139-688: The Proof of the Truthful ( Arabic : برهان الصديقين , romanized : burhān al-ṣiddīqīn , also translated Demonstration of the Truthful or Proof of the Veracious , among others) is a formal argument for proving the existence of God introduced by the Islamic philosopher Avicenna (also known as Ibn Sina, 980–1037). Avicenna argued that there must be a "necessary existent" (Arabic: واجب الوجود , romanized: wājib al-wujūd ), an entity that cannot not exist. The argument says that
278-422: A proof by contradiction , or reductio , showing that a contradiction would follow if one supposes that there were more than one necessary existent. If one postulates two necessary existents, A and B, a simplified version of the argument considers two possibilities: if A is distinct from B as a result of something implied from necessity of existence, then B would share it, too (being a necessary existent itself), and
417-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
556-423: A cause of other existents, which can be shown to imply a range of positive relations (e.g. knowing and powerful). Present-day historian of philosophy Peter Adamson called this argument one of the most influential medieval arguments for God's existence, and Avicenna's biggest contribution to the history of philosophy. Generations of Muslim philosophers and theologians took up the proof and its conception of God as
695-575: A collection of related dialects that constitute the precursor of Arabic, first emerged during the Iron Age . Previously, the earliest attestation of Old Arabic was thought to be a single 1st century CE inscription in Sabaic script at Qaryat al-Faw , in southern present-day Saudi Arabia. However, this inscription does not participate in several of the key innovations of the Arabic language group, such as
834-435: A corpus of poetic texts, in addition to Qur'an usage and Bedouin informants whom he considered to be reliable speakers of the ʿarabiyya . Arabic spread with the spread of Islam . Following the early Muslim conquests , Arabic gained vocabulary from Middle Persian and Turkish . In the early Abbasid period , many Classical Greek terms entered Arabic through translations carried out at Baghdad's House of Wisdom . By
973-425: A critic might reject Avicenna's conception of contingency, a starting point in the original proof, by saying that the universe could just happen to exist without being necessary or contingent on an external cause. German philosopher Immanuel Kant (1724–1804) divided arguments for the existence of God into three groups: ontological , cosmological , or teleological . Scholars disagree on whether Avicenna's "Proof of
1112-1081: A dialect of Arabic and written in the Latin alphabet . The Balkan languages, including Albanian, Greek , Serbo-Croatian, and Bulgarian , have also acquired many words of Arabic origin, mainly through direct contact with Ottoman Turkish . Arabic has influenced languages across the globe throughout its history, especially languages where Islam is the predominant religion and in countries that were conquered by Muslims. The most markedly influenced languages are Persian , Turkish , Hindustani ( Hindi and Urdu ), Kashmiri , Kurdish , Bosnian , Kazakh , Bengali , Malay ( Indonesian and Malaysian ), Maldivian , Pashto , Punjabi , Albanian , Armenian , Azerbaijani , Sicilian, Spanish, Greek, Bulgarian, Tagalog , Sindhi , Odia , Hebrew and African languages such as Hausa , Amharic , Tigrinya , Somali , Tamazight , and Swahili . Conversely, Arabic has borrowed some words (mostly nouns) from other languages, including its sister-language Aramaic, Persian, Greek, and Latin and to
1251-485: A distinction between two types of proof for the existence of God: the first is derived from reflection on nothing but existence itself; the second requires reflection on things such as God's creations or God's acts. Avicenna says that the first type is the proof for "the truthful", which is more solid and nobler than the second one, which is proof for a certain "group of people". According to the professor of Islamic philosophy Shams C. Inati , by "the truthful" Avicenna means
1390-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
1529-669: 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)
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#17327661784701668-487: A lesser extent and more recently from Turkish, English, French, and Italian. Arabic is spoken by as many as 380 million speakers, both native and non-native, in the Arab world, making it the fifth most spoken language in the world, and the fourth most used language on the internet in terms of users. It also serves as the liturgical language of more than 2 billion Muslims . In 2011, Bloomberg Businessweek ranked Arabic
1807-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
1946-422: 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
2085-690: A millennium before the modern period . Early lexicographers ( لُغَوِيُّون lughawiyyūn ) sought to explain words in the Quran that were unfamiliar or had a particular contextual meaning, and to identify words of non-Arabic origin that appear in the Quran. They gathered shawāhid ( شَوَاهِد 'instances of attested usage') from poetry and the speech of the Arabs—particularly the Bedouin ʾaʿrāb [ ar ] ( أَعْراب ) who were perceived to speak
2224-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . 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
2363-473: A necessary existent with approval and sometimes with modifications. The phrase wajib al-wujud (necessary existent) became widely used to refer to God, even in the works of Avicenna's staunch critics, a sign of the proof's influence. Outside the Muslim tradition, it is also "enthusiastically" received, repeated, and modified by later philosophers such as Thomas Aquinas (1225–1274) and Duns Scotus (1266–1308) of
2502-423: A piecemeal approach to prove the necessary existent, and then derives God's traditional attributes from it one at a time. This makes each of the arguments subject to separate assessments. Some might accept the proof for the necessary existent while rejecting the other arguments; such a critic could still reject the existence of God. Another type of criticism might attack the proof of the necessary existent itself. Such
2641-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
2780-594: A result, many European languages have borrowed words from it. Arabic influence, mainly in vocabulary, is seen in European languages (mainly Spanish and to a lesser extent Portuguese , Catalan , and Sicilian ) owing to the proximity of Europe and the long-lasting Arabic cultural and linguistic presence, mainly in Southern Iberia, during the Al-Andalus era. Maltese is a Semitic language developed from
2919-462: A script derived from ASA attest to a language known as Hasaitic . On the northwestern frontier of Arabia, various languages known to scholars as Thamudic B , Thamudic D, Safaitic , and Hismaic are attested. The last two share important isoglosses with later forms of Arabic, leading scholars to theorize that Safaitic and Hismaic are early forms of Arabic and that they should be considered Old Arabic . Linguists generally believe that "Old Arabic",
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#17327661784703058-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
3197-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
3336-470: A single language, despite mutual incomprehensibility among differing spoken versions. From a linguistic standpoint, it is often said that the various spoken varieties of Arabic differ among each other collectively about as much as the Romance languages . This is an apt comparison in a number of ways. The period of divergence from a single spoken form is similar—perhaps 1500 years for Arabic, 2000 years for
3475-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . 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
3614-607: A thing that is guaranteed to exist by its essence or intrinsic nature – a necessary existent . The argument tries to prove that there is indeed a necessary existent. It does this by first considering whether the opposite could be true: that everything that exists is contingent. Each contingent thing will need something other than itself to bring it into existence, which will in turn need another cause to bring it into existence, and so on. Because this seemed to lead to an infinite regress , cosmological arguments before Avicenna concluded that some necessary cause (such as God)
3753-507: A type of Arabic. Cypriot Arabic is recognized as a minority language in Cyprus. The sociolinguistic situation of Arabic in modern times provides a prime example of the linguistic phenomenon of diglossia , which is the normal use of two separate varieties of the same language, usually in different social situations. Tawleed is the process of giving a new shade of meaning to an old classical word. For example, al-hatif lexicographically means
3892-507: A variety of regional vernacular Arabic dialects , which are not necessarily mutually intelligible. Classical Arabic is the language found in the Quran , used from the period of Pre-Islamic Arabia to that of the Abbasid Caliphate . Classical Arabic is prescriptive, according to the syntactic and grammatical norms laid down by classical grammarians (such as Sibawayh ) and the vocabulary defined in classical dictionaries (such as
4031-477: 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 . 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
4170-476: A wider audience." In the wake of the industrial revolution and European hegemony and colonialism , pioneering Arabic presses, such as the Amiri Press established by Muhammad Ali (1819), dramatically changed the diffusion and consumption of Arabic literature and publications. Rifa'a al-Tahtawi proposed the establishment of Madrasat al-Alsun in 1836 and led a translation campaign that highlighted
4309-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
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4448-760: Is cosmological . The argument is outlined in Avicenna's various works. The most concise and influential form is found in the fourth "class" of his Remarks and Admonitions ( Al-isharat wa al-tanbihat ). It is also present in Book II, Chapter 12 of the Book of Salvation ( Kitab al-najat ) and throughout the Metaphysics section of the Book of Healing ( al-Shifa ). The passages in Remarks and Admonitions draw
4587-532: Is God. Avicenna is aware of this limitation, and his works contain numerous arguments to show the necessary existent must have the attributes associated with God identified in Islam. For example, Avicenna gives a philosophical justification for the Islamic doctrine of tawhid (oneness of God) by showing the uniqueness and simplicity of the necessary existent. He argues that the necessary existent must be unique, using
4726-737: Is a Central Semitic language of the Afroasiatic language family spoken primarily in the Arab world . The ISO assigns language codes to 32 varieties of Arabic , including its standard form of Literary Arabic, known as Modern Standard Arabic , which is derived from Classical Arabic . This distinction exists primarily among Western linguists; Arabic speakers themselves generally do not distinguish between Modern Standard Arabic and Classical Arabic, but rather refer to both as al-ʿarabiyyatu l-fuṣḥā ( اَلعَرَبِيَّةُ ٱلْفُصْحَىٰ "the eloquent Arabic") or simply al-fuṣḥā ( اَلْفُصْحَىٰ ). Arabic
4865-471: Is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
5004-590: Is a minimum level of comprehension between all Arabic dialects, this level can increase or decrease based on geographic proximity: for example, Levantine and Gulf speakers understand each other much better than they do speakers from the Maghreb. The issue of diglossia between spoken and written language is a complicating factor: A single written form, differing sharply from any of the spoken varieties learned natively, unites several sometimes divergent spoken forms. For political reasons, Arabs mostly assert that they all speak
5143-559: Is a sister language rather than their direct ancestor. Arabia had a wide variety of Semitic languages in antiquity. The term "Arab" was initially used to describe those living in the Arabian Peninsula , as perceived by geographers from ancient Greece . In the southwest, various Central Semitic languages both belonging to and outside the Ancient South Arabian family (e.g. Southern Thamudic) were spoken. It
5282-478: Is believed that the ancestors of the Modern South Arabian languages (non-Central Semitic languages) were spoken in southern Arabia at this time. To the north, in the oases of northern Hejaz , Dadanitic and Taymanitic held some prestige as inscriptional languages. In Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested. In eastern Arabia, inscriptions in
5421-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
5560-408: Is credited with establishing the rules of Arabic prosody . Al-Jahiz (776–868) proposed to Al-Akhfash al-Akbar an overhaul of the grammar of Arabic, but it would not come to pass for two centuries. The standardization of Arabic reached completion around the end of the 8th century. The first comprehensive description of the ʿarabiyya "Arabic", Sībawayhi's al - Kitāb , is based first of all upon
5699-472: Is credited with standardizing Arabic grammar , or an-naḥw ( النَّحو "the way" ), and pioneering a system of diacritics to differentiate consonants ( نقط الإعجام nuqaṭu‿l-i'jām "pointing for non-Arabs") and indicate vocalization ( التشكيل at-tashkīl ). Al-Khalil ibn Ahmad al-Farahidi (718–786) compiled the first Arabic dictionary, Kitāb al-'Ayn ( كتاب العين "The Book of the Letter ع "), and
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5838-513: Is defined by the set of all similar objects and the properties that these objects must have. 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. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
5977-407: 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. 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
6116-493: 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. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
6255-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
6394-617: Is needed to end the infinite chain. However, Avicenna's argument does not preclude the possibility of an infinite regress. Instead, the argument considers the entire collection ( jumla ) of contingent things, the sum total of every contingent thing that exists, has existed, or will exist. Avicenna argues that this aggregate, too, must obey the rule that applies to a single contingent thing; in other words, it must have something outside itself that causes it to exist. This cause has to be either contingent or necessary. It cannot be contingent, though, because if it were, it would already be included within
6533-574: Is not present in the spoken varieties, but deletes Classical words that sound obsolete in MSA. In addition, MSA has borrowed or coined many terms for concepts that did not exist in Quranic times, and MSA continues to evolve. Some words have been borrowed from other languages—notice that transliteration mainly indicates spelling and not real pronunciation (e.g., فِلْم film 'film' or ديمقراطية dīmuqrāṭiyyah 'democracy'). The current preference
6672-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
6811-1192: 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." 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. 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 , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
6950-855: Is official in Mali and recognized as a minority language in Morocco, while the Senegalese government adopted the Latin script to write it. Maltese is official in (predominantly Catholic ) Malta and written with the Latin script . Linguists agree that it is a variety of spoken Arabic, descended from Siculo-Arabic , though it has experienced extensive changes as a result of sustained and intensive contact with Italo-Romance varieties, and more recently also with English. Due to "a mix of social, cultural, historical, political, and indeed linguistic factors", many Maltese people today consider their language Semitic but not
7089-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . 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. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
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#17327661784707228-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
7367-515: Is sometimes mistranslated as a condemnation of mathematicians. 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
7506-572: Is the third most widespread official language after English and French, one of six official languages of the United Nations , and the liturgical language of Islam . Arabic is widely taught in schools and universities around the world and is used to varying degrees in workplaces, governments and the media. During the Middle Ages , Arabic was a major vehicle of culture and learning, especially in science, mathematics and philosophy. As
7645-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
7784-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
7923-590: Is the variety used in most current, printed Arabic publications, spoken by some of the Arabic media across North Africa and the Middle East, and understood by most educated Arabic speakers. "Literary Arabic" and "Standard Arabic" ( فُصْحَى fuṣḥá ) are less strictly defined terms that may refer to Modern Standard Arabic or Classical Arabic. Some of the differences between Classical Arabic (CA) and Modern Standard Arabic (MSA) are as follows: MSA uses much Classical vocabulary (e.g., dhahaba 'to go') that
8062-413: Is to avoid direct borrowings, preferring to either use loan translations (e.g., فرع farʻ 'branch', also used for the branch of a company or organization; جناح janāḥ 'wing', is also used for the wing of an airplane, building, air force, etc.), or to coin new words using forms within existing roots ( استماتة istimātah ' apoptosis ', using the root موت m/w/t 'death' put into
8201-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
8340-429: Is trying to prove. Avicenna remarks, "in a certain way, this is the very thing that is sought". The limitation of the argument so far is that it only shows the existence of a necessary existent, and that is different from showing the existence of God as worshipped in Islam. An atheist might agree that a necessary existent exists, but it could be the universe itself, or there could be many necessary existents, none of which
8479-524: Is used to denote concepts that have arisen in the industrial and post-industrial era , especially in modern times. Due to its grounding in Classical Arabic, Modern Standard Arabic is removed over a millennium from everyday speech, which is construed as a multitude of dialects of this language. These dialects and Modern Standard Arabic are described by some scholars as not mutually comprehensible. The former are usually acquired in families, while
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#17327661784708618-445: The Lisān al-ʻArab ). Modern Standard Arabic (MSA) largely follows the grammatical standards of Classical Arabic and uses much of the same vocabulary. However, it has discarded some grammatical constructions and vocabulary that no longer have any counterpart in the spoken varieties and has adopted certain new constructions and vocabulary from the spoken varieties. Much of the new vocabulary
8757-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. 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
8896-660: 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. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
9035-511: The Pythagoreans appeared to have considered it a subject in its own right. 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. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
9174-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
9313-585: The Western Christian tradition, as well by Jewish philosophers such as Maimonides (d. 1204). Adamson said that one reason for its popularity is that it matches "an underlying rationale for many people's belief in God", which he contrasted with Anselm 's ontological argument , formulated a few years later, which read more like a "clever trick" than a philosophical justification of one's faith. Professor of medieval philosophy Jon McGinnis said that
9452-568: The Xth form , or جامعة jāmiʻah 'university', based on جمع jamaʻa 'to gather, unite'; جمهورية jumhūriyyah 'republic', based on جمهور jumhūr 'multitude'). An earlier tendency was to redefine an older word although this has fallen into disuse (e.g., هاتف hātif 'telephone' < 'invisible caller (in Sufism)'; جريدة jarīdah 'newspaper' < 'palm-leaf stalk'). Colloquial or dialectal Arabic refers to
9591-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
9730-494: The northern Hejaz . These features are evidence of common descent from a hypothetical ancestor , Proto-Arabic . The following features of Proto-Arabic can be reconstructed with confidence: On the other hand, several Arabic varieties are closer to other Semitic languages and maintain features not found in Classical Arabic, indicating that these varieties cannot have developed from Classical Arabic. Thus, Arabic vernaculars do not descend from Classical Arabic: Classical Arabic
9869-399: The philosophers , and the "group of people" means the theologians and others who seek to demonstrate God's existence through his creations. The proof then became known in the Arabic tradition as the "Proof of the Truthful" ( burhan al-siddiqin ). Ibn Sina distinguishes between a thing that needs an external cause in order to exist – a contingent thing – and
10008-419: The "learned" tradition (Classical Arabic). This variety and both its classicizing and "lay" iterations have been termed Middle Arabic in the past, but they are thought to continue an Old Higazi register. It is clear that the orthography of the Quran was not developed for the standardized form of Classical Arabic; rather, it shows the attempt on the part of writers to record an archaic form of Old Higazi. In
10147-716: The "purest," most eloquent form of Arabic—initiating a process of jamʿu‿l-luɣah ( جمع اللغة 'compiling the language') which took place over the 8th and early 9th centuries. Kitāb al-'Ayn ( c. 8th century ), attributed to Al-Khalil ibn Ahmad al-Farahidi , is considered the first lexicon to include all Arabic roots ; it sought to exhaust all possible root permutations —later called taqālīb ( تقاليب ) — calling those that are actually used mustaʿmal ( مستعمَل ) and those that are not used muhmal ( مُهمَل ). Lisān al-ʿArab (1290) by Ibn Manzur gives 9,273 roots, while Tāj al-ʿArūs (1774) by Murtada az-Zabidi gives 11,978 roots. Mathematics Mathematics
10286-454: The 11th and 12th centuries in al-Andalus , the zajal and muwashah poetry forms developed in the dialectical Arabic of Cordoba and the Maghreb. The Nahda was a cultural and especially literary renaissance of the 19th century in which writers sought "to fuse Arabic and European forms of expression." According to James L. Gelvin , " Nahda writers attempted to simplify the Arabic language and script so that it might be accessible to
10425-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
10564-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
10703-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
10842-571: The 4th to the 6th centuries, the Nabataean script evolved into the Arabic script recognizable from the early Islamic era. There are inscriptions in an undotted, 17-letter Arabic script dating to the 6th century CE, found at four locations in Syria ( Zabad , Jebel Usays , Harran , Umm el-Jimal ). The oldest surviving papyrus in Arabic dates to 643 CE, and it uses dots to produce the modern 28-letter Arabic alphabet. The language of that papyrus and of
10981-779: The 8th century, knowledge of Classical Arabic had become an essential prerequisite for rising into the higher classes throughout the Islamic world, both for Muslims and non-Muslims. For example, Maimonides , the Andalusi Jewish philosopher, authored works in Judeo-Arabic —Arabic written in Hebrew script . Ibn Jinni of Mosul , a pioneer in phonology , wrote prolifically in the 10th century on Arabic morphology and phonology in works such as Kitāb Al-Munṣif , Kitāb Al-Muḥtasab , and Kitāb Al-Khaṣāʾiṣ [ ar ] . Ibn Mada' of Cordoba (1116–1196) realized
11120-637: The Middle Ages and made available in Europe. 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 by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
11259-412: The Middle East and North Africa have become a badge of sophistication and modernity and ... feigning, or asserting, weakness or lack of facility in Arabic is sometimes paraded as a sign of status, class, and perversely, even education through a mélange of code-switching practises." Arabic has been taught worldwide in many elementary and secondary schools, especially Muslim schools. Universities around
11398-690: The Qur'an is referred to by linguists as "Quranic Arabic", as distinct from its codification soon thereafter into " Classical Arabic ". In late pre-Islamic times, a transdialectal and transcommunal variety of Arabic emerged in the Hejaz , which continued living its parallel life after literary Arabic had been institutionally standardized in the 2nd and 3rd century of the Hijra , most strongly in Judeo-Christian texts, keeping alive ancient features eliminated from
11537-576: The Romance languages. Also, while it is comprehensible to people from the Maghreb , a linguistically innovative variety such as Moroccan Arabic is essentially incomprehensible to Arabs from the Mashriq , much as French is incomprehensible to Spanish or Italian speakers but relatively easily learned by them. This suggests that the spoken varieties may linguistically be considered separate languages. With
11676-456: The Truthful" is ontological, that is, derived through sheer conceptual analysis, or cosmological, that is, derived by invoking empirical premises (e.g. "a contingent thing exists"). Scholars Herbert A. Davidson, Lenn E. Goodman, Michael E. Marmura, M. Saeed Sheikh, and Soheil Afnan argued that it was cosmological. Davidson said that Avicenna did not regard "the analysis of the concept necessary existent by virtue of itself as sufficient to establish
11815-402: The actual existence of anything in the external world" and that he had offered a new form of cosmological argument. Others, including Parviz Morewedge, Gary Legenhausen , Abdel Rahman Badawi , Miguel Cruz Hernández, and M. M. Sharif , argued that Avicenna's argument was ontological. Morewedge referred to the argument as "Ibn Sina's ontological argument for the existence of God", and said that it
11954-408: The aggregate. Thus the only remaining possibility is that an external cause is necessary, and that cause must be a necessary existent. Avicenna anticipates that one could reject the argument by saying that the collection of contingent things may not be contingent. A whole does not automatically share the features of its parts; for example, in mathematics a set of numbers is not a number. Therefore,
12093-423: The argument include Averroes , who objected to its methodology, Al-Ghazali , who disagreed with its characterization of God, and modern critics who state that its piecemeal derivation of God's attributes allows people to accept parts of the argument but still reject God's existence. There is no consensus among modern scholars on the classification of the argument; some say that it is ontological while others say it
12232-425: The argument one of the most influential medieval arguments for God's existence, and Avicenna's biggest contribution to the history of philosophy. It was enthusiastically received and repeated (sometimes with modification) by later philosophers, including generations of Muslim philosophers, Western Christian philosophers such as Thomas Aquinas and Duns Scotus , and Jewish philosophers such as Maimonides . Critics of
12371-442: The argument requires only a few premises, namely, the distinction between the necessary and the contingent, that "something exists", and that a set subsists through their members (an assumption McGinnis said to be "almost true by definition"). The Islamic Andalusi philosopher Averroes or Ibn Rushd (1126–1198) criticized the argument on its methodology. Averroes, an avid Aristotelian , argued that God's existence has to be shown on
12510-428: The attributes' derivations, Adamson commented that "a complete consideration of Avicenna's derivation of all the attributes ... would need a book-length study". In general, Avicenna derives the attributes based on two aspects of the necessary existent: (1) its necessity, which can be shown to imply its sheer existence and a range of negations (e.g. not being caused, not being multiple), and (2) its status as
12649-477: The basis of the natural world, as Aristotle had done. According to Averroes, a proof should be based on physics , and not on metaphysical reflections as in the "Proof of the Truthful". Other Muslim philosophers such as Al-Ghazali (1058–1111) attacked the argument over its implications that seemed incompatible with God as known through the Islamic revelation. For example, according to Avicenna, God can have no features or relations that are contingent, so his causing of
12788-583: The beginnings of algebra (Diophantus, 3rd 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 . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
12927-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
13066-574: The conversion of Semitic mimation to nunation in the singular. It is best reassessed as a separate language on the Central Semitic dialect continuum. It was also thought that Old Arabic coexisted alongside—and then gradually displaced— epigraphic Ancient North Arabian (ANA), which was theorized to have been the regional tongue for many centuries. ANA, despite its name, was considered a very distinct language, and mutually unintelligible, from "Arabic". Scholars named its variant dialects after
13205-399: 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
13344-569: 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. 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. The Pythagoreans were likely
13483-587: The emergence of Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include: There are several features which Classical Arabic, the modern Arabic varieties, as well as the Safaitic and Hismaic inscriptions share which are unattested in any other Central Semitic language variety, including the Dadanitic and Taymanitic languages of
13622-412: The entire set of contingent things must have a cause that is not contingent because otherwise it would be included in the set. Furthermore, through a series of arguments, he derived that the necessary existent must have attributes that he identified with God in Islam , including unity, simplicity, immateriality, intellect, power, generosity, and goodness. Historian of philosophy Peter Adamson called
13761-728: The eve of the conquests: Northern and Central (Al-Jallad 2009). The modern dialects emerged from a new contact situation produced following the conquests. Instead of the emergence of a single or multiple koines, the dialects contain several sedimentary layers of borrowed and areal features, which they absorbed at different points in their linguistic histories. According to Veersteegh and Bickerton, colloquial Arabic dialects arose from pidginized Arabic formed from contact between Arabs and conquered peoples. Pidginization and subsequent creolization among Arabs and arabized peoples could explain relative morphological and phonological simplicity of vernacular Arabic compared to Classical and MSA. In around
13900-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
14039-607: The fact that they participate in the innovations common to all forms of Arabic. The earliest attestation of continuous Arabic text in an ancestor of the modern Arabic script are three lines of poetry by a man named Garm(')allāhe found in En Avdat, Israel , and dated to around 125 CE. This is followed by the Namara inscription , an epitaph of the Lakhmid king Imru' al-Qays bar 'Amro, dating to 328 CE, found at Namaraa, Syria. From
14178-567: 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. 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",
14317-510: The fourth most useful language for business, after English, Mandarin Chinese , and French. Arabic is written with the Arabic alphabet , an abjad script that is written from right to left . Arabic is usually classified as a Central Semitic language . Linguists still differ as to the best classification of Semitic language sub-groups. The Semitic languages changed between Proto-Semitic and
14456-597: The inclusion of new words into their published standard dictionaries. They also publish old and historical Arabic manuscripts. In 1997, a bureau of Arabization standardization was added to the Educational, Cultural, and Scientific Organization of the Arab League . These academies and organizations have worked toward the Arabization of the sciences, creating terms in Arabic to describe new concepts, toward
14595-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
14734-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
14873-613: The language. Software and books with tapes are an important part of Arabic learning, as many of Arabic learners may live in places where there are no academic or Arabic language school classes available. Radio series of Arabic language classes are also provided from some radio stations. A number of websites on the Internet provide online classes for all levels as a means of distance education; most teach Modern Standard Arabic, but some teach regional varieties from numerous countries. The tradition of Arabic lexicography extended for about
15012-604: The late 6th century AD, a relatively uniform intertribal "poetic koine" distinct from the spoken vernaculars developed based on the Bedouin dialects of Najd , probably in connection with the court of al-Ḥīra . During the first Islamic century, the majority of Arabic poets and Arabic-writing persons spoke Arabic as their mother tongue. Their texts, although mainly preserved in far later manuscripts, contain traces of non-standardized Classical Arabic elements in morphology and syntax. Abu al-Aswad al-Du'ali ( c. 603 –689)
15151-420: The latter is taught in formal education settings. However, there have been studies reporting some degree of comprehension of stories told in the standard variety among preschool-aged children. The relation between Modern Standard Arabic and these dialects is sometimes compared to that of Classical Latin and Vulgar Latin vernaculars (which became Romance languages ) in medieval and early modern Europe. MSA
15290-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
15429-883: The many national or regional varieties which constitute the everyday spoken language. Colloquial Arabic has many regional variants; geographically distant varieties usually differ enough to be mutually unintelligible , and some linguists consider them distinct languages. However, research indicates a high degree of mutual intelligibility between closely related Arabic variants for native speakers listening to words, sentences, and texts; and between more distantly related dialects in interactional situations. The varieties are typically unwritten. They are often used in informal spoken media, such as soap operas and talk shows , as well as occasionally in certain forms of written media such as poetry and printed advertising. Hassaniya Arabic , Maltese , and Cypriot Arabic are only varieties of modern Arabic to have acquired official recognition. Hassaniya
15568-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
15707-428: The necessary existent in multiple texts in order to justify its identification with God. He shows that the necessary existent must also be immaterial, intellective , powerful, generous, of pure good ( khayr mahd ), willful ( irada ), "wealthy" or "sufficient" ( ghani ), and self-subsistent ( qayyum ), among other qualities. These attributes often correspond to the epithets of God found in the Quran . In discussing some of
15846-539: The necessary existent must be simple (not a composite) by a similar reductio strategy. If it were a composite, its internal parts would need a feature that distinguishes each from the others. The distinguishing feature cannot be solely derived from the parts' necessity of existence, because then they would both have the same feature and not be distinct: a contradiction. But it also cannot be accidental , or requiring an outside cause, because this would contradict its necessity of existence. Avicenna derives other attributes of
15985-782: The need for a lexical injection in Arabic, to suit concepts of the industrial and post-industrial age (such as sayyārah سَيَّارَة 'automobile' or bākhirah باخِرة 'steamship'). In response, a number of Arabic academies modeled after the Académie française were established with the aim of developing standardized additions to the Arabic lexicon to suit these transformations, first in Damascus (1919), then in Cairo (1932), Baghdad (1948), Rabat (1960), Amman (1977), Khartum [ ar ] (1993), and Tunis (1993). They review language development, monitor new words and approve
16124-412: The objection goes, the step in the argument that assumes that the collection of contingent things is also contingent, is wrong. However, Avicenna dismisses this counter-argument as a capitulation, and not an objection at all. If the entire collection of contingent things is not contingent, then it must be necessary. This also leads to the conclusion that there is a necessary existent, the very thing Avicenna
16263-536: 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
16402-424: The one whose sound is heard but whose person remains unseen. Now the term al-hatif is used for a telephone. Therefore, the process of tawleed can express the needs of modern civilization in a manner that would appear to be originally Arabic. In the case of Arabic, educated Arabs of any nationality can be assumed to speak both their school-taught Standard Arabic as well as their native dialects, which depending on
16541-549: The overhaul of Arabic grammar first proposed by Al-Jahiz 200 years prior. The Maghrebi lexicographer Ibn Manzur compiled Lisān al-ʿArab ( لسان العرب , "Tongue of Arabs"), a major reference dictionary of Arabic, in 1290. Charles Ferguson 's koine theory claims that the modern Arabic dialects collectively descend from a single military koine that sprang up during the Islamic conquests; this view has been challenged in recent times. Ahmad al-Jallad proposes that there were at least two considerably distinct types of Arabic on
16680-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . 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. Evidence for more complex mathematics does not appear until around 3000 BC , when
16819-658: The proof of numerous theorems. 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 . 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 has since been greatly extended, and there has been
16958-410: The region may be mutually unintelligible. Some of these dialects can be considered to constitute separate languages which may have "sub-dialects" of their own. When educated Arabs of different dialects engage in conversation (for example, a Moroccan speaking with a Lebanese), many speakers code-switch back and forth between the dialectal and standard varieties of the language, sometimes even within
17097-458: The same sentence. The issue of whether Arabic is one language or many languages is politically charged, in the same way it is for the varieties of Chinese , Hindi and Urdu , Serbian and Croatian , Scots and English, etc. In contrast to speakers of Hindi and Urdu who claim they cannot understand each other even when they can, speakers of the varieties of Arabic will claim they can all understand each other even when they cannot. While there
17236-458: The sole example of Medieval linguist Abu Hayyan al-Gharnati – who, while a scholar of the Arabic language, was not ethnically Arab – Medieval scholars of the Arabic language made no efforts at studying comparative linguistics, considering all other languages inferior. In modern times, the educated upper classes in the Arab world have taken a nearly opposite view. Yasir Suleiman wrote in 2011 that "studying and knowing English or French in most of
17375-563: The standardization of these new terms throughout the Arabic-speaking world, and toward the development of Arabic as a world language . This gave rise to what Western scholars call Modern Standard Arabic. From the 1950s, Arabization became a postcolonial nationalist policy in countries such as Tunisia, Algeria, Morocco, and Sudan. Arabic usually refers to Standard Arabic, which Western linguists divide into Classical Arabic and Modern Standard Arabic. It could also refer to any of
17514-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
17653-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
17792-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
17931-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
18070-501: The towns where the inscriptions were discovered (Dadanitic, Taymanitic, Hismaic, Safaitic). However, most arguments for a single ANA language or language family were based on the shape of the definite article, a prefixed h-. It has been argued that the h- is an archaism and not a shared innovation, and thus unsuitable for language classification, rendering the hypothesis of an ANA language family untenable. Safaitic and Hismaic, previously considered ANA, should be considered Old Arabic due to
18209-411: The two are not distinct after all. If, on the other hand, the distinction resulted from something not implied by necessity of existence, then this individuating factor will be a cause for A, and this means that A has a cause and is not a necessary existent after all. Either way, the opposite proposition resulted in contradiction, which to Avicenna proves the correctness of the argument. Avicenna argued that
18348-456: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
18487-470: The universe must be necessary. Al-Ghazali disputed this as incompatible with the concept of God's untrammelled free will as taught in Al-Ghazali's Asharite theology. He further argued that God's free choice can be shown by the arbitrary nature of the exact size of the universe or the time of its creation. Peter Adamson offered several more possible lines of criticism. He pointed out that Avicenna adopts
18626-451: The world have classes that teach Arabic as part of their foreign languages , Middle Eastern studies , and religious studies courses. Arabic language schools exist to assist students to learn Arabic outside the academic world. There are many Arabic language schools in the Arab world and other Muslim countries. Because the Quran is written in Arabic and all Islamic terms are in Arabic, millions of Muslims (both Arab and non-Arab) study
18765-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
18904-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
19043-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
19182-501: Was purely based on his analytic specification of this concept [the Necessary Existent]." Steve A. Johnson and Toby Mayer said the argument was a hybrid of the two. Arabic language Arabic (endonym: اَلْعَرَبِيَّةُ , romanized : al-ʿarabiyyah , pronounced [al ʕaraˈbijːa] , or عَرَبِيّ , ʿarabīy , pronounced [ˈʕarabiː] or [ʕaraˈbij] )
19321-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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