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This article is about the field of study. Mathematics_sentence_0

For other uses, see Mathematics (disambiguation) and Math (disambiguation). Mathematics_sentence_1

Mathematics (from Greek: , máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). Mathematics_sentence_2

It has no generally accepted definition. Mathematics_sentence_3

Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. Mathematics_sentence_4

When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Mathematics_sentence_5

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Mathematics_sentence_6

Practical mathematics has been a human activity from as far back as written records exist. Mathematics_sentence_7

The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Mathematics_sentence_8

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics_sentence_9

Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics_sentence_10

Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day. Mathematics_sentence_11

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Mathematics_sentence_12

Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematics_sentence_13

Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later. Mathematics_sentence_14

History Mathematics_section_0

Main article: History of mathematics Mathematics_sentence_15

The history of mathematics can be seen as an ever-increasing series of abstractions. Mathematics_sentence_16

The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members. Mathematics_sentence_17

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time—days, seasons, or years. Mathematics_sentence_18

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. Mathematics_sentence_19

The most ancient mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Mathematics_sentence_20

Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Mathematics_sentence_21

It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. Mathematics_sentence_22

The Babylonians also possessed a place-value system, and used a sexagesimal numeral system which is still in use today for measuring angles and time. Mathematics_sentence_23

Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Mathematics_sentence_24

Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. Mathematics_sentence_25

His textbook Elements is widely considered the most successful and influential textbook of all time. Mathematics_sentence_26

The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. Mathematics_sentence_27

He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Mathematics_sentence_28

Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD). Mathematics_sentence_29

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Mathematics_sentence_30

Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series. Mathematics_sentence_31

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. Mathematics_sentence_32

The most notable achievement of Islamic mathematics was the development of algebra. Mathematics_sentence_33

Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Mathematics_sentence_34

Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. Mathematics_sentence_35

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. Mathematics_sentence_36

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Mathematics_sentence_37

Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Mathematics_sentence_38

Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. Mathematics_sentence_39

In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics_sentence_40

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematics_sentence_41

Mathematical discoveries continue to be made today. Mathematics_sentence_42

According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. Mathematics_sentence_43

The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematics_sentence_44

Etymology Mathematics_section_1

The word mathematics comes from Ancient Greek máthēma (), meaning "that which is learnt," "what one gets to know," hence also "study" and "science". Mathematics_sentence_45

The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Mathematics_sentence_46

Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." Mathematics_sentence_47

In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art." Mathematics_sentence_48

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. Mathematics_sentence_49

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. Mathematics_sentence_50

This has resulted in several mistranslations. Mathematics_sentence_51

For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians. Mathematics_sentence_52

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek. Mathematics_sentence_53

In English, the noun mathematics takes a singular verb. Mathematics_sentence_54

It is often shortened to maths or, in North America, math. Mathematics_sentence_55

Definitions of mathematics Mathematics_section_2

Main article: Definitions of mathematics Mathematics_sentence_56

Mathematics has no generally accepted definition. Mathematics_sentence_57

Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. Mathematics_sentence_58

However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. Mathematics_sentence_59

In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Mathematics_sentence_60

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Mathematics_sentence_61

There is not even consensus on whether mathematics is an art or a science. Mathematics_sentence_62

Some just say, "Mathematics is what mathematicians do." Mathematics_sentence_63

Three leading types Mathematics_section_3

Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. Mathematics_sentence_64

All have severe flaws, none has widespread acceptance, and no reconciliation seems possible. Mathematics_sentence_65

Logicist definitions Mathematics_section_4

An early definition of mathematics in terms of logic was that of Benjamin Peirce (1870): "the science that draws necessary conclusions." Mathematics_sentence_66

In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic. Mathematics_sentence_67

A logicist definition of mathematics is Russell's (1903) "All Mathematics is Symbolic Logic." Mathematics_sentence_68

Intuitionist definitions Mathematics_section_5

Formalist definitions Mathematics_section_6

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Mathematics_sentence_69

Haskell Curry defined mathematics simply as "the science of formal systems". Mathematics_sentence_70

A formal system is a set of symbols, or tokens, and some rules on how the tokens are to be combined into formulas. Mathematics_sentence_71

In formal systems, the word axiom has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system. Mathematics_sentence_72

Mathematics as science Mathematics_section_7

The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". Mathematics_sentence_73

More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery". Mathematics_sentence_74

The philosopher Karl Popper observed that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Mathematics_sentence_75

Popper also noted that "I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience." Mathematics_sentence_76

Several authors consider that mathematics is not a science because it does not rely on empirical evidence. Mathematics_sentence_77

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Mathematics_sentence_78

Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Mathematics_sentence_79

Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. Mathematics_sentence_80

The opinions of mathematicians on this matter are varied. Mathematics_sentence_81

Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. Mathematics_sentence_82

One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). Mathematics_sentence_83

In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. Mathematics_sentence_84

This is one of many issues considered in the philosophy of mathematics. Mathematics_sentence_85

Inspiration, pure and applied mathematics, and aesthetics Mathematics_section_8

Main article: Mathematical beauty Mathematics_sentence_86

Mathematics arises from many different kinds of problems. Mathematics_sentence_87

At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Mathematics_sentence_88

For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics. Mathematics_sentence_89

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. Mathematics_sentence_90

But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. Mathematics_sentence_91

A distinction is often made between pure mathematics and applied mathematics. Mathematics_sentence_92

However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. Mathematics_sentence_93

This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what Eugene Wigner has called "the unreasonable effectiveness of mathematics". Mathematics_sentence_94

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages. Mathematics_sentence_95

Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. Mathematics_sentence_96

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Mathematics_sentence_97

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Mathematics_sentence_98

Simplicity and generality are valued. Mathematics_sentence_99

There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. Mathematics_sentence_100

G. Mathematics_sentence_101 H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Mathematics_sentence_102

He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematics_sentence_103

Mathematical research often seeks critical features of a mathematical object. Mathematics_sentence_104

A theorem expressed as a characterization of the object by these features is the prize. Mathematics_sentence_105

Examples of particularly succinct and revelatory mathematical arguments have been published in Proofs from THE BOOK. Mathematics_sentence_106

The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. Mathematics_sentence_107

And at the other social extreme, philosophers continue to find problems in philosophy of mathematics, such as the nature of mathematical proof. Mathematics_sentence_108

Notation, language, and rigor Mathematics_section_9

Main article: Mathematical notation Mathematics_sentence_109

Most of the mathematical notation in use today was not invented until the 16th century. Mathematics_sentence_110

Before that, mathematics was written out in words, limiting mathematical discovery. Mathematics_sentence_111

Euler (1707–1783) was responsible for many of the notations in use today. Mathematics_sentence_112

Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. Mathematics_sentence_113

According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language. Mathematics_sentence_114

Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog. Mathematics_sentence_115

Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas. Mathematics_sentence_116

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematics_sentence_117

Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Mathematics_sentence_118

Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. Mathematics_sentence_119

There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematics_sentence_120

Mathematicians refer to this precision of language and logic as "rigor". Mathematics_sentence_121

Mathematical proof is fundamentally a matter of rigor. Mathematics_sentence_122

Mathematicians want their theorems to follow from axioms by means of systematic reasoning. Mathematics_sentence_123

This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject. Mathematics_sentence_124

The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Mathematics_sentence_125

Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Mathematics_sentence_126

Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Mathematics_sentence_127

Today, mathematicians continue to argue among themselves about computer-assisted proofs. Mathematics_sentence_128

Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous. Mathematics_sentence_129

On the other hand, proof assistants allow verifying all details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the Feit–Thompson theorem. Mathematics_sentence_130

Axioms in traditional thought were "self-evident truths", but that conception is problematic. Mathematics_sentence_131

At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. Mathematics_sentence_132

It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Mathematics_sentence_133

Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. Mathematics_sentence_134

Fields of mathematics Mathematics_section_10

See also: Areas of mathematics and Glossary of areas of mathematics Mathematics_sentence_135

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). Mathematics_sentence_136

In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. Mathematics_sentence_137

While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory. Mathematics_sentence_138

Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous. Mathematics_sentence_139

Foundations and philosophy Mathematics_section_11

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematics_sentence_140

Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Mathematics_sentence_141

The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Mathematics_sentence_142

Some disagreement about the foundations of mathematics continues to the present day. Mathematics_sentence_143

The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer–Hilbert controversy. Mathematics_sentence_144

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. Mathematics_sentence_145

As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Mathematics_sentence_146

Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Mathematics_sentence_147

Therefore, no formal system is a complete axiomatization of full number theory. Mathematics_sentence_148

Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science, as well as to category theory. Mathematics_sentence_149

In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem. Mathematics_sentence_150

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Mathematics_sentence_151

Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model—the Turing machine. Mathematics_sentence_152

Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. Mathematics_sentence_153

A famous problem is the "P = NP?" Mathematics_sentence_154

problem, one of the Millennium Prize Problems. Mathematics_sentence_155

Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy. Mathematics_sentence_156

Pure mathematics Mathematics_section_12

Main article: Pure mathematics Mathematics_sentence_157

Number systems and number theory Mathematics_section_13

Main articles: Arithmetic, Number system, and Number theory Mathematics_sentence_158

Structure Mathematics_section_14

Main article: Algebra Mathematics_sentence_159

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics_sentence_160

Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Mathematics_sentence_161

Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Mathematics_sentence_162

Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. Mathematics_sentence_163

By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Mathematics_sentence_164

Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. Mathematics_sentence_165

This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Mathematics_sentence_166

Combinatorics studies ways of enumerating the number of objects that fit a given structure. Mathematics_sentence_167

Space Mathematics_section_15

Main article: Geometry Mathematics_sentence_168

The study of space originates with geometry—in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. Mathematics_sentence_169

Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. Mathematics_sentence_170

The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Mathematics_sentence_171

Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Mathematics_sentence_172

Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Mathematics_sentence_173

Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Mathematics_sentence_174

Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Mathematics_sentence_175

Lie groups are used to study space, structure, and change. Mathematics_sentence_176

Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. Mathematics_sentence_177

In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Mathematics_sentence_178

Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the Hodge conjecture. Mathematics_sentence_179

Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proven only with the help of computers. Mathematics_sentence_180


Change Mathematics_section_16

Main article: Calculus Mathematics_sentence_181

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a tool to investigate it. Mathematics_sentence_182

Functions arise here, as a central concept describing a changing quantity. Mathematics_sentence_183

The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Mathematics_sentence_184

Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. Mathematics_sentence_185

One of many applications of functional analysis is quantum mechanics. Mathematics_sentence_186

Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Mathematics_sentence_187

Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Mathematics_sentence_188

Applied mathematics Mathematics_section_17

Main article: Applied mathematics Mathematics_sentence_189

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Mathematics_sentence_190

Thus, "applied mathematics" is a mathematical science with specialized knowledge. Mathematics_sentence_191

The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice. Mathematics_sentence_192

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Mathematics_sentence_193

Thus, the activity of applied mathematics is vitally connected with research in pure mathematics. Mathematics_sentence_194

Statistics and other decision sciences Mathematics_section_18

Main article: Statistics Mathematics_sentence_195

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Mathematics_sentence_196

Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). Mathematics_sentence_197

When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on new data. Mathematics_sentence_198

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. Mathematics_sentence_199

In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Mathematics_sentence_200

Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics. Mathematics_sentence_201

Computational mathematics Mathematics_section_19

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Mathematics_sentence_202

Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretisation broadly with special concern for rounding errors. Mathematics_sentence_203

Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Mathematics_sentence_204

Other areas of computational mathematics include computer algebra and symbolic computation. Mathematics_sentence_205

Mathematical awards Mathematics_section_20

Arguably the most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years (except around World War II) to as many as four individuals. Mathematics_sentence_206

The Fields Medal is often considered a mathematical equivalent to the Nobel Prize. Mathematics_sentence_207

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was instituted in 2003. Mathematics_sentence_208

The Chern Medal was introduced in 2010 to recognize lifetime achievement. Mathematics_sentence_209

These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field. Mathematics_sentence_210

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. Mathematics_sentence_211

This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. Mathematics_sentence_212

A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Mathematics_sentence_213

Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. Mathematics_sentence_214

A solution to any of these problems carries a 1 million dollar reward. Mathematics_sentence_215

Currently, only one of these problems, the Poincaré Conjecture, has been solved. Mathematics_sentence_216

See also Mathematics_section_21

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