Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. 

We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance. 

Pythagoras's father was Mnesarchus ([12] and [13]), while his mother was Pythais [8] and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and there is a story ([12] and [13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father. 

Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer. There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras. 

The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Thales's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views. 

In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry ([12] and [13]) Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission. 

It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt. Porphyry in [12] and [13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander. 

In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras (see [8]):- 

... 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... 

In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide or as the result of an accident. The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return. This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there. 

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus [8] writes in the third century AD that:- 

... 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... 

Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier). Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods:- 

... 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. 

This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:- 

... 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. 

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heel of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:- 

(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. 

Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians. 

Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems. 

Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. As Brumbaugh writes in [3]:- 

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. 

In fact today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc. There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house. 

Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:- 

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. 

This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. In fact Pythagoras made remarkable contributions to the mathematical theory of music. He was a fine musician, playing the lyre, and he used music as a means to help those who were ill. 

Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [3]:- 

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. 

Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [7]):- 

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. 

Again Proclus, writing of geometry, said:- 

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. 

Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. 

(i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles. 

(ii) 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. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square. 

(iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means. 

(iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. 

(v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two. 

(vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. 

Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry and astronomy described above, he held [2]:- 

... 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 [3] their practical ethics are also described:- 

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

Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris and there is certainly some suggestions that Pythagoras became involved in the dispute. Then in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society. Iamblichus in [8] quotes one version of events:- 

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. 

This seems accepted by most but Iamblichus himself does not accept this version and argues that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly the Pythagorean Society thrived for many years after this and spread from Croton to many other Italian cities. Gorman [6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around 100 at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after 480 BC. 

The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt into a number of factions. In 460 BC the Society [2]:- 

... 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. 


Article by: J J O'Connor and E F Robertson

Click on this link to see a list of the Glossary entries for this page 
List of References (27 books/articles)

Some Quotations (7)

A Poster of Pythagoras

Mathematicians born in the same country

Some pages from publications

Pythagorus in competition with Boethius in Margaista Philosophica (1504)


Cross-references to History Topics

Greek Astronomy

Perfect numbers

Prime numbers

The Indian Sulbasutras

The history of cartography

Pythagoras's theorem in Babylonian mathematics

The Golden ratio

Mathematics and Architecture

Infinity

Christianity and the Mathematical Sciences - the Heliocentric Hypothesis

A history of time: Classical time

Mathematics and the physical world

Overview of Chinese mathematics

The Ten Mathematical Classics

Nine Chapters on the Mathematical Art


Other references in MacTutor

Pythagoras's theorem 
Chronology: 30000BC to 500BC 


Theosophy Online (Pythagoras and his School)

Internet Encyclopedia of Philosophy

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The Pythagoreans 

Because of the Persian domination, philosophy moves from Ionia to the coasts of Magna Graeca, southern Italy and Sicily -- to form what Aristotle calls the Italian school. 

Pythagoras is a highly obscure figure. He apparently came from the island of Samos, settled in Croton (Magna Graeca). Several journeys are attributed to him, including one to Persia where he is said to have met the Magus Zaratas [= Zoroaster/Zarathustra]. 

He is further associated with the Orphics and the revival of the worship of Dionysus. 

[CE] Indeed, it should be emphasized that the insights gained in this "philosophy" qua "theory" of the physical order are not designed so much for manipulating the environment as for saving one's soul. 

The Pythagoreans settled in a number of cities on the Italian mainland and Sicily, and from thence to Greece proper. 

They formed a league or a sect. They did not eat meat or beans; the could not wear clothes made of wool; the could not pick up anything that had fallen, stir a fire with iron, etc. 

The sect was divided between the akousmatikoi (hearers) and the mathematikoi (learned). The local democrats frowned on this aristocracy, if not on the sect as such, and many were killed. 

The Pythagoreans formed the first "school" (from schole, "leisure"), defined as a way of life. And, perhaps because of their situation as foreigners, they understood themselves as following the spectator's way of life (in contrast with those who buy and sell, and those who run in the stadium). This is the bios theoretikos, the contemplative or theoretic life. 

The main difficulty to overcome: the body and its necessities which subdue man. It is necessary to free oneself from these. The body is a tomb -- one must triumph over it, but not lose it. To so so requires that one attain the state of enthusiasm (en- theos). (This seems to suggest a connection with the Orphics and their rites, founded on mania, "orgy" -- though the Pythagoreans apparently moderated this somewhat.) 

In this way, one attains a self-sufficient, theoretic life -- a life not tied to the necessities of the body, a divine life. 

Such a man is a wise man, a sophos. 

(The term philosophia, "love of wisdom," is first used in Pythagorean circles.) 

MATHEMATICS 

Greek mathematics began in the Milesian school (cf. Thales, Anaximander), inheriting the knowledge of Egypt and Asia Minor (Babylonia). The Pythagoreans transform it into an autonomous and rigorous science. 

In mathematics, the Pythagoreans discovered a type of entity -- numbers and geometric figures -- which is not corporeal, but which seems to have non-arbitrary features of its own (in contrast with the arbitrary, changing whim of fancy, imagination, dream). Marias suggests that this discovery perhaps leads to the further claim that Being is not simply corporeal, material being -- in which case, we would now have a problem. A development of the concept of being is called for[?]. 

In any case, for the Pythagoreans, Being means the being of mathematical objects: 
Numbers and figures are the essence of things; 

Entities which exist are imitations of mathematical forms 

[anticipates Plato's alleged theory of forms] 

Pythagorean mathematics is not an operative technique: it is the discovery and construction of new entities, which are changeless, eternal -- in contrast with things which are variable and transitory. 

Aristotle gives this account - and critique - of the Pythagoreans: 
Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being - more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity - and similarly almost all other things being numerically expressible); since again they saw that the attributes and the ratios of the musical scales were expressible in numbers; since, then, all other things seemed in their whole nature to be modelled after numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangment of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth the 'counter-earth.' (Metaphysics A5, 985b23) 

PreParmenidean Pythagoreanism 

Beyond the insight first articulated by Pythagoras -- that the universe, on analogy with the lyre, is built out of numbers and a harmony expressible in numbers -- the dualism of Limit and Unlimited is expanded: 

Limit 	Unlimited 	
odd 	even 	
one 	plurality 	
right 	left 	
male 	female 	
resting 	moving 	
straight 	curved 	
light 	darkness 	
good 	bad 	
square 	oblong 	

(see Jones) 

Aristotle further reports that the Pythagoreans -- evidently in contrast with all other Greeks -- regarded the unit to have spatial magnitude (thus confusing "the point of geometry with the unit of arithmetic.") 

It is against such an assumption that Zeno's paradoxes have their greatest force. 

These unit-points functioned also as the basis of physical matter: they were regarded in fact as a primitive form of atom. Concrete objects are literally composed of aggregations of unit-point-atoms 

Hence Aristotle: 
Further, how are we to combine the belief that the modifications of number, and number itself, are causes of what exists and happens in the heavens both from the beginning and now, and that there is no other number than this number out of which the world is composed? (Met. A8, 990al8) 
But the Pythagoreans, because they saw many attributes of numbers belonging to sensible bodies, supposed real things to be numbers - not separable numbers, however, but numbers of which real things consist. (Met.. N3, lO9Oa20) 

While this may seem bizarre to us, Plato seems to have been the first Greek to have consciouslv thought that anything could exist otherwise than in space, and he was followed in this respect by Aristotle. 

COSMOGONY 

"When the one had been constructed, either out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by limit." 

Cf. modern accounts of the "Big Bang," etc. 

This is apparently a biologically-based conception, one which 
(a) further recalls the basic similarity between the mythopoetic and philosophical/scientific structures of explanation, and 

(b) jibes with placing the male principle under limit and the female under unlimited in the table of opposites: 

"The early Pythagoreans may well, therefore, have initiated the cosmogonical process by representing the male principle of Limit as somehow implanting in the midst of the surrounding Unlimited the seed which, by progressive growth, was to develop into the visible universe." (K&R 251) 

In this process, the void exists and functions to differentiate things: 
Apparently the first unit, like other living things, began at once to grow, and somehow as the result of its growth burst asunder into two; whereupon the void, fulfilling its proper function, keeps the two units apart, and thus, owing to the confusion of the units of arithmetic with the points of geometry, brings into existence not only the number 2 but also the line. So the process is begun which, continuing indefinitely, is to result in the visible universe as we know it. 

Notice here as well that the leap from the lyre to the conception of the entire universe as number and the harmony of the spheres rests on the Milesian tendency to draw analogies between the human and the natural -- a tendency in keeping with the attempt to uncover an original unity which accounts for both the human and the physical orders. 

Indeed, as already noted above, Aristotle chastizes the Pythagoreans in regard to their theory of a counter-earth (so as to complete the nine bodies in the heavens with a perfect 10th) this way "In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accomodate them to certain theories and opinions of their own." (see K&R, pp. 257ff.) 

Student Comments on the Pythagoreans 

The pythagoreans "... infused all nature with mathematical concepts." Mathematics became abstract and deductive. "The early Pythagorean community was a mystical, religious group; researches into the science of mathematics were part of a larger philosophy." An idea that was very important to the Pythagoreans was an idea of a natural harmony in the universe. Janet L. 

I found that Alioto had an interesting quote about the Pythagoreans: "In other words, with Thales, mathematics became deductive and therfore abstract. The Pythagoreans extended this process of abstraction and in turn infused all of nature with mathematical concepts. It seems that they were the first to stress the idea of number and geometry underlying diverse natural phenomena. The result, adapted and enshrined in Plato's later philosophy along with an ethical, transcendental corollary, was the important recognition that numbers are abstractions, mental concepts, suggested by material things but independent of them. For the early Pythagoreans, however, the physical world was actually constructed from numbers." (36). This, I believe, was a major step for philosophy because of the use of abstraction to relate to reality.  - Robert-
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The Father of Numbers
Pythagoras (582 BC  496 BC, Greek: ???a???a?) was an Ionian mathematician and philosopher, known best for formulating the Pythagorean theorem. Known as "the father of numbers", he made influential contributions to Greek philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. Pythagoras and his students believed that everything was related to mathematics, and felt that everything could be predicted and measured in rhythmic cycles. 

Pythagoras was born on the island of Samos, off the coast of Asia Minor. As a young man he left his native city for Croton in Southern Italy to escape the tyrannical government of Polycrates. Many writers credit him with visits to the sages of Egypt and of Babylon before going west; but such visits feature stereotypically in the biographies of Greek wise men, and may express legend rather than fact. 

In any case, Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and forming an elite circle of followers around himself. Very strict rules of conduct governed this cultural center. He opened his school to men and women students alike. 

According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. We may infer from this that participants required some degree of wealth and leisure to join the inner circle. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory. 
Doctrine and Teaching
Pythagoras as portrayed on Roman coins from SamosPythagoras has the reputation of having taught a doctrine of reincarnation. His other teachings appear framed in pithy sayings, or sumbola, often in question-and-answer format. Some of these teachings took a simple form: "What is wisest?" "Number"; "What is truest?" "Most men are bad." Others were more cryptic: "What is the Delphic oracle?" "The tetraktys, in which the Sirens sing." Other sumbola related to sexual, dietary and other taboos, including the proper way to stir a fire or place one's shoes before going to sleep. The Idea that Pythagoras forbade his disciples to eat beans has been questioned by some recent writers, who understand the phrase, "Abstain from beans" (kyamon apechete), to refer to a measure of practical prudence, and not to a dietary restriction. Beans, black and white, were, according to this interpretation, the means of voting in Magna Grcia (lower east Italy), and "Abstain from beans" would, therefore, mean merely "Avoid politics". 

Later Pythagoreans divided into two camps. The akousmatikoi held to these sumbola as the whole of their master's teaching. The mathematikoi added research into geometry, musical theory, astronomy, mechanics and other sciences. The mathematikoi held that the akousmatikoi knew only the outer form of the doctrine, but they themselves claimed to know the inner as well. The akousmatikoi accused the mathematikoi of adding extraneous material to the original teaching. Even today, scholars cannot definitively identify the "real" Pythagoreans. 

The subsequent biographical traditions of Pythagoras reflect this split: they portray him alternately as a down-to-earth political reformer, a pioneering scientist, or a wild shaman-figure. The truth no doubt lies somewhere in between. 
Literary works
No texts by Pythagoras survive, although forgeries under his name  a few of which remain extant  did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. And ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said")  emphasizing the essentially oral nature of his teaching.

Scientific Contributions
Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training. Evidence certainly suggests that the Egyptians had advanced further than the Greeks of their time in mathematics and astronomy, and many scholars now believe that the Egyptians used the Pythagorean theorem in some of their architectural projects before the 6th century BC. 

In astronomy, the Pythagoreans were well aware of the periodic numerical relations of the planets, moon, and sun. The celestial spheres of the planets were thought to produce a harmony called the music of the spheres. 
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These ideas, as well as the ideas of the perfect solids, would later be used by Johannes Kepler in his attempt to formulate a model of the solar system in his work The Harmony of the Worlds. Pythagoreans also believed that the earth itself was in motion. 

It is sometimes difficult to determine which ideas Pythagoras taught originally, as opposed to the ideas his followers later added. While he clearly attached great importance to geometry, classical Greek writers tended to cite Thales as the great pioneer of this science rather than Pythagoras. The later tradition of Pythagoras as the inventor of mathematics stems largely from the Roman period. 

Whether or not we attribute the Pythagorean theorem to Pythagoras, it seems fairly certain that he had the pioneering insight into the numerical ratios which determine the musical scale, since this plays a key role in many other areas of the Pythagorean tradition and since no evidence remains of earlier Greek or Egyptian musical theories. Another important discovery of this school -- which upset Greek mathematics, as well as the Pythagoreans' own belief that whole numbers and their ratios could account for geometrical properties -- was the incommensurability of the diagonal of a square with its side. This result showed the existence of irrational numbers. 

The influence of Pythagoras has transcended the field of mathematics, and the Hippocratic Oath  with its central commitment to First do no harm  has its roots in the oath of the Pythagorean Brotherhood.