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Mathematics From Wikipedia, the free encyclopedia Jump to: navigation, search "Maths" and "Math" redirect here. For other uses see Mathematics (disambiguation ) and Math (disambiguation). Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this deta il from The School of Athens.[1] Mathematics is the abstract study of topics such as quantity (numbers),[2] struc ture,[3] space,[2] and change.[4][5][6] There is a range of views among mathemat icians and philosophers as to the exact scope and definition of mathematics.[7][ 8] Mathematicians seek out patterns[9][10] and use them to formulate new conjecture s. Mathematicians resolve the truth or falsity of conjectures by mathematical pr oof. When mathematical structures are good models of real phenomena, then mathem atical reasoning can provide insight or predictions about nature. Through the us e of abstraction and logic, mathematics developed from counting, calculation, me asurement, and the systematic study of the shapes and motions of physical object s. Practical mathematics has been a human activity for as far back as written re cords exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. 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 beco me customary to view mathematical research as establishing truth by rigorous ded uction from appropriately chosen axioms and definitions. Mathematics developed a t a relatively slow pace until the Renaissance, when mathematical innovations in teracting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] Galileo Galilei (1564 1642) said, "The universe cannot be read until we have learn ed the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles a nd other geometrical figures, without which means it is humanly impossible to co mprehend a single word. Without these, one is wandering about in a dark labyrint h."[12] Carl Friedrich Gauss (1777 1855) referred to mathematics as "the Queen of the Sciences".[13] Benjamin Peirce (1809 1880) called mathematics "the science tha t draws necessary conclusions".[14] David Hilbert said of mathematics: "We are n ot speaking here of arbitrariness in any sense. Mathematics is not like a game w hose tasks are determined by arbitrarily stipulated rules. Rather, it is a conce ptual system possessing internal necessity that can only be so and by no means o therwise."[15] Albert Einstein (1879 1955) stated that "as far as the laws of math ematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[16] French mathematician Claire Voisin states "Th ere is creative drive in mathematics, it's all about movement trying to express itself." [17] Mathematics is used throughout the world as an essential tool in many fields, in cluding natural science, engineering, medicine, finance and the social sciences. Applied mathematics, the branch of mathematics concerned with application of ma thematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disc iplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practi cal applications for what began as pure mathematics are often discovered.[18]

Contents 1 History 1.1 Evolution 1.2 Etymology 2 Definitions of mathematics 3 Inspiration, pure and applied mathematics, and aesthetics 4 Notation, language, and rigor 5 Fields of mathematics 5.1 Foundations and philosophy 5.2 Pure mathematics 5.2.1 Quantity 5.2.2 Structure 5.2.3 Space 5.2.4 Change 5.3 Applied mathematics 5.3.1 Statistics and other decision sciences 5.3.2 Computational mathematics 6 Mathematics as profession 7 Mathematics as science 8 See also 9 Notes 10 References 11 Further reading 12 External links History Evolution Main article: History of mathematics Greek mathematician Pythagoras (c. 570 c. 495 BC), commonly credited with discov ering the Pythagorean theorem The evolution of mathematics might be seen as an ever-increasing series of abstr actions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[19] was probably that of numbers: the realizat ion that a collection of two apples and a collection of two oranges (for example ) have something in common, namely quantity of their members.

Mayan numerals Evidenced by tallies found on bone, in addition to recognizing how to count phys ical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time days, seasons, years.[20] More complex mathematics did not appear until around 3000 BC, when the Babylonia ns and Egyptians began using arithmetic, algebra and geometry for taxation and o ther financial calculations, for building and construction, and for astronomy.[2 1] The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. In Babylonian mathematics elementary arithmetic (addition, subtraction, multipli cation and division) first appears in the archaeological record. Numeracy pre-da ted writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]

Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematic s in its own right with Greek mathematics.[22] Mathematics has since been greatly extended, and there has been a fruitful inter action between mathematics and science, to the benefit of both. Mathematical dis coveries continue to be made today. According to Mikhail B. Sevryuk, in the Janu ary 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (t he 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 o f works in this ocean contain new mathematical theorems and their proofs."[23] Etymology The word mathematics comes from the Greek µ???µa (máthema), which, in the ancient Gree k language, means "that which is learnt",[24] "what one gets to know", hence als o "study" and "science", and in modern Greek just "lesson". The word máthema is de rived from µa????? (manthano), while the modern Greek equivalent is µa?a??? (mathain o), both of which mean "to learn". In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Class ical times.[25] Its adjective is µa??µat???? (mathematikós), meaning "related to learn ing" or "studious", which likewise further came to mean "mathematical". In parti cular, µa??µat??? t???? (mathematik? tékhne), Latin: ars mathematica, meant "the mathe matical art". In Latin, and in English until around 1700, the term mathematics more commonly m eant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meani ng gradually changed to its present one from about 1500 to 1800. This has result ed in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.[citation needed ] 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 th e Latin neuter plural mathematica (Cicero), based on the Greek plural ta µa??µat??? (ta mathematiká), used by Aristotle (384 322 BC), and meaning roughly "all things ma thematical"; although it is plausible that English borrowed only the adjective m athematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[26] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in Eng lish-speaking North America, math.[27] Definitions of mathematics Main article: Definitions of mathematics Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.[28] Starting in the 19th century, when the stu dy of mathematics increased in rigor and began to address abstract topics such a s group theory and projective geometry, which have no clear-cut relation to quan tity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[29] Some of these definitions emphasize the deductive chara cter of much of mathematics, some emphasize its abstractness, some emphasize cer tain topics within mathematics. Today, no consensus on the definition of mathema tics prevails, even among professionals.[7] There is not even consensus on wheth er mathematics is an art or a science.[8] A great many professional mathematicia ns take no interest in a definition of mathematics, or consider it undefinable.[ 7] Some just say, "Mathematics is what mathematicians do."[7] Three leading types of definition of mathematics are called logicist, intuitioni

st, and formalist, each reflecting a different philosophical school of thought.[ 30] All have severe problems, none has widespread acceptance, and no reconciliat ion seems possible.[30] An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870).[31] In the Principia Mathemati ca, Bertrand Russell and Alfred North Whitehead advanced the philosophical progr am known as logicism, and attempted to prove that all mathematical concepts, sta tements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Sym bolic Logic" (1903).[32] Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an i ntuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[30] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other d efinitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionis m allows only mathematical objects that one can actually construct. Formalist definitions identify mathematics with its symbols and the rules for op erating on them. Haskell Curry defined mathematics simply as "the science of for mal systems".[33] A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the wo rd axiom has a special meaning, different from the ordinary meaning of "a self-e vident truth". In formal systems, an axiom is a combination of tokens that is in cluded in a given formal system without needing to be derived using the rules of the system. Inspiration, pure and applied mathematics, and aesthetics Main article: Mathematical beauty Isaac Newton Gottfried Wilhelm von Leibniz Isaac Newton (left) and Gottfried Wilhelm Leibniz (right), developers of infinit esimal calculus Mathematics arises from many different kinds of problems. At first these were fo und in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise wit hin mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematic al 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.[34] Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that are a. But often mathematics inspired by one area proves useful in many areas, and j oins the general stock of mathematical concepts. A distinction is often made bet ween pure mathematics and applied mathematics. However pure mathematics topics o ften turn out to have applications, e.g. number theory in cryptography. This rem arkable fact that even the "purest" mathematics often turns out to have practica l applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".[35] As in most areas of study, the explosion of knowledge in th e scientific age has led to specialization: there are now hundreds of specialize d areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[36] Several areas of applied mathematics have merged with related tra ditions outside of mathematics and become disciplines in their own right, includ ing statistics, operations research, and computer science. For those who are mathematically inclined, there is often a definite aesthetic a

spect to much of mathematics. Many mathematicians talk about the elegance of mat hematics, its intrinsic aesthetics and inner beauty. Simplicity and generality a re valued. 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 metho d that speeds calculation, such as the fast Fourier transform. G.H. Hardy in A M athematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He iden tified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[37] Mathematicians ofte n strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdos.[38][39] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. Notation, language, and rigor Main article: Mathematical notation Leonhard Euler, who created and popularized much of the mathematical notation us ed today Most of the mathematical notation in use today was not invented until the 16th c entury.[40] Before that, mathematics was written out in words, a painstaking pro cess that limited mathematical discovery.[41] Euler (1707 1783) was responsible fo r many of the notations in use today. Modern notation makes mathematics much eas ier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical nota tion, modern mathematical notation has a strict syntax (which to a limited exten t varies from author to author and from discipline to discipline) and encodes in formation that would be difficult to write in any other way. Mathematical language can be difficult to understand for beginners. Words such a s or and only have more precise meanings than in everyday speech. Moreover, word s such as open and field have been given specialized mathematical meanings. Tech nical terms such as homeomorphism and integrable have precise meanings in mathem atics. Additionally, shorthand phrases such as iff for "if and only if" belong t o mathematical jargon. There is a reason for special notation and technical voca bulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor". Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avo id mistaken "theorems", based on fallible intuitions, of which many instances ha ve occurred in the history of the subject.[42] The level of rigor expected in ma thematics has varied over time: the Greeks expected detailed arguments, but at t he time of Isaac Newton the methods employed were less rigorous. Problems inhere nt in the definitions used by Newton would lead to a resurgence of careful analy sis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians cont inue to argue among themselves about computer-assisted proofs. Since large compu tations are hard to verify, such proofs may not be sufficiently rigorous.[43] Axioms in traditional thought were "self-evident truths", but that conception is problematic. 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 axioma tic system. 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 (suffi ciently powerful) axiomatic system has undecidable formulas; and so a final axio matization of mathematics is impossible. Nonetheless mathematics is often imagin ed to be (as far as its formal content) nothing but set theory in some axiomatiz ation, in the sense that every mathematical statement or proof could be cast int

o formulas within set theory.[44] Fields of mathematics See also: Areas of mathematics and Glossary of areas of mathematics An abacus, a simple calculating tool used since ancient times Mathematics can, broadly speaking, be subdivided into the study of quantity, str ucture, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to explo ring links from the heart of mathematics to other fields: to logic, to set theor y (foundations), to the empirical mathematics of the various sciences (applied m athematics), and more recently to the rigorous study of uncertainty. Foundations and philosophy In order to clarify the foundations of mathematics, the fields of mathematical l ogic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathemati cs; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical struc tures and relationships between them, is still in development. The phrase "crisi s of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[45] Some disagreement about th e foundations of mathematics continues to the present day. The crisis of foundat ions was stimulated by a number of controversies at the time, including the cont roversy over Cantor's set theory and the Brouwer Hilbert controversy. Mathematical logic is concerned with setting mathematics within a rigorous axiom atic framework, and studying the implications of such a framework. As such, it i s home to Gödel's incompleteness theorems which (informally) imply that any effect ive formal system that contains basic arithmetic, if sound (meaning that all the orems that can be proven are true), is necessarily incomplete (meaning that ther e are true theorems which cannot be proved in that system). Whatever finite coll ection 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 d oes not follow from those axioms. Therefore no formal system is a complete axiom atization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer sc ience,[citation needed] as well as to category theory. Theoretical computer science includes computability theory, computational comple xity theory, and information theory. Computability theory examines the limitatio ns of various theoretical models of the computer, including the most well-known model the Turing machine. Complexity theory is the study of tractability by comp uter; some problems, although theoretically solvable by computer, are so expensi ve in terms of time or space that solving them is likely to remain practically u nfeasible, even with the rapid advancement of computer hardware. A famous proble m is the "P = NP?" problem, one of the Millennium Prize Problems.[46] Finally, i nformation theory is concerned with the amount of data that can be stored on a g iven medium, and hence deals with concepts such as compression and entropy. p \Rightarrow q \, Venn A intersect B.svg Commutative diagram for morphism .svg DFAexample.svg Mathematical logic Set theory Category theory Theory of computation Pure mathematics Quantity

The study of quantity starts with numbers, first the familiar natural numbers an d integers ("whole numbers") and arithmetical operations on them, which are char acterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory. As the number system is further developed, the integers are recognized as a subs et of the rational numbers ("fractions"). These, in turn, are contained within t he real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the conc ept of "infinity". Another area of study is size, which leads to the cardinal nu mbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets. 1, 2, 3,\ldots\! \ldots,-2, -1, 0, 1, 2\,\ldots\! -2, \frac{2}{3} , 1.21\,\! -e, \sqrt{2}, 3, \pi\,\! 2, i, -2+3i, 2e^{i\frac{4\pi}{3} }\,\! Natural numbers Integers Rational numbers Real numbers Complex numbers Structure Many mathematical objects, such as sets of numbers and functions, exhibit intern al structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the s et of integers that can be expressed in terms of arithmetic operations. Moreover , it frequently happens that different such structured sets (or structures) exhi bit similar properties, which makes it possible, by a further step of abstractio n, to state axioms for a class of structures, and then study at once the whole c lass of structures satisfying these axioms. Thus one can study groups, rings, fi elds and other abstract systems; together such studies (for structures defined b y algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problem s; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field the ory and group theory. 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. This is one example of the phenomenon that the originally unrelated a reas of geometry and algebra have very strong interactions in modern mathematics . Combinatorics studies ways of enumerating the number of objects that fit a giv en structure. \begin{matrix} (1,2,3) matrix} Elliptic curve 6.svg Lattice of the Combinatorics Number Algebra Space & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{ simple.svg Rubik's cube.svg Group diagdram D divisibility of 60.svg Braid-modular-group-cover.svg theory Group theory Graph theory Order theory

The study of space originates with geometry in particular, Euclidean geometry. T rigonometry is the branch of mathematics that deals with relationships between t he sides and the angles of triangles and with the trigonometric functions; it co mbines space and numbers, and encompasses the well-known Pythagorean theorem. Th e modern study of space generalizes these ideas to include higher-dimensional ge ometry, non-Euclidean geometries (which play a central role in general relativit y) and topology. Quantity and space both play a role in analytic geometry, diffe

rential geometry, and algebraic geometry. Convex and discrete geometry were deve loped to solve problems in number theory and functional analysis but now are pur sued with an eye on applications in optimization and computer science. Within di fferential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the des cription of geometric objects as solution sets of polynomial equations, combinin g the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure , and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-the oretic topology, algebraic topology and differential topology. In particular, in stances of modern day topology are metrizability theory, axiomatic set theory, h omotopy theory, and Morse theory. Topology also includes the now solved Poincaré c onjecture, and the still unsolved areas of the Hodge conjecture. Other results i n geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers. Illustration to Euclid's proof of the Pythagorean theorem.svg Sinusvåg 400px.png Hyperbolic triangle.svg Torus.png Mandel zoom 07 satellite.jpg Measure illustration.png Geometry Trigonometry Differential geometry Topology Fractal geometry Measure theory Change Further information: Finite difference Understanding and describing change is a common theme in the natural sciences, a nd calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with c omplex analysis the equivalent field for the complex numbers. Functional analysi s focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be des cribed by dynamical systems; chaos theory makes precise the ways in which many o f these systems exhibit unpredictable yet still deterministic behavior. Integral as region under curve.svg Vector field.svg Airflow-Obstruct ed-Duct.png Limitcycle.svg Lorenz attractor.svg Conformal grid after Möbiu s transformation.svg Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis Applied mathematics Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematic s" is a mathematical science with specialized knowledge. The term applied mathem atics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathe matics focuses on the "formulation, study, and use of mathematical models" in sc ience, engineering, and other areas of mathematical practice. In the past, practical applications have motivated the development of mathematic al theories, which then became the subject of study in pure mathematics, where m athematics is developed primarily for its own sake. Thus, the activity of applie d mathematics is vitally connected with research in pure mathematics. Statistics and other decision sciences Applied mathematics has significant overlap with the discipline of statistics, w

hose theory is formulated mathematically, especially with probability theory. St atisticians (working as part of a research project) "create data that makes sens e" with random sampling and with randomized experiments;[47] the design of a sta tistical sample or experiment specifies the analysis of the data (before the dat a be available). When reconsidering data from experiments and samples or when an alyzing 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.[48] Statistical theory studies decision problems such as minimizing the risk (expect ed loss) of a statistical action, such as using a procedure in, for example, par ameter estimation, hypothesis testing, and selecting the best. In these traditio nal areas of mathematical statistics, a statistical-decision problem is formulat ed by minimizing an objective function, like expected loss or cost, under specif ic constraints: For example, designing a survey often involves minimizing the co st of estimating a population mean with a given level of confidence.[49] Because of its use of optimization, the mathematical theory of statistics shares concer ns with other decision sciences, such as operations research, control theory, an d mathematical economics.[50] Computational mathematics Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical an alysis studies methods for problems in analysis using functional analysis and ap proximation theory; numerical analysis includes the study of approximation and d iscretization broadly with special concern for rounding errors. Numerical analys is and, more broadly, scientific computing also study non-analytic topics of mat hematical science, especially algorithmic matrix and graph theory. Other areas o f computational mathematics include computer algebra and symbolic computation. Gravitation space source.png BernoullisLawDerivationDiagram.svg Composit e trapezoidal rule illustration small.svg Maximum boxed.png Two red dice 01.svg Oldfaithful3.png Caesar3.svg Mathematical physics Fluid dynamics Numerical analysis Optimization Probability theory Statistics Cryptography Market Data Index NYA on 20050726 202628 UTC.png Arbitrary-gametree-solve d.svg Signal transduction pathways.svg CH4-structure.svg GDP PPP Per Capita IMF 2008.svg Simple feedback control loop2.svg Mathematical finance Game theory Mathematical biology Mathematical che mistry Mathematical economics Control theory Mathematics as profession Arguably the most prestigious award in mathematics is the Fields Medal,[51][52] established in 1936 and now awarded every four years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achieveme nt, and another major international award, the Abel Prize, was introduced in 200 3. The Chern Medal was introduced in 2010 to recognize lifetime achievement. The se 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 establish ed field. A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity a mong mathematicians, and at least nine of the problems have now been solved. A n ew list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. A solution to each of these problems carries a $1 million re

ward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems. Mathematics as science

Carl Friedrich Gauss, known as the "prince of mathematicians"[53] Gauss referred to mathematics as "the Queen of the Sciences".[13] In the origina l Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the w ord corresponding to science means a "field of knowledge", and this was the orig inal meaning of "science" in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natura l science follows the rise of Baconian science, which contrasted "natural scienc e" to scholasticism, the Aristotelean method of inquiring from first principles. The role of empirical experimentation and observation is negligible in mathemat ics, compared to natural sciences such as psychology, biology, or physics. Alber t Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." [16] More recently, Marcus du Sautoy has called mathematics "the Queen of Scienc e ... the main driving force behind scientific discovery".[54] Many philosophers believe that mathematics is not experimentally falsifiable, an d thus not a science according to the definition of Karl Popper.[55] However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians[who?] tha t mathematics cannot be reduced to logic alone, and Karl Popper concluded that " most mathematical theories are, like those of physics and biology, hypothetico-d eductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[56] Ot her thinkers, notably Imre Lakatos, have applied a version of falsificationism t o mathematics itself. An alternative view is that certain scientific fields (such as theoretical physi cs) are mathematics with axioms that are intended to correspond to reality. The theoretical physicist J.M. Ziman proposed that science is public knowledge, and thus includes mathematics.[57] Mathematics shares much in common with many field s in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulatio n of conjectures in both mathematics and the (other) sciences. Experimental math ematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. The opinions of mathematicians on this matter are varied. Many mathematicians[wh o?] 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[wh o?] 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. One way this differ ence of viewpoint plays out is in the philosophical debate as to whether mathema tics is created (as in art) or discovered (as in science). It is common to see u niversities divided into sections that include a division of Science and Mathema tics, indicating that the fields are seen as being allied but that they do not c oincide. In practice, mathematicians are typically grouped with scientists at th e gross level but separated at finer levels. This is one of many issues consider ed in the philosophy of mathematics.[citation needed] See also Main article: Lists of mathematics topics Mathematics and art Mathematics education

Nuvola apps edu mathematics blue-p.svgMathematics portal Notes ^ No likeness or description of Euclid's physical appearance made during his lif etime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid). ^ a b "mathematics, n.''". Oxford English Dictionary. Oxford University Press. 2 012. Retrieved June 16, 2012. "The science of space, number, quantity, and arran gement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis." ^ Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. pp. 4. ISBN 0-486-41712-3. "Mathematics ... is si mply the study of abstract structures, or formal patterns of connectedness." ^ LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cy nthia R Harris (2011). Calculus Concepts: An Informal Approach to the Mathematic s of Change. Cengage Learning. pp. 2. ISBN 1-4390-4957-2. "Calculus is the study of change how things change, and how quickly they change." ^ Ramana (2007). Applied Mathematics. Tata McGraw Hill Education. p. 2.10. ISBN 007-066753-5. "The mathematical study of change, motion, growth or decay is calcu lus." ^ Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. pp. 7. ISBN 3-642-19532-6. ^ a b c d Mura, Robert (Dec 1993). "Images of Mathematics Held by University Tea chers of Mathematical Sciences". Educational Studies in Mathematics 25 (4): 375 38 5. ^ a b Tobies, Renate and Helmut Neunzert (2012). Iris Runge: A Life at the Cross roads of Mathematics, Science, and Industry. Springer. pp. 9. ISBN 3-0348-0229-3 . "It is first necessary to ask what is meant by mathematics in general. Illustr ious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form." ^ Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611 616. And summarized at Association for Supervision and Curriculum Development, www.ascd. org. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in L ife, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 97 8-0-7167-5047-5 ^ Eves ^ Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and Leibniz, BBC R adio 4, 27 September 2010. ^ a b Waltershausen ^ Peirce, p. 97. ^ Hilbert, D. (1919 20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919 1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an E nglish introduction by David E. Rowe), Basel, Birkhäuser (1992). ^ a b Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is inde pendent of experience, is so admirably appropriate to the objects of reality?" H e, too, is concerned with The Unreasonable Effectiveness of Mathematics in the N atural Sciences. ^ "Claire Voisin, Artist of the Abstract". .cnrs.fr. Retrieved 2013-10-13. ^ Peterson ^ Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (Aug 1998). "A bstract representations of numbers in the animal and human brain". Trends in Neu roscience 21 (8): 355 361. doi:10.1016/S0166-2236(98)01263-6 . PMID 9720604. ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an El ementary Study, passim ^ Kline 1990, Chapter 1. ^ "A History of Greek Mathematics: From Thales to Euclid". Thomas Little Heath ( 1981). ISBN 0-486-24073-8

^ Sevryuk ^ "mathematic". Online Etymology Dictionary. ^ Both senses can be found in Plato. µa??µat???. Liddell, Henry George; Scott, Rober t; A Greek English Lexicon at the Perseus Project ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "ma thematics", "mathematic", "mathematics" ^ "maths, n." and "math, n.3". Oxford English Dictionary, on-line version (2012) . ^ James Franklin, "Aristotelian Realism" in Philosophy of Mathematics", ed. A.D. Irvine, p. 104. Elsevier (2009). ^ Cajori, Florian (1893). A History of Mathematics. American Mathematical Societ y (1991 reprint). pp. 285 6. ISBN 0-8218-2102-4. ^ a b c Snapper, Ernst (September 1979). "The Three Crises in Mathematics: Logic ism, Intuitionism, and Formalism". Mathematics Magazine 52 (4): 207 16. doi:10.230 7/2689412 . JSTOR 2689412. ^ Peirce, Benjamin (1882). Linear Associative Algebra. p. 1. ^ Bertrand Russell, The Principles of Mathematics, p. 5. University Press, Cambr idge (1903) ^ Curry, Haskell (1951). Outlines of a Formalist Philosophy of Mathematics. Else vier. pp. 56. ISBN 0-444-53368-0. ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynma n's Operational Calculus. Oxford University Press. ISBN 0-8218-2413-9. ^ Wigner, Eugene (1960). "The Unreasonable Effectiveness of Mathematics in the N atural Sciences". Communications on Pure and Applied Mathematics 13 (1): 1 14. doi :10.1002/cpa.3160130102 . ^ "Mathematics Subject Classification 2010" (PDF). Retrieved 2010-11-09. ^ Hardy, G.H. (1940). A Mathematician's Apology. Cambridge University Press. ISB N 0-521-42706-1. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA. ^ Aigner, Martin; Ziegler, Günter M. (2001). Proofs from The Book. Springer. ISBN 3-540-40460-0. ^ Earliest Uses of Various Mathematical Symbols (Contains many further reference s). ^ Kline, p. 140, on Diophantus; p. 261, on Vieta. ^ See false proof for simple examples of what can go wrong in a formal proof. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p . 4 "A few complain that the computer program can't be verified properly", (in r eference to the Haken Apple proof of the Four Color Theorem). ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, " Among the many branches of modern mathematics set theory occupies a unique place : with a few rare exceptions the entities which are studied and analyzed in math ematics may be regarded as certain particular sets or classes of objects." ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford Universit y Press, 2005. ^ Clay Mathematics Institute, P=NP, claymath.org ^ Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientifi c. ISBN 981-02-3111-3 ^ Like other mathematical sciences such as physics and computer science, statist ics is an autonomous discipline rather than a branch of applied mathematics. Lik e research physicists and computer scientists, research statisticians are mathem atical scientists. Many statisticians have a degree in mathematics, and some sta tisticians are also mathematicians. ^ Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah. Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statist ics. New York: Wiley. pp. vii viii. ISBN 0-471-08073-X. MR 607328. ^ Whittle (1994, pp. 10 11 and 14 18): Whittle, Peter (1994). "Almost home". In Kell y, F.P.. Probability, statistics and optimisation: A Tribute to Peter Whittle (p reviously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revi sed 2002)" ed.). Chichester: John Wiley. pp. 1 28. ISBN 0-471-94829-2.

^ "The Fields Medal is now indisputably the best known and most influential awar d in mathematics." Monastyrsky ^ Riehm ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxfo rd University Press. p. 1188. ISBN 0-19-850763-1. ^ Marcus du Sautoy, A Brief History of Mathematics: 10. Nicolas Bourbaki, BBC Ra dio 4, 1 October 2010. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. ^ Popper 1995, p. 56 ^ Ziman References Courant, Richard and H. Robbins, What Is Mathematics? : An Elementary Approach t o Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). IS BN 0-19-510519-2. Einstein, Albert (1923). Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D ).. E.P. Dutton & Co., New York. du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010). Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saun ders, 1990, ISBN 0-03-029558-0. Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford Univers ity Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7. Monastyrsky, Michael (2001). Some Trends in Modern Mathematics and the Fields Me dal (PDF). Canadian Mathematical Society. Retrieved 2006-07-28. Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, C larendon Press, 1989, ISBN 0-19-861186-2. The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9. Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition ( June 1989). ISBN 0-933174-65-9. Peirce, Benjamin (1881). "Linear associative algebra". In Peirce, Charles Sander s. American Journal of Mathematics (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.) (Johns Hopkins University) 4 (1 4): 97 229. doi:10.2307/2369153 . Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce a nd annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google Eprin t and as an extract, D. Van Nostrand, 1882, Google Eprint.. Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathe matics, Owl Books, 2001, ISBN 0-8050-7159-8. Popper, Karl R. (1995). "On knowledge". In Search of a Better World: Lectures an d Essays from Thirty Years. Routledge. ISBN 0-415-13548-6. Riehm, Carl (August 2002). "The Early History of the Fields Medal" (PDF). Notice s of the AMS (AMS) 49 (7): 778 782. Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the Americ an Mathematical Society 43 (1): 101 109. doi:10.1090/S0273-0979-05-01069-4 . Retri eved 2006-06-24. Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sän dig Reprint Verlag H. R. Wohlwend. ASIN B0000BN5SQ. ISBN 3-253-01702-8. Further reading Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford Universi ty Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4. Boyer, Carl B., A History of Mathematics, Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7. A concise history of mathematics fr om the Concept of Number to contemporary Mathematics. Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7. Gullberg, Jan, Mathematics From the Birth of Numbers. W. W. Norton & Company; 1s t edition (October 1997). ISBN 0-393-04002-X.

Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publish ers 2000. A translated and expanded version of a Soviet mathematics encyclopedia , in ten (expensive) volumes, the most complete and authoritative work available . Also in paperback and on CD-ROM, and online. Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover Publications, 2003, ISBN 0-486-43268-8. Maier, Annaliese, At the Threshold of Exact Science: Selected Writings of Annali ese Maier on Late Medieval Natural Philosophy, edited by Steven Sargent, Philade lphia: University of Pennsylvania Press, 1982. External links Find more about Mathematics at Wikipedia's sister projects Definitions and translations from Wiktionary Media from Commons Quotations from Wikiquote Source texts from Wikisource Textbooks from Wikibooks Learning resources from Wikiversity Library resources about Mathematics Resources in your library Wikiversity At Wikiversity you can learn more and teach others about Mathematics at: The School of Mathematics Mathematics on In Our Time at the BBC. (listen now) Free Mathematics books Free Mathematics books collection. Encyclopaedia of Mathematics online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in ma thematics. HyperMath site at Georgia State University FreeScience Library The mathematics section of FreeScience library Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.) Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resour ce focusing on algebraic, ordinary differential, partial differential (mathemati cal physics), integral, and other mathematical equations. Cain, George: Online Mathematics Textbooks available free online. Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques. Mathematical Structures, list information about classes of mathematical structur es. Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians. Metamath. A site and a language, that formalize mathematics from its foundations . Nrich, a prize-winning site for students from age five from Cambridge University Open Problem Garden, a wiki of open problems in mathematics Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article ex change with Wikipedia. Uses TeX markup. Some mathematics applets, at MIT Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics. Patrick Jones' Video Tutorials on Mathematics Citizendium: Theory (mathematics). du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010). MathOverflow A Q&A site for research-level mathematics v t e

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