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nfinite sets 51 but by showing that this implies different sizes of infinity per cantor s diagonal argument this led to the controversy over cantor s set theory 52 in the same period various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigor 53 this became the foundational crisis of mathematics 54 it was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory roughly speaking each mathematical object is defined by the set of all similar objects and the properties that these objects must have 16 for example in peano arithmetic the natural numbers are defined by zero is a number each number has a unique successor each number but zero has a unique predecessor and some rules of reasoning 55 this mathematical abstraction from reality is embodied in the modern philosophy of formalism as founded by david hilbert around 1910 56 the nature of the objects defined this way is a philosophical problem that mathematicians leave to philosophers even if many mathematicians have opinions on this nature and use their opinion sometimes called intuition to guide their study and proofs the approach allows considering logics that is sets of allowed deducing rules theorems proofs etc as mathematical objects and to prove theorems about them for example gödel s incompleteness theorems assert roughly speaking that in every consistent formal system that contains the natural numbers there are theorems that are true that is provable in a stronger system but not provable inside the system 57 this approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by brouwer who promoted intuitionistic logic which explicitly lacks the law of excluded middle 58 59 these problems and debates led to a wide expansion of mathematical logic with subareas such as model theory modeling some logical theories inside other theories proof theory type theory computability theory and computational complexity theory 9 although these aspects of mathematical logic were introduced before the rise of computers their use in compiler design formal verification program analysis proof assistants and other aspects of computer science contributed in turn to the expansion of these logical theories 60 computational mathematics main article computational mathematics computational mathematics is the study of mathematical problems that are typically too large for human computing capacity 61 62 part of computational mathematics involves numerical analysis which is devoted to computation with approximations of real numbers floating point arithmetic numerical analysis provides methods for problems in analysis using functional analysis and approximation theory numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors 63 numerical analysis and more broadly scientific computing also study non analytic topics of mathematical science especially algorithmic matrix and graph theory another area of computational mathematics is computer algebra which consists mainly of the manipulation of mathematical formulas on a computer automated theorem proving which include proof assistants and proof checkers is another area of computational mathematics that is used for formal verification and allowed formal proof of difficult theorems of mathematics such feit thompson theorem on finite groups algorithmics a large part of discrete mathematics and public key cryptography are also often considered as a part of computational mathematics history main article history of mathematics etymology the word mathematics comes from the ancient greek word máthēma μάθημα meaning something learned knowledge mathematics and the derived expression mathēmatikḗ tékhnē μαθηματικὴ τέχνη meaning mathematical science it entered the english language during the late middle english period through french and latin 64 traditionally 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 the pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry by the time of aristotle 384 322 bc this meaning was fully established 65 in latin and 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 this change has resulted in several mistranslations for example saint augustine s warning that christians should beware of mathematici meaning astrologers is sometimes mistranslated as a condemnation of mathematicians 66 the apparent plural form in english goes back to the latin neuter plural mathematica cicero based on the greek plural ta mathēmatiká τὰ μαθηματικά and means 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 inherited from greek 67 in english the noun mathematics takes a singular verb it is often shortened to maths 68 or in north america math 69 ancient in addition to recognizing how to count physical objects prehistoric peoples may have also known how to count abstract quantities like time days seasons or years 70 71 image of problem 14 from the egyptian moscow mathematical papyrus the problem includes a diagram indicating the dimensions of the truncated pyramid archaeological evidence has suggested that the ancient egyptian counting system had origins in sub saharan africa 72 also fractal geometry designs which are widespread among sub saharan african cultures are also found in egyptian architecture and cosmological signs 73 the ishango bone according to scholar alexander marshack may have influenced the later development of mathematics in egypt as like some entries on the ishango bone egyptian arithmetic also made use of multiplication by 2 this however is disputed 74 megalithic structures located in nabta playa upper egypt featured astronomy calendar arrangements in alignment with the heliacal rising of sirius and supported calibration of the yearly calendar for the annual nile flood 75 ancient nubians established a system of geometric rules which served as the basis for initial sunclocks nubians also exercised a trigonometric methodology comparable to their egyptian counterparts 76 77 78 79 the babylonian mathematical tablet plimpton 322 dated to 1800 bc 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 80 the oldest mathematical texts from mesopotamia and egypt are from 2000 to 1800 bc 81 many early texts mention pythagorean triples and so by inference the pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry it is in babylonian mathematics that elementary arithmetic addition subtraction multiplication and division first appear in the archaeological record 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 82 by the 5th century bc greek mathematics began to emerge as a distinct discipline and some ancient greeks such as the pythagoreans appeared to have considered it a subject in its own right 83 around 300 bc euclid organized mathematical knowledge by way of postulates and first principles which evolved into the axiomatic method that is used in mathematics today consisting of definition axiom theorem and proof 84 his book elements covers geometry and number theory and is widely considered the most successful and influential textbook of all time 85 another notable mathematician of antiquity is archimedes of syracuse c 287 c 212 bc 86 he developed methods for calculating the surface area and volume of solids of revolution including using the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series in a manner reminiscent of modern calculus 87 other notable achievements of greek mathematics are conic sections apollonius of perga 3rd century bc 88 trigonometry hipparchus of nicaea 2nd century bc 89 and the beginnings of pre modern algebra diophantus 3rd century ad 90 the numerals used in the bakhshali manuscript dated between the 2nd century bc and the 2nd century ad 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 91 other notable developments of indian mathematics include the modern definition and approximation of sine and cosine and an early form of infinite series 92 93 medieval and later a page from al khwarizmi s al jabr during the golden age of islam especially during the 9th and 10th centuries mathematics saw many important innovations building on greek mathematics the most notable achievement of islamic mathematics was the development of algebra other achievements of the islamic period include advances in spherical trigonometry and the addition of the decimal point to the arabic numeral system 94 many notable mathematicians from this period were persian such as al khwarizmi omar khayyam and sharaf al dīn al ṭūsī 95 the greek and arabic mathematical texts were in turn translated to latin during the middle ages and made available in europe 96 during the early modern period mathematics began to develop at an accelerating pace in western europe with innovations that revolutionized mathematics such as the introduction of variables and symbolic notation attributed to françois viète 1540 1603 97 the introduction of logarithms by john napier in 1614 98 which greatly simplified numerical calculations especially for astronomy and marine navigation the introduction of cartesian coordinates by rené descartes 1596 1650 for reducing geometry to algebra 99 and the development of calculus by isaac newton 1643 1727 and gottfried leibniz 1646 1716 100 leonhard euler 1707 1783 unified and built upon these innovations with a standardized terminology and complemented them with the discovery and proofs of numerous theorems 101 carl friedrich gauss perhaps the foremost mathematician of the 19th century was the german mathematician carl gauss who made numerous contributions to fields such as algebra analysis differential geometry matrix theory number theory and statistics 102 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 57 mathematics has since been greatly extended and there has been a fruitful interaction between mathematics and science to the benefit of both mathematical discoveries continue to be made to this very day according to mikhail b sevryuk in the january 2006 issue of the bulletin of the american mathematical society the number of papers and books included in the mathematical reviews mr database since 1940 the first year of operation of mr is now more than 1 9 million and more than 75 thousand items are added to the database each year the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs 103 symbolic notation and terminology main articles mathematical notation language of mathematics and glossary of mathematics an explanation of the sigma σ summation notation mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise unambiguous and accurate way this notation consists of symbols used for representing operations unspecified numbers relations and any other mathematical objects and then assembling them into expressions and formulas 104 more precisely numbers and other mathematical objects are represented by symbols called variables which are generally latin or greek letters and often include subscripts operation and relations are generally represented by specific symbols or glyphs 105 such as plus multiplication textstyle int integral equal and less than 106 all these symbols are generally grouped according to specific rules to form expressions and formulas 107 normally expressions and formulas do not appear alone but are included in sentences of the current language where expressions play the role of noun phrases and formulas play the role of clauses mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract idealized objects and how they interact it is based on rigorous definitions that provide a standard foundation for communication an axiom or postulate is a mathematical statement that is taken to be true without need of proof if a mathematical statement has yet to be proven or disproven it is termed a conjecture through a series of rigorous arguments employing deductive reasoning a statement that is proven to be true becomes a theorem a specialized theorem that is mainly used to prove another theorem is called a lemma a proven instance that forms part of a more general finding is termed a corollary 108 numerous technical terms used in mathematics are neologisms such as polynomial and homeomorphism 109 other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning for example in mathematics or means one the other or both while in common language it is either ambiguous or means one or the other but not both in mathematics the latter is called exclusive or finally many mathematical terms are common words that are used with a completely different meaning 110 this may lead to sentences that are correct and true mathematical assertions but appear to be nonsense to people who do not have the required background for example every free module is flat and a field is always a ring relationship with sciences mathematics is used in most sciences for modeling phenomena which then allows predictions to be made from experimental laws 111 the independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model 112 inaccurate predictions rather than being caused by invalid mathematical concepts imply the need to change the mathematical model used 113 for example the perihelion precession of mercury could only be explained after the emergence of einstein s general relativity which replaced new...
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