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Pythagoras

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Pythagoras of Samos (Greek: Πυθαγόρα&sigmaf;; between 580 and 572 BC–between 500 BC and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician and scientist; however, careful scholarship in the past three decades has found no evidence of his contributions to mathematics or natural philosophy.[2] His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.[3]
He is best known for the Pythagorean theorem which bears his name. Known as "the father of numbers," Pythagoras made influential contributions to 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. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and demons."
He was the first man to call himself a philosopher, or lover of wisdom. Many of the accomplishments of Plato, Aristotle and Copernicus were based on the ideas of Pythagoras. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.
 
<h2>Life</h2>
Born on the island of Samos (a Greek island in the Eastern Aegean), off the coast of Asia Minor. He was born to Pythais (his mother, a native of Samos) and Mnesarchus (his father, a Phoenician merchant from Tyre). As a young man, he left his native city for Croton, Calabria, in Southern Italy, to escape the tyrannical government of Polycrates. According to Iamblichus, Thales, impressed with his abilities, advised Pythagoras to head to Memphis in Egypt and study with the priests there who were renowned for their wisdom. He also was discipled in the temples of Tyre and Byblos in Phoenicia. It may have been in Egypt where he learned some geometric principles which eventually inspired his formulation of the theorem that is now called by his name. This possible inspiration is presented as an example problem in the Berlin Papyrus.
Upon his migration from Samos to Croton, Calabria, Italy, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult.
Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself called Pythagoreans. Very strict rules of conduct governed this cultural center. He opened his school to male and female students alike. Those who joined the inner circle of Pythagoras's society called themselves the Mathematikoi. They lived at the school, owned no personal possessions and were required to assume a vegetarian diet. Other students who lived in neighboring areas were also permitted to attend Pythagoras's school. Known as Akousmatikoi, these students were permitted to eat meat and own personal belongings.
According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. 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.
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Bust of Pythagoras, Vatican</div>
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Flavius Josephus relates that, according to Hermippus of Smyrna, Pythagoras was familiar with and an admirer of Jewish customs and wisdom (De Pythagora, Contra Apionem I, 162/165). Hermippus is quoted as saying about Pythagoras: "In practicing and repeating these precepts he was imitating and appropriating the doctrines of Jews and Thracians. In fact, it is actually said that that great man introduced many points of Jewish law into his philosophy." (trans. H. St. J. Thackeray, The Loeb Classical Library, Cambridge (Mass.)-London)
Pythagoras is commonly given credit for discovering the Pythagorean theorem, a theorem in trigonometry that states that in a right-angled triangle the area of the square whose side is the hypotenuse (the side opposite the right angle), c, is equal to the sum of the areas of the squares of the other two sides, b and a, that is, a2+b2=c2.
The history of the Pythagorean theorem that bears his name is complex. There is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. The myth seems to have been carefully constructed by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. [4] The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch. There are many ancient references to the facts stated in the Pythagorean theorem; Egyptian and Chinese tablets and writings show that they knew the theorem.
Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato.
 
<h2>Pythagoreans</h2>
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The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon Pythagoras’ religious teachings and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism and vegetarianism, and devote themselves to the mathematics of nature.
Pythagoras's followers were commonly called "Pythagoreans." For the most part we remember them as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.
The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all. (The Secret Teachings of All Ages by Manly P. Hall).
In his biography of Pythagoras (written seven centuries after Pythagoras's time), Porphyry stated that this silence was "of no ordinary kind." The Pythagoreans were divided into an inner circle called the mathematikoi ("mathematicians") and an outer circle called the akousmatikoi ("listeners"). Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborate version of this knowledge, the akousmatikoi (were) those which had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition." According to Iamblichus, the akosmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil.
The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult. Instead they were taught laws of behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of Pythagoras and a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi.
Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the principle of the golden mean. In a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society.
The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos ("mystical speech") was hurled at Pythagoras even in ancient times to discredit him. The prohibition on beans could be linked to favism, which is relatively widespread around the Mediterranean.
The key here is that akousmata means "rules", so that the superstitious taboos primarily applied to the akousmatikoi, and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi (arguably the real preservers of the Pythagorean tradition). The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi, even to the extent of dispensing with certain rules and ritual practices. For the mathematikoi, being a Pythagorean was a question of innate quality and inner understanding.
There was also another way of dealing with the akousmata — by allegorizing them. We have a few examples of this, one being Aristotle's explanations of them: "'step not over a balance', i.e. be not covetous; 'poke not the fire with a sword', i.e. do not vex with sharp words a man swollen with anger, 'eat not heart', i.e. do not vex yourself with grief," etc. We have evidence for Pythagoreans allegorizing in this way at least as far back as the early fifth century BC. This suggests that the strange sayings were riddles for the initiated.
The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods.
Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics advocated various purificatory rites and practices as well as incubatory rites of descent into the underworld. Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks", or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (ugieia).
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Pythagoras, the man in the center with the book, teaching music, in The School of Athens by Raphael</div>
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Pythagoras was interested in music and the Pythagoreans were musicians as well as mathematicians. He wanted to improve the music of his day, which he believed was not harmonious enough and was too chaotic.
According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was one day as he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools. He discovered that it was because the anvils were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on." The Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the harmony of the spheres. He believed that since planets and the stars all moved in the universe according to mathematical equations that these mathematical equations could be translated into musical notes and thus produce a symphony.[5]
 

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<th colspan="2">Academic Genealogy</th>
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<th width="50%">Notable teachers</th>
<th width="50%">Notable students</th>
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<td style="TEXT-ALIGN: center" width="50%">Anaximander 
Pherecydes of Samos 
Hermodamas of Samos 
Thales 

</td>
<td style="TEXT-ALIGN: center" width="50%">Ameinias 
Bathyllus 
Brontinus, husband of Theano 
Calliphon 
Cercops 
Echecrates 
Empedocles 
Eurytus 
Hippasus 
Leon 
Lysis of Tarentum 
Milon, whose house was used as a Pythagorean meeting place 
Parmeniscus 
Petron 
Philolaus of Croton 
Theano, Pythagoras' daughter 
Xenophilus of Chaldice 
Zalmoxis, Pythagoras' slave 

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<h2>Religion and science</h2>
Pythagoras’ religious and scientific views were, in his opinion, inseparably interconnected. However, they are looked at separately in the 21st century. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were probably borrowed from Hinduism. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog.
 
<h2>Literary works</h2>
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. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints. Pythagoras has been quoted as saying, "No man is free who cannot command himself."
 
<h2>Other accomplishments</h2>
One of his major accomplishments was the discovery that music was based on proportional intervals of four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. He assigned roles for the numbers as follows: one was reason, two was opinion, four was justice, five was marriage because it was the sum of the first odd and the first even numbers (one was disregarded), seven was virgin because it neither factors or produces among the numbers one through ten. Odd numbers were masculine and even were feminine. He discovered the theory of mathematical proportions, constructed from three to five geometrical solids. One of his order also discovered irrational numbers, but the idea was unthinkable to Pythagoras, and he had this member executed[citation needed]. He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central “fire” that he never identified as the sun. He also believed that the moon was another planet that he called a “counter-Earth” – furthering his belief in the Limited-Unlimited and bringing the number of planets to ten.
 
<h2>Groups influenced by Pythagoras</h2>
 
<h3>Influence on Plato</h3>
Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". c) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.[6]
Plato's harmonics were clearly influenced by the work of Archytas, a genuine Pythagorean of the third generation, who made important contributions to geometry, reflected in Book VIII of Euclid's Elements.
 
<h3>Roman influence</h3>
In the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap.
 
<h3>Influence on esoteric groups</h3>
Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".
Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. The 8th-century Islamic alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.
 
<h2>See also</h2>

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<li>Hippasus </li>
<li>Pythagoreanism </li>
<li>Pythagorean comma </li>
<li>Pythagorean cup </li>
<li>Pythagorean theorem </li>
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<li>Sacred geometry </li>
<li>Heliopolis-Pythagoras connection </li>
<li>Lute of Pythagoras </li>
<li>Pythagoras tree </li>
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<h2>References</h2>
 
<h3>Primary sources</h3>
Only a few relevant source texts deal with Pythagoras and the Pythagoreans, most are available in different translations. Other texts usually build solely on information from these four books.
<ul>
<li>Diogenes Laertius, Vitae philosophorum VIII (Lives of Eminent Philosophers), c. 200 AD, which in turn reference the lost work Successions of Philosophers by Alexander Polyhistor) — Pythagoras, Translation by C.D. Yonge </li>
<li>Porphyry, Vita Pythagorae (Life of Pythagoras), c. 270 AD </li>
<li>Iamblichus, De Vita Pythagorica (On the Pythagorean Life), c. 300 AD </li>
<li>Apuleius also writes about Pythagoras in Apologia, including a story of him being taught by Babylonian disciples of Zoroaster, c. 150 AD </li>
<li>Hierocles of Alexandria, Golden Verses of Pythagoras, Concord Grove Pr., 1983 </li>
</ul>
 
<h3>Secondary sources</h3>
<ul>
<li>Burkert, Walter. Lore and Science in Ancient Pythagoreanism, Harvard University Press, June 1, 1972. ISBN 0-674-53918-4 </li>
<li>Guthrie W. K. 1979. A History of Geek Philosophy - Earlier Presocratics and the Pythagoreans, Cambridge University Press ISBN 0-521-29420-7 </li>
<li>Kingsley, P. 1995. Ancient Philosophy, Mystery, and Magic - Empedocles and the Pythagorean Tradition, Oxford University Press </li>
<li>Hermann, Arnold (2005). To Think Like God: Pythagoras and Parmenides-The Origins of Philosophy. Parmenides Publishing. ISBN 978-1-930972-00-1 </li>
<li>O'Meara, Dominic J. Pythagoras Revived, Clarendon Press, Oxford, 1989. ISBN 0-19-823913-0 (paperback), ISBN 0-19-824485-1 (hardcover) </li>
<li>M.F. Burnyeat. The Truth about Pythagoras, London Review of Books, 22 February 2007. </li>
</ul>
 
<h2>Notes</h2>
<ol class="references">
<li id="_note-0">^ According to Diogenes Laertius, ”Pythagoras was the first person who invented the term philosophy, and called himself a philosopher” (Φιλοσοφίαν δὲ πρῶτο&sigmaf; ὠνόμασε Πυθαγόρα&sigmaf; καὶ ἑαυτὸν φιλόσοφον: Lives of Philosophers 1.12 (Greek). </li>
<li id="_note-1">^ Walter Burkert seminal work "Lore and Science in Ancient Pythagoreanism" (see sources) sheds considerable doubt on the widely held traditions of late Classical Greece, accepted without scrutiny until the beginning of the 20th century, that Pythagoras made substantial contributions to mathematics and science. </li>
<li id="_note-2">^ Christoph Riedweg, Pythagoras: His Life, Teaching and Influence, trans. Steven Rendall (Cornell UP, 2005), pp. 5-6, 59, 73. </li>
<li id="_note-3">^ From Christoph Reidwig , Pythagoras, His Life, Teaching and Influence, Cornell: Cornell University Press, 2005: "Had Pythagoras and his teachings not been since the early Academy overwritten with Plato’s philosophy, and had this ‘palimpsest’ not in the course of the Roman Empire achieved unchallenged authority among Platonists, it would be scarcely conceivable that scholars from the Middle Ages and modernity down to the present would have found the Presocratic charismatic from Samos so fascinating. In fact, as a rule it was the image of Pythagoras elaborated by Neopythagoreans and Neoplatonists that determined the idea of what was Pythagorean over the centuries." </li>
<li id="_note-4">^ Christoph Reidwig , Pythagoras, His Life, Teaching and Influence, Cornell: Cornell University Press, 2005 . </li>
<li id="_note-5">^ R.M. Hare, Plato in C.C.W. Taylor, R.M. Hare and Jonathan Barnes, Greek Philosophers, Socrates, Plato, and Aristotle, Oxford: Oxford University Press, 1999 (1982), 103-189, here 117-9. </li>
</ol>
 
<h2>External links</h2>
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<li>
<div style="MARGIN-LEFT: 60px">[http://biosophy.org Biosophy.org]</div>
</li>
<li>
<div style="MARGIN-LEFT: 60px">[http://biophilosophy.org Biophilosophy.org]</div>
</li>
<li>
<div style="MARGIN-LEFT: 60px">“<a class="external text" title="http://www.lrb.co.uk/v29/n04/burn02_.html" rel="nofollow" href="http://www.lrb.co.uk/v29/n04/burn02_.html">Other Lives</a>” by M.F. Burnyeat, an article about current Pythagoras scholarship </div>
</li>
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<li><a class="external text" title="http://users.ucom.net/~vegan" rel="nofollow" href="http://users.ucom.net/~vegan">Pythagoreanism Web Site</a> </li>
<li><a class="external text" title="http://www-history.mcs.st-andrews.ac.uk/References/Pythagoras.html" rel="nofollow" href="http://www-history.mcs.st-andrews.ac.uk/References/Pythagoras.html">References for Pythagoras</a> </li>
<li><a class="external text" title="http://www.utm.edu/research/iep/p/pythagor.htm" rel="nofollow" href="http://www.utm.edu/research/iep/p/pythagor.htm">Pythagoras, Internet Encyclopedia of Philosophy</a> </li>
<li><a class="external text" title="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pythagoras.html" rel="nofollow" href="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pythagoras.html">Pythagoras of Samos, The MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland</a> </li>
<li><a class="external text" title="http://history.hanover.edu/texts/presoc/pythagor.htm" rel="nofollow" href="http://history.hanover.edu/texts/presoc/pythagor.htm">Pythagoras and the Pythagoreans, Fragments and Commentary, Arthur Fairbanks Hanover Historical Texts Project, Hanover College Department of History</a> </li>
<li><a class="external text" title="http://www.completepythagoras.net/" rel="nofollow" href="http://www.completepythagoras.net/">The Complete Pythagoras</a>, an on-line book containing all survived biographies and Pythagorean fragments. </li>
<li><a class="external text" title="http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html" rel="nofollow" href="http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html">Pythagoras and the Pythagoreans, Department of Mathematics, Texas A&M University</a> </li>
<li><a class="external text" title="http://www.newadvent.org/cathen/12587b.htm" rel="nofollow" href="http://www.newadvent.org/cathen/12587b.htm">Pythagoras and Pythagoreanism, The Catholic Encyclopedia</a> </li>
<li><a class="external text" title="http://cyberspacei.com/jesusi/inlight/philosophy/western/Pythagoreanism.htm" rel="nofollow" href="http://cyberspacei.com/jesusi/inlight/philosophy/western/Pythagoreanism.htm">Pythagoreanism Web Article</a> </li>
<li><a class="external text" title="http://groups.yahoo.com/group/Pythagorean-L" rel="nofollow" href="http://groups.yahoo.com/group/Pythagorean-L">Pythagoreanism Discussion Group</a> </li>
<li><a class="external text" title="http://www.geocities.com/go_darkness/god-pythagorean-pentacle.html" rel="nofollow" href="http://www.geocities.com/go_darkness/god-pythagorean-pentacle.html">Occult conception of Pythagoreanism</a> </li>
<li><a class="external text" title="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html" rel="nofollow" href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html">Pythagoras of Samos</a> </li>
<li><a class="external text" title="http://plato.stanford.edu/entries/pythagoras/" rel="nofollow" href="http://plato.stanford.edu/entries/pythagoras/">Stanford Encyclopedia of Philosophy entry</a> </li>
<li><a class="external text" title="http://www.organelle.org/organelle/tetra/tetraktys.html" rel="nofollow" href="http://www.organelle.org/organelle/tetra/tetraktys.html">Tetraktys</a> </li>
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