Kurt Friedrich Gödel was an Austrian, American, logician and philosopher. Considered along with Aristotle, Alfred Tarski and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna; the first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs, he made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, modal logic. Gödel was born April 28, 1906, in Brünn, Austria-Hungary into the German family of Rudolf Gödel, the manager of a textile factory, Marianne Gödel. Throughout his life, Gödel would remain close to his mother. At the time of his birth the city had a German-speaking majority, his father was Catholic and his mother was Protestant and the children were raised Protestant. The ancestors of Kurt Gödel were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the Brünner Männergesangverein. Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I.
In his family, young Kurt was known as Herr Warum because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever. Beginning at age four, Gödel suffered from "frequent episodes of poor health," which would continue for his entire life. Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects in mathematics and religion. Although Kurt had first excelled in languages, he became more interested in history and mathematics, his interest in mathematics increased when in 1920 his older brother Rudolf left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, the writings of Immanuel Kant. At the age of 18, Gödel entered the University of Vienna. By that time, he had mastered university-level mathematics.
Although intending to study theoretical physics, he attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism, he read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, participated in the Vienna Circle with Moritz Schlick, Hans Hahn, Rudolf Carnap. Gödel studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik, an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement, true in all models of the system?
This problem became the topic. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding the first-order predicate calculus, he was awarded his doctorate in 1930, his thesis was published by the Vienna Academy of Science. Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time.... The subject of logic has completely changed its nature and possibilities with Gödel's achievement. In 1931 and while still in Vienna, Gödel published
Mathematics and art
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, painting, architecture and textiles; this article focuses, however, on mathematics in the visual arts. Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, in his paintings; the engraver Albrecht Dürer made many references to mathematics in his work Melencolia I.
In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, cross-stitch, embroidery, weaving and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry, mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Escher. Computer art makes use of fractals including the Mandelbrot set, sometimes explores other mathematical objects such as cellular automata.
Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes. Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional batiks from different regions of Java have distinct fractal dimensions, stimuli to mathematics research Filippo Brunelleschi's theory of perspective, which led to Girard Desargues's projective geometry. A persistent view, based on the Pythagorean notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, that therefore the world's geometry is sacred, as seen in artworks such as William Blake's The Ancient of Days. Polykleitos the elder was a Greek sculptor from the school of Argos, a contemporary of Phidias, his works and statues consisted of bronze and were of athletes. According to the philosopher and mathematician Xenocrates, Polykleitos is ranked as one of the most important sculptors of classical antiquity for his work on the Doryphorus and the statue of Hera in the Heraion of Argos.
While his sculptures may not be as famous as those of Phidias, they are much admired. In the Canon of Polykleitos, a treatise he wrote designed to document the "perfect" anatomical proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body. Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Polykleitos multiplies the length of the distal phalanx by the square root of two to get the distance of the second phalanges and multiplies the length again by √2 to get the length of the third phalanges. Next, he takes the finger length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna; this geometric series of measurements progresses until Polykleitos has formed the arm, body, so on. The influence of the Canon of Polykleitos is immense in Classical Greek and Renaissance sculpture, many sculptors following Polykleitos's prescription.
While none of Polykleitos's original works survive, Roman copies demonstrate his ideal of physical perfection and mathematical precision. Some scholars argue; the Canon applies the basic mathematical concepts of Greek geometry, such as the ratio and symmetria and turns it into a system capable of describing the human form through a series of continuous geometric progressions. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects and figures according to their thematic importance. In the Middle Ages, some artists used reverse perspective for special emphasis; the Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, but never applied it to art. The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts. Two major motives drove artists in the Renaissance towards mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas.
Second and artists alike were convinced that mathematics was the true essence of the physical world and that the entire
Zen is a school of Mahayana Buddhism that originated in China during the Tang dynasty as the Chan school of Chinese Buddhism and developed into various schools. Chán Buddhism was influenced by Taoist philosophy Neo-Daoist thought. From China, Chán spread south to Vietnam and became Vietnamese Thiền, northeast to Korea to become Seon Buddhism, east to Japan, becoming Japanese Zen; the term Zen is derived from the Japanese pronunciation of the Middle Chinese word 禪, which traces its roots to the Indian practice of dhyāna. Zen emphasizes rigorous self-control, meditation-practice, insight into the nature of things, the personal expression of this insight in daily life for the benefit of others; as such, it de-emphasizes mere knowledge of sutras and doctrine and favors direct understanding through spiritual practice and interaction with an accomplished teacher. The teachings of Zen include various sources of Mahayana thought Yogachara, the Tathāgatagarbha sūtras and the Huayan school, with their emphasis on Buddha-nature and the Bodhisattva-ideal.
The Prajñāpāramitā literature as well as Madhyamaka thought have been influential in the shaping of the apophatic and sometimes iconoclastic nature of Zen rhetoric. The word Zen is derived from the Japanese pronunciation of the Middle Chinese word 禪, which in turn is derived from the Sanskrit word dhyāna, which can be translated as "absorption" or "meditative state"; the actual Chinese term for the "Zen school" is Chánzong, while "Chan" just refers to the practice of meditation itself or the study of meditation though it is used as an abbreviated form of Chánzong. The practice of dhyana or meditation sitting meditation is a central part of Zen Buddhism; the practice of Buddhist meditation first entered China through the translations of An Shigao, Kumārajīva, who both translated Dhyāna sutras, which were influential early meditation texts based on the Yogacara teachings of the Kashmiri Sarvāstivāda circa 1st-4th centuries CE. Among the most influential early Chinese meditation texts include the Anban Shouyi Jing, the Zuochan Sanmei Jing and the Damoduolo Chan Jing.
While dhyāna in a strict sense refers to the four dhyānas, in Chinese Buddhism, dhyāna may refer to various kinds of meditation techniques and their preparatory practices, which are necessary to practice dhyāna. The five main types of meditation in the Dhyāna sutras are ānāpānasmṛti. According to the modern Chan master Sheng Yen, these practices are termed the "five methods for stilling or pacifying the mind" and serve to focus and purify the mind, can lead to the dhyana absorptions. Chan shares the practice of the four foundations of mindfulness and the Three Gates of Liberation with early Buddhism and classic Mahayana. Early Chan texts teach forms of meditation that are unique to Mahayana Buddhism, for example, the Treatise on the Essentials of Cultivating the Mind which depicts the teachings of the 7th-century East Mountain school teaches a visualization of a sun disk, similar to that taught in the Sutra of the Contemplation of the Buddha Amitáyus. Chinese Buddhists developed their own meditation manuals and texts, one of the most influential being the works of the Tiantai patriarch, Zhiyi.
His works seemed to have exerted some influence on the earliest meditation manuals of the Chán school proper, an early work being the imitated and influential Tso-chan-i. During sitting meditation, practitioners assume a position such as the lotus position, half-lotus, Burmese, or seiza using the dhyāna mudrā. A square or round cushion placed on a padded mat is used to sit on. To regulate the mind, Zen students are directed towards counting breaths. Either both exhalations and inhalations are counted; the count can be up to ten, this process is repeated until the mind is calmed. Zen teachers like Omori Sogen teach a series of long and deep exhalations and inhalations as a way to prepare for regular breath meditation. Attention is placed on the energy center below the navel. Zen teachers promote diaphragmatic breathing, stating that the breath must come from the lower abdomen, that this part of the body should expand forward as one breathes. Over time the breathing should become smoother and slower.
When the counting becomes an encumbrance, the practice of following the natural rhythm of breathing with concentrated attention is recommended. Another common form of sitting meditation is called "Silent illumination"; this practice was traditionally promoted by the Caodong school of Chinese Chan and is associated with Hongzhi Zhengjue who wrote various works on the practice. This method derives from the Indian Buddhist practice of the union of śamatha and vipaśyanā. In Hongzhi's practice of "nondual objectless meditation" the mediator strives to be aware of the totality of phenomena instead of focusi
An ant colony is the basic unit around which ants organize their lifecycle. Ant colonies are eusocial, are much like those found in other social Hymenoptera, though the various groups of these developed sociality independently through convergent evolution; the typical colony consists of one or more egg-laying queens, a large number of sterile females and, seasonally, a large number of winged sexual males and females. In order to establish new colonies, ants undertake nuptial flights that occur at species-characteristic times of the day. Swarms of the winged sexuals depart the nest in search of other nests; the males die shortly thereafter, along with most of the females. A small percentage of the females survive to initiate new nests; the term "ant colony" refers to the collections of workers, reproductive individuals, brood that live together and treat one another non-aggressively. This comprises the genetically related progeny from a single queen, although this is not universal across ants; the name "ant farm" is given to ant nests that are kept in formicarium.
Another name is "formicary". The word derives from formica. "Ant nests" are the physical spaces. These can be underground, in trees, under rocks, or inside a single acorn; the name "anthill" applies to underground nests where the workers pile sand or soil outside the entrance, forming a large mound. Colony size is important to ants: it can affect how they forage, how they defend their nests, how they mate, their physical appearances. Body size is seen as the most important factor in shaping the natural history of non-colonial organisms. However, colony sizes are different in different ant species: some are just several ants living in a twig, while others are super colonies with many millions of workers. Looking at a single ant colony, seasonal variation can be huge. For example, in the ant Dolichoderus mariae, one colony can shift from around 300 workers in the summer to over 2,000 workers per queen in the winter. Genetics and environmental factors can cause the variation among colonies of a particular species to be bigger.
Zooming out further, within a related group of different ant species, the differences can be enormous: Formica yessensis has colony sizes that are reported to be 306 million workers while Formica fusca colonies sometimes comprise only 500 workers. A supercolony occurs, they still continue to recognize genetic differences in order to mate, but the different colonies within the super colony avoid aggression. Until 2000, the largest known ant supercolony was on the Ishikari coast of Japan; the colony was estimated to contain 306 million worker ants and one million queen ants living in 45,000 nests interconnected by underground passages over an area of 2.7 km2. In 2000, an enormous supercolony of Argentine ants was found in Southern Europe. Of 33 ant populations tested along the 6,004-kilometre stretch along the Mediterranean and Atlantic coasts in Southern Europe, 30 belonged to one supercolony with estimated millions of nests and billions of workers, interspersed with three populations of another supercolony.
The researchers claim that this case of unicoloniality cannot be explained by loss of their genetic diversity due to the genetic bottleneck of the imported ants. In 2009, it was demonstrated that the largest Japanese and European Argentine ant supercolonies were in fact part of a single global "megacolony"; this intercontinental megacolony represents the most populous recorded animal society on earth, other than humans. Another supercolony, measuring 100 km wide, was found beneath Melbourne, Australia in 2004; the following terminology is used among myrmecologists to describe the behaviors demonstrated by ants when founding and organizing colonies: Monogyny Establishment of an ant colony under a single egg-laying queen. Polygyny Establishment of an ant colony under multiple egg-laying queens. Oligogyny Establishment of a polygynous colony where the multiple egg-laying queens remain far apart from one another in the nest. Haplometrosis Establishment of a colony by a single queen. Pleometrosis Establishment of a colony by multiple queens.
Monodomy Establishment of a colony at a single nest site. Polydomy Establishment of a colony across multiple nest sites. Ant colonies have a complex hierarchical social structure. Ants jobs can be changed by age; as ants grow older their jobs move them further from the center of the colony. Younger ants work within the nest protecting the queen and young. Sometimes, a queen is replaced by egg-laying workers; these worker ants can only lay haploid eggs producing sterile offspring. Despite the title of queen, she doesn't delegate the tasks to the worker ants. Ants as a colony work as a collective "super mind". Ants can compare areas and solve complex problems by using information gained by each member of the colony to find the best nesting site or to find food; some social-parasitic species of ants, known as the Slave-making ant and steal larvae from neighboring colonies. Ant hill art is a growing collecting hobby, it involves pouring molten metal, plaster or cement down an ant colony mound acting as a mold and upon hardeni
Egbert B. Gebstadter
Egbert B. Gebstadter is a fictional author. For each Hofstadter book, there is a corresponding Gebstadter book, his name is derived from "GEB", the abbreviation for Hofstadter's first book Gödel, Bach: An Eternal Golden Braid. From Gebstadter's brief 1985 biography: Having spent the last several years in the Psychology Department of Pakistania University in Wiltington, Pakistania, he has joined the faculty of the Computer Science Department of the University of Mishuggan in Tom Treeline, where he occupies the Rexall Chair in the College of Art and Letters, his current research projects in IA are called Quest-Essence, Mind-Pattern and Studio. His focus is on deterministic sequential models of digital emotion; the equivalent section of Hofstadter's brief 1985 biography: Having spent the last several years in the Computer Science Department of Indiana University in Bloomington, Indiana, he has joined the faculty of the Psychology Department of the University of Michigan in Ann Arbor, where he occupies the Walgreen Chair in the College of Literature and the Arts.
His current research projects in AI are called Seek-Whence, Letter Spirit and Jumbo. His focus is on stochastic parallel models of analogical thought. Most of Gebstadter's books are published by a fictional publisher in Perth, Australia. Gebstadter's first book is described in GEB as follows: Gebstadter, Egbert B. Copper, Gold: an Indestructible Metallic Alloy. Perth: Acidic Books, 1979. A formidable hodge-podge and confused -- yet remarkably similar to the present work. Professor Gebstadter's Shandean digressions include some excellent examples of indirect self-reference. Of particular interest is a reference in its well-annotated bibliography to an isomorphic, but imaginary, book. CSG is referred to in the text of GEB itself as Giraffes, Baboons: An Equatorial Grasslands Bestiary, which maintains the "GEB: An EGB" acronym. Or this is a different work by Gebstadter altogether: the evidence is incomplete. CSG is referred to in Aria with Diverse Variations, a dialogue in GEB. Achilles and the Tortoise are trying to remember the name of an amateur mathematician, Achilles suggests "Kupfergödel" and "Silberescher" Mr Tortoise recalls Goldbach".
Half-translated from German, these names are "copper Gödel", "silver Escher" and "gold Bach", respectively. Gebstadter's second book is an anthology co-edited with the Australian philosopher Denial E. Dunnitt entitled The Brain's U: Phantoms and Mirror-Images of Selfless Souls, appears in the bibliography of The Mind's I: Fantasies and Reflections on Self and Soul, edited by Hofstadter and Daniel Dennett; the Gebstadter bibliography on the website for Stepford Ninniversity's "Residential Pictures in Mundanities and Artifice" gives the subtitle as Fairy Tales and Rotations on Ego and Anima. Gebstadter's third book appears in the bibliography to Hofstadter's third book, Metamagical Themas: Questing for the Essence of Mind and Pattern; this is a collection of Hofstadter's monthly columns from Scientific American with postscripts written specially for the book. Gebstadter, Egbert B. Thetamagical Memas: Seeking the Whence of Letter and Spirit. Perth: Acidic Books, 1985. A curious pot-pourri and muddled -- yet remarkably similar to the present work.
This is a collection of Gebstadter's monthly rows in Literary Australian together with a few other articles, all with prescripts. Gebstadter is well known for his love of twisty analogies, such as this one: "Egbert B. Gebstadter is the Egbert B. Gebstadter of indirect self-reference."Gebstadter's fourth book in Italian, is entitled Ambifoni: un minimondo ottimo per lo studio della scopertività, published by Hopeless Mobster, Tokyo, in 1987. This book is referenced by Hofstadter in his Italian book Ambigrammi: un microcosmo ideale per lo studio della creatività, published by Hopeful Monster, Florence in the same year. Hofstadter's book contains a fictional dialogue between Gebstadter. Curiously, there is no mention of Gebstadter's fifth book in Hofstadter's Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought. However, rumor has it that it is entitled Stagnant Notions and Derivative Metaphors: Anthropomorphic Models of the Mechanical Fundaments of Computation, a collection of articles on the research of Gebstadter and his colleagues at the University of Mishuggan's Stagnant Metaphors User Group.
Reviewers have described it as "an eclectic jumble and unfocused," and it has been suggested by some that Hofstadter's Fluid Analogies Research Group was reluctant to give recognition to the work of a rival lab which they sometimes referred to as "those SMUG copycats." Gebstadter's sixth book is called The Graced Tone of Clément: A la louange de la mélodie des mots, is referred to in Hofstadter's book Le Ton beau de Marot: In Praise of the Music of Language. Hofstadter's corresponding book is in English despite its French title, we may conjecture that Gebstadter's book is in French despite its English title; this is supported by the fact that this book was published not by Acidic Books but by the Éditions Noitide in Cahors, France. Gebstadter's seventh book is U Are an Odd Ball, referenced in Ho
Jinn Romanized as djinn or Anglicized as genies, are supernatural creatures in early pre-Islamic Arabian and Islamic mythology and theology. Jinn are not a Islamic concept. Since jinn are not evil, Islam was able to adapt spirits from other religions during its expansion. Besides the jinn, Islam acknowledges the existence of demons; the lines between demons and jinn are blurred, since malevolent jinn are called shayāṭīn. However both Islam and non-Islamic scholarship distinguishes between angels and demons as three different types of spiritual entities in Islamic traditions; the jinn are distinguished from demons in that they can be both evil or good, while genuine demons are evil. Some academic scholars assert that demons are related to monotheistic traditions and jinn to polytheistic traditions. In an Islamic context, the term jinn is used for both a collective designation for any supernatural creature and to refer to a specific type of supernatural creature. Jinn is an Arabic collective noun deriving from the Semitic root JNN, whose primary meaning is "to hide" or "to conceal".
Some authors interpret the word to mean "beings that are concealed from the senses". Cognates include the Arabic majnūn, janīn. Jinn is properly treated as a plural, with the singular being jinnī; the origin of the word Jinn remains uncertain. Some scholars relate the Arabic term jinn to the Latin genius, as a result of syncretism during the reign of the Roman empire under Tiberius Augustus, but this derivation is disputed. Another suggestion holds that jinn may be derived from Aramaic "ginnaya" with the meaning of "tutelary deity", or "garden". Others claim a Persian origin of the word, in the form of a wicked spirit. Jaini were among various creatures in the even pre-Zoroastrian mythology of peoples of Iran; the Anglicized form genie is a borrowing of the French génie, from the Latin genius, a guardian spirit of people and places in Roman religion. It first appeared in 18th-century translations of the Thousand and One Nights from the French, where it had been used owing to its rough similarity in sound and sense and further applies to benevolent intermediary spirits, in contrast to the malevolent spirits called demon and heavenly angels, in literature.
In Assyrian art, creatures ontologically between humans and divinities are called genie. Jinn were worshipped by many Arabs during the Pre-Islamic period, unlike gods, jinn were not regarded as immortal. In ancient Arabia, the term jinn applied to all kinds of supernatural entities among various religions and cults; the exact origins of belief in jinn are not clear. Some scholars of the Middle East hold that they originated as malevolent spirits residing in deserts and unclean places, who took the forms of animals. According to common Arabian belief, pre-Islamic philosophers, poets were inspired by the jinn. However, jinn were feared and thought to be responsible for causing various diseases and mental illnesses. Julius Wellhausen observed that such spirits were thought to inhabit desolate and dark places and that they were feared. One had to protect oneself from them. In the Islamic sense, the term jinn is used in two different ways: An invisible entity, who roamed the earth before Adam, created by God out of a "mixture of fire" or "smokeless fire".
They are believed to resemble humans in that they eat and drink, have children and die, are subject to judgment, so will either be sent to heaven or hell according to their deeds. But they were stronger than humans. Jinn are related to heavenly beings, a sub-category of angels or a tribe of angelic beings, able to sin and created from fire, unlike their light-created counterpart; however these jinn must be distinguished, from the pre-Adamite jinn-race, who share many characteristics with human, instead of angels. As the opposite of al-Ins referring to any object that cannot be detected by human sensory organs, including angels and the interior of human beings, thus every demon and every angel is a jinn, but not every jinn is an angel or a demon. Belief in jinn is not included among the six articles of Islamic faith, as belief in angels is, however at least some Muslims believe it essential to the Islamic faith. Jinn are mentioned 29 times in the Quran together with humans, the 72 surah named after them.
They are mentioned in collections of Ṣaḥīḥ ahadith. One hadith divides them with one type flying through the air. In Islamic tradition, Muhammad was sent as a prophet to both human and jinn communities, that prophets and messengers were sent to both communities. Traditionally Surah 72 is held to tell about the revelation to jinn and several stories mention one of Muhammad's followers accompanied him, witnessing the revelation to the jinn. Another Islamic prophet, related to interactions with jinn, is Solomon. In Quran, he is said to be a king in ancient Israel and was gifted by God to talk to animals and jin
In music, counterpoint is the relationship between voices that are harmonically interdependent yet independent in rhythm and contour. It has been most identified in the European classical tradition developing during the Renaissance and in much of the common practice period in the Baroque; the term originates from the Latin punctus contra punctum meaning "point against point". Counterpoint has been used to designate a voice or an entire composition. Counterpoint focuses on melodic interaction—only secondarily on the harmonies produced by that interaction. In the words of John Rahn: It is hard to write a beautiful song, it is harder to write several individually beautiful songs that, when sung sound as a more beautiful polyphonic whole. The internal structures that create each of the voices separately must contribute to the emergent structure of the polyphony, which in turn must reinforce and comment on the structures of the individual voices; the way, accomplished in detail is...'counterpoint'.
Counterpoint theory has been given a mathematical foundation in the work initiated by Guerino Mazzola. In particular, his model gives a structural foundation of forbidden parallels of fifths and the dissonant fourth; the model has been extended to microtonal contexts by Octavio Agustin. Some examples of related compositional techniques include: the round, the canon, the most complex contrapuntal convention: the fugue. All of these are examples of imitative counterpoint. In 1725 Johann Joseph Fux published Gradus ad Parnassum, in which he described five species: Note against note. A succession of theorists quite imitated Fux's seminal work with some small and idiosyncratic modifications in the rules. Many of Fux's rules concerning the purely linear construction of melodies have their origin in solfeggi. Concerning the common practice era, alterations to the melodic rules were introduced to enable the function of certain harmonic forms; the combination of these melodies produced the figured bass.
The following rules apply to melodic writing in each species, for each part: The final must be approached by step. If the final is approached from below the leading tone must be raised in a minor key, but not in Phrygian or Hypophrygian mode. Thus, in the Dorian mode on D, a C♯ is necessary at the cadence. Permitted melodic intervals are the perfect fourth and octave, as well as the major and minor second and minor third, ascending minor sixth; the ascending minor sixth must be followed by motion downwards. If writing two skips in the same direction—something that must be only done—the second must be smaller than the first, the interval between the first and the third note may not be dissonant; the three notes should be from the same triad. In general, do not write more than two skips in the same direction. If writing a skip in one direction, it is best to proceed after the skip with motion in the other direction; the interval of a tritone in three notes should be avoided as is the interval of a seventh in three notes.
There must be a climax or high point in the line countering the cantus firmus. This occurs somewhere in the middle of exercise and must occur on a strong beat. An outlining of a seventh is avoided within a single line moving in the same direction. And, in all species, the following rules govern the combination of the parts: The counterpoint must begin and end on a perfect consonance. Contrary motion should predominate. Perfect consonances must be approached by contrary motion. Imperfect consonances may be approached by any type of motion; the interval of a tenth should not be exceeded between two adjacent parts. Build from the bass, upward. In first species counterpoint, each note in every added part sounds against one note in the cantus firmus. Notes in all parts are sounded and move against each other simultaneously. Since all notes in First species counterpoint are whole notes, rhythmic independence is not available. In the present context, a "step" is a melodic interval of whole step. A "skip" is an interval of a fourth.
An interval of a fifth or larger is referred to as a "leap". A few further rules given by Fux, by study of the Palestrina style, given in the works of counterpoint pedagogues, are as follows. Begin and end on either the unison, octave, or fifth, unless the added part is underneath, in which case begin and end only on unison or octave. Use no unisons except at the beginning or end. Avoid parallel fifths or octaves between any two parts. Avoid moving in parallel fourths. Avoid moving in parallel thirds or sixths for long. Attempt to keep any two adjacent parts within a tenth of each other, unless an exceptionally pleasing line can be written by moving outside that range. Avoid having any two parts move in the same direction by skip Attempt to have as much contrary motion as possible. Avoid dissonant inter