Mathematics - Knowledge

2367: 11941: 2831: 11075: 9648: 4373: 14544: 14480: 4103: 2466: 209: 2822: 971: 14568: 1807: 1533: 3306: 3892: 1978: 2541: 3487:—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. 14710: 3491:

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: 14580: 14492: 2794:. 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 14556: 2319: 14608: 14676: 14693: 14659: 3996: 728: 14727: 14642: 1647: 5868: 14516: 14625: 14504: 2984:. 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: 3561:

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.

3731:, 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 3195: 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 3752:. 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 3510:

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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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

3114: 2722:"). 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

2518:(MR) database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." 3800:

expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetics.

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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

3577:. 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. 2853:

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,

2656:). 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 3724:

schools in the 16th and 17th century. Most mathematical curricula remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics 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

3402:(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. 10454: 3980:

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

3811:, 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 10334: 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. 4072:", 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. 11183:

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 3121:

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 3229:, 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 2778:

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

10446: 3434:. 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. 3144:

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 2667:

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

6311: 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 2862:

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

3943: 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 12220:

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

12601:. 22nd Biennial Conference of The Australian Association of Mathematics Teachers. Fremantle, Western Australia: Australian Association of Mathematics Teachers. pp. 125–126. 3763:

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 11447: 10288: 1342: 3026:

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

2714:. 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, " 2940:

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:

13409:(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. 3088: 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, 10854: 12516: 9639:. Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers. Vol. 66. pp. S1–S13. 2929: 12876: 5093:

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.

2718:" 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 3053:

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

13639: 13435:. 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. 10855:"Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. | The Online Books Page" 2634: 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°.

13594: 2506:, 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 11735: 1857:. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring 5049:... 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 10880: 9356:

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 4350: 2499:, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. 13041: 4083:, 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.

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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).

4671: 3541:. Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' 13111: 4476: 2905:, 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. 10824: 9432: 9224: 6300: 11874: 8658: 3398:

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

2961: 13628:(previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. 12451:

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

13940: 13342: 12659: 4418: 12757: 10806: 10384: 1093:, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: 13575: 13398: 12625: 4367: 3805:

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

2508: 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: 10985: 8456: 4694: 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: 4451: 3354:. 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, 3834:(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 12978: 8719: 5396: 5298: 4295:

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".

4923: 3369:, which explains phases of economic growth or crisis. Towards the end of the 19th century, mathematicians extended their analysis into 12117: 11416: 7664: 7582: 6749: 2580:

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 13813: – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and 10502: 2977: 9773:"Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy" 8801: 12905: 12543: 9928: 3748:

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

11474:, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Apple proof of the 9951: 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

7970: 7275: 3094:, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments. 3030:. 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. 12724: 11709: 3090:

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

13547: 11918: 9895: 8988: 4333:(in German and English). Vol. 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. 4260:

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 13908: 13886: 13860: 13838: 13792: 13766: 13731: 13703: 13633: 13521: 13377: 13316: 13259: 13227: 13205: 13175: 13084: 12972: 12669: 12635: 12500: 12468: 12190: 12153: 11892: 11886: 11756: 11734:

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

11640: 11596: 11553: 11283: 11246: 11216: 11165: 11069: 10920: 10787: 10743: 10706: 10599: 10524: 10149: 9833: 9745:. International Congress on Mathematical Software 2014. Lecture Notes in Computer Science. Vol. 8592. Berlin: Springer. 9470: 9365: 9333: 9266: 9194: 9169: 9134: 9098: 9065: 9034: 8903: 8757: 8495: 8365: 8329: 8246: 8101: 8025: 7940: 7792: 7439: 7362: 6989: 6697: 6627: 6581: 6397: 6117: 5189: 5070: 5026: 4891: 4730: 4315:

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: 12787: 12573: 9682: 2496: 1889: 14303: 13896: 12865: 12843: 9088: 6711: 6619: 3185: 415:, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary 172: 11203:. Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer. pp. 63–77. 4800: 14596: 13137: 9814:"Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity" 7128: 7076: 5412:

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

4883: 2945: 1854: 12813: 11919:"Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career" 7175: 14448: 13933: 13617: 7637: 3826:. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are 2863: 1144: 690: 682: 416: 408: 404: 13567: 12411: 14520: 13471: 13008: 10410:

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.).

6531: 5954: 5899: 5745: 5608: 5548: 5491: 5483: 5165: 5125: 4501:(September 2011). "The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?". 3648: 2937: 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

13984: 5450: 2928:, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the 2756: 2593: 2565: 468:), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, 10872: 8316:. Sources and Studies in the History of Mathematics and Physical Sciences. Singapore: Springer. pp. 438–461. 3822:

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

9497: 8699: 8565: 7512: 4268:. Even with this modern technology, it may take years of human work for writing down a completely detailed proof. 3554: 3157: 2331: 1932: 1865: 744: 559:) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective 14614: 10260: 9717: 7881:. Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage. World Scientific. p. 28. 7699: 3913: 3674:

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.

3497: 3346:. This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual 3236: 2768: 2502:

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

1616: 14716: 12492: 10944: 9498:"The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account" 8402: 4992: 4878:(1995). "What Augustine Didn't Say About Mathematicians". In Alexanderson, Gerald L.; Mugler, Dale H. (eds.). 3619:, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the 443: 14682: 13926: 13875: 13613: 13033: 12694:, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics." 11201:"What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it 11105:

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: 14699: 8213: 7043:. STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing. pp. 603–618. 3501:

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 14762: 14508: 13806: 12206: 11322: 11232: 6798: 6389: 6070: 3796: 3623:

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

3230: 3141: 3131: 3004: 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: 17: 13818: 10185: 7845: 1128:). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. 14423: 13979: 13721: 13621: 10694: 10655: 9010: 4795: 4767: 4734: 4666: 4132: 4122: 3990: 2024: 65: 13781: 13755: 13739: 13107: 11816:"Introduction: the history of mathematics teaching. Indicators for modernization processes in societies" 10105:

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

9675:"Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century" 6235: 4466: 2849:

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

2830: 14534: 13994: 13678: 9401: 9214: 8060: 7932: 7406: 7338: 6973: 5117: 4726: 4018: 3612: 3478: 2782:, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a 1777: 710: 13692: 13190: 10624: 8201: 3740:

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 14631: 14572: 14408: 14380: 14017: 10959:"Modern Macroeconomic Models as Tools for Economic Policy | Federal Reserve Bank of Minneapolis" 10786:

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

8797: 8647: 8085: 6078: 4983: 4952: 4929: 4471: 4327:

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

4322:

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.

3411: 3316:

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

3135: 2840: 2775: 2092: 1690: 1594: 826: 761: 296: 187: 177: 167: 13327: 12256:'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his 9894:. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. 9856:

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

2577: 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: 3063: 14338: 14328: 14298: 14232: 13967: 10235: 9631: 9601: 8511: 8425: 7071: 6681: 5995: 4988: 4957: 4413: 4127: 3550: 3260:

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

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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

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

12927: 10977: 10210: 8090: 6666: 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

14648: 14436: 14333: 14313: 14308: 14237: 13962: 13394: 11623:; Ross, Micah (2014). "Mathematics Education in Antiquity". In Karp, A.; Schubring, G. (eds.). 10065: 7787:. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. 6957: 6276: 3812: 3431: 3358:

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

3339: 3281: 3189: 2573: 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: 13439: 11580: 11273: 11236: 10733: 9813: 9321: 9250: 9014: 7399:""Clarifying the nature of the infinite": the development of metamathematics and proof theory" 6961: 6874: 6521: 4084: 3568:) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the 3553:

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

2991:

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

14463: 14393: 14270: 14194: 14133: 14118: 14113: 14090: 13972: 12954: 12749: 12143: 11062:

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.).

7398: 6613: 6271: 5339: 5309: 4880:

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

4824: 4152: 4142: 4137: 4069: 4050: 3604: 3594: 2867: 2561: 2527: 2480: 2431: 2211: 2014: 1998: 1773: 1730: 1527: 1353: 1306: 1190: 1113: 1070: 883: 855: 644: 627: 281: 192: 106: 35: 12596: 12453:

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

11543: 11155: 10380: 10139: 7465:(September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". 7429: 6571: 6299: 5885: 14752: 14443: 14323: 14318: 14242: 14143: 13509: 12434: 12371: 12084: 10112: 9285:(1976). "Mathematical vs. Experimental Traditions in the Development of Physical Science". 9157: 8749: 8570:. MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA. 7958: 7810: 7754: 7657: 7574: 6737: 6689: 6243: 4627: 4530: 4427: 3977: 3807: 3702: 3565: 3386: 3222: 3153: 3035: 2668: 2514: 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: 13067: 11196: 8177: 8165: 5790:

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

8: 14458: 14368: 14290: 14189: 14123: 14080: 14070: 14050: 13185: 12901: 12517:"Lecture 8: Translation Symmetry | Physics III: Vibrations and Waves | Physics" 10671: 10506: 10184:. Proceedings of the International Congress of Mathematicians. p. 31. Archived from 9254: 8789: 6885:. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. Vol. 36. 5305: 4147: 3971: 3848: 3735:

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

3616: 3599:

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

3530: 3343: 3285: 3226: 2988: 2807: 2787: 2731: 2715: 2688: 2680: 2613: 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: 13513: 12535: 12375: 10116: 9920: 8153: 4431: 14548: 14484: 14403: 14343: 14275: 14265: 14204: 14179: 14055: 14012: 14007: 13802: 13428: 13281: 12403: 12340: 12332: 12297: 12289: 12109: 11994: 11967: 11837: 11796: 11681: 11673: 11524: 11475: 11439: 11354: 11346: 11130: 11122: 11027: 10326: 10085: 9998: 9865: 9794: 9640: 9570: 9519: 9424: 9302: 8947: 8874: 8703: 8628: 8620: 8531: 8448: 8335: 8294: 8286: 8189: 8142: 7907: 7540: 7532: 7490: 7350: 7246: 7222: 7198: 7052: 7011: 6932: 6848: 6446: 6414: 6204: 6172: 6151:"Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria" 6106:

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

6044: 6012: 5946: 5849: 5791: 5767: 5706: 5674: 5630: 5570: 5513: 5475: 5364: 5252: 5211: 5177: 4841: 4639: 4607: 4591: 4542: 4467:"Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion" 4443: 4243:

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

4108: 4080: 3753: 3526: 3362: 3257: 3194: 3161: 3047: 3043: 2764: 2751: 2735: 2619: 2335: 2203: 2102: 2056: 2044: 2036: 2020: 2006: 1972: 1797: 1678: 1628: 1606: 1511: 1457: 1382: 1295: 1232: 1212: 1170: 1136: 1125: 1094: 1039: 1002: 930: 910: 706: 265: 261: 212: 116: 13079:. Logic and Computation in Philosophy series. Oxford University Press. pp. 3–27. 7123: 5867: 3857:

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: 14560: 14543: 14479: 14199: 14184: 14128: 14075: 13904: 13882: 13856: 13834: 13788: 13762: 13727: 13699: 13629: 13517: 13502: 13485: 13477: 13467: 13373: 13312: 13305: 13285: 13273: 13265: 13255: 13223: 13201: 13197: 13171: 13080: 12968: 12665: 12631: 12496: 12464: 12395: 12387: 12344: 12301: 12186: 12149: 12113: 11999: 11937: 11882: 11841: 11752: 11685: 11636: 11592: 11549: 11528: 11443: 11385: 11358: 11279: 11242: 11212: 11161: 11134: 11065: 11054: 11031: 10926: 10916: 10910: 10739: 10702: 10675: 10530: 10520: 10361: 10330: 10318: 10145: 10047: 9829: 9798: 9740: 9615: 9596: 9550: 9523: 9466: 9379: 9371: 9361: 9329: 9262: 9210: 9190: 9165: 9130: 9094: 9030: 8927: 8899: 8878: 8866: 8858: 8850: 8771: 8763: 8753: 8711: 8632: 8491: 8452: 8371: 8361: 8339: 8325: 8298: 8242: 8134: 8097: 8031: 8021: 8013: 7996: 7962: 7946: 7936: 7923: 7915: 7882: 7814: 7798: 7788: 7758: 7742: 7732: 7618: 7605: 7482: 7435: 7376: 7368: 7358: 7271: 7238: 7230: 7214: 7149: 7141: 7133: 7097: 7089: 7081: 7003: 6995: 6985: 6924: 6916: 6908: 6898: 6840: 6832: 6824: 6703: 6693: 6641: 6633: 6623: 6587: 6577: 6545: 6537: 6527: 6450: 6438: 6430: 6422: 6363: 6355: 6347: 6208: 6196: 6188: 6180: 6131: 6123: 6113: 6048: 6036: 6028: 6020: 5968: 5960: 5950: 5913: 5905: 5895: 5891: 5853: 5841: 5833: 5759: 5751: 5741: 5698: 5690: 5682: 5634: 5622: 5614: 5604: 5574: 5562: 5554: 5544: 5517: 5505: 5497: 5487: 5388: 5380: 5372: 5276: 5268: 5260: 5247: 5230: 5215: 5203: 5195: 5185: 5139: 5131: 5121: 5084: 5076: 5066: 5040: 5032: 5022: 4905: 4897: 4887: 4875: 4857: 4849: 4833: 4631: 4615: 4599: 4534: 4518: 4342: 4334: 4248: 4231: 4117: 4102: 3816: 3660: 3656: 3632: 3492: 3309: 3169: 2941: 2925: 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: 13532: 12716: 11700: 11008:

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

7544: 7250: 7056: 7015: 6936: 5771: 5710: 4546: 3449: 2475:

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)

10667: 10512: 10376: 10308: 10300: 10120: 10077: 10037: 9990: 9884: 9821: 9784: 9746: 9610: 9562: 9509: 9416: 9294: 9022: 8980: 8976: 8951: 8939: 8840: 8610: 8523: 8440: 8317: 8278: 8126: 7645: 7613: 7570: 7524: 7474: 7342: 7206: 7190: 7044: 6977: 6890: 6852: 6814: 6685: 6479: 6406: 6337: 6162: 6004: 5881: 5825: 5733: 5729: 5664: 5596: 5536: 5354: 5242: 5169: 5014: 4643: 4587: 4583: 4510: 4447: 4435: 4244: 4088: 3749: 3712:, mathematics education in Europe was provided by religious schools as part of the 3644: 3557:

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

3465:

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

3440:

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

3366: 3199: 2996: 2921: 2890: 2886: 2879: 2875: 2811: 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: 13814: 13673: 13589: 13446: 13368:

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

2821: 1935:. Although these aspects of mathematical logic were introduced before the rise of 14353: 14280: 14209: 14002: 13848: 13750: 13746: 13713: 13390: 13366: 13241: 13163: 13074: 12779: 12068: 12051: 12035: 12018: 9892:

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".

7954: 7806: 7750: 7280: 6981: 6878: 5815: 4740: 4623: 4526: 4381: 4265: 4021:) to up to four individuals. It is considered the mathematical equivalent of the 3959: 3545:, rigorous proofs can require hundreds of pages to express, such as the 255-page 3445: 3399: 3318: 3031: 3016: 2791: 2755:

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: 12835: 12460: 11632: 11208: 10508:

Beyond the Molecular Frontier: Challenges for Chemistry and Chemical Engineering

10124: 8748:. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. 8358:

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

6670: 6576:. Milano, Italy: Polimetrica International Scientific Publisher. pp. 3–39. 1001:

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

14584: 14431: 14358: 14065: 13361: 13168:

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

12491:. Princeton Science Library. Vol. 47. Princeton University Press. p. 12285: 12180: 11748: 11584: 11576: 11520: 11269: 10172: 9789: 9772: 9397: 9219: 8426:"Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages" 7911: 7633: 7036: 6819: 6333: 5829: 4790: 4498: 4316: 3946: 3903: 3694: 3675: 3600: 3534: 3252: 3240: 3203: 3173: 2783: 2653: 2585: 2581: 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: 14665: 13251: 13133: 11832: 11815: 11435: 10081: 9566: 9420: 9383: 8943: 8444: 8232: 7380: 6894: 6410: 5600: 5207: 5173: 4762: 14746: 14219: 14151: 14103: 13870: 13822: 12809: 12391: 12272:

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

11151: 10906: 10679: 10322: 10051: 9162:

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

8854: 8707: 8624: 8353: 8138: 8035: 7966: 7780: 7641: 7486: 7462: 7394: 7372: 7354: 7218: 7202: 7137: 7085: 7007: 6912: 6844: 6828: 6707: 6591: 6426: 6418: 6351: 6184: 6176: 6167: 6150: 6135: 6024: 6016: 5938: 5811: 5686: 5678: 5376: 5368: 5264: 5256: 5199: 5181: 5157: 4909: 4837: 4722: 4661: 4603: 4595: 4574:(December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". 4522: 4409: 4346: 4338: 4227: 3858: 3853:

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

3664: 3608: 3395: 3374: 3149: 3008: 2981: 2933: 2710: 2637: 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: 13489: 13277: 12930:. School of Mathematics and Statistics, University of St Andrews, Scotland. 10930: 9927:. School of Mathematics and Statistics, University of St Andrews, Scotland. 9820:. Mathematics for Industry. Vol. 5. Tokyo: Springer. pp. 393–399. 8775: 8715: 8375: 8130: 7852:. School of Mathematics and Statistics, University of St Andrews, Scotland. 7762: 7242: 7153: 7101: 6928: 6645: 6549: 6367: 6040: 5987: 5972: 5917: 5763: 5588: 5566: 5143: 5088: 3802: 3305: 2492: 14161: 14156: 14060: 13776: 13717: 13300: 12484: 12359: 12003: 10573: 10534: 10042: 10025: 10021: 9251:"Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss" 9127:

Mathematical Enculturation: A Cultural Perspective on Mathematics Education

8870: 8845: 8824: 7818: 7653: 6442: 6200: 5702: 5626: 5509: 5392: 5280: 4861: 4822:

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

4635: 4538: 4439: 4065: 4014: 4000: 3907: 3823: 3736: 3698: 3679: 3624: 3484: 3437: 3378: 3347: 3265: 3218: 3165: 3145: 3058: 3020: 2909: 2855: 2719: 2684: 2540: 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

13918: 13000: 12399: 11656:

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

11157:

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

10785: 8269:

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

7346: 7048: 5845: 5540: 5044: 2691:. 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. 14363: 14027: 13497: 13215: 12176: 12085:"A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)" 11984: 9282: 8972: 8615: 8598: 7919: 7649: 7528: 7311: 7303: 6008: 5533:

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

5335: 5109: 5105: 4978: 4947: 4040: 4022: 3891: 3854: 3757: 3529:. This means that the definitions must be absolutely unambiguous and the 3441: 3370: 3269: 3261: 2779: 2727: 2723: 2657: 2645: 2553: 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: 11917:

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

11677: 10138:

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

10002: 9869: 9514: 9259:

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

7536: 5439: 4845: 2913:

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

319:

to already established results. These results include previously proved

14348: 14227: 14022: 13432: 12336: 12293: 12105: 11879:

International Association for the Evaluation of Educational Achievement

11800: 11350: 11126: 11118: 10984:(Spring 2016 ed.). Metaphysics Research Lab, Stanford University. 10902: 10313: 10304: 9952:"Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry" 8535: 7878:

Grand Challenges, High Performance Computing, and Computational Science

7494: 7226: 7210: 6965: 5669: 5652: 4611: 4414:"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" 4033: 4007: 3787: 3713: 3640: 3483:

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

3422: 3391: 3299: 2957: 2850: 2704: 2676: 2564:

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

616: 604: 498: 388: 351: 285: 130: 120: 14579: 11488:

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

10089: 9818:

A Mathematical Approach to Research Problems of Science and Technology

9705: 9644: 9306: 9129:. Norwell, Massachusetts: Kluwer Academic Publishers. pp. 20–59. 8514:(January 1949). "History of Mathematics After the Sixteenth Century". 8290: 8146: 7690: 4043:

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

3663:

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

12383: 12238: 12221: 9994: 7724: 6776: 6526:(2nd revised ed.). Germany: Birkhäuser Basel. pp. 247–252. 6517: 5593:

Number Theory: An Approach Through History From Hammurapi to Legendre

5359: 3950: 3775: 3636: 3569: 3542: 3538: 3295: 3291: 3039: 2858:

introduced infinitesimal calculus for explaining the movement of the

2699: 2692: 2672: 2605: 2589: 2545: 2439: 2191: 1806: 1663: 1659: 1473: 1018: 991: 620: 517: 513: 502: 455: 396: 13626:

Probability, statistics and optimisation: A Tribute to Peter Whittle

13414: 12328: 11814:

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

11792: 11708:. History and Pedagogy of Mathematics conference, July 16–20, 2012. 11342: 11326: 8527: 7478: 7194: 6771: 2980:

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

1658:

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

1286: 14252: 14171: 14098: 11779:

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

11548:. Practical Career Guides. Rowman & Littlefield. pp. 1–3. 11023: 10516: 9298: 8836: 8390: 8282: 7776: 7513:"On the Philosophical Relevance of Gödel's Incompleteness Theorems" 5065:(1st ed.). New Delhi, India: Mittal Publications. p. 27. 4514: 4261: 4004: 3995: 3955: 3791: 3743: 3717: 3709: 3574: 3091: 3054: 3027: 2992: 2956:

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

2893:

increasingly focused their research on internal problems, that is,

2760: 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: 13504:

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

13269: 11743:. Series on Mathematics Education. Vol. 1. pp. 157–185. 11278:. American Mathematical Society (1991 reprint). pp. 285–286. 10485:

The Use of Statistical Methods in the Analysis of Clinical Studies

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7950: 7145: 7093: 6999: 6836: 6637: 6541: 6359: 6127: 5964: 5796: 5755: 5384: 4017:, established in 1936 and awarded every four years (except around 2671:

that provide a standard foundation for communication. An axiom or

2576:

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

14037: 13757:

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

12964: 10625:"Using Mathematical Models to Investigate Planetary Habitability" 10287:

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

9328:. Oxford Handbooks in Mathematics. OUP Oxford. pp. 107–132. 6803:"Discrete and Continuous: A Fundamental Dichotomy in Mathematics" 6236:"Figurate Numbers: A Historical Survey of an Ancient Mathematics" 5737: 4197: 3895: 3628: 3214: 3113: 3000: 2747: 2557: 2503: 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: 14733: 13481: 11042:. Vol. 4. New York: Simon and Schuster. pp. 2348–2364. 9742:

Cylindrical Algebraic Decomposition in the RegularChains Library

8862: 8391:"Contributions of Islamic scholars to the scientific enterprise" 7886: 7802: 7746: 7234: 6920: 6618:. Science Networks – Historical Studies. Vol. 32. Germany: 6434: 6192: 6032: 5909: 5837: 5694: 5618: 5558: 5501: 5272: 5135: 5080: 5036: 4901: 4853: 4619: 3603:, the practice of mathematics has a social side, which includes 2592:. Operation and relations are generally represented by specific 2055:, a statistical-decision problem is formulated by minimizing an 1646: 13851:(2003). "The Nature of Mathematics". In James R. Newman (ed.). 11966:

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

9551:"The History Of Applied Mathematics And The History Of Society" 8928:"Mathematical Models and Reality: A Constructivist Perspective" 7825: 5653:"From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem" 3779: 3764: 3721: 2871: 2859: 2661: 2391:. Other achievements of the Islamic period include advances in 2351: 2250: 2164: 1704: 1334: 1026: 987: 867: 752: 546: 532: 378: 304: 12315:

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

10789:

Handbook of Behavioral Economics: Applications and Foundations

8311: 3312:

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

2874:

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

14590: 13068:"Deciding the undecidable: Wrestling with Hilbert's problems" 11870: 11866: 7632: 6772:"Calculus (Differential and Integral Calculus with Examples)" 3520: 3426: 3140:

Computing is closely related to mathematics in several ways.

3046:

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

2597: 1819: 1767: 1763: 1553: 1105: 1022: 983: 473: 463: 300: 8898:. Mathematical Modeling. Vol. 1. CRC Press. p. 1. 8683:

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

6962:"Turing's Legacy: Developments from Turing's Ideas in Logic" 6615:

Tool and Object: A History and Philosophy of Category Theory

6301:"Where the x we use in algebra came from, and the X in Xmas" 3221:. 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 12222:"Reflections on Paul Erdős on His Birth Centenary, Part II" 12219: 11859:

TIMSS 2019 International Results in Mathematics and Science

11581:"Mathematics education in an Old Babylonian scribal school" 10945:"Biden's Misleading Unemployment Statistic – FactCheck.org" 10632: 10415: 10408:

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

9261:. Springer Science & Business Media. pp. 235–268. 8593:

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

8592: 8542: 7658:"On the Unusual Effectiveness of Logic in Computer Science" 3906:. For example, the group underlying mirror symmetry is the 3701:(618–907 CE), a mathematics curriculum was adopted for the 3620: 3012: 2347: 2288: 2060: 2028: 1986: 11965: 11702:

Indian pedagogy and problem solving in ancient Thamizhakam

11545:

Mathematicians and Statisticians: A Practical Career Guide

8042: 5728:. Series in Pure Mathematics. Vol. 4 (2nd ed.). 3461:

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

3217:

uses probability extensively in fields such as ecology or

2930:

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,

13464:

The Mathematical Tourist: Snapshots of Modern Mathematics

12697: 12685: 11854: 11506: 10735:

Moral Markets: The Critical Role of Values in the Economy

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Modelling Mathematical Methods and Scientific Computation

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5162:

Leonhard Euler and the Foundations of Celestial Mechanics

3338:). 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

13328:"Some Trends in Modern Mathematics and the Fields Medal" 12052:"The Frontage of Creativity and Mathematical Creativity" 12050:

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

11496:(167 (4)). Revue Internationale de Philosophie: 500–508. 9983:

Proceedings of the American Academy of Arts and Sciences

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3767:. This aspect of mathematical activity is emphasized in 2932:, a problem of pure mathematics that was proved true by 2202:

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

13831:

A History of Mathematics: From Mesopotamia to Modernity

12717:"Fields Medal | International Mathematical Union (IMU)" 12019:"The Outlook of the Mathematicians' Creative Processes" 11916: 11813: 10427: 8599:"Knowledge of Mathematical Symbols Goes Beyond Numbers" 8000:

Evolution of Mathematical Concepts; an Elementary Study

7606:"At the Heart of Analysis: Intuitionism and Philosophy" 7337:. Cambridge Elements in the Philosophy of Mathematics. 7304:"Wittgenstein's analysis on Cantor's diagonal argument" 3272:

also use mathematics due to their heavy use of models.

3038:

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 11199:; Loos, Andreas (November 2, 2017). Kaiser, G. (ed.). 9185:

Nickles, Thomas (2013). "The Problem of Demarcation".

8892:

Bellomo, Nicola; Preziosi, Luigi (December 22, 1994).

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2622: 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 14532: 13726:(Reprint ed.). Boston; New York: Mariner Books. 9771:

Pérez-Escobar, José Antonio; Sarikaya, Deniz (2021).

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A History of Greek Mathematics: From Thales to Euclid

6735: 6068: 5299:"Splash Talk: The Foundational Crisis of Mathematics" 4387: 3916: 3867: 3533:

must be reducible to a succession of applications of

3156:

is useful in many areas of computer science, such as

3066: 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

12744: 12742: 12630:. Springer Science & Business Media. p. 2. 12323:(377). Cambridge University Press ({CUP}): 204–215. 12148:. American Mathematical Society. pp. xiii–xiv. 10825:"Kondratiev, Nikolai Dmitrievich | Encyclopedia.com" 8222:, "Revival and Decline of Greek Mathematics" p. 180. 8018:

Africa Counts: Number and Pattern in African Culture

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6671:"The Newton–Leibniz Calculus Controversy, 1708–1730" 5988:"Reconstructing the Unity of Mathematics circa 1900" 4213:, Book 6 Section 510c. However, Plato did not use a 4098: 4075:

A new list of seven important problems, titled the "

2552:

Mathematical notation is widely used in science and

2521: 1826:

and was not specifically studied by mathematicians.

1791: 1209:, the study of distance properties in curved spaces. 280:(the study of shapes and spaces that contain them), 13761:(2nd ed.). New York: Oxford University Press. 10873:"Cliodynamics: a science for predicting the future" 10285: 9322:"The two cultures of mathematics in ancient Greece" 9257:; Schappacher, Norbert; Schwermer, Joachim (eds.). 9123:"Environmental activities and mathematical culture" 8746:

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8685:"Constituent Structure in Mathematical Expressions" 7506: 7504: 6877:. In Rosenstein, Joseph G.; Franzblau, Deborah S.; 6069:O'Connor, J. J.; Robertson, E. F. (February 1996). 5440:"MSC2020-Mathematics Subject Classification System" 4060:, also for lifetime achievement, instituted in 1978 2679:. Through a series of rigorous arguments employing 2137:. Other areas of computational mathematics include 2117:; numerical analysis broadly includes the study of 1235:, the use in topology of algebraic methods, mainly 936:Number theory includes several subareas, including 933:, it remains unproven despite considerable effort. 440: 13874: 13780: 13754: 13691: 13501: 13365: 13304: 13189: 12049: 10901: 9324:. In Robson, Eleanor; Stedall, Jacqueline (eds.). 9244: 9242: 8089: 7041:The History and Status of the P versus NP Question 5061:Tiwari, Sarju (1992). "A Mirror of Civilization". 3937: 3880: 3782:, it is hard to define, it is commonly related to 3506:guided by a variety of general insights and ideas. 3082: 3015:with planes). It is almost 2,000 years later that 2628: 2512:, "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 13881:. San Carlos, California: Wide World Publishing. 13307:Mathematical Thought from Ancient to Modern Times 12739: 11618: 11589:The Oxford Handbook of the History of Mathematics 10766:Levin, Jonathan; Milgrom, Paul (September 2004). 10373:The Oxford Handbook of Probability and Philosophy 10371:. In Hájek, Alan; Hitchcock, Christopher (eds.). 10070:The British Journal for the Philosophy of Science 10068:(November 1961). "Discovering the Positron (I)". 9326:The Oxford Handbook of the History of Mathematics 8518:. Part 2: Outline of the History of Mathematics. 8119:The British Journal for the Philosophy of Science 6270:Overbay, Shawn; Schorer, Jimmy; Conger, Heather. 6148: 4168:Science, technology, engineering, and mathematics 4013:The most prestigious award in mathematics is the 2698: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. 14744: 12358:Enquist, Magnus; Arak, Anthony (November 1994). 11625:Handbook on the History of Mathematics Education 11305: 10261:"Real-Life Applications of Discrete Mathematics" 10076:(47). The University of Chicago Press: 194–214. 9739:Chen, Changbo; Maza, Marc Moreno (August 2014). 7874: 7501: 6875:"What is Discrete Mathematics? The Many Answers" 6269: 5531:Goldman, Jay R. (1998). "The Founding Fathers". 4079:", was published in 2000. Only one of them, the 3744:Psychology (aesthetic, creativity and intuition) 3176: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), 13442:and as an extract, D. Van Nostrand, 1882, 12953:Chern, S. S.; Hirzebruch, F. (September 2000). 12952: 12082: 11875:Lynch School of Education and Human Development 9239: 8891: 6520:(2004). "Emmy Noether: Ideals and Structures". 6149:Christianidis, Jean; Oaks, Jeffrey (May 2013). 5785: 5783: 5781: 5021:(2nd ed.). Dover Publications. p. 3. 4087:, has been solved by the Russian mathematician 3949:is a figure invariant by this symmetry, as are 3794:, completeness, and generality. G. H. Hardy in 3466: 2999:. A second historical example is the theory of 2801: 2790:and results (theorems) are often obtained from 2759: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. 11150: 10969: 10211:"Course 18C Mathematics with Computer Science" 9465:. Princeton University Press. pp. 17–26. 9203: 8210:, "Greek Trigonometry and Mensuration" p. 162. 7276:"Dispute over Infinity Divides Mathematicians" 5292: 5290: 4566: 4564: 4562: 4560: 4558: 4556: 4419:Communications on Pure and Applied Mathematics 4036:, instituted in 2002 and first awarded in 2003 4028:Other prestigious mathematics awards include: 3898: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 13934: 13576:Bulletin of the American Mathematical Society 13220:An Introduction to the History of Mathematics 11509:International Journal of Educational Research 11179:t is first necessary to ask what is meant by 10447:"1.4: The Lotka-Volterra Predator-Prey Model" 10366:"Probability in Biology: The Case of Fitness" 10137: 9502:Bulletin of the American Mathematical Society 9093:. London: Taylor & Francis. p. 100. 8971: 6676:. In Schliesser, Eric; Smeenk, Chris (eds.). 5108:(1992). "Mathematics from the Ground Up". In 2971: 2741: 2509: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: 13694:The Moment of Proof: Mathematical Epiphanies 12226:Notices of the American Mathematical Society 10765: 9710:Notices of the American Mathematical Society 9355: 9178: 9156: 9058:"The science checklist applied: Mathematics" 8823:Silver, Daniel S. (November–December 2017). 8790:"Earliest Uses of Some Words of Mathematics" 8439:(1–2). Cambridge University Press: 313–331. 7729:Statistics and Truth: Putting Chance to Work 7434:. Cambridge University Press. pp. 3–4. 7176:"The Road to Modern Logic—An Interpretation" 6665: 5778: 5347:Notices of the American Mathematical Society 4721: 4404: 4402: 4163:Relationship between mathematics and physics 3984: 3537:, without any use of empirical evidence and 3452:and Albert Einstein that support his views. 3225:. Ecology heavily uses modeling to simulate 3109:Relationship between mathematics and physics 2885: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 13948: 13325: 12691: 12357: 11972:Psychology Research and Behavior Management 11321: 11195: 10738:. Princeton University Press. p. 158. 9980: 9681:. Frankfurt am Main: Verlag Harri Deutsch. 9677:. In Schlote, K. H.; Schneider, M. (eds.). 9358:Emerging Security Algorithms and Techniques 9114: 8314:Studies in Indian Mathematics and Astronomy 8096:. New York: Dover Publications. p. 1. 7569: 7510: 5535:. Wellesley, MA: A K Peters. pp. 2–3. 5287: 4937:"mathematics", "mathematic", "mathematics". 4553: 3323: 2035:) of a statistical action, such as using a 1187:and independent from the concept of length. 554: 540: 13941: 13927: 12904:. International Mathematical Union (IMU). 10693:Batchelder, William H. (January 1, 2015). 10692: 9890:. In Bos, Joppe W.; Stam, Martijn (eds.). 9777:European Journal for Philosophy of Science 8490:. Vol. 2. Bibliotex Digital Library. 7393: 5880: 5333: 4924:The Oxford Dictionary of English Etymology 4582:(5). Taylor & Francis, Ltd.: 291–314. 3938:{\displaystyle \mathbb {Z} /2\mathbb {Z} } 3050:is a (curved) manifold of dimension four. 2936:, 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: 13588: 13422: 13246:. Springer Science & Business Media. 12237: 12067: 12056:Procedia – Social and Behavioral Sciences 12034: 12023:Procedia – Social and Behavioral Sciences 11993: 11983: 11831: 11225: 11146: 11144: 10600:"HOW MATH IS USED IN WEATHER FORECASTING" 10360: 10312: 10289:"A Formal Proof of the Kepler Conjecture" 10041: 10016: 10014: 10012: 9950:Mukunth, Vasudevan (September 10, 2015). 9788: 9629: 9614: 9513: 9248: 9209: 8844: 8614: 8510: 8012: 7906: 7617: 7270: 7173: 7121: 6818: 6507:, "History of Linear Algebra" pp. 79–101. 6224:, "History of Classical Algebra" pp. 3–5. 6166: 5795: 5668: 5358: 5340:"Mathematics Subject Classification 2020" 5246: 4497: 4399: 3931: 3918: 3342:, and always makes optimal choices using 2978: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 13903:. Sändig Reprint Verlag H. R. Wohlwend. 13461: 13062: 12627:Mathematics Today Twelve Informal Essays 12432: 11787:(1). Taylor & Francis, Ltd.: 38–55. 11487: 11471: 11231: 11100: 11098: 11096: 10975: 10475: 9882: 9855: 9738: 9492: 9287:The Journal of Interdisciplinary History 7779:(1981). "Foreword". In Arthanari, T.S.; 7427: 7332: 6972:. Lecture Notes in Logic. Vol. 42. 6797: 6390:"François Viète's revolution in algebra" 5156: 4393: 4314: 3994: 3890: 3583: 3304: 3193: 3179: 3112: 2539: 2464: 2365: 2317: 2163: 1976: 1892:assert, roughly speaking that, in every 1805: 1645: 1531: 1517: 1369:), and polynomial equations in a single 1300: 1285: 969: 735:, which illustrates the distribution of 726: 13828: 13642:from the original on December 19, 2013. 13612: 13565: 13533:"The Early History of the Fields Medal" 13236: 13162: 12784:MacTutor History of Mathematics Archive 12594: 12533: 12141: 12016: 11698: 11664:(4). 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Canadian Mathematical Society. 13140:from the original on December 20, 2018 13028: 13026: 12981:from the original on February 21, 2022 12934:from the original on November 17, 2022 12790:from the original on December 17, 2021 12727:from the original on December 26, 2018 12657: 12450: 12414:from the original on December 28, 2022 12314: 12274:Journal of Research in Music Education 12271: 12145:Famous Puzzles of Great Mathematicians 12142:Petkovi, Miodrag (September 2, 2009). 12123:from the original on September 6, 2015 11947:from the original on November 22, 2023 11898:from the original on November 29, 2022 11715:from the original on November 28, 2022 11655: 11627:. New York: Springer. pp. 27–53. 11575: 11414: 11268: 11141: 11052: 11037: 10883:from the original on December 29, 2022 10727: 10725: 10701:. Oxford: Elsevier. pp. 808–815. 10653: 10457:from the original on December 29, 2022 10356: 10354: 10352: 10171: 10064: 10009: 9962:from the original on November 20, 2022 9901:from the original on November 20, 2022 9811: 9703: 9672: 9530:from the original on November 20, 2022 9456: 9120: 9086: 9064:. University of California, Berkeley. 8991:from the original on November 17, 2022 8925: 8822: 8725:from the original on November 16, 2022 8574:from the original on November 16, 2022 8567:Mathematical Notation: Past and Future 8462:from the original on December 17, 2022 8405:from the original on November 14, 2022 8401:(4). Shannon Research Press: 391–399. 8388: 8352: 7925:The Princeton Companion to Mathematics 7856:from the original on November 13, 2022 7785:Mathematical programming in statistics 7769: 7692:Probability, Statistics and Estimation 7688: 7610:Philosophia Scientiæ, Cahier spécial 6 7585:from the original on December 16, 2022 7551:from the original on November 12, 2022 7397:; Reck, Erich H. (December 11, 2001). 7335:Mathematical Rigour and Informal Proof 7035: 6872: 6611: 6314:from the original on December 22, 2018 6103: 5434: 5432: 5430: 5428: 5426: 5424: 5422: 5420: 5060: 4803:from the original on November 26, 2022 4788: 4760: 4703:from the original on February 24, 2021 4659: 4408: 4217:word; Aristotle did, commenting on it. 3842: 2878:, now widely used for the security of 2495:transformed mathematics by publishing 1963:Statistics and other decision sciences 1183:, the study of properties relative to 1158:Today's subareas of geometry include: 1151:under specific transformations of the 1104:Analytic geometry allows the study of 858:. Number theory dates back to ancient 13922: 13897:Waltershausen, Wolfgang Sartorius von 13553:from the original on October 26, 2006 13530: 13311:. New York: Oxford University Press. 13299: 13184: 13044:from the original on January 23, 2022 13011:from the original on January 12, 2020 12928:"The Leroy P Steele Prize of the AMS" 12846:from the original on January 26, 2020 12703: 12623: 12175: 11778: 11453:from the original on December 5, 2022 11093: 11007: 10988:from the original on January 30, 2022 10381:10.1093/oxfordhb/9780199607617.013.27 10337:from the original on December 4, 2020 10020: 9931:from the original on October 14, 2022 9630:Peressini, Anthony (September 1999). 9548: 9463:The Best Writing on Mathematics, 2016 9402:"How applied mathematics became pure" 9396: 9319: 9068:from the original on October 27, 2019 8743: 8664:from the original on December 8, 2022 8423: 8268: 8219: 8207: 8195: 8183: 8171: 8159: 8084: 8072: 8048: 7124:"The Early Development of Set Theory" 7069: 6956: 6752:from the original on November 9, 2022 6717:from the original on November 9, 2022 6620:Springer Science & Business Media 6569: 6516: 6460:from the original on November 8, 2022 6398:Archive for History of Exact Sciences 6233: 6085:from the original on November 6, 2022 5985: 5937: 5228: 4789:Harper, Douglas (December 22, 2018). 4356:from the original on November 7, 2022 4209:This meaning can be found in Plato's 2897:. This led to split mathematics into 2446:, the introduction of coordinates by 2361: 2187: 14597: 14503: 13566:Sevryuk, Mikhail B. (January 2006). 13348:from the original on August 13, 2006 12908:from the original on August 25, 2010 12661:The Best Writing on Mathematics 2016 12483: 11104: 11081:from the original on August 17, 2023 10672:10.1146/annurev.soc.28.110601.140942 10597: 10144:. PHI Learning Pvt. Ltd. p. 3. 9654:from the original on January 2, 2024 9597:"On the role of applied mathematics" 9438:from the original on August 12, 2017 9281: 9215:"Are There 'Other' Ways of Knowing?" 8639: 8480: 7705:from the original on October 9, 2022 7072:"The Emergence of First-Order Logic" 7070:Ewald, William (November 17, 2018). 6690:10.1093/oxfordhb/9780199930418.013.9 6387: 6332: 5723: 5587: 5456:from the original on January 2, 2024 5013: 4874: 4868: 4650: 3186:Mathematical and theoretical biology 3097: 2924:for validating computations done in 2916: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 14515: 13023: 12816:from the original on April 14, 2022 12760:from the original on March 22, 2019 11733: 11365:from the original on March 23, 2023 10731: 10722: 10598:Hill, Mullica (September 7, 2022). 10476:Salsburg, David (August 17, 1992). 10349: 9720:from the original on March 13, 2014 9685:from the original on March 23, 2023 9594: 9090:Dictionary of Scientific Quotations 8985:Stanford Encyclopedia of Philosophy 8231: 7775: 7723: 7579:Stanford Encyclopedia of Philosophy 7517:Revue Internationale de Philosophie 7129:Stanford Encyclopedia of Philosophy 7077:Stanford Encyclopedia of Philosophy 6883:Discrete Mathematics in the Schools 6297: 5595:. Birkhäuser Boston. pp. 2–3. 5417: 5402:from the original on August 3, 2021 4884:Mathematical Association of America 2946: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: 13648: 13600:from the original on July 23, 2006 13372:. New York: Penguin Random House. 12882:from the original on June 17, 2009 12605:from the original on March 7, 2023 12576:from the original on March 2, 2023 12546:from the original on March 7, 2023 11325:; Porter, Timothy (January 2000). 11107:Educational Studies in Mathematics 10695:"Mathematical Psychology: History" 10390:from the original on March 7, 2023 8186:, "Archimedes of Syracuse" p. 130. 8174:, "Archimedes of Syracuse" p. 120. 7995: 7976:from the original on March 7, 2023 7670:from the original on March 3, 2021 7511:Raatikainen, Panu (October 2005). 7301: 7122:Ferreirós, José (June 18, 2020) . 5296: 5114:Mathematics in Society and History 4995:from the original on April 4, 2020 4761:Harper, Douglas (April 20, 2022). 4674:from the original on March 7, 2013 4660:Harper, Douglas (March 28, 2019). 4003:with an illustration of the Greek 3837: 3275: 3068: 2948:that became a fundamental tool in 2870:, for example, which goes back to 2864: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: 14774: 13114:from the original on July 3, 2015 13034:"Hilbert's Problems: 23 and Math" 12536:"Fractals – Chaos & Fractals" 11781:The American Mathematical Monthly 10835:from the original on July 1, 2016 9706:"Model theory and exponentiation" 9227:from the original on May 13, 2020 8825:"The New Language of Mathematics" 8516:The American Mathematical Monthly 7910:(2008). "Numerical Analysis". In 7846:"G I Marchuk's plenary: ICM 1970" 7333:Tanswell, Fenner Stanley (2024). 7174:Ferreirós, José (December 2001). 6807:Journal of Humanistic Mathematics 5941:(2004) . "Fermat and Descartes". 5484:Addison-Wesley Publishing Company 5166:Springer International Publishing 4692: 4479:from the original on June 1, 2019 3965: 3246: 2843:developed infinitesimal calculus. 2522:Symbolic notation and terminology 2430:(1540–1603), the introduction of 1792:Mathematical logic and set theory 14725: 14708: 14691: 14674: 14657: 14640: 14623: 14606: 14578: 14566: 14554: 14542: 14514: 14502: 14491: 14490: 14478: 13126: 13100: 13056: 12993: 12946: 12920: 12894: 12858: 12836:"Abel Prize | mathematics award" 12828: 12802: 12772: 12709: 12651: 12617: 12588: 12558: 12527: 12509: 12477: 12444: 12426: 12360:"Symmetry, beauty and evolution" 12351: 12308: 12265: 12246: 12213: 12169: 12135: 12076: 12043: 12017:Yaftian, Narges (June 2, 2015). 12010: 11959: 11938:10.1002/j.2161-0045.2012.00001.x 11926:The Career Development Quarterly 11910: 11848: 11807: 11772: 11727: 11692: 11649: 11612: 11569: 11535: 11500: 11481: 11465: 11408: 11377: 11327:"The Methodology of Mathematics" 11315: 11299: 11189: 11046: 11000: 10951: 10937: 10895: 10865: 10847: 10817: 10799: 10779: 10759: 10686: 10647: 10617: 10591: 10566: 10541: 10495: 10469: 10439: 10402: 10279: 10253: 10228: 10203: 10165: 10131: 10096: 10058: 9974: 9943: 9913: 9883:Wagstaff, Samuel S. Jr. (2021). 9876: 9849: 9805: 9764: 9732: 9697: 9666: 9623: 9588: 9542: 9486: 9459:"In Defense of Pure Mathematics" 9450: 9390: 9349: 9313: 9275: 9150: 9080: 9050: 9003: 8965: 8919: 8885: 8816: 8782: 8737: 8676: 8586: 8554: 8504: 8162:, "Euclid of Alexandria" p. 119. 7619:10.4000/philosophiascientiae.411 5866: 4464: 4371: 4289: 4280: 4271: 4254: 4101: 3786:, which involves qualities like 3525:Mathematical reasoning requires 2829: 2820: 2771:as a better mathematical model. 2757:perihelion precession of Mercury 2544: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: 14399:Computational complexity theory 13403:American Journal of Mathematics 13240:(2007). Kleiner, Israel (ed.). 13108:"The Millennium Prize Problems" 10549:"Catastrophe Models (Property)" 9015:"Mathematics, Maps, and Models" 8700:University of California Merced 8474: 8417: 8395:International Education Journal 8382: 8346: 8305: 8262: 8225: 8198:, "Apollonius of Perga" p. 145. 8110: 8078: 8054: 8006: 7988: 7900: 7868: 7837: 7717: 7689:Rouaud, Mathieu (April 2017) . 7682: 7626: 7597: 7563: 7421: 7387: 7326: 7295: 7264: 7167: 7115: 7063: 7029: 6950: 6866: 6791: 6764: 6729: 6659: 6605: 6573:Perspectives on Universal Logic 6563: 6510: 6472: 6381: 6326: 6298:Lim, Lisa (December 21, 2018). 6291: 6263: 6227: 6142: 6097: 6062: 5979: 5931: 5874: 5804: 5717: 5641: 5581: 5524: 5468: 5327: 5222: 5150: 5098: 5054: 5017:(1945) . "General Prospectus". 5007: 4971: 4940: 4916: 4815: 4782: 4754: 4715: 4237: 4220: 4203: 3705: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} ),} 13136:. Clay Mathematics Institute. 13110:. Clay Mathematics Institute. 12664:. Princeton University Press. 12185:. Cambridge University Press. 12093:The Mathematical Intelligencer 11670:10.1080/10724117.2002.11975154 11417:"Mathematical Rigor and Proof" 11308:Mathematics, form and function 11241:. Elsevier. pp. 104–106. 10654:Edling, Christofer R. (2002). 10236:"Theoretical Computer Science" 10030:The Mathematical Intelligencer 10026:"Mathematics: Art and Science" 9885:"History of Integer Factoring" 8603:Journal of Numerical Cognition 7183:The Bulletin of Symbolic Logic 6110:Johns Hopkins University Press 5019:The Development of Mathematics 4696:Republic, Book 6, Section 510c 4686: 4588:10.1080/0025570X.1991.11977625 4491: 4458: 4308: 4186: 3881:{\displaystyle {\frac {3}{2}}} 3498:Mathematics, form and function 3237:Statistical hypothesis testing 2660: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: 13829:Hodgkin, Luke Howard (2005). 13811:. Kluwer Academic Publishers. 13787:. W.W. Norton & Company. 13590:10.1090/S0273-0979-05-01069-4 13466:. W. H. Freeman and Company. 13326:Monastyrsky, Michael (2001). 13243:A History of Abstract Algebra 12259:History of Western Philosophy 11737:How Chinese Learn Mathematics 10980:. In Zalta, Edward N. (ed.). 10859:onlinebooks.library.upenn.edu 10769:Introduction to Choice Theory 10697:. In Wright, James D. (ed.). 9360:. CRC Press. pp. 59–60. 8655:American Mathematical Society 8360:. New York University Press. 8238:Number Theory and Its History 8063:pg 10. Retrieved June 1, 2024 6887:American Mathematical Society 6678:The Oxford Handbook of Newton 6343:Encyclopaedia of Islam Online 5884:(2000). "Euclid's Geometry". 5822:Open Court Publishing Company 5480:Fundamentals of Number Theory 4230:, which are intersections of 4174: 3683: 3668: 3405: 3023: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 13808:Encyclopaedia of Mathematics 13679:Resources in other libraries 13399:"Linear associative algebra" 13395:Peirce, Charles Sanders 12595:Kissane, Barry (July 2009). 12083:van der Poorten, A. (1979). 12069:10.1016/j.sbspro.2013.07.101 12036:10.1016/j.sbspro.2015.04.617 11873:International Study Center, 11424:The Review of Symbolic Logic 11055:"Why are laws mathematical?" 10178:On the Work of Edward Witten 9826:10.1007/978-4-431-55060-0_29 9751:10.1007/978-3-662-44199-2_65 9616:10.1016/0001-8708(76)90024-4 9409:The Review of Symbolic Logic 9027:10.1007/978-3-319-72478-2_18 8424:Lorch, Richard (June 2001). 8322:10.1007/978-981-13-7326-8_20 6982:10.1017/CBO9781107338579.001 5943:History of Analytic Geometry 5248:10.1016/0315-0860(79)90074-0 5231:"The roots of combinatorics" 4302: 3588: 3365:discerned the ~50-year-long 3142:Theoretical computer science 3132:Theoretical computer science 3125: 3083:{\displaystyle \Omega ^{-}.} 3005:ancient Greek mathematicians 2960:departments, such as at the 2802:Pure and applied mathematics 2746: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: 13833:. Oxford University Press. 13723:The Mathematical Experience 13698:. Oxford University Press. 13531:Riehm, Carl (August 2002). 12461:10.1002/9781119547167.ch131 12455:. Wiley. pp. 367–376. 11968:"Spotlight on math anxiety" 11633:10.1007/978-1-4614-9155-2_3 11415:Hamami, Yacin (June 2022). 11209:10.1007/978-3-319-62597-3_5 11154:; Neunzert, Helmut (2012). 11064:. Ashgate. pp. 31–66. 10511:. NAP.edu. pp. 71–73. 10125:10.1016/j.shpsb.2015.12.001 9461:. In Pitici, Mircea (ed.). 9160:; Lazere, Cathy A. (1998). 8389:Faruqi, Yasmeen M. (2006). 7274:, ed. (November 26, 2013). 6873:Maurer, Stephen B. (1997). 5887:Geometry: Euclid and Beyond 5817:The Foundations of Geometry 4796:Online Etymology Dictionary 4768:Online Etymology Dictionary 4667:Online Etymology Dictionary 4319:; Neges, Katharina (eds.). 4133:Lists of mathematics topics 4123:List of mathematical jargon 4094: 3778:value to mathematics. Like 3774:Mathematicians can find an 3728:L'Enseignement Mathématique 3011:(that is, intersections of 3003:. They were studied by the 2995:communications through the 2786:. Similarly as in science, 2769:Newton's law of gravitation 2588:letters, and often include 2497:his incompleteness theorems 2332:Hindu–Arabic numeral system 1073:. This constituted a major 959: 905:, who used tools including 567:anew, after the pattern of 10: 14779: 14449:Films about mathematicians 13690:Benson, Donald C. (1999). 13222:(6th ed.). Saunders. 13196:(2nd ed.). New York: 13155: 12286:10.1177/002242945900700205 11749:10.1142/9789812562241_0006 11521:10.1016/j.ijer.2018.10.002 10978:"Platonism in Metaphysics" 10660:Annual Review of Sociology 10656:"Mathematics in Sociology" 9790:10.1007/s13194-021-00435-9 9704:Marker, Dave (July 1996). 9595:Lin, C. C . (March 1976). 9457:Silver, Daniel S. (2017). 9293:(1). The MIT Press: 1–31. 8926:Hennig, Christian (2010). 8744:Rossi, Richard J. (2006). 8075:, "Mesopotamia" pp. 24–27. 7933:Princeton University Press 7428:Hamilton, Alan G. (1982). 7407:Carnegie Mellon University 7339:Cambridge University Press 6974:Cambridge University Press 6820:10.5642/jhummath.201702.18 6622:. pp. xxi–xxv, 1–91. 5830:10.1126/science.16.399.307 5118:Kluwer Academic Publishers 5116:. Episteme. Vol. 20. 3988: 3969: 3846: 3592: 3518: 3479:Definitions of mathematics 3476: 3467:unreasonable effectiveness 3425:. The ancient philosopher 3416: 3409: 3279: 3250: 3208:reaction–diffusion systems 3206:, which can be modeled by 3183: 3129: 3106: 3102: 2972:Unreasonable effectiveness 2805: 2742:Relationship with sciences 2525: 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: 14472: 14422: 14379: 14289: 14251: 14218: 14170: 14142: 14089: 14036: 14018:Philosophy of mathematics 13993: 13958: 13674:Resources in your library 13252:10.1007/978-0-8176-4685-1 12956:Wolf Prize in Mathematics 12207:A Mathematician's Apology 12182:A Mathematician's Apology 11833:10.1007/s11858-012-0445-7 11820:ZDM Mathematics Education 11490:Philosophy of Mathematics 11436:10.1017/S1755020319000443 11306:Saunders Maclane (1986). 11238:Philosophy of Mathematics 10503:National Research Council 9633:Applying Pure Mathematics 9421:10.1017/S1755020308080027 9164:. Springer. p. 228. 8944:10.1007/s10699-009-9167-x 8798:University of St. Andrews 8445:10.1017/S0269889701000114 7604:McCarty, Charles (2006). 6411:10.1007/s00407-018-0208-0 6388:Oaks, Jeffery A. (2018). 6079:University of St. Andrews 5601:10.1007/978-0-8176-4571-7 5229:Biggs, N. L. (May 1979). 5174:10.1007/978-3-031-12322-1 4984:Oxford English Dictionary 4953:Oxford English Dictionary 4930:Oxford English Dictionary 4472:The University of Georgia 4380:November 5, 2022, at the 4158:Philosophy of mathematics 4077:Millennium Prize Problems 4058:Wolf Prize in Mathematics 3985:Awards and prize problems 3797:A Mathematician's Apology 3412:Philosophy of mathematics 3136:Computational mathematics 2987:A notable example is the 2841:Gottfried Wilhelm Leibniz 2556: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 14454:Recreational mathematics 13853:The World of Mathematics 13742:(registration required). 13500:(1995). "On knowledge". 13462:Peterson, Ivars (1988). 13192:A History of Mathematics 12433:Hestenes, David (1999). 12317:The Mathematical Gazette 11331:The Mathematical Gazette 11275:A History of Mathematics 11040:The World of Mathematics 10412:15–17 janvier 1996, CNRS 10293:Forum of Mathematics, Pi 8752:. pp. 1–14, 47–48. 8512:Archibald, Raymond Clare 8020:. Chicago Review Press. 7834:, pp. 10–11, 14–18. 6746:University of St Andrews 6307:South China Morning Post 6168:10.1016/j.hm.2012.09.001 6071:"Non-Euclidean geometry" 5986:Stump, David J. (1997). 5478:(1977). "Introduction". 4179: 3769:recreational mathematics 3716:. Formal instruction in 3551:computer-assisted proofs 3514: 3231:Lotka–Volterra equations 2839: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), 14339:Mathematical statistics 14329:Mathematical psychology 14299:Engineering mathematics 14233:Algebraic number theory 13895: 13508:. New York: Routledge. 13424:2027/hvd.32044030622997 13335:CMS – Notes – de la SMC 12840:Encyclopedia Britannica 12658:Pitici, Mircea (2017). 12534:Bradley, Larry (2010). 11542:Endsley, Kezia (2021). 11384:Strauss, Danie (2011). 11160:. Springer. p. 9. 10976:Balaguer, Mark (2016). 10082:10.1093/bjps/xiii.49.54 10066:Hanson, Norwood Russell 9602:Advances in Mathematics 9567:10.1023/A:1020823608217 9549:Stolz, Michael (2002). 8481:Kent, Benjamin (2022). 6682:Oxford University Press 5996:Perspectives on Science 5726:The Goldbach Conjecture 5657:Elemente der Mathematik 4989:Oxford University Press 4958:Oxford University Press 4736:A Greek–English Lexicon 4128:Lists of mathematicians 2966:University of Cambridge 2950:real algebraic geometry 2918:theory of distributions 2798:on empirical evidence. 2572:, unspecified numbers, 2536:Glossary of mathematics 2532: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 14485:Mathematics portal 14334:Mathematical sociology 14314:Mathematical economics 14309:Mathematical chemistry 14238:Analytic number theory 14119:Differential equations 13877:The Joy Of Mathematics 13855:. Dover Publications. 13849:Jourdain, Philip E. B. 13821:December 20, 2012, at 12810:"About the Abel Prize" 12780:"Honours/Prizes Index" 11583:. In Robson, Eleanor; 11386:"Defining mathematics" 11053:Dorato, Mauro (2005). 11038:Newman, J. R. (1956). 10963:www.minneapolisfed.org 10451:Mathematics LibreTexts 9494:Parshall, Karen Hunger 9320:Asper, Markus (2009). 9249:Ferreirós, J. (2007). 9087:Mackay, A. L. (1991). 8932:Foundations of Science 8846:10.1511/2017.105.6.364 8829:The American Scientist 7575:"Intuitionistic Logic" 6612:Krömer, Ralph (2007). 6277:University of Kentucky 5164:. History of Physics. 4440:10.1002/cpa.3160130102 4010: 3999:The front side of the 3939: 3899: 3882: 3813:fast Fourier transform 3508: 3485:epistemological status 3463: 3313: 3282:Mathematical economics 3239:, is run on data from 3211: 3190:Mathematical chemistry 3118: 3084: 2962:Faculty of Mathematics 2683:, a statement that is 2630: 2568:used for representing 2549: 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 14464:Mathematics education 14394:Theory of computation 14114:Hypercomplex analysis 13901:Gauss zum Gedächtniss 13134:"Millennium Problems" 13076:In the Light of Logic 12624:Steen, L. A. (2012). 11011:Philosophy of Science 10915:. New York: Picador. 10732:Zak, Paul J. (2010). 10364:(September 8, 2016). 10191:on September 28, 2013 10141:Essentials of Physics 10036:(4). Springer: 9–17. 9158:Shasha, Dennis Elliot 9121:Bishop, Alan (1991). 9062:Understanding Science 8750:John Wiley & Sons 8131:10.1093/bjps/20.4.289 7573:(September 4, 2018). 7347:10.1017/9781009325110 7049:10.1145/129712.129771 6895:10.1090/dimacs/036/13 6667:Guicciardini, Niccolo 6234:Shane, David (2022). 5541:10.1201/9781439864623 4825:Pi Mu Epsilon Journal 4791:"Mathematical (adj.)" 4723:Liddell, Henry George 4153:Mathematics education 4143:Mathematical sciences 4138:Mathematical constant 4051:Leroy P. Steele Prize 3998: 3940: 3894: 3883: 3595:Mathematics education 3584:Training and practice 3547:Feit–Thompson theorem 3503: 3454: 3308: 3280:Further information: 3197: 3180:Biology and chemistry 3130:Further information: 3117:Diagram of a pendulum 3116: 3085: 2868:integer factorization 2631: 2543: 2528: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) 14444:Informal mathematics 14324:Mathematical physics 14319:Mathematical finance 14304:Mathematical biology 14243:Diophantine geometry 13186:Boyer, Carl Benjamin 11985:10.2147/PRBM.S141421 10912:Fashionable Nonsense 10829:www.encyclopedia.com 10375:. pp. 601–622. 9255:Goldstein, Catherine 8979:(February 4, 2020). 8616:10.5964/jnc.v6i3.293 8597:(December 3, 2020). 8086:Heath, Thomas Little 7935:. pp. 604–615. 7777:Rao, C. Radhakrishna 7725:Rao, C. Radhakrishna 7529:10.3917/rip.234.0513 7467:Mathematics Magazine 6976:. pp. 260–261. 6889:. pp. 121–124. 6680:. Oxford Handbooks. 6244:Methodist University 6155:Historia Mathematica 6112:. pp. 219–232. 6009:10.1162/posc_a_00532 5235:Historia Mathematica 4576:Mathematics Magazine 4064:A famous list of 23 4053:, awarded since 1970 3978:mathematical anxiety 3914: 3865: 3808:Proofs from THE BOOK 3566:Weierstrass function 3473:Proposed definitions 3387:Fashionable Nonsense 3223:evolutionary fitness 3154:Discrete mathematics 3064: 3036:theory of relativity 3019:discovered that the 2620: 2515: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 14763:Main topic articles 14459:Mathematics and art 14369:Operations research 14124:Functional analysis 13803:Hazewinkel, Michiel 13514:1992sbwl.book.....P 13384:A Mind for Numbers. 12902:"Chern Medal Award" 12866:"Chern Medal Award" 12706:, pp. 778–782. 12598:Popular mathematics 12376:1994Natur.372..169E 12261:. 1919. p. 60. 11585:Stedall, Jacqueline 10453:. January 5, 2022. 10424:, pp. 282–283. 10117:2016SHPMP..53...20G 9812:Takase, M. (2014). 9673:Lützen, J. (2011). 8981:"Models in Science" 8551:, pp. 101–109. 7908:Trefethen, Lloyd N. 6480:"Variable in Maths" 5949:. pp. 74–102. 5724:Wang, Yuan (2002). 5476:LeVeque, William J. 5306:Columbia University 4432:1960CPAM...13....1W 4148:Mathematics and art 4085:Poincaré conjecture 3972:Popular mathematics 3849:Mathematics and art 3843:Artistic expression 3649:computer consultant 3549:. The emergence of 3344:perfect information 3286:Historical dynamics 3227:population dynamics 2989:prime factorization 2903:applied mathematics 2808:Applied mathematics 2681: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 14573:History of science 14404:Numerical analysis 14013:Mathematical logic 14008:Information theory 13540:Notices of the AMS 12812:. The Abel Prize. 12521:MIT OpenCourseWare 12106:10.1007/BF03028234 11476:Four Color Theorem 11197:Ziegler, Günter M. 11119:10.1007/BF01273907 10604:www.mathnasium.com 10362:Millstein, Roberta 10305:10.1017/fmp.2017.1 10043:10.4171/news/103/8 9211:Pigliucci, Massimo 8484:History of Science 8433:Science in Context 8014:Zaslavsky, Claudia 7997:Wilder, Raymond L. 7994:See, for example, 7916:Barrow-Green, June 7272:Wolchover, Natalie 5947:Dover Publications 5670:10.1007/PL00000079 4763:"Mathematics (n.)" 4232:circular cylinders 4109:Mathematics portal 4081:Riemann hypothesis 4070:Hilbert's problems 4011: 3991:Mathematics awards 3935: 3900: 3878: 3703:civil service exam 3363:Nikolai Kondratiev 3314: 3258:Structural geology 3212: 3162:information theory 3119: 3080: 3048:general relativity 3044:special relativity 2765:general relativity 2629:{\textstyle \int } 2626: 2550: 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 14530: 14529: 14129:Harmonic analysis 13910:978-3-253-01702-5 13888:978-0-933174-65-8 13862:978-0-486-43268-7 13840:978-0-19-152383-0 13794:978-0-393-04002-9 13768:978-0-19-510519-3 13733:978-0-395-92968-1 13705:978-0-19-513919-8 13655:Library resources 13635:978-0-471-94829-2 13523:978-0-415-13548-1 13379:978-0-399-16524-5 13318:978-0-19-506135-2 13261:978-0-8176-4684-4 13229:978-0-03-029558-4 13207:978-0-471-54397-8 13177:978-2-7384-8125-2 13086:978-0-19-508030-8 13064:Feferman, Solomon 13038:Simons Foundation 12974:978-981-02-3945-9 12873:www.mathunion.org 12721:www.mathunion.org 12671:978-1-4008-8560-2 12637:978-1-4613-9435-8 12566:"Self-similarity" 12502:978-1-4008-7434-7 12470:978-1-119-05751-2 12435:"Symmetry Groups" 12370:(6502): 169–172. 12252:See, for example 12192:978-0-521-42706-7 12155:978-0-8218-4814-2 11888:978-1-889938-54-7 11758:978-981-256-014-8 11642:978-1-4614-9154-5 11621:Proust, Christine 11598:978-0-19-921312-2 11555:978-1-5381-4517-3 11285:978-0-8218-2102-2 11248:978-0-08-093058-9 11218:978-3-319-62596-6 11167:978-3-0348-0229-1 11071:978-0-7546-3994-7 10922:978-0-312-19545-8 10745:978-1-4008-3736-6 10708:978-0-08-097087-5 10526:978-0-309-16839-7 10151:978-81-203-4642-0 9921:"Curves: Ellipse" 9835:978-4-431-55059-4 9515:10.1090/bull/1754 9472:978-0-691-17529-4 9367:978-0-8153-6145-9 9335:978-0-19-921312-2 9268:978-3-540-34720-0 9196:978-0-226-05182-6 9171:978-0-387-98269-4 9136:978-0-7923-1270-3 9100:978-0-7503-0106-0 9036:978-3-319-72478-2 8977:Hartmann, Stephan 8905:978-0-8493-8331-1 8759:978-0-470-04295-3 8648:"AMS Style Guide" 8497:978-1-984668-67-7 8367:978-0-8147-7962-0 8331:978-981-13-7325-1 8248:978-0-486-65620-5 8103:978-0-486-24073-2 8027:978-1-61374-115-3 7942:978-0-691-11880-2 7794:978-0-471-08073-2 7646:Kolaitis, Phokion 7571:Moschovakis, Joan 7441:978-0-521-28761-6 7364:978-1-00-949438-0 7302:Zhuang, Chaohui. 6991:978-1-107-04348-0 6699:978-0-19-993041-8 6629:978-3-7643-7523-2 6583:978-88-7699-077-9 6119:978-0-8018-9012-3 5894:. pp. 9–13. 5892:Springer New York 5882:Hartshorne, Robin 5732:. pp. 1–18. 5486:. pp. 1–30. 5315:on March 22, 2023 5191:978-3-031-12321-4 5072:978-81-7099-404-6 5028:978-0-486-27239-9 4893:978-0-88385-323-8 4662:"Mathematic (n.)" 4249:generating series 4118:Law (mathematics) 3910:of two elements, 3876: 3861:multiplies it by 3817:harmonic analysis 3661:Greco-Roman world 3659:and then for the 3657:ancient Near East 3633:financial analyst 3493:Saunders Mac Lane 3381:since the 1990s. 3310:Supply and demand 3198:The skin of this 3170:Kepler conjecture 3158:complexity theory 3098:Specific sciences 2926:quantum mechanics 2908:The aftermath of 2880:computer networks 2866:. The problem of 2774:There is still a 2767:, 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 14770: 14738: 14730: 14729: 14728: 14721: 14720:from Wikiversity 14713: 14712: 14711: 14704: 14696: 14695: 14694: 14687: 14679: 14678: 14677: 14670: 14662: 14661: 14660: 14653: 14645: 14644: 14643: 14636: 14628: 14627: 14626: 14619: 14611: 14610: 14609: 14599: 14583: 14582: 14571: 14570: 14569: 14559: 14558: 14557: 14547: 14546: 14538: 14518: 14517: 14506: 14505: 14494: 14493: 14483: 14482: 14414:Computer algebra 14389:Computer science 14109:Complex analysis 13943: 13936: 13929: 13920: 13919: 13914: 13892: 13880: 13866: 13844: 13812: 13798: 13786: 13772: 13760: 13751:Robbins, Herbert 13747:Courant, Richard 13737: 13714:Davis, Philip J. 13709: 13697: 13643: 13609: 13607: 13605: 13599: 13592: 13572: 13562: 13560: 13558: 13552: 13537: 13527: 13507: 13493: 13457: 13455: 13453: 13426: 13391:Peirce, Benjamin 13386: 13371: 13357: 13355: 13353: 13347: 13332: 13322: 13310: 13296: 13294: 13292: 13233: 13211: 13195: 13181: 13164:Bouleau, Nicolas 13150: 13149: 13147: 13145: 13130: 13124: 13123: 13121: 13119: 13104: 13098: 13097: 13095: 13093: 13072: 13060: 13054: 13053: 13051: 13049: 13030: 13021: 13020: 13018: 13016: 13001:"The Wolf Prize" 12997: 12991: 12990: 12988: 12986: 12950: 12944: 12943: 12941: 12939: 12924: 12918: 12917: 12915: 12913: 12898: 12892: 12891: 12889: 12887: 12881: 12875:. 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Scotland, UK: 8786: 8780: 8779: 8741: 8735: 8734: 8732: 8730: 8724: 8689: 8680: 8674: 8673: 8671: 8669: 8663: 8652: 8643: 8637: 8636: 8618: 8595:LeFevre, Jo-Anne 8590: 8584: 8583: 8581: 8579: 8564:(October 2000). 8562:Wolfram, Stephan 8558: 8552: 8546: 8540: 8539: 8508: 8502: 8501: 8489: 8478: 8472: 8471: 8469: 8467: 8461: 8430: 8421: 8415: 8414: 8412: 8410: 8386: 8380: 8379: 8350: 8344: 8343: 8309: 8303: 8302: 8266: 8260: 8259: 8257: 8255: 8229: 8223: 8217: 8211: 8205: 8199: 8193: 8187: 8181: 8175: 8169: 8163: 8157: 8151: 8150: 8114: 8108: 8107: 8095: 8082: 8076: 8070: 8064: 8058: 8052: 8046: 8040: 8039: 8010: 8004: 8003: 7992: 7986: 7985: 7983: 7981: 7975: 7930: 7904: 7898: 7897: 7895: 7893: 7872: 7866: 7865: 7863: 7861: 7841: 7835: 7829: 7823: 7822: 7773: 7767: 7766: 7721: 7715: 7714: 7712: 7710: 7704: 7697: 7686: 7680: 7679: 7677: 7675: 7669: 7662: 7630: 7624: 7623: 7621: 7601: 7595: 7594: 7592: 7590: 7567: 7561: 7560: 7558: 7556: 7508: 7499: 7498: 7459: 7453: 7452: 7450: 7448: 7425: 7419: 7418: 7416: 7414: 7403: 7391: 7385: 7384: 7330: 7324: 7323: 7321: 7319: 7308: 7299: 7293: 7292: 7290: 7288: 7268: 7262: 7261: 7259: 7257: 7180: 7171: 7165: 7164: 7162: 7160: 7119: 7113: 7112: 7110: 7108: 7067: 7061: 7060: 7033: 7027: 7026: 7024: 7022: 6958:Hales, Thomas C. 6954: 6948: 6947: 6945: 6943: 6879:Roberts, Fred S. 6870: 6864: 6863: 6861: 6859: 6822: 6795: 6789: 6788: 6786: 6784: 6768: 6762: 6761: 6759: 6757: 6744:. Scotland, UK: 6738:"Leonhard Euler" 6733: 6727: 6726: 6724: 6722: 6716: 6675: 6663: 6657: 6656: 6654: 6652: 6609: 6603: 6602: 6600: 6598: 6567: 6561: 6560: 6558: 6556: 6514: 6508: 6502: 6496: 6495: 6493: 6491: 6486:. April 24, 2024 6476: 6470: 6469: 6467: 6465: 6459: 6394: 6385: 6379: 6378: 6376: 6374: 6346:(3rd ed.). 6330: 6324: 6323: 6321: 6319: 6303: 6295: 6289: 6288: 6286: 6284: 6267: 6261: 6260: 6254: 6252: 6240: 6231: 6225: 6219: 6213: 6212: 6170: 6146: 6140: 6139: 6108:(2nd ed.). 6101: 6095: 6094: 6092: 6090: 6077:. Scotland, UK: 6066: 6060: 6059: 6057: 6055: 5992: 5983: 5977: 5976: 5935: 5929: 5928: 5926: 5924: 5878: 5872: 5871: 5870: 5864: 5862: 5860: 5808: 5802: 5801: 5799: 5787: 5776: 5775: 5730:World Scientific 5721: 5715: 5714: 5672: 5645: 5639: 5638: 5585: 5579: 5578: 5528: 5522: 5521: 5472: 5466: 5465: 5463: 5461: 5455: 5444: 5436: 5415: 5414: 5409: 5407: 5401: 5362: 5360:10.1090/noti2052 5344: 5331: 5325: 5324: 5322: 5320: 5314: 5308:. Archived from 5303: 5294: 5285: 5284: 5250: 5226: 5220: 5219: 5154: 5148: 5147: 5102: 5096: 5095: 5058: 5052: 5051: 5011: 5005: 5004: 5002: 5000: 4975: 4969: 4968: 4966: 4964: 4944: 4938: 4920: 4914: 4913: 4872: 4866: 4865: 4819: 4813: 4812: 4810: 4808: 4786: 4780: 4779: 4777: 4775: 4758: 4752: 4751: 4749: 4747: 4719: 4713: 4712: 4710: 4708: 4690: 4684: 4683: 4681: 4679: 4657: 4648: 4647: 4568: 4551: 4550: 4495: 4489: 4488: 4486: 4484: 4462: 4456: 4455: 4406: 4397: 4391: 4385: 4376: 4375: 4365: 4363: 4361: 4355: 4332: 4312: 4296: 4293: 4287: 4284: 4278: 4275: 4269: 4266:proof assistants 4258: 4252: 4245:complex analysis 4241: 4235: 4224: 4218: 4207: 4201: 4190: 4111: 4106: 4105: 4089:Grigori Perelman 3944: 3942: 3941: 3936: 3934: 3926: 3921: 3887: 3885: 3884: 3879: 3877: 3869: 3750:computer program 3692: 3688: 3685: 3673: 3670: 3645:commodity trader 3555:quasi-empiricist 3469:of mathematics. 3367:Kondratiev cycle 3337: 3334: 3331: 3328: 3325: 3200:giant pufferfish 3089: 3087: 3086: 3081: 3076: 3075: 2997:RSA cryptosystem 2922:Laurent Schwartz 2920:, introduced by 2899:pure mathematics 2895:pure mathematics 2891:Richard Dedekind 2887:Karl Weierstrass 2876:RSA cryptosystem 2833: 2824: 2812:Pure mathematics 2651: 2643: 2635: 2633: 2632: 2627: 2611: 2603: 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: 14778: 14777: 14773: 14772: 14771: 14769: 14768: 14767: 14758:Formal sciences 14743: 14742: 14741: 14731: 14726: 14724: 14714: 14709: 14707: 14697: 14692: 14690: 14686:from Wikisource 14680: 14675: 14673: 14663: 14658: 14656: 14646: 14641: 14639: 14629: 14624: 14622: 14618:from Wiktionary 14612: 14607: 14605: 14602: 14598:sister projects 14595:at Knowledge's 14589: 14577: 14567: 14565: 14555: 14553: 14541: 14533: 14531: 14526: 14477: 14468: 14418: 14375: 14354:Systems science 14285: 14281:Homotopy theory 14247: 14214: 14166: 14138: 14085: 14032: 14003:Category theory 13989: 13954: 13947: 13917: 13911: 13889: 13869: 13863: 13847: 13841: 13801: 13795: 13775: 13769: 13745: 13734: 13712: 13706: 13689: 13685: 13684: 13683: 13663: 13662: 13658: 13651: 13649:Further reading 13646: 13636: 13603: 13601: 13597: 13570: 13556: 13554: 13550: 13535: 13524: 13498:Popper, Karl R. 13496: 13474: 13451: 13449: 13415:10.2307/2369153 13389: 13380: 13362:Oakley, Barbara 13360: 13351: 13349: 13345: 13330: 13319: 13290: 13288: 13262: 13238:Kleiner, Israel 13230: 13214: 13208: 13178: 13170:. 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May 6, 2020. 13032: 13031: 13024: 13014: 13012: 13005:Wolf Foundation 12999: 12998: 12994: 12984: 12982: 12975: 12951: 12947: 12937: 12935: 12926: 12925: 12921: 12911: 12909: 12900: 12899: 12895: 12885: 12883: 12879: 12868: 12864: 12863: 12859: 12849: 12847: 12834: 12833: 12829: 12819: 12817: 12808: 12807: 12803: 12793: 12791: 12778: 12777: 12773: 12763: 12761: 12748: 12747: 12740: 12730: 12728: 12715: 12714: 12710: 12702: 12698: 12690: 12686: 12676: 12674: 12672: 12656: 12652: 12642: 12640: 12638: 12622: 12618: 12608: 12606: 12593: 12589: 12579: 12577: 12564: 12563: 12559: 12549: 12547: 12532: 12528: 12515: 12514: 12510: 12503: 12482: 12478: 12471: 12449: 12445: 12437: 12431: 12427: 12417: 12415: 12356: 12352: 12329:10.2307/3613237 12313: 12309: 12270: 12266: 12257: 12251: 12247: 12218: 12214: 12197: 12195: 12193: 12174: 12170: 12160: 12158: 12156: 12140: 12136: 12126: 12124: 12120: 12087: 12081: 12077: 12048: 12044: 12015: 12011: 11964: 11960: 11950: 11948: 11944: 11921: 11915: 11911: 11901: 11899: 11895: 11889: 11862: 11853: 11849: 11812: 11808: 11793:10.2307/2314867 11777: 11773: 11763: 11761: 11759: 11740: 11732: 11728: 11718: 11716: 11712: 11705: 11697: 11693: 11654: 11650: 11643: 11617: 11613: 11603: 11601: 11599: 11577:Robson, Eleanor 11574: 11570: 11560: 11558: 11556: 11540: 11536: 11505: 11501: 11486: 11482: 11470: 11466: 11456: 11454: 11450: 11419: 11413: 11409: 11399: 11397: 11382: 11378: 11368: 11366: 11343:10.2307/3618304 11320: 11316: 11304: 11300: 11290: 11288: 11286: 11270:Cajori, Florian 11267: 11263: 11253: 11251: 11249: 11233:Franklin, James 11230: 11226: 11219: 11194: 11190: 11172: 11170: 11168: 11149: 11142: 11103: 11094: 11084: 11082: 11078: 11072: 11057: 11051: 11047: 11005: 11001: 10991: 10989: 10974: 10970: 10957: 10956: 10952: 10943: 10942: 10938: 10923: 10900: 10896: 10886: 10884: 10871: 10870: 10866: 10853: 10852: 10848: 10838: 10836: 10823: 10822: 10818: 10805: 10804: 10800: 10792: 10784: 10780: 10772: 10764: 10760: 10750: 10748: 10746: 10730: 10723: 10713: 10711: 10709: 10691: 10687: 10652: 10648: 10638: 10636: 10627: 10623: 10622: 10618: 10608: 10606: 10596: 10592: 10582: 10580: 10574:"MAM2001 Essay" 10572: 10571: 10567: 10557: 10555: 10547: 10546: 10542: 10527: 10500: 10496: 10480: 10474: 10470: 10460: 10458: 10445: 10444: 10440: 10432: 10428: 10420: 10416: 10407: 10403: 10393: 10391: 10387: 10368: 10359: 10350: 10340: 10338: 10284: 10280: 10270: 10268: 10267:. April 8, 2024 10259: 10258: 10254: 10244: 10242: 10234: 10233: 10229: 10219: 10217: 10209: 10208: 10204: 10194: 10192: 10188: 10181: 10173:Atiyah, Michael 10170: 10166: 10156: 10154: 10152: 10136: 10132: 10101: 10097: 10063: 10059: 10019: 10010: 9989:(11): 389–507. 9979: 9975: 9965: 9963: 9948: 9944: 9934: 9932: 9919: 9918: 9914: 9904: 9902: 9898: 9887: 9881: 9877: 9858:Current Science 9854: 9850: 9840: 9838: 9836: 9810: 9806: 9769: 9765: 9755: 9753: 9737: 9733: 9723: 9721: 9702: 9698: 9688: 9686: 9671: 9667: 9657: 9655: 9651: 9636: 9628: 9624: 9593: 9589: 9579: 9577: 9547: 9543: 9533: 9531: 9491: 9487: 9477: 9475: 9473: 9455: 9451: 9441: 9439: 9435: 9404: 9395: 9391: 9368: 9354: 9350: 9340: 9338: 9336: 9318: 9314: 9283:Kuhn, Thomas S. 9280: 9276: 9269: 9247: 9240: 9230: 9228: 9208: 9204: 9197: 9183: 9179: 9172: 9155: 9151: 9141: 9139: 9137: 9119: 9115: 9105: 9103: 9101: 9085: 9081: 9071: 9069: 9056: 9055: 9051: 9041: 9039: 9037: 9008: 9004: 8994: 8992: 8970: 8966: 8956: 8954: 8924: 8920: 8910: 8908: 8906: 8890: 8886: 8821: 8817: 8807: 8805: 8788: 8787: 8783: 8760: 8742: 8738: 8728: 8726: 8722: 8687: 8681: 8677: 8667: 8665: 8661: 8650: 8644: 8640: 8591: 8587: 8577: 8575: 8559: 8555: 8547: 8543: 8528:10.2307/2304570 8509: 8505: 8498: 8487: 8479: 8475: 8465: 8463: 8459: 8428: 8422: 8418: 8408: 8406: 8387: 8383: 8368: 8351: 8347: 8332: 8310: 8306: 8267: 8263: 8253: 8251: 8249: 8230: 8226: 8218: 8214: 8206: 8202: 8194: 8190: 8182: 8178: 8170: 8166: 8158: 8154: 8115: 8111: 8104: 8083: 8079: 8071: 8067: 8059: 8055: 8047: 8043: 8028: 8011: 8007: 7993: 7989: 7979: 7977: 7973: 7943: 7928: 7912:Gowers, Timothy 7905: 7901: 7891: 7889: 7873: 7869: 7859: 7857: 7842: 7838: 7830: 7826: 7795: 7774: 7770: 7739: 7722: 7718: 7708: 7706: 7702: 7695: 7687: 7683: 7673: 7671: 7667: 7660: 7634:Halpern, Joseph 7631: 7627: 7602: 7598: 7588: 7586: 7568: 7564: 7554: 7552: 7509: 7502: 7479:10.2307/2689412 7460: 7456: 7446: 7444: 7442: 7426: 7422: 7412: 7410: 7401: 7392: 7388: 7365: 7331: 7327: 7317: 7315: 7306: 7300: 7296: 7286: 7284: 7281:Quanta Magazine 7269: 7265: 7255: 7253: 7195:10.2307/2687794 7178: 7172: 7168: 7158: 7156: 7120: 7116: 7106: 7104: 7068: 7064: 7037:Sipser, Michael 7034: 7030: 7020: 7018: 6992: 6970:Turing's Legacy 6955: 6951: 6941: 6939: 6905: 6871: 6867: 6857: 6855: 6799:Franklin, James 6796: 6792: 6782: 6780: 6770: 6769: 6765: 6755: 6753: 6734: 6730: 6720: 6718: 6714: 6700: 6673: 6664: 6660: 6650: 6648: 6630: 6610: 6606: 6596: 6594: 6584: 6568: 6564: 6554: 6552: 6534: 6515: 6511: 6503: 6499: 6489: 6487: 6478: 6477: 6473: 6463: 6461: 6457: 6392: 6386: 6382: 6372: 6370: 6334:Berntjes, Sonja 6331: 6327: 6317: 6315: 6296: 6292: 6282: 6280: 6268: 6264: 6250: 6248: 6238: 6232: 6228: 6220: 6216: 6147: 6143: 6120: 6102: 6098: 6088: 6086: 6067: 6063: 6053: 6051: 5990: 5984: 5980: 5957: 5936: 5932: 5922: 5920: 5902: 5879: 5875: 5865: 5858: 5856: 5809: 5805: 5788: 5779: 5748: 5722: 5718: 5649:Kleiner, Israel 5646: 5642: 5611: 5586: 5582: 5551: 5529: 5525: 5494: 5473: 5469: 5459: 5457: 5453: 5442: 5438: 5437: 5418: 5405: 5403: 5399: 5342: 5334:Dunne, Edward; 5332: 5328: 5318: 5316: 5312: 5301: 5295: 5288: 5227: 5223: 5192: 5155: 5151: 5128: 5103: 5099: 5073: 5059: 5055: 5029: 5012: 5008: 4998: 4996: 4977: 4976: 4972: 4962: 4960: 4946: 4945: 4941: 4921: 4917: 4894: 4886:. p. 257. 4873: 4869: 4820: 4816: 4806: 4804: 4787: 4783: 4773: 4771: 4759: 4755: 4745: 4743: 4741:Clarendon Press 4720: 4716: 4706: 4704: 4691: 4687: 4677: 4675: 4658: 4651: 4572:Kleiner, Israel 4569: 4554: 4499:Alexander, Amir 4496: 4492: 4482: 4480: 4463: 4459: 4407: 4400: 4392: 4388: 4382:Wayback Machine 4370: 4368:at ResearchGate 4359: 4357: 4353: 4330: 4317:Mitterer, Josef 4313: 4309: 4305: 4300: 4299: 4294: 4290: 4285: 4281: 4276: 4272: 4259: 4255: 4242: 4238: 4225: 4221: 4208: 4204: 4191: 4187: 4182: 4177: 4172: 4107: 4100: 4097: 3993: 3989:Main category: 3987: 3974: 3968: 3960:self-similarity 3930: 3922: 3917: 3915: 3912: 3911: 3868: 3866: 3863: 3862: 3851: 3845: 3840: 3838:Cultural impact 3830:(as in art) or 3754:Apery's theorem 3746: 3690: 3686: 3672: 1650 BCE 3671: 3597: 3591: 3586: 3535:inference rules 3523: 3517: 3481: 3475: 3446:Charles Hermite 3419: 3414: 3408: 3400:complex systems 3352:Homo economicus 3335: 3332: 3329: 3326: 3319:Homo economicus 3288: 3278: 3276:Social sciences 3255: 3249: 3241:clinical trials 3192: 3184:Main articles: 3182: 3168:. In 1998, the 3138: 3128: 3111: 3105: 3100: 3071: 3067: 3065: 3062: 3061: 3032:Albert Einstein 3017:Johannes Kepler 2974: 2944:introduced the 2847: 2846: 2845: 2844: 2836: 2835: 2834: 2826: 2825: 2814: 2806:Main articles: 2804: 2792:experimentation 2744: 2649: 2641: 2621: 2618: 2617: 2609: 2601: 2538: 2526:Main articles: 2524: 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: 14776: 14766: 14765: 14760: 14755: 14740: 14739: 14722: 14705: 14703:from Wikibooks 14688: 14671: 14669:from Wikiquote 14654: 14637: 14620: 14591: 14588: 14587: 14575: 14563: 14551: 14528: 14527: 14525: 14524: 14512: 14500: 14488: 14473: 14470: 14469: 14467: 14466: 14461: 14456: 14451: 14446: 14441: 14440: 14439: 14432:Mathematicians 14428: 14426: 14424:Related topics 14420: 14419: 14417: 14416: 14411: 14406: 14401: 14396: 14391: 14385: 14383: 14377: 14376: 14374: 14373: 14372: 14371: 14366: 14361: 14359:Control theory 14351: 14346: 14341: 14336: 14331: 14326: 14321: 14316: 14311: 14306: 14301: 14295: 14293: 14287: 14286: 14284: 14283: 14278: 14273: 14268: 14263: 14257: 14255: 14249: 14248: 14246: 14245: 14240: 14235: 14230: 14224: 14222: 14216: 14215: 14213: 14212: 14207: 14202: 14197: 14192: 14187: 14182: 14176: 14174: 14168: 14167: 14165: 14164: 14159: 14154: 14148: 14146: 14140: 14139: 14137: 14136: 14134:Measure theory 14131: 14126: 14121: 14116: 14111: 14106: 14101: 14095: 14093: 14087: 14086: 14084: 14083: 14078: 14073: 14068: 14063: 14058: 14053: 14048: 14042: 14040: 14034: 14033: 14031: 14030: 14025: 14020: 14015: 14010: 14005: 13999: 13997: 13991: 13990: 13988: 13987: 13982: 13977: 13976: 13975: 13970: 13959: 13956: 13955: 13946: 13945: 13938: 13931: 13923: 13916: 13915: 13909: 13893: 13887: 13871:Pappas, Theoni 13867: 13861: 13845: 13839: 13826: 13805:, ed. (2000). 13799: 13793: 13773: 13767: 13743: 13732: 13710: 13704: 13686: 13682: 13681: 13676: 13671: 13665: 13664: 13653: 13652: 13650: 13647: 13645: 13644: 13634: 13614:Whittle, Peter 13610: 13583:(1): 101–109. 13568:"Book Reviews" 13563: 13546:(7): 778–782. 13528: 13522: 13494: 13472: 13459: 13387: 13378: 13358: 13323: 13317: 13297: 13260: 13234: 13228: 13212: 13206: 13182: 13176: 13159: 13157: 13154: 13152: 13151: 13125: 13099: 13085: 13055: 13022: 12992: 12973: 12945: 12919: 12893: 12857: 12827: 12801: 12771: 12750:"Fields Medal" 12738: 12708: 12696: 12684: 12670: 12650: 12636: 12616: 12587: 12557: 12526: 12508: 12501: 12476: 12469: 12443: 12425: 12350: 12307: 12280:(2): 197–220. 12264: 12245: 12232:(3): 226–247. 12212: 12191: 12168: 12154: 12134: 12100:(4): 195–203. 12075: 12042: 12009: 11958: 11909: 11887: 11847: 11826:(4): 457–459. 11806: 11771: 11757: 11726: 11691: 11648: 11641: 11611: 11597: 11591:. OUP Oxford. 11568: 11554: 11534: 11499: 11480: 11464: 11430:(2): 409–449. 11407: 11390:Acta Academica 11376: 11314: 11298: 11284: 11261: 11247: 11224: 11217: 11188: 11166: 11152:Tobies, Renate 11140: 11092: 11070: 11045: 11024:10.1086/286957 11018:(4): 289–303. 10999: 10968: 10950: 10936: 10921: 10894: 10864: 10846: 10816: 10798: 10795:. Vol. 2. 10778: 10758: 10744: 10721: 10707: 10685: 10666:(1): 197–220. 10646: 10616: 10590: 10578:ww2.amstat.org 10565: 10540: 10525: 10517:10.17226/10633 10494: 10468: 10438: 10436:, p. 285. 10426: 10414: 10401: 10348: 10278: 10252: 10227: 10202: 10164: 10150: 10130: 10095: 10057: 10008: 9973: 9942: 9912: 9875: 9864:(3): 415–423. 9848: 9834: 9804: 9763: 9731: 9716:(7): 753–759. 9696: 9665: 9622: 9609:(3): 267–288. 9587: 9541: 9508:(3): 405–427. 9485: 9471: 9449: 9389: 9366: 9348: 9334: 9312: 9299:10.2307/202372 9274: 9267: 9238: 9220:Philosophy Now 9202: 9195: 9177: 9170: 9149: 9135: 9113: 9099: 9079: 9049: 9035: 9002: 8964: 8918: 8904: 8884: 8815: 8781: 8758: 8736: 8675: 8657:. p. 75. 8638: 8609:(3): 322–354. 8585: 8553: 8541: 8503: 8496: 8473: 8416: 8381: 8366: 8354:Saliba, George 8345: 8330: 8304: 8283:10.1086/368443 8261: 8247: 8224: 8212: 8200: 8188: 8176: 8164: 8152: 8125:(4): 289–309. 8109: 8102: 8077: 8065: 8053: 8041: 8026: 8005: 7987: 7941: 7899: 7867: 7836: 7824: 7793: 7781:Dodge, Yadolah 7768: 7737: 7716: 7698:. p. 10. 7681: 7642:Immerman, Neil 7638:Harper, Robert 7625: 7596: 7562: 7523:(4): 513–534. 7500: 7473:(4): 207–216. 7463:Snapper, Ernst 7454: 7440: 7420: 7395:Avigad, Jeremy 7386: 7363: 7325: 7294: 7263: 7189:(4): 441–484. 7166: 7114: 7062: 7028: 6990: 6949: 6903: 6865: 6813:(2): 355–378. 6790: 6763: 6728: 6698: 6658: 6628: 6604: 6582: 6562: 6532: 6509: 6497: 6471: 6405:(3): 245–302. 6380: 6325: 6290: 6272:"Al-Khwarizmi" 6262: 6226: 6214: 6161:(2): 127–163. 6141: 6118: 6096: 6061: 6003:(3): 383–417. 5978: 5955: 5939:Boyer, Carl B. 5930: 5900: 5873: 5812:Hilbert, David 5803: 5777: 5746: 5716: 5651:(March 2000). 5640: 5609: 5580: 5549: 5523: 5492: 5467: 5416: 5353:(3): 410–411. 5338:(March 2020). 5326: 5297:Warner, Evan. 5286: 5241:(2): 109–136. 5221: 5190: 5158:Musielak, Dora 5149: 5126: 5120:. p. 14. 5097: 5071: 5053: 5027: 5006: 4979:"Math (Noun³)" 4970: 4948:"Maths (Noun)" 4939: 4915: 4892: 4876:Boas, Ralph P. 4867: 4814: 4781: 4753: 4714: 4685: 4649: 4552: 4515:10.1086/661620 4509:(3): 475–480. 4490: 4457: 4410:Wigner, Eugene 4398: 4386: 4306: 4304: 4301: 4298: 4297: 4288: 4279: 4270: 4253: 4236: 4228:conic sections 4226:This includes 4219: 4202: 4184: 4183: 4181: 4178: 4176: 4173: 4171: 4170: 4165: 4160: 4155: 4150: 4145: 4140: 4135: 4130: 4125: 4120: 4114: 4113: 4112: 4096: 4093: 4062: 4061: 4054: 4044: 4037: 3986: 3983: 3970:Main article: 3967: 3966:Popularization 3964: 3947:Rorschach test 3933: 3929: 3925: 3920: 3904:symmetry group 3875: 3872: 3847:Main article: 3844: 3841: 3839: 3836: 3745: 3742: 3708:Following the 3695:Imperial China 3691: 500 BCE 3676:oral tradition 3617:popularization 3601:human activity 3593:Main article: 3590: 3587: 3585: 3582: 3516: 3513: 3477:Main article: 3474: 3471: 3460: 3450:Henri Poincaré 3418: 3415: 3410:Main article: 3407: 3404: 3277: 3274: 3253:Geomathematics 3251:Main article: 3248: 3247:Earth sciences 3245: 3204:Turing pattern 3181: 3178: 3174:sphere packing 3127: 3124: 3107:Main article: 3104: 3101: 3099: 3096: 3079: 3074: 3070: 3034:developed the 3009:conic sections 2973: 2970: 2942:George Collins 2838: 2837: 2828: 2827: 2819: 2818: 2817: 2816: 2815: 2803: 2800: 2797: 2784:counterexample 2743: 2740: 2625: 2614:multiplication 2523: 2520: 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: 9: 6: 4: 3: 2: 14775: 14764: 14761: 14759: 14756: 14754: 14751: 14750: 14748: 14737:from Wikidata 14736: 14735: 14723: 14719: 14718: 14706: 14702: 14701: 14689: 14685: 14684: 14672: 14668: 14667: 14655: 14652:from Wikinews 14651: 14650: 14638: 14634: 14633: 14621: 14617: 14616: 14604: 14603: 14600: 14594: 14586: 14581: 14576: 14574: 14564: 14562: 14552: 14550: 14545: 14540: 14539: 14536: 14523: 14522: 14513: 14511: 14510: 14501: 14499: 14498: 14489: 14487: 14486: 14481: 14475: 14474: 14471: 14465: 14462: 14460: 14457: 14455: 14452: 14450: 14447: 14445: 14442: 14438: 14435: 14434: 14433: 14430: 14429: 14427: 14425: 14421: 14415: 14412: 14410: 14407: 14405: 14402: 14400: 14397: 14395: 14392: 14390: 14387: 14386: 14384: 14382: 14381:Computational 14378: 14370: 14367: 14365: 14362: 14360: 14357: 14356: 14355: 14352: 14350: 14347: 14345: 14342: 14340: 14337: 14335: 14332: 14330: 14327: 14325: 14322: 14320: 14317: 14315: 14312: 14310: 14307: 14305: 14302: 14300: 14297: 14296: 14294: 14292: 14288: 14282: 14279: 14277: 14274: 14272: 14269: 14267: 14264: 14262: 14259: 14258: 14256: 14254: 14250: 14244: 14241: 14239: 14236: 14234: 14231: 14229: 14226: 14225: 14223: 14221: 14220:Number theory 14217: 14211: 14208: 14206: 14203: 14201: 14198: 14196: 14193: 14191: 14188: 14186: 14183: 14181: 14178: 14177: 14175: 14173: 14169: 14163: 14160: 14158: 14155: 14153: 14152:Combinatorics 14150: 14149: 14147: 14145: 14141: 14135: 14132: 14130: 14127: 14125: 14122: 14120: 14117: 14115: 14112: 14110: 14107: 14105: 14104:Real analysis 14102: 14100: 14097: 14096: 14094: 14092: 14088: 14082: 14079: 14077: 14074: 14072: 14069: 14067: 14064: 14062: 14059: 14057: 14054: 14052: 14049: 14047: 14044: 14043: 14041: 14039: 14035: 14029: 14026: 14024: 14021: 14019: 14016: 14014: 14011: 14009: 14006: 14004: 14001: 14000: 13998: 13996: 13992: 13986: 13983: 13981: 13978: 13974: 13971: 13969: 13966: 13965: 13964: 13961: 13960: 13957: 13952: 13944: 13939: 13937: 13932: 13930: 13925: 13924: 13921: 13912: 13906: 13902: 13898: 13894: 13890: 13884: 13879: 13878: 13872: 13868: 13864: 13858: 13854: 13850: 13846: 13842: 13836: 13832: 13827: 13824: 13823:archive.today 13820: 13816: 13810: 13809: 13804: 13800: 13796: 13790: 13785: 13784: 13778: 13777:Gullberg, Jan 13774: 13770: 13764: 13759: 13758: 13752: 13748: 13744: 13741: 13735: 13729: 13725: 13724: 13719: 13718:Hersh, Reuben 13715: 13711: 13707: 13701: 13696: 13695: 13688: 13687: 13680: 13677: 13675: 13672: 13670: 13667: 13666: 13661: 13656: 13641: 13637: 13631: 13627: 13623: 13619: 13618:"Almost home" 13615: 13611: 13596: 13591: 13586: 13582: 13578: 13577: 13569: 13564: 13549: 13545: 13541: 13534: 13529: 13525: 13519: 13515: 13511: 13506: 13505: 13499: 13495: 13491: 13487: 13483: 13479: 13475: 13473:0-7167-1953-3 13469: 13465: 13460: 13448: 13445: 13441: 13438: 13434: 13430: 13425: 13420: 13416: 13412: 13408: 13404: 13400: 13396: 13392: 13388: 13385: 13381: 13375: 13370: 13369: 13363: 13359: 13344: 13340: 13336: 13329: 13324: 13320: 13314: 13309: 13308: 13302: 13301:Kline, Morris 13298: 13287: 13283: 13279: 13275: 13271: 13267: 13263: 13257: 13253: 13249: 13245: 13244: 13239: 13235: 13231: 13225: 13221: 13217: 13213: 13209: 13203: 13199: 13194: 13193: 13187: 13183: 13179: 13173: 13169: 13165: 13161: 13160: 13139: 13135: 13129: 13113: 13109: 13103: 13088: 13082: 13078: 13077: 13069: 13065: 13059: 13043: 13039: 13035: 13029: 13027: 13010: 13006: 13002: 12996: 12980: 12976: 12970: 12966: 12962: 12958: 12957: 12949: 12933: 12929: 12923: 12907: 12903: 12897: 12878: 12874: 12867: 12861: 12845: 12841: 12837: 12831: 12815: 12811: 12805: 12789: 12785: 12781: 12775: 12759: 12755: 12754:Maths History 12751: 12745: 12743: 12726: 12722: 12718: 12712: 12705: 12700: 12693: 12688: 12673: 12667: 12663: 12662: 12654: 12639: 12633: 12629: 12628: 12620: 12604: 12600: 12599: 12591: 12575: 12571: 12567: 12561: 12545: 12541: 12540:www.stsci.edu 12537: 12530: 12522: 12518: 12512: 12504: 12498: 12494: 12490: 12486: 12485:Weyl, Hermann 12480: 12472: 12466: 12462: 12458: 12454: 12447: 12436: 12429: 12413: 12409: 12405: 12401: 12397: 12393: 12389: 12385: 12381: 12377: 12373: 12369: 12365: 12361: 12354: 12346: 12342: 12338: 12334: 12330: 12326: 12322: 12318: 12311: 12303: 12299: 12295: 12291: 12287: 12283: 12279: 12275: 12268: 12260: 12255: 12249: 12240: 12235: 12231: 12227: 12223: 12216: 12209: 12208: 12194: 12188: 12184: 12183: 12178: 12172: 12157: 12151: 12147: 12146: 12138: 12119: 12115: 12111: 12107: 12103: 12099: 12095: 12094: 12086: 12079: 12070: 12065: 12061: 12057: 12053: 12046: 12037: 12032: 12029:: 2519–2525. 12028: 12024: 12020: 12013: 12005: 12001: 11996: 11991: 11986: 11981: 11977: 11973: 11969: 11962: 11943: 11939: 11935: 11931: 11927: 11920: 11913: 11894: 11890: 11884: 11880: 11876: 11872: 11868: 11861: 11860: 11851: 11843: 11839: 11834: 11829: 11825: 11821: 11817: 11810: 11802: 11798: 11794: 11790: 11786: 11782: 11775: 11760: 11754: 11750: 11746: 11739: 11738: 11730: 11711: 11704: 11703: 11695: 11687: 11683: 11679: 11675: 11671: 11667: 11663: 11659: 11658:Math Horizons 11652: 11644: 11638: 11634: 11630: 11626: 11622: 11615: 11600: 11594: 11590: 11586: 11582: 11578: 11572: 11557: 11551: 11547: 11546: 11538: 11530: 11526: 11522: 11518: 11514: 11510: 11503: 11495: 11491: 11484: 11477: 11473: 11472:Peterson 1988 11468: 11449: 11445: 11441: 11437: 11433: 11429: 11425: 11418: 11411: 11395: 11391: 11387: 11380: 11364: 11360: 11356: 11352: 11348: 11344: 11340: 11336: 11332: 11328: 11324: 11323:Brown, Ronald 11318: 11309: 11302: 11287: 11281: 11277: 11276: 11271: 11265: 11250: 11244: 11240: 11239: 11234: 11228: 11220: 11214: 11210: 11206: 11202: 11198: 11192: 11185: 11182: 11169: 11163: 11159: 11158: 11153: 11147: 11145: 11136: 11132: 11128: 11124: 11120: 11116: 11113:(4): 375–85. 11112: 11108: 11101: 11099: 11097: 11077: 11073: 11067: 11063: 11056: 11049: 11041: 11033: 11029: 11025: 11021: 11017: 11013: 11012: 11003: 10987: 10983: 10979: 10972: 10964: 10960: 10954: 10946: 10940: 10932: 10928: 10924: 10918: 10914: 10913: 10908: 10907:Jean Bricmont 10904: 10898: 10882: 10878: 10874: 10868: 10860: 10856: 10850: 10834: 10830: 10826: 10820: 10812: 10808: 10807:"Mathematics" 10802: 10791: 10790: 10782: 10771: 10770: 10762: 10747: 10741: 10737: 10736: 10728: 10726: 10714:September 30, 10710: 10704: 10700: 10696: 10689: 10681: 10677: 10673: 10669: 10665: 10661: 10657: 10650: 10635: 10634: 10626: 10620: 10605: 10601: 10594: 10579: 10575: 10569: 10554: 10550: 10544: 10536: 10532: 10528: 10522: 10518: 10514: 10510: 10509: 10505:(2003). "8". 10504: 10498: 10490: 10486: 10479: 10472: 10456: 10452: 10448: 10442: 10435: 10430: 10423: 10418: 10411: 10405: 10386: 10382: 10378: 10374: 10367: 10363: 10357: 10355: 10353: 10336: 10332: 10328: 10324: 10320: 10315: 10310: 10306: 10302: 10298: 10294: 10290: 10282: 10266: 10265:GeeksforGeeks 10262: 10256: 10241: 10237: 10231: 10216: 10212: 10206: 10187: 10180: 10179: 10174: 10168: 10153: 10147: 10143: 10142: 10134: 10126: 10122: 10118: 10114: 10110: 10106: 10099: 10091: 10087: 10083: 10079: 10075: 10071: 10067: 10061: 10053: 10049: 10044: 10039: 10035: 10031: 10027: 10023: 10022:Borel, Armand 10017: 10015: 10013: 10004: 10000: 9996: 9992: 9988: 9984: 9977: 9961: 9957: 9953: 9946: 9930: 9926: 9922: 9916: 9897: 9893: 9886: 9879: 9871: 9867: 9863: 9859: 9852: 9837: 9831: 9827: 9823: 9819: 9815: 9808: 9800: 9796: 9791: 9786: 9782: 9778: 9774: 9767: 9752: 9748: 9744: 9743: 9735: 9719: 9715: 9711: 9707: 9700: 9684: 9680: 9676: 9669: 9650: 9646: 9642: 9635: 9634: 9626: 9617: 9612: 9608: 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7734: 7730: 7726: 7720: 7701: 7694: 7693: 7685: 7666: 7659: 7655: 7654:Vianu, Victor 7651: 7647: 7643: 7639: 7635: 7629: 7620: 7615: 7611: 7607: 7600: 7584: 7580: 7576: 7572: 7566: 7550: 7546: 7542: 7538: 7534: 7530: 7526: 7522: 7518: 7514: 7507: 7505: 7496: 7492: 7488: 7484: 7480: 7476: 7472: 7468: 7464: 7458: 7443: 7437: 7433: 7432: 7424: 7409: 7408: 7400: 7396: 7390: 7382: 7378: 7374: 7370: 7366: 7360: 7356: 7352: 7348: 7344: 7340: 7336: 7329: 7314: 7313: 7305: 7298: 7283: 7282: 7277: 7273: 7267: 7252: 7248: 7244: 7240: 7236: 7232: 7228: 7224: 7220: 7216: 7212: 7208: 7204: 7200: 7196: 7192: 7188: 7184: 7177: 7170: 7155: 7151: 7147: 7143: 7139: 7135: 7131: 7130: 7125: 7118: 7103: 7099: 7095: 7091: 7087: 7083: 7079: 7078: 7073: 7066: 7058: 7054: 7050: 7046: 7042: 7039:(July 1992). 7038: 7032: 7017: 7013: 7009: 7005: 7001: 6997: 6993: 6987: 6983: 6979: 6975: 6971: 6967: 6963: 6959: 6953: 6938: 6934: 6930: 6926: 6922: 6918: 6914: 6910: 6906: 6904:0-8218-0448-0 6900: 6896: 6892: 6888: 6884: 6880: 6876: 6869: 6854: 6850: 6846: 6842: 6838: 6834: 6830: 6826: 6821: 6816: 6812: 6808: 6804: 6801:(July 2017). 6800: 6794: 6779: 6778: 6773: 6767: 6751: 6747: 6743: 6739: 6732: 6713: 6709: 6705: 6701: 6695: 6691: 6687: 6683: 6679: 6672: 6668: 6662: 6647: 6643: 6639: 6635: 6631: 6625: 6621: 6617: 6616: 6608: 6593: 6589: 6585: 6579: 6575: 6574: 6566: 6551: 6547: 6543: 6539: 6535: 6533:3-7643-7002-5 6529: 6525: 6524: 6519: 6513: 6506: 6501: 6485: 6484:GeeksforGeeks 6481: 6475: 6456: 6452: 6448: 6444: 6440: 6436: 6432: 6428: 6424: 6420: 6416: 6412: 6408: 6404: 6400: 6399: 6391: 6384: 6369: 6365: 6361: 6357: 6353: 6349: 6345: 6344: 6339: 6335: 6329: 6313: 6309: 6308: 6302: 6294: 6279: 6278: 6273: 6266: 6259: 6258:progressions. 6246: 6245: 6237: 6230: 6223: 6218: 6210: 6206: 6202: 6198: 6194: 6190: 6186: 6182: 6178: 6174: 6169: 6164: 6160: 6156: 6152: 6145: 6137: 6133: 6129: 6125: 6121: 6115: 6111: 6107: 6100: 6084: 6080: 6076: 6072: 6065: 6050: 6046: 6042: 6038: 6034: 6030: 6026: 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4528: 4524: 4520: 4516: 4512: 4508: 4504: 4500: 4494: 4478: 4474: 4473: 4468: 4465:Wise, David. 4461: 4453: 4449: 4445: 4441: 4437: 4433: 4429: 4425: 4421: 4420: 4415: 4411: 4405: 4403: 4396:, p. 12. 4395: 4394:Peterson 1988 4390: 4383: 4379: 4374: 4369: 4352: 4348: 4344: 4340: 4336: 4328: 4324: 4323: 4318: 4311: 4307: 4292: 4283: 4274: 4267: 4263: 4257: 4250: 4246: 4240: 4233: 4229: 4223: 4216: 4212: 4206: 4199: 4195: 4189: 4185: 4169: 4166: 4164: 4161: 4159: 4156: 4154: 4151: 4149: 4146: 4144: 4141: 4139: 4136: 4134: 4131: 4129: 4126: 4124: 4121: 4119: 4116: 4115: 4110: 4104: 4099: 4092: 4090: 4086: 4082: 4078: 4073: 4071: 4067: 4066:open problems 4059: 4055: 4052: 4049: 4045: 4042: 4038: 4035: 4031: 4030: 4029: 4026: 4024: 4020: 4016: 4009: 4006: 4002: 3997: 3992: 3982: 3979: 3973: 3963: 3961: 3957: 3952: 3948: 3927: 3923: 3909: 3905: 3897: 3893: 3889: 3873: 3870: 3860: 3859:perfect fifth 3856: 3850: 3835: 3833: 3829: 3825: 3820: 3818: 3814: 3810: 3809: 3804: 3799: 3798: 3793: 3789: 3785: 3781: 3777: 3772: 3770: 3766: 3761: 3759: 3755: 3751: 3741: 3738: 3732: 3730: 3729: 3723: 3719: 3715: 3711: 3706: 3704: 3700: 3696: 3681: 3677: 3667:, dated from 3666: 3665:Rhind papyrus 3662: 3658: 3652: 3650: 3646: 3642: 3638: 3634: 3630: 3626: 3622: 3618: 3614: 3610: 3606: 3602: 3596: 3581: 3578: 3576: 3571: 3567: 3562: 3558: 3556: 3552: 3548: 3544: 3540: 3536: 3532: 3528: 3522: 3512: 3507: 3502: 3500: 3499: 3494: 3488: 3486: 3480: 3470: 3468: 3462: 3458: 3453: 3451: 3447: 3443: 3439: 3435: 3433: 3428: 3424: 3413: 3403: 3401: 3397: 3393: 3389: 3388: 3382: 3380: 3376: 3375:Peter Turchin 3372: 3368: 3364: 3359: 3355: 3353: 3349: 3345: 3341: 3340:self-interest 3321: 3320: 3311: 3307: 3303: 3301: 3297: 3293: 3287: 3283: 3273: 3271: 3267: 3263: 3259: 3254: 3244: 3242: 3238: 3234: 3232: 3228: 3224: 3220: 3216: 3209: 3205: 3201: 3196: 3191: 3187: 3177: 3175: 3171: 3167: 3163: 3159: 3155: 3151: 3150:coding theory 3147: 3143: 3137: 3133: 3123: 3115: 3110: 3095: 3093: 3077: 3072: 3060: 3056: 3051: 3049: 3045: 3041: 3037: 3033: 3029: 3024: 3022: 3018: 3014: 3010: 3006: 3002: 2998: 2994: 2990: 2985: 2983: 2982:Eugene Wigner 2979: 2969: 2967: 2963: 2959: 2953: 2951: 2947: 2943: 2939: 2935: 2934:Alfred Tarski 2931: 2927: 2923: 2919: 2914: 2911: 2906: 2904: 2900: 2896: 2892: 2888: 2883: 2881: 2877: 2873: 2869: 2865: 2861: 2857: 2852: 2842: 2832: 2823: 2813: 2809: 2799: 2795: 2793: 2789: 2785: 2781: 2777: 2776:philosophical 2772: 2770: 2766: 2762: 2758: 2753: 2749: 2739: 2737: 2733: 2729: 2725: 2721: 2717: 2713: 2712: 2711:homeomorphism 2707: 2706: 2701: 2696: 2694: 2690: 2686: 2682: 2678: 2674: 2670: 2665: 2663: 2659: 2655: 2647: 2639: 2623: 2615: 2607: 2599: 2595: 2591: 2587: 2583: 2579: 2575: 2571: 2567: 2563: 2559: 2555: 2547: 2542: 2537: 2533: 2529: 2519: 2517: 2516: 2511: 2510: 2505: 2500: 2498: 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: 14732: 14715: 14698: 14681: 14664: 14647: 14635:from Commons 14630: 14613: 14592: 14519: 14507: 14495: 14476: 14409:Optimization 14271:Differential 14195:Differential 14162:Order theory 14157:Graph theory 14061:Group theory 13950: 13900: 13876: 13852: 13830: 13807: 13782: 13756: 13722: 13693: 13669:Online books 13659: 13625: 13602:. 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Retrieved 8237: 8233:Ore, Øystein 8227: 8215: 8203: 8191: 8179: 8167: 8155: 8122: 8118: 8112: 8091: 8080: 8068: 8056: 8051:, Chapter 1. 8044: 8017: 8008: 7999: 7990: 7980:February 15, 7978:. Retrieved 7924: 7920:Leader, Imre 7902: 7892:November 13, 7890:. Retrieved 7877: 7870: 7860:November 13, 7858:. Retrieved 7849: 7839: 7832:Whittle 1994 7827: 7784: 7771: 7728: 7719: 7709:February 13, 7707:. Retrieved 7691: 7684: 7672:. Retrieved 7650:Vardi, Moshe 7628: 7609: 7599: 7589:November 12, 7587:. Retrieved 7578: 7565: 7555:November 12, 7553:. Retrieved 7520: 7516: 7470: 7466: 7457: 7447:November 12, 7445:. Retrieved 7430: 7423: 7411:. Retrieved 7405: 7389: 7334: 7328: 7316:. Retrieved 7310: 7297: 7285:. Retrieved 7279: 7266: 7254:. Retrieved 7186: 7182: 7169: 7157:. Retrieved 7127: 7117: 7105:. Retrieved 7075: 7065: 7040: 7031: 7019:. Retrieved 6969: 6952: 6940:. Retrieved 6882: 6868: 6856:. Retrieved 6810: 6806: 6793: 6781:. Retrieved 6775: 6766: 6754:. Retrieved 6741: 6731: 6719:. 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Springer. 11181:mathematics 11085:December 5, 10903:Sokal, Alan 10314:2066/176365 9783:(1): 1–22. 9072:October 27, 8839:: 364–371. 8808:February 3, 8729:February 3, 8668:February 3, 8578:February 3, 8466:December 5, 8277:: 606–628. 8061:Mesopotamia 7674:January 15, 7312:PhilArchive 7211:11441/38373 7021:February 9, 6966:Downey, Rod 6942:February 9, 6858:February 9, 6756:February 9, 6721:February 9, 6651:February 8, 6597:February 8, 6555:February 8, 6464:February 8, 6318:February 8, 6089:February 8, 6054:February 8, 5923:February 7, 5859:February 6, 5589:Weil, André 5460:February 3, 5406:February 3, 5319:February 3, 5015:Bell, E. T. 4999:January 25, 4963:January 25, 4807:January 25, 4774:February 2, 4746:February 2, 4707:February 2, 4678:January 25, 4483:January 18, 4426:(1): 1–14. 4360:January 17, 4247:applied to 4234:and planes. 4041:Chern Medal 4023:Nobel Prize 3855:frequencies 3758:Roger Apery 3720:began with 3697:during the 3687: 1500 3613:recognition 3442:G. H. Hardy 3371:geopolitics 3270:planetology 3262:meteorology 3202:exhibits a 2780:falsifiable 2724:free module 2669:definitions 2578:expressions 2554: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 14747:Categories 14666:Quotations 14561:Arithmetic 14349:Statistics 14228:Arithmetic 14190:Arithmetic 14056:Elementary 14023:Set theory 13738:Available 13557:October 2, 13270:2007932362 12704:Riehm 2002 12677:January 3, 12643:January 3, 11515:: 91–100. 11312:, page 409 10751:January 3, 10157:January 3, 9384:1082226900 9376:2019010556 8768:2006041609 8220:Boyer 1991 8208:Boyer 1991 8196:Boyer 1991 8184:Boyer 1991 8172:Boyer 1991 8160:Boyer 1991 8073:Boyer 1991 8049:Kline 1990 7951:2008020450 7381:1418750041 7146:sn97004494 7094:sn97004494 7000:2014000240 6837:2011202231 6638:2007920230 6542:2004556211 6518:Corry, Leo 6360:2007238847 6128:2008011322 5965:2004056235 5756:2003268597 5385:sf77000404 5208:1332780664 4175:References 4068:, called " 4034:Abel Prize 4008:Archimedes 3832:discovered 3803:Paul Erdős 3788:simplicity 3714:Quadrivium 3689: – c. 3678:since the 3641:accountant 3519:See also: 3423:Pythagoras 3406:Philosophy 3377:developed 3300:psychology 2958:university 2851:technology 2705:polynomial 2702:, such as 2700:neologisms 2677:conjecture 2600:, such as 2590:subscripts 2570:operations 2562: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 14717:Resources 14700:Textbooks 14276:Geometric 14266:Algebraic 14205:Euclidean 14180:Algebraic 14076:Universal 13899:(1965) . 13286:117392219 12392:1476-4687 12345:126119711 12302:220636812 12204:See also 12114:121589323 11842:145507519 11686:126067145 11529:149753721 11444:209980693 11396:(4): 1–28 11359:178923299 11135:122351146 11034:. 189303; 11032:119887253 10680:0360-0572 10331:216912822 10323:2050-5086 10111:: 20–27. 10052:1027-488X 9799:245465895 9561:: 43–57. 9524:249561106 9398:Maddy, P. 9106:March 19, 8938:: 29–48. 8879:125455764 8855:0003-0996 8708:1069-7977 8633:228085700 8625:2363-8761 8453:146539132 8340:190176726 8299:144760421 8139:0007-0882 8088:(1981) . 8036:843204342 8002:. passim. 7967:227205932 7727:(1997) . 7612:: 81–94. 7487:0025-570X 7373:2514-3808 7355:2399-2883 7219:1079-8986 7203:1943-5894 7138:1095-5054 7086:1095-5054 7008:867717052 6913:1052-1798 6845:700943261 6829:2159-8118 6708:975829354 6592:647049731 6451:125704699 6427:0003-9519 6419:1432-0657 6352:1573-3912 6338:"Algebra" 6209:121346342 6185:0315-0860 6177:1090-249X 6136:213765703 6049:117709681 6025:1063-6145 6017:1530-9274 5854:238499430 5797:1409.1140 5687:0013-6018 5679:1420-8962 5635:117789303 5575:118934517 5518:118560854 5377:0002-9920 5369:1088-9477 5265:0315-0860 5257:1090-249X 5216:253240718 5200:2730-7549 5182:2730-7557 4910:633018890 4838:0031-952X 4604:0025-570X 4596:1930-0980 4523:0021-1753 4347:236026294 4339:1022-3398 4303:Citations 4262:computers 3951:butterfly 3776:aesthetic 3710:Dark Ages 3637:economist 3605:education 3589:Education 3570:apodictic 3543:concision 3539:intuition 3432:Platonism 3296:sociology 3292:economics 3126:Computing 3073:− 3069:Ω 3040:spacetime 3028:manifolds 2938:implement 2730:" and "a 2693:corollary 2673:postulate 2654:less than 2624:∫ 2574:relations 2546: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 14497:Category 14253:Topology 14200:Discrete 14185:Analytic 14172:Geometry 14144:Discrete 14099:Calculus 14091:Analysis 14046:Abstract 13985:Glossary 13968:Timeline 13873:(1986). 13819:Archived 13779:(1997). 13753:(1996). 13720:(1999). 13640:Archived 13616:(1994). 13604:June 24, 13595:Archived 13548:Archived 13490:17202382 13482:87033078 13393:(1881). 13364:(2014). 13352:July 28, 13343:Archived 13303:(1990). 13278:76935733 13218:(1990). 13188:(1991). 13166:(1999). 13138:Archived 13112:Archived 13066:(1998). 13042:Archived 13009:Archived 12979:Archived 12932:Archived 12906:Archived 12877:Archived 12844:Archived 12814:Archived 12788:Archived 12758:Archived 12725:Archived 12603:Archived 12574:Archived 12544:Archived 12489:Symmetry 12487:(2015). 12412:Archived 12179:(1940). 12118:Archived 12004:30123014 11942:Archived 11893:Archived 11710:Archived 11678:25678363 11587:(eds.). 11579:(2009). 11448:Archived 11363:Archived 11291:June 20, 11272:(1893). 11254:June 20, 11235:(2009). 11173:June 20, 11076:Archived 11036:also in 10992:April 2, 10986:Archived 10931:39605994 10909:(1998). 10881:Archived 10833:Archived 10535:25032300 10455:Archived 10385:Archived 10335:Archived 10175:(1990). 10024:(1983). 10003:20022840 9960:Archived 9956:The Wire 9929:Archived 9925:MacTutor 9896:Archived 9870:24110208 9718:Archived 9683:Archived 9649:Archived 9575:34271623 9555:Synthese 9528:Archived 9496:(2022). 9433:Archived 9429:18122406 9400:(2008). 9231:April 6, 9225:Archived 9213:(2014). 9142:April 5, 9066:Archived 9013:(2018). 8989:Archived 8863:43020253 8837:Sigma Xi 8802:Archived 8794:MacTutor 8776:64085024 8720:Archived 8716:68713073 8659:Archived 8572:Archived 8457:Archived 8403:Archived 8376:28723059 8356:(1994). 8235:(1988). 8016:(1999). 7971:Archived 7922:(eds.). 7887:91018998 7854:Archived 7850:MacTutor 7803:80021637 7783:(eds.). 7763:36597731 7747:97010349 7700:Archived 7665:Archived 7656:(2001). 7583:Archived 7549:Archived 7545:52083793 7537:23955909 7413:June 14, 7318:June 14, 7287:June 14, 7256:June 14, 7251:43258676 7243:31616719 7235:95652899 7159:June 14, 7154:37550526 7107:June 14, 7102:37550526 7057:11678884 7016:19315498 6960:(2014). 6937:67358543 6929:37141146 6921:97023277 6881:(eds.). 6783:June 13, 6750:Archived 6742:MacTutor 6712:Archived 6669:(2017). 6646:85242858 6550:51234417 6490:June 13, 6455:Archived 6435:63024699 6373:June 13, 6368:56713464 6312:Archived 6283:June 13, 6251:June 13, 6193:75642280 6083:Archived 6075:MacTuror 6041:26085129 6033:94657506 5973:56317813 5918:42290188 5910:99044789 5838:02019303 5814:(1902). 5772:14555830 5764:51533750 5711:53319514 5695:66083524 5619:83011857 5591:(1983). 5567:30437959 5559:94020017 5502:76055645 5451:Archived 5397:Archived 5273:75642280 5160:(2022). 5144:92013695 5136:25709270 5089:28115124 5081:92909575 5037:45010599 4993:Archived 4902:94078313 4854:58015848 4846:24338341 4801:Archived 4729:(1940). 4701:Archived 4672:Archived 4620:47003192 4547:21629993 4539:22073771 4477:Archived 4452:Archived 4412:(1960). 4378:Archived 4351:Archived 4211:Republic 4198:formulas 4095:See also 4005:polymath 3958:possess 3956:Fractals 3792:symmetry 3784:elegance 3718:pedagogy 3575:pleonasm 3396:Bricmont 3390:(1997), 3092:particle 3057:and the 3055:positron 3001:ellipses 2993:internet 2788:theories 2761:Einstein 2752:modeling 2748:sciences 2638:integral 2558:concepts 2548: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 14585:Science 14535:Portals 14509:Commons 14291:Applied 14261:General 14038:Algebra 13963:History 13624:(ed.). 13510:Bibcode 13433:2369153 13397:(ed.). 13156:Sources 12408:4310147 12400:7969448 12372:Bibcode 12337:3613237 12294:3344215 11995:6087017 11801:2314867 11351:3618304 11127:3482762 10639:May 19, 10609:May 19, 10583:May 19, 10558:May 19, 10271:May 19, 10245:June 1, 10220:June 1, 10113:Bibcode 8952:6229200 8871:1480717 8536:2304570 7959:2467561 7819:6707805 7811:0607328 7755:1474730 7495:2689412 7227:2687794 6968:(ed.). 6853:6945363 6443:1482042 6201:2240703 5703:1567783 5627:9576587 5510:3519779 5393:1480366 5281:2240703 5112:(ed.). 4862:1762376 4693:Plato. 4644:7787171 4636:1756877 4628:1141557 4612:2690647 4531:2884913 4448:6112252 4428:Bibcode 4194:algebra 3896:Fractal 3828:created 3765:puzzles 3629:actuary 3609:careers 3459:feeling 3417:Reality 3327: 3215:Biology 3103:Physics 2964:at the 2860:planets 2796:relying 2662:clauses 2648:), and 2594:symbols 2566:symbols 2504: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 14210:Finite 14066:Linear 13973:Future 13949:Major 13907: 13885: 13859: 13837: 13815:online 13791: 13765: 13740:online 13730: 13702: 13657:about 13632: 13520: 13488: 13480: 13470: 13447:Eprint 13444:Google 13440:Eprint 13437:Google 13431: 13376: 13315: 13284: 13276: 13268: 13258: 13226: 13204: 13174: 13083: 12971: 12668: 12634: 12499: 12467: 12406: 12398: 12390: 12364:Nature 12343: 12335: 12300: 12292: 12189: 12152: 12112: 12002: 11992: 11885: 11869:& 11840: 11799: 11755: 11684: 11676: 11639: 11595: 11552: 11527: 11442: 11357: 11349: 11282: 11245: 11215: 11164: 11133: 11125: 11068: 11030: 10929: 10919: 10742: 10705: 10678: 10533: 10523: 10329: 10321: 10299:: e2. 10148: 10090:685207 10088: 10050: 10001: 9868: 9832: 9797: 9645:188757 9643: 9573: 9522: 9469: 9427: 9382: 9374: 9364: 9332: 9307:202372 9305: 9265: 9193: 9168: 9133: 9097: 9033: 8950: 8902: 8877: 8869: 8861: 8853: 8774: 8766: 8756: 8714: 8706: 8631: 8623: 8534: 8494: 8451: 8374: 8364: 8338: 8328: 8297: 8291:301627 8289: 8271:Osiris 8245: 8147:686258 8145: 8137: 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3780:beauty 3722:Jesuit 3693:). 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In 13598:(PDF) 13571:(PDF) 13551:(PDF) 13536:(PDF) 13429:JSTOR 13346:(PDF) 13331:(PDF) 13282:S2CID 13198:Wiley 13071:(PDF) 12880:(PDF) 12869:(PDF) 12438:(PDF) 12404:S2CID 12341:S2CID 12333:JSTOR 12298:S2CID 12290:JSTOR 12121:(PDF) 12110:S2CID 12088:(PDF) 11945:(PDF) 11922:(PDF) 11896:(PDF) 11871:PIRLS 11867:TIMSS 11863:(PDF) 11838:S2CID 11797:JSTOR 11741:(PDF) 11713:(PDF) 11706:(PDF) 11682:S2CID 11674:JSTOR 11525:S2CID 11451:(PDF) 11440:S2CID 11420:(PDF) 11355:S2CID 11347:JSTOR 11184:form. 11131:S2CID 11123:JSTOR 11079:(PDF) 11058:(PDF) 11028:S2CID 10877:ZDNET 10793:(PDF) 10773:(PDF) 10628:(PDF) 10491:: 17. 10481:(PDF) 10388:(PDF) 10369:(PDF) 10327:S2CID 10189:(PDF) 10182:(PDF) 10086:JSTOR 9999:JSTOR 9899:(PDF) 9888:(PDF) 9866:JSTOR 9795:S2CID 9652:(PDF) 9641:JSTOR 9637:(PDF) 9571:S2CID 9520:S2CID 9436:(PDF) 9425:S2CID 9405:(PDF) 9303:JSTOR 9253:. 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