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, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from '' The School Of Athens ''.No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (''see Euclid '').]] Mathematics (colloquially, '''maths''' or '''math''') is the body of knowledge centered on such concepts as Quantity , Structure , Space , and Change , and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".Peirce, p.97 Other practitioners of mathematicss seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new Conjecture s and establish their truth by Rigorous Deduction from appropriately chosen Axiom s and Definition s.Jourdain Through the use of Abstraction and Logic al Reasoning , mathematics evolved from Counting , Calculation , Measurement , and the systematic study of the Shape s and Motion s of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in Ancient Egypt , Mesopotamia , Ancient India , Ancient China , and Ancient Greece . Rigorous arguments first appear in Euclid 's ''Elements'' . The development continued in fitful bursts until the Renaissance period of the 16th Century , when mathematical innovations interacted with new Scientific Discoveries , leading to an acceleration in research that continues to the present day.Eves Today, mathematics is used throughout the world in many fields, including Natural Science , Engineering , Medicine , and the Social Science s such as Economics . Applied Mathematics , the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in Pure Mathematics , or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.Peterson ETYMOLOGY The word "mathematics" (Greek: μαθηματικά or ''mathēmatiká'') comes from the Greek μάθημα (''máthēma''), which means ''learning'', ''study'', ''science'', and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (''mathēmatikós''), ''related to learning'', or ''studious'', which likewise further came to mean ''mathematical''. In particular, (''mathēmatikḗ tékhnē''), in Latin ''ars mathematica'', meant ''the mathematical art''. 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 τα μαθηματικά (''ta mathēmatiká''), used by Aristotle , and meaning roughly "all things mathematical".'' The Oxford Dictionary Of English Etymology '', '' Oxford English Dictionary '' In English, however, ''mathematics'' is a singular noun, often shortened to ''math'' in English-speaking North America and ''maths'' elsewhere. HISTORY , a counting device used by the Inca .]] See Also: History of mathematics The evolution of mathematics might be seen as an ever-increasing series of Abstraction s, or alternatively an expansion of subject matter. The first abstraction was probably that of Number s. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to Count ''physical'' objects, Prehistoric peoples also recognized how to count ''abstract'' quantities, like Time — Day s, Season s, Year s. Arithmetic ( Addition , Subtraction , Multiplication and Division ), naturally followed. Monolithic monuments testify to knowledge of Geometry . Further steps need Writing or some other system for recording numbers such as Tallies or the knotted strings called Quipu used by the Inca Empire to store numerical data. Numeral System s have been many and diverse. From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to Taxation and Commerce , to understand the relationships among numbers, to Measure Land , and to predict Astronomical Events . These needs can be roughly related to the broad subdivision of mathematics into the studies of ''quantity'', ''structure'', ''space'', and ''change''. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin Of The American Mathematical Society , "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical Theorem s and their Proof s."Sevryuk INSPIRATION, PURE AND APPLIED MATHEMATICS, AND AESTHETICS (1643-1727), an inventor of Infinitesimal Calculus .]] See Also: Mathematical beauty Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in Commerce , Land Measurement and later Astronomy ; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton was one of the Infinitesimal Calculus inventors, Feynman invented the Feynman Path Integral using a combination of reasoning and physical insight, and today's String Theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The 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 ." As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between Pure Mathematics and Applied Mathematics . 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 . For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the ''elegance'' of mathematics, its intrinsic Aesthetics and inner Beauty . Simplicity and Generality are valued. There is beauty in a simple and elegant proof, such as Euclid 's proof that there are infinitely many Prime Number s, and in an elegant numerical method that speeds calculation, such as the Fast Fourier Transform . G. 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. Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. The popularity of Recreational Mathematics is another sign of the pleasure many find in solving mathematical questions. NOTATION, LANGUAGE, AND RIGOR of Cos ( ''y'' arccos sin|''x''| + ''x'' arcsin cos|''y''| ). ]] See Also: Mathematical notation Most of the mathematical notation in use today was not invented until the 16th Century . Earliest Uses of Various Mathematical Symbols (Contains many further references) Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way. Mathematical Language also is hard for beginners. Words such as ''or'' and ''only'' have more precise meanings than in everyday speech. Also confusing to beginners, words such as '' Open '' and '' Field '' have been given specialized mathematical meanings. Mathematical Jargon includes technical terms such as '' Homeomorphism '' and '' Integrable ''. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor". the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about Computer-assisted Proof s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous. 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 reference to the Haken-Apple proof of the Four Color Theorem). Axiom s 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 Axiomatic 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 (sufficiently powerful) axiomatic system has Undecidable formulas; and so a final Axiomatization of mathematics is impossible. 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. 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 mathematics may be regarded as certain particular sets or classes of objects." MATHEMATICS AS SCIENCE , himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".]] ''. Many philosophers believe that mathematics is not experimentally , have applied a version of falsificationism to mathematics itself. An alternative view is that certain scientific fields (such as Theoretical Physics ) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman , proposed that science is ''public knowledge'' and thus includes mathematics.Ziman In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and Experiment ation also play a role in the formulation of Conjecture s in both mathematics and the (other) sciences. Experimental Mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the Scientific Method . In his 2002 book '' A New Kind Of Science '', Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right. The opinions of mathematicians on this matter are varied. While some in Applied Mathematics feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to Logic and that they are, hence, fundamentally Philosophers . 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. 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). It is common to see Universities divided into sections that include a division of ''Science and Mathematics'', indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the Philosophy Of Mathematics . Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal ,"''The Fields Medal is now indisputably the best known and most influential award in mathematics.''" MonastyrskyRiehm established in 1936 and now awarded every 4 years. It is often considered, misleadingly, the equivalent of science's Nobel Prize s. The Wolf Prize In Mathematics , instituted in 1979, recognizes lifetime achievement, and another major international award, the Abel Prize , was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called " Hilbert's Problems ", was compiled in 1900 by German mathematician David Hilbert . This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the " Millennium Prize Problems ", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann Hypothesis ) is duplicated in Hilbert's problems. FIELDS OF MATHEMATICS , a simple calculating tool used since ancient times]] As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict , to Set Theory ( Foundations ), to the empirical mathematics of the various sciences ( Applied Mathematics ), and more recently to the rigorous study of Uncertainty . Quantity The study of quantity starts with and Goldbach's Conjecture . As the number system is further developed, the integers are recognized as a s, which allow meaningful comparison of the size of infinitely large sets.
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