6639: 6637: 6606: 6600: 6599: 6597: 6595: 6553: 6547: 6541: 6535: 6534: 6532: 6530: 6525:. April 24, 2024 6515: 6509: 6508: 6506: 6504: 6498: 6433: 6424: 6418: 6417: 6415: 6413: 6385:(3rd ed.). 6369: 6363: 6362: 6360: 6358: 6342: 6334: 6328: 6327: 6325: 6323: 6306: 6300: 6299: 6293: 6291: 6279: 6270: 6264: 6258: 6252: 6251: 6209: 6185: 6179: 6178: 6147:(2nd ed.). 6140: 6134: 6133: 6131: 6129: 6116:. Scotland, UK: 6105: 6099: 6098: 6096: 6094: 6031: 6022: 6016: 6015: 5974: 5968: 5967: 5965: 5963: 5917: 5911: 5910: 5909: 5903: 5901: 5899: 5847: 5841: 5840: 5838: 5826: 5815: 5814: 5769:World Scientific 5760: 5754: 5753: 5711: 5684: 5678: 5677: 5624: 5618: 5617: 5567: 5561: 5560: 5511: 5505: 5504: 5502: 5500: 5494: 5483: 5475: 5454: 5453: 5448: 5446: 5440: 5401: 5399:10.1090/noti2052 5383: 5370: 5364: 5363: 5361: 5359: 5353: 5347:. Archived from 5342: 5333: 5324: 5323: 5289: 5265: 5259: 5258: 5193: 5187: 5186: 5141: 5135: 5134: 5097: 5091: 5090: 5050: 5044: 5043: 5041: 5039: 5014: 5008: 5007: 5005: 5003: 4983: 4977: 4959: 4953: 4952: 4911: 4905: 4904: 4858: 4852: 4851: 4849: 4847: 4825: 4819: 4818: 4816: 4814: 4797: 4791: 4790: 4788: 4786: 4758: 4752: 4751: 4749: 4747: 4729: 4723: 4722: 4720: 4718: 4696: 4687: 4686: 4607: 4590: 4589: 4534: 4528: 4527: 4525: 4523: 4501: 4495: 4494: 4445: 4436: 4430: 4424: 4415: 4414: 4404: 4402: 4400: 4394: 4371: 4351: 4335: 4332: 4326: 4323: 4317: 4314: 4308: 4305:proof assistants 4297: 4291: 4284:complex analysis 4280: 4274: 4263: 4257: 4246: 4240: 4229: 4150: 4145: 4144: 4128:Grigori Perelman 3983: 3981: 3980: 3975: 3973: 3965: 3960: 3926: 3924: 3923: 3918: 3916: 3908: 3789:computer program 3731: 3727: 3724: 3712: 3709: 3684:commodity trader 3594:quasi-empiricist 3508:of mathematics. 3365:Kondratiev cycle 3335: 3332: 3329: 3326: 3323: 3198:giant pufferfish 3087: 3085: 3084: 3079: 3074: 3073: 2995:RSA cryptosystem 2920:Laurent Schwartz 2918:, introduced by 2897:pure mathematics 2893:pure mathematics 2889:Richard Dedekind 2885:Karl Weierstrass 2874:RSA cryptosystem 2831: 2822: 2810:Pure mathematics 2649: 2641: 2633: 2631: 2630: 2625: 2609: 2601: 2454:(1643–1727) and 2274: 2270: 2267: 2189: 2139:computer algebra 1989:(x̄) tends to a 1957:computer science 1953:proof assistants 1949:program analysis 1880:, as founded by 1870:Peano arithmetic 1760:continuous games 1713:geometric shapes 1603:potential theory 1573:complex analysis 1404:abstract algebra 1383:modular integers 1361:linear equations 1265:Complex geometry 1167:Girard Desargues 901:only in 1994 by 876:Pierre de Fermat 853: 851: 850: 845: 837: 822:rational numbers 819: 817: 816: 811: 806: 788: 786: 785: 780: 772: 687:axiomatic method 558: 557: 544: 477: 476: 475:μαθηματικὴ τέχνη 467: 466: 448: 447: 446: 413:axiomatic method 364:pure mathematics 344:computer science 328:natural sciences 293:abstract objects 239: 232: 225: 211: 75: 74: 43: 42: 21: 14881: 14880: 14876: 14875: 14874: 14872: 14871: 14870: 14861:Formal sciences 14846: 14845: 14844: 14834: 14829: 14827: 14817: 14812: 14810: 14800: 14795: 14793: 14789:from Wikisource 14783: 14778: 14776: 14766: 14761: 14759: 14749: 14744: 14742: 14732: 14727: 14725: 14721:from Wiktionary 14715: 14710: 14708: 14705: 14701:sister projects 14698:at Knowledge's 14692: 14680: 14670: 14668: 14658: 14656: 14644: 14636: 14634: 14629: 14580: 14571: 14521: 14478: 14457:Systems science 14388: 14384:Homotopy theory 14350: 14317: 14269: 14241: 14188: 14135: 14106:Category theory 14092: 14057: 14050: 14020: 14014: 13992: 13972: 13966: 13950: 13944: 13904: 13898: 13878: 13872: 13848: 13837: 13815: 13809: 13792: 13788: 13787: 13786: 13766: 13765: 13761: 13754: 13752:Further reading 13749: 13739: 13706: 13704: 13700: 13673: 13659: 13657: 13653: 13638: 13627: 13601:Popper, Karl R. 13599: 13577: 13554: 13552: 13518:10.2307/2369153 13492: 13483: 13465:Oakley, Barbara 13463: 13454: 13452: 13448: 13433: 13422: 13393: 13391: 13365: 13341:Kleiner, Israel 13333: 13317: 13311: 13281: 13273:. 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May 6, 2020. 13135: 13134: 13127: 13117: 13115: 13108:Wolf Foundation 13102: 13101: 13097: 13087: 13085: 13078: 13054: 13050: 13040: 13038: 13029: 13028: 13024: 13014: 13012: 13003: 13002: 12998: 12988: 12986: 12982: 12971: 12967: 12966: 12962: 12952: 12950: 12937: 12936: 12932: 12922: 12920: 12911: 12910: 12906: 12896: 12894: 12881: 12880: 12876: 12866: 12864: 12851: 12850: 12843: 12833: 12831: 12818: 12817: 12813: 12805: 12801: 12793: 12789: 12779: 12777: 12775: 12759: 12755: 12745: 12743: 12741: 12725: 12721: 12711: 12709: 12696: 12692: 12682: 12680: 12667: 12666: 12662: 12652: 12650: 12635: 12631: 12618: 12617: 12613: 12606: 12585: 12581: 12574: 12552: 12548: 12540: 12534: 12530: 12520: 12518: 12459: 12455: 12432:10.2307/3613237 12416: 12412: 12373: 12369: 12360: 12354: 12350: 12321: 12317: 12300: 12298: 12296: 12277: 12273: 12263: 12261: 12259: 12243: 12239: 12229: 12227: 12223: 12190: 12184: 12180: 12151: 12147: 12118: 12114: 12067: 12063: 12053: 12051: 12047: 12024: 12018: 12014: 12004: 12002: 11998: 11992: 11965: 11956: 11952: 11915: 11911: 11896:10.2307/2314867 11880: 11876: 11866: 11864: 11862: 11843: 11835: 11831: 11821: 11819: 11815: 11808: 11800: 11796: 11757: 11753: 11746: 11720: 11716: 11706: 11704: 11702: 11680:Robson, Eleanor 11677: 11673: 11663: 11661: 11659: 11643: 11639: 11608: 11604: 11589: 11585: 11573: 11569: 11559: 11557: 11553: 11522: 11516: 11512: 11502: 11500: 11485: 11481: 11471: 11469: 11446:10.2307/3618304 11423: 11419: 11407: 11403: 11393: 11391: 11389: 11373:Cajori, Florian 11370: 11366: 11356: 11354: 11352: 11336:Franklin, James 11333: 11329: 11322: 11297: 11293: 11275: 11273: 11271: 11252: 11245: 11206: 11197: 11187: 11185: 11181: 11175: 11160: 11154: 11150: 11108: 11104: 11094: 11092: 11077: 11073: 11063: 11061: 11048: 11047: 11043: 11033: 11031: 11029: 11013: 11009: 10996: 10995: 10991: 10982: 10981: 10977: 10962: 10939: 10935: 10925: 10923: 10910: 10909: 10905: 10892: 10891: 10887: 10877: 10875: 10862: 10861: 10857: 10844: 10843: 10839: 10831: 10823: 10819: 10811: 10803: 10799: 10789: 10787: 10785: 10769: 10762: 10752: 10750: 10748: 10730: 10726: 10691: 10687: 10677: 10675: 10666: 10662: 10661: 10657: 10647: 10645: 10635: 10631: 10621: 10619: 10613:"MAM2001 Essay" 10611: 10610: 10606: 10596: 10594: 10586: 10585: 10581: 10566: 10539: 10535: 10519: 10513: 10509: 10499: 10497: 10484: 10483: 10479: 10471: 10467: 10459: 10455: 10446: 10442: 10432: 10430: 10426: 10407: 10398: 10389: 10379: 10377: 10323: 10319: 10309: 10307: 10306:. April 8, 2024 10298: 10297: 10293: 10283: 10281: 10273: 10272: 10268: 10258: 10256: 10248: 10247: 10243: 10233: 10231: 10227: 10220: 10212:Atiyah, Michael 10209: 10205: 10195: 10193: 10191: 10175: 10171: 10140: 10136: 10102: 10098: 10058: 10049: 10028:(11): 389–507. 10018: 10014: 10004: 10002: 9987: 9983: 9973: 9971: 9958: 9957: 9953: 9943: 9941: 9937: 9926: 9920: 9916: 9897:Current Science 9893: 9889: 9879: 9877: 9875: 9849: 9845: 9808: 9804: 9794: 9792: 9776: 9772: 9762: 9760: 9741: 9737: 9727: 9725: 9710: 9706: 9696: 9694: 9690: 9675: 9667: 9663: 9632: 9628: 9618: 9616: 9586: 9582: 9572: 9570: 9530: 9526: 9516: 9514: 9512: 9494: 9490: 9480: 9478: 9474: 9443: 9434: 9430: 9407: 9393: 9389: 9379: 9377: 9375: 9357: 9353: 9322:Kuhn, Thomas S. 9319: 9315: 9308: 9286: 9279: 9269: 9267: 9247: 9243: 9236: 9222: 9218: 9211: 9194: 9190: 9180: 9178: 9176: 9158: 9154: 9144: 9142: 9140: 9124: 9120: 9110: 9108: 9095: 9094: 9090: 9080: 9078: 9076: 9047: 9043: 9033: 9031: 9009: 9005: 8995: 8993: 8963: 8959: 8949: 8947: 8945: 8929: 8925: 8860: 8856: 8846: 8844: 8827: 8826: 8822: 8799: 8781: 8777: 8767: 8765: 8761: 8726: 8720: 8716: 8706: 8704: 8700: 8689: 8683: 8679: 8630: 8626: 8616: 8614: 8598: 8594: 8586: 8582: 8567:10.2307/2304570 8548: 8544: 8537: 8526: 8518: 8514: 8504: 8502: 8498: 8467: 8461: 8457: 8447: 8445: 8426: 8422: 8407: 8390: 8386: 8371: 8349: 8345: 8306: 8302: 8292: 8290: 8288: 8269: 8265: 8257: 8253: 8245: 8241: 8233: 8229: 8221: 8217: 8209: 8205: 8197: 8193: 8154: 8150: 8143: 8122: 8118: 8110: 8106: 8098: 8094: 8086: 8082: 8067: 8050: 8046: 8032: 8028: 8018: 8016: 8012: 7982: 7967: 7951:Gowers, Timothy 7944: 7940: 7930: 7928: 7912: 7908: 7898: 7896: 7881: 7877: 7869: 7865: 7834: 7813: 7809: 7778: 7761: 7757: 7747: 7745: 7741: 7734: 7726: 7722: 7712: 7710: 7706: 7699: 7673:Halpern, Joseph 7670: 7666: 7641: 7637: 7627: 7625: 7607: 7603: 7593: 7591: 7548: 7541: 7518:10.2307/2689412 7499: 7495: 7485: 7483: 7481: 7465: 7461: 7451: 7449: 7440: 7431: 7427: 7404: 7370: 7366: 7356: 7354: 7345: 7339: 7335: 7325: 7323: 7320:Quanta Magazine 7308: 7304: 7294: 7292: 7234:10.2307/2687794 7217: 7211: 7207: 7197: 7195: 7159: 7155: 7145: 7143: 7107: 7103: 7076:Sipser, Michael 7073: 7069: 7059: 7057: 7031: 7009:Turing's Legacy 6994: 6990: 6980: 6978: 6944: 6910: 6906: 6896: 6894: 6838:Franklin, James 6835: 6831: 6821: 6819: 6809: 6808: 6804: 6794: 6792: 6773: 6769: 6759: 6757: 6753: 6739: 6712: 6703: 6699: 6689: 6687: 6669: 6649: 6645: 6635: 6633: 6623: 6607: 6603: 6593: 6591: 6573: 6554: 6550: 6542: 6538: 6528: 6526: 6517: 6516: 6512: 6502: 6500: 6496: 6431: 6425: 6421: 6411: 6409: 6373:Berntjes, Sonja 6370: 6366: 6356: 6354: 6335: 6331: 6321: 6319: 6307: 6303: 6289: 6287: 6277: 6271: 6267: 6259: 6255: 6186: 6182: 6159: 6141: 6137: 6127: 6125: 6106: 6102: 6092: 6090: 6029: 6023: 6019: 5996: 5975: 5971: 5961: 5959: 5941: 5918: 5914: 5904: 5897: 5895: 5848: 5844: 5827: 5818: 5787: 5761: 5757: 5688:Kleiner, Israel 5685: 5681: 5650: 5625: 5621: 5590: 5568: 5564: 5533: 5512: 5508: 5498: 5496: 5492: 5481: 5477: 5476: 5457: 5444: 5442: 5438: 5381: 5373:Dunne, Edward; 5371: 5367: 5357: 5355: 5351: 5340: 5334: 5327: 5266: 5262: 5231: 5194: 5190: 5167: 5142: 5138: 5112: 5098: 5094: 5068: 5051: 5047: 5037: 5035: 5016: 5015: 5011: 5001: 4999: 4985: 4984: 4980: 4960: 4956: 4933: 4925:. p. 257. 4912: 4908: 4859: 4855: 4845: 4843: 4826: 4822: 4812: 4810: 4798: 4794: 4784: 4782: 4780:Clarendon Press 4759: 4755: 4745: 4743: 4730: 4726: 4716: 4714: 4697: 4690: 4611:Kleiner, Israel 4608: 4593: 4538:Alexander, Amir 4535: 4531: 4521: 4519: 4502: 4498: 4446: 4439: 4431: 4427: 4421:Wayback Machine 4409: 4407:at ResearchGate 4398: 4396: 4392: 4369: 4356:Mitterer, Josef 4352: 4348: 4344: 4339: 4338: 4333: 4329: 4324: 4320: 4315: 4311: 4298: 4294: 4281: 4277: 4264: 4260: 4247: 4243: 4230: 4226: 4221: 4216: 4211: 4146: 4139: 4136: 4032: 4028:Main category: 4026: 4013: 4007: 3999:self-similarity 3969: 3961: 3956: 3954: 3951: 3950: 3907: 3905: 3902: 3901: 3890: 3884: 3879: 3877:Cultural impact 3869:(as in art) or 3793:Apery's theorem 3785: 3729: 3725: 3711: 1650 BCE 3710: 3636: 3630: 3625: 3574:inference rules 3562: 3556: 3520: 3514: 3485:Charles Hermite 3458: 3453: 3447: 3437:a science, but 3434:Theodor Zwinger 3412:; for example, 3406: 3398:complex systems 3350:Homo economicus 3333: 3330: 3327: 3324: 3317:Homo economicus 3286: 3276: 3274:Social sciences 3253: 3247: 3239:clinical trials 3190: 3182:Main articles: 3180: 3166:. In 1998, the 3136: 3126: 3109: 3103: 3098: 3069: 3065: 3063: 3060: 3059: 3030:Albert Einstein 3015:Johannes Kepler 2972: 2942:introduced the 2845: 2844: 2843: 2842: 2834: 2833: 2832: 2824: 2823: 2812: 2804:Main articles: 2802: 2790:experimentation 2742: 2647: 2639: 2619: 2616: 2615: 2607: 2599: 2536: 2524:Main articles: 2522: 2364: 2356:infinite series 2272: 2268: 2162: 2157: 2151: 2127:rounding errors 2095: 2089: 2011:random sampling 1975: 1967:Main articles: 1965: 1939:, their use in 1904:, who promoted 1804: 1796:Main articles: 1794: 1644: 1638: 1577:complex numbers 1538:Cauchy sequence 1530: 1522:Main articles: 1520: 1500:category theory 1480:Boolean algebra 1284: 1278: 1269:complex numbers 1253:, the study of 1251:Convex geometry 1201:Manifold theory 1181:Affine geometry 1124:(which spawned 1060:Euclidean space 1052:Euclidean plane 968: 962: 915:category theory 833: 828: 825: 824: 802: 797: 794: 793: 768: 763: 760: 759: 757:natural numbers 725: 719: 672:solid mechanics 597: 522:Saint Augustine 484:ars mathematica 452:Classical times 425: 393:natural numbers 348:social sciences 317:deductive rules 243: 198: 197: 148: 140: 139: 135:Decision theory 83: 39: 28: 23: 22: 15: 12: 11: 5: 14879: 14869: 14868: 14863: 14858: 14843: 14842: 14825: 14808: 14806:from Wikibooks 14791: 14774: 14772:from Wikiquote 14757: 14740: 14723: 14694: 14691: 14690: 14678: 14666: 14654: 14631: 14630: 14628: 14627: 14615: 14603: 14591: 14576: 14573: 14572: 14570: 14569: 14564: 14559: 14554: 14549: 14544: 14543: 14542: 14535:Mathematicians 14531: 14529: 14527:Related topics 14523: 14522: 14520: 14519: 14514: 14509: 14504: 14499: 14494: 14488: 14486: 14480: 14479: 14477: 14476: 14475: 14474: 14469: 14464: 14462:Control theory 14454: 14449: 14444: 14439: 14434: 14429: 14424: 14419: 14414: 14409: 14404: 14398: 14396: 14390: 14389: 14387: 14386: 14381: 14376: 14371: 14366: 14360: 14358: 14352: 14351: 14349: 14348: 14343: 14338: 14333: 14327: 14325: 14319: 14318: 14316: 14315: 14310: 14305: 14300: 14295: 14290: 14285: 14279: 14277: 14271: 14270: 14268: 14267: 14262: 14257: 14251: 14249: 14243: 14242: 14240: 14239: 14237:Measure theory 14234: 14229: 14224: 14219: 14214: 14209: 14204: 14198: 14196: 14190: 14189: 14187: 14186: 14181: 14176: 14171: 14166: 14161: 14156: 14151: 14145: 14143: 14137: 14136: 14134: 14133: 14128: 14123: 14118: 14113: 14108: 14102: 14100: 14094: 14093: 14091: 14090: 14085: 14080: 14079: 14078: 14073: 14062: 14059: 14058: 14049: 14048: 14041: 14034: 14026: 14019: 14018: 14012: 13996: 13990: 13974:Pappas, Theoni 13970: 13964: 13948: 13942: 13929: 13908:, ed. (2000). 13902: 13896: 13876: 13870: 13846: 13835: 13813: 13807: 13789: 13785: 13784: 13779: 13774: 13768: 13767: 13756: 13755: 13753: 13750: 13748: 13747: 13737: 13717:Whittle, Peter 13713: 13686:(1): 101–109. 13671:"Book Reviews" 13666: 13649:(7): 778–782. 13631: 13625: 13597: 13575: 13562: 13490: 13481: 13461: 13426: 13420: 13400: 13363: 13337: 13331: 13315: 13309: 13285: 13279: 13262: 13260: 13257: 13255: 13254: 13228: 13202: 13188: 13158: 13125: 13095: 13076: 13048: 13022: 12996: 12960: 12930: 12904: 12874: 12853:"Fields Medal" 12841: 12811: 12799: 12787: 12773: 12753: 12739: 12719: 12690: 12660: 12629: 12611: 12604: 12579: 12572: 12546: 12528: 12453: 12410: 12383:(2): 197–220. 12367: 12348: 12335:(3): 226–247. 12315: 12294: 12271: 12257: 12237: 12203:(4): 195–203. 12178: 12145: 12112: 12061: 12012: 11990: 11950: 11929:(4): 457–459. 11909: 11874: 11860: 11829: 11794: 11751: 11744: 11714: 11700: 11694:. OUP Oxford. 11671: 11657: 11637: 11602: 11583: 11567: 11533:(2): 409–449. 11510: 11493:Acta Academica 11479: 11417: 11401: 11387: 11364: 11350: 11327: 11320: 11291: 11269: 11255:Tobies, Renate 11243: 11195: 11173: 11148: 11127:10.1086/286957 11121:(4): 289–303. 11102: 11071: 11041: 11027: 11007: 10989: 10975: 10960: 10933: 10903: 10885: 10855: 10837: 10834:. Vol. 2. 10817: 10797: 10783: 10760: 10746: 10724: 10705:(1): 197–220. 10685: 10655: 10629: 10617:ww2.amstat.org 10604: 10579: 10564: 10556:10.17226/10633 10533: 10507: 10477: 10475:, p. 285. 10465: 10453: 10440: 10387: 10317: 10291: 10266: 10241: 10203: 10189: 10169: 10134: 10096: 10047: 10012: 9981: 9951: 9914: 9903:(3): 415–423. 9887: 9873: 9843: 9802: 9770: 9755:(7): 753–759. 9735: 9704: 9661: 9648:(3): 267–288. 9626: 9580: 9547:(3): 405–427. 9524: 9510: 9488: 9428: 9405: 9387: 9373: 9351: 9338:10.2307/202372 9313: 9306: 9277: 9259:Philosophy Now 9241: 9234: 9216: 9209: 9188: 9174: 9152: 9138: 9118: 9088: 9074: 9041: 9003: 8957: 8943: 8923: 8854: 8820: 8797: 8775: 8714: 8696:. p. 75. 8677: 8648:(3): 322–354. 8624: 8592: 8580: 8542: 8535: 8512: 8455: 8420: 8405: 8393:Saliba, George 8384: 8369: 8343: 8322:10.1086/368443 8300: 8286: 8263: 8251: 8239: 8227: 8215: 8203: 8191: 8164:(4): 289–309. 8148: 8141: 8116: 8104: 8092: 8080: 8065: 8044: 8026: 7980: 7938: 7906: 7875: 7863: 7832: 7820:Dodge, Yadolah 7807: 7776: 7755: 7737:. p. 10. 7720: 7681:Immerman, Neil 7677:Harper, Robert 7664: 7635: 7601: 7562:(4): 513–534. 7539: 7512:(4): 207–216. 7502:Snapper, Ernst 7493: 7479: 7459: 7434:Avigad, Jeremy 7425: 7402: 7364: 7333: 7302: 7228:(4): 441–484. 7205: 7153: 7101: 7067: 7029: 6988: 6942: 6904: 6852:(2): 355–378. 6829: 6802: 6767: 6737: 6697: 6667: 6643: 6621: 6601: 6571: 6548: 6536: 6510: 6444:(3): 245–302. 6419: 6364: 6329: 6311:"Al-Khwarizmi" 6301: 6265: 6253: 6200:(2): 127–163. 6180: 6157: 6135: 6100: 6042:(3): 383–417. 6017: 5994: 5978:Boyer, Carl B. 5969: 5939: 5912: 5851:Hilbert, David 5842: 5816: 5785: 5755: 5690:(March 2000). 5679: 5648: 5619: 5588: 5562: 5531: 5506: 5455: 5392:(3): 410–411. 5377:(March 2020). 5365: 5336:Warner, Evan. 5325: 5280:(2): 109–136. 5260: 5229: 5197:Musielak, Dora 5188: 5165: 5159:. p. 14. 5136: 5110: 5092: 5066: 5045: 5018:"Math (Noun³)" 5009: 4987:"Maths (Noun)" 4978: 4954: 4931: 4915:Boas, Ralph P. 4906: 4853: 4820: 4792: 4753: 4724: 4688: 4591: 4554:10.1086/661620 4548:(3): 475–480. 4529: 4496: 4449:Wigner, Eugene 4437: 4425: 4345: 4343: 4340: 4337: 4336: 4327: 4318: 4309: 4292: 4275: 4267:conic sections 4265:This includes 4258: 4241: 4223: 4222: 4220: 4217: 4215: 4212: 4210: 4209: 4204: 4199: 4194: 4189: 4184: 4179: 4174: 4169: 4164: 4159: 4153: 4152: 4151: 4135: 4132: 4101: 4100: 4093: 4083: 4076: 4025: 4022: 4009:Main article: 4006: 4005:Popularization 4003: 3986:Rorschach test 3972: 3968: 3964: 3959: 3943:symmetry group 3914: 3911: 3886:Main article: 3883: 3880: 3878: 3875: 3784: 3781: 3747:Following the 3734:Imperial China 3730: 500 BCE 3715:oral tradition 3656:popularization 3640:human activity 3632:Main article: 3629: 3626: 3624: 3621: 3555: 3552: 3516:Main article: 3513: 3510: 3499: 3489:Henri Poincaré 3457: 3454: 3449:Main article: 3446: 3443: 3405: 3402: 3275: 3272: 3251:Geomathematics 3249:Main article: 3246: 3245:Earth sciences 3243: 3202:Turing pattern 3179: 3176: 3172:sphere packing 3125: 3122: 3105:Main article: 3102: 3099: 3097: 3094: 3077: 3072: 3068: 3032:developed the 3007:conic sections 2971: 2968: 2940:George Collins 2836: 2835: 2826: 2825: 2817: 2816: 2815: 2814: 2813: 2801: 2798: 2795: 2782:counterexample 2741: 2738: 2623: 2612:multiplication 2521: 2518: 2460:Leonhard Euler 2448:René Descartes 2428:François Viète 2420:Western Europe 2363: 2360: 2301:conic sections 2243:Ancient Greeks 2224:multiplication 2161: 2158: 2153:Main article: 2150: 2147: 2123:discretization 2091:Main article: 2088: 2085: 2077:control theory 2013:or randomized 1964: 1961: 1793: 1790: 1789: 1788: 1771: 1753: 1748: 1743: 1737: 1733:and a part of 1724: 1715: 1668:implementation 1640:Main article: 1637: 1634: 1633: 1632: 1626: 1620: 1614: 1599:measure theory 1592: 1586: 1519: 1516: 1488: 1487: 1477: 1467: 1461: 1443: 1437: 1434:linear algebra 1427: 1421: 1400:modern algebra 1366:linear algebra 1350:François Viète 1321:and formulas. 1280:Main article: 1277: 1274: 1273: 1272: 1262: 1248: 1242: 1241: 1240: 1220: 1210: 1204: 1198: 1188: 1178: 1175:parallel lines 1081:as lengths of 1067:René Descartes 1056:plane geometry 1038:The resulting 964:Main article: 961: 958: 880:Leonhard Euler 843: 840: 836: 832: 809: 805: 801: 778: 775: 771: 767: 721:Main article: 718: 715: 696: 596: 593: 561:mathematic(al) 551:ta mathēmatiká 491:Pythagoreanism 424: 421: 245: 244: 242: 241: 234: 227: 219: 216: 215: 204: 203: 200: 199: 196: 195: 190: 185: 180: 175: 170: 165: 160: 155: 149: 146: 145: 142: 141: 138: 137: 128: 123: 114: 109: 100: 95: 90: 84: 79: 78: 71: 70: 69: 68: 63: 55: 54: 48: 47: 26: 18:Mathematically 9: 6: 4: 3: 2: 14878: 14867: 14864: 14862: 14859: 14857: 14854: 14853: 14851: 14840:from Wikidata 14839: 14838: 14826: 14822: 14821: 14809: 14805: 14804: 14792: 14788: 14787: 14775: 14771: 14770: 14758: 14755:from Wikinews 14754: 14753: 14741: 14737: 14736: 14724: 14720: 14719: 14707: 14706: 14703: 14697: 14689: 14684: 14679: 14677: 14667: 14665: 14655: 14653: 14648: 14643: 14642: 14639: 14626: 14625: 14616: 14614: 14613: 14604: 14602: 14601: 14592: 14590: 14589: 14584: 14578: 14577: 14574: 14568: 14565: 14563: 14560: 14558: 14555: 14553: 14550: 14548: 14545: 14541: 14538: 14537: 14536: 14533: 14532: 14530: 14528: 14524: 14518: 14515: 14513: 14510: 14508: 14505: 14503: 14500: 14498: 14495: 14493: 14490: 14489: 14487: 14485: 14484:Computational 14481: 14473: 14470: 14468: 14465: 14463: 14460: 14459: 14458: 14455: 14453: 14450: 14448: 14445: 14443: 14440: 14438: 14435: 14433: 14430: 14428: 14425: 14423: 14420: 14418: 14415: 14413: 14410: 14408: 14405: 14403: 14400: 14399: 14397: 14395: 14391: 14385: 14382: 14380: 14377: 14375: 14372: 14370: 14367: 14365: 14362: 14361: 14359: 14357: 14353: 14347: 14344: 14342: 14339: 14337: 14334: 14332: 14329: 14328: 14326: 14324: 14323:Number theory 14320: 14314: 14311: 14309: 14306: 14304: 14301: 14299: 14296: 14294: 14291: 14289: 14286: 14284: 14281: 14280: 14278: 14276: 14272: 14266: 14263: 14261: 14258: 14256: 14255:Combinatorics 14253: 14252: 14250: 14248: 14244: 14238: 14235: 14233: 14230: 14228: 14225: 14223: 14220: 14218: 14215: 14213: 14210: 14208: 14207:Real analysis 14205: 14203: 14200: 14199: 14197: 14195: 14191: 14185: 14182: 14180: 14177: 14175: 14172: 14170: 14167: 14165: 14162: 14160: 14157: 14155: 14152: 14150: 14147: 14146: 14144: 14142: 14138: 14132: 14129: 14127: 14124: 14122: 14119: 14117: 14114: 14112: 14109: 14107: 14104: 14103: 14101: 14099: 14095: 14089: 14086: 14084: 14081: 14077: 14074: 14072: 14069: 14068: 14067: 14064: 14063: 14060: 14055: 14047: 14042: 14040: 14035: 14033: 14028: 14027: 14024: 14015: 14009: 14005: 14001: 13997: 13993: 13987: 13982: 13981: 13975: 13971: 13967: 13961: 13957: 13953: 13949: 13945: 13939: 13935: 13930: 13927: 13926:archive.today 13923: 13919: 13913: 13912: 13907: 13903: 13899: 13893: 13888: 13887: 13881: 13880:Gullberg, Jan 13877: 13873: 13867: 13862: 13861: 13855: 13851: 13847: 13844: 13838: 13832: 13828: 13827: 13822: 13821:Hersh, Reuben 13818: 13814: 13810: 13804: 13799: 13798: 13791: 13790: 13783: 13780: 13778: 13775: 13773: 13770: 13769: 13764: 13759: 13744: 13740: 13734: 13730: 13726: 13722: 13721:"Almost home" 13718: 13714: 13699: 13694: 13689: 13685: 13681: 13680: 13672: 13667: 13652: 13648: 13644: 13637: 13632: 13628: 13622: 13618: 13614: 13609: 13608: 13602: 13598: 13594: 13590: 13586: 13582: 13578: 13576:0-7167-1953-3 13572: 13568: 13563: 13551: 13548: 13544: 13541: 13537: 13533: 13528: 13523: 13519: 13515: 13511: 13507: 13503: 13499: 13495: 13491: 13488: 13484: 13478: 13473: 13472: 13466: 13462: 13447: 13443: 13439: 13432: 13427: 13423: 13417: 13412: 13411: 13405: 13404:Kline, Morris 13401: 13390: 13386: 13382: 13378: 13374: 13370: 13366: 13360: 13356: 13352: 13348: 13347: 13342: 13338: 13334: 13328: 13324: 13320: 13316: 13312: 13306: 13302: 13297: 13296: 13290: 13286: 13282: 13276: 13272: 13268: 13264: 13263: 13242: 13238: 13232: 13216: 13212: 13206: 13191: 13185: 13181: 13180: 13172: 13168: 13162: 13146: 13142: 13138: 13132: 13130: 13113: 13109: 13105: 13099: 13083: 13079: 13073: 13069: 13065: 13061: 13060: 13052: 13036: 13032: 13026: 13010: 13006: 13000: 12981: 12977: 12970: 12964: 12948: 12944: 12940: 12934: 12918: 12914: 12908: 12892: 12888: 12884: 12878: 12862: 12858: 12857:Maths History 12854: 12848: 12846: 12829: 12825: 12821: 12815: 12808: 12803: 12796: 12791: 12776: 12770: 12766: 12765: 12757: 12742: 12736: 12732: 12731: 12723: 12707: 12703: 12702: 12694: 12678: 12674: 12670: 12664: 12648: 12644: 12643:www.stsci.edu 12640: 12633: 12625: 12621: 12615: 12607: 12601: 12597: 12593: 12589: 12588:Weyl, Hermann 12583: 12575: 12569: 12565: 12561: 12557: 12550: 12539: 12532: 12516: 12512: 12508: 12504: 12500: 12496: 12492: 12488: 12484: 12480: 12476: 12472: 12468: 12464: 12457: 12449: 12445: 12441: 12437: 12433: 12429: 12425: 12421: 12414: 12406: 12402: 12398: 12394: 12390: 12386: 12382: 12378: 12371: 12363: 12358: 12352: 12343: 12338: 12334: 12330: 12326: 12319: 12312: 12311: 12297: 12291: 12287: 12286: 12281: 12275: 12260: 12254: 12250: 12249: 12241: 12222: 12218: 12214: 12210: 12206: 12202: 12198: 12197: 12189: 12182: 12173: 12168: 12164: 12160: 12156: 12149: 12140: 12135: 12132:: 2519–2525. 12131: 12127: 12123: 12116: 12108: 12104: 12099: 12094: 12089: 12084: 12080: 12076: 12072: 12065: 12046: 12042: 12038: 12034: 12030: 12023: 12016: 11997: 11993: 11987: 11983: 11979: 11975: 11971: 11964: 11963: 11954: 11946: 11942: 11937: 11932: 11928: 11924: 11920: 11913: 11905: 11901: 11897: 11893: 11889: 11885: 11878: 11863: 11857: 11853: 11849: 11842: 11841: 11833: 11814: 11807: 11806: 11798: 11790: 11786: 11782: 11778: 11774: 11770: 11766: 11762: 11761:Math Horizons 11755: 11747: 11741: 11737: 11733: 11729: 11725: 11718: 11703: 11697: 11693: 11689: 11685: 11681: 11675: 11660: 11654: 11650: 11649: 11641: 11633: 11629: 11625: 11621: 11617: 11613: 11606: 11598: 11594: 11587: 11580: 11576: 11575:Peterson 1988 11571: 11552: 11548: 11544: 11540: 11536: 11532: 11528: 11521: 11514: 11498: 11494: 11490: 11483: 11467: 11463: 11459: 11455: 11451: 11447: 11443: 11439: 11435: 11431: 11427: 11426:Brown, Ronald 11421: 11412: 11405: 11390: 11384: 11380: 11379: 11374: 11368: 11353: 11347: 11343: 11342: 11337: 11331: 11323: 11317: 11313: 11309: 11305: 11301: 11295: 11288: 11285: 11272: 11266: 11262: 11261: 11256: 11250: 11248: 11239: 11235: 11231: 11227: 11223: 11219: 11216:(4): 375–85. 11215: 11211: 11204: 11202: 11200: 11180: 11176: 11170: 11166: 11159: 11152: 11144: 11136: 11132: 11128: 11124: 11120: 11116: 11115: 11106: 11090: 11086: 11082: 11075: 11059: 11056:(in French). 11055: 11051: 11045: 11030: 11024: 11020: 11019: 11011: 11003: 10999: 10993: 10985: 10979: 10971: 10967: 10963: 10957: 10953: 10952: 10947: 10946:Jean Bricmont 10943: 10937: 10921: 10917: 10913: 10907: 10899: 10895: 10889: 10873: 10869: 10865: 10859: 10851: 10847: 10846:"Mathematics" 10841: 10830: 10829: 10821: 10810: 10809: 10801: 10786: 10780: 10776: 10775: 10767: 10765: 10753:September 30, 10749: 10743: 10739: 10735: 10728: 10720: 10716: 10712: 10708: 10704: 10700: 10696: 10689: 10674: 10673: 10665: 10659: 10644: 10640: 10633: 10618: 10614: 10608: 10593: 10589: 10583: 10575: 10571: 10567: 10561: 10557: 10553: 10549: 10548: 10544:(2003). "8". 10543: 10537: 10529: 10525: 10518: 10511: 10495: 10491: 10487: 10481: 10474: 10469: 10462: 10457: 10450: 10444: 10425: 10421: 10417: 10413: 10406: 10402: 10396: 10394: 10392: 10375: 10371: 10367: 10363: 10359: 10354: 10349: 10345: 10341: 10337: 10333: 10329: 10321: 10305: 10304:GeeksforGeeks 10301: 10295: 10280: 10276: 10270: 10255: 10251: 10245: 10226: 10219: 10218: 10213: 10207: 10192: 10186: 10182: 10181: 10173: 10165: 10161: 10157: 10153: 10149: 10145: 10138: 10130: 10126: 10122: 10118: 10114: 10110: 10106: 10100: 10092: 10088: 10083: 10078: 10074: 10070: 10066: 10062: 10061:Borel, Armand 10056: 10054: 10052: 10043: 10039: 10035: 10031: 10027: 10023: 10016: 10000: 9996: 9992: 9985: 9969: 9965: 9961: 9955: 9936: 9932: 9925: 9918: 9910: 9906: 9902: 9898: 9891: 9876: 9870: 9866: 9862: 9858: 9854: 9847: 9839: 9835: 9830: 9825: 9821: 9817: 9813: 9806: 9791: 9787: 9783: 9782: 9774: 9758: 9754: 9750: 9746: 9739: 9723: 9719: 9715: 9708: 9689: 9685: 9681: 9674: 9673: 9665: 9656: 9651: 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7777:981-02-3111-3 7773: 7769: 7765: 7759: 7740: 7733: 7732: 7724: 7705: 7698: 7694: 7693:Vianu, Victor 7690: 7686: 7682: 7678: 7674: 7668: 7659: 7654: 7650: 7646: 7639: 7623: 7619: 7615: 7611: 7605: 7589: 7585: 7581: 7577: 7573: 7569: 7565: 7561: 7557: 7553: 7546: 7544: 7535: 7531: 7527: 7523: 7519: 7515: 7511: 7507: 7503: 7497: 7482: 7476: 7472: 7471: 7463: 7448: 7447: 7439: 7435: 7429: 7421: 7417: 7413: 7409: 7405: 7399: 7395: 7391: 7387: 7383: 7379: 7375: 7368: 7353: 7352: 7344: 7337: 7322: 7321: 7316: 7312: 7306: 7291: 7287: 7283: 7279: 7275: 7271: 7267: 7263: 7259: 7255: 7251: 7247: 7243: 7239: 7235: 7231: 7227: 7223: 7216: 7209: 7194: 7190: 7186: 7182: 7178: 7174: 7170: 7169: 7164: 7157: 7142: 7138: 7134: 7130: 7126: 7122: 7118: 7117: 7112: 7105: 7097: 7093: 7089: 7085: 7081: 7078:(July 1992). 7077: 7071: 7056: 7052: 7048: 7044: 7040: 7036: 7032: 7026: 7022: 7018: 7014: 7010: 7006: 7002: 6998: 6992: 6977: 6973: 6969: 6965: 6961: 6957: 6953: 6949: 6945: 6943:0-8218-0448-0 6939: 6935: 6931: 6927: 6923: 6919: 6915: 6908: 6893: 6889: 6885: 6881: 6877: 6873: 6869: 6865: 6860: 6855: 6851: 6847: 6843: 6840:(July 2017). 6839: 6833: 6818: 6817: 6812: 6806: 6790: 6786: 6782: 6778: 6771: 6752: 6748: 6744: 6740: 6734: 6730: 6726: 6722: 6718: 6711: 6707: 6701: 6686: 6682: 6678: 6674: 6670: 6664: 6660: 6656: 6655: 6647: 6632: 6628: 6624: 6618: 6614: 6613: 6605: 6590: 6586: 6582: 6578: 6574: 6572:3-7643-7002-5 6568: 6564: 6563: 6558: 6552: 6545: 6540: 6524: 6523:GeeksforGeeks 6520: 6514: 6495: 6491: 6487: 6483: 6479: 6475: 6471: 6467: 6463: 6459: 6455: 6451: 6447: 6443: 6439: 6438: 6430: 6423: 6408: 6404: 6400: 6396: 6392: 6388: 6384: 6383: 6378: 6374: 6368: 6352: 6348: 6347: 6341: 6333: 6318: 6317: 6312: 6305: 6298: 6297:progressions. 6285: 6284: 6276: 6269: 6262: 6257: 6249: 6245: 6241: 6237: 6233: 6229: 6225: 6221: 6217: 6213: 6208: 6203: 6199: 6195: 6191: 6184: 6176: 6172: 6168: 6164: 6160: 6154: 6150: 6146: 6139: 6123: 6119: 6115: 6111: 6104: 6089: 6085: 6081: 6077: 6073: 6069: 6065: 6061: 6057: 6053: 6049: 6045: 6041: 6037: 6036: 6028: 6021: 6013: 6009: 6005: 6001: 5997: 5995:0-486-43832-5 5991: 5987: 5983: 5979: 5973: 5958: 5954: 5950: 5946: 5942: 5940:0-387-98650-2 5936: 5932: 5928: 5927: 5922: 5916: 5908: 5894: 5890: 5886: 5882: 5878: 5874: 5870: 5866: 5863:. p. 1. 5862: 5858: 5857: 5852: 5846: 5837: 5832: 5825: 5823: 5821: 5812: 5808: 5804: 5800: 5796: 5792: 5788: 5786:981-238-159-7 5782: 5778: 5774: 5770: 5766: 5759: 5751: 5747: 5743: 5739: 5735: 5731: 5727: 5723: 5719: 5715: 5710: 5705: 5701: 5697: 5693: 5689: 5683: 5675: 5671: 5667: 5663: 5659: 5655: 5651: 5649:0-8176-3141-0 5645: 5641: 5637: 5633: 5629: 5623: 5615: 5611: 5607: 5603: 5599: 5595: 5591: 5589:1-56881-006-7 5585: 5581: 5577: 5573: 5566: 5558: 5554: 5550: 5546: 5542: 5538: 5534: 5532:0-201-04287-8 5528: 5524: 5520: 5516: 5510: 5491: 5487: 5480: 5474: 5472: 5470: 5468: 5466: 5464: 5462: 5460: 5452: 5437: 5433: 5429: 5425: 5421: 5417: 5413: 5409: 5405: 5400: 5395: 5391: 5387: 5380: 5376: 5369: 5350: 5346: 5339: 5332: 5330: 5321: 5317: 5313: 5309: 5305: 5301: 5297: 5293: 5288: 5283: 5279: 5275: 5271: 5264: 5256: 5252: 5248: 5244: 5240: 5236: 5232: 5226: 5222: 5218: 5214: 5210: 5206: 5202: 5198: 5192: 5184: 5180: 5176: 5172: 5168: 5166:0-7923-1765-3 5162: 5158: 5154: 5150: 5146: 5140: 5133: 5129: 5125: 5121: 5117: 5113: 5107: 5103: 5096: 5089: 5085: 5081: 5077: 5073: 5069: 5063: 5059: 5055: 5049: 5033: 5029: 5025: 5024: 5019: 5013: 4998: 4994: 4993: 4988: 4982: 4975: 4971: 4970: 4965: 4964: 4958: 4950: 4946: 4942: 4938: 4934: 4928: 4924: 4920: 4916: 4910: 4902: 4898: 4894: 4890: 4886: 4882: 4878: 4874: 4870: 4866: 4865: 4857: 4841: 4837: 4836: 4831: 4824: 4809: 4808: 4803: 4796: 4781: 4777: 4776: 4771: 4767: 4766:Scott, Robert 4763: 4757: 4741: 4737: 4736: 4728: 4712: 4708: 4707: 4702: 4695: 4693: 4684: 4680: 4676: 4672: 4668: 4664: 4660: 4656: 4652: 4648: 4644: 4640: 4636: 4632: 4628: 4624: 4620: 4616: 4612: 4606: 4604: 4602: 4600: 4598: 4596: 4587: 4583: 4579: 4575: 4571: 4567: 4563: 4559: 4555: 4551: 4547: 4543: 4539: 4533: 4517: 4513: 4512: 4507: 4504:Wise, David. 4500: 4492: 4488: 4484: 4480: 4476: 4472: 4468: 4464: 4460: 4459: 4454: 4450: 4444: 4442: 4435:, p. 12. 4434: 4433:Peterson 1988 4429: 4422: 4418: 4413: 4408: 4391: 4387: 4383: 4379: 4375: 4367: 4363: 4362: 4357: 4350: 4346: 4331: 4322: 4313: 4306: 4302: 4296: 4289: 4285: 4279: 4272: 4268: 4262: 4255: 4251: 4245: 4238: 4234: 4228: 4224: 4208: 4205: 4203: 4200: 4198: 4195: 4193: 4190: 4188: 4185: 4183: 4180: 4178: 4175: 4173: 4170: 4168: 4165: 4163: 4160: 4158: 4155: 4154: 4149: 4143: 4138: 4131: 4129: 4125: 4121: 4117: 4112: 4110: 4106: 4105:open problems 4098: 4094: 4091: 4088: 4084: 4081: 4077: 4074: 4070: 4069: 4068: 4065: 4063: 4059: 4055: 4048: 4045: 4041: 4036: 4031: 4021: 4018: 4012: 4002: 4000: 3996: 3991: 3987: 3966: 3962: 3948: 3944: 3936: 3932: 3928: 3912: 3909: 3899: 3898:perfect fifth 3895: 3889: 3874: 3872: 3868: 3864: 3859: 3857: 3853: 3849: 3848: 3843: 3838: 3837: 3832: 3828: 3824: 3820: 3816: 3811: 3809: 3805: 3800: 3798: 3794: 3790: 3780: 3777: 3771: 3769: 3768: 3762: 3758: 3754: 3750: 3745: 3743: 3739: 3735: 3720: 3716: 3706:, dated from 3705: 3704:Rhind papyrus 3701: 3697: 3691: 3689: 3685: 3681: 3677: 3673: 3669: 3665: 3661: 3657: 3653: 3649: 3645: 3641: 3635: 3620: 3617: 3615: 3610: 3606: 3601: 3597: 3595: 3591: 3587: 3583: 3579: 3575: 3571: 3567: 3561: 3551: 3546: 3541: 3539: 3538: 3533: 3527: 3525: 3519: 3509: 3507: 3501: 3497: 3492: 3490: 3486: 3482: 3478: 3474: 3472: 3467: 3463: 3452: 3442: 3440: 3439:pseudoscience 3435: 3431: 3427: 3423: 3419: 3415: 3411: 3401: 3399: 3395: 3391: 3387: 3386: 3380: 3378: 3374: 3373:Peter Turchin 3370: 3366: 3362: 3357: 3353: 3351: 3347: 3343: 3339: 3338:self-interest 3319: 3318: 3309: 3305: 3301: 3299: 3295: 3291: 3285: 3281: 3271: 3269: 3265: 3261: 3257: 3252: 3242: 3240: 3236: 3232: 3230: 3226: 3222: 3218: 3214: 3207: 3203: 3199: 3194: 3189: 3185: 3175: 3173: 3169: 3165: 3161: 3157: 3153: 3149: 3148:coding theory 3145: 3141: 3135: 3131: 3121: 3113: 3108: 3093: 3091: 3075: 3070: 3058: 3054: 3049: 3047: 3043: 3039: 3035: 3031: 3027: 3022: 3020: 3016: 3012: 3008: 3004: 3000: 2996: 2992: 2988: 2983: 2981: 2980:Eugene Wigner 2977: 2967: 2965: 2961: 2957: 2951: 2949: 2945: 2941: 2937: 2933: 2932:Alfred Tarski 2929: 2925: 2921: 2917: 2912: 2909: 2904: 2902: 2898: 2894: 2890: 2886: 2881: 2879: 2875: 2871: 2867: 2863: 2859: 2855: 2850: 2840: 2830: 2821: 2811: 2807: 2797: 2793: 2791: 2787: 2783: 2779: 2775: 2774:philosophical 2770: 2768: 2764: 2760: 2756: 2751: 2747: 2737: 2735: 2731: 2727: 2723: 2719: 2715: 2711: 2710: 2709:homeomorphism 2705: 2704: 2699: 2694: 2692: 2688: 2684: 2680: 2676: 2672: 2668: 2663: 2661: 2657: 2653: 2645: 2637: 2621: 2613: 2605: 2597: 2593: 2589: 2585: 2581: 2577: 2573: 2569: 2565: 2561: 2557: 2553: 2545: 2540: 2535: 2531: 2527: 2517: 2514: 2513: 2508: 2507: 2502: 2497: 2494: 2490: 2486: 2485:matrix theory 2482: 2478: 2471: 2467: 2463: 2461: 2458:(1646–1716). 2457: 2453: 2449: 2445: 2441: 2437: 2433: 2429: 2425: 2421: 2417: 2412: 2410: 2406: 2402: 2398: 2397:decimal point 2394: 2390: 2386: 2379: 2378: 2373: 2368: 2359: 2357: 2353: 2349: 2345: 2341: 2340:Western world 2337: 2333: 2325: 2320: 2316: 2314: 2310: 2306: 2302: 2298: 2294: 2290: 2286: 2283:and used the 2282: 2278: 2273: 212 BC 2263: 2259: 2258: 2252: 2248: 2244: 2240: 2235: 2233: 2229: 2225: 2221: 2217: 2213: 2209: 2205: 2201: 2197: 2193: 2185: 2181: 2173: 2172: 2166: 2156: 2146: 2144: 2140: 2136: 2132: 2128: 2124: 2120: 2119:approximation 2116: 2112: 2108: 2104: 2100: 2094: 2084: 2082: 2078: 2074: 2070: 2066: 2062: 2058: 2054: 2050: 2046: 2042: 2038: 2034: 2033:expected loss 2030: 2026: 2022: 2018: 2016: 2012: 2008: 2000: 1996: 1992: 1988: 1984: 1979: 1974: 1970: 1960: 1958: 1954: 1950: 1946: 1942: 1938: 1934: 1930: 1926: 1922: 1918: 1913: 1911: 1907: 1903: 1898: 1897:formal system 1895: 1891: 1885: 1884:around 1910. 1883: 1882:David Hilbert 1879: 1875: 1871: 1867: 1862: 1860: 1856: 1852: 1848: 1844: 1840: 1836: 1835:infinite sets 1832: 1827: 1825: 1821: 1813: 1808: 1803: 1799: 1787: 1783: 1779: 1775: 1772: 1770:are discrete) 1769: 1765: 1761: 1757: 1754: 1752: 1749: 1747: 1744: 1741: 1738: 1736: 1732: 1728: 1727:Coding theory 1725: 1723: 1719: 1716: 1714: 1710: 1706: 1702: 1701:Combinatorics 1699: 1698: 1697: 1694: 1692: 1688: 1684: 1680: 1675: 1673: 1669: 1665: 1661: 1653: 1648: 1643: 1630: 1627: 1624: 1621: 1618: 1615: 1612: 1608: 1604: 1600: 1596: 1593: 1590: 1587: 1585: 1582: 1581: 1580: 1578: 1574: 1570: 1566: 1565:real analysis 1561: 1559: 1555: 1551: 1547: 1539: 1534: 1529: 1525: 1515: 1513: 1509: 1505: 1501: 1497: 1493: 1485: 1481: 1478: 1475: 1471: 1468: 1465: 1462: 1459: 1455: 1451: 1447: 1444: 1441: 1438: 1435: 1431: 1430:vector spaces 1428: 1425: 1422: 1419: 1416: 1415: 1414: 1411: 1409: 1405: 1401: 1396: 1392: 1388: 1384: 1380: 1376: 1372: 1368: 1367: 1362: 1357: 1355: 1351: 1346: 1344: 1340: 1336: 1332: 1328: 1324: 1320: 1312: 1308: 1303: 1297: 1293: 1288: 1283: 1270: 1266: 1263: 1260: 1256: 1252: 1249: 1246: 1243: 1238: 1234: 1231: 1230: 1228: 1224: 1221: 1218: 1214: 1211: 1208: 1205: 1202: 1199: 1196: 1192: 1189: 1186: 1182: 1179: 1176: 1172: 1168: 1164: 1161: 1160: 1159: 1156: 1154: 1150: 1149:do not change 1146: 1142: 1138: 1134: 1129: 1127: 1123: 1119: 1115: 1111: 1107: 1102: 1100: 1096: 1092: 1088: 1084: 1083:line segments 1080: 1076: 1072: 1068: 1063: 1061: 1057: 1053: 1049: 1045: 1041: 1036: 1034: 1033: 1028: 1024: 1020: 1016: 1012: 1008: 1004: 999: 997: 993: 989: 985: 981: 972: 967: 957: 955: 951: 947: 943: 939: 934: 932: 928: 927:prime numbers 924: 920: 916: 912: 908: 907:scheme theory 904: 900: 896: 891: 889: 885: 881: 877: 873: 869: 865: 862:and probably 861: 857: 841: 823: 792: 776: 758: 754: 746: 742: 738: 737:prime numbers 734: 729: 724: 723:Number theory 717:Number theory 714: 712: 708: 704: 700: 699:number theory 694: 692: 688: 684: 679: 677: 676:combinatorics 673: 669: 665: 661: 657: 653: 652: 647: 646: 641: 637: 633: 629: 624: 622: 618: 614: 613:pseudoscience 610: 606: 602: 592: 590: 586: 582: 578: 577: 572: 571: 566: 562: 556:τὰ μαθηματικά 552: 548: 543: 538: 534: 531:The apparent 529: 527: 523: 519: 515: 511: 506: 504: 500: 496: 492: 487: 485: 481: 471: 461: 457: 453: 449: 445: 437: 434: 433:Ancient Greek 430: 420: 418: 414: 410: 406: 402: 398: 394: 390: 386: 385: 380: 376: 372: 367: 365: 361: 357: 353: 349: 345: 341: 337: 333: 329: 324: 322: 318: 314: 310: 306: 302: 298: 294: 289: 287: 283: 279: 275: 271: 270:number theory 267: 263: 259: 255: 251: 240: 235: 233: 228: 226: 221: 220: 218: 217: 214: 210: 206: 205: 194: 191: 189: 186: 184: 181: 179: 176: 174: 171: 169: 166: 164: 161: 159: 156: 154: 151: 150: 144: 143: 136: 132: 129: 127: 124: 122: 118: 115: 113: 110: 108: 104: 101: 99: 96: 94: 91: 89: 88:Number theory 86: 85: 82: 77: 76: 73: 72: 67: 64: 62: 59: 58: 57: 56: 53: 50: 49: 45: 44: 41: 37: 33: 19: 14835: 14818: 14801: 14784: 14767: 14750: 14738:from Commons 14733: 14716: 14695: 14622: 14610: 14598: 14579: 14512:Optimization 14374:Differential 14298:Differential 14265:Order theory 14260:Graph theory 14164:Group theory 14053: 14003: 13979: 13955: 13933: 13910: 13885: 13859: 13825: 13796: 13772:Online books 13762: 13728: 13705:. 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Retrieved 10489: 10480: 10473:Bouleau 1999 10468: 10461:Bouleau 1999 10456: 10448: 10443: 10433:December 29, 10431:. Retrieved 10411: 10380:February 25, 10378:. Retrieved 10335: 10331: 10320: 10308:. Retrieved 10303: 10294: 10282:. Retrieved 10279:math.mit.edu 10278: 10269: 10257:. Retrieved 10254:math.mit.edu 10253: 10244: 10234:December 29, 10232:. Retrieved 10225:the original 10216: 10206: 10194:. Retrieved 10179: 10172: 10147: 10143: 10137: 10112: 10108: 10099: 10072: 10068: 10025: 10021: 10015: 10005:November 20, 10003:. Retrieved 9994: 9984: 9974:November 20, 9972:. Retrieved 9963: 9954: 9944:November 20, 9942:. Retrieved 9930: 9917: 9900: 9896: 9890: 9880:November 20, 9878:. Retrieved 9856: 9846: 9819: 9815: 9805: 9795:November 19, 9793:. Retrieved 9780: 9773: 9763:November 19, 9761:. Retrieved 9752: 9748: 9738: 9728:November 19, 9726:. Retrieved 9717: 9707: 9697:November 30, 9695:. Retrieved 9671: 9664: 9645: 9639: 9629: 9619:November 20, 9617:. 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Retrieved 8437: 8433: 8423: 8396: 8387: 8352: 8346: 8313: 8309: 8303: 8293:November 14, 8291:. Retrieved 8276: 8272:Ore, Øystein 8266: 8254: 8242: 8230: 8218: 8206: 8194: 8161: 8157: 8151: 8130: 8119: 8107: 8095: 8090:, Chapter 1. 8083: 8056: 8047: 8038: 8029: 8019:February 15, 8017:. Retrieved 7963: 7959:Leader, Imre 7941: 7931:November 13, 7929:. Retrieved 7916: 7909: 7899:November 13, 7897:. Retrieved 7888: 7878: 7871:Whittle 1994 7866: 7823: 7810: 7767: 7758: 7748:February 13, 7746:. Retrieved 7730: 7723: 7711:. Retrieved 7689:Vardi, Moshe 7667: 7648: 7638: 7628:November 12, 7626:. Retrieved 7617: 7604: 7594:November 12, 7592:. Retrieved 7559: 7555: 7509: 7505: 7496: 7486:November 12, 7484:. Retrieved 7469: 7462: 7450:. Retrieved 7444: 7428: 7373: 7367: 7355:. Retrieved 7349: 7336: 7324:. Retrieved 7318: 7305: 7293:. Retrieved 7225: 7221: 7208: 7196:. Retrieved 7166: 7156: 7144:. Retrieved 7114: 7104: 7079: 7070: 7058:. Retrieved 7008: 6991: 6979:. Retrieved 6921: 6907: 6895:. 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Retrieved 4365: 4360: 4349: 4330: 4321: 4312: 4295: 4278: 4261: 4253: 4249: 4244: 4232: 4227: 4113: 4102: 4066: 4058:World War II 4054:Fields Medal 4051: 4040:Fields Medal 4014: 3947:cyclic group 3940: 3891: 3870: 3866: 3863:liberal arts 3860: 3845: 3834: 3822: 3812: 3801: 3786: 3776:math anxiety 3772: 3765: 3746: 3738:Tang dynasty 3719:Vedic period 3692: 3664:statistician 3637: 3618: 3602: 3598: 3563: 3548: 3543: 3535: 3528: 3521: 3503: 3494: 3477:Armand Borel 3475: 3459: 3410:astrologists 3407: 3383: 3381: 3377:cliodynamics 3358: 3354: 3349: 3346:calculations 3331:economic man 3315: 3313: 3287: 3264:oceanography 3254: 3233: 3217:neurobiology 3211: 3164:graph theory 3144:cryptography 3137: 3118: 3050: 3023: 3019:trajectories 2984: 2973: 2952: 2913: 2908:World War II 2905: 2900: 2896: 2892: 2882: 2854:Isaac Newton 2846: 2771: 2743: 2732:is always a 2718:exclusive or 2707: 2701: 2695: 2664: 2656:noun phrases 2549: 2510: 2504: 2498: 2474: 2452:Isaac Newton 2413: 2405:Omar Khayyam 2401:Al-Khwarizmi 2382: 2375: 2372:al-Khwarizmi 2370:A page from 2329: 2309:trigonometry 2255: 2247:Pythagoreans 2245:such as the 2236: 2177: 2171:Plimpton 322 2169: 2135:graph theory 2096: 2065:optimization 2019: 2004: 1983:distribution 1921:proof theory 1917:model theory 1914: 1886: 1863: 1833:'s study of 1828: 1817: 1812:Venn diagram 1776:, including 1735:cryptography 1729:, including 1718:Graph theory 1695: 1676: 1657: 1652:Markov chain 1569:real numbers 1562: 1543: 1489: 1424:field theory 1418:group theory 1412: 1408:Emmy Noether 1399: 1374: 1364: 1358: 1347: 1338: 1330: 1327:al-Khwarizmi 1316: 1311:group theory 1259:optimization 1157: 1130: 1103: 1090: 1079:real numbers 1064: 1046:from lines, 1037: 1030: 1006: 1000: 996:architecture 977: 935: 903:Andrew Wiles 892: 750: 741:independence 731:This is the 680: 649: 643: 625: 598: 588: 584: 580: 574: 568: 564: 560: 550: 530: 525: 509: 507: 495:mathēmatikoi 494: 488: 469: 460:mathēmatikós 459: 439: 435: 428: 426: 382: 368: 325: 312: 297:abstractions 290: 249: 248: 51: 40: 14856:Mathematics 14718:Definitions 14696:Mathematics 14652:Mathematics 14624:WikiProject 14467:Game theory 14447:Probability 14184:Homological 14174:Multilinear 14154:Commutative 14131:Type theory 14098:Foundations 14054:mathematics 13763:Mathematics 13725:Kelly, F.P. 13394:February 8, 13247:January 23, 13221:January 23, 13151:January 23, 13118:January 23, 13015:January 23, 12953:January 23, 12923:January 23, 12673:math.bu.edu 12165:: 344–350. 12081:: 311–322. 12035:(1): 2–15. 11413:. Springer. 11284:mathematics 11188:December 5, 10942:Sokal, Alan 10353:2066/176365 9822:(1): 1–22. 9111:October 27, 8878:: 364–371. 8847:February 3, 8768:February 3, 8707:February 3, 8617:February 3, 8505:December 5, 8316:: 606–628. 8100:Mesopotamia 7713:January 15, 7351:PhilArchive 7250:11441/38373 7060:February 9, 7005:Downey, Rod 6981:February 9, 6897:February 9, 6795:February 9, 6760:February 9, 6690:February 8, 6636:February 8, 6594:February 8, 6503:February 8, 6357:February 8, 6128:February 8, 6093:February 8, 5962:February 7, 5898:February 6, 5628:Weil, André 5499:February 3, 5445:February 3, 5358:February 3, 5054:Bell, E. T. 5038:January 25, 5002:January 25, 4846:January 25, 4813:February 2, 4785:February 2, 4746:February 2, 4717:January 25, 4522:January 18, 4465:(1): 1–14. 4399:January 17, 4286:applied to 4273:and planes. 4080:Chern Medal 4062:Nobel Prize 3894:frequencies 3797:Roger Apery 3759:began with 3736:during the 3726: 1500 3652:recognition 3481:G. H. Hardy 3418:Regiomantus 3369:geopolitics 3268:planetology 3260:meteorology 3200:exhibits a 2778:falsifiable 2722:free module 2667:definitions 2576:expressions 2552:engineering 2436:John Napier 2414:During the 2383:During the 2232:sexagesimal 2220:subtraction 2196:Mesopotamia 2192:Babylonians 2190:, when the 2184:prehistoric 2015:experiments 1925:type theory 1847:enumeration 1756:Game theory 1722:hypergraphs 1707:of a given 1595:Integration 1470:Lie algebra 1454:polynomials 1440:ring theory 1363:(presently 1255:convex sets 1217:polynomials 1185:parallelism 1091:coordinates 1087:number line 1044:constructed 1011:measurement 755:, that is, 733:Ulam spiral 711:foundations 695:sixty-three 601:Renaissance 599:Before the 581:mathematics 576:metaphysics 565:mathematics 542:mathematica 526:mathematici 510:mathematics 465:μαθηματικός 431:comes from 429:mathematics 356:game theory 332:engineering 250:Mathematics 178:Linguistics 168:Computation 163:Geosciences 126:Probability 52:Mathematics 14850:Categories 14769:Quotations 14664:Arithmetic 14452:Statistics 14331:Arithmetic 14293:Arithmetic 14159:Elementary 14126:Set theory 13841:Available 13660:October 2, 13373:2007932362 12807:Riehm 2002 12780:January 3, 12746:January 3, 11618:: 91–100. 11415:, page 409 11034:January 3, 10790:January 3, 10196:January 3, 9423:1082226900 9415:2019010556 8807:2006041609 8259:Boyer 1991 8247:Boyer 1991 8235:Boyer 1991 8223:Boyer 1991 8211:Boyer 1991 8199:Boyer 1991 8112:Boyer 1991 8088:Kline 1990 7990:2008020450 7420:1418750041 7185:sn97004494 7133:sn97004494 7039:2014000240 6876:2011202231 6677:2007920230 6581:2004556211 6557:Corry, Leo 6399:2007238847 6167:2008011322 6004:2004056235 5795:2003268597 5424:sf77000404 5247:1332780664 4214:References 4107:, called " 4073:Abel Prize 4047:Archimedes 3871:discovered 3842:Paul Erdős 3827:simplicity 3753:Quadrivium 3728: – c. 3717:since the 3680:accountant 3558:See also: 3462:Pythagoras 3445:Philosophy 3375:developed 3298:psychology 2956:university 2849:technology 2703:polynomial 2700:, such as 2698:neologisms 2675:conjecture 2598:, such as 2588:subscripts 2568:operations 2560:properties 2493:Kurt Gödel 2489:statistics 2477:Carl Gauss 2432:logarithms 2271: – c. 2269: 287 2262:Archimedes 2071:, such as 1969:Statistics 1894:consistent 1824:philosophy 1802:Set theory 1758:(although 1693:problems. 1664:Algorithms 1323:Diophantus 1019:postulates 899:was proved 872:Diophantus 617:numerology 615:, such as 605:arithmetic 499:arithmetic 389:arithmetic 352:statistics 346:, and the 286:set theory 188:Philosophy 131:Statistics 121:Set theory 14820:Resources 14803:Textbooks 14379:Geometric 14369:Algebraic 14308:Euclidean 14283:Algebraic 14179:Universal 14002:(1965) . 13389:117392219 12495:1476-4687 12448:126119711 12405:220636812 12307:See also 12217:121589323 11945:145507519 11789:126067145 11632:149753721 11547:209980693 11499:(4): 1–28 11462:178923299 11238:122351146 11137:. 189303; 11135:119887253 10719:0360-0572 10370:216912822 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4635:1930-0980 4562:0021-1753 4386:236026294 4378:1022-3398 4342:Citations 4301:computers 3990:butterfly 3815:aesthetic 3749:Dark Ages 3676:economist 3644:education 3628:Education 3609:apodictic 3582:concision 3578:intuition 3471:Platonism 3294:sociology 3290:economics 3124:Computing 3071:− 3067:Ω 3038:spacetime 3026:manifolds 2936:implement 2728:" and "a 2691:corollary 2671:postulate 2652:less than 2622:∫ 2572:relations 2544:summation 2440:astronomy 2295:with the 2037:procedure 1937:computers 1878:formalism 1660:countable 1611:continuum 1484:computers 1474:Lie group 1319:equations 1173:at which 992:surveying 664:variables 621:astrology 518:astronomy 514:astrology 503:Aristotle 456:adjective 427:The word 423:Etymology 397:fractions 193:Education 183:Economics 158:Chemistry 14600:Category 14356:Topology 14303:Discrete 14288:Analytic 14275:Geometry 14247:Discrete 14202:Calculus 14194:Analysis 14149:Abstract 14088:Glossary 14071:Timeline 13976:(1986). 13922:Archived 13882:(1997). 13856:(1996). 13823:(1999). 13743:Archived 13719:(1994). 13707:June 24, 13698:Archived 13651:Archived 13593:17202382 13585:87033078 13496:(1881). 13467:(2014). 13455:July 28, 13446:Archived 13406:(1990). 13381:76935733 13321:(1990). 13291:(1991). 13269:(1999). 13241:Archived 13215:Archived 13169:(1998). 13145:Archived 13112:Archived 13082:Archived 13035:Archived 13009:Archived 12980:Archived 12947:Archived 12917:Archived 12891:Archived 12861:Archived 12828:Archived 12706:Archived 12677:Archived 12647:Archived 12592:Symmetry 12590:(2015). 12515:Archived 12282:(1940). 12221:Archived 12107:30123014 12045:Archived 11996:Archived 11813:Archived 11781:25678363 11690:(eds.). 11682:(2009). 11551:Archived 11466:Archived 11394:June 20, 11375:(1893). 11357:June 20, 11338:(2009). 11276:June 20, 11179:Archived 11139:also in 11095:April 2, 11089:Archived 11058:Archived 10970:39605994 10948:(1998). 10920:Archived 10872:Archived 10574:25032300 10494:Archived 10424:Archived 10374:Archived 10214:(1990). 10063:(1983). 10042:20022840 9999:Archived 9995:The Wire 9968:Archived 9964:MacTutor 9935:Archived 9909:24110208 9757:Archived 9722:Archived 9688:Archived 9614:34271623 9594:Synthese 9567:Archived 9535:(2022). 9472:Archived 9468:18122406 9439:(2008). 9270:April 6, 9264:Archived 9252:(2014). 9181:April 5, 9105:Archived 9052:(2018). 9028:Archived 8902:43020253 8876:Sigma Xi 8841:Archived 8833:MacTutor 8815:64085024 8759:Archived 8755:68713073 8698:Archived 8611:Archived 8496:Archived 8442:Archived 8415:28723059 8395:(1994). 8274:(1988). 8055:(1999). 8010:Archived 7961:(eds.). 7926:91018998 7893:Archived 7889:MacTutor 7842:80021637 7822:(eds.). 7802:36597731 7786:97010349 7739:Archived 7704:Archived 7695:(2001). 7622:Archived 7588:Archived 7584:52083793 7576:23955909 7452:June 14, 7357:June 14, 7326:June 14, 7295:June 14, 7290:43258676 7282:31616719 7274:95652899 7198:June 14, 7193:37550526 7146:June 14, 7141:37550526 7096:11678884 7055:19315498 6999:(2014). 6976:67358543 6968:37141146 6960:97023277 6920:(eds.). 6822:June 13, 6789:Archived 6781:MacTutor 6751:Archived 6708:(2017). 6685:85242858 6589:51234417 6529:June 13, 6494:Archived 6474:63024699 6412:June 13, 6407:56713464 6351:Archived 6322:June 13, 6290:June 13, 6232:75642280 6122:Archived 6114:MacTuror 6080:26085129 6072:94657506 6012:56317813 5957:42290188 5949:99044789 5877:02019303 5853:(1902). 5811:14555830 5803:51533750 5750:53319514 5734:66083524 5658:83011857 5630:(1983). 5606:30437959 5598:94020017 5541:76055645 5490:Archived 5436:Archived 5312:75642280 5199:(2022). 5183:92013695 5175:25709270 5128:28115124 5120:92909575 5076:45010599 5032:Archived 4941:94078313 4893:58015848 4885:24338341 4840:Archived 4768:(1940). 4740:Archived 4711:Archived 4659:47003192 4586:21629993 4578:22073771 4516:Archived 4491:Archived 4451:(1960). 4417:Archived 4390:Archived 4250:Republic 4237:formulas 4134:See also 4044:polymath 3997:possess 3995:Fractals 3831:symmetry 3823:elegance 3757:pedagogy 3614:pleonasm 3430:John Dee 3394:Bricmont 3388:(1997), 3090:particle 3055:and the 3053:positron 2999:ellipses 2991:internet 2786:theories 2759:Einstein 2750:modeling 2746:sciences 2636:integral 2556:concepts 2546:notation 2293:parabola 2277:Syracuse 2257:Elements 2228:division 2216:addition 2107:analysis 2023:studies 1995:variance 1991:Gaussian 1943:design, 1941:compiler 1843:infinity 1558:function 1524:Calculus 1379:matrices 1223:Topology 1120:, often 1032:Elements 1015:theorems 966:Geometry 960:Geometry 791:integers 640:Calculus 636:formulas 609:geometry 384:Elements 336:medicine 321:theorems 282:analysis 278:geometry 258:theorems 254:theories 107:Analysis 103:Calculus 93:Geometry 14688:Science 14638:Portals 14612:Commons 14394:Applied 14364:General 14141:Algebra 14066:History 13727:(ed.). 13613:Bibcode 13536:2369153 13500:(ed.). 13259:Sources 12511:4310147 12503:7969448 12475:Bibcode 12440:3613237 12397:3344215 12098:6087017 11904:2314867 11454:3618304 11230:3482762 10678:May 19, 10648:May 19, 10622:May 19, 10597:May 19, 10310:May 19, 10284:June 1, 10259:June 1, 10152:Bibcode 8991:6229200 8910:1480717 8575:2304570 7998:2467561 7858:6707805 7850:0607328 7794:1474730 7534:2689412 7266:2687794 7007:(ed.). 6892:6945363 6482:1482042 6240:2240703 5742:1567783 5666:9576587 5549:3519779 5432:1480366 5320:2240703 5151:(ed.). 4901:1762376 4732:Plato. 4683:7787171 4675:1756877 4667:1141557 4651:2690647 4570:2884913 4487:6112252 4467:Bibcode 4233:algebra 3935:Fractal 3867:created 3804:puzzles 3668:actuary 3648:careers 3498:feeling 3456:Reality 3422:Cardano 3414:Ptolemy 3325: 3213:Biology 3101:Physics 2962:at the 2858:planets 2794:relying 2660:clauses 2646:), and 2592:symbols 2564:symbols 2501:science 2389:algebra 2377:Al-Jabr 2160:Ancient 2149:History 1902:Brouwer 1829:Before 1740:Matroid 1705:subsets 1550:Leibniz 1476:theory; 1371:unknown 1339:al-jabr 1331:algebra 1282:Algebra 1276:Algebra 988:circles 860:Babylon 753:numbers 632:algebra 630:led to 570:physics 539:plural 436:máthēma 340:finance 274:algebra 173:Biology 153:Physics 98:Algebra 61:History 14313:Finite 14169:Linear 14076:Future 14052:Major 14010: 13988: 13962: 13940: 13918:online 13894: 13868: 13843:online 13833: 13805: 13760:about 13735: 13623: 13591: 13583: 13573: 13550:Eprint 13547:Google 13543:Eprint 13540:Google 13534: 13479: 13418: 13387: 13379: 13371: 13361: 13329: 13307: 13277: 13186: 13074: 12771: 12737: 12602: 12570: 12509: 12501: 12493: 12467:Nature 12446: 12438: 12403: 12395: 12292: 12255: 12215: 12105: 12095: 11988: 11972:& 11943: 11902: 11858: 11787: 11779: 11742: 11698: 11655: 11630: 11545: 11460: 11452: 11385: 11348: 11318: 11267: 11236: 11228: 11171: 11133: 11025: 10968: 10958: 10781: 10744: 10717: 10572: 10562: 10368: 10360: 10338:: e2. 10187: 10129:685207 10127: 10089: 10040: 9907: 9871: 9836: 9684:188757 9682: 9612: 9561: 9508: 9466: 9421: 9413: 9403: 9371: 9346:202372 9344: 9304: 9232: 9207: 9172: 9136: 9072: 8989: 8941: 8916: 8908: 8900: 8892: 8813: 8805: 8795: 8753: 8745: 8670: 8662: 8573: 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