128-484: Kispu or kispum was an ancient Mesopotamian ritual in which the ancestors were venerated , nourished and cared. The ritual included regular offering of food and drinks to the deads. Textual evidence for the kispu were found from as early as the 3rd millennium BC ; these evidence indicate the ceremony included sacrifices and offerings. The textual evidence allow identification of archaeological findings, such as masses of dining vessels near secondary burial skeletons
256-595: A sar being 3,600 years, this would indicate that at least some of the Mesopotamians believed that the Earth would only last 518,400 years. Berossus does not report what was thought to follow this event, however. As with most dead religions, many aspects of the common practices and intricacies of the doctrine have been lost and forgotten over time. However, much of the information and knowledge has survived, and great work has been done by historians and scientists, with
384-591: 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
512-648: A certain deity; this practice appeared to have begun in the third millennium BC among the Sumerians, but also was later adopted by the Akkadians, Assyrians and Babylonians as well. Initially, the pantheon was not ordered, but later Mesopotamian theologians came up with the concept of ranking the deities in order of importance. A Sumerian list of around 560 deities that did this was uncovered at Farm and Tell Abû Ṣalābīkh and dated to circa 2600 BC, ranking five primary deities as being of particular importance. One of
640-399: A divine act of creation, and the gods were believed to be the source of life, and held power over sickness and health, as well as the destinies of men. Personal names show that each child was considered a gift from divinity. Man was believed to have been created to serve the gods, or perhaps wait on them: the god is lord ( belu ) and man is servant or slave ( ardu ), and was to fear ( puluhtu )
768-669: 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)
896-461: A giant altar. Other theories treat the tower as an image of the cosmic mountain where a dying and rising god "lay buried." Some temples, such as the temple of Enki in Eridu contained a holy tree ( kiskanu ) in a holy grove, which was the central point of various rites performed by the king, who functioned as a "master gardener." Mesopotamian temples were originally built to serve as dwelling places for
1024-425: A great extent, an oligarchy rather than a monarchy. Authority was considered to lie with "the city", and the polity had three main centres of power—an assembly of elders, a hereditary ruler, and an eponym . The ruler presided over the assembly and carried out its decisions. He was not referred to with the usual Akkadian term for "king", šarrum ; that was instead reserved for the city's patron deity Ashur , of whom
1152-404: 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
1280-422: 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
1408-402: 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
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#17327730226871536-535: 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
1664-411: 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
1792-424: 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
1920-409: A stable for the god's chariot and draft animals. Generally, the god's well-being was maintained through service, or work ( dullu ). The image was dressed and served banquets twice a day. It is not known how the god was thought to consume the food, but a curtain was drawn before the table while he or she "ate", just as the king himself was not allowed to be seen by the masses while he ate. Occasionally,
2048-418: 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
2176-477: 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
2304-703: 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
2432-403: 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
2560-513: 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
2688-628: Is depicted as a system which, according to the author, continued to hold sway in the first century AD, eventually to be utterly annihilated (17:5; 18:9). According to some interpretations, this is believed to refer to the Roman Empire , but according to other interpretations, this system remains extant in the world until the Second Coming . Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for
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#17327730226872816-407: 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
2944-429: Is far-reaching; his word is sublime and holy. His decisions are unalterable; he decides fate forever! His eyes scrutinize the entire world!" A prayer to the god Enlil. Each Mesopotamian city was home to a deity, and each of the prominent deities was the patron of a city, and all known temples were located in cities, though there may have been shrines in the suburbs. The temple itself was constructed of mud brick in
3072-493: 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
3200-586: 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
3328-404: 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,
3456-1192: 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
3584-547: 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
3712-433: 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
3840-567: 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
3968-418: 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,
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4096-405: 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
4224-508: 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
4352-586: 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
4480-768: 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
4608-511: 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
4736-536: 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
4864-446: 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
4992-640: The creation myth , the Garden of Eden , the flood myth , the Tower of Babel , figures such as Nimrod and Lilith and the Book of Esther . It has also inspired various contemporary neo-pagan groups. In the New Testament Book of Revelation , Babylonian religion is associated with religious apostasy of the lowest order, the archetype of a political/religious system heavily tied to global commerce, and it
5120-420: The worship of forces of nature as providers of sustenance. In the 3rd millennium BC , objects of worship were personified and became an expansive cast of divinities with particular functions. The last stages of Mesopotamian polytheism, which developed in the 2nd and 1st millennia BC, introduced greater emphasis on personal religion and structured the gods into a monarchical hierarchy, with the national god being
5248-425: The "conscious everyday world into the trance world of spiritual ecstasy." Mesopotamians believed in an afterlife that took place in a region below the surface of the earth inhabited by living humans. This was the ancient Mesopotamian underworld , known by many names including Arallû , Ganzer or Irkallu ("Great Below"). It was believed everyone went to this region after death, irrespective of social status or
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5376-400: 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
5504-405: 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
5632-532: 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
5760-846: The Achaemenids in 522 and 482 BC. During this period the Syriac language and Syriac alphabet evolved in Assyria among the Assyrian people, and were centuries later to be the vehicle for the spread of Syriac Christianity throughout the near east. Then, two centuries later in 330 BC, the Macedonian emperor Alexander the Great overthrew the Persians and took control of Mesopotamia itself. After Alexander's death, increased Hellenistic influence
5888-493: The Adapa myth resignedly relate that, due to a blunder, all men must die and that true everlasting life is the sole property of the gods. There are no known Mesopotamian tales about the end of the world , although it has been speculated that they believed that this would eventually occur. This is largely because Berossus wrote that the Mesopotamians believed the world to last "twelve times twelve sars " in his Babyloniaca ; with
6016-674: The Assyrians are still present today. During the Parthian Empire there was a major revival in Assyria between the second century BC and fourth century AD, with temples once more being dedicated to gods such as Assur , Sin , Shamash , Hadad and Ishtar in various Parthian vassal states in Mesopotamia. In the third century AD, Manichaeism , which incorporated elements of Christianity, Judaism, Buddhism , Zoroastrianism , and local Mesopotamian religion, developed. With
6144-616: The Mesopotamian world. A number of scholars once argued that defining a single Mesopotamian religion was not possible, and as such, a systematic exposition of Mesopotamian religion should not be produced. Other have rebutted that this is a mistaken approach, insofar as it would fracture the study of religion among social divisions (such as private religion, religion of the educated), individual cities and provinces ( Ebla , Mari , Assyria ), and time periods ( Seleucid , Achaemenid , etc), and that this fracture would be counterproductive as
6272-637: 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 ,
6400-737: The Sumerian moon god Nanna became the Akkadian Sîn while the Sumerian sun god Utu became the Akkadian Shamash . One of the most notable goddesses was the Sumerian sex and war deity Inanna . With the later rise to power of the Babylonians in the 18th century BC, the king, Hammurabi , declared Marduk, a deity who before then had not been of significant importance, to a position of supremacy alongside Anu and Enlil in southern Mesopotamia. Although Marduk may have been modelled after
6528-648: The Sumerian storm-god Ninurta ; whose exploits share great similarity. Perhaps the most significant legend to survive from Mesopotamian religion is the Epic of Gilgamesh , which tells the story of the heroic king Gilgamesh and his wild friend Enkidu , and the former's search for immortality which is entwined with all the gods and their approval. It also contains the earliest reference to The Great Flood . Akkadian religion sometimes took inspiration from influential Sumerian religious leaders and beliefs, and deified Sumerian kings at some points. " Enlil ! his authority
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#17327730226876656-493: The actions performed during their lifetime. Unlike Christian hell , the Mesopotamians considered the underworld neither a punishment nor a reward. Nevertheless, the condition of the dead was hardly considered the same as the life previously enjoyed on earth: they were considered merely weak and powerless ghosts. The myth of Ishtar's descent into the underworld relates that "dust is their food and clay their nourishment, they see no light, where they dwell in darkness." Stories such as
6784-583: 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
6912-578: The behavior of birds ( augury ) and observing celestial and meteorological phenomena ( astrology ), as well as through interpretation of dreams. Often interpretation of these phenomena required the need for two classes of priests: askers (sa'ilu) and observer ( baru ), and also sometimes a lower class of ecstatic seer (mahhu) that was also associated with witchcraft . "Do not return evil to the man who disputes with you, requite with kindness your evil-doer, maintain justice to your enemy... Let not your heart be induced to do evil... Give food to eat, beer to drink,
7040-493: The city, which included the power to detain people and confiscate property. The institution of the eponym as well as the formula iššiak Assur lingered on as ceremonial vestiges of this early system throughout the history of the Assyrian monarchy. The religion of the Neo-Assyrian Empire centered around the Assyrian king as the king of their lands as well. However, kingship at the time was linked very closely with
7168-429: The civilizations of ancient Mesopotamia , particularly Sumer , Akkad , Assyria and Babylonia between circa 6000 BC and 400 AD. The religious development of Mesopotamia and Mesopotamian culture in general, especially in the south, were not particularly influenced by the movements of the various peoples into and throughout the area. Rather, Mesopotamian religion was a consistent and coherent tradition, which adapted to
7296-506: The claim that the Mesopotamian's "entire existence was infused by their religiosity, just about everything they have passed on to us can be used as a source of knowledge about their religion." The modern study of Mesopotamia ( Assyriology ) is still a fairly young science, beginning only in the middle of the Nineteenth century, and the study of Mesopotamian religion can be a complex and difficult subject because, by nature, their religion
7424-511: 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,
7552-399: 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
7680-401: The deity approved, it would accept the image and agree to "inhabit" it. These images were also entertained, and sometime escorted on hunting expeditions. In order to service the gods, the temple was equipped with a household with kitchens and kitchenware, sleeping rooms with beds and side rooms for the deity's family, as well as a courtyard with a basin and water for cleansing visitors, as well as
7808-569: 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
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#17327730226877936-556: 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
8064-583: The exception of the Enuma Elish , there are no surviving records that systematically explain Mesopotamian cosmology. Nonetheless, modern scholars have reconstructed a roughly accurate depiction from the surviving evidence, including Sumerian and Akkadian cuneiform and the Hebrew Bible . In the Enuma Elish (ca. 12th century BC), the god Marduk kills Tiamat , the mother of the gods, and, from
8192-428: 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
8320-651: The first records, and are believed to have been the founders of the civilization of the Ubaid period (6500 BC to 3800 BC) in Upper Mesopotamia . By historical times they resided in southern Mesopotamia, which was known as Sumer (and much later, Babylonia ), and had considerable influence on the Akkadian speakers and their culture. Akkadian speakers are believed to have entered the region at some point between 3500 BC and 3000 BC, with Akkadian names first appearing in
8448-567: 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",
8576-415: The form of a ziggurat , which rose to the sky in a series of stairstep stages. Its significance and symbolism have been the subject of much discussion, but most regard the tower as a kind of staircase or ladder for the god to descend from and ascend to the heavens, though there are signs which point towards an actual cult having been practiced in the upper temple, so the entire temple may have been regarded as
8704-469: The god Marduk with Babylon . Though the full number of gods and goddesses found in Mesopotamia is not known, K. Tallqvist, in his Akkadische Götterepitheta (1938) counted around 2,400 that scholars know, most of which had Sumerian names. In the Sumerian language , the gods were referred to as dingir , while in the Akkadian language they were known as ilu and it seems that there was syncretism between
8832-474: The god that they are describing above all others. The historian J. Bottéro stated that these poems display "extreme reverence, profound devotion, [and] the unarguable emotion that the supernatural evoked in the hearts of those ancient believers" but that they showed a people who were scared of their gods rather than openly celebrating them. They were thought to offer good luck, success, and protection from disease and demons, and one's place and success in society
8960-689: The god, who was thought to reside and hold court on earth for the good of the city and kingdom. His presence was symbolized by an image of the god in a separate room. The god's presence within the image seems to have been thought of in a very concrete way, as instruments for the presence of the deity." This is evident from the poem How Erra Wrecked the World , in which Erra deceived the god Marduk into leaving his cult statue . Once constructed, idols were consecrated through special nocturnal rituals where they were given "life", and their mouth "was opened" ( pet pî ) and washed ( mes pî ) so they could see and eat. If
9088-415: The gods and have the appropriate attitude towards them. Duties seem to have been primarily of a cultic and ritual nature, although some prayers express a positive psychological relationship, or a sort of conversion experience in regard to a god. Generally the reward to mankind is described as success and long life. Every man also had duties to his fellow man which had some religious character, particularly
9216-413: The gods as well. Besides demons, there were also spirits of the dead , (etimmu) who could also cause mischief. Amulets were occasionally used, and sometimes a special priest or exorcist ( āšipu or mašmašu) was required. Incantations and ceremonies were also used to cure diseases which were also thought to be associated with demonic activity, sometimes making use of sympathetic magic . Sometimes an attempt
9344-553: The gods worshipped by the two groups, adopting one another's deities. The Mesopotamian gods bore many similarities with humans, and were anthropomorphic , thereby having humanoid form. Similarly, they often acted like humans, requiring food and drink, as well as drinking alcohol and subsequently suffering the effects of drunkenness , but were thought to have a higher degree of perfection than common men. They were thought to be more powerful, all-seeing and all-knowing, unfathomable, and, above all, immortal. One of their prominent features
9472-437: The gods, and how kings were used as a tool for deliverance. Sumerian myths suggest a prohibition against premarital sex. Marriages were often arranged by the parents of the bride and groom; engagements were usually completed through the approval of contracts recorded on clay tablets. These marriages became legal as soon as the groom delivered a bridal gift to his bride's father. Nonetheless, evidence suggests that premarital sex
9600-444: The gods. Punishment came through sickness or misfortune, which inevitably lead to the common reference to unknown sins, or the idea that one can transgress a divine prohibition without knowing it—psalms of lamentation rarely mention concrete sins. This idea of retribution was also applied to the nation and history as a whole. A number of examples of Mesopotamian literature show how war and natural disasters were treated as punishment from
9728-593: The head of the pantheon. Mesopotamian religion finally declined with the spread of Iranian religions during the Achaemenid Empire and with the Christianization of Mesopotamia. The very earliest undercurrents of Mesopotamian religious thought are believed to have developed in the first half of the sixth millennium BC, at the time people first began to permanently settle in Mesopotamia owing to improved irrigation. The early religious developments of
9856-487: The help of religious scholars and translators, to re-construct a working knowledge of the religious history, customs, and the role these beliefs played in everyday life in Sumer, Akkad, Assyria, Babylonia, Ebla and Chaldea during this time. Mesopotamian religion is thought to have been an influence on subsequent religions throughout the world, including Canaanite /Israelite, Aramean , and ancient Greek . Mesopotamian religion
9984-453: The idea of divine mandate. The Assyrian king, while not being a god himself, was acknowledged as the chief servant of the chief god, Ashur . In this manner, the king's authority was seen as absolute so long as the high priest reassured the peoples that the gods, or in the case of the henotheistic Assyrians, the god, was pleased with the current ruler. For the Assyrians who lived in Assur and
10112-422: The influence of Babylonian mythology on Jewish mythology and other Near Eastern mythologies, albeit indirect. Indeed, similarities between both religious traditions may draw from even older sources. In March 2020, archaeologists announced the discovery of a 5,000-year-old cultic area filled with more than 300 broken ceremonial ceramic cups, bowls, jars, animal bones and ritual processions dedicated to Ningirsu at
10240-491: 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
10368-411: The internal needs of its adherents over millennia of development. The earliest undercurrents of Mesopotamian religious thought are believed to have developed in Mesopotamia in the 6th millennium BC , coinciding with when the region began to be permanently settled. The earliest evidence of Mesopotamian religion dates to the mid- 4th millennium BC , coincides with the invention of writing , and involved
10496-400: 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
10624-420: The king shared in these meals, and the priests may have had some share in the offerings as well. Incense was also burned before the image, because it was thought that the gods enjoyed the smell. Sacrificial meals were also set out regularly, with a sacrificial animal seen as a replacement ( pūhu ) or substitute ( dinānu ) for a man, and it was considered that the anger of the gods or demons was then directed towards
10752-533: The king's duties to his subjects. It was thought that one of the reasons the gods gave power to the king was to exercise justice and righteousness, described as mēšaru and kettu , literally "straightness, rightness, firmness, truth". Examples of this include not alienating and causing dissension between friends and relatives, setting innocent prisoners free, being truthful, being honest in trade, respecting boundary lines and property rights, and not putting on airs with subordinates. Some of these guidelines are found in
10880-409: 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
11008-458: The most important of these early Mesopotamian deities was the god Enlil , who was originally a Sumerian divinity viewed as a king of the gods and a controller of the world, who was later adopted by the Akkadians. Another was the Sumerian god An, who served a similar role to Enlil and became known as Anu among the Akkadians. The Sumerian god Enki was later also adopted by the Akkadians, initially under his original name, and later as Éa . Similarly
11136-400: 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
11264-409: 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 a foundation for all mathematics). Mathematics involves
11392-536: 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
11520-408: The one begging for alms honor, clothe; in this a man's god takes pleasure, it is pleasing to Shamash, who will repay him with favour. Be helpful, do good" Incantation from the Šurpu series. Although ancient paganism tended to focus more on duty and ritual than morality, a number of general moral virtues can be gleaned from surviving prayers and myths. It was believed that man originated as
11648-521: 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
11776-537: The personal gods changed over time and little is known about early practice as they are rarely named or described. In the mid-third millennium BC, some rulers regarded a particular god or gods as being their personal protector. In the second millennium BC, personal gods began to function more on behalf of the common man, with whom he had a close, personal relationship, maintained through prayer and maintenance of his god's statue. A number of written prayers have survived from ancient Mesopotamia, each of which typically exalt
11904-658: 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
12032-426: The purification of the temple ( mašmašu ), priests to appease the wrath of the gods with song and music ( kalū ), as well as female singers ( nāru ), male singers ( zammeru ), craftsmen ( mārē ummāni ), swordbearers ( nāš paṭri ), masters of divination ( bārû ), penitents ( šā'ilu ), and others. Besides the worship of the gods at public rituals, individuals also paid homage to a personal deity. As with other deities,
12160-475: The region are unknown since they preceded the invention of writing. The first evidence for what is recognisably Mesopotamian religion can be seen with the invention in Mesopotamia of writing circa 3500 BC. The people of Mesopotamia originally consisted of two groups, East Semitic speakers of Akkadian and the people of Sumer , who spoke Sumerian , a language isolate . These peoples were members of various city-states and small kingdoms . The Sumerians left
12288-547: The region, particularly numerous literary sources, which are usually written in Sumerian , Akkadian ( Assyro-Babylonian ) or Aramaic using cuneiform script on clay tablets and which describe both mythology and cultic practices. Other artifacts can also be useful when reconstructing Mesopotamian religion. As is common with most ancient civilizations, the objects made of the most durable and precious materials, and thus more likely to survive, were associated with religious beliefs and practices. This has prompted one scholar to make
12416-600: The regnal lists of these states c. 29th century BC. The Sumerians were advanced: as well as inventing writing, they developed early forms of mathematics , early wheeled vehicles/ chariots , astronomy , astrology , written code of law , organised medicine , advanced agriculture and architecture , and the calendar . They created the first city-states such as Uruk , Ur , Lagash , Isin , Kish , Umma , Eridu , Adab , Akshak , Sippar , Nippur and Larsa , each of them ruled by an ensí . The Sumerians remained largely dominant in this synthesised culture, however, until
12544-718: The religions of the Near East were rooted in Babylonian astral science- including the Hebrew Bible and Judaism. This theory of a Babylonian-derived Bible originated from the discovery of a stele in the acropolis of Susa bearing a Babylonian flood myth with many similarities to the flood of Genesis, the Epic of Gilgamesh . However, flood myths appear in almost every culture around the world, including cultures that never had contact with Mesopotamia. The fundamental tenets of Panbabylonism were eventually dismissed as pseudoscientific, however Assyriologists and biblical scholars recognize
12672-467: The rise of the Akkadian Empire under Sargon of Akkad circa 2335 BC, which united all of Mesopotamia under one ruler. There was increasing syncretism between the Sumerian and Akkadian cultures and deities, with the Akkadians typically preferring to worship fewer deities but elevating them to greater positions of power. Like many nations in Mesopotamian history, Assyria was originally, to
12800-433: The royal hypogeum at Qatna , with the kispu. This Ancient Near East –related article is a stub . You can help Misplaced Pages by expanding it . This death -related article is a stub . You can help Misplaced Pages by expanding it . Ancient Mesopotamian religion Mesopotamian religion refers to the religious beliefs (concerning the gods, creation and the cosmos , the origin of man, and so forth) and practices of
12928-425: The ruler was the high priest . The ruler himself was only designated as "steward of Assur" ( iššiak Assur ), where the term for steward is a borrowing from Sumerian ensí . The third centre of power was the eponym ( limmum ), who gave the year his name, similarly to the eponymous archon and Roman consuls of classical antiquity . He was annually elected by lot and was responsible for the economic administration of
13056-530: The sacrificial animal. Additionally, certain days required extra sacrifices and ceremonies for certain gods, and every day was sacred to a particular god. The king was thought, in theory, to be the religious leader ( enu or šangū ) of the cult and exercised a large number of duties within the temple, with a large number of specialists whose task was to mediate between men and gods: a supervising or "watchman" priest ( šešgallu ), priests for individual purification against demons and magicians ( āšipu ), priests for
13184-401: The second tablet of the Šurpu incantation series. Sin, on the other hand, was expressed by the words hitu (mistake, false step), annu or arnu (rebellion), and qillatu (sin or curse), with strong emphasis on the idea of rebellion, sometimes with the idea that sin is man's wishing to "live on his own terms" (ina ramanisu). Sin also was described as anything which incited the wrath of
13312-608: The site of Girsu . One of the remains was a duck-shaped bronze figurine with eyes made from bark which is thought to be dedicated to Nanshe . While Mesopotamian religion had almost completely died out by approximately 400–500 AD after its indigenous adherents had largely become Assyrian Christians , it has still had an influence on the modern world, predominantly because many biblical stories that are today found in Judaism , Christianity , Islam and Mandaeism were possibly based upon earlier Mesopotamian myths, in particular that of
13440-657: 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
13568-568: 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
13696-432: The succession of ancient near eastern states did not impact the presence of a broadly shared religious system across them. According to Panbabylonism , a school of thought founded by Hugo Winckler and held in the early 20th century among primarily German Assyriologists, there was a common cultural system extending over the ancient Near East which was overwhelmingly influenced by the Babylonians. According to this theory
13824-551: The surrounding lands, this system was the norm. For the conquered peoples, however, it was novel, particularly to the people of smaller city-states. In time, Ashur was promoted from being the local deity of Assur to the overlord of the vast Assyrian domain, which spread from the Caucasus and Armenia in the north to Egypt, Nubia and the Arabian Peninsula in the south, and from Cyprus and the eastern Mediterranean Sea in
13952-672: 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,
14080-487: 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
14208-462: The two halves of her carcass, constructs the heavens and the earth to shape the modern observable cosmos. A document from a similar period stated that the heavens and the earth can each be divided into three layers. The gods dwell in the higher levels of heaven, with the stars being fixed into the lowermost heavenly layer. Mesopotamian religion was polytheistic , thereby accepting the existence of many different deities, both male and female, though it
14336-508: 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
14464-497: The west to central Iran in the east. Assur , the patron deity of the city of Assur from the late Bronze Age , was in constant rivalry with the patron deity of Babylon, Marduk . Worship was conducted in his name throughout the lands dominated by the Assyrians. With the worship of Assur across much of the Fertile Crescent , the Assyrian king could command the loyalty of his fellow servants of Assur. In 539 BC, Mesopotamia
14592-411: Was polytheistic , worshipping over 2,100 different deities, many of which were associated with a specific state within Mesopotamia, such as Sumer, Akkad, Assyria or Babylonia, or a specific Mesopotamian city. Mesopotamian religion has historically the oldest body of recorded literature of any religious tradition. What is known about Mesopotamian religion comes from archaeological evidence uncovered in
14720-425: Was a common, but surreptitious, occurrence. The worship of Inanna/Ishtar, which was prevalent in Mesopotamia could involve wild, frenzied dancing and bloody ritual celebrations of social and physical abnormality. It was believed that "nothing is prohibited to Inanna", and that by depicting transgressions of normal human social and physical limitations, including traditional gender definition, one could cross over from
14848-475: Was a strong belief in demons in Mesopotamia, and private individuals, like the temple priests, also participated in incantations (šiptu) to ward them off. Although there was no collective term for these beings either in Sumerian or Akkadian, they were merely described as harmful or dangerous beings or forces, and they were used as a logical way to explain the existence of evil in the world. They were thought to be countless in number, and were thought to even attack
14976-534: Was a terrifying brightness ( melammu ) which surrounded them, producing an immediate reaction of awe and reverence among men. In many cases, the various deities were family relations of one another, a trait found in many other polytheistic religions. The historian J. Bottéro was of the opinion that the gods were not viewed mystically , but were instead seen as high-up masters who had to be obeyed and feared, as opposed to loved and adored. Nonetheless, many Mesopotamians, of all classes, often had names that were devoted to
15104-419: Was also henotheistic , with certain gods being viewed as superior to others by their specific devotees. These devotees were often from a particular city or city-state that held that deity as its tutelary deity ; for instance, Enki was often associated with the city of Eridu in Sumer, Assur with Assur and Assyria , Enlil with the Sumerian city of Nippur , Ishtar with the Assyrian city of Arbela , and
15232-402: Was believed that the gods expressed their will through "words" (amatu) and "commandments" (qibitu) which were not necessarily spoken, but were thought to manifest in the unfolding routine of events and things. There were countless ways to divine the future, such as observing oil dropped into a cup of water ( lecanomancy ), observing the entrails of sacrificial animals ( extispicy ), observation of
15360-596: Was brought to the region by the Seleucid Empire . Assyria and Babylonia later became provinces under the Parthian Empire ( Achaemenid Assyria and province of Babylonia ), Rome (province of Assyria ) and Sasanian Empire (province of Asoristan ). Babylonia was dissolved as an entity during the Parthian Empire, though Assyria endured as a geopolitical entity until the Muslim conquest of Persia , while
15488-478: Was conquered by the Achaemenid Empire (539–332 BC), then ruled by Cyrus the Great . This brought to an end over 3000 years of Mesopotamian dominance of the Near East. The Persians maintained and did not interfere in the native culture and religion and Assyria and Babylon continued to exist as entities (although Chaldea and the Chaldeans disappeared), and Assyria was strong enough to launch major rebellions against
15616-462: 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
15744-410: Was governed only by usage, not by any official decision, and by nature it was neither dogmatic nor systematic. Deities, characters, and their actions within myths changed in character and importance over time, and occasionally depicted different, sometimes even contrasting images or concepts. This is further complicated by the fact that scholars are not entirely certain what role religious texts played in
15872-414: 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"
16000-505: Was made to capture a demon by making an image of it, placing it above the head of a sick person, then destroying the image, which the demon was somehow likely to inhabit. Images of protecting spirits were also made and placed at gates to ward off disaster. Divination was also employed by private individuals, with the assumption that the gods have already determined the destinies of men and these destinies could be ascertained through observing omens and through rituals (e.g., casting lots). It
16128-437: 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
16256-571: 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
16384-433: Was thought to depend on his personal deity, including the development of his certain talents and even his personality. This was even taken to the point that everything he experienced was considered a reflection of what was happening to his personal god. When a man neglected his god, it was assumed that the demons were free to inflict him, and when he revered his god, that god was like a shepherd who seeks food for him. There
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