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Encyclopedia > Pythagoreanism

Bust of Pythagoras
Bust of Pythagoras

Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. Image File history File links Kapitolinischer_Pythagoras. ... Image File history File links Kapitolinischer_Pythagoras. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Etymology Esoteric is an adjective originating during Hellenic Greece under the domain of the Roman Empire; it comes from the Greek esôterikos, from esôtero, the comparative form of esô: within. It is a word meaning anything that is inner and occult, a latinate word meaning hidden (from which... Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ...


Later resurgence of ideas similar to those held by the early Pythagoreans are collected under the term Neopythagoreanism.


The Pythagoreans were called mathematikoi, which means "those that study all."[1]

Contents

Pythagorean natural philosophy

A pentagram from Agrippa's book; this one has the Pythagorean letters inscribed around the circle.
A pentagram from Agrippa's book; this one has the Pythagorean letters inscribed around the circle.

Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. But lots of other mathmaticians had the idea too. In the area of sex there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. The Pythagorean conception of substance, on the other hand, is of unknown origin, partly because various accounts of his teachings are conflicting. The Pythagorean account actually begins with Anaximander's teaching that the ultimate substance of things is "the boundless," or what Anaximander called the "apeiron." The Pythagorean account holds that it is only through the notion of the "limit" that the "boundless" takes form. Image File history File links Download high resolution version (800x800, 17 KB)Inverted pentagram (rotated version of Image:Pentacle-circumscribed. ... Image File history File links Download high resolution version (800x800, 17 KB)Inverted pentagram (rotated version of Image:Pentacle-circumscribed. ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Cornelius Agrippa, as portrayed in Libri tres de occulta philosophia. ... This article or section does not cite its references or sources. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Metempsychosis is a philosophical term in the Greek language referring to the belief of transmigration of the soul, especially its reincarnation after death. ... Anaximander Possibly what Anaximanders map looked like Anaximander (Greek: Αναξίμανδρος)(c. ... The apeiron is a cosmological theory created by Anaximander in the 6th century BC. Anaximanders work is mostly lost. ...


Pythagoras wrote nothing down, and relying on the writings of Parmenides, Empedocles, Philolaus and Plato (people either considered Pythagoreans, or whose works are thought deeply indebted to Pythagoreanism) results in a very diverse picture in which it is difficult to ascertain what the common unifying Pythagorean themes were. Relying on Philolaus, whom most scholars agree is highly representative of the Pythagorean school, one has a very intricate picture. Aristotle explains how the Pythagoreans (by which he meant the circle around Philolaus) developed Anaximander's ideas about the apeiron and the peiron, the unlimited and limited, by writing that: Parmenides of Elea (Greek: , early 5th century BC) was an ancient Greek philosopher born in Elea, a Hellenic city on the southern coast of Italy. ... Empedocles (Greek: , ca. ... Philolaus (circa 480 BC – circa 405 BC) was a Greek mathematician and philosopher. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Anaximander Possibly what Anaximanders map looked like Anaximander (Greek: Αναξίμανδρος)(c. ... The apeiron is a cosmological theory created by Anaximander in the 6th century BC. Anaximanders work is mostly lost. ...

... for they [the Pythagoreans] plainly say that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by the limit.

Continuing with the Pythagoreans:

The Pythagoreans, too, held that void exists, and that it enters the heaven from the unlimited breath – it, so to speak, breathes in void. The void distinguishes the natures of things, since it is the thing that separates and distinguishes the successive terms in a series. This happens in the first case of numbers; for the void distinguishes their nature.

Pythagoreans celebrate sunrise by Fyodor Bronnikov
Pythagoreans celebrate sunrise by Fyodor Bronnikov

When the apeiron is inhaled by the peiron it causes separation, which also apparently means that it "separates and distinguishes the successive terms in a series." Instead of an undifferentiated whole we have a living whole of inter-connected parts separated by "void" between them. This inhalation of the apeiron is also what makes the world mathematical, not just possible to describe using maths, but truly mathematical since it shows numbers and reality to be upheld by the same principle. Both the continuum of numbers (that is yet a series of successive terms, separated by void) and the field of reality, the cosmos — both are a play of emptiness and form, apeiron and peiron. What really sets this apart from Anaximander's original ideas is that this play of apeiron and peiron must take place according to harmonia (harmony), about which Stobaeus commentated: Image File history File links Size of this preview: 800 × 492 pixelsFull resolution (2112 × 1300 pixel, file size: 417 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Size of this preview: 800 × 492 pixelsFull resolution (2112 × 1300 pixel, file size: 417 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... The apeiron is a cosmological theory created by Anaximander in the 6th century BC. Anaximanders work is mostly lost. ... Joannes Stobaeus, so called from his native place Stobi in Macedonia, was the compiler of a valuable series of extracts from Greek authors. ...

About nature and harmony this is the position. The being of the objects, being eternal, and nature itself admit of divine, not human, knowledge – except that it was not possible for any of the things that exist and are known by us to have come into being, without there existing the being of those things from which the universe was composed, the limited and the unlimited. And since these principles existed being neither alike nor of the same kind, it would have been impossible for them to be ordered into a universe if harmony had not supervened – in whatever manner this came into being. Things that were alike and of the same kind had no need of harmony, but those that were unlike and not of the same kind and of unequal order – it was necessary for such things to have been locked together by harmony, if they are to be held together in an ordered universe.

A musical scale presupposes an unlimited continuum of pitches, which must be limited in some way in order for a scale to arise. The crucial point is that not just any set of limiters will do. One may not simply choose pitches at random along the continuum and produce a scale that will be musically pleasing. The diatonic scale, also known as "Pythagorean," is such that the ratio of the highest to the lowest pitch is 2:1, which produces the interval of an octave. That octave is in turn divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively and which, when added, make an octave. If we go up a fifth from the lowest note in the octave and then up a fourth from there, we will reach the upper note of the octave. Finally the fifth can be divided into three whole tones, each corresponding to the ratio of 9:8 and a remainder with a ratio of 256:243 and the fourth into two whole tones with the same remainder. This is a good example of a concrete applied use of Philolaus’ reasoning. In Philolaus' terms the fitting together of limiters and unlimiteds involves their combination in accordance with ratios of numbers (harmony). Similarly the cosmos and the individual things in the cosmos do not arise by a chance combination of limiters and unlimiteds; the limiters and unlimiteds must be fitted together in a "pleasing" (harmonic) way in accordance with number for an order to arise.


This teaching was recorded by Philolaus' pupil Archytas in a lost work entitled On Harmonics or On Mathematics, and this is the influence that can be traced in Plato. Plato's pupil Aristotle made a distinction in his Metaphysics between Pythagoreans and "so-called" Pythagoreans. He also recorded the Table of Opposites, and commented that it might be due to Alcmaeon of the medical school at Croton, who defined health as a harmony of the elements in the body. Crotone is a city in Calabria, southern Italy, on the Gulf of Taranto. ...


After attacks on the Pythagorean meeting-places at Croton, the movement dispersed, but regrouped in Tarentum, also in Southern Italy. A collection of Pythagorean writings on ethics collected by Taylor show a creative response to the troubles. Taranto is a coastal city in Apulia, southern Italy. ...


The legacy of Pythagoras, Socrates and Plato was claimed by the wisdom tradition of the Hellenized Jews of Alexandria, on the ground that their teachings derived from those of Moses. Through Philo of Alexandria this tradition passed into the Medieval culture, with the idea that groups of things of the same number are related or in sympathy. This idea evidently influenced Hegel in his concept of internal relations.


The ancient Pythagorean pentagram was drawn with two points up and represented the doctrine of Pentemychos. Pentemychos means "five recesses" or "five chambers," also known as the pentagonas — the five-angle, and was the title of a work written by Pythagoras' teacher and friend Pherecydes of Syros.[2] Pherecydes of Syros (in Greek: Φερεχύδης) was a Greek thinker from the island of Siros, Magna Graecia of the 6th century BC. Pherecydes authored the Heptamychia, one of the first attested prose works in Greek literature, which formed an important bridge between...


The Pythagorean symbols are central to the mystery in the novel The Oxford Murders (Crímenes imperceptibles, 2003) by Guillermo Martinez. Guillermo Martínez (born 29 July 1962) is an Argentinian writer. ...


Pythagorean cosmology

For more details on this topic, see Iamblichus of Chalcis.
The Monad was a symbol referred by the Greek philosophers as "The First," "The Seed," "The Essence," "The Builder," and "The Foundation"
The Monad was a symbol referred by the Greek philosophers as "The First," "The Seed," "The Essence," "The Builder," and "The Foundation"

The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods. Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics included various purifactory rites and practices as well as incubatory rites of descent into the underworld. Apart from being linked with this, Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks," or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (eugieia). Iamblichus, also known as Iamblichus Chalcidensis, (ca. ... Image File history File links Monad. ... Image File history File links Monad. ... The Pythagorean Monad Monad, according to the Pythagoreans, was a term for God or the first being, or the totality of all beings. ... Transmigration can has several meanings: Transmigration of the soul is a common term for reincarnation. ... The head of Orpheus, from an 1865 painting by Gustave Moreau. ... Pherecydes of Syros (in Greek: Φερεχύδης) was a Greek thinker from the island of Siros, Magna Graecia of the 6th century BC. Pherecydes authored the Heptamychia, one of the first attested prose works in Greek literature, which formed an important bridge between... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ...


Pythagorean vegetarianism

The Pythagoreans were well-known in antiquity for their vegetarianism, which they practised for religious, ethical and ascetic reasons. "Pythagorean diet" was a common name for the abstention from eating meat and fish, until the coining of "vegetarian" in the nineteenth century.[3] Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, which begins roughly with the earliest-recorded Greek poetry of Homer (7th century BC), and continues through the rise of Christianity and the fall of the Western Roman Empire (5th century AD... This article refers to human nutrition and diet. ...


The Pythagorean code further restricted the diet of its followers, prohibiting the consumption or even touching any sort of bean. The reason is unclear: perhaps the flatulence they cause, perhaps as protection from potential favism, but most likely for magico-religious reasons,[4] such as the belief that beans and humans were created from the same material.[5] Flatulence is the presence of a mixture of gases known as flatus in the digestive tract of mammals expelled from the rectum. ... Glucose-6-Phosphate Dehydrogenase (G6PD) deficiency is an X-linked recessive hereditary disease featuring nonimmune hemolytic anemia in response to a number of causes. ...


Pythagorean view of women

Women were given equal opportunity to study as Pythagoreans; however, they learned practical domestic skills in addition to philosophy.[6] Women were held to be different from men, but sometimes in good ways.[7]


Neo-Pythagoreanism

Neo-Pythagoreanism was a revival in the 2nd century BC2nd century AD period, of various ideas traditionally associated with the followers of Pythagoras, the Pythagoreans. (2nd millennium BC - 1st millennium BC - 1st millennium) The 2nd century BC started on January 1, 200 BC and ended on December 31, 101 BC. // Coin of Antiochus IV. Reverse shows Apollo seated on an omphalos. ... (1st century - 2nd century - 3rd century - other centuries) Events Roman Empire governed by the Five Good Emperors (96–180) – Nerva, Trajan, Hadrian, Antoninus Pius, Marcus Aurelius. ...


Notable Neo-Pythagoreans include first century Apollonius of Tyana. Middle and Neo-Platonists such as Numenius and Plotinus also exhibited some Neo-Pythagorean influence. Apollonius of Tyana (Greek: ; 16—ca. ... Plotinus (Greek: ) (ca. ...


In 1915 a subterranean basilica was discovered near Porta Maggiore on Via Praenestina, Rome where Neo-Pythagoreans held their meetings in the 1st century. The groundplan shows a basilica with three naves and an apse similarly to early Christian basilicas that appeared only much later, in the 4th century. The vaults are decorated with white stuccoes symbolizing Neo-Pythagorean beliefs but its exact meaning remains a subject of debate.[8] Oblique view of the Porta Maggiore, showing the aqueduct channels through the gate. ... Palestrina (ancient Praeneste) was and is a very ancient city of Latium (modern Lazio) 23 miles (37 km) east of Rome, and was reached by the Via Praenestina (see below). ... For other uses, see Rome (disambiguation). ...


Further Neo-Pythagorean sentiments exist in modern philosophy, with the Lowenheim-Skolem theorem, which indicates that a valid interpretation of the world can be restricted to numbers. Hilary Putnam has a similar Realist thesis, "Internal Realism," whereby one could be a Pythagorean in this way. In mathematical logic, the classic Löwenheim-Skolem theorem states that any infinite model M has a countably infinite submodel N that satisfies exactly the same set of first-order sentences that M satisfies. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ...


Influences

  • The Pythagorean idea that whole numbers and harmonic (pleasing) sounds are intimately connected in music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantitated and described. This may have paved the way to his son's crucial insight that all physical phenomena may be described quantitatively in mathematical language (as physical "laws"), thus beginning and defining the era of modern physics.
  • Pythagoreanism has had a clear and obvious influence on the texts found in the hermetica corpus and thus flows over into hermeticism, gnosticism and alchemy.
  • The Pythagorean cosmology also inspired the Arabic gnostic Monoimus to combine this system with monism and other things to form his own cosmology.
  • The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans, which is often seen as being related to the elements theorized by Empedocles to comprise all matter.
  • The Pythagoreans were advised to "speak the truth in all situations," which Pythagoras said he learned from the Magi of Babylon.

Vincenzo Galilei (1520 – July 2, 1591) was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer Galileo Galilei. ... Galileo redirects here. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Hermetica refers to a category of popular Late Antique literature purporting to contain secret wisdom, and generally attributed to Hermes Trismegistus. ... Hermeticism should not be confused with the concept of a hermit. ... This article or section is in need of attention from an expert on the subject. ... For other uses, see Alchemy (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Monoimus (lived somewhere between 150 - 210) was an arabic gnostic (arabic name: Munim), who was known to us only from one account in Theodoret (Haereticarum Fabularum Compendium i. ... For other uses, see Monist (disambiguation). ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Empedocles (Greek: , ca. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... For other uses, see Magi (disambiguation). ... For other uses, see Babylon (disambiguation). ...

References

  1. ^ Hemmenway, Pryia – Divine Proportion pp66, Sterling Publishing, ISBN 1-4027-3522-7
  2. ^ This is actually a lost book whose contents are preserved in Damascius, de principiis, quoted in Kirk and Raven, The Pre-Socratic Philosophers, Cambridge Univ. Press, 1956, page 55.
  3. ^ See for instance the popular treatise by Antonio Cocchi, Del vitto pitagorico per uso della medicina, Firenze 1743, which initiated a debate on the "Pythagorean diet".
  4. ^ Gabrielle Hatfield, review of Frederick J. Simoons, Plants of Life, Plants of Death, University of Wisconsin Press, 1999. ISBN 0-299-15904-3. In Folklore 111:317-318 (2000).
  5. ^ Riedweg, Christoph, Pythagoras: his life, teaching, and influence. Ithaca : Cornell University Press, pp. 39, 70. (2005), ISBN 0-8014-4240-0
  6. ^ Glenn, Cheryl, Rhetoric Retold: Regendering the Tradition from Antiquity Through the Renaissance. Southern Illinois University, 1997. 30-31.
  7. ^ Glenn, Cheryl, Rhetoric Retold: Regendering the Tradition from Antiquity Through the Renaissance. Southern Illinois University, 1997. 31.
  8. ^ Ball Platner, Samuel. Basilicae. penelope.uchicago.edu.

Damascius, the last of the Neoplatonists, was born in Damascus about AD 480. ...

Further reading

  • O'Meara, Dominic J. Pythagoras Revived: Mathematics and Philosophy in Late Antiquity , Clarendon Press, Oxford, 1989. ISBN 0-19-823913-0
  • Riedweg, Christoph Pythagoras : his life, teaching, and influence ; translated by Steven Rendall in collaboration with Christoph Riedweg and Andreas Schatzmann, Ithaca : Cornell University Press, (2005), ISBN 0-8014-4240-0

See also

Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... Esoteric cosmology is cosmology that is an intrinsic part of an esoteric or occult system of thought. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... Numerology is any of many systems, traditions or beliefs in a mystical or esoteric relationship between numbers and physical objects or living things. ...

Pythagorean symbols

The Pythagorean Monad Monad, according to the Pythagoreans, was a term for God or the first being, or the totality of all beings. ... The Dyad Dyad, according to the Pythagoreans, is the principle of twoness or otherness, .[1] Monad Triad Tetrad Pentad Decad Vesica piscis Ichthys Iamblichus of Chalcis ^ Hemenway, Priya – Divine Proportion pp. ... The Triad Enclosing the greater area with the smallest perimeter, the triangle, derived from the vesica piscis, the Triad was considered by the Pythagoreans as the most beautiful number, as it is the only number to equal the sum of all the terms below it, and the only number whose... The Tetrad The tetrad or number four is the first number formed by the addition and multiplication of equals. ... The Pentad The pentad, was used by the Pythagoreans as a secret sign to recognize each other. ... The Decad The number ten or decad was not considered a number by the Pythagoreans, it was seen as an assembly point and a symbol of earth and heaven. ... The Tetractys, also known as the decad, is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row. ... The Vesica Piscis The vesica piscis is a symbol made from two circles of the same radius, intersecting in such a way that the center of each circle lies on the circumference of the other. ...

External links

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In philosophy generally, empiricism is a theory of knowledge emphasizing the role of experience, especially sensory perception, in the formation of ideas, while discounting the notion of innate ideas. ... Epicureanism is a system of philosophy based upon the teachings of Epicurus (c. ... Existentialism is a philosophical movement which claims that individual human beings create the meanings and essence of their own lives. ... Hegelianism is a philosophy developed by Georg Wilhelm Friedrich Hegel which can be summed up by a favorite motto by Hegel, the rational alone is real, which means that all reality is capable of being expressed in rational categories. ... Hermeneutics may be described as the development and study of theories of the interpretation and understanding of texts. ... See also the specific life stance known as Humanism For the Renaissance liberal arts movement, see Renaissance humanism Humanism is a broad category of ethical philosophies that affirm the dignity and worth of all people, based on the ability to determine right and wrong by appeal to universal human qualities... This section may require cleanup to meet Wikipedias quality standards. ... Kant redirects here. ... Logical positivism grew from the discussions of Moritz Schlicks Vienna Circle and Hans Reichenbachs Berlin Circle in the 1920s and 1930s. ... Marxism is both the theory and the political practice (that is, the praxis) derived from the work of Karl Marx and Friedrich Engels. ... In philosophy, materialism is that form of physicalism which holds that the only thing that can truly be said to exist is matter; that fundamentally, all things are composed of material and all phenomena are the result of material interactions; that matter is the only substance. ... For other uses, see Monist (disambiguation). ... Neoplatonism (also Neo-Platonism) is the modern term for a school of religious and mystical philosophy that took shape in the 3rd century AD, founded by Plotinus and based on the teachings of Plato and earlier Platonists. ... The New Philosophers (French nouveaux philosophes) were a group of French philosophers (for example, André Glucksmann and Bernard Henri-Lévy) who appeared in the early 1970s, as critics of the previously-fashionable philosophers (roughly speaking, the post-structuralists). ... This article is about the philosophical position. ... This article or section does not adequately cite its references or sources. ... This article is about the philosophical movement. ... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... Positivism is a philosophy that states that the only authentic knowledge is scientific knowledge, and that such knowledge can only come from positive affirmation of theories through strict scientific method. ... Postmodern philosophy is an eclectic and elusive movement characterized by its criticism of Western philosophy. ... Post-structuralism is a body of work that followed in the wake of structuralism, and sought to understand the Western world as a network of structures, as in structuralism, but in which such structures are ordered primarily by local, shifting differences (as in deconstruction) rather than grand binary oppositions and... Pragmatism is a philosophic school that originated in the late nineteenth century with Charles Sanders Peirce, who first stated the pragmatic maxim. ... The Pre-Socratic philosophers were active before Socrates or contemporaneously, but expounding knowledge developed earlier. ... In epistemology and in its broadest sense, rationalism is any view appealing to reason as a source of knowledge or justification (Lacey 286). ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ... For the physics theory with a similar name, see Theory of Relativity. ... Scholasticism comes from the Latin word scholasticus, which means that [which] belongs to the school, and is the school of philosophy taught by the academics (or schoolmen) of medieval universities circa 1100–1500. ... Philosophical scepticism (UK spelling, scepticism) is both a philosophical school of thought and a method that crosses disciplines and cultures. ... A restored Stoa in Athens. ... Structuralism as a term refers to various theories across the humanities, social sciences and economics many of which share the assumption that structural relationships between concepts vary between different cultures/languages and that these relationships can be usefully exposed and explored. ... This article discusses utilitarian ethical theory. ... This article or section does not cite its references or sources. ...


  Results from FactBites:
 
Pythagorean tuning - Wikipedia, the free encyclopedia (937 words)
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2.
Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave.
In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.
Pythagorean triple - Wikipedia, the free encyclopedia (1837 words)
A primitive Pythagorean triple is one in which a, b and c are coprime.
The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution.
In every Pythagorean triple, the radius of the in-circle and the radii of the three ex-circles are natural numbers.
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