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137-417: Already during Pythagoras' life it is likely that the distinction between the akousmatikoi ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who learn") existed. The ancient biographers of Pythagoras, Iamblichus (c.245-c.325 AD) and his master Porphyry (c.234–c.305 AD) seem to make

274-678: A celestial sphere . This invention undoubtedly made him the first to realize the obliquity of the Zodiac as the Roman philosopher Pliny the Elder reports in Natural History (II, 8). It is a little early to use the term ecliptic , but his knowledge and work on astronomy confirm that he must have observed the inclination of the celestial sphere in relation to the plane of the Earth to explain

411-571: A religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism . Even the early Christian cenobitic traditions may be seen in light of the matematikoi. The biblical Greek name for 'disciple' is 'mathetes' [1] . Pythagoras was already well known in ancient times for his supposed mathematical achievement of the Pythagorean theorem . Pythagoras had been credited with discovering that in

548-410: A body distinct from the elements). the infinite, and not air or water, in order that the other things may not be destroyed by their infinity. They are in opposition one to another. air is cold, water moist, and fire hot. and therefore, if any one of them were infinite, the rest would have ceased to be by this time. Accordingly they say that what is infinite is something other than the elements, and from it

685-460: A boundless stock from which the waste of existence is continually made good, "elements.". That is only the natural development of the thought we have ascribed to Thales, and there can be no doubt that Anaximander at least formulated it distinctly. Indeed, we can still follow to some extent the reasoning which led him to do so. Thales had regarded water as the most likely thing to be that of which all others are forms; Anaximander appears to have asked how

822-409: A common standard of law and justice for both the family and the city. Whoever follows the paths within and searches will discover; for within is law and justice, which is the proper arrangement of the soul. Pythagoreans believed that body and soul functioned together, and a healthy body required a healthy psyche. Early Pythagoreans conceived of the soul as the seat of sensation and emotion. They regarded

959-635: A counterculture in the dominant meat-eating culture. The philosopher Empedocles is said to have refused the customary blood sacrifice by offering a substitute sacrifice after his victory in a horse race in Olympia . Late-Pythagorean philosophers were absorbed into the Platonic school of philosophy and in the 4th century BC the head of the Platonic Academy Polemon included vegetarianism in his concept of living according to nature. In

1096-475: A horizontal plane. The position of its shadow on the plane indicated the time of day. As it moves through its apparent course, the Sun draws a curve with the tip of the projected shadow, which is shortest at noon, when pointing due south. The variation in the tip's position at noon indicates the solar time and the seasons; the shadow is longest on the winter solstice and shortest on the summer solstice. The invention of

1233-405: A less active fire to break free. Thunderbolts are the result of a thicker and more violent air flow. He saw the sea as a remnant of the mass of humidity that once surrounded Earth. A part of that mass evaporated under the Sun's action, thus causing the winds and even the rotation of the celestial bodies, which he believed were attracted to places where water is more abundant. He explained rain as

1370-561: A life which is pleasing to his soul, if Pythagoras the wise truly achieved knowledge and understanding beyond that of all men. Empedocles described Pythagoras as "a man of surpassing knowledge, master especially of all kinds of wise works, who had acquired the upmost wealth of understanding." In the 4th century BC the Sophist Alcidamas wrote that Pythagoras was widely honored by Italians. Today scholars typically distinguish two periods of Pythagoreanism: early-Pythagoreanism, from

1507-752: A movable bridge could be used to demonstrate the relationship of musical intervals. Much of the surviving sources on Pythagoras originated with Aristotle and the philosophers of the Peripatetic school , which founded historiographical academic traditions such as biography , doxography and the history of science . The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoras' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander , Andron of Ephesus, Heraclides and Neanthes had access to historical written sources as well as

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1644-460: A pre-Socratic effort to demystify physical processes. His major contribution to history was writing the oldest prose document about the Universe and the origins of life; for this he is often called the "Father of Cosmology " and founder of astronomy. However, pseudo-Plutarch states that he still viewed celestial bodies as deities. He placed the celestial bodies in the wrong order. He thought that

1781-460: A product of the humidity pumped up from Earth by the sun. For him, the Earth was slowly drying up and water only remained in the deepest regions, which someday would go dry as well. According to Aristotle's Meteorology (II, 3), Democritus also shared this opinion. Anaximander speculated about the beginnings and origin of animal life, and that humans came from other animals in waters. According to his evolutionary theory , animals sprang out of

1918-567: A religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism . The akousmatikoi believed that humans had to act in appropriate ways. The Akousmata (translated as "oral saying") was the collection of all the sayings of Pythagoras as divine dogma. The tradition of the akousmatikoi resisted any reinterpretation or philosophical evolution of Pythagoras' teachings. Individuals who strictly followed most akousmata were regarded as wise. The akousmatikoi philosophers refused to recognise that

2055-493: A religious community they relied on oral teachings and worshiped the Pythian Apollo , the oracular god of Delphic Oracle . Pythagoreans preached an austere life. They believed that the soul was buried in the body, which acted as a tomb for the soul in this life. The highest reward a human could attain was for the soul to join in the life of the gods and thus escaped the cycle of reincarnation in another human body. Like

2192-399: A right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals also credit him as the inventor of the monochord , a straight rod on which a string and

2329-437: A rigorous life of the intellect and strict rules on diet, clothing and behavior. Their burial rites were tied to their belief in the immortality of the soul. Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five-year initiation period of listening to the teachings ( akousmata ) in silence. Initiates could through

2466-468: A slightly rounded metal surface. The centre or “navel” of the world ( ὀμφαλός γῆς omphalós gẽs ) could have been Delphi , but is more likely in Anaximander's time to have been located near Miletus. The Aegean Sea was near the map's centre and enclosed by three continents, themselves located in the middle of the ocean and isolated like islands by sea and rivers. Europe was bordered on the south by

2603-490: A sort of primal chaos . According to him, the Universe originates in the separation of opposites in the primordial matter. It embraces the opposites of hot and cold, wet and dry, and directs the movement of things; an entire host of shapes and differences then grow that are found in "all the worlds" (for he believed there were many). "Anaximander taught, then, that there was an eternal. The indestructible something out of which everything arises, and into which everything returns;

2740-413: A system of hollow concentric wheels, filled with fire, with the rims pierced by holes like those of a flute. Consequently, the Sun was the fire that one could see through a hole the same size as the Earth on the farthest wheel, and an eclipse corresponded with the occlusion of that hole. The diameter of the solar wheel was twenty-seven times that of the Earth (or twenty-eight, depending on the sources) and

2877-471: A test become members of the inner circle. However, Pythagoreans could also leave the community if they wished. Iamblichus listed 235 Pythagoreans by name, among them 17 women whom he described as the "most famous" women practitioners of Pythagoreanism. It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic tradition that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras

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3014-745: A third sphere between the two worlds, separating and uniting them. The identification of nous with the demiurge in the neoplatonic tradition was adopted and developed in Christian gnosticism . Augustine of Hippo follows Plotinus, identifying the nous with logos (the creative principle) as part of the Trinity . Iamblichus multiplied the number of divine entities according to universal mathematical theorems. He conceived of gods, angels, demons and heroes: twelve heavenly gods (whose number increases to 36 or 360), 72 other gods proceeding from them, 21 chiefs and 42 nature-gods. His divine realm extends from

3151-475: A vegetarian diet fostered a healthy body and enhanced the search for Arete . The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conductive to the best in a human being. Pythagoreans advanced a grounded theory on the treatment of animals. They believed that any being that experienced pain or suffering should not have pain inflicted on it unnecessarily. Because it

3288-407: A visual comprehension of mathematics and allowed for a geometrical exploration of numerical relationships. Pythagorean philosophers investigated the relationship of numbers extensively. They defined perfect numbers as those that were equal to the sum of all their divisors. For example: 28 = 1 + 2 + 4 + 7 + 14. The theory of odd and even numbers was central to Pythagorean arithmetic . This distinction

3425-407: A wisdom or made a wisdom of his own: a polymathy, an imposture. Two other surviving fragments of ancient sources on Pythagoras are by Ion of Chios and Empedocles . Both were born in the 490s, after Pythagoras' death. By that time, he was known as a sage and his fame had spread throughout Greece. According to Ion, Pythagoras was: ... distinguished for his manly virtue and modesty, even in death has

3562-600: A world map comes from the late Babylonian Map of the World later than 9th century BC but is based probably on a much older map. These maps indicated directions, roads, towns, borders, and geological features. Anaximander's innovation was to represent the entire inhabited land known to the ancient Greeks. Such an accomplishment is more significant than it at first appears. Anaximander most likely drew this map for three reasons. First, it could be used to improve navigation and trade between Miletus 's colonies and other colonies around

3699-450: Is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2 . This ratio, also known as the " pure " perfect fifth , is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." The fact that mathematics could explain

3836-517: Is derived in the last resort from Theophrastos, who certainly knew his book. He seems once at least to have quoted Anaximander's own words, and he criticised his style. Here are the remains of what he said of him in the First Book: "Anaximander of Miletos, son of Praxiades, a fellow-citizen and associate of Thales, said that the material cause and first element of things was the Infinite, he being

3973-711: Is not an independent authority, and the only question is what Theophrastos wrote." For him, it became no longer a mere point in time, but a source that could perpetually give birth to whatever will be. The indefiniteness is spatial in early usages as in Homer (indefinite sea) and as in Xenophanes (6th century BC) who said that the Earth went down indefinitely (to apeiron ) i.e. beyond the imagination or concept of men. Burnet (1930) in Early Greek Philosophy says: "Nearly all we know of Anaximander's system

4110-459: Is the first philosopher known to have written down his studies, although only one fragment of his work remains. Fragmentary testimonies found in documents after his death provide a portrait of the man. Anaximander was an early proponent of science and tried to observe and explain different aspects of the universe, with a particular interest in its origins , claiming that nature is ruled by laws, just like human societies, and anything that disturbs

4247-654: The Suda and Iamblichus' biographer, Eunapius , Iamblichus was born in Chalcis (later called Qinnašrīn ) in Coele , now in northwest Syria . Iamblichus was descended from the Emesene dynasty . He initially studied under Anatolius of Laodicea and later studied under Porphyry , a pupil of Plotinus (the founder of neoplatonism). Iamblichus disagreed with Porphyry about theurgy , reportedly responding to Porphyry's criticism of

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4384-589: The Greek mythical tradition, and by some ideas of Thales  – the father of Western philosophy  – as well as by observations made by older civilizations in the Near East, especially Babylon. All these were developed rationally. In his desire to find some universal principle, he assumed, like traditional religion, the existence of a cosmic order; and his ideas on this used the old language of myths which ascribed divine control to various spheres of reality. This

4521-852: The Mediterranean Sea and was separated from Asia by the Black Sea, the Lake Maeotis , and, further east, either by the Phasis River (now called the Rioni in Georgia ) or the Tanais . The Nile flowed south into the ocean, separating Libya (which was the name for the part of the then-known African continent) from Asia. The Suda relates that Anaximander explained some basic notions of geometry. It also mentions his interest in

4658-478: The gnomon in Greece. He created a map of the world that contributed greatly to the advancement of geography . Anaximander was involved in the politics of Miletus and was sent as a leader to one of its colonies. Anaximander, son of Praxiades, was born in the third year of the 42nd Olympiad (610 BC). According to Apollodorus of Athens , Greek grammarian of the 2nd century BC, he was sixty-four years old during

4795-437: The pentagram , as each diagonal divides the two others at the golden ratio . When linear geometrical figures replaced the dots, the combination of Babylonian algebra and Pythagorean arithmetic provided the basis for Greek geometric algebra. By attempting to establish a system of concrete and permanent rules, Pythagoreans helped to establish strict axiomatic procedures of solving mathematical problems. Pythagoras pioneered

4932-415: The " intermediate " with the something " distinct from " the elements ." "It is certain that he [Anaximander] cannot have said anything about elements, which no one thought of before Empedokles, and no one could think of before Parmenides. The question has only been mentioned because it has given rise to a lengthy controversy, and because it throws light on the historical value of Aristotle's statements. From

5069-596: The "Infinite" with a "material cause", Theophrastos is following the Aristotelian tradition of "nearly always discussing the facts from the point of view of his own system". Aristotle writes ( Metaphysics , I.III 3–4) that the Pre-Socratics were searching for the element that constitutes all things. While each pre-Socratic philosopher gave a different answer as to the identity of this element ( water for Thales and air for Anaximenes), Anaximander understood

5206-421: The "immense ocean from which everything is born and upon which the Earth floats." Anaximander was then able to envisage the Earth at the centre of an infinite space, in which case it required no support as there was nowhere "down" to fall. In Rovelli's view, the shape – a cylinder or a sphere – is unimportant compared to the appreciation of a "finite body that floats free in space." Anaximander's realization that

5343-461: The 1st century AD Ovid identified Pythagoras as the first opponent to meat-eating. But the fuller argument Pythagoreans advanced against the maltreatment of animals did not sustain. Pythagoreans had argued that certain types of food arouse the passions and hindered spiritual ascent. Thus Porphyry would rely on the teachings of the Pythagoreans when arguing that abstinence from eating meat for

5480-574: The 4th century BC as significant mendicant school of philosophy by the Cynics . Mathēmatikoi philosophers were in the 4th century BC absorbed into the Platonic school of Speusippus , Xenocrates and Polemon . As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism . The worship of Pythagoras continued in Italy in the two intervening centuries. As

5617-504: The 4th century BC, when the philosopher Archytas engaged in advanced mathematics as part of his devotion to Pythagoras' teachings. Today, Pythagoras is mostly remembered for his mathematical ideas, and by association with the work early Pythagoreans did in advancing mathematical concepts and theories on harmonic musical intervals , the definition of numbers , proportion and mathematical methods such as arithmetic and geometry . The mathēmatikoi philosophers claimed that numbers were at

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5754-531: The 6th until the 5th century BC, and late-Pythagoreanism, from the 4th until the 3rd century BC. The Spartan colony of Taranto in Italy became the home for many practitioners of Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in Crotone and Metaponto , both of which were Achaean colonies. Early-Pythagorean sects lived in Croton and throughout Magna Graecia . They espoused to

5891-654: The Boundless " intermediate between the elements " than to say that it is " distinct from the elements." Indeed, if once we introduce the elements at all, the former description is the more adequate of the two. At any rate, if we refuse to understand these passages as referring to Anaximander, we shall have to say that Aristotle paid a great deal of attention to some one whose very name has been lost, and who not only agreed with some of Anaximander's views, but also used some of his most characteristic expressions. We may add that in one or two places Aristotle certainly seems to identify

6028-407: The Earth floats free without falling and does not need to be resting on something has been indicated by many as the first cosmological revolution and the starting point of scientific thinking. Karl Popper calls this idea "one of the boldest, most revolutionary, and most portentous ideas in the whole history of human thinking." Such a model allowed the concept that celestial bodies could pass under

6165-501: The Earth, opening the way to Greek astronomy. Rovelli suggests that seeing the stars circling the Pole star , and both vanishing below the horizon on one side and reappearing above it on the other, would suggest to the astronomer that there was a void both above and below the Earth. Anaximander's bold use of non- mythological explanatory hypotheses considerably distinguishes him from previous cosmology writers such as Hesiod . It indicates

6302-603: The Mediterranean Sea and Black Sea. Second, Thales would probably have found it easier to convince the Ionian city-states to join in a federation in order to push the Median threat away if he possessed such a tool. Finally, the philosophical idea of a global representation of the world simply for the sake of knowledge was reason enough to design one. Surely aware of the sea's convexity, he may have designed his map on

6439-559: The Monad to material nature, where the soul descends into matter and becomes embodied in human form. These superhuman beings influence natural events and communicate knowledge about the future, and are accessible with prayers and offerings. Iamblichus posited that numbers are independent, occupying a middle realm between the limited and unlimited. He believed that nature was bound by fate , differing from divine things which are not subject to fate and turn evil and imperfection to good ends; evil

6576-691: The Mysteries of the Egyptians, Chaldeans, and Assyrians , also known as The Theurgia . Although stylistic and doctrinal differences exist between this book and Iamblichus' other works, it originated from his school at least. Iamblichus also completed a coherent polytheist theological system under the Egyptian pseudonym Abammon . Iamblichus was praised by his followers, and contemporaries credited him with miraculous powers. The Roman emperor Julian , not content with Eunapius' modest eulogy that Iamblichus

6713-457: The Orphics, who considered metempsychosis a cycle of grief that could be escaped by attaining liberation from it, Pythagoras seems to postulate an eternal, endless reincarnation where subsequent lives would not be conditioned by any action done in the previous. Some Medieval authors refer to a "Pythagorean diet", entailing the abstention from eating meat, beans or fish. Pythagoreans believed that

6850-521: The Pythagorean community; subsequently did the Greek philosophical traditions become more diverse. The Platonic Academy was arguably a Pythagorean cenobitic institution, outside the city walls of Athens in the 4th century BC. As a sacred grove dedicated to Athena, and Hecademos (Academos). The Academy, the sacred grove of Academos, may have existed, as the contemporaries seem to have believed, since

6987-433: The Pythagorean meeting place in Croton. The anti-Pythagorean attacks in c.  508 BC were headed by Cylon of Croton . Pythagoras escaped to Metapontium. After these initial attacks and the death of Pythagoras, Pythagorean communities in Croton and elsewhere continued to flourish. At around 450 BC attacks on Pythagorean communities were carried out across Magna Graecia . In Croton, a house where Pythagoreans gathered

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7124-437: The Pythagorean philosophers with being the first to attempt a clarification of the planet sequence. The early-Pythagorean philosopher Philolaus believed that limited and unlimited things were the components of the cosmos, and these had existed ever since. The center of the universe, according to Philolaus, was the number one ( hēn ), which equated to the unity of Monism . Philolaus called the number one an "even-odd" because it

7261-456: The Western world. By the time of Plato , his philosophy was almost forgotten, and Aristotle , his successor Theophrastus , and a few doxographers provide us with the little information that remains. However, we know from Aristotle that Thales, also from Miletus, precedes Anaximander. It is debatable whether Thales actually was the teacher of Anaximander, but there is no doubt that Anaximander

7398-417: The agreement of unlike spirits". In Pythagoreanism, numeric harmony was applied in mathematical, medical, psychological, aesthetic, metaphysical and cosmological problems. For Pythagorean philosophers, the basic property of numbers was expressed in the harmonious interplay of opposite pairs. Harmony assured the balance of opposite forces. Pythagoras had in his teachings named numbers and the symmetries of them as

7535-441: The assembly of demos in the agora which is lying in the middle of the city. The same rational way of thought led him to introduce the abstract apeiron (indefinite, infinite, boundless, unlimited ) as an origin of the universe, a concept that is probably influenced by the original Chaos (gaping void, abyss, formless state) from which everything else appeared in the mythical Greek cosmogony . It also takes notice of

7672-457: The balance of nature does not last long. Like many thinkers of his time, Anaximander's philosophy included contributions to many disciplines. In astronomy , he attempted to describe the mechanics of celestial bodies in relation to the Earth. In physics, his postulation that the indefinite (or apeiron ) was the source of all things, led Greek philosophy to a new level of conceptual abstraction . His knowledge of geometry allowed him to introduce

7809-406: The beginning or first principle to be an endless, unlimited primordial mass ( apeiron ), subject to neither old age nor decay, that perpetually yielded fresh materials from which everything we perceive is derived. He proposed the theory of the apeiron in direct response to the earlier theory of his teacher, Thales, who had claimed that the primary substance was water. The notion of temporal infinity

7946-531: The bronze age, even pre-existing the Trojan War. Yet according to Plutarch it was the Athenian strategos (general) Kimon Milkiadou (c.510–450 BC) who converted this, "waterless and arid spot into a well watered grove, which he provided with clear running-tracks and shady walks". Plato (less known as Aristocles) lived almost a hundred years later, circa 427 to 348 BC. On the other hand it seems likely that this

8083-400: The chief representative of Syrian Neoplatonism ". In Chisholm, Hugh (ed.). Encyclopædia Britannica . Vol. 14 (11th ed.). Cambridge University Press. pp. 213–215. Anaximander Anaximander ( / æ ˌ n æ k s ɪ ˈ m æ n d ər / an- AK -sih- MAN -dər ; ‹See Tfd› Greek : Ἀναξίμανδρος Anaximandros ; c.  610  – c.  546 BC )

8220-561: The continuous development of mathematical and scientific research conducted by the mathēmatikoi was in line with Pythagoras's intention. Until the demise of Pythagoreanism in the 4th century BC, the akousmatikoi continued to engage in a pious life by practicing silence, dressing simply and avoiding meat, for the purpose of attaining a privileged afterlife . The akousmatikoi engaged deeply in questions of Pythagoras' moral teachings, concerning matters such as harmony , justice , ritual purity and moral behavior. The mathēmatikoi acknowledged

8357-451: The cosmic order is not monarchic but geometric , and that this causes the equilibrium of the Earth, which is lying in the centre of the universe. This is the projection on nature of a new political order and a new space organized around a centre which is the static point of the system in the society as in nature. In this space there is isonomy (equal rights) and all the forces are symmetrical and transferable. The decisions are now taken by

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8494-690: The countless worlds. This theory places Anaximander close to the Atomists and the Epicureans who, more than a century later, also claimed that an infinity of worlds appeared and disappeared. In the timeline of the Greek history of thought , some thinkers conceptualized a single world (Plato, Aristotle, Anaxagoras and Archelaus ), while others instead speculated on the existence of a series of worlds, continuous or non-continuous ( Anaximenes , Heraclitus , Empedocles and Diogenes ). Anaximander attributed some phenomena, such as thunder and lightning, to

8631-456: The death of the body, the soul would be punished or rewarded. Humans could, through their conduct, ensure that their soul was admitted to another world. The reincarnation in this world equated to a punishment. In Pythagoreanism life in this world is social and in the realm of society justice existed when each part of society received its due. The Pythagorean tradition of universal justice was later referenced by Plato . For Pythagorean philosophers

8768-515: The development of two philosophical traditions within Pythagoreanism in Italy : akousmatikoi and mathēmatikoi . The mathēmatikoi recognised the akousmatikoi as fellow Pythagoreans, but because the mathēmatikoi allegedly followed the teachings of Hippasus , the akousmatikoi philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism. The akousmatikoi were superseded in

8905-523: The distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, akousmatikoi , following initation became mathematikoi . It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinction mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for

9042-539: The early 5th century BC Croton acquired great military and economic importance. Pythagoras emphasised moderation, piety, respect for elders and of the state, and advocated a monogamous family structure. The Croton Council appointed him to official positions. Among others Pythagoras was in charge of education in the city. His influence as political reformer reputedly extended to other Greek colonies in southern Italy and in Sicily. Pythagoras died shortly after an arson attack on

9179-603: The element of origin was often revisited afterwards, notably by Aristotle, and by the Greek tragedian Euripides : "what comes from earth must return to earth." Friedrich Nietzsche , in his Philosophy in the Tragic Age of the Greeks , stated that Anaximander viewed "... all coming-to-be as though it were an illegitimate emancipation from eternal being, a wrong for which destruction is the only penance." Physicist Max Born , in commenting upon Werner Heisenberg 's arriving at

9316-407: The elements arise.'⁠—Aristotle Physics. F, 5 204 b 22 (Ritter and Preller (1898) Historia Philosophiae Graecae, section 16 b)." Anaximander maintains that all dying things are returning to the element from which they came ( apeiron ). The one surviving fragment of Anaximander's writing deals with this matter. Simplicius transmitted it as a quotation, which describes the balanced and mutual changes of

9453-439: The elements: Whence things have their origin, Thence also their destruction happens, According to necessity; For they give to each other justice and recompense For their injustice In conformity with the ordinance of Time. Simplicius mentions that Anaximander said all these "in poetic terms", meaning that he used the old mythical language. The goddess Justice ( Dike ) keeps the cosmic order. This concept of returning to

9590-525: The first Greek philosophers believed made up the universe in fact represent the primordial forces imagined in earlier ways of thinking. Their collision produced what the mythical tradition had called cosmic harmony. In the old cosmogonies – Hesiod (8th – 7th century BC) and Pherecydes (6th century BC) – Zeus establishes his order in the world by destroying the powers which were threatening this harmony (the Titans ). Anaximander claimed that

9727-434: The first principle and called these numeric symmetries harmony. This numeric harmony could be discovered in rules throughout nature. Numbers governed the properties and conditions of all beings and were regarded the causes of being in everything else. Pythagorean philosophers believed that numbers were the elements of all beings and the universe as a whole was composed of harmony and numbers. The philosopher Philolaus , one of

9864-574: The first to introduce this name of the material cause. He says it is neither water nor any other of the so-called elements, but a substance different from them which is infinite" [apeiron, or ἄπειρον ] "from which arise all the heavens and the worlds within them.—Phys, Op. fr. 2 (Dox. p. 476; R. P. 16)." Burnet's quote from the "First Book" is his translation of Theophrastos' Physic Opinion fragment 2 as it appears in p. 476 of Historia Philosophiae Graecae (1898) by Ritter and Preller and section 16 of Doxographi Graeci (1879) by Diels. By ascribing

10001-536: The geographer Eratosthenes , Anaximander was the first to publish a map of the world . The map probably inspired the Greek historian Hecataeus of Miletus to draw a more accurate version. Strabo viewed both as the first geographers after Homer . Maps were produced in ancient times, also notably in Egypt , Lydia , the Middle East , and Babylon . Only some small examples survived until today. The unique example of

10138-473: The gnomon itself cannot be attributed to Anaximander because its use, as well as the division of days into twelve parts, came from the Babylonians . It is they, according to Herodotus ' Histories (II, 109), who gave the Greeks the art of time measurement. It is likely that he was not the first to determine the solstices, because no calculation is necessary. On the other hand, equinoxes do not correspond to

10275-501: The goodness of each person was believed to be threefold: prudence, drive and good fortune. Pythagoreans thought numbers existed "outside of [human] minds" and separate from the world. They had many mystical and magical interpretations of the roles of numbers in governing existence. The Pythagoreans engaged with geometry as a liberal philosophy which served to establish principles and allowed theorems to be explored abstractly and rationally. Pythagorean philosophers believed that there

10412-472: The heart of everything and constructed a new view of the cosmos . In the mathēmatikoi tradition of Pythagoreanism the Earth was removed from the center of the universe . The mathēmatikoi believed that the Earth, along with other celestial bodies, orbited around a central fire. This, they believed, constituted a celestial harmony. Pythagoreanism was a philosophic tradition as well as a religious practice. As

10549-433: The human sentimental world had a profound impact on the Pythagorean philosophy. Pythagoreanism became the quest for establishing the fundamental essences of reality. Pythagorean philosophers advanced the unshakable belief that the essence of all things are numbers and that the universe was sustained by harmony. According to ancient sources music was central to the lives of those practicing Pythagoreanism. They used medicines for

10686-421: The idea that humans had to spend part of this transition inside the mouths of big fish to protect themselves from the Earth's climate until they could come out in open air and lose their scales. He thought that, considering humans' extended infancy, we could not have survived in the primeval world in the same manner we do presently. Both Strabo and Agathemerus (later Greek geographers) claim that, according to

10823-426: The idea that the elementary particles of quantum mechanics are to be seen as different manifestations, different quantum states, of one and the same "primordial substance,"' proposed that this primordial substance be called apeiron . Anaximander was the first to conceive a mechanical model of the world. In his model, the Earth floats very still in the centre of the infinite, not supported by anything. It remains "in

10960-420: The intervention of elements, rather than to divine causes. In his system, thunder results from the shock of clouds hitting each other; the loudness of the sound is proportionate with that of the shock. Thunder without lightning is the result of the wind being too weak to emit any flame, but strong enough to produce a sound. A flash of lightning without thunder is a jolt of the air that disperses and falls, allowing

11097-496: The lunar wheel, whose fire was less intense, eighteen (or nineteen) times. Its hole could change shape, thus explaining lunar phases . The stars and the planets, located closer, followed the same model. Anaximander was the first astronomer to consider the Sun as a huge mass, and consequently, to realize how far from Earth it might be, and the first to present a system where the celestial bodies turned at different distances. Furthermore, according to Diogenes Laertius (II, 2), he built

11234-511: The majority of Pythagorean philosophers had left Italy. Archytas remained in Italy and ancient sources record that he was visited there by young Plato in the early 4th century BC. The Pythagorean schools and societies died out from the 4th century BC. Pythagorean philosophers continued to practice, albeit no organised communities were established. According to surviving sources by the Neopythagorean philosopher Nicomachus , Philolaus

11371-452: The majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy . Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school . As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism . The worship of Pythagoras continued in Italy and as

11508-486: The mathematical and experimental study of music. He objectively measured physical quantities, such as the length of a string and discovered quantitative mathematical relationships of music through arithmetic ratios. Pythagoras attempted to explain subjective psychological and aesthetic feelings, such as the enjoyment of musical harmony. Pythagoras and his students experimented systematically with strings of varying length and tension, with wind instruments , with brass discs of

11645-462: The measurement of time and associates him with the introduction in Greece of the gnomon. In Lacedaemon , he participated in the construction, or at least in the adjustment, of sundials to indicate solstices and equinoxes . Indeed, a gnomon required adjustments from a place to another because of the difference in latitude . In his time, the gnomon was simply a vertical pillar or rod mounted on

11782-512: The middle point between the positions during solstices, as the Babylonians thought. As the Suda seems to suggest, it is very likely that with his knowledge of geometry, he became the first Greek to determine accurately the equinoxes. In his philosophical work De Divinatione (I, 50, 112), Cicero states that Anaximander convinced the inhabitants of Lacedaemon to abandon their city and spend

11919-408: The most prominent figures in Pythagoreanism, was the precursor of Copernicus in moving the earth from the center of the cosmos and making it a planet. According to Aristotle's student Eudemus of Cyprus , the first philosopher to determine quantitatively the size of the known planets and the distance between them was Anaximander , a teacher to Pythagoras, in the 6th century BC. Historic sources credit

12056-439: The mutual changes between the four elements. Origin, then, must be something else unlimited in its source, that could create without experiencing decay, so that genesis would never stop. The Refutation attributed to Hippolytus of Rome (I, 5), and the later 6th century Byzantine philosopher Simplicius of Cilicia , attribute to Anaximander the earliest use of the word apeiron ( ἄπειρον "infinite" or "limitless") to designate

12193-547: The odd was limited. Aristotle recorded in the 4th century BC on the Pythagorean astronomical system: It is not known whether Philolaus believed Earth to be round or flat, but he did not believe the earth rotated, so that the Counter-Earth and the Central Fire were both not visible from Earth's surface, or at least not from the hemisphere where Greece was located. But the conclusion of Pythagorean philosophers that

12330-467: The opposites found in nature — for example, water can only be wet, never dry — and therefore cannot be the one primary substance; nor could any of the other candidates. He postulated the apeiron as a substance that, although not directly perceptible to us, could explain the opposites he saw around him. "If Thales had been right in saying that water was the fundamental reality, it would not be easy to see how anything else could ever have existed. One side of

12467-650: The opposition, the cold and moist, would have had its way unchecked, and the warm and dry would have been driven from the field long ago. We must, then, have something not itself one of the warring opposites, something more primitive, out of which they arise, and into which they once more pass away." Anaximander explains how the four elements of ancient physics ( air , earth , water and fire ) are formed, and how Earth and terrestrial beings are formed through their interactions. Unlike other Pre-Socratics, he never defines this principle precisely, and it has generally been understood (e.g., by Aristotle and by Saint Augustine ) as

12604-447: The oral tradition about Pythagoreanism, which by the 4th century BC was in decline. Neopythagorean philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy. The earliest surviving ancient source on Pythagoras and his followers is a satire by Xenophanes , on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that: Once they say that he

12741-431: The original principle. He was the first philosopher to employ, in a philosophical context, the term archē ( ἀρχή ), which until then had meant beginning or origin. "That Anaximander called this something by the name of Φύσις is the natural interpretation of what Theophrastos says; the current statement that the term ἀρχή was introduced by him appears to be due to a misunderstanding." And "Hippolytos, however,

12878-637: The point of view of his own system, these may be justified; but we shall have to remember in other cases that, when he seems to attribute an idea to some earlier thinker, we are not bound to take what he says in an historical sense." For Anaximander, the principle of things, the constituent of all substances, is nothing determined and not an element such as water in Thales' view. Neither is it something halfway between air and water, or between air and fire, thicker than air and fire, or more subtle than water and earth. Anaximander argues that water cannot embrace all of

13015-504: The practice in On the Mysteries of the Egyptians, Chaldeans, and Assyrians . He returned to Coele Syria around 304 to found a school in Apamea (near Antioch ), a city known for its neoplatonic philosophers. Iamblichus designed a curriculum for studying Plato and Aristotle , and wrote commentaries on the two which survive only in fragments. Pythagoras was his supreme authority, and he wrote

13152-616: The practitioners of Orphism , a religious tradition that developed in parallel to Pythagorean religious practice, Pythagoreanism held that the soul was buried in the body as a punishment for a committed offense and that the soul could be purified. Aside from conducting their daily lives according to strict rules Pythagorean also engaged in rituals to attain purity. The 4th century Greek historian and sceptic philosopher Hecataeus of Abdera asserted that Pythagoras had been inspired by ancient Egyptian philosophy in his use of ritual regulations and his belief in reincarnation . Early Pythagoreanism

13289-457: The primary substance could be one of these particular things. His argument seems to be preserved by Aristotle, who has the following passage in his discussion of the Infinite: "Further, there cannot be a single, simple body which is infinite, either, as some hold, one distinct from the elements, which they then derive from it, or without this qualification. For there are some who make this. (i.e.

13426-422: The psychic force that pervaded the cosmos. Pythagoreans reasoned that the logic of this argument could not be avoided by killing an animal painlessly. The Pythagoreans also thought that animals were sentient and minimally rational. The arguments advanced by Pythagoreans convinced numerous of their philosopher contemporaries to adopt a vegetarian diet. The Pythagorean sense of kinship with non-humans positioned them as

13563-423: The purification ( katharsis ) of the body and, according to Aristoxenus , music for the purification of the soul. Pythagoreans used different types of music to arouse or calm their souls, and certain stirring songs could have notes that existed in the same ratio as the "distances of the heavenly bodies from the centre of" Earth. For Pythagoreans, harmony signified the "unification of a multifarious composition and

13700-686: The purpose of spiritual purification should be practiced only by philosophers, whose aim was to reach a divine state. The biographical tradition on Pythagoras holds that his mother, wife and daughters were part of his inner circle. Women were given equal opportunity to study as Pythagoreans and learned practical domestic skills in addition to philosophy. Iamblichus Iamblichus ( / aɪ ˈ æ m b l ɪ k ə s / eye- AM -blik-əs ; ‹See Tfd› Greek : Ἰάμβλιχος , translit.   Iámblichos ; Arabic : يَمْلِكُ, romanized: Yamlīkū; Aramaic : 𐡉𐡌𐡋𐡊𐡅 , romanized:  Yamlīkū ; c.  245  – c.  325 )

13837-459: The reign of Constantine the Great . Iamblichus detailed Plotinus ' neoplatonic formal divisions, applied Pythagorean number symbolism more systematically, and (influenced by other Asian systems) interpreted neoplatonic concepts mythically. Unlike Plotinus, who broke from platonic tradition by positing a separate soul, Iamblichus re-affirmed the soul's embodiment in matter and believed that matter

13974-445: The relations of string length: the octave (1/2), the fifth (2/3) and the fourth (3/4). The sum of those numbers 1 + 2 + 3 + 4 = 10 was for Pythagoreans the perfect number, because it contained in itself "the whole essential nature of numbers". Werner Heisenberg has called this formulation of musical arithmetic as "among the most powerful advances of human science" because it enables the measurement of sound in space. Pythagorean tuning

14111-495: The religious underpinning of Pythagoreanism and engaged in mathēma (translated as "learning" or "studying") as part of their practice. While their scientific pursuits were largely mathematical, they also promoted other fields of scientific study in which Pythagoras had engaged during his lifetime. A sectarianism developed between the dogmatic akousmatikoi and the mathēmatikoi , who in their intellectual activism became regarded as increasingly progressive. This tension persisted until

14248-441: The revolution of the stars produced a harmonic sound. The early-Pythagorean philosopher Philolaus argued that the structure of the cosmos was determined by the musical numerical proportions of the diatonic octave , which contained the fifth and fourth harmonic intervals. Pythagoreans equated justice with geometrical proportion, because proportion ensured that each part receives what it is due. Early-Pythagoreans believed that after

14385-452: The same diameter but different thickness, and with identical vases filled with different levels of water. Early Pythagoreans established quantitative ratios between the length of a string or pipe and the pitch of notes and the frequency of string vibration. Pythagoras is credited with discovering that the most harmonious musical intervals are created by the simple numerical ratio of the first four natural numbers which derive respectively from

14522-457: The same place because of its indifference", a point of view that Aristotle considered ingenious, in On the Heavens . Its curious shape is that of a cylinder with a height one-third of its diameter. The flat top forms the inhabited world. Carlo Rovelli suggests that Anaximander took the idea of the Earth's shape as a floating disk from Thales , who had imagined the Earth floating in water,

14659-516: The sea long ago, born trapped in a spiny bark, but as they got older, the bark would dry up and animals would be able to break it. The 3rd century Roman writer Censorinus reports: Anaximander of Miletus considered that from warmed up water and earth emerged either fish or entirely fishlike animals. Inside these animals, men took form and embryos were held prisoners until puberty; only then, after these animals burst open, could men and women come out, now able to feed themselves. Anaximander put forward

14796-443: The seasons. The doxographer and theologian Aetius attributes to Pythagoras the exact measurement of the obliquity. According to Simplicius, Anaximander already speculated on the plurality of worlds , similar to atomists Leucippus and Democritus , and later philosopher Epicurus . These thinkers supposed that worlds appeared and disappeared for a while, and that some were born when others perished. They claimed that this movement

14933-422: The second year of the 58th Olympiad (547–546 BC) and died shortly afterwards. Establishing a timeline of his work is impossible, since no document provides chronological references. Themistius , a 4th-century Byzantine rhetorician , mentions that he was the "first of the known Greeks to publish a written document on nature." Therefore, his texts would be amongst the earliest written in prose , at least in

15070-515: The significance of numerology , he believed that numbers themselves explained the true nature of the Universe. Numbers were in the Greek world of Pythagoras' days natural numbers – that is positive integers (there was no zero ). But unlike their Greek contemporaries, the Pythagorean philosophers represented numbers graphically, not symbolically through letters. Pythagoreans used dots, also known as psiphi (pebbles), to represent numbers in triangles, squares, rectangles and pentagons. This enabled

15207-410: The soul as distinct from the intellect. However, only fragments of the early Pythagorean texts have survived, and it is not certain whether they believed the soul was immortal. The surviving texts of the Pythagorean philosopher Philolaus indicate that while early Pythagoreans did not believe that the soul contained all psychological faculties, the soul was life and a harmony of physical elements. As such

15344-469: The soul passed away when certain arrangements of these elements ceased to exist. However, the teaching most securely identified with Pythagoras is metempsychosis , or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. Pythagorean metempsychosis resembles the teachings of the Orphics , although its version contains substantial differences. Unlike

15481-443: The soul was the source of justice and through the harmony of the soul, divinity could be achieved. Injustice inverted the natural order. According to the 4th century BC philosopher Heraclides Ponticus , Pythagoras taught that "happiness consists in knowledge of the perfection of the numbers of the soul. A surviving fragment from the 3rd century BC by the late-Pythagorean philosopher Aesara reasoned that: I think human nature provides

15618-652: The stars were nearest to the Earth, then the Moon, and the Sun farthest away. His scheme is compatible with the Indo-Iranian philosophical traditions contained in the Iranian Avesta and the Indian Upanishads . At the origin, after the separation of hot and cold, a ball of flame appeared that surrounded Earth like bark on a tree. This ball broke apart to form the rest of the Universe. It resembled

15755-574: The ten-volume Collection of Pythagorean Doctrines with extracts from several ancient philosophers; only the first four volumes and fragments of the fifth survive. Iamblichus wrote the Exhortation to Philosophy in Apamea during the early fourth century. Considered a man of great culture and learning, he was renowned for his charity and self-denial and had a number of students. According to Johann Albert Fabricius , he died sometime before 333 during

15892-585: The universe is not geocentric was not based on empirical observation . Instead, as Aristotle noted, the Pythagorean view of the astronomical system was grounded in a fundamental reflection on the value of individual things and the hierarchical order of the universe. Pythagoreans believed in a musica universalis . They reasoned that stars must produce a sound because they were large swiftly moving bodies. Pythagoreans also determined that stars revolved at distances and speeds that were proportional to each other. They reasoned that because of this numerical proportion

16029-504: Was a Syrian neoplatonic philosopher . He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras . In addition to his philosophical contributions, his Protrepticus is important for the study of the sophists because it preserved about ten pages of an otherwise unknown sophist known as the Anonymus Iamblichi. According to

16166-540: Was a pre-Socratic Greek philosopher who lived in Miletus , a city of Ionia (in modern-day Turkey). He belonged to the Milesian school and learned the teachings of his master Thales . He succeeded Thales and became the second master of that school where he counted Anaximenes and, arguably, Pythagoras amongst his pupils. Little of his life and work is known today. According to available historical documents, he

16303-446: Was a close relationship between numbers and geometrical forms. Early-Pythagorean philosophers proved simple geometrical theorems, including "the sum of the angles of a triangle equals two right angles". Pythagoreans also came up with three of the five platonic solids : the tetrahedron , the cube and the dodecahedron . The sides of a regular dodecahedron are regular pentagons , which for Pythagoreans symbolised health. They also revered

16440-554: Was a common practice for the Greek philosophers in a society which saw gods everywhere, and therefore could fit their ideas into a tolerably elastic system. Some scholars see a gap between the existing mythical and the new rational way of thought which is the main characteristic of the archaic period (8th to 6th century BC) in the Greek city-states . This has given rise to the phrase "Greek miracle". But there may not have been such an abrupt break as initially appears. The basic elements of nature ( water , air , fire , earth ) which

16577-780: Was a part of the re-building of Athens led by Kimon Milkiadou and Themosticles, following the Achaemenid destruction of Athens in 480-479 BC - The Second Persian War . Kimon is at least associated with the building of the southern Wall of Themosticles , the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos. Following political instability in Magna Graecia, some Pythagorean philosophers moved to mainland Greece while others regrouped in Rhegium . By about 400 BC

16714-502: Was a prominent citizen. Indeed, Various History (III, 17) explains that philosophers sometimes also dealt with political matters. It is very likely that leaders of Miletus sent him there as a legislator to create a constitution or simply to maintain the colony's allegiance. Anaximander lived the final few years of his life as a subject of the Persian Achaemenid Empire . Anaximander's theories were influenced by

16851-404: Was able to generate both even and odd numbers. When one was added to an odd number it produced an even number, and when added to an even number it produced an odd number. Philolaus further reasoned that the fitting together of the earth and the universe corresponded to the construction of the number one out of the even and the odd. Pythagorean philosophers believed that the even was unlimited and

16988-607: Was as divine as the rest of the cosmos. Iamblichus placed the Monad at the head of his system, from which emanates the Nous (intellect, or demiurge ) and the psyche . Plotinus represented the Nous as three stages: objective being, subjective life, and realized intellect. Iamblichus divided them into two spheres: intelligible (the objects of thought) and intellective (the domain of thought). Iamblichus and Proclus may have introduced

17125-412: Was based on research and the accumulation of knowledge from the books written by other philosophers. Pythagoras' philosophic teachings made direct reference to the philosophy of Anaximander , Anaximenes of Miletus and Pherecydes of Syros . Of the Pythagorean philosophers, Hippasus , Alcmaeon , Hippon , Archytas and Theodorus , written sources have survived. Pythagoras, in his teachings focused on

17262-445: Was eternal, "for without movement, there can be no generation, no destruction". In addition to Simplicius, Hippolytus reports Anaximander's claim that from the infinite comes the principle of beings, which themselves come from the heavens and the worlds (several doxographers use the plural when this philosopher is referring to the worlds within, which are often infinite in quantity). Cicero writes that he attributes different gods to

17399-410: Was familiar to the Greek mind from remote antiquity in the religious concept of immortality, and Anaximander's description was in terms appropriate to this conception. This archē is called "eternal and ageless". (Hippolytus (?), Refutation , I,6,I;DK B2) " Aristotle puts things in his own way regardless of historical considerations, and it is difficult to see that it is more of an anachronism to call

17536-409: Was for the Pythagorean philosophers direct and visual, as they arranged triangular dots so that the even and odd numbers successively alternate: 2, 4, 6, ... 3, 5, 7, ... Early-Pythagorean philosophers such as Philolaus and Archytas held the conviction that mathematics could help in addressing important philosophical problems. In Pythagoreanism numbers became related to intangible concepts. The one

17673-453: Was generated accidentally in the conflict between the finite and the infinite . Only a fraction of Iamblichus' books have survived; knowledge of his system is preserved in fragments of writings preserved by Stobaeus and others: notes by his successors (especially Proclus ), his five extant books and sections of his work on Pythagoreanism . In addition to these, Proclus attributed to him the On

17810-528: Was inferior to Porphyry only in style, regarded him as second only to Plato and said that he would give all the gold in Lydia for one of his letters. During the 15th- and 16th-century revival of interest in his philosophy, Iamblichus' name was rarely mentioned without the epithet "divine" or "most divine". [REDACTED]   This article incorporates text from a publication now in the public domain :  Sorley, William Ritchie (1911). " Iamblichus,

17947-676: Was influenced by Thales' theory that everything is derived from water. One thing that is not debatable is that even the ancient Greeks considered Anaximander to be from the Monist school which began in Miletus, with Thales followed by Anaximander and which ended with Anaximenes . 3rd-century Roman rhetorician Aelian depicts Anaximander as leader of the Milesian colony to Apollonia on the Black Sea coast, and hence some have inferred that he

18084-412: Was known as the site of mysteries. Pythagoras had been born on the island of Samos at around 570 BC and left his homeland at around 530 BC in opposition to the policies of Polycrates . Before settling in Croton, Pythagoras had traveled throughout Egypt and Babylonia . In Croton, Pythagoras established the first Pythagorean community, described as a secret society, and attained political influence. In

18221-611: Was not necessary to inflict pain on animals for humans to enjoy a healthy diet, they believed that animals should not be killed for the purpose of eating them. The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill an animal and that doing so would diminish the moral status of a human. By failing to show justice to the animal, humans diminish themselves. Pythagoreans believed that human beings were animals, but with an advanced intellect and therefore humans had to purify themselves through training. Through purification humans could join

18358-408: Was passing by when a puppy was being whipped, And he took pity and said: "Stop! Do not beat it! For it is the soul of a friend That I recognised when I heard it giving tongue." In a surviving fragment from Heraclitus , Pythagoras and his followers are described as follows: Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selecting of these writings made for himself

18495-410: Was related to the intellect and being, the two to thought, the number four was related to justice because 2 * 2 = 4 and equally even. A dominant symbolism was awarded to the number three , Pythagoreans believed that the whole world and all things in it are summed up in this number, because end, middle and beginning give the number of the whole. The triad had for Pythagoreans an ethical dimension, as

18632-424: Was set on fire and all but two of the Pythagorean philosophers burned alive. Pythagorean meeting places in other cities were also attacked and philosophic leaders killed. These attacks occurred in the context of widespread violence and destruction in Magna Graecia. Following the political instability in the region, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium . By about 400 BC

18769-592: Was the successor of Pythagoras. According to Cicero ( de Orat. III 34.139), Philolaus was teacher of Archytas . According to the Neoplatonist philosopher Iamblichus , Archytas in turn became the head of the Pythagorean school about a century after the Pythagoras' death. Philolaus, Eurytus and Xenophilus are identified by Aristoxenus as the teachers of the last generation of Pythagoreans. Following Pythagoras' death, disputes about his teachings led to

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