biography of pythagoras in 300 words

 MacTutor

Pythagoras of samos.

... was transported by the followers of Cambyses as a prisoner of war. Whilst he was there he gladly associated with the Magoi ... and was instructed in their sacred rites and learnt about a very mystical worship of the gods. He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians...

... he formed a school in the city [ of Samos ] , the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings. They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics...

... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner.

... Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs. ... He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method.

(1) that at its deepest level, reality is mathematical in nature, (2) that philosophy can be used for spiritual purification, (3) that the soul can rise to union with the divine, (4) that certain symbols have a mystical significance, and (5) that all brothers of the order should observe strict loyalty and secrecy.

It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution.

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle.

After [ Thales , etc. ] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

... the following philosophical and ethical teachings: ... the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification ( particularly through the intellectual life of the ethically rigorous Pythagoreans ) ; and the understanding ...that all existing objects were fundamentally composed of form and not of material substance. Further Pythagorean doctrine ... identified the brain as the locus of the soul; and prescribed certain secret cultic practices.

In their ethical practices, the Pythagorean were famous for their mutual friendship, unselfishness, and honesty.

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

... was violently suppressed. Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places.

References ( show )

• K von Fritz, Biography in Dictionary of Scientific Biography ( New York 1970 - 1990) . See THIS LINK .
• Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Pythagoras
• R S Brumbaugh, The philosophers of Greece ( Albany, N.Y., 1981) .
• M Cerchez, Pythagoras ( Romanian ) ( Bucharest, 1986) .
• Diogenes Laertius, Lives of eminent philosophers ( New York, 1925) .
• P Gorman, Pythagoras, a life (1979) .
• T L Heath, A history of Greek mathematics 1 ( Oxford, 1931) .
• Iamblichus, Life of Pythagoras ( translated into English by T Taylor ) ( London, 1818) .
• I Levy, La légende de Pythagore de Grèce en Ralestine ( Paris, 1927) .
• L E Navia, Pythagoras : An annotated bibliography ( New York, 1990) .
• D J O'Meara, Pythagoras revived : Mathematics and philosophy in late antiquity ( New York, 1990) .
• Porphyry, Vita Pythagorae ( Leipzig, 1886) ,
• Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods ( London, 1965) ..
• E S Stamatis, Pythagoras of Samos ( Greek ) ( Athens, 1981) .
• B L van der Waerden, Science Awakening ( New York, 1954) .
• C J de Vogel, Pythagoras and Early Pythagoreanism (1966) .
• H Wussing, Pythagoras, in H Wussing and W Arnold, Biographien bedeutender Mathematiker ( Berlin, 1983) .
• L Ya Zhmud', Pythagoras and his school ( Russian ) , From the History of the World Culture 'Nauka' ( Leningrad, 1990) .
• C Byrne, The left-handed Pythagoras, Math. Intelligencer 12 (3) (1990) , 52 - 53 .
• H S M Coxeter, Polytopes, kaleidoscopes, Pythagoras and the future, C. R. Math. Rep. Acad. Sci. Canada 7 (2) (1985) , 107 - 114 .
• W K C Guthrie, A History of Greek Philosophy I (1962) , 146 - 340 .
• F Lleras, The theorem of Pythagoras ( Spanish ) , Mat. Ense nanza Univ. 19 (1981) , 3 - 12 .
• B Russell, History of Western Philosophy ( London, 1961) , 49 - 56 .
• G Tarr, Pythagoras and his theorem, Nepali Math. Sci. Rep. 4 (1) (1979) , 35 - 45 .
• B L van der Waerden, Die Arithmetik der Pythagoreer, Math. Annalen 120 (1947 - 49) , 127 - 153 , 676 - 700 .
• L Zhmud, Pythagoras as a Mathematician, Historia Mathematica 16 (1989) , 249 - 268 .
• L Ya Zhmud', Pythagoras as a mathematician ( Russian ) , Istor.-Mat. Issled. 32 - 33 (1990) , 300 - 325 .

Additional Resources ( show )

Other pages about Pythagoras:

• See Pythagoras on a timeline
• Pythagoras's theorem
• Pythagorus in competition with Boethius in Margaista Philosophica (1504)
• An entry in The Mathematical Gazetteer of the British Isles
• Astronomy: The Structure of the Solar System
• Heinz Klaus Strick biography
• Miller's postage stamps

Other websites about Pythagoras:

• Dictionary of Scientific Biography
• Encyclopaedia Britannica
• G Don Allen
• Internet Encyclopedia of Philosophy
• Google books
• Sci Hi blog

Honours ( show )

Honours awarded to Pythagoras

• Lunar features Crater Pythagoras
• Popular biographies list Number 2

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P ythagoras is often referred to as the first pure mathematician. He was born on the island of Samos, Greece in 569 BC. Various writings place his death between 500 BC and 475 BC in Metapontum, Lucania, Italy. His father, Mnesarchus, was a gem merchant. His mother's name was Pythais. Pythagoras had two or three brothers.

Some historians say that Pythagoras was married to a woman named Theano and had a daughter Damo, and a son named Telauges, who succeeded Pythagoras as a teacher and possibly taught Empedocles. Others say that Theano was one of his students, not his wife, and say that Pythagoras never married and had no children.

Pythagoras was well educated, and he played the lyre throughout his lifetime, knew poetry and recited Homer. He was interested in mathematics, philosophy, astronomy and music, and was greatly influenced by Pherekydes (philosophy), Thales (mathematics and astronomy) and Anaximander (philosophy, geometry).

Pythagoras left Samos for Egypt in about 535 B.C. to study with the priests in the temples. Many of the practices of the society he created later in Italy can be traced to the beliefs of Egyptian priests, such as the codes of secrecy, striving for purity, and refusal to eat beans or to wear animal skins as clothing.

Ten years later, when Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon (in what is now Iraq), where he met the Magoi, priests who taught him sacred rites. Iamblichus (250-330 AD), a Syrian philosopher, wrote about Pythagoras, "He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians..."

In 520 BC, Pythagoras, now a free man, left Babylon and returned to Samos, and sometime later began a school called The Semicircle. His methods of teaching were not popular with the leaders of Samos, and their desire for him to become involved in politics did not appeal to him, so he left.

Pythagoras settled in Crotona, a Greek colony in southern Italy, about 518 BC, and founded a philosophical and religious school where his many followers lived and worked. The Pythagoreans lived by rules of behavior, including when they spoke, what they wore and what they ate. Pythagoras was the Master of the society, and the followers, both men and women, who also lived there, were known as mathematikoi. They had no personal possessions and were vegetarians. Another group of followers who lived apart from the school were allowed to have personal possessions and were not expected to be vegetarians. They all worked communally on discoveries and theories. Pythagoras believed:

• All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can understood through mathematics.
• The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of reincarnations called transmigration until it becomes pure. Pythagoras believed that both mathematics and music could purify.
• Numbers have personalities, characteristics, strengths and weaknesses.
• The world depends upon the interaction of opposites, such as male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow.
• Certain symbols have a mystical significance.
• All members of the society should observe strict loyalty and secrecy.

Because of the strict secrecy among the members of Pythagoras' society, and the fact that they shared ideas and intellectual discoveries within the group and did not give individuals credit, it is difficult to be certain whether all the theorems attributed to Pythagoras were originally his, or whether they came from the communal society of the Pythagoreans. Some of the students of Pythagoras eventually wrote down the theories, teachings and discoveries of the group, but the Pythagoreans always gave credit to Pythagoras as the Master for:

• The sum of the angles of a triangle is equal to two right angles.
• The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it.
• Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means.
• The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers.
• The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two.
• Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus.

Pythagoras studied odd and even numbers, triangular numbers, and perfect numbers. Pythagoreans contributed to our understanding of angles, triangles, areas, proportion, polygons, and polyhedra.

Pythagoras also related music to mathematics. He had long played the seven string lyre, and learned how harmonious the vibrating strings sounded when the lengths of the strings were proportional to whole numbers, such as 2:1, 3:2, 4:3. Pythagoreans also realized that this knowledge could be applied to other musical instruments.

The reports of Pythagoras' death are varied. He is said to have been killed by an angry mob, to have been caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans, or been burned out of his school in Crotona and then went to Metapontum where he starved himself to death. At least two of the stories include a scene where Pythagoras refuses to trample a crop of bean plants in order to escape, and because of this, he is caught.

The Pythagorean Theorem is a cornerstone of mathematics, and continues to be so interesting to mathematicians that there are more than 400 different proofs of the theorem, including an original proof by President Garfield.

Other biographies on this site

Pythagoras was a Greek mathematician known for formulating the Pythagorean Theorem. He was also a philosopher who taught that numbers were the essence of all things. He associated numbers with virtues, colors, music and other qualities. He also believed that the human soul is immortal and after death it moves into another living being, sometimes an animal.

The Pythagorean Theorem

Pythagoras believed the earth was round and that the sun, moon, and other planets had their own movements. His beliefs eventually led to the Copernican theory of the universe.

The principles of the Pythagorean Theorem had already been known before they were formulated by Pythagoras. The Egyptians used a form of the Pythagorean Theorem to lay out their fields and the Greeks borrowed it from the Egyptians.

The theorem says that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. A right triangle is a triangle where one angle equals 90 degrees and the hypotenuse is the side opposite the right angle. If you know the values of two sides of a right triangle, you can easily calculate the missing side.

The Pythagorean Theorem has many proofs. One of the most famous was that of Euclid , the Greek mathematician who was born around 300 B.C. Pythagoras also developed a method of tuning instruments called the Pythagorean tuning. If Pythagoras committed any of his theorems or thoughts to paper, no one has yet found them, though there were forgeries. Pythagoras seemed to have favored oral teaching.

Pythagoras’ Life

Not too much is known about Pythagoras’ early life. Some scholars believe he was born on the island of Samos and that his father was either a merchant of some kind or a lapidist. His mother’s name might have been Pythais. An oracle told her she would give birth to a well-favored son. It is possible that Pythagoras had a variety of teachers as a boy and a young man. Some of these teachers were Greek and some might have been Egyptian or from the East. It is even said that he might have been taught ethics by a woman called Themistoclea, who was a priestess from the oracle at Delphi.

Around 529 BC, Pythagoras settled in Crotona, an Italian colony, and founded a school, or a brotherhood, among the city’s aristocrats. No one is quite sure what was studied at the school because its members were very secretive about it. They seemed to also have followed an ascetic lifestyle.

Some scholars believe that the school was actually a religious sect. However, the citizens of Crotona were suspicious of the brotherhood and killed most of its members during political uprisings. Scholars still are not sure whether Pythagoras left the city before the violence broke out, whether he survived it, or whether he was himself killed during the uprising. It is likely that he Pythagoras died in Metapontum, another Greek colony.

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The Life of Pythagoras

The Father of Numbers

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Pythagoras, a Greek mathematician and philosopher, is best known for his work developing and proving the theorem of geometry that bears his name. Most students remember it as follows: the square of the hypotenuse is equal to the sum of the squares of the other two sides. It's written as: a 2 + b 2 = c 2 .

Pythagoras was born on the island of Samos, off the coast of Asia Minor (what is now mostly Turkey), about 569 BCE. Not much is known of his early life. There is evidence that he was well educated, and learned to read and play the lyre. As a youth, he may have visited Miletus in his late teenage years to study with the philosopher Thales, who was a very old man, Thales's student, Anaximander was giving lectures on Miletus and quite possibly, Pythagoras attended these lectures. Anaximander took a great interest in geometry and cosmology, which influenced the young Pythagoras.

Odyssey to Egypt

The next phase of Pythagoras's life is a bit confusing. He went to Egypt for some time and visited, or at least tried to visit, many of the temples. When he visited Diospolis, he was accepted into the priesthood after completing the rites necessary for admission. There, he continued his education, especially in mathematics and geometry.

From Egypt in Chains

Ten years after Pythagoras arrived in Egypt, relations with Samos fell apart. During their war, Egypt lost and Pythagoras was taken as a prisoner to Babylon. He wasn't treated as a prisoner of war as we would consider it today. Instead, he continued his education in mathematics and music and delved into the teachings of the priests, learning their sacred rites. He became extremely proficient in his studies of mathematics and sciences as taught by the Babylonians.

A Return Home Followed by Departure

Pythagoras eventually returned to Samos, then went to Crete to study their legal system for a short time. In Samos, he founded a school called the Semicircle. In about 518 BCE, he founded another school in Croton (now known as Crotone, in southern Italy). With Pythagoras at the head, Croton maintained an inner circle of followers known as mathematikoi (priests of mathematics). These mathematikoi lived permanently within the society, were allowed no personal possessions and were strict vegetarians. They received training only from Pythagoras, following very strict rules. The next layer of the society was called the akousmatics . They lived in their own houses and only came to the society during the day. The society contained both men and women. 

The Pythagoreans were a highly secretive group, keeping their work out of public discourse. Their interests lay not just in math and "natural philosophy", but also in metaphysics and religion. He and his inner circle believed that souls migrated after death into the bodies of other beings. They thought that animals could contain human souls. As a result, they saw eating animals as cannibalism. 

Contributions

Most scholars know that Pythagoras and his followers didn't study mathematics for the same reasons as people do today. For them, numbers had a spiritual meaning. Pythagoras taught that all things are numbers and saw mathematical relationships in nature, art, and music.

There are a number of theorems attributed to Pythagoras, or at least to his society, but the most famous one,  the Pythagorean theorem , may not be entirely his invention. Apparently, the Babylonians had realized the relationships between the sides of a right triangle more than a thousand years before Pythagoras learned about it. However, he spent a great deal of time working on a proof of the theorem. 

Besides his contributions to mathematics, Pythagoras's work was essential to astronomy. He felt the sphere was the perfect shape. He also realized the orbit of the Moon was inclined to Earth's equator, and deduced that the evening star ( Venus) was the same as the morning star. His work influenced later astronomers such as Ptolemy and Johannes Kepler (who formulated the laws of planetary motion).

Final Flight 

During the later years of the society, it came into conflict with supporters of democracy. Pythagoras denounced the idea, which resulted in attacks against his group. Around 508 BCE, Cylon, a Croton noble attacked the Pythagorean Society and vowed to destroy it. He and his followers persecuted the group, and Pythagoras fled to Metapontum.

Some accounts claim that he committed suicide. Others say that Pythagoras returned to Croton a short time later since the society was not wiped out and continued for some years. Pythagoras may have lived at least beyond 480 BCE, possibly to age 100. There are conflicting reports of both his birth and death dates. Some sources think he was born in 570 BCE and died in 490 BCE. 

Pythagoras Fast Facts

• Born : ~569 BCE on Samos
• Died: ~475 BCE
• Parents : Mnesarchus (father), Pythias (mother)
• Education :  Thales, Anaximander
• Key Accomplishments:  first mathematician
• Britannica: Pythagoras-Greek Philosopher and Mathematician
• University of St. Matthews: Pythagoras Biography

Edited by Carolyn Collins Petersen.

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Some others say that Theano was not his wife, but one of his students, and that Pythagoras was never married and hence had no children. He was well educated, and played the lyre throughout his lifetime. He also knew poetry and recited Homer. Pythagoras was very much interested in mathematics, astronomy, music and philosophy, and was greatly influenced by Thales for mathematics and astronomy, Pherekydes for philosophy and Anaximander for philosophy and geometry. He left Samos in about 535 B.C. for Egypt to study with the priests in the temples.

Pythagoras' contributions to math and science:

When Pythagoras went to southern Italy, in 512 B. C., there he tried to use his symbolic method for teaching which was very similar in all respects to the lessons that he had learnt in Egypt. The Samians treated him in a rude and improper manner as they were not very keen on this method. This was the reason which led Pythagoras to leave Samos. He was dragged into all kinds of diplomatic missions by the citizens of Samos, and he was forced to participate in the public affairs. After this Pythagoras founded a religious and philosophical school in Croton (now known as Crotone) that led to many followers.

Pythagoras was the head of this society. There was an inner circle of followers which was known as mathematikoi. These mathematikoi lived permanently with the Society. They had no personal possessions and they were vegetarians. Both women and men were permitted to become members of this Society. The outer circle of followers of the Society was known as the akousmatics. They on contrary to the mathematikoi lived in their own houses, and come to the Society only during the day. They were also allowed to own their own possessions and were not vegetarians.

Pythagoras' contribution towards Pythagorean Theorem:

Pythagoras has been given credit for discovering the Pythagorean Theorem, since the fourth century AD. This is a theorem in geometry which states that the area of the square on the hypotenuse i.e. the side opposite to the right angle, in a right-angled triangle is equal to the sum of areas of the squares of base and the height i.e. the other two sides corresponding to the right angle of the right-angled triangle.

This means if 'a' and 'b' are the two sides corresponding to the angle that is 90°, and 'c' is the side opposite to the right angle then this means the Pythagoras Theorem states that a2 + b2 = c2.

This theorem was previously utilized by the Indians and the Babylonians. Pythagoras and his students was said to have constructed the first proof. But due to the secretive nature of Pythagoras' school and the tradition of the students to attribute everything towards their teacher, there's no evidence that Pythagoras himself had worked on or proved this theorem. But since it was first discovered by Pythagoras, hence the Theorem is named after him, although the theorem occurred even five centuries after his death, in the writings of Plutarch and Cicero.

Rumor Has It …

… that the Pythagorean hypotenuse started off as a hippopotamus triangulated by two moose outside the Bronx zoo creating the first hippopotanuse moose in history.

Written by Kevin Lepton

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Pythagoras, one of the most famous and controversial ancient Greek philosophers, lived from ca. 570 to ca. 490 BCE. He spent his early years on the island of Samos, off the coast of modern Turkey. At the age of forty, however, he emigrated to the city of Croton in southern Italy and most of his philosophical activity occurred there. Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries. By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato’s and Aristotle’s mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view. See the entry on Pythagoreanism.

The Pythagorean question, then, is how to get behind this false glorification of Pythagoras in order to determine what the historical Pythagoras actually thought and did. In order to obtain an accurate appreciation of Pythagoras’ achievement, it is important to rely on the earliest evidence before the distortions of the later tradition arose. The popular modern image of Pythagoras is that of a master mathematician and scientist. The early evidence shows, however, that, while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested. Pythagoras was famous (1) as an expert on the fate of the soul after death, who thought that the soul was immortal and went through a series of reincarnations; (2) as an expert on religious ritual; (3) as a wonder-worker who had a thigh of gold and who could be two places at the same time; (4) as the founder of a strict way of life that emphasized dietary restrictions, religious ritual and rigorous self discipline.

It remains controversial whether he also engaged in the rational cosmology that is typical of the Presocratic philosopher/scientists and whether he was in any sense a mathematician. The early evidence suggests, however, that Pythagoras presented a cosmos that was structured according to moral principles and significant numerical relationships and may have been akin to conceptions of the cosmos found in Platonic myths, such as those at the end of the Phaedo and Republic . In such a cosmos, the planets were seen as instruments of divine vengeance (“the hounds of Persephone”), the sun and moon are the isles of the blessed where we may go, if we live a good life, while thunder functioned to frighten the souls being punished in Tartarus. The heavenly bodies also appear to have moved in accordance with the mathematical ratios that govern the concordant musical intervals in order to produce a music of the heavens, which in the later tradition developed into “the harmony of the spheres.” It is doubtful that Pythagoras himself thought in terms of spheres, and the mathematics of the movements of the heavens was not worked out in detail. There is evidence that he valued relationships between numbers such as those embodied in the so-called Pythagorean theorem, though it is not likely that he proved the theorem. In recent scholarship this consensus view has received strong challenges, which will be discussed below.

Pythagoras’ cosmos was developed in a more scientific and mathematical direction by his successors in the Pythagorean tradition, Philolaus and Archytas. Pythagoras succeeded in promulgating a new more optimistic view of the fate of the soul after death and in founding a way of life that was attractive for its rigor and discipline and that drew to him numerous devoted followers.

1. The Pythagorean Question

2.1 chronological chart of sources for pythagoras, 2.2 post-aristotelian sources for pythagoras, 2.3 plato and aristotle as sources for pythagoras, 3. life and works, 4.1 the fate of the soul—metempsychosis, 4.2 pythagoras as a wonder-worker, 4.3 the pythagorean way of life, 5. was pythagoras a mathematician or cosmologist, primary sources and commentaries, secondary sources, other internet resources, related entries.

What were the beliefs and practices of the historical Pythagoras? This apparently simple question has become the daunting Pythagorean question for several reasons. First, Pythagoras himself wrote nothing, so our knowledge of Pythagoras’ views is entirely derived from the reports of others. Second, there was no extensive or authoritative contemporary account of Pythagoras. No one did for Pythagoras what Plato and Xenophon did for Socrates. Third, only fragments of the first detailed accounts of Pythagoras, written about 150 years after his death, have survived. Fourth, it is clear that these accounts disagreed with one another on significant points. These four points would already make the problem of determining Pythagoras’ philosophical beliefs more difficult than determining those of almost any other ancient philosopher, but a fifth factor complicates matters even more. By the third century CE, when the first detailed accounts of Pythagoras that survive intact were written, Pythagoras had come to be regarded, in some circles, as the master philosopher, from whom all that was true in the Greek philosophical tradition derived. By the end of the first century BCE, a large collection of books had been forged in the name of Pythagoras and other early Pythagoreans, which purported to be the original Pythagorean texts from which Plato and Aristotle derived their most important ideas. A treatise forged in the name of Timaeus of Locri was the supposed model for Plato’s Timaeus , just as forged treatises assigned to Archytas were the supposed model for Aristotle’s Categories . Pythagoras himself was widely presented as having anticipated Plato’s later metaphysics, in which the one and the indefinite dyad are first principles. Thus, not only is the earliest evidence for Pythagoras’ views meager and contradictory, it is overshadowed by the hagiographical presentation of Pythagoras, which became dominant in late antiquity. Given these circumstances, the only reliable approach to answering the Pythagorean question is to start with the earliest evidence, which is independent of the later attempts to glorify Pythagoras, and to use the picture of Pythagoras which emerges from this early evidence as the standard against which to evaluate what can be accepted and what must be rejected in the later tradition. Following such an approach, Walter Burkert, in his epoch-making book (1972a), revolutionized our understanding of the Pythagorean question, and all modern scholarship on Pythagoras, including this article, stands on his shoulders. For a detailed discussion of the source problems that generate the Pythagorean Question see 2. Sources, below.

The problems regarding the sources for the life and philosophy of Pythagoras are quite complicated, but it is impossible to understand the Pythagorean Question without an accurate appreciation of at least the general nature of these problems. It is best to start with the extensive but problematic later evidence and work back to the earlier reliable evidence. The most detailed, extended and hence most influential accounts of Pythagoras’ life and thought date to the third century CE, some 800 years after he died. Diogenes Laertius (ca. 200–250 CE) and Porphyry (ca. 234–305 CE) each wrote a Life of Pythagoras , while Iamblichus (ca. 245–325 CE) wrote On the Pythagorean Life , which includes some biography but focuses more on the way of life established by Pythagoras for his followers. All of these works were written at a time when Pythagoras’ achievements had become considerably exaggerated. Diogenes may have some claim to objectivity, but both Iamblichus and Porphyry have strong agendas that have little to do with historical accuracy. Iamblichus presents Pythagoras as a soul sent from the gods to enlighten mankind (O’Meara 1989, 35–40). Iamblichus’ work was just the first in a ten volume work, which in effect Pythagoreanized Neoplatonism, but the Pythagoreanism involved was Iamblichus’ own conception of Pythagoras as particularly concerned with mathematics rather than an account of Pythagoreanism based on the earliest evidence. Porphyry also emphasizes Pythagoras’ divine aspects and may be setting him up as a rival to Jesus (Iamblichus 1991, 14). These three third-century accounts of Pythagoras were in turn based on earlier sources, which are now lost. Some of these earlier sources were heavily contaminated by the Neopythagorean view of Pythagoras as the source of all true philosophy, whose ideas Plato, Aristotle and all later Greek philosophers plagiarized. Iamblichus cites biographies of Pythagoras by Nicomachus of Gerasa and a certain Apollonius ( VP 251 and 254) and appears to have used them extensively even where they are not cited (Burkert 1972a, 98 ff.). Nicomachus (ca. 50–ca.150 CE) assigns Pythagoras a metaphysics that is patently Platonic and Aristotelian and that employs distinctive Platonic and Aristotelian terminology ( Introduction to Arithmetic I.1). If the Apollonius cited by Iamblichus is Apollonius of Tyana (1st CE), his account will be influenced by his veneration of Pythagoras as the model for his ascetic life, but some scholars argue that Iamblichus is using an otherwise unknown Apollonius (Flinterman 2014, 357). Porphyry ( VP 48–53) explicitly cites Moderatus of Gades as one of his sources. Moderatus was an “aggressive” Neopythagorean of the first century CE, who reports that Plato, Aristotle, and their pupils Speusippus, Aristoxenus and Xenocrates took for their own everything that was fruitful in Pythagoreanism, leaving only what was superficial and trivial to be ascribed to the school (Dillon 1977, 346). Diogenes Laertius, who appears to have less personal allegiance to the Pythagorean legend, bases his primary account of Pythagoras’ philosophy (VIII. 24–33) on the Pythagorean Memoirs excerpted by Alexander Polyhistor, which are a forgery dating sometime around 200 BCE and which assign not just Platonic but also Stoic ideas to Pythagoras (Burkert 1972a, 53; Kahn 2001, 79–83).

In the Pythagorean Memoirs , Pythagoras is said to have adopted the Monad and the Indefinite Dyad as incorporeal principles, from which arise first the numbers, then plane and solid figures and finally the bodies of the sensible world (Diogenes Laertius VIII. 25). This is the philosophical system that is most commonly ascribed to Pythagoras in the post-Aristotelian tradition, and it is found in Sextus Empiricus’ (2nd century CE) detailed accounts of Pythagoreanism (e.g., M . X. 261) and most significantly in the influential handbook of the differing opinions of the Greek philosophers, which was compiled by Aetius in the first century CE and is based on the Tenets of the Natural Philosophers of Aristotle’s pupil Theophrastus (e.g., H. Diels, Doxographi Graeci I. 3.8). The testimony of Aristotle makes completely clear, however, that this was the philosophical system of Plato in his later years and not that of Pythagoras or even the later Pythagoreans. Aristotle is explicit that, although Plato’s system has similarities to the earlier Pythagorean philosophy of limiters and unlimiteds, the indefinite dyad is unique to Plato ( Metaphysics 987b26 ff.) and the Pythagoreans recognized only the sensible world and hence did not derive it from immaterial principles. In the Philebus , Plato himself tells a story that is very much in agreement with Aristotle’s report. While acknowledging a debt to the philosophy of limiters and unlimiteds, which is found in Aristotle’s accounts of Pythagoreanism and in the fragments of the fifth-century Pythagorean Philolaus, Plato makes clear that this is a considerably earlier philosophy, which he is completely reworking for his own purposes (16c ff.; see Huffman 2001). How are we to explain the later tradition’s divergence from this testimony of Aristotle and Plato? The most convincing suggestion points to evidence that, for reasons which are not entirely clear, Plato’s successors in the Academy, Speusippus, Xenocrates and Heraclides, chose to present Pythagoreanism not just as a precursor of late Platonic metaphysics but as having anticipated its central theses. Thus the tradition which falsely ascribes Plato’s late metaphysics to Pythagoras begins not with the Neopythagoreans in the first centuries BCE and CE but already in the fourth century BCE among Plato’s own pupils (Burkert 1972a, 53–83; Dillon 2003, 61–62 and 153–154). This view of Pythagoreanism finds its way into the doxography of Aetius either because Theophrastus followed the early Academy rather than his teacher Aristotle (Burkert 1972a, 66) or because the Theophrastan doxography on Pythagoras was rewritten in the first century BCE under the influence of Neopythagoreanism (Diels 1958, 181; Zhmud 2012a, 455). Aristotle’s careful distinctions between Plato and fifth-century Pythagoreanism, which make excellent sense in terms of the general development of Greek philosophy, are largely ignored in the later tradition in favor of the more sensational ascription of mature Platonism to Pythagoras. The evidence for the early Academy is, however, very limited and some reject the thesis that its members assigned late Platonic metaphysics to Pythagoras (Zhmud 2012, 415–432). The key text is found in Proclus’ Commentary on the Parmenides (pp. 38.32–40.7 Klibansky). Proclus quotes a passage in which Speusippus assigns to the ancients, who in this context are the Pythagoreans, the One and the Indefinite Dyad. Some scholars argue that this is not a genuine fragment of Speusippus but rather a later fabrication (see Zhmud 2012a, 424–425 and for a response Dillon 2014, 251). If the Academy did not assign the One and the Dyad to Pythagoras, however, it becomes less clear how these principles came to be assigned to him. Theophrastus assigns them to the Pythagoreans ( Metaphysics 11a27), but since Aristotle distinguishes the Pythagoreans from Plato on this point, Zhmud’s suggestion (2012a, 455) that he is following his teacher and just taking “the next step” does not work. Theophrastus’ evidence makes best sense if we accept the traditional view and suppose that it is on the authority of Plato’s successors in the Academy that he bases his departure from his teacher’s, Aristotle’s, view.

If we step back for a minute and compare the sources for Pythagoras with those available for other early Greek philosophers, the extent of the difficulties inherent in the Pythagorean Question becomes clear. When trying to reconstruct the philosophy of Heraclitus, for example, modern scholars rely above all on the actual quotations from Heraclitus’ book preserved in later authors. Since Pythagoras wrote no books, this most fundamental of all sources is denied us. In dealing with Heraclitus, the modern scholar turns with reluctance next to the doxographical tradition, the tradition represented by Aetius in the first century CE, which preserves in handbook form a systematic account of the beliefs of the Greek philosophers on a series of topics having to do with the physical world and its first principles. Aetius’ work was first reconstructed by Hermann Diels (1958) and more recently by Mansfeld and Runia (2020) on the basis of two later works, which were derived from it, the Selections of Stobaeus (5th century CE) and the Opinions of Philosophers by pseudo-Plutarch (2nd century CE). Scholars’ faith in this evidence is largely based on the assumption that most of it goes back to Aristotle’s school and in particular to Theophrastus’ Tenets of the Natural Philosophers . Here again the case of Pythagoras is exceptional. Pythagoras is represented in this tradition but, as we have seen, Theophrastus in this case either adopted the view that, against all historical plausibility, assigns Plato’s later metaphysics to Pythagoras or Theophrastus’ doxography on the Pythagoreans was rewritten in the first century BCE. Thus, the second standard source for evidence for early Greek philosophy is, in the case of Pythagoras, corrupted. Whatever views Pythagoras might have had are replaced by late Platonic metaphysics in the doxographical tradition.

A third source of evidence for early Greek philosophy is regarded with great skepticism by most scholars and, in the case of most early Greek philosophers, used only with great caution. This is the biographical tradition represented by the Lives of the Philosophers written by Diogenes Laertius. In this case we at first sight appear to be in luck, at least with regard to the amount of evidence for Pythagoras, since, as we have seen, two major accounts of the life of Pythagoras and the Pythagorean way of life survive in addition to Diogenes’ life. Unfortunately, these two additional lives are written by authors (Iamblichus and Porphyry) whose goal is explicitly non-historical, and all three of the lives rely heavily on authors in the Neopythagorean tradition, whose goal was to show that all later Greek philosophy, insofar as it was true, had been stolen from Pythagoras. There are, however, some sections in these three lives that derive from sources that go back beyond the distorting influence of Neopythagoreanism, to sources in the fourth-century BCE, sources which are also independent of the early Academy’s attempt to assign Platonic metaphysics to the Pythagoreans. The most important of these sources are the fragments of Aristotle’s lost treatises on the Pythagoreans and the fragments of works on Pythagoreanism or of works which dealt in passing with Pythagoreanism written by Aristotle’s pupils Dicaearchus and Aristoxenus, in the second half of the fourth century BCE. The historian Timaeus of Tauromenium (ca. 350–260 BCE), who wrote a history of Sicily, which included material on southern Italy where Pythagoras was active, is also important. In some cases, the fragments of these early works are clearly identified in the later lives, but in other cases we may suspect that they are the source of a given passage without being able to be certain. Large problems remain even in the case of these sources. They were all written 150–250 years after the death of Pythagoras; given the lack of written evidence for Pythagoras, they are based largely on oral traditions. Aristoxenus, who grew up in the southern Italian town of Tarentum, where the Pythagorean Archytas was the dominant political figure, and who was himself a Pythagorean before joining Aristotle’s school, undoubtedly had a rich set of oral traditions upon which to draw. It is clear, nonetheless, that 150 years after his death conflicting traditions regarding Pythagoras’ beliefs had arisen on even the most central issues. Thus, Aristoxenus is emphatic that Pythagoras was not a strict vegetarian and ate a number of types of meat (Diogenes Laertius VIII. 20), whereas Aristoxenus’ contemporary, the mathematician Eudoxus, portrays him not only as avoiding all meat but as even refusing to associate with butchers (Porphyry, VP 7). Even among fourth-century authors that had at least some pretensions to historical accuracy and who had access to the best information available, there are widely divergent presentations, simply because such contradictions were endemic to the evidence available in the fourth century. What we can hope to obtain from the evidence presented by Aristotle, Aristoxenus, Dicaearchus, and Timaeus is thus not a picture of Pythagoras that is consistent in all respects but rather a picture that at least defines the main areas of his achievement. This picture can then be tested by the most fundamental evidence of all, the testimony of authors that precede even Aristotle, testimony in some cases that derives from Pythagoras’ own contemporaries. This testimony is extremely limited, about twenty brief references, but this dearth of evidence is not unique to Pythagoras. The pre-Aristotelian testimony for Pythagoras is more extensive than for most other early Greek philosophers and is thus testimony to his fame.

In reconstructing the thought of early Greek philosophers, scholars often turn to Aristotle’s and Plato’s accounts of their predecessors, although Plato’s accounts are embedded in the literary structure of his dialogues and thus do not pretend to historical accuracy, while Aristotle’s apparently more historical presentation masks a considerable amount of reinterpretation of his predecessors’ views in terms of his own thought. In the case of Pythagoras, what is striking is the essential agreement of Plato and Aristotle in their presentation of his significance. Aristotle frequently discusses the philosophy of Pythagoreans, whom he dates to the middle and second half of the fifth century and who posited limiters and unlimiteds as first principles. He sometimes refers to these Pythagoreans as the “so-called Pythagoreans,” suggesting that he had some reservations about the application of the label “Pythagorean” to them. Aristotle strikingly may never refer to Pythagoras himself in his extant writings (of the three possible mentions Metaph . 986a29 is probably an interpolation; Rh. 1398b14 is a quotation from Alcidamas; MM 1182a11 may not be by Aristotle and, if it is, may well be a case where “Pythagoreans” have been turned into “Pythagoras” in the transmission). In the fragments of his now lost two-book treatise on the Pythagoreans, Aristotle does discuss Pythagoras himself, but the references are all to Pythagoras as a founder of a way of life, who forbade the eating of beans (Fr. 195), and to Pythagoras as a wonder-worker, who had a golden thigh and bit a snake to death (Fr. 191). Zhmud (2012a, 259–260) argues that in one place Aristotle also describes Pythagoras as a mathematician (Fr. 191) and in another as studying nature ( Protrepticus Fr. 20) but in neither case are the words likely to belong to Aristotle (see paragraph 9 of section 5 below and Huffman 2014b, 281, n.7). If Aristotle only found evidence for Pythagoras as a wonder-worker and founder of a way of life, it becomes clear why he never mentions Pythagoras in his account of his philosophical predecessors and why he uses the expression “so-called Pythagoreans” to refer to the Pythagoreanism of the fifth-century. For Aristotle Pythagoras did not belong to the succession of thinkers starting with Thales, who were attempting to explain the basic principles of the natural world, and hence he could not see what sense it made to call a fifth-century thinker like Philolaus, who joined that succession by positing limiters and unlimiteds as first principles, a Pythagorean. Álvarez Salas, accepting the three apparent references to Pythagoras in Aristotle’s extant writings which are mentioned above as genuine, suggests that scholars are mistaken to regard Aristotle as slighting Pythagoras, because Aristotle also refers very infrequently to other early figures such as Thales. Aristotle’s three references to Pythagoras are just his “average treatment of the sages of old” (Álvarez Salas 2021, 256). However, even if we accept all three references to Pythagoras as genuine, they are not on a par with what he has to say about the Milesians, since they are not part of his account of the development of Presocratic philosophy in Book One of the Metaphysics and elsewhere. The Milesians are mentioned prominently, while Pythagoras is conspicuously missing. Aristotle does mention the Pythagoreans later this account but they are presented as active around the time of the atomists, so that the reference is clearly to Philolaus and Pythagoreans of his time. Álvarez Salas also argues that Aristotle’s use of the plural Pythagoreans is meant to include Pythagoras, but while this may be plausible in the discussion of metempsychosis in De Anima (407b22), it is problematic when Aristotle explicitly dates the Pythagoreans to the middle of the fifth-century as he does in the Metaphysics (985b24). Aristotle simply does not include Pythagoras in his account of the Presocratic philosopher/scientists. Plato is often thought to be heavily indebted to the Pythagoreans, but he is almost as parsimonious in his references to Pythagoras as Aristotle and mentions him only once in his writings. Plato’s one reference to Pythagoras ( R . 600a) treats him as the founder of a way of life, just as Aristotle does, and, when Plato traces the history of philosophy prior to his time in the Sophist , (242c-e), there is no allusion to Pythagoras. In the Philebus , Plato does describe the philosophy of limiters and unlimiteds, which Aristotle assigns to the so-called Pythagoreans of the fifth century and which is found in the fragments of Philolaus, but like Aristotle he does not ascribe this philosophy to Pythagoras himself. Scholars, both ancient and modern, under the influence of the later glorification of Pythagoras, have supposed that the Prometheus, whom Plato describes as hurling the system down to men, was Pythagoras (e.g., Kahn 2002: 13–14), but careful reading of the passage shows that Prometheus is just Prometheus and that Plato, like Aristotle, assigns the philosophical system to a group of men (Huffman 2001). The fragments of Philolaus show that he was the primary figure of this group. When Plato refers to Philolaus in the Phaedo (61d-e), he does not identify him as a Pythagorean, so that once again Plato agrees with Aristotle in distancing the “so-called Pythagoreans” of the fifth century from Pythagoras himself. For both Plato and Aristotle, then, Pythagoras is not a part of the cosmological and metaphysical tradition of Presocratic philosophy nor is he closely connected to the metaphysical system presented by fifth-century Pythagoreans like Philolaus; he is instead the founder of a way of life.

References to Pythagoras by Xenophanes (ca. 570–475 BCE) and Heraclitus (fl. ca. 500 BCE) show that he was a famous figure in the late sixth and early fifth centuries. For the details of his life we have to rely on fourth-century sources such as Aristoxenus, Dicaearchus and Timaeus of Tauromenium. There is a great deal of controversy about his origin and early life, but there is agreement that he grew up on the island of Samos, near the birthplace of Greek philosophy, Miletus, on the coast of Asia Minor. There are a number of reports that he traveled widely in the Near East while living on Samos, e.g., to Babylonia, Phoenicia and Egypt. To some extent reports of these trips are an attempt to claim the ancient wisdom of the east for Pythagoras and some scholars totally reject them (Zhmud 2012, 83–91), but relatively early sources such as Herodotus (II. 81) and Isocrates ( Busiris 28) associate Pythagoras with Egypt and one of the oral sayings that may go back to him mentions the Egyptian goddess Isis (Huffman 2019), which at least shows interest in Egypt. Aristoxenus says that he left Samos at the age of forty, when the tyranny of Polycrates, who came to power ca. 535 BCE, became unbearable (Porphyry, VP 9). This chronology would suggest that he was born ca. 570 BCE. He then emigrated to the Greek city of Croton in southern Italy ca. 530 BCE; it is in Croton that he first seems to have attracted a large number of followers to his way of life. There are a variety of stories about his death, but the most reliable evidence (Aristoxenus and Dicaearchus) suggests that violence directed against Pythagoras and his followers in Croton ca. 510 BCE, perhaps because of the exclusive nature of the Pythagorean way of life, led him to flee to another Greek city in southern Italy, Metapontum, where he died around 490 BCE (Porphyry, VP 54–7; Iamblichus, VP 248 ff.; On the chronology, see Minar 1942, 133–5). There is little else about his life of which we can be confident.

The evidence suggests that Pythagoras did not write any books. No source contemporaneous with Pythagoras or in the first two hundred years after his death, including Plato, Aristotle and their immediate successors in the Academy and Lyceum, quotes from a work by Pythagoras or gives any indication that any works written by him were in existence. Several later sources explicitly assert that Pythagoras wrote nothing (e.g., Lucian [ Slip of the Tongue , 5], Josephus, Plutarch and Posidonius in DK 14A18; see Burkert 1972a, 218–9). Diogenes Laertius tried to dispute this tradition by quoting Heraclitus’ assertion that “Pythagoras, the son of Mnesarchus, practiced inquiry most of all men and, by selecting these things which have been written up, made for himself a wisdom, a polymathy, an evil conspiracy” (Fr. 129). This fragment shows only that Pythagoras read the writings of others, however, and says nothing about him writing something of his own. The wisdom and evil conspiracy that Pythagoras constructs from these writings need not have been in writing, and Heraclitus’ description of it as an “evil conspiracy” rather suggests that it was not (for the translation and interpretation of Fr. 129, see Huffman 2008b). In the later tradition several books came to be ascribed to Pythagoras, but such evidence as exists for these books indicates that they were forged in Pythagoras’ name and belong with the large number of pseudo-Pythagorean treatises forged in the name of early Pythagoreans such as Philolaus and Archytas. In the third century BCE a group of three books were circulating in Pythagoras’ name, On Education , On Statesmanship , and On Nature (Diogenes Laertius, VIII. 6). A letter from Plato to Dion asking him to purchase these three books from Philolaus was forged in order to “authenticate” them (Burkert 1972a, 223–225). Heraclides Lembus in the second century BCE gives a list of six books ascribed to Pythagoras (Diogenes Laertius, VIII. 7; Thesleff 1965, 155–186 provides a complete collection of the spurious writings assigned to Pythagoras). The second of these is a Sacred Discourse , which some have wanted to trace back to Pythagoras himself. The idea that Pythagoras wrote such a Sacred Discourse seems to arise from a misreading of the early evidence. Herodotus says that the Pythagoreans agreed with the Egyptians in not allowing the dead to be buried in wool and then asserts that there is a sacred discourse about this (II. 81). Herodotus’ focus here is the Egyptians and not the Pythagoreans, who are introduced as a Greek parallel, so that the Sacred Discourse to which he refers is Egyptian and not Pythagorean, as similar passages elsewhere in Book II of Herodotus show (e.g., II. 62; see Burkert 1972a, 219).Various lines of hexameter verse were already circulating in Pythagoras’ name in the third century BCE and were later combined into a compilation known as the Golden Verses , which marks the culmination of the tradition of a Sacred Discourse assigned to Pythagoras (Burkert 1972a, 219, Thesleff 1965, 158–163; and most recently Thom 1995, although his dating of the compilation before 300 BCE is questionable). The lack of any viable written text which could be reasonably ascribed to Pythagoras is shown most clearly by the tendency of later authors to quote either Empedocles or Plato, when they needed to quote “Pythagoras” (e.g., Sextus Empiricus, M . IX. 126–30; Nicomachus, Introduction to Arithmetic I. 2). For an interesting but ultimately unconvincing attempt to argue that the historical Pythagoras did write books, see Riedweg 2005, 42–43 and the response by Huffman 2008a, 205–207.

4. The Philosophy of Pythagoras

One of the manifestations of the attempt to glorify Pythagoras in the later tradition is the report that he, in fact, invented the word ‘philosophy’. This story goes back to the early Academy, since it is first found in Heraclides of Pontus (Cicero, Tusc. V 3.8; Diogenes Laertius, Proem ). The historical accuracy of the story is called into question by its appearance not in a historical or biographical text but rather in a dialogue that recounted Empedocles’ revival of a woman who had stopped breathing. Moreover, the story depends on a conception of a philosopher as having no knowledge but being situated between ignorance and knowledge and striving for knowledge. Such a conception is thoroughly Platonic, however (see, e.g., Symposium 204A), and Burkert demonstrated that it could not belong to the historical Pythagoras (1960). For a recent attempt to defend at least the partial accuracy of the story, see Riedweg 2005: 90–97 and the response by Huffman 2008a, 207–208; see also Zhmud 2012a, 428–430.

Even if he did not invent the word, what can we say about the philosophy of Pythagoras? For the reasons given in 1. The Pythagorean Question and 2. Sources above, any responsible account of Pythagoras’ philosophy must be based in the first place on the evidence prior to Aristotle and in the second place on evidence that our sources explicitly identify as deriving from Aristotle’s books on the Pythagoreans as well as from the books of his pupils such as Aristoxenus and Dicaearchus. There is general agreement as to what the pre-Aristotelian evidence is, although there are differences in interpretation of it. There is less agreement as to what should be included in Aristotle’s, Dicaearchus’ and Aristoxenus’ evidence. What one includes as evidence from these authors will have a significant effect on one’s picture of Pythagoras. One particularly pressing question is whether both chapters 18 and 19 of Porphyry’s Life of Pythagoras should be regarded as deriving from Dicaearchus, as the most recent editor proposes (Mirhady Fr. 40), or whether only chapter 18 should be included, as in the earlier edition of Wehrli (Fr. 33). It is crucial to decide this question before developing a picture of the philosophy of Pythagoras since chapter 19, if it is by Dicaearchus, is our earliest summary of Pythagorean philosophy. Porphyry is very reliable about quoting his sources. He explicitly cites Dicaearchus at the beginning of Chapter 18 and names Nicomachus as his source at the beginning of chapter 20. The material in chapter 19 follows seamlessly on chapter 18: the description of the speeches that Pythagoras gave upon his arrival in Croton in chapter 18 is followed in chapter 19 by an account of the disciples that he gained as the result of those speeches and a discussion of what he taught these disciples. Thus, the onus is on anyone who would claim that Porphyry changes sources before the explicit change at the beginning of chapter 20. Chapter 19 provides a very restrained account of what can be reliably known about Pythagoras’ teachings and that very restraint is one of the strongest supporting arguments for its being based on Dicaearchus, since Porphyry or anyone else in the luxuriant later tradition would be expected to give a much more expanisve presentation of Pythagoras in accordance with the Neopythagorean view of him (Burkert 1972a, 122–123). Wehrli gives no reason for not including chapter 19 and the great majority of scholars accept it as being based on Dicaearchus (see the references in Burkert 1972a, 122, n.7). Zhmud (2012a, 157) following Philip (1966, 139) argues that the passage cannot derive from Dicaearchus, since it presents immortality of the soul with approval, whereas Dicaearchus did not accept its immortality. However, the passage merely reports that Pythagoras introduced the notion of the immortality of the soul without expressing approval or disapproval. Zhmud lists other features of the chapter that he regards as unparalleled in fourth-century sources (2012a, 157) but, since the evidence is so fragmentary, such arguments from silence can carry little weight. Nothing in the chapter is demonstrably late or inconsistent with Dicaearchus’ authorship so we must follow what is suggested by the context in Porphyry and regard it as derived from Dicaearchus.

In the face of the Pythagorean question and the problems that arise even regarding the early sources, it is reasonable to wonder if we can say anything about Pythagoras. A minimalist might argue that the early evidence only allows us to conclude that Pythagoras was a historical figure who achieved fame for his wisdom but that it is impossible to determine in what that wisdom consisted. We might say that he was interested in the fate of the soul and taught a way of life, but we can say nothing precise about the nature of that life or what he taught about the soul (Lloyd 2014). There is some reason to believe, however, that something more than this can be said.

The earliest evidence makes clear that above all Pythagoras was known as an expert on the fate of our soul after death. Herodotus tells the story of the Thracian Zalmoxis, who taught his countrymen that they would never die but instead go to a place where they would eternally possess all good things (IV. 95). Among the Greeks the tradition arose that this Zalmoxis was the slave of Pythagoras. Herodotus himself thinks that Zalmoxis lived long before Pythagoras, but the Greeks’ willingness to portray Zalmoxis as Pythagoras’ slave shows that they thought of Pythagoras as the expert from whom Zalmoxis derived his teaching. Ion of Chios (5 th c. BCE) says of Phercydes of Syros that “although dead he has a pleasant life for his soul, if Pythagoras is truly wise, who knew and learned wisdom beyond all men.” Here Pythagoras is again the expert on the life of the soul after death. A famous fragment of Xenophanes, Pythagoras’ contemporary, provides some more specific information on what happens to the soul after death. He reports that “once when he [Pythagoras] was present at the beating of a puppy, he pitied it and said ‘stop, don’t keep hitting him, since it is the soul of a man who is dear to me, which I recognized, when I heard it yelping’” (Fr. 7). Although Xenophanes clearly finds the idea ridiculous, the fragment shows that Pythagoras believed in metempsychosis or reincarnation, according to which human souls were reborn into other animals after death. This early evidence is emphatically confirmed by Dicaearchus in the fourth century, who first comments on the difficulty of determining what Pythagoras taught and then asserts that his most recognized doctrines were “that the soul is immortal and that it transmigrates into other kinds of animals” (Porphyry, VP 19). Unfortunately we can say little more about the details of Pythagoras’ conception of metempsychosis. According to Herodotus, the Egyptians believed that the soul was reborn as every sort of animal before returning to human form after 3,000 years. Without naming names, he reports that some Greeks both earlier and later adopted this doctrine; this seems very likely to be a reference to Pythagoras (earlier) and perhaps Empedocles (later). Many doubt that Herodotus is right to assign metempsychosis to the Egyptians, since none of the other evidence we have for Egyptian beliefs supports his claim, but it is nonetheless clear that we cannot assume that Pythagoras accepted the details of the view Herodotus ascribes to them. Similarly both Empedocles (see Inwood 2001, 55–68) and Plato (e.g., Republic X and Phaedrus ) provide a more detailed account of transmigration of souls, but neither of them ascribes these details to Pythagoras nor should we. Did he think that we ever escape the cycle of reincarnations? We simply do not know. The fragment of Ion quoted above may suggest that the soul could have a pleasant existence after death between reincarnations or even escape the cycle of reincarnation altogether, but the evidence is too weak to be confident in such a conclusion. In the fourth century several authors report that Pythagoras remembered his previous human incarnations, but the accounts do not agree on the details. Dicaearchus (Aulus Gellius IV. 11.14) and Heraclides (Diogenes Laertius VIII. 4) agree that he was the Trojan hero Euphorbus and Hermes’ son Aethalides in previous lives and Heraclides reports that he was also reborn as Pyrrhus, a fisherman from the island of Delos. Dicaearchus continues the tradition of savage satire begun by Xenophanes, when he suggests that Pythagoras was the beautiful prostitute, Alco, in another incarnation (Huffman 2014b, 281–285). It has been suggested that one of the most important features of this wide variety of reincarnations may have been the experience and hence wisdom that Pythagoras gained from them (Pellò 2018).

It is not clear how Pythagoras conceived of the nature of the transmigrating soul but a few tentative conjectures can be made (Huffman 2009). Transmigration does not require that the soul be immortal; it could go through several incarnations before perishing. Dicaearchus explicitly says that Pythagoras regarded the soul as immortal, however, and this agrees with Herodotus’ description of Zalmoxis’ view. Horky (2021) rejects Dicaearchus’ testimony on the grounds that he is assigning Platonic views back to Pythagoras. This is a common practice among the Neopythaogreans and appears frequently in the pseudo-Pythagorean writings of the first century BCE and later. However, there is no evidence that Aristotle or his school followed this practice and, as Burkert has shown (1972a, 15, 28, and 79), Aristotle was in fact careful to distinguish what is Pythagorean from what is Platonic. Most importantly the goal of such a practice is to glorify Pythagoras and the Pythagoreans by assigning them developed Platonic (and Aristotelian) doctrine. If this were the goal of Dicaerarchus he would hardly have assigned just a general doctrine of the immortality of the soul to Pythagoras without also assigning to him the tripartite Platonic soul as pseudo-Pythagorean texts do (Huffman 1993, 310). So Dicaearchus’ report that Pythagoras believed in the immortality of the soul should be taken at face value. It is likely that Pythagoras used the Greek word psychê to refer to the transmigrating soul, since this is the word used by all sources reporting his views, unlike Empedocles, who used daimon . His successor, Philoalus, uses psychê to refer not to a comprehensive soul but rather to just one psychic faculty, the seat of emotions, which is located in the heart along with the faculty of sensation (Philolaus, Fr. 13). This psychê is explicitly said by Philolaus to be shared with animals. Herodotus uses psychê in a similar way to refer to the seat of emotions. Thus it seems likely that Pythagoras too thought of the transmigrating psychê in this way. If so, it is unlikely that Pythagoras thought that humans could be reincarnated as plants, since psychê is not assigned to plants by Philolaus. It has often been assumed that the transmigrating soul is immaterial, but Philolaus seems to have a materialistic conception of soul and he may be following Pythagoras. Similarly, it is doubtful that Pythagoras thought of the transmigrating soul as a comprehensive soul that includes all psychic faculties. His ability to recognize something distinctive of his friend in the puppy (if this is not pushing the evidence of a joke too far) and to remember his own previous incarnations show that personal identity was preserved through incarnations. This personal identity could well be contained in the pattern of emotions, that constitute a person’s character and that is preserved in the psychê and need not presuppose all psychic faculties. In Philolaus this psychê explicitly does not include the nous (intellect), which is not shared with animals. Thus, it would appear that what is shared with animals and which led Pythagoras to suppose that they had special kinship with human beings (Dicaearchus in Porphyry, VP 19) is not intellect, as some have supposed (Sorabji 1993, 78 and 208) but rather the ability to feel emotions such as pleasure and pain.

There are significant points of contact between the Greek religious movement known as Orphism and Pythagoreanism, but the evidence for Orphism is at least as problematic as that for Pythagoras and often complicates rather than clarifies our understanding of Pythagoras (Betegh 2014; Burkert 1972a, 125 ff.; Kahn 2002, 19–22; Riedweg 2002). There is some evidence that the Orphics also believed in metempsychosis and considerable debate has arisen as to whether they borrowed the doctrine from Pythagoras (Burkert 1972a, 133; Bremmer 2002, 24) or he borrowed it from them (Zhmud 2012a, 221–238). Dicaearchus says that Pythagoras was the first to introduce metempsychosis into Greece (Porphyry VP 19). Moreover, while Orphism presents a heavily moralized version of metempsychosis in accordance with which we are born again for punishment in this life so that our body is the prison of the soul while it undergoes punishment, it is not clear that the same was true in Pythagoreanism. It may be that rebirths in a series of animals and people were seen as a natural cycle of the soul (Zhmud 2012a, 232–233). One would expect that the Pythagorean way of life was connected to metempsychosis, which would in turn suggest that a certain reincarnation is a reward or punishment for following or not following the principles set out in that way of life. However, there is no unambiguous evidence connecting the Pythagorean way of life with metempsychosis.

It is crucial to recognize that most Greeks followed Homer in believing that the soul was an insubstantial shade, which lived a shadowy existence in the underworld after death, an existence so bleak that Achilles famously asserts that he would rather be the lowest mortal on earth than king of the dead (Homer, Odyssey XI. 489). Pythagoras’ teachings that the soul was immortal, would have other physical incarnations and might have a good existence after death were striking innovations that must have had considerable appeal in comparison to the Homeric view. According to Dicaearchus, in addition to the immortality of the soul and reincarnation, Pythagoras believed that “after certain periods of time the things that have happened once happen again and nothing is absolutely new” (Porphyry, VP 19). This doctrine of “eternal recurrence” is also attested by Aristotle’s pupil Eudemus, although he ascribes it to the Pythagoreans rather than to Pythagoras himself. (Fr. 88 Wehrli). The doctrine of transmigration thus seems to have been extended to include the idea that we and indeed the whole world will be reborn into lives that are exactly the same as those we are living and have already lived.

Some have wanted to relegate the more miraculous features of Pythagoras’ persona to the later tradition, but these characteristics figure prominently in the earliest evidence and are thus central to understanding Pythagoras. Aristotle emphasized his superhuman nature in the following ways: there was a story that Pythagoras had a golden thigh (a sign of divinity); the Pythagoreans taught that “of rational beings, one sort is divine, one is human, and another such as Pythagoras” (Iamblichus, VP 31); Pythagoras was seen on the same day at the same time in both Metapontum and Croton; he killed a deadly snake by biting it; as he was crossing a river it spoke to him (all citations are from Aristotle, Fr. 191, unless otherwise noted). Aristotle reports that the people of Croton called Pythagoras the “Hyperborean Apollo” and Iamblichus’ report ( VP 140) that a priest from the land of the Hyperboreans, Abaris, visited Pythagoras and presented him with his arrow, a token of power, may well also go back to Aristotle (Burkert 1972a, 143). Kingsley argues that the visit of Abaris is the key to understanding the identity and significance of Pythagoras. Abaris was a shaman from Mongolia (part of what the Greeks called Hyperborea), who recognized Pythagoras as an incarnation of Apollo. The stillness of ecstacy practiced by Abaris and handed on to Pythagoras is the foundation of all civilization. Abaris’ visit to Pythagoras thus becomes the central moment when civilizing power is passed from East to West (Kingsley 2010).

Whether or not one accepts this account of Pythagoras and his relation to Abaris, there is a clear parallel for some of the remarkable abilities of Pythagoras in the later figure of Empedocles, who promises to teach his pupils to control the winds and bring the dead back to life (Fr. 111). There are recognizable traces of this tradition about Pythagoras even in the pre-Aristotelian evidence, and his wonder-working clearly evoked diametrically opposed reactions. Heraclitus’ description of Pythagoras as “the chief of the charlatans” (Fr. 81) and of his wisdom as “fraudulent art” (Fr. 129) is most easily understood as an unsympathetic reference to his miracles. Empedocles, on the other hand, is clearly sympathetic to Pythagoras, when he describes him as “ a man who knew remarkable things” and who “possessed the greatest wealth of intelligence” and again probably makes reference to his wonder-working by calling him “accomplished in all sorts of wise deeds ”(Fr. 129). In Herodotus’ report, Zalmoxis, whom some of the Greeks identified as the slave and pupil of Pythagoras, tried to gain authority for his teachings about the fate of the soul by claiming to have journeyed to the next world (IV. 95). The skeptical tradition represented in Herodotus’ report treats this as a ruse on Zalmoxis’ part; he had not journeyed to the next world but had in reality hidden in an underground dwelling for three years. Similarly Pythagoras may have claimed authority for his teachings concerning the fate of our soul on the basis of his remarkable abilities and experiences, and there is some evidence that he too claimed to have journeyed to the underworld and that this journey may have been transferred from Pythagoras to Zalmoxis (Burkert 1972a, 154 ff.).

The testimony of both Plato ( R . 600a) and Isocrates ( Busiris 28) shows that Pythagoras was above all famous for having left behind him a way of life, which still had adherents in the fourth century over 100 years after his death. It is plausible to assume that many features of this way of life were designed to insure the best possible future reincarnations, but it is important to remember that nothing in the early evidence connects the way of life to reincarnation in any specific fashion.

One of the clearest strands in the early evidence for Pythagoras is his expertise in religious ritual. Isocrates emphasizes that “he more conspicuously than others paid attention to sacrifices and rituals in temples” ( Busiris 28). Herodotus gives an example: the Pythagoreans agree with the Egyptians in not allowing the dead to be buried in wool (II. 81). It is not surprising that Pythagoras, as an expert on the fate of the soul after death, should also be an expert on the religious rituals surrounding death. A significant part of the Pythagorean way of life thus consisted in the proper observance of religious ritual. One major piece of evidence for this emphasis on ritual is the symbola or acusmata (“things heard”), short maxims that were handed down orally. The earliest source to quote acusmata is Aristotle, in the fragments of his now lost treatise on the Pythagoreans. It is not always possible to be certain which of the acusmata quoted in the later tradition go back to Aristotle and which of the ones that do go back to Pythagoras. Most of Iamblichus’ examples in sections 82–86 of On the Pythagorean Life , however, appear to derive from Aristotle (Burkert 1972a, 166 ff.), and many are in accord with the early evidence we have for Pythagoras’ interest in ritual. Thus the acusmata advise Pythagoreans to pour libations to the gods from the ear (i.e., the handle) of the cup, to refrain from wearing the images of the gods on their fingers, not to sacrifice a white cock, and to sacrifice and enter the temple barefoot. A number of these practices can be paralleled in Greek mystery religions of the day (Burkert 1972a, 177). Indeed, it is important to emphasize that Pythagoreanism was not a religion and there were no specific Pythagorean rites (Burkert 1985, 302). Pythagoras rather taught a way of life that emphasized certain aspects of traditional Greek religion.

A second characteristic of the Pythagorean way of life was the emphasis on dietary restrictions. There is no direct evidence for these restrictions in the pre-Aristotelian evidence, but both Aristotle and Aristoxenus discuss them extensively. Unfortunately the evidence is contradictory and it is difficult to establish any points with certainty. One might assume that Pythagoras advocated vegetarianism on the basis of his belief in metempsychosis, as did Empedocles after him (Fr. 137). Indeed, the fourth-century mathematician and philosopher Eudoxus says that “he not only abstained from animal food but would also not come near butchers and hunters” (Porphyry, VP 7). According to Dicaearchus, one of Pythagoras’ most well-known doctrines was that “all animate beings are of the same family” (Porphyry, VP 19), which suggests that we should be as hesitant about eating other animals as other humans. Unfortunately, Aristotle reports that “the Pythagoreans refrain from eating the womb and the heart, the sea anemone and some other such things but use all other animal food” (Aulus Gellius IV. 11. 11–12). This makes it sound as if Pythagoras forbade the eating of just certain parts of animals and certain species of animals rather than all animals; such specific prohibitions are easy to parallel elsewhere in Greek ritual (Burkert 1972a, 177). Aristoxenus asserts that Pythagoras only refused to eat plough oxen and rams (Diogenes Laertius VIII. 20) and that he was fond of young kids and suckling pigs as food (Aulus Gellius IV. 11. 6). Some have tried to argue that Aristoxenus is refashioning Pythagoreanism in order to make it more rational (e.g., Kahn 2001, 70; Zhmud 2012b, 228), but Aristoxenus, in fact, recognizes the non-rational dimension of Pythagoreanism and Pythagoras’ eating of kids and suckling pigs may itself have religious motivations (Huffman 2012b). Moreover, even if Aristoxenus’ evidence were set aside Aristotle’s testimony and many of the acusmata indicate that Pythagoras ate some meat. Certainly animal sacrifice was the central act of Greek religious worship and to abolish it completely would be a radical step. The acusma reported by Aristotle, in response to the question “what is most just?” has Pythagoras answer “to sacrifice” (Iamblichus, VP 82). Based on the direct evidence for Pythagoras’ practice in Aristotle and Aristoxenus, it seems most prudent to conclude that he did not forbid the eating of all animal food. The later tradition proposes a number of ways to reconcile metempsychosis with the eating of some meat. Pythagoras may have adopted one of these positions, but no certainty is possible. For example, he may have argued that it was legitimate to kill and eat sacrificial animals, on the grounds that the souls of men do not enter into these animals (Iamblichus, VP 85). Perhaps the most famous of the Pythagorean dietary restrictions is the prohibition on eating beans, which is first attested by Aristotle and assigned to Pythagoras himself (Diogenes Laertius VIII. 34). Aristotle suggests a number of explanations including one that connects beans with Hades, hence suggesting a possible connection with the doctrine of metempsychosis. A number of later sources suggest that it was believed that souls returned to earth to be reincarnated through beans (Burkert 1972a, 183). There is also a physiological explanation. Beans, which are difficult to digest, disturb our abilities to concentrate. Moreover, the beans involved are a European vetch ( Vicia faba ) rather than the beans commonly eaten today. Certain people with an inherited blood abnormality develop a serious disorder called favism, if they eat these beans or even inhale their pollen. Aristoxenus interestingly denies that Pythagoras forbade the eating of beans and says that “he valued it most of all vegetables, since it was digestible and laxative” (Aulus Gellius IV. 11.5). The discrepancies between the various fourth-century accounts of the Pythagorean way of life suggest that there were disputes among fourth-century Pythagoreans as to the proper way of life and as to the teachings of Pythagoras himself.

The acusmata indicate that the Pythagorean way of life embodied a strict regimen not just regarding religious ritual and diet but also in a wide variety of aspects of life. The surviving collection of acusmata , however, is not extensive enough to cover all features of human life (Thom 2020, 16). Some of the restrictions appear to be largely arbitrary taboos, e.g., “one must put the right shoe on first” or “one must not travel the public roads” (Iamblichus, VP 83, probably from Aristotle). On the other hand, some aspects of the Pythagorean life involved a moral discipline that was greatly admired, even by outsiders. Pythagorean silence is an important example. Isocrates reports that even in the fourth century people “marvel more at the silence of those who profess to be his pupils than at those who have the greatest reputation for speaking” ( Busiris 28). The ability to remain silent was seen as important training in self-control, and the later tradition reports that those who wanted to become Pythagoreans had to observe a five-year silence (Iamblichus, VP 72). Isocrates is contrasting the marvelous self-control of Pythagorean silence with the emphasis on public speaking in traditional Greek education. Pythagoreans also displayed great loyalty to their friends as can be seen in Aristoxenus’ story of Damon who is willing to stand surety for his friend Phintias, who has been sentenced to death (Iamblichus, VP 233 ff.). In addition to silence as a moral discipline, there is evidence that secrecy was kept about certain of the teachings of Pythagoras. Aristoxenus reports that the Pythagoreans thought that “not all things were to be spoken to all people” (Diogenes Laertius, VIII. 15), but this may only apply to teaching and mean that children should not be taught all things (Zhmud 2012a, 155). Clearer evidence is found in Dicaearchus’ complaint that it is not easy to say what Pythagoras taught his pupils because they observed no ordinary silence about it (Porphyry, VP 19). Indeed, one would expect that an exclusive society such as that of the Pythagoreans would have secret doctrines and symbols. Aristotle says that the Pythagoreans “guarded among their very secret doctrines that one type of rational being is divine, one human, and one such as Pythagoras” (Iamblichus, VP 31). That there should be secret teachings about the special nature and authority of the master is not surprising. This does not mean, however, that all Pythagorean philosophy was secret. Aristotle discusses the fifth-century metaphysical system of Philolaus in some detail with no hint that there was anything secret about it, and Plato’s discussion of Pythagorean harmonic theory in Book VII of the Republic gives no suggestion of any secrecy. Aristotle singles out the acusma quoted above (Iamblichus, VP 31) as secret, but this statement in itself implies that others were not. The idea that all of Pythagoras’ teachings were secret was used in the later tradition to explain the lack of Pythagorean writings and to try to validate forged documents as recently discovered secret treatises. For a sceptical evaluation of Pythagorean secrecy see Zhmud 2012a, 150–158.

There is some controversy as to whether Pythagoras, in fact, taught a way of life governed in great detail by the acusmata as described above. Plato praises the Pythagorean way of life in the Republic (600b), but it is hard to imagine him admiring the set of taboos found in the acusmata (Lloyd 2014, 44; Zhmud 2012a). Although acusmata were collected already by Anaximander of Miletus the younger (ca. 400 BCE) and by Aristotle in the fourth century, Zhmud (2012a, 177–178 and 192–205) argues that very few of these embody specifically Pythagorean ideas and that it is difficult to imagine anyone following this bewildering set of rules literally as Burkert argues (1972a, 191). However, the early evidence suggests that Pythagoras largely constructed the acusmata out of ideas collected from others (Thom 2013; Huffman 2008b: Gemelli Marciano 2002), so it is no surprise that many of them are not uniquely Pythagorean. Moreover, Thom suggests a middle ground between Zhmud and Burkert whereby, contra Zhmud, most of the acusmata were followed by the Pythagoreans but contra Burkert, they were subject to interpretation from the beginning and not followed literally, so that it is possible to imagine people living according to them (Thom, 2013). It is true that there is little if any fifth- and fourth-century evidence for Pythagoreans living according to the acusmata and Zhmud argues that the undeniable political impact of the Pythagoreans would be inexplicable if they lived the heavily ritualized life of the acusmata , which would inevitably isolate them from society (Zhmud 2012a, 175–183). He suggests that the Pythagorean way of life differed little from standard aristocratic morality (Zhmud 2012a, 175). If, however, the Pythagorean way of life was little out of the ordinary, why do Plato and Isocrates specifically comment on how distinctive those who followed it were? The silence of fifth-century sources about people practicing acusmata is not terribly surprising given the very meager sources for the Greek cities in southern Italy in the period. Why not suppose that the vast majority of names in Aristoxenus’ catalogue of Pythagoreans, who are not associated with any political, philosophical or scientific accomplishment, who are just names to us, are precisely those who were Pythagoreans because they followed the Pythagorean way of life? We would then have lots of people who followed the acusmata (166 of the 222 name in the catalogue appear nowhere else). This suspicion is confirmed by the fact that one of the names from Aristoxenus’ catalogue (Hippomedon of Argos) is elsewhere (Iamblichus, On the Pythagorean Life , 87) explicitly said to belong the acusmatici . Moreover, other scholars argue that archaic Greek society in southern Italy was pervaded by religion and the presence of similar precepts in authors such as Hesiod show that adherence to taboos such as are found in the acusmata would not have caused a scandal and adherence to many of them would have gone unobserved by outsiders (Gemelli Marciano 2014, 133–134).

Once again a problem of source criticism raises its head. Zhmud argues that the split between acusmatici who blindly followed the acusmata and the mathematici who learned the reasons for them (see the fifth paragraph of section 5 below) is a creation of the later tradition, appearing first in Clement of Alexandria and disappearing after Iamblichus (Zhmud 2012a, 169–192). He also notes that the term acusmata appears first in Iamblichus ( On the Pythagorean Life 82–86) and suggests that it is also a creation of the later tradition. The Pythagorean maxims did exist earlier, as the testimony of Aristotle shows, but they were known as symbola , were originally very few in number and were mainly a literary phenomena rather than being tied to people who actually practiced them (Zhmud 2012a, 192–205). However, several scholars have argued that the passages in which the split between the acusmatici and mathematici is described as well as the passage in which the term acusmata is used, in fact, go back to Aristotle (Burkert 1972a, 196; see Burkert 1998, 315 where he comments that the Aristotelian provenance of the text is “as obvious as it is unprovable”) and even Zhmud recognizes that a large part of the material in Iamblichus is derived from Aristotle (2012a, 170). Indeed, the description of the split in what is likely to be the original version (Iamblichus, On General Mathematical Science 76.16–77.18 Festa) uses language in describing the Pythagoreans that is almost an Aristotelian signature, “There are two forms of the Italian philosophy which is called Pythagorean” (76.16). Aristotle famously describes the Pythagoreans as “those called Pythagoreans” and also describes them as “the Italians” (e.g., Mete. 342b30, Cael. 293a20). So the question of whether Pythagoras taught a way of life tightly governed by the acusmata turns again on whether key passages in Iamblichus ( On the Pythagorean Life 81–87, On General Mathematical Science 76.16–77.18 Festa) go back to Aristotle. If they do, we have very good reason to believe that Pythagoras taught such a life, if they do not the issue is less clear.

The testimony of fourth-century authors such as Aristoxenus and Dicaearchus indicates that the Pythagoreans also had an important impact on the politics and society of the Greek cities in southern Italy. Dicaearchus reports that, upon his arrival in Croton, Pythagoras gave a speech to the elders and that the leaders of the city then asked him to speak to the young men of the town, the boys and the women (Porphyry, VP 18). Women, indeed, may have played an unusually large role in Pythagoreanism (see Rowett 2014, 122–123), since both Timaeus and Dicaearchus report on the fame of Pythagorean women including Pythagoras’ daughter (Porphyry, VP 4 and 19). The acusmata teach men to honor their wives and to beget children in order to insure worship for the gods (Iamblichus, VP 84–6). Dicaearchus reports that the teaching of Pythagoras was largely unknown, so that Dicaearchus cannot have known of the content of the speech to the women or of any of the other speeches; the speeches presented in Iamblichus ( VP 37–57) are thus likely to be later forgeries (Burkert 1972a, 115), but there is early evidence that he gave different speeches to different groups (Antisthenes V A 187). The attacks on the Pythagoreans both in Pythagoras’ own day and in the middle of the fifth century are presented by Dicaearchus and Aristoxenus as having a wide-reaching impact on Greek society in southern Italy; the historian Polybius (II. 39) reports that the deaths of the Pythagoreans meant that “the leading citizens of each city were destroyed,” which clearly indicates that many Pythagoreans had positions of political authority. On the other hand, it is noteworthy that Plato explicitly presents Pythagoras as a private rather than a public figure ( R . 600a). It seems most likely that the Pythagorean societies were in essence private associations but that they also could function as political clubs (see Zhmud 2012a, 141–148), while not being a political party in the modern sense; their political impact should perhaps be better compared to modern fraternal organizations such as the Masons. Thus, the Pythagoreans did not rule as a group but had political impact through individual members who gained positions of authority in the Greek city-states in southern Italy. See further Burkert 1972a, 115 ff., von Fritz 1940, Minar 1942 and Rowett 2014.

In the modern world Pythagoras is most of all famous as a mathematician, because of the theorem named after him, and secondarily as a cosmologist, because of the striking view of a universe ascribed to him in the later tradition, in which the heavenly bodies produce “the music of the spheres” by their movements. It should be clear from the discussion above that, while the early evidence shows that Pythagoras was indeed one of the most famous early Greek thinkers, there is no indication in that evidence that his fame was primarily based on mathematics or cosmology. Neither Plato nor Aristotle treats Pythagoras as having contributed to the development of Presocratic cosmology, although Aristotle in particular discusses the topic in some detail in the first book of the Metaphysics and elsewhere. Aristotle evidently knows of no cosmology of Pythagoras that antedates the cosmological system of the “so-called Pythagoreans,” which he dates to the middle of the fifth century, and which is found in the fragments of Philolaus. There is also no mention of Pythagoras’ work in geometry or of the Pythagorean theorem in the early evidence. Dicaearchus comments that “what he said to his associates no one can say reliably,” but then identifies four doctrines that became well known: (1) that the soul is immortal; (2) that it transmigrates into other kinds of animals; (3) that after certain intervals the things that have happened once happen again, so that nothing is completely new; (4) that all animate beings belong to the same family (Porphyry, VP 19). Thus, for Dicaearchus too, it is not as a mathematician or Presocratic writer on nature that Pythagoras is famous. It might not be too surprising that Plato, Aristotle and Dicaearchus do not mention Pythagoras’ work in mathematics, because they are not primarily dealing with the history of mathematics. On the other hand, Aristotle’s pupil Eudemus did write a history of geometry in the fourth century and what we find in Eudemus is very significant. A substantial part of Eudemus’ overview of the early history of Greek geometry is preserved in the prologue to Proclus’ commentary on Book One of Euclid’s Elements (p. 65, 12 ff.), which was written much later, in the fifth century CE. At first sight, it appears that Eudemus did assign Pythagoras a significant place in the history of geometry. Eudemus is reported as beginning with Thales and an obscure figure named Mamercus, but the third person mentioned by Proclus in this report is Pythagoras, immediately before Anaxagoras. There is no mention of the Pythagorean theorem, but Pythagoras is said to have transformed the philosophy of geometry into a form of liberal education, to have investigated its theorems in an immaterial and intellectual way and specifically to have discovered the study of irrational magnitudes and the construction of the five regular solids. Unfortunately close examination of the section on Pythagoras in Proclus’ prologue reveals numerous difficulties and shows that it comes not from Eudemus but from Iamblichus with some additions by Proclus himself (Burkert 1972a, 409 ff.). The first clause is taken word for word from Iamblichus’ On Common Mathematical Science (p. 70.1 Festa). Proclus elsewhere quotes long passages from Iamblichus and is doing the same here. As Burkert points out, however, as soon as we recognize that Proclus has inserted a passage from Iamblichus into Eudemus’ history, we must also recognize that Proclus was driven to do so by the lack of any mention of Pythagoras in Eudemus. Even those who want to assign Pythagoras a larger role in early Greek mathematics recognize that most of what Proclus says here cannot go back to Eudemus (Zhmud 2012a, 263–266). Thus, not only is Pythagoras not commonly known as a geometer in the time of Plato and Aristotle, but also the most authoritative history of early Greek geometry assigns him no role in the history of geometry in the overview preserved in Proclus. According to Proclus, Eudemus did report that two propositions, which are later found in Euclid’s Elements , were discoveries of the Pythagoreans (Proclus 379 and 419). Eudemus does not assign the discoveries to any specific Pythagorean, and they are hard to date. The discoveries might be as early as Hippasus in the middle of the fifth century, who is associated with a group of Pythagoreans known as the mathematici , who arose after Pythagoras’ death (see below). The crucial point to note is that Eudemus does not assign these discoveries to Pythagoras himself. The first Pythagorean whom we can confidently identify as an accomplished mathematician is Archytas in the late fifth and the first half of the fourth century.

Are we to conclude, then, that Pythagoras had nothing to do with mathematics or cosmology? The evidence is not quite that simple. The tradition regarding Pythagoras’ connection to the Pythagorean theorem reveals the complexity of the problem. None of the early sources, including Plato, Aristotle and their pupils shows any knowledge of Pythagoras’ connection to the theorem. Almost a thousand years later, in the fifth century CE, Proclus, in his commentary on Euclid’s proof of the theorem ( Elements I. 47), gives the following report: “If we listen to those who wish to investigate ancient history, it is possible to find them referring this theorem back to Pythagoras and saying that he sacrificed an ox upon its discovery” (426.6). Proclus gives no indication of his source, but a number of other late reports (Diogenes Laertius VIII. 12; Athenaeus 418f; Plutarch, Moralia 1094b) show that it ultimately relied on two lines of verse whose context is unknown: “When Pythagoras found that famous diagram, in honor of which he offered a glorious sacrifice of oxen...” The author of these verses is variously identified as Apollodorus the calculator or Apollodorus the arithmetician. This Apollodorus probably dates before Cicero, who alludes to the story ( On the Nature of the Gods III. 88), and, if he can be identified with Apollodorus of Cyzicus, the follower of Democritus, the story would go back to the fourth century BCE (Burkert 1972a, 428). Two lines of poetry of indeterminate date are obviously a very slender support upon which to base Pythagoras’ reputation as a geometer, but they cannot be simply ignored. Several things need to be noted about this tradition, however, in order to understand its true significance. First, Proclus does not ascribe a proof of the theorem to Pythagoras but rather goes on to contrast Pythagoras as one of those “knowing the truth of the theorem” with Euclid who not only gave the proof found in Elements I.47 but also a more general proof in VI. 31. Although a number of modern scholars have speculated on what sort of proof Pythagoras might have used (e.g., Heath 1956, 352 ff.), it is important to note that there is not a jot of evidence for a proof by Pythagoras; what we know of the history of Greek geometry makes such a proof by Pythagoras improbable, since the first work on the elements of geometry, upon which a rigorous proof would be based, is not attested until Hippocrates of Chios, who was active after Pythagoras in the latter part of the fifth century (Proclus, A Commentary on the First Book of Euclid’s Elements , 66). All that this tradition ascribes to Pythagoras, then, is discovery of the truth contained in the theorem. The truth may not have been in general form but rather focused on the simplest such triangle (with sides 3, 4 and 5), pointing out that such a triangle and all others like it will have a right angle. Modern scholarship has shown, moreover, that long before Pythagoras the Babylonians were aware of the basic Pythagorean rule and could generate Pythagorean triples (integers that satisfy the Pythagorean rule such as 3, 4 and 5), although they never formulated the theorem in explicit form or proved it (Høyrup 1999, 401–2, 405; cf. Robson 2001). Thus, it is likely that Pythagoras and other Greeks first encountered the truth of the theorem as a Babylonian arithmetical technique (Høyrup 1999, 402; Burkert 1972a,429). It is possible, then, that Pythagoras just passed on to the Greeks a truth that he learned from the East. The emphasis in the two lines of verse is not just on Pythagoras’ discovery of the truth of the theorem, it is as much or more on his sacrifice of oxen in honor of the discovery. We are probably supposed to imagine that the sacrifice was not of a single ox; Apollodorus describes it as “a famous sacrifice of oxen” and Diogenes Laertius paraphrases this as a hecatomb , which need not be, as it literally says, a hundred oxen, but still suggests a large number. Some have wanted to doubt the whole story, including the discovery of the theorem, because it conflicts with Pythagoras’ supposed vegetarianism, but it is far from clear to what extent he was a vegetarian (see above). If the story is to have any force and if it dates to the fourth century, it shows that Pythagoras was famous for a connection to a certain piece of geometrical knowledge, but it also shows that he was famous for his enthusiastic response to that knowledge, as evidenced in his sacrifice of oxen, not for any geometric proof. What emerges from this evidence, then, is not Pythagoras as the master geometer, who provides rigorous proofs, but rather Pythagoras as someone who recognizes and celebrates with a ritual act thanking the gods certain geometrical relationships as of high importance.

It is striking that a very similar picture of Pythagoras emerges from the evidence for his cosmology. A famous discovery is attributed to Pythagoras in the later tradition, i.e., that the central musical concords (the octave, fifth and fourth) correspond to the whole number ratios 2 : 1, 3 : 2 and 4 : 3 respectively (e.g., Nicomachus, Handbook 6 = Iamblichus, On the Pythagorean Life 115). The only early source to associate Pythagoras with the whole number ratios that govern the concords is Xenocrates (Fr. 9) in the early Academy, but the early Academy is precisely one source of the later exaggerated tradition about Pythagoras (see above). One story has it that Pythagoras passed by a blacksmith’s shop and heard the concords in the sounds of the hammers striking the anvil and then discovered that the sounds made by hammers whose weights are in the ratio 2 : 1 will be an octave apart, etc. Unfortunately, the stories of Pythagoras’ discovery of these relationships are clearly false, since none of the techniques for the discovery ascribed to him would, in fact, work (e.g., the pitch of sounds produced by hammers is not directly proportional to their weight: see Burkert 1972a, 375). An experiment ascribed to Hippasus, who was active in the first half of the fifth century, after Pythagoras’ death, would have worked, and thus we can trace the scientific verification of the discovery at least to Hippasus; knowledge of the relation between whole number ratios and the concords is clearly found in the fragments of Philolaus (Fr. 6a, Huffman), in the second half of the fifth century. There is some evidence that the truth of the relationship was already known to Pythagoras’ contemporary, Lasus, who was not a Pythagorean (Burkert 1972a, 377). It may be once again that Pythagoras knew of the relationship without either having discovered it or having demonstrated it scientifically. The relationship was probably first discovered by instrument makers, and specifically makers of wind instruments rather than stringed instruments (Barker 2014, 202). The acusmata reported by Aristotle, which may go back to Pythagoras, report the following question and answer “What is the oracle at Delphi? The tetraktys , which is the harmony in which the Sirens sing” (Iamblichus, On the Pythagorean Life , 82, probably derived from Aristotle). The tetraktys , literally “the four,” refers to the first four numbers, which when added together equal the number ten, which was regarded as the perfect number in fifth-century Pythagoreanism. Here in the acusmata , these four numbers are identified with one of the primary sources of wisdom in the Greek world, the Delphic oracle. In the later tradition the tetraktys is treated as the summary of all Pythagorean wisdom, since the Pythagoreans swore oaths by Pythagoras as “the one who handed down the tetraktys to our generation.” The tetraktys can be connected to the music which the Sirens sing in that all of the ratios that correspond to the basic concords in music (octave, fifth and fourth) can be expressed as whole number ratios of the first four numbers. This acusma thus seems to be based on the knowledge of the relationship between the concords and the whole number ratios. The picture of Pythagoras that emerges from the evidence is thus not of a mathematician, who offered rigorous proofs, or of a scientist, who carried out experiments to discover the nature of the natural world, but rather of someone who sees special significance in and assigns special prominence to mathematical relationships that were in general circulation. This is the context in which to understand Aristoxenus’ remark that “Pythagoras most of all seems to have honored and advanced the study concerned with numbers, having taken it away from the use of merchants and likening all things to numbers” (Fr. 23, Wehrli). Some might suppose that this is a reference to a rigorous treatment of arithmetic, such as that hypothesized by Becker (1936), who argued that Euclid IX. 21–34 was a self-contained unit that represented a deductive theory of odd and even numbers developed by the Pythagoreans (see Mueller 1997, 296 ff. and Burkert 1972a, 434 ff.). It is crucial to recognize, however, that Becker’s reconstruction is rejected in some recent scholarship (e.g., Netz 2014, 179) and no ancient source assigns it even to the Pythagoreans, let alone to Pythagoras himself. There is, moreover, no talk of mathematical proof or a deductive system in the passage from Aristoxenus just quoted. Pythagoras is known for the honor he gives to number and for removing it from the practical realm of trade and instead pointing to correspondences between the behavior of number and the behavior of things. Such correspondences were highlighted in Aristotle’s book on the Pythagoreans, e.g., the female is likened to the number two and the male to the number three and their sum, five, is likened to marriage (Aristotle, Fr. 203).

What then was the nature of Pythagoras’ cosmos? The doxographical tradition reports that Pythagoras discovered the sphericity of the earth, the five celestial zones and the identity of the evening and morning star (Diogenes Laertius VIII. 48, Aetius III.14.1, Diogenes Laertius IX. 23). In each case, however, Burkert has shown that these reports seem to be false and the result of the glorification of Pythagoras in the later tradition, since the earliest and most reliable evidence assigns these same discoveries to someone else (1972a, 303 ff.). Thus, Theophrastus, who is the primary basis of the doxographical tradition, says that it was Parmenides who discovered the sphericity of the earth (Diogenes Laertius VIII. 48). Parmenides is also identified as the discoverer of the identity of the morning and evening star (Diogenes Laertius IX. 23), and Pythagoras’ claim appears to be based on a poem forged in his name, which was rejected already by Callimachus in the third century BCE (Burkert 1972a, 307). The identification of the five celestial zones depends on the discovery of the obliquity of the ecliptic, and some of the doxography duly assigns this discovery to Pythagoras as well and claims that Oenopides stole it from Pythagoras (Aetius II.12.2); the history of astronomy by Aristotle’s pupil Eudemus, our most reliable source, seems to attribute the discovery to Oenopides (there are problems with the text), however (Eudemus, Fr. 145 Wehrli). It thus appears that the later tradition, finding no evidence for Pythagoras’ cosmology in the early evidence, assigned the discoveries of Parmenides back to Pythagoras, encouraged by traditions which made Parmenides the pupil of Pythagoras. In the end, there is no evidence for Pythagoras’ cosmology in the early evidence, beyond what can be reconstructed from acusmata . As was shown above, Pythagoras saw the cosmos as structured according to number insofar as the tetraktys is the source of all wisdom. His cosmos was also imbued with a moral significance, which is in accordance with his beliefs about reincarnation and the fate of the soul (West 1971, 215–216; Huffman 2013, 60–68). Thus, in answer to the question “What are the Isles of the Blest?” (where we might hope to go, if we lived a good life), the answer is “the sun and the moon.” Again “the planets are the hounds of Persephone,” i.e., the planets are agents of vengeance for wrong done (Aristotle in Porphyry VP 41). Aristotle similarly reports that for the Pythagoreans thunder “is a threat to those in Tartarus, so that they will be afraid” ( Posterior Analytics 94b) and another acusma says that “an earthquake is nothing other than a meeting of the dead” (Aelian, Historical Miscellany , IV. 17). Zhmud calls these cosmological acusmata into question (2012a, 329–330), noting that some only appear in Porphyry, but Porphyry explicitly identifies Aristotle as his source and we have no reason to doubt him ( VP 41). Pythagoras’ cosmos embodied mathematical relationships that had a basis in fact and combined them with moral ideas tied to the fate of the soul. The best analogy for the type of account of the cosmos which Pythagoras gave might be some of the myths which appear at the end of Platonic dialogues such as the Phaedo , Gorgias or Republic , where cosmology has a primarily moral purpose. Should the doctrine of the harmony of the spheres be assigned to Pythagoras? Certainly the acusma which talks of the sirens singing in the harmony represented by the tetraktys suggests that there might have been a cosmic music and that Pythagoras may well have thought that the heavenly bodies, which we see move across the sky at night, made music by their motions. On the other hand, there is no evidence for “the spheres,” if we mean by that a cosmic model according to which each of the heavenly bodies is associated with a series of concentric circular orbits, a model which is at least in part designed to explain celestical phenomena. The first such cosmic model in the Pythagorean tradition is that of Philolaus in the second half of the fifth century, a model which still shows traces of the connection to the moral cosmos of Pythagoras in its account of the counter-earth and the central fire (see Philolaus ).

If Pythagoras was primarily a figure of religious and ethical significance, who left behind an influential way of life and for whom number and cosmology primarily had significance in this religious and moral context, how are we to explain the prominence of rigorous mathematics and mathematical cosmology in later Pythagoreans such as Philolaus and Archytas? It is important to note that this is not just a question asked by modern scholars but was already a central question in the fourth century BCE. What is the connection between Pythagoras and fifth-century Pythagoreans? The question is implicit in Aristotle’s description of the fifth-century Pythagoreans such as Philolaus as “the so-called Pythagoreans.” This expression is most easily understood as expressing Aristotle’s recognition that these people were called Pythagoreans and at the same time his puzzlement as to what connection there could be between the wonder-worker who promulgated the acusmata , which his researches show Pythagoras to have been, and the philosophy of limiters and unlimiteds put forth in fifth-century Pythagoreanism. The tradition of a split between two groups of Pythagoreans in the fifth century, the mathematici and the acusmatici , points to the same puzzlement. The evidence for this split is quite confused in the later tradition, but Burkert (1972a, 192 ff.) has shown that the original and most objective account of the split is found in a passage of Aristotle’s book on the Pythagoreans, which is preserved in Iamblichus ( On Common Mathematical Science , 76.19 ff). The acusmatici , who are clearly connected by their name to the acusmata , are recognized by the other group, the mathematici , as genuine Pythagoreans, but the acusmatici do not regard the philosophy of the mathematici as deriving from Pythagoras but rather from Hippasus. The mathematici appear to have argued that, while the acusmatici were indeed Pythagoreans, it was the mathematici who were the true Pythagoreans; Pythagoras gave the acusmata to those who did not have the time to study the mathematical sciences, so that they would at least have moral guidance, while to those who had the time to fully devote themselves to Pythagoreanism he gave training in the mathematical sciences, which explained the reasons for this guidance. This tradition thus shows that all agreed that the acusmata represented the teaching of Pythagoras, but that some regarded the mathematical work associated with the mathematici as not deriving from Pythagoras himself, but rather from Hippasus (on the controversy about the evidence for this split into two groups of Pythagoreans see the fifth paragraph of section 4.3 above). For fourth-century Greeks as for modern scholars, the question is whether the mathematical and scientific side of later Pythagoreanism derived from Pythagoras or not. If there were no intelligible way to understand how later Pythagoreanism could have arisen out of the Pythagoreanism of the acusmata , the puzzle of Pythagoras’ relation to the later tradition would be insoluble. The cosmos of the acusmata , however, clearly shows a belief in a world structured according to mathematics, and some of the evidence for this belief may have been drawn from genuine mathematical truths such as those embodied in the “Pythagorean” theorem and the relation of whole number ratios to musical concords. Even if Pythagoras’ cosmos was of primarily moral and symbolic significance, these strands of mathematical truth, which were woven into it, would provide the seeds from which later Pythagoreanism grew. Philolaus’ cosmos and his metaphysical system, in which all things arise from limiters and unlimiteds and are known through numbers, are not stolen from Pythagoras. They embody a conception of mathematics, which owes much to the more rigorous mathematics of Hippocrates of Chios in the middle of the fifth century; the contrast between limiter and unlimited makes most sense after Parmenides’ emphasis on the role of limit in the first part of the fifth century. Philolaus’ system is nonetheless an intelligible development of the reverence for mathematical truth found in Pythagoras’ own cosmological scheme, which is embodied in the acusmata .

The picture of Pythagoras presented above is inevitably based on crucial decisions about sources and has been recently challenged in a searching critique (Zhmud 2012a). Zhmud argues that the consensus view of Pythagoras’ cosmos as presented above is based on the faulty assumption that there was a progression from myth and religion to reason and science in Pythagoreanism. In many cases, he argues, the evidence suggests that early Pythagoreanism was more scientific and that religious and mythic elements only gained in importance in the later tradition. The consensus picture of Pythagoras’ cosmos assigns number symbolism a central role and treats the tetraktys , the first four numbers, which total to the perfect number ten, as a central concept. Zhmud argues that the tetraktys and the importance of the number ten do not go back to Pythagoras but flourish in the Neopythagorean tradition, while having roots in number speculation in the Academy associated with such figures as Plato’s successor Speusippus. One of the central pieces of evidence for this view is that the tetraktys does not first appear until late in the tradition, in Aetius in the first century CE (DK 1.3.8). However, the tetraktys does appear in one of the acusmata in a section (82) of Iamblichus’ On the Pythagorean Life that is commonly regarded as deriving from Aristotle. Zhmud himself agrees that sections 82–86 of On the Pythagorean Life as a whole go back to Aristotle but suggests that the acusma about the tetraktys was a post-Aristotelian addition (2012a, 300–303). Once again source criticism is crucial. If the acusma in question goes back to Aristotle then there is good evidence for the tetraktys in early Pythagoreanism. If we regard it as a later insertion into Aristotelian material, the early Pythagorean credentials of the tetraktys are less clear.

Zhmud supports Pythagoras’ position as genuine mathematician rather than someone interested only in number symbolism by pointing to gaps in the development of early Greek mathematics. Although there is no explicit evidence, Pythagoras is the most likely candidate to fill these gaps. Thus between Thales, whom Eudemus identifies as the first geometer, and Hippocrates of Chios, who produced the first Elements , someone turned geometry into a deductive science (Zhmud 2012a, 256). Similarly, Hippasus’ experiment with bronze disks to show that the concordant intervals of the octave, fifth and fourth were governed by whole number ratios is too complex to be a first attempt so that once again someone must have discovered the ratios in a simpler way earlier (Zhmud 2012a, 291). In each case Zhmud suggests that Pythagoras is that someone. Finally, the study of proportion ties together arithmetic, geometry and harmonics and Zhmud argues that, although there is no explicit fourth-century evidence, later reports which assign Pythagoras the discovery of the first three proportions (Iamblichus, Commentary on Nicomachus’ Introduction to Arithmetic 100.19–101.11) are likely to go back to Eudemus (2012a, 265–266). Such speculations have some plausibility but they highlight even more the puzzle as to why, if Pythagoras played this central role in early Greek mathematics, no early source explicitly ascribes it to him. Of course, some scholars argue that the majority have overlooked key passages that do assign mathematical achievements to Pythagoras. In order to gain a rounded view of the Pythagorean question it is thus appropriate to look at the most controversial of these passages.

Some scholars who regard Pythagoras as a mathematician and rational cosmologist, such as Guthrie, admit that the earliest evidence does not support this view (Lloyd 2014, 25), but maintain that the prominence of Pythagoras the mathematician in the late tradition must be based on something early. Others maintain that there is evidence in the sixth- and fifth-century BCE for Pythagoras as a mathematician and cosmologist. They argue that Herodotus’ reference to Pythagoras as a wise man ( sophistês ) and Heraclitus’ description of him as pursuing inquiry ( historiê ), show that he was regarded as practicing rational cosmology (Kahn 2002, 16–17; Zhmud 2012a, 33–43). The concept of a wise man in Herodotus’ time was very broad, however, and includes poets and sages as well as Ionian cosmologists; the same is true of the concept of inquiry. Historiê peri physeos (inquiry concerning nature) is later used to refer specifically to the inquiry into nature practiced by the Presocratic cosmologists, but Herodotus’ usage shows that at Heraclitus’ time historiê referred to inquiry in a quite general sense and has no specific reference to the cosmological inquiry of the Presocratics (Huffman 2008b). In one instance in Herodotus it refers to inquiry into the stories of Menelaus’ and Helen’s adventures in Egypt (II. 118). Heraclitus may be thinking of Pythagoras’ inquiry into and collection of the mythical and religious lore that is found in the acusmata (Huffman 2008b; see also Gemelli Marciano 2002, 96–103). Thus the description of Pythagoras as a wise man who practiced inquiry is simply too general to aid in deciding what sort of figure Herodotus and Heraclitus saw him as being. It is certainly true that Empedocles shows that the roles of rational cosmologist and wonder-working religious teacher could be combined in one figure, but this does not prove these roles were combined in Pythagoras’ case. The only thing that could prove this in Pythagoras’ case is reliable early evidence for a rational cosmology and that is precisely what is lacking.

There is more controversy about the fourth-century evidence. Zhmud argues that Isocrates regards Pythagoras as a philosopher and mathematician (2012a, 50). However, it is hard to see how the passage in question ( Busiris 28–29) supports this view. Nowhere in it does Isocrates ascribe mathematical work or a rational cosmology to Pythagoras. He reports in general terms that Pythagoras brought “ other philosophy” to Greece from Egypt but what he emphasizes is that Pythagoras was “more clearly interested than others in sacrificial rites and temple rituals.” It is true that earlier, in a passage that does not mention Pythagoras ( Busiris 22–23), Isocrates had said that some of the Egyptian priests studied mathematics but if Isocrates thought Pythagoras also brought mathematical learning from Egypt he has chosen not to say so explicitly. What Isocrates emphasizes about Pythagoras is what the rest of the early tradition emphasizes, his interest in religious rites. Fr. 191 from Aristotle’s lost work on the Pythagoreans reports that Pythagoras “dedicated himself to the study of mathematical sciences, especially numbers” and Fragment 20 from Aristotle’s Proptrepticus says that Pythagoras said that human beings were born to contemplate the heavens and described himself as an observer of nature (Zhmud 2012a, 56 and 259–260). Unfortunately, in neither case are the words in question likely to be Aristotle’s. Fr. 191 comes from a book on marvels by Apollonius (2nd BCE?). The words in question come before Apollonius mentions Aristotle and, as Burkert pointed out (1972a, 412), are overwhelming likely to be by Apollonius himself, since they serve as the transition sentence between Apollonius’ account of Pherecydes and his account of Pythagoras. In the face of the huge extant corpus of Aristotle’s works in which he never ascribes any mathematical work to Pythagoras, a single sentence that is not ascribed directly to Aristotle and that, in terms of function, appears to be the work of Apollonius and not Aristotle cannot with any confidence be used as evidence that Aristotle regarded Pythagoras as a mathematician. The same situation arises with Fr. 20 of the Protrepticus. If the words in question were by Aristotle they would be his sole statement that Pythagoras was a natural philosopher. The case of Fr. 20 is even more tenuous than that of Fr. 191. Fr. 20 comes from Iamblichus’ Protrepticus , large parts of which are likely to derive from Aristotle’s lost Protrepticus but, as is his practice, Iamblichus does not make any explicit reference to Aristotle. The further problem with Fr. 20, as Burkert noted (1960, 166–168), is that the same story is told first about Pythagoras and then immediately afterwards about Anaxagoras: both are asked why human beings were born and both answer “to contemplate the heavens” (Iamblichus, Protrepticus 51.8–15). This awkward repetition of the same story about two different people immediately suggests that only one story was in the original and the other was added in the later tradition. This suggestion is strikingly confirmed by the fact that Aristotle does tell this story about Anaxagoras in his extant works ( Eudemian Ethics 1216a11–16) but not about Pythagoras. Thus, if the passage in Iamblichus’ Protrepticus is, in fact, from Aristotle, it is very likely that only Anaxagoras appeared in Aristotle’s version and that Pythagoras was added in the later tradition, perhaps by Iamblichus himself. Since these two passages are unlikely to be from Aristotle, there are no references to Pythagoras as a mathematician or as a natural philosopher either in Aristotle’s extant works or in the fragments of his works. Aristotle only knows Pythagoras as a wonder working sage and teacher of a way of life (Fr. 191). Aristotle’s attitude is similar to his predecessors in the earlier fourth century: Plato’s sole reference to Pythagoras is as the founder of a way of life and Isocrates emphasizes both the way of life and the interest in religious ritual.

What about the pupils of Plato and Aristotle? As discussed in the second paragraph of section 5 above, Eudemus, who wrote a series of histories of mathematics never mentions Pythagoras by name. Arguments from silence are perilous but, when the most well-informed source of the fourth-century fails to mention Pythagoras in works explicitly directed towards the history of mathematics, the silence means something. There are only two passages in which Pythagoras is explicitly associated with anything mathematical or scientific by pupils of Plato and Aristotle. First, Aristotle’s pupil Aristoxenus reports that Pythagoras “most of all valued the pursuit ( pragmateia ) of number and brought it forward, taking it away from the use of traders, by likening all things to numbers” (Fr. 23). Zhmud translates pragmateia as “science” (2012a, 216) so that he has Aristoxenus attributing the invention of the science of number to Pythagoras but, while Aristoxenus does use pragmateia to mean science in some contexts, it more commonly simply means “pursuit” (Huffman 2014b, 292). Here surely it must mean “pursuit,” because Pythagoras is presented as taking it away from the traders and we can hardly suppose that the traders were engaged in the theoretical science of arithmetic. Moreover, Aristoxenus explains what he means in the final participial phrase. He is not ascribing rigorous mathematics with proofs to Pythagoras but rather says that Pythagoras was “likening all things to numbers”. This is consistent with the moralized cosmos of Pythagoras sketched above in which numbers have symbolic significance. The second important passage is Plato’s pupil Xenocrates’ assertion that Pythagoras “discovered that the intervals in music, too, do not arise in separation from number” (Fr. 9). Xenocrates is being quoted here in a fragment of a work by a Heraclides (Barker 1989, 235–236), perhaps Heraclides of Pontus. There is controversy whether the quotation of Xenocrates is limited just to what has been quoted in the previous sentence or whether the whole fragment of Heraclides is a quotation of Xenocrates. Burkert (1972a, 381) and Barker (1989, 235) argue that it is probably just the first sentence that Heraclides ascribes to Xenocrates, while Zhmud would include at least a second sentence in which Heraclides presents Pythagoras as pursuing a program of research into “the conditions under which concordant and discordant intervals arise” (Zhmud 2012a, 258). If the second sentence is accepted then Xenocrates clearly presents Pythagoras as an acoustic scientist. It seems most reasonable, however, to accept only the first sentence as belonging to Xenocrates. If the quotation from Xenocrates does not break off at that point, there is no other obvious breaking point in the fragment and the whole two pages of text must be ascribed to Xenocrates. The problem with ascribing it all to Xenocrates is that Porphyry introduces the passage as a quotation from Heraclides, which would be strange if everything quoted, in fact, belongs to Xenocrates. If just the first sentence comes from Xenocrates, then all he is ascribing to Pythagoras is the recognition that the concordant intervals are connected to numbers. It is easy to assume, as Zhmud does, that Xenocrates is saying that Pythagoras was the first to discover that the concordant intervals are governed by whole number ratios but Xenocrates’ remarks need not mean this. Xenocrates’ comments might well come from a context like that in the fragment of Aristoxenus, above, i.e., a context in which Pythagoras is presented as likening all things to numbers and arguing that numbers in some sense explain or control things. In such a context Xenocrates would not be making the point that Pythagoras discovered the whole number ratios but rather that he found out that concords arose in accordance with whole number ratios, perhaps from musicians (who discovered them first not being the issue), and used this fact as another illustration of how things are like numbers. Thus, the fragments of Aristoxenus and Xenocrates show that Pythagoras likened things to numbers and took the concordant musical intervals as a central example, but do not suggest that he founded arithmetic as a rigorous mathematical discipline or carried out a program of scientific research in harmonics.

Controversy concerning Pythagoras’ role as a scientist and mathematician will continue. Indeed, Hahn has recently endorsed many of Zhmud’s arguments and argues that Pythagoras was a rational cosmologist, who was further developing a project begun by Thales to construct the cosmos out of right triangles. Hahn admits, however, that his thesis is “speculative” and “a circumstantial case at best” (2017, xi). It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras’ cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus.

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• Festugière, A.-J., 1945, ‘Les Mémoires Pythagoriques cités par Alexandre Polyhistor’, REG , 58: 1–65.
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• –––, 2014, ‘The Pythagorean Way of Life and Pythagorean ethics’, in Huffman 2014a, 131–148.
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• –––, 1999, ‘The Pythagorean Tradition’, in The Cambridge Companion to Early Greek Philosophy , A. A. Long (ed.), Cambridge: Cambridge University Press, 66–87.
• –––, 2001, ‘The Philolaic Method: The Pythagoreanism behind the Philebus ’, in Essays in Ancient Greek Philosophy VI: Before Plato , A. Preus (ed.), Albany: State University of New York Press, 67–85.
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• –––, 2008b, ‘Heraclitus’ Critique of Pythagoras’ Enquiry in Fragment 129’, Oxford Studies in Ancient Philosophy , 35: 19–47.
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We explain who Pythagoras was, his contributions and the Pythagorean brotherhood. In addition, we explore his theories and main characteristics.

Who was Pythagoras?

Pythagoras of Samos, also simply known as Pythagoras, was an Ancient Greek philosopher and mathematician known for his contributions to the advancement of Hellenistic arithmetic, geometry and mathematics and for his influence on both Plato and Aristotle .

He is considered to be the first pure mathematician , as it was this discipline that mainly (though not exclusively) occupied his interest. Some of his theorems and postulates, especially in geometry and arithmetic, have survived to the present time.

His contributions to Western thought were key and decisive though no authored text has survived, and even if it is difficult to discern his thought from that of his followers.

• See also: Socrates

Life of Pythagoras

Pythagoras was born in the Greek city of Samos, Ionia, around 569 BC . Though there is no reliable information about the dates of his life, we know from the accounts of the Greek philosopher Aristoxenus that Pythagoras left Samos at the age of forty, which is the reason he is believed to have been born around the proposed date. He allegedly died around the age of seventy or eighty.

His father was Mnesarchus, a gem-engraver, a merchant originally from the city of Tyre and his mother was a local woman named Pythais.

Pythagoras visited Egypt and Babylon , which would have provided him the opportunity to cultivate himself from a very early age: he learned to play the lyre, recite Homer and write poetry. It is believed that Polycrates gave him a letter of introduction to Pharaoh Amasis, friend and ally of the Greek tyrant.

Pythagoras emigrated to Croton in southern Italy, possibly encouraged by Democedes , a physician at the court of Polycrates. There he founded his school and had a great impact and influence on regional rulers, thinkers and even artists, mathematicians and craftsmen, being Pythagoras himself an experienced artist.

The school’s influence immediately spread to the neighboring cities . In many of them, several members of the Pythagorean community even came to hold leadership and official positions. This led to what is known as "the Cylon conspiracy", which resulted in the murder of many Pythagoreans and in Pythagoras’ exile and his subsequent refuge in the temple of the Muses in Metapontum, where he allegedly died of starvation.

Little more is known about Pythagoras' general education beyond what has been said. He is believed to have been a disciple of Chaldeans and sages of Syria . Also mentioned in major sources are Pherecydes of Syros, a quite elderly Thales of Miletus and lastly, the latter’s disciple, Anaximander. Accounts of Pythagoras’ life are filled with myths and legends, which were compiled by Neoplatonic and Neopythagorean philosophers.

The most comprehensive work on his life dates from the third century AD and was written by Diogenes Laertius and Porphyry. The Life of Pythagoras , by Iamblichus, is also a valuable text on the philosopher’s life.

Much remains unknown about the death of Pythagoras, although it presumably occurred around 532 BC, after the Pythagorean brotherhood was attacked by their political rivals. According to the most widespread account, Pythagoras would have died in the city of Metapontum , where his tomb was displayed in Roman times. He would have taken refuge in the temple of the Muses, where he had presumably hidden following the attack on his school until he died of starvation and dehydration.

The Pythagorean brotherhood

Pythagoras settled in the Italian city of Croton around the year 522 BC , after having been a prisoner of war of the Persian emperor Cambyses II, who invaded Egypt in 525 BC.

In this Italian city he founded his school, known as "the Pythagorean brotherhood", in which both men and women were admitted . They referred to themselves as "mathematicians" ( matematikoi ) since they believed reality is essentially mathematical in nature, and they applied and promoted the study of numbers beyond their use in commerce and political matters.

The brotherhood came to have 300 followers known as "the Pythagoreans" , who owned no possessions, practiced vegetarianism and were divided between core members of the order and listeners (called "acousmatics", who did not live in the temple).

Initiates were sworn to secrecy, a source of suspicion and envy among its enemies and dissidents . Communal life was promoted and everyone who was part of it was demanded to observe strict loyalty. Asceticism and metempsychosis (the belief in the transmigration of certain psychic elements from one body to another) were common practices among its members.

Around 500 BC, the Pythagoreans assumed a certain political role in the region until they were attacked and defeated in 460 BC by their political opponents.

Pythagorean principles

Pythagoras and his disciples observed the following philosophical principles:

• Reality, in its deepest perception, is mathematical. All things are numbers.
• Philosophy can be a path to spiritual purification.
• The human soul can rise high enough to merge with the divine.
• Certain symbols of a mystical nature are considered sacred signs.
• All members of the Pythagorean brotherhood must swear to absolute secrecy regarding their beliefs and practices.

Contributions of Pythagoras

The most significant contributions of Pythagoras were:

• Philosophy . Pythagoras was the first Greek thinker to provide a non-mystical or religious explanation for the origin of all that is. His idea of a physical or natural principle, in his case water as underlying of all things, paved the way for a rational and discursive understanding of the world as we know it.
• Mathematics . Pythagoras formulated the eponymous theorem, according to which "the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse". He is also credited with the geometric construction of the first perfect solids and the discovery of perfect and amicable numbers as well as polygonal numbers. His work with triangles and the square root is a cornerstone of this discipline.
• Astronomy . He was among the first to point out that the morning and evening stars are the same planet: Venus. He also taught that the Earth was the center of the universe (geocentric model) and that the orbit of the Moon was inclined to the equator of the Earth, although these discoveries are also ascribed to Parmenides.
• Music . He is credited with the discovery of the laws of regular musical intervals and the invention of the monochord, as well as with the teaching of an ethical and medicinal use of music. From this, emerged the idea that there is a reciprocal harmony between the various systems of the universe and that in this sense, astronomy, music, health and other areas of thought are related.

Acknowledgement of Pythagoras

Beyond the eponymous theories, Pythagoras' name is honored with a lunar crater (Pythagoras) and an asteroid (6143) in the solar system.

• De la Fuente, H., & David, A. (2011). Vidas de Pitágoras . Atalanta.
• Guthrie, W. (1984). Historia de la filosofía griega, vol. I. Los primeros presocráticos y los pitagóricos. Gredos.
• Guthrie, W. (1953). Los filósofos griegos. De Tales a Aristóteles . FCE.

Related articles:

• René Descartes
• Jean Monnet
• Lech Walesa
• Thales of Miletus

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The Life And Philosophy of Pythagoras

• 12 January 2024
• Biography , History

Introduction – The Life And Philosophy of Pythagoras

Pythagoras of Samos was an ancient Ionian Greek philosopher who lived in the 6th century BC. An influential yet mysterious figure shrouded in legends, he authored no writings himself but imparted extensive mathematical and philosophical wisdom. Today Pythagoras is hailed primarily for the Pythagorean theorem relating triangle sides in geometry, which transformed mathematics and physics. However, through the life And philosophy of Pythagoras and his Pythagorean society, he made seminal contributions across areas like numeric theory, symbolic logic, mathematical notation, astronomy models, and music theory that profoundly impacted ancient and modern understanding of mathematics as a descriptor of universal truths. To learn more about the relation of astronomy and mathematics, read Role of Astronomy in Mathematics

Early Life and Education of Pythagoras

Pythagoras was born around 570 BC on the Greek island of Samos, situated off the western coast of Asia Minor. Little concrete evidence exists about his early childhood, though legends suggest his birth was prophesied by oracles.

At the time, Samos harbored immigrants of various origins including Egyptians, Jews, Phoenicians and Babylonians. This cosmopolitan harbor town facilitated a melting pot of cultures and belief systems that likely influenced the life And philosophy of Pythagoras and his intellectual development.

Wide-Ranging Studies and Influences

From Greek writers to Eastern mystics across Egypt and Mesopotamia, Pythagoras accumulated extensive knowledge through his travels. He is said to have studied under masters like Thales and Anaximander expanding his mathematics and astronomy expertise in addition to absorbing religious philosophies on issues of morality, unity of opposites, harmony and more. This confluence of teachers and schools of thought all contributed to the life And philosophy of Pythagoras, synthesizing uniquely in his worldview.

Pythagoras’ Journey to Egypt

Having exhausted the knowledge limits of his various mentors in Samos, legends contend that sometime around 550 BC an impatient Pythagoras embarked for Egypt seeking higher mathematical revelations said to be guarded by ancient priests.

In Egypt, Pythagoras found a welcoming intellectual refuge and is said to have intensely studied sacred geometry and astronomy with temple priests for over two decades. From the Egyptians, he acquired advanced knowledge on topics like triangular numbers, geometric algebra, and practical accounting techniques among other insights.

This lengthy sojourn in Egypt proved formative, sparking Pythagoras’ innate mathematical genius by exposing him to novel algebraic concepts he later built upon to derive his eponymous theorem.

To learn about basic algebra, read Introductory Algebra .

The Pythagorean Brotherhood

The Pythagorean Brotherhood, also known as the Pythagorean School or Pythagorean Community, was a philosophical and religious community established by Pythagoras, the ancient Greek mathematician and philosopher, in the 6th century BCE. The Brotherhood played a significant role in the development of both mathematics and philosophy, driven by the life And philosophy of Pythagoras in ancient Greece.

Curriculum and Tenets of The Pythagorean Brotherhood

Pythagoras instructed members of the society in a curriculum fusing arithmetic, music, astronomy and geometry with philosophical-religious principles like metempsychosis (reincarnation), respect for beans, and worship of numbers themselves. The society pursued understanding reality through the certainty of math calculations and logic.

The tight-knit community shared property, adopted vegetarianism, wore white clothes, and perpetuated various taboos and rituals under the authoritarian rule of the life And philosophy of Pythagoras. They expanded the society’s influence across Southern Italy, though were ultimately persecuted. Nonetheless, the Pythagoreans indelibly shaped philosophical and mathematical inquiry for centuries. 

Pythagoras’ Contributions to Mathematics

The pythagorean theorem.

Pythagoras’ most celebrated legacy is his theorem establishing the fundamental relation between the sides in any right triangle – where the square of the hypotenuse equals the sum of squares of opposite sides. While no written record of Pythagoras’ original proof survives, it enabled quantitative analysis in geometry. 

The deceptively simple Pythagorean theorem became a building block underlying important mathematical feats – from calculating triangular area and volume, determining distances in surveying, to navigation by stars. 

Additional Mathematical Breakthroughs

Beyond his famous theorem, Pythagoras analyzed number sequences leading him to identify special integer-based right triangles called Pythagorean triples obeying the pattern. This hinted at number theory as a foundation of mathematical reality. 

Pythagoras also devised theories connecting music intervals to numeric ratios, sparking mathematical musicology to explain harmony via quantifiable periodic vibrations. This embodied his view of numbers as the substrate linking all disciplines.

Pythagoras’ Influence on Geometry

Pythagoras pioneered the concept of mathematical proofs, elevating geometry from a practical measurement tool to an abstract deductive science. Though his original writings are lost, his systematic approach became the model for logically demonstrating mathematical truths via rigorous deduction-based proofs – a landmark in mathematical theory.

Additionally, his geometric insights seeded further pioneering theorems by followers like the Pythagorean philosopher Hippasus on irrational numbers and incommensurability of diagonal and side lengths – expanding mathematical theory beyond just rational numbers.

Foundations for Euclid’s Elements

In the 3rd century BC, Euclid built upon Pythagoras’ Proposition proofs as the foundation for his seminal 13-volume work Elements deriving fundamental geometric axioms still taught globally. Pythagoras’ use of elemental geometric proofs provided the early template.

Pythagoras’ Fueling Greek Math

More broadly, Pythagoras inspired Plato, Aristotle and Archimedes to also frame mathematics as underpinning all scientific inquiry, catalyzing Greek mathematics as a whole. His quantitative philosophy persisted for centuries.

Pythagoras’ Contributions to Philosophy

Pythagoras pioneered melding mathematics with grander explanations of reality’s inner workings. To him, mathematical relationships like harmonic ratios and numeric patterns in nature revealed the orderly, quantitative structure behind the physical and spiritual worlds alike.

This viewpoint of an intrinsically mathematical cosmos led Pythagoras to posit all elements of existence being fundamentally number-based – from celestial orbits obeying predictable cycles to musical tones conforming to fraction-based frequencies. Mathematics governed everything.

Though Pythagorean metaphysics faded, his mathematical mysticism directly fertilized Platonic philosophy. The idea of an abstract realm of math forms and absolutes underlying flawed earthly matter persisted for centuries to influence thinkers from Gnostics to Kepler during the Scientific Revolution, underscoring Pythagoras’ immense legacy.

Controversies and Mysteries about Pythagoras

Pythagoras imposed strict secrecy and mystical rites in his society, forcing members to adhere to codes of silence outside while allegedly only gradually welcoming them into higher echelons of hidden knowledge. This esotericism left no extant record from his or early followers’ own hand, obscuring authorship. 

As such scholars still debate if innovations like the golden ratio or the Pythagorean theorem itself originated from Pythagoras personally versus the wider school. Such uncertainty persists around separating the historical Pythagoras from teachings of the society he founded but may not have personally derived.  

In the centuries after his death, legends grew of Pythagoras possessing supernatural powers. Neo-Pythagorean followers propagated far-fetched myths based on forged ‘sacred writings’ to elevate Pythagoras as a miracle-worker eroding objective assessments. In truth, besides the theorem bearing his name, historians can only tenuously connect assorted discoveries to Pythagoras the man rather than the later Pythagoreans who continued expounding his mathematical philosophy.

Pythagoras’ Later Years and Legacy

Around 530 BC political upheavals likely forced Pythagoras to abandon Samos for the Greek colony of Croton in Southern Italy. There he fully devoted himself to spreading mathematical philosophy through his society as an exile revered sage.

Pythagoras died around 495 BC aged 75 with conflicting reports of him perishing in an arson attack or fleeing persecution to starve himself. Regardless of the exact circumstances, the Pythagoreans faced growing hostility in Italy. But the numerical mystic’s unique synthesis of mathematical principles with cosmological ideas already began immortalizing his insights.

Over two millennia since his lifetime, the life And philosophy of Pythagoras still remains revered as the driving force linking abstract mathematics to universal order. From Euclid’s proofs to Kepler’s celestial models, Pythagoras mathematized of the cosmos inspired the evolution of geometry, physics, astronomy and philosophy itself through the ages.

Conclusion – The Life And Philosophy of Pythagoras

Pythagoras of Samos pioneered numerating the cosmos itself. Inspired by polymathic studies from Ancient Greece to Egypt, he contended numbers and quantifiable logic comprised the divine code behind existence’s very fabric. From approximating 600BC until death, Pythagoras helmed an esoteric brotherhood spreading his mathematical mysticism through the life And philosophy of Pythagoras and geometric explorations interweaving fields as diverse as music, astronomy and metaphysics. Through deductive geometry, number theory and more, Pythagoras’ fusion of mathematics with cosmology and spirituality ignited inquiries into natural law that still burn brightly.

Frequently Asked Questions (FAQs)

Where did pythagoras live most of his life.

Pythagoras is believed to have lived most of his life in ancient Greece, primarily in the city of Croton (modern-day Crotone) in southern Italy. He was born around 570 BCE and is thought to have died around 495 BCE.

How many years ago did pythagoras live?

Pythagoras lived approximately 2,500 years ago. Please note that the exact dates of his birth and death are not known with certainty, and these are rough estimates based on historical accounts.

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Pythagoras: Life, work and achievements

Although famous throughout the world, Pythagoras’ life is shrouded in mystery.

Pythagoras Theory

In his footsteps, additional resources, bibliography.

Born in Samos in around 570 B.C, Pythagoras is commonly said to be the first pure mathematician who proposed that everything is a number.

Although he is most famous for his mathematical theorem, Pythagoras also made extraordinary developments in astronomy and geometry. He also developed a theory of music while and founded a philosophical and religious school in Croton, Italy. It was here he taught that "the whole cosmos is a scale and a number", according to the University of St Andrews .

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While playing on his lyre, which was an ancient Greek stringed instrument, Pythagoras discovered that the vibrating strings created a beautiful sound when the ratios of the lengths of the wires were whole numbers, and that this was also true of other instruments. He combined this discovery with his understanding of the planets, conceiving the theory that when the planets were in harmony, it created beautiful music that man was incapable of hearing. 

Pythagoras concluded that mathematics and music were interconnected and that knowledge of one area led to an understanding of the other, according to the University of Connecticut . He also believed that music had healing properties and would often play his lyre for the sick and dying.

Little is known about the life of Pythagoras and, as a result, many bizarre myths have sprung up around the man.

It was claimed amongst other things that he had taken part in the Olympics and was awarded laurels for pugilism, or boxing, when he was a young man. It was also said that he had fought in the Trojan Wars during a previous life.

This last myth reflects Pythagoras' genuine belief in metempsychosis, which argues that all souls are everlasting and, when the physical body dies, it simply floats away and finds a new body to live in, according to Stanford University . Later reports stated that he had been able to clearly recall four previous lives.

His fascination with astronomy ,as with many ancient Greeks, combined with his deep understanding of numbers led Pythagoras to confirm that the Earth was in fact a sphere and, through patient study, he discovered that the Evening Star and the Morning Star were the same planet, Venus .

Pythagoras' Theory states that in a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides, according to Encyclopedia Britannica. 

In other words, when a triangle has a right angle and squares are made of each of the three sides, then the biggest square has the same area as the other two squares combined. The equation can be used to work out the length of a third side if only two measurements are given. 

The Babylonians discovered this mathematical phenomena circa 1900 – 1600 BC but Pythagoras may have been first to prove it, according to New Scientist .

Although Pythagoras’ Theory is still taught in every classroom today, no one would recognise his original school of thought as it combined his mathematical teachings with philosophy and religion. His followers, the Pythagoreans 

created a secret commune, filled with strange rules and regulations, according to Encyclopaedia Britannica . 

Much of his written work was stored in the Great Library of Alexandria . Far from being the master mathematician that we think of today, Pythagoras was known for his belief in reincarnation, religious rituals and almost magical abilities, according to Stanford University. For example, it was said that he could be in two places at the same time. Today, these mystical elements have been almost forgotten and he is now looked upon as a founding father of science and mathematics .

Greek philosopher Plato created the world's first university, known as the Platonic Academy, in ancient Athens. Although different from a modern day university, the Academy was a place where people could meet and share their academic beliefs. Plato based a large proportion of his teachings on the thoughts of Pythagoras and his Pythagorean disciples, according to Stanford University.

Like Pythagoras, Aristotle was interested in the concept of a soul, according to the University of Washington . He wrote "On the Soul", which set out to examine the psychology of mankind, the principles of which are still referred to by psychologists today. Aristotle combined metaphysics with scientific investigation just as Pythagoras had achieved with metaphysics and the Number Theory. He was also inspired by Pythagoras’ interest in astronomy, ultimately developing the physical model of the heavens.

To find out more about Pythagoras, check out “ Pythagoras: His Life, Teaching, and Influence ”, by Christoph Riedweg and “ P ythagoras: His Lives And The Legacy Of A Rational Universe ”, by Kitty Ferguson.

• Mickaël Launay & Stephen S. Wilson, " It All Adds Up: The Story of People and Mathematics ", William Collins, 2019. 
• NRICH, " All is Number ", University of Cambridge, 2017. 
• Michael Marshall, " Babylonians calculated with triangles centuries before Pythagoras ", New Scientist, August 2021. 
• Stanford Encyclopedia of Philosophy, " Pythagoras ", University of Stanford,  October 2018.
• Holger Thesleff, " Pythagoreanism ", Encyclopedia Britannica, May 2020. 
• Encyclopedia Britannica, " Pythagorean theorem ", May 2020.
• University of Connecticut, " 3.7 Music of the Spheres and the Lessons of Pythagoras ", accessed in March 2022.  
• Silvano Leonessi, " The Pythagorean Philosophy of Numbers ", Rosicrucian Digest, Volume 1, 2009. 
• J. J. O'Connor & E. F. Robertson, " Pythagoras of Samos ", University of St Andrews, January 1999. 
• Brent Swancer, " The Great Pythagoras and his Mystical Cult ", Mysterious Universe, January 2021. 
• Dimtry Sudakov, " Pythagoras and his theory of reincarnation ", Pravda.ru, May 2013. 

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Introduction

Quest of knowledge, cult or school, the olympian example, life after death, the cult figure, rise of the pythagorean brotherhood, meeting with theano of crotona and marriage, reasons for the uprising, the flashpoint.

• Last days and death

“The oldest, shortest words - ' yes' and ' no' - are those which require the most thought,” said Pythagoras , the most famous yet controversial ancient Greek thinker, more renowned for his mathematical conundrum- Pythagoras Theorem- and other scientific achievements. Pythagoras words encapsulate the entire gamut of human wisdom because to say yes or no requires the deepest thoughts and a multitude of considerations which in most cases, are required to be taken instantly.

To date, psychologists are unable to unlock the mysteries of the human thought that makes them reply ‘yes’ or ‘no’. For example, saying ‘yes’ to a simple offer to have coffee may sound uncomplicated. Yet, the thoughts causing this simple response are perplexing and influenced by a variety of factors, all processed by the mind in split seconds to deliver an almost instant reply. Indeed, a ‘yes’ or ‘no’ sometimes defines the mental status of a person. These two simple words can mean the difference between death and life for a few.

While this statement by Pythagoras is labyrinthine, so is the author’s life which is equally abstruse due to an intermingling of fact, fiction and myths about his life. Till date, psychologists and scholars are confronted by his legacy, aptly called the ‘ Pythagorean Question’ about who was this great thinker and what were his doctrines.

Pythagoras was born in 570BC on the island of Samos , in the Aegean Sea off modern day Greece . Ancient historians including Herodotus and Isocrates claim, Pythagoras was born in a wealthy family and was the son of Mnesarchus , an affluent trader who imported expensive gemstones and created exquisite jewellery for the rich and famous of ancient Greece. His mother’s name was Pythais .

According to later thinker Aristippus , the name Pythagoras is derived from two sources- his mother Pythais and a legend involving his parents, who went to pay obeisance to the Oracle of Delphi in Samos. The Oracle, or high-priestess named Pythia prophesized to the pregnant Pythais that she would be blessed with a son who will bring unimaginable good repute to the family through his intelligence which would benefit the world. Hence, claims Aristippus, the name Pythagoras was derived from both Pythais- his mother and Pythia- the Oracle.

Renowned Greek historian and chronicler, Diogenes Laertius also wrote extensively about the life and doctrines of Pythagoras in his famous compilation ‘The Lives and Thoughts of Eminent Philosophers’ after seemingly intensive research, albeit conducted much later. Yet, Diogenes Laertius’ work is considered as relatively authoritative since he has painstakingly separated the proverbial chaff of myth and fiction from the grain- or- some indubitable facts about Pythagoras.

Laertius notes, Pythagoras was well educated and urbane, due to his wealthy and sophisticated lineage. Further records about his early education do not exist. Despite, Pythagoras’ works clearly indicate he was interested in cosmology, mathematics and psychology. Some historians claim Pythagoras studied under Anaximander of Miletus , who propounded several geometrical and cosmic theories during the era and was the first to propound that all terrestrial animals, including humans originated from the seas, attempted to explain meteorological phenomenon and draw an intricate theory about cosmology in a non-mythical manner.

This theory gains credence from Pythagorean doctrines which bear the unmistakable mark of the Milesian school of mathematics, cosmology and science, though it cannot be conclusively proved he was a student of Anaximander or his successors and the school.

Eminent ancient historian and chronicler Diogenes Laertius claims, Pythagoras in his quest for knowledge, undertook a long journey that lasted for over a decade. He travelled to various parts of the Athenian empire, Turkey, Libya, Egypt, Mesopotamia and Persia, India and possibly, some part of China. Once again, the ubiquitous Pythagorean Question raises its head here since some historians claim his travels were to study religious rites and mystic traditions while others assert, his journeys were purely aimed at learning more about the human nature, mathematics, cosmology, origins of life and other worldly issues rather than esoteric empyrean pursuit. Yet, some of his teachings are apparently based on Oriental religions- Hinduism and Buddhism - which may have led ancient writers to attribute his journeys to seek supernatural powers.

Returning to native Samos, the now widely travelled and learned Pythagoras formed a school to propound his doctrines which were an exotic blend of mathematics and ascetic lifestyle.

“As long as humans slaughter weaker beings, they will continue to remain in conflict, bereft of inner peace and good health. If they can kill helpless animals, they are capable of destroying one-another,” said Pythagoras. This particular axiom is not found, to date, in any Western spiritual text. Rather, these teachings date back to an era when Buddhism, founded by Gautama Buddha, was gaining ground in ancient India. The dates of Pythagoras visit to ancient India and the proliferation of Buddhism in India coincide.

Pythagorean students maintained a strict code of discipline, which remains unknown to the world to date. This mysterious school thus created a popular perception that Pythagoras had formed some exquisite cult of people who studied sciences unknown to others and lived a strange lifestyle. His quote regarding animal slaughter continues to be debated to date whether Pythagoras indeed propounded vegetarianism and if his students were debarred from consuming meat.

It is now widely accepted however that Pythagoras did not form a cult but a unique school whose teachings were based purely on mathematics and psychology as was known to that era while the institution was run on some spiritual basis that was a blend of then extant Greek mores and Oriental philosophy- a sort of ‘brotherhood’.

Despite, Pythagoras did study human behaviour and formulated independent doctrines that formed the basis of modern psychology. One of his axioms was drawn from the Olympic Games .

Pythagoras once visited Athens during the Olympic Games and upon witnessing the different types of people visiting the city-state, drew a conclusion that humans can broadly be divided into three categories, based upon their behaviour. It is debatable whether his classification was based on the ancient Indian caste system but the division bears a striking resemblance to the Oriental tradition of division of labour, albeit in a different perspective.

The three types of individuals described at the Olympics by Pythagoras are:

The Highest - are people who take pains to travel merely to witness the games, cheer the athletes and derive simple pleasure through these activities while taking an opportunity to learn from their experiences.

The Middle - he said consists of athletes and sportsmen who compete in the Olympics for their personal and national honour. Such participants train well and are looked up to by their families and other citizens of their native lands.

The Low - describes Pythagoras, are individuals who converge upon the venue with the sole objective of profiting from the mammoth event by trading their myriad wares to visitors from all over Greece and neighbouring kingdoms. They also utilize the opportunity to buy goods from distant lands that are on offer by traders from those places and sell them in their hometowns for a good profit.

Further, Pythagoras states, every individual possesses these three traits. Any of these characteristics crops up according to the circumstances in which an individual is placed. He does not deride an individual for these traits since they are intrinsically bound with the human nature and the primordial quest for a happy life.

Is there life after death? What happens to the soul after an individual dies? These questions remain unanswered to date and in all probability, will never be. Various religious beliefs propound different theories about the afterlife- but none have ever offered a conclusive proof about the ultimate fate of the soul.

Pythagoras, based upon Oriental teachings learnt during his travels, tried to solve this mystery. He propounded, the soul does not perish along with the individual and nor does it undergo imminent peace or turmoil based upon one’s actions. Instead, a soul merely migrates from one body into another. Such transmigration of the soul is not limited to humans since the soul can reappear in any other newborn creature- human or animal.

Coincidentally, this Pythagorean doctrine is strikingly similar to the Hindu philosophy about the soul’s immortality and transmigration. Therefore, it is most likely that Pythagoras developed his thoughts about the soul after studying under some Indian thinkers.

A few ancient Greek writers believe Pythagoras evolved his theory about afterlife based upon ancient lore. But this claim is debatable since most ancient Greek fables concerning the soul were based upon heroes, whose souls were claimed to have migrated to some paradise.

Pythagoras was merely trying to explain the simple concept that everything in this universe is intrinsically bound and hence all things recycle themselves. But his doctrine about rebirth, the brotherhood of mathematicians in his school with strange rules and his overall knowledge about a wide range of topics made Pythagoras a cult figure of sorts, with commoners and rulers attributing him with supernatural powers.

With Pythagoras being considered as a divinely anointed cult figure, his fame in native Samos and neighbouring areas spread like wildfire: Inadvertently, he found himself becoming a consultant for a myriad of issues ranging from counselling in marital discords to advising rulers on governance, helping students understand complex studies and attempting to cure the sick through celestial intervention.

But this popularity had its flip-side: Those envious of his fame and reverence soon began defaming Pythagoras as a charlatan and plotted to charge him with impiety for invoking powers from alien deities. Thankfully for Pythagoras, his clique included the rich and famous and followers included the sick and poor. Hence, plotters were wary of levelling charges against him fearing they may boomerang.

Overwhelmed by incessant demands he could least handle, Pythagoras and some pupils agreed to flee Samos into a self-imposed exile. They also wished to establish more schools that would study mathematics and psychology to further Pythagorean doctrines and hence, embarked on a journey that took them to various parts of ancient Greece.

As Pythagoras and his students travelled and propagated their theories and serene, austere lifestyle, they found several supporters and enthusiasts. Small and large Pythagorean brotherhoods mushroomed in various towns and cities in Greece.

Their journey also took them to Crotona , now in Italy , which was renowned for its citizens who loved learning. Here, Pythagoras’ life was to take an abrupt yet pleasant twist that would shape his future forever and give a newer direction to his esoteric school.

This happy incident occurred at the home of a local, highly respected, wealthy physician and aristocrat, Brontinus of Crotona . Being fond of Orphic mysticism, which deals with the mythical Greek demigod Orpheus, Brontinus would often host debates between local thinkers, aristocrats and foreign scholars transiting Crotona.

Brontinus’ daughter, Theano, was also fond of these debates and would often listen to them silently and occasionally, accompany her father to some of these events hosted by other nobles at their palatial homes or for the proletariat, in public places. She had heard of Pythagoras and was eager to see and hear the great thinker and mathematician due to myths associated with this man from Samos.

During the debate hosted by Brontinus, a silent spectator was overawed by Pythagoras- Theano of Crotona. She chose to become his disciple and was willingly accepted by Pythagoras as a pupil. Indeed, Brontinus and other aficionados of Pythagorean teachings helped the great mathematician and psychologist to establish his famous school in Crotona. Pythagoras and Theano soon became lovers and eventually married. The couple had three daughters, Damo, Myia and Arignote and two sons, Mnesarchus- named after Pythagoras’ father and Teleuges.

The Pythagorean school in Crotona remained popular for several years. However, its teachings, especially those related to cosmology- or the origins of cosmos and its inherent bearings on human life, divine beings and other controversial teachings- led to a popular revolt against its followers all over ancient Greece, around 500BC.

Pythagoras prescribed strict rules for his students or the brotherhood. Those who violated any of these norms would be expelled from the school disgracefully. Hence, Pythagoras was rather strict in selecting students who could join the brotherhood. Pythagoras was hence accused of forming a cult that digressed from ancient Grecian religion and traditions. Further, people who could not join the school due to its strict admission policies bore resentment against Pythagoras, his brotherhood or school and its disciples.

Among the edicts of the Pythagorean school or brotherhood was one famous edict where students were strictly prohibited from consuming all types of beans, especially the Foul or Fava beans- consumed widely in Mediterranean countries since ages. Pythagoras claimed, eating beans causes severe discomfort in the alimentary canal and causes digestive tract disturbances- which he likened to upheavals in the cosmos.

Unverifiable records claim, he likened bean consumption to cannibalism- which pollutes the body and impedes with the purity of the human soul. Hence, candidates who were capable of living well despite the rigorous regimen were admitted by Pythagoras to his brotherhood. Without realizing the extent of resentments he had caused, Pythagoras continued to enforce his strict rules regardless of the social standing of a student.

While in Croton, the great mathematician and psychologist was approached by Kylon , the son of a local aristocrat, who wanted to study under Pythagoras. Kylon, despite his wealth and noble lineage, had a bad reputation for indulging in unabashed violence against anyone who would not cede his demands.

Kylon reportedly approached Pythagoras and sought admission to the brotherhood. The teacher, knowing that Kylon would never assimilate into the school and its teachings, refused admission. Angered at this denial, Kylon vowed revenge and gathered a bunch of local ruffians who ransacked the Pythagorean school in Croton, setting it ablaze.

Pythagoras and his family were rescued by his students and sheltered in neighbouring forests. The group, along with their master, later embarked on a journey to Pythagoras’ native Samos. However, ill health forced Pythagoras to seek shelter in Metapontum.

However this incident marked the end of the widespread Pythagorean brotherhood. By 500BC, every Pythagorean school in ancient Greece was attacked and destroyed for reasons ranging from impiety to violation of local rules.

The rampage at Crotona and wanton violence against the Pythagorean brotherhood left the teacher in great distress. He isolated himself, spending time with wife Theano and a few select students. Despite his own achievements in mathematics, cosmology and psychology, Pythagoras did not write any treatise, perhaps fearing he would be persecuted for it. He died a dejected man five years later, in 495BC in Metapontum. A monument commemorating the great master was built in Metapontum by his ardent followers.

One would have expected the Pythagorean brotherhood to become extinct after the death of its founder and the systematic destruction of its schools. However, history proved otherwise.

Theano, the wife of Pythagoras painstakingly penned her husband’s studies, discourses and doctrines.  His disciples spread the Pythagorean doctrines across Greece and other parts of the world. The popularity of his teachings reached zenith when most later thinkers began working on further developing the doctrines preached by Pythagoras.

Further, Pythagoras is also reputed for several quotes, though their sources are debatable considering the great master never wrote anything. Some of these quotes are:

“Strength of mind rests in sobriety; for this keeps your reason unclouded by passion”.

“As soon as laws are necessary for men, they are no longer fit for freedom.”

“There is nothing so easy but that it becomes difficult when you do it reluctantly.”

But the most popular legacy of Pythagoras is the geometrical apothegm that bears his name ‘Pythagoras Theorem’.

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Pythagoras of Samos  (Greek: Πυθαγόρας ὁ Σάμιος , or just Πυθαγόρας ; Πυθαγόρης in Ionian Greek; Samos, c. 570 – Metapontus, c. 495 BC ) was a Greek philosopher and mathematician called the Ionian movement credited with founding the Pitagorism movement. Most information about Pythagoras was written down centuries after his death, so there is little reliable information about him. He was born on the island of Samos and traveled through Egypt and Greece. In 520 BC he returned to Samos. Around 530 BC he moved to Crotona in Magna Graecia.

Born on the Greek island of Samos, his mother would have been called Pythals and his father Mnesarco, supposedly a merchant from the city of Tyre, who in addition to Pythagoras would have had two or three other children. Pythagoras spent his childhood on Samos although he traveled extensively with his father; he was trained by the best teachers, some of them philosophers. He played the lyre, learned arithmetic, geometry, astronomy and poetry.

Herodotus, Isocrates and other early writers all agree that Pythagoras was the son of Mnesarchus and that he was born on the Greek island of Samos in the eastern Aegean Sea. His father is said to have been a gem engraver or a wealthy merchant, but his ancestry is controversial and unclear. Pythagoras' name led him to be associated with Apollo Python; Aristippus of Cyrene explained his name by saying, "He spoke ( ἀγορεύω , agoreúō ) the truth no less than the Pythia [  sic  ] ( Πῡθῐ́ᾱ , Pūthíā )". Only a late source is given the name of Pythagoras' mother as  Pythaïs  .Iamblichus tells the story that the Pythia prophesied to her while she was pregnant that she would give birth to a man supremely beautiful, wise, and beneficial to mankind. As for the date of his birth, Aristoxenes stated that Pythagoras left Samos in the reign of Polycrates, at age 40, which would give a birth date around 570 BC.

During Pythagoras' formative years, Samos was a thriving cultural center known for its feats of advanced architectural engineering, including the construction of the Eupalinos Tunnel and its festive culture. It was an important trading center in the Aegean, where traders brought goods from the Near East. According to Christiane L. Joost-Gaugier, these traders almost certainly brought with them ideas and traditions from the Near East. The early life of Pythagoras also coincided with the flowering of early Ionian natural philosophy. He was a contemporary of the philosophers Anaximander, Anaximenes, and the historian Hecataeus, all of whom lived in Miletus, across the Sea from Samos.

Alleged trips

Pythagoras is traditionally believed to have received most of his education in the Near East. Modern studies have shown that the culture of archaic Greece was heavily influenced by the culture of the Near East. Like many other important thinkers in Greece, Pythagoras would have studied in Egypt, where he would have traveled in about 535 BC - a few years after the occupation of Samos by the tyrant Polycrates - there, he got to know the temples and learned about the local priests. By the time of Isocrates, in the 4th century BC, the supposed studies of Pythagoras in Egypt were already taken for granted. The writer Antiphon, who may have lived during the Hellenic Age, stated in his lost work  On Men of Exceptional Merit , used as a source by Porphyry, that Pythagoras learned to speak Egyptian from Pharaoh Amosis II himself, that he studied with the Egyptian priests at Diopolis (Thebes), and that he was the only foreigner to receive the privilege of participating in their cult. The Middle Platonic biographer Plutarch (c. 46 – c. 120 CE) writes in his treatise  On Isis and Osiris  that, during his visit to Egypt, Pythagoras received instructions from the Egyptian priest Enuphis of Heliopolis (meanwhile, Solon received lectures from a Sonchis of Sais). According to the Christian theologian Clement of Alexandria (c. 150 – c. 215 AD), "Pythagoras was a disciple of Soches, an Egyptian archprophet, as Plato was of Secnuphis of Heliopolis."Some ancient writers claimed that Pythagoras learned geometry and the doctrine of metempsychosis from the Egyptians.

Other ancient writers, however, claimed that Pythagoras had learned these teachings from the magicians of Persia or even from Zoroaster himself. Diogenes Laertius claims that Pythagoras later visited Crete, where he went to the Cave of Ida with Epimenides. It is said that the Phoenicians taught Pythagoras arithmetic and that the Chaldeans taught him astronomy. As early as the third century BC, it was said that Pythagoras had already studied under the Jews. Contrary to all these accounts, the novelist Antonio Diogenes, writing in the 2nd century BC, reports that Pythagoras discovered all his doctrines by interpreting dreams. The 3rd century AD sophist Philostratus claims that in addition to the Egyptians, Pythagoras also studied with Hindu sages in India.Iamblichus expands this list even further, stating that Pythagoras would also have studied with the Celts and the Iberians.

In 525 BC, the Persian king Cambyses I attacked Egypt and a legend tells that Pythagoras would have been captured and sent to Babylon, where he would have received spiritual teachings of oriental influence, in anecdotal attempts to associate him with a possible contact with the so-called Persian, Babylonian or even Hindu magicians.

By 522 BC both Polycrates and Cambyses had died, so Pythagoras returned to Samos where he founded a school of philosophy called the  Semicircle  .

Around 518 BC, to avoid political conflicts, he traveled to southern Italy, to the city of Crotona, where he founded a spiritual school; there he would have married.

Family and friends

Diogenes Laërtius claims that Pythagoras "did not indulge in the pleasures of love" and that he warned others to have sex only "when you are willing to become weaker than you are". According to Porphyry, Pythagoras married Theano, a Cretan lady and daughter of Pitonax of Crete, and had several children with her. Porphyry writes that Pythagoras had two sons named Telauge and Arignot and a daughter named Myia who "took precedence among the maidens of Crotona, and, when espoused, among married women". Iamblichus does not mention any of these sons , and instead only mentions a son named Mnesarchus in honor of his grandfather.This son would have been raised by Pythagoras' appointed successor, Aristaeus, and ended up taking over the school when Aristaeus was too old to continue running it.

The fighter Milo of Croton was said to be a close associate of Pythagoras and was credited with saving the philosopher's life when a roof was about to collapse. This association may have been the result of confusion with a different man named Pythagoras, who was an athletics coach. Diogenes Laertius records the name of Milo's wife as Myia. Iamblichus mentions Theano as the wife of Brontino of Crotona. Diogenes Laertius states that this same Theano was a student of Pythagoras and that Pythagoras' wife, Theano, was her daughter. Diogenes Laertius also records that the works supposedly written by Theano still existed during her own lifetime , and cites several opinions attributed to her.These writings are now known to be pseudepigraphs.

Pythagoras' emphasis on dedication and asceticism is credited with aiding Crotona's decisive victory over the neighboring colony of Sybaris in 510 BC. After the victory, some prominent citizens of Crotona proposed a democratic constitution, which the Pythagoreans rejected. The partisans of democracy, led by Cylon and Nynon, the former of whom is said to have been irritated by their exclusion from the brotherhood of Pythagoras, roused the populace against them. Followers of Cylon and Nyno attacked the Pythagoreans during one of their meetings, either at Milan's house or at some other meeting place. Accounts of the attack are often contradictory and many have likely confused it with later anti-Pythagorean rebellions. The building was apparently set on fireand many of the assembled members died; only the youngest and most active members managed to escape.

Sources disagree on whether Pythagoras was present when the attack took place and, if he was, whether or not he managed to escape. In some accounts, Pythagoras was not at the meeting when the Pythagoreans were attacked because he was in Delos, tending to Pherecydes on his deathbed. According to another account by Dicéarchus, Pythagoras was at the meeting and managed to escape, taking a small group of followers to the nearby town of Locris, where they sought refuge but were denied. They reached the city of Metaponto, where they took shelter in the temple of the Muses and died of hunger after forty days without food.Another tale recorded by Porphyry states that while Pythagoras' enemies were burning the house, his devoted students lay on the ground to make a way for him to escape, walking over their bodies through the flames like a bridge. Pythagoras managed to escape, but was so disheartened by the death of his beloved students that he would have committed suicide. A different legend related by Diogenes Laertius and Iamblichus states that Pythagoras almost managed to escape, but that he came to a field of broad beans and refused to go through it, as doing so would violate his teachings, he stopped then and was killed. This story appears to have originated with the writer Neantes, who spoke about the later Pythagoreans, not Pythagoras himself.

The pythagorean school

The Pythagoreans were interested in the study of the properties of numbers. For them, the number, synonymous with harmony, is constituted by the sum of even and odd numbers (the even and odd numbers expressing the relationships that are in a permanent process of mutation), being considered as the essence of things, creating opposing notions (limited and unlimited) and the basis of the theory of the harmony of the spheres. The Pythagorean school was connected with esoteric conceptions and Pythagorean morals emphasized the concept of  harmony  , ascetic practices and advocated metempsychosis.

According to the Pythagoreans, the cosmos is governed by mathematical relationships. The observation of the stars suggested to them that an order dominates the universe. Evidence of this would be in the day and night, in the changing of the seasons and in the circular and perfect movement of the stars. That is why the world could be called cosmos, a term that contains the ideas of order, correspondence and beauty. In this worldview they also concluded that the Earth is spherical, a star among the stars that move around a Central Fire. Some Pythagoreans even spoke of the Earth's rotation on its axis, but the greatest discovery of Pythagoras or his disciples (since there are obscurities around Pythagoreanism, due to the esoteric and secret character of the school) took place in the domain of geometry. and refers to the relationships between the sides of the right triangle.

Pythagoras was expelled from Crotona and took up residence at Metapontum, where he died, probably in 496 BC or 497 BC. According to Pythagoreanism, the essence, which is the fundamental principle that forms all things, is the  number  . The Pythagoreans did not distinguish between form, law, and substance, considering number to be the link between these elements. For this school there were four elements: earth, water, air and fire.

Thus, Pythagoras and the Pythagoreans investigated mathematical relationships and discovered various foundations of physics and mathematics.

The symbol used by the school was the pentagram, which, as Pythagoras discovered, has some interesting properties. A pentagram is obtained by tracing the diagonals of a regular pentagon; by the intersections of the segments of this diagonal, a new regular pentagon is obtained, which is exactly proportional to the original by the golden ratio.

Pythagoras discovered in what proportions a string must be divided to obtain musical notes at the beginning, without a defined pitch, being one taken as a fundamental (let us think of a long string attached to two ends that, when plucked, will give us the lowest sound) - and from it, the fifth and third will be generated through harmonic reverberation. Harmonic sounds. By attaching half of the string, then the third part and then the fifth part, we will obtain the fifth and third intervals in relation to the fundamental. The so-called harmonic series. As we subdivide the string we get louder sounds and the intervals will be different. And so on. He further discovered that simple fractions of notes, played along with the original note, produce pleasant sounds. The more complicated fractions, played with the original note,

The name is mainly linked to the important theorem that states: In every right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

During the 4th century BC, there was a revival of religious life in the Greek world. According to some historians, one of the factors that contributed to this phenomenon was the political line adopted by the tyrants: to guarantee the role of popular leaders and to weaken the old aristocracy, the tyrants encouraged the expansion of popular or foreign cults.

Among these cults, one had great diffusion: Orphism (of Orpheus), originating in Thrace, and which was an essentially esoteric religion. The followers of this doctrine believed in the immortality of the soul, that is, while the body degenerated, the soul migrated to another body, several times, in order to effect purification. Dionysus would guide this cycle of reincarnations, being able to help man to free himself from it.

Pythagoras followed a different doctrine. He would have arrived at the conception that all things are numbers and the process of liberating the soul would be the result of a basically intellectual effort. Purification would result from an intellectual work, which discovers the numerical structure of things and thus makes the soul a harmonious unity. The numbers would not be, in this case, symbols, but the values ​​of magnitudes, that is, the world would not be composed of the numbers 0, 1, 2, etc., but of the values ​​they express. So, therefore, a thing would externally manifest the numerical structure, this thing being what it is because of this value.

Metempsychosis

Although the exact details of Pythagoras' teachings are uncertain, it is possible to reconstruct a general outline of his main ideas. Aristotle writes at length about the teachings of the Pythagoreans, but without directly mentioning Pythagoras. One of Pythagoras' main doctrines appears to have been  metempsychosis  , the belief that all souls are immortal and that, after death, a soul is transferred into a new body. This teaching is referenced by Xenophanes, Ion of Chio and Herodotus. However, nothing is known about the nature or mechanism by which Pythagoras believed that metempsychosis occurred.

Empedocles alludes in one of his poems that Pythagoras may have claimed to possess the ability to remember his previous incarnations. Diogenes Laertius reports an account by Heraclides of Pontus that Pythagoras told people that he had lived four previous lives that he could remember in detail. The first of these lives was as Ethalides, son of Hermes, who granted him the ability to remember all of his past incarnations. He then incarnated as Euphorbus, a minor hero of the Trojan War mentioned briefly in the  Iliad  . He then became the philosopher Hermotimus, who recognized the shield of Euphorbus in the temple of Apollo. His final incarnation was as Pyrrhus, a fisherman from Delos.One of her past lives, as reported by Dicearco, was as a beautiful courtesan. By the way, in his book  The Life of Apollonius of Tyana  , Philostratus also cites that Pythagoras knew who he had been.

Another belief attributed to Pythagoras was that of the "harmony of the spheres", which held that the planets and stars move according to mathematical equations, which correspond to musical notes and therefore produce an inaudible symphony. According to Porphyry, Pythagoras taught that the seven Muses were actually the seven planets singing together. In his philosophical dialogue  Protrepticus  , Aristotle says for his literary double:

When Pythagoras was asked [why human beings exist], he said, "to observe the heavens", and used to claim that he himself was an observer of nature and that was why he passed into life.

Pythagoras was said to have practiced divination and prophecy. On visits to various places in Greece - Delos, Sparta, Phlius, Crete, etc. - attributed to him, he usually appears in his religious or priestly guise, or as a lawgiver.

Communal lifestyle

Both Plato and Isocrates claim that, above all, Pythagoras was known as the founder of a new way of life. The organization Pythagoras founded at Croton was called a "school", but in many ways it looked like a monastery. The adepts were bound by a vow to Pythagoras and to one another, for the purpose of pursuing religious and ascetic observances, and of studying his religious and philosophical theories. The sect members shared all their possessions in common and were devoted to one another, excluding foreigners. Ancient sources record that the Pythagoreans had meals together, in the manner of the Spartans. A Pythagorean maxim was "  koinà tà phílōn  " ("All things in common among friends").Iamblichus and Porphyry provide detailed accounts of the school's organization, although the primary interest of both writers is not historical accuracy, but presenting Pythagoras as a divine figure, sent by the gods to benefit mankind. Iamblichus, in particular, presents the "Pythagorean Way of Life" as a pagan alternative to the Christian monastic communities of his own time.

Two groups existed in early Pythagoreanism: the  mathematikoi  ("learners") and the  akousmatikoi  ("listeners"). Akousmatikoi are  traditionally  identified by scholars as "old believers" in mysticism, numerology, and religious teachings; while the  mathematikoi  are traditionally identified as a modernist and more intellectual, more rationalist and scientific faction. Gregory warns that there was probably no sharp distinction between them and that many Pythagoreans probably believed the two approaches were compatible. The study of mathematics and music may have been related to the cult of Apollo.The Pythagoreans believed that music was a purification for the soul, just as medicine was a purification for the body. An anecdote from Pythagoras relates that when he found some drunken youths trying to break into the home of a virtuous woman, he sang a solemn tune with long spondees, and the boys' "furious will" was suppressed. The Pythagoreans also particularly emphasized the importance of physical exercise; therapeutic dancing, daily morning walks along scenic routes, and athletics were major components of the Pythagorean lifestyle. Moments of contemplation at the beginning and end of each day were also advised.

Prohibitions and regulations

Pythagoras' teachings were known as "symbols" (  symbola  ) and members took a vow of silence that they would not reveal these symbols to non-members. Those who did not obey the community's laws were expelled and the remaining members erected tombstones for them as if they had died. A number of "oral sayings" (  akoúsmata  ) attributed to Pythagoras have survived, dealing with how members of the Pythagorean community should perform sacrifices, how they should honor the gods, how they should "move" here, and how they should be buried. Many of these sayings emphasize the importance of ritual purity and avoiding contamination.For example, a saying that Leonid Zhmud concludes can probably be traced genuinely to Pythagoras himself, forbidding his followers to wear woolen clothing. Other existing oral sayings forbid the Pythagoreans from breaking bread, poking at fires with swords or picking up crumbs and teaching that a person should always put the right sandal before the left. The exact meanings of these sayings, however, are often obscure. Iamblichus preserves Aristotle's descriptions of the original ritualistic intentions behind some of these sayings, but these apparently later fell out of fashion because Porphyry provides markedly different ethico-philosophical interpretations for them:

Allegedly, new initiates were not allowed to meet with Pythagoras until they had completed a five-year initiation period, during which they were to remain silent. Sources indicate that Pythagoras himself was extraordinarily progressive in his attitudes towards women and the women of the Pythagorean school appear to have played an active role in his operations. Iamblichus provides a list of 235 famous Pythagoreans, seventeen of whom are women. In later times, many prominent philosophers contributed to the development of Neo-Pythagoreanism.

Pythagoreanism also involved a series of food prohibitions. It is more or less in agreement that Pythagoras issued a ban against the consumption of fava beans and the meat of non-sacrifice animals such as fish and poultry. Both assumptions, however, have since been contradicted or associated with symbolic meaning. Pythagorean dietary restrictions may have been motivated by belief in the doctrine of metempsychosis. Some ancient writers present Pythagoras as enforcing a strictly vegetarian diet.Eudoxus of Cnidus, a student of Archytas, writes: "Pythagoras distinguished himself by this purity, and therefore avoided killing and those who killed, so that he not only abstained from food of animal origin, but also kept his distance from cooks and hunters". Other authorities contradict this claim. According to Aristoxenes, Pythagoras allowed the use of all types of animal feed, except the meat of oxen used for plowing and sheep. According to Heraclides Ponticus, Pythagoras ate the meat of sacrifices and established a meat-dependent diet for athletes.

Main findings

In addition to being great mystics, the Pythagoreans were great mathematicians. They discovered interesting and curious properties about numbers.

Figured numbers

The Pythagoreans studied and demonstrated various properties of figurative numbers. Among these the most important was the triangular number 10, called by the Pythagoreans  tetraktys  , tetrad in Portuguese. This number was seen as a mystical number as it contained the four elements fire, water, air and earth: 10=1 + 2 + 3 + 4, and served as a representation for the completeness of the whole.αα αα α αα α α α

The tetrad, which the Pythagoreans drew with an α above, two below, then three and finally four at the base, was one of the main symbols of their advanced knowledge of theoretical realities.

Perfect numbers

The sum of the divisors of a given number, with the exception of itself, is the number itself. Examples:

Pythagorean theorem

An unsolved problem in Pythagoras' time was to determine the relationships between the sides of a right triangle. Pythagoras proved that the sum of the squares of the legs is equal to the square of the hypotenuse.

The Greeks did not know the square root symbol and simply said: "the number that multiplied by itself is 2".

From the discovery of the root of 2 many other irrational numbers were discovered.

rector of the first university

The word Mathematics (Mathematike, in Greek) came up with Pythagoras, who was the first to conceive of it as a system of thought, based on deductive proofs.

There are, however, indications that the so-called Pythagorean Theorem (c²= a²+b²) was already known to the Babylonians in 1600 BC with an empirical scope. These used sexagesimal notation systems for measuring time (1h=60min) and measuring angles (60º, 120º, 180º, 240º, 360º).

Pythagoras traveled for 30 years through Egypt, Babylon, Syria, Phoenicia and perhaps India and Persia, where he accumulated eclectic knowledge: astronomy, mathematics, science, philosophy, mysticism and religion. He was a contemporary of Thales of Miletus, Buddha, Confucius and Lao-Tzu.

When he returned to Samos, he fell out with the tyrant Polycrates and emigrated to Crotona in southern Italy. There he founded the Pythagorean School, which is given the glory of being the "first University in the world".

The  Pythagorean  School and its activities have since been shrouded in a veil of legends. It was a partially secret entity with hundreds of students who made up a religious and intellectual brotherhood. Among the concepts they defended, the following stand out:

• practice of purification rituals and belief in the doctrine of metempsychosis, that is, in the transmigration of the soul after death, from one body to another. Therefore, they advocated the reincarnation and immortality of the soul;
• loyalty among members and community distribution of material goods;
• austerity, asceticism and obedience to the hierarchy of the School;
• prohibition of drinking wine and eating meat (so the information that the disciples had 100 oxen killed when demonstrating the so-called Pythagorean Theorem is false);
• purification of the mind by the study of Geometry, Arithmetic, Music and Astronomy;
• arithmetic classification of numbers into even, odd, prime and factorable;
• "the creation of a model of definitions, axioms, theorems and proofs, according to which the intricate structure of Geometry is obtained from a small number of explicitly made statements and the action of rigorous deductive reasoning" (George Simmons);
• a great stir arose among the disciples of Pythagoras about the irrationality of the 'root of 2'. Using algebraic notation, the Pythagoreans did not accept any numerical solution for x² = 2, as they only admitted rational numbers. Given the mystical connotation attributed to the numbers, it is said that when the unfortunate Hippasus of Metapontum proposed a solution to the impasse, the other disciples expelled him from the School and drowned him in the sea;
• in Astronomy, innovative ideas, although not always true: the Earth is spherical, the planets move at different speeds in the various orbits around the Earth. Through the careful observation of the stars, the idea was crystallized that there is an order that dominates the Universe;
• the Pythagoreans are probably due to the construction of the cube, tetrahedron, octahedron, dodecahedron and the well-known "golden section";
• in Music, a remarkable discovery that musical intervals are so placed that they admit of expressions through arithmetical proportions. Pythagoras - as well as other pre-Socratic Greek philosophers - also described the power of sound and its effects on the human psyche. This music therapy experience was possibly later used by Aristotle as a theoretical basis for his definition of music, which, according to him, was a "medicinal art".

Pythagoras is the first pure mathematician. However, it is difficult to separate the historical from the legendary, since he must be considered a historically inaccurate figure, since everything we know about him is due to oral tradition. He left nothing written, and the first works on the same are due to Philolaus, almost 100 years after the death of Pythagoras. But it is not easy to deny the Pythagoreans - asserts Carl Boyer - "the primordial role in the establishment of mathematics as a rational discipline". Despite some exaggeration, centuries ago a phrase was coined: "If there were no 'Pythagoras theorem', Geometry would not exist".

When biography of Pythagoras, Iamblichus (c. 300 AD) records that the master kept repeating to his disciples: “all things resemble numbers”.

Later influence in antiquity

The Pythagorean School had a strong influence on the powerful words of Euclid, Archimedes and Plato, in the ancient Christian era, in the Middle Ages, in the Renaissance and even in our days with Neo-Pythagoreanism.

in greek philosophy

Considerable Pythagorean communities existed in Magna Graecia, Fliunte, and Thebes during the early 4th century BC. At the same time, the Pythagorean philosopher Archytas had a great influence on the politics of the city of Tarentum, in Magna Graecia. According to later tradition, Archytas was elected  strategos  ("general") seven times, although others were banned from serving for more than a year. Archytas was also a renowned mathematician and musician. He was a close friend of Plato and is quoted in Plato's  Republic  . Aristotle claims that Plato's philosophy was heavily dependent on the teachings of the Pythagoreans. Cicero repeats this statement, noting that Platonem ferunt didicisse Pythagorea omnia  ("It is said that Plato learned all things Pythagorean"). According to Charles H. Kahn, Plato's middle dialogues, including the  Meno  ,  Phaedo  , and  The Republic  , have a strong "Pythagorean coloring", and his later dialogues (particularly  Philebus  and  Timaeus  ) are extremely Pythagorean in character.

According to RM Hare, Plato's  Republic  may be partly based on the "strongly organized community of like-minded thinkers" established by Pythagoras at Croton. Furthermore, Plato may have borrowed from Pythagoras the idea that mathematics and abstract thought are a secure foundation for philosophy, science, and morality. Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world" and it is likely that both were influenced by Orphism. The historian of philosophy Frederick Copleston claims that Plato probably borrowed his tripartite theory of the soul from the Pythagoreans. Bertrand Russell, in  A History of Western Philosophy , claims that Pythagoras' influence on Plato and others was so great that he should be considered the most influential philosopher of all time. He concludes that "I know of no other man who has been as influential as he was in the school of thought".

A Table of Opposites was attributed by Aristotle to Pythagoras, which shows that it shares the pre-Socratic Greek thought of investigating dualities in nature.

A revival of Pythagorean teachings took place in the first century BC when Middle Platonist philosophers such as Eudorus and Philo of Alexandria welcomed the emergence of a "new" Pythagoreanism in Alexandria. Around the same time, Neo-Pythagoreanism became prominent. The 1st century AD philosopher. Apollonius of Tyana, sought to imitate Pythagoras and live up to the teachings of Pythagoras. The Neo-Pythagorean philosopher Moderato of Gades, at the end of the first century, expanded the Pythagorean philosophy of numbers and probably understood the soul as a "type of mathematical harmony". The Neo-Pathagorean mathematician and musicologist Nicomachus also expanded upon Pythagorean numerology and music theory.Numenius of Apameia interpreted Plato's teachings in the light of Pythagorean doctrines.

About art and architecture

Greek sculpture sought to represent the permanent reality behind surface appearances. Early archaic sculpture represents life in simple ways and may have been influenced by early Greek natural philosophies. The Greeks generally believed that nature expressed itself in ideal forms and was represented by a type ( εἶδος ), which was calculated mathematically. As dimensions shifted, architects sought to relay permanence through mathematics. Maurice Bowra believes that these ideas influenced the theory of Pythagoras and his students, who believed that "all things are numbers".

During the 6th century BC, the numerical philosophy of the Pythagoreans sparked a revolution in Greek sculpture. Greek sculptors and architects tried to find the mathematical relationship (canon) behind aesthetic perfection. Possibly taking advantage of the ideas of Pythagoras, the sculptor Polycletus writes in his  canon  that beauty consists in proportion, not in the elements (materials), but in the interrelation of the parts with each other and with the whole. In Greek architectural orders, every element was calculated and constructed by mathematical relationships. Rhys Carpenter states that the 2:1 ratio was "the generative ratio of the Doric order, and in Hellenistic times an ordinary Doric colonnade, trumps a rhythm of notes".

The oldest known building, designed according to the teachings of Pythagoras, is the Porta Maggiore Basilica, an underground basilica that was built during the reign of Roman Emperor Nero as a secret place of worship for the Pythagoreans. The basilica was built underground because of the Pythagorean emphasis on secrecy and also because of the legend that Pythagoras had isolated himself in a cave on Samos. The basilica's apse is on the east and its atrium on the west out of respect for the rising sun. It has a narrow entrance that leads to a small pool where initiates can purify themselves. The building is also designed according to Pythagoras' numerology, with each table in the sanctuary offering seating for seven people.Three aisles lead to a single altar, symbolizing the three parts of the soul that approach the unity of Apollo. The apse shows a scene of the poet Sappho jumping off the Leucadian cliffs, holding her lyre to her chest, while Apollo stands beneath it, extending his right hand in a protective gesture symbolizing the Pythagorean teachings on the immortality of the soul. The interior of the shrine is almost entirely white because the color white was considered sacred by the Pythagoreans.

Emperor Hadrian's pantheon in Rome was also built on Pythagorean numerology. The temple's circular plan, central axis, hemispherical dome and alignment with the four cardinal directions symbolize the Pythagorean visions of the order of the universe. The single oculus at the top of the dome symbolizes the monad and the sun god Apollo. The twenty-eight ribs extending from the oculus symbolize the moon, because twenty-eight were the same number of months in Pythagoras' lunar calendar. The five rings boxed below the ribs represent the marriage of the sun and moon.

at the beginning of christianity

Pythagoras. Eusebius ( c. 260 - c. 340 CE), bishop of Caesarea, praises Pythagoras in his book  Against Hierocles  for his mastery of silence, his frugality, his "extraordinary" morals and his wise teachings. In another work, Eusebius compares Pythagoras to Moses. In one of his letters, Church Father Jerome ( c. 347 - 420 AD) praises Pythagoras for his wisdom , and in another letter he credits Pythagoras for his belief in the immortality of the soul, which he suggests Christians inherited from him. . Augustine of Hippo (AD 354 - 430) rejected Pythagoras' teachings on metempsychosis without explicitly naming him, but expressed admiration for him. In On the Trinity  , Augustine praises the fact that Pythagoras was humble enough to call himself a  philosopher  or "lover of wisdom" rather than "sage". In another passage, Augustine defends Pythagoras' reputation, arguing that Pythagoras certainly never taught the doctrine of metempsychosis.

Influence after antiquity

In the middle ages.

During the Middle Ages, Pythagoras was revered as the founder of mathematics and music, two of the Seven Liberal Arts. It appears in numerous medieval depictions, in illuminated manuscripts, and in the relief carvings on the portal of Chartres Cathedral. The  Timaeus  was the only dialogue of Plato to survive in Latin translation in western Europe, which led William of Conches (c. 1080-1160) to declare Plato to be a Pythagorean. In the 1430s, the Camaldolese friar Ambrose Traversari translated the  Lives and Doctrines of the Illustrious Philosophers  of Diogenes Laertius from Greek into Latin , and in the 1460s the philosopher Marsilio Ficino also translated the  Lives of Pythagoras  by Porphyry and Iamblichus.thus allowing them to be read and studied by Western academics. In 1494 the Greek Neo-Pythagorean scholar Constantine Láscaris published  The Golden Verses of Pythagoras  , translated into Latin, with a printed edition of his  Grammatica  , bringing them to a widespread audience. In 1499 he published the first Renaissance biography of Pythagoras in his work  Vitae illustrium philosophorum siculorum et calabrorum  , published in Messina.

in modern science

Em seu prefácio ao livro  A revolução das esferas celestiais  (1543), Nicolau Copérnico cita vários pitagóricos como as influências mais importantes no desenvolvimento de seu modelo heliocêntrico do universo, omitindo deliberadamente a menção de Aristarco de Samos, um astrônomo não pitagórico que havia desenvolvido um modelo totalmente heliocêntrico no século IV a.C., em um esforço para retratar seu modelo como fundamentalmente pitagórico. Johannes Kepler considerava-se um pitagórico. Ele acreditava na doutrina pitagórica da  musica universalis  e foi sua busca pelas equações matemáticas por trás dessa doutrina que levou à descoberta das leis do movimento planetário. Kepler intitulou seu livro sobre o assunto  Harmonices Mundi  ( Harmônicas do Mundo ), em homenagem aos ensinamentos pitagóricos que o inspiraram. Perto da conclusão do livro, Kepler descreve-se adormecendo ao som da música celestial, "aquecido por ter bebido um gole generoso ... do copo de Pitágoras".

Isaac Newton was a firm believer in the Pythagorean teaching of the mathematical harmony and order of the universe. Although Newton was notorious for rarely giving other people credit for his discoveries, he attributed the discovery of the Law of Universal Gravitation to Pythagoras. Albert Einstein believed that a scientist can also be "a Platonist or a Pythagorean, insofar as he regards the point of view of logical simplicity as an indispensable and effective tool of his research". The English philosopher Alfred North Whitehead argued that "In some ways Plato and Pythagoras are closer to modern physical science than Aristotle. The first two were mathematicians, while Aristotle was the son of a physician."By this measure, Whitehead declared that Einstein and other modern scientists like him are "following the pure Pythagorean tradition."

about vegetarianism

A fictionalized portrayal of Pythagoras appears in Book XV of  Ovid  's Metamorphoses, in which he gives a speech imploring his followers to adhere to a strictly vegetarian diet. It was through Arthur Golding's 1567 English translation of  Ovid  's Metamorphoses that Pythagoras was best known to English speakers during the early modern period. John Donne 's  Progress of the Soul  discusses the implications of the doctrines expounded in the speech and Michel de Montaigne quoted the speech at least three times in his treatise "On Cruelty" to express his moral objections against the mistreatment of animals. William Shakespeare references the speech in his play The Merchant of Venice  . John Dryden included a translation of the scene with Pythagoras in his 1700 work  Fables, Ancient and Modern,  and John Gay's 1726 fable "Pythagoras and the Peasant" reiterates its main themes, linking carnivorism with tyranny. Lord Chesterfield records that his conversion to vegetarianism was motivated by reading Pythagoras' speech in  Ovid  's Metamorphoses. Until the word  vegetarianism  was coined in the 1840s, vegetarians were referred to in English as "Pythagoreans". Percy Bysshe Shelley wrote an ode entitled "To the Pythagorean Diet" and Leo Tolstoy adopted the Pythagorean Diet.

About Western Esotericism

Modern European esotericism drew heavily from the teachings of Pythagoras. The German humanist scholar Johannes Reuchlin (1455-1522) synthesized Pythagoreanism with Christian theology and Jewish Kabbalah, arguing that Kabbalah and Pythagoreanism were both inspired by the Mosaic tradition and that Pythagoras was therefore a Kabbalist. In his dialogue  De verbo mirifico  (1494), Reuchlin compared the Pythagorean tetractyls to the ineffable divine name YHWH, attributing to each of the four letters of the tetragrammaton a symbolic meaning in keeping with the mystical teachings of Pythagoras.

Heinrich Cornelius Agrippa's popular and influential three-volume treatise  De Occulta Philosophia  cites Pythagoras as a "religious magician" and indicates that Pythagoras' mystical numerology operates on a supercelestial level. Freemasons deliberately modeled their society on the community founded by Pythagoras at Crotona. Rosicrucianism used Pythagorean symbolism , as did Robert Fludd (1574-1637), who believed that his own musical writings had been inspired by Pythagoras. John Dee was heavily influenced by Pythagorean ideology, particularly the teaching that all things are made of numbers. Adam Weishaupt, the founder of the Illuminati, was a strong admirer of Pythagorasand, in his book  Pythagoras  (1787), he argued that society should be reformed to resemble more like the community of Pythagoras at Crotona. Wolfgang Amadeus Mozart incorporated Masonic and Pythagorean symbolism into his opera  The Magic Flute  . Sylvain Maréchal, in his 1799 six-volume biography  The Voyages of Pitthagoras  , declared that all revolutionaries in all time periods are the "heirs of Pythagoras".

In literature

Dante Alighieri was fascinated by Pythagorean numerology and based his descriptions of hell, purgatory and heaven on Pythagorean numbers. Dante wrote that Pythagoras saw Unity as good and plurality as evil , and in  Paradise  XV, 56–57, he declares: "five and six, if understood, radiate from unity". The number eleven and its multiples are found throughout the  Divine Comedy  , each book having thirty-three cantos, with the exception of  Inferno  , having thirty-four, the first of which serves as a general introduction. Dante describes the ninth and tenth bolgias in the eighth circle of hell as being twenty-two miles and eleven miles respectively, which corresponds to the fraction of227, which was Pythagoras' approximation of pi. Hell, Purgatory and Heaven are all described as circular and Dante compares the wonder of God's majesty to the mathematical puzzle of squaring the circle. The number three also features prominently: the  Divine Comedy  has three parts and Beatrice is associated with the number nine, which is equal to three times three.

Transcendentalists read the ancient  Lives of Pythagoras  as guides on how to live a model life. Henry David Thoreau was impacted by Thomas Taylor's translations of Iamblichus '  Life of Pythagoras and Stobeeus'  Pythagorean Sayings  and his views on nature may have been influenced by the Pythagorean idea of ​​images corresponding to archetypes. The Pythagorean teaching of  universal music  is a recurring theme throughout Thoreau  's magnum opus  Walden  .

The Brazilian philosopher Mário Ferreira dos Santos wrote more than one work on Pythagoreanism, from a very particular perspective and called himself "Pythagorean".

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Pythagoras was a Greek philosopher known for many things. Among his accomplishments in life was the founding of the religion known as Pythagoreanism. The works of Pythagoras continue to influence and impact math and, in particular, geometry, to a great deal even well into the modern era. A brief look at his life reveals he was both a historically important and controversial figure.

The Life of Pythagoras

Pythagoras was born in Samos, Greece, near Turkey around 570 B.C. However, there are no clear and accurate records of the life of Pythagoras that exist. What does exist are biographical sketches crafted after his death. Many of his followers shrouded his life in secrecy and there are more than a few outright myths surrounding his life. One of the most outrageous noted that he was the son of the god, Apollo.

Based on the limited sketches we have, his father was Mnesarchus, a gem engraver and merchant. Pythagoras is believed to have eventually left Samos when he was about 40 years old during the reign of Polycrates. Throughout the course of his life, Pythagoras was able to study from Greeks, Egyptians, and various people in Asia. His formal education was in areas related to arithmetic, geometry, morals, astronomy, and religion.

Pythagoras was a sort of sojourner having traveled quite a bit of the world for the purpose of increasing his knowledge. Considering the vast array of knowledge he amassed in his life as evidenced by his teachings, his time traveling yielded great results.

Works of Pythagoras

It was Aristotle who said the Pythagoreans, the followers of Pythagoras, were the true fathers of mathematics. He also noted it was they who believed the notion that all things in life related back to mathematics.

The Pythagorean theorem (a^2 + b^2 = c^2) is the masterwork attributed to Pythagoras. The theorem states that a right-angled triangle has an area on the side opposite the right angle (the hypotenuse) is equal to the complete sum of the squares of the other two sides.

Research shoes this theorem existed prior to Pythagoras and in cultures outside of Greece. However, it is Pythagoras who is credited with drawing the most attention to it and establishing it as a valid component to geometric thought. Ironically, the school of Pythagoras was one so rooted in secrecy, rumors exist that it may even have been one of his students who really came up with the theorem.

The very first mention of the connection between Pythagoras and his theorem are found in the writings of Cicero and Plutarch. These writings, however, did not appear until a full 500 years after the death of the Pythagoras. The origins of the development of the theorem may be forever lost, but Pythagoras does get all the credit surrounding it nonetheless.

The Religion of Pythagoreanism

Pythagoras established a strange metaphysical religion known as Pythagoreanism. The religion was influenced heavily by mathematics, astronomy, and even music. Eventually two different schools of thought emerged in this religion with one school being the learners and the other school being made up of the listeners. A key component to this religion and its two schools was the notion that mathematics was mystical, hence, appropriate as an influential basis for a religion.

Within this secretive religion was a symbol of the tetractys. The tetractys was a triangle figure made from four rows. The number 10 is arrived at when all the rows are added. The tetractys was an original invention of Pythagoras himself.

Last Years of Pythagoras

Later in his life, Pythagoras moved to Metapontum, Italy. He passed away around 495 B.C. According to legend, he did have a wife and children, though records about them are typically sketchy.

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Pythagoras Biography

Encyclopedia of World Biography on Pythagoras

Born on the island of Samos, Pythagoras was the son of Mnesarchus. He fled to southern Italy to escape the tyranny of Polycrates, who came to power about 538, and he is said to have traveled to Egypt and Babylon. He and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect. It is probable that the Pythagoreans took positions in the local government in order to lead men to the pure life which their teachings set forth. Eventually, however, a rival faction launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely annihilated. Pythagoras either had been banished...

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