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	Topics Pythagoreanism Topic Home Pythagoreanism is a term used for the esoteric and metaphysicalMetaphysics Metaphysics is the branch of philosophy concerned with explaining the nature of the world.... beliefs held by PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... and his followers, the Pythagoreans, who were much influenced by mathematicsMathematics Summary Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... and probably a main inspirational source for PlatoPlato Plato , whose real name is believed to have been Aristocles, was an immensely influential ancient Greek philosopher, ... and PlatonismPlatonism Summary Platonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it.... . Later resurgence of ideas similar to those held by the early Pythagoreans are collected under the term NeopythagoreanismNeopythagoreanism Neopythagoreanism was a Graeco-Alexandrian school of philosophy, reviving Pythagorean doctrines, which became prominent in ... . Two rival schools According to tradition, Pythagoreanism developed at some point into two separate schools of thought, the akousmatikoi ("listeners") and the mathematikoi ("learners"). The mathematikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras, while the akousmatikoi focused on the more religious and ritualistic aspects of his teachings. The akousmatikoi claimed that the mathematikoi were not genuinely Pythagorean, but followers of the "renegade" Pythagorean HippasusHippasus Hippasus of Metapontum, born circa 500 B.C.... . The mathematikoi, on the other hand, allowed that the akousmatikoi were Pythagorean but felt that they were more representative of Pythagoras. Pythagorean natural philosophy Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the area of cosmologyCosmology Cosmology, from the Greek:??sµ?????a is the study of the Universe in its totality, and by extension, humanity's place in it... there is less agreement about what PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soulMetempsychosis Metempsychosis is a philosophical term in the Greek language referring to the belief of transmigration of the soul, especial... 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 AnaximanderAnaximander Anaximander , also known as Aniximander, was the second of the physical philosophers of Ionia, a citizen of Miletus, a... '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. Pythagoras wrote nothing down, and relying on the writings of ParmenidesParmenides Parmenides of Elea was an ancient Greek philosopher born in Elea, a Hellenic city on the southern coast of Italy.... , EmpedoclesEmpedocles Empedocles was a Greek presocratic philosopher and a citizen of Agrigentum, a Greek colony in Sicily.... , PhilolausPhilolaus Philolaus was a Greek Pythagorean and Presocratic.... and PlatoPlato Plato , whose real name is believed to have been Aristocles, was an immensely influential ancient Greek philosopher, ... (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 PhilolausPhilolaus Philolaus was a Greek Pythagorean and Presocratic.... , 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 AnaximanderAnaximander Anaximander , also known as Aniximander, was the second of the physical philosophers of Ionia, a citizen of Miletus, a... 's ideas about the apeiron and the peiron, the unlimited and limited, by writing that: Continuing with the Pythagoreans: 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 StobaeusStobaeus Joannes Stobaeus, so called from his native place Stobi in Macedonia, was the compiler of a valuable series of extracts from... commentated: 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 CrotonCrotone Crotone is a city in Calabria, southern Italy, on the Gulf of Taranto.... , who defined health as a harmony of the elements in the body. After attacks on the Pythagorean meeting-places at Croton, the movement dispersed, but regrouped in TarentumTaranto Taranto is a coastal city in Apulia, southern Italy.... , also in Southern Italy. A collection of Pythagorean writings on ethics collected by Taylor show a creative response to the troubles. 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 SyrosPherecydes of Syros Overview Pherecydes of Syros was a Greek thinker from the island of Syros, of the 6th century BC.... . The Pythagorean symbols are central to the mystery in the novel The Oxford Murders (Crímenes imperceptibles, 2003) by Guillermo MartinezGuillermo Martínez Guillermo Mart?nez is an Argentinian novelist and short story writer.... . Pythagorean cosmology 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 SyrosPherecydes of Syros Pherecydes of Syros was a Greek thinker from the island of Syros, of the 6th century BC.... , 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. Pherecydes 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 pentagramPentagram A pentagram is a five-pointed star drawn with five straight strokes.... , used by them as a symbol of recognition among members and as a symbol of inner health (eugieia). Pythagorean vegetarianism The Pythagoreans were well-known in antiquityClassical antiquity *Classical antiquity* is a broad term for a long period of cultural history centered on the Mediterranean Sea, which begin... 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 "vegetarianVegetarianism Vegetarianism is the practice of not consuming meat, with or without the use of other animal derivatives, such as dairy prod... " in the nineteenth century. 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 flatulenceFlatulence Flatulence is the presence of gas under some degree of pressure, in a confined space.... they cause, perhaps as protection from potential favism, but most likely for magico-religious reasons, such as the belief that beans and humans were created from the same material. Pythagorean view of women Women were given equal opportunity to study as Pythagoreans; however, they learned practical domestic skills in addition to philosophy. Women were held to be different from men, but sometimes in good ways. Neo-Pythagoreanism Neopythagoreanism was a revival in the 2nd century BC—2nd century AD period, of various ideas traditionally associated with the followers of Pythagoras, the Pythagoreans. Notable Neopythagoreans include first century Apollonius of TyanaApollonius of Tyana Apollonius of Tyana was a Greek Pythagorean philosopher and teacher.... and Moderatus of GadesModeratus of Gades Moderatus of Gades was a Greek philosopher of the Neopythagorean school, contemporary with Apollonius of Tyana.... . Middle and Neo-Platonists such as NumeniusNumenius of Apamea Numenius of Apamea was a Greek philosopher, who lived in Apamea in Syria and flourished during the latter half of the second... and PlotinusPlotinus Plotinus was a major philosopher in the ancient world and is widely considered the father of Neoplatonism.... also exhibited some Neo-Pythagorean influence. In 1915 a subterranean basilica was discovered near Porta MaggiorePorta Maggiore The Porta Maggiore, or Porta Prenestina, is one of the eastern gates in the ancient but well-preserved 3rd century Aurelian ... on Via Praenestina, RomeRome Overview Rome is the capital of Italy and of its region, called Latium.... 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 Neopythagorean beliefs but its exact meaning remains a subject of debate. Sentiments similar to Neopythagoreanism can be found in modern philosophy, such as Hilary PutnamHilary Putnam Hilary Whitehall Putnam is an American philosopher who has been a central figure in Western philosophy since the 1960s, espe... 's Realist thesis, "Internal RealismPhilosophical realism Contemporary philosophical realism is the belief in and allegiance to a reality that exists independently of observers.... ," whereby one could be a Pythagorean in this way. 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 GalileiVincenzo Galilei Overview Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicis... , father of Galileo GalileiGalileo Galilei Galileo Galilei was an Italian physicist, astronomer, astrologer and philosopher who is closely associated with the scienti... . 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 physicsPhysics Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world.... . Pythagoreanism has had a clear and obvious influence on the texts found in the hermeticaHermetica Hermetica is a category of popular Late Antique literature purporting to contain secret wisdom, and generally attributed to ... corpus and thus flows over into hermeticismHermeticism Hermeticism is a set of philosophical and religious beliefs based primarily upon the writings attributed to Hermes Trismegis... , gnosticismGnosticism Gnosticism is a term created by modern scholars to describe a collection of religious groups, many of which thought of thems... and alchemyAlchemy Alchemy refers to both an early form of the investigation of nature and an early philosophical and spiritual discipline, bot... . The Pythagorean cosmology also inspired the Arabic gnosticGnosticism Gnosticism is a term created by modern scholars to describe a collection of religious groups, many of which thought of thems... MonoimusMonoimus Monoimus was an Arab gnostic, who was known only from one account in Theodoret until a lost work of anti-heretical writings ... to combine this system with monismMonism Overview Monism is the metaphysical and theological view that all is of one essential essence, principle, substance or energy.... and other things to form his own cosmology. The pentagramPentagram A pentagram is a five-pointed star drawn with five straight strokes.... (five-pointed star) was an important religious symbol used by the Pythagoreans, which is often seen as being related to the elements theorized by EmpedoclesEmpedocles Empedocles was a Greek presocratic philosopher and a citizen of Agrigentum, a Greek colony in Sicily.... to comprise all matter. The Pythagoreans were advised to "speak the truth in all situations," which PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... said he learned from the MagiMagi The Magi was a tribe from ancient Media, who - prior to the absorption of the Medes into the Persian Empire in 550 BC - we... of BabylonBabylon Overview Babylon was an ancient city in Mesopotamia, the ruins of which can be found in present-day Babil Province, Iraq, about 50 mi... . 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 PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... Pythagorean tuningPythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the rati... Esoteric cosmologyEsoteric cosmology or spiritual disciplines and techniques of self-transformation, and references to [[mysticism\|mys... Sacred geometrySacred geometry Sacred geometry can be described as a belief system attributing a religious or cultural value to many of the fundamental for... NumerologyNumerology Numerology refers to any of several systems, traditions or beliefs in a mystical or esoteric relationship between numbers an... Incommensurable magnitudesFacts About Incommensurable magnitudes The Greek discovery of incommensurable magnitudes changed the face of mathematics.... Pythagorean symbols Monad DyadDyad (symbol) Dyad, according to the Pythagoreans, is the principle of "twoness" or "otherness", '.Hemenway, Priya Divine Proportion ... TriadTriad (symbol) Enclosing the greater area with the smallest perimeterm, the triangle, derived from the vesica piscis, the Triad was conside... TetradTetrad (symbol) Summary The tetrad or number four is the first number formed by the addition and multiplication of equals.... PentadPentad The pentad, was used by the as a secret sign to recognize each other.... DecadDecad The number ten or decad was not considered a number by the Pythagoreans, it was seen as an "assembly point" and a symbol of ... TetractysTetractys The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in ea... Vesica piscisVesica 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 c... External links