{{Short description|Philosophical system based on the teachings of Pythagoras}}
[[File:Cropped image of Pythagoras from Raphael's School of Athens.jpg|thumb|upright=1.3|In [[Raphael]]'s fresco ''[[The School of Athens]]'', Pythagoras is shown writing in a book as a young man presents him with a tablet showing a [[diagram]]matic representation of [[music theory]] on a [[lyre]] above a drawing of the sacred ''[[tetractys]]''.]]
'''Pythagoreanism''' originated in the 6th century BC, based on and around the teachings and beliefs held by [[Pythagoras]] and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the [[Ancient Greece|ancient Greek]] colony of [[Crotone|Croton]], in modern [[Calabria]] (Italy) circa 530 BC. Early Pythagorean communities spread throughout [[Magna Graecia]].
Already during Pythagoras's life it is likely that the distinction between the ''akousmatikoi'' ("those who listen"), who are conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the ''mathematikoi'' ("those who learn") existed. The ancient biographers of Pythagoras, [[Iamblichus]] ({{circa|245|AD 325}}) and his master [[Porphyry (philosopher)|Porphyry]] ({{circa|234|AD 305}}) seem to make the distinction between the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the [[Greco-Roman mysteries|mystery school]]s of antiquity, the adherents, ''akousmatikoi'', following initiation, became ''mathematikoi''. It is wrong to say that the Pythagoreans were superseded by the [[Cynicism (philosophy)|Cynics]] in the 4th century BC, but it seems to be a distinctive mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community. The Greek philosophical traditions subsequently became more diverse. The [[Platonic Academy]] was arguably a Pythagorean [[Cenobitic monasticism|cenobitic]] institution, outside the city walls of Athens in the 4th century BC; a sacred grove dedicated to Athena, and [[Academus|Hecademos]] (Academos). As contemporaries seem to have believed, the academy may have existed since the Bronze Age, even prior to the time of the [[Trojan War]]. Yet according to Plutarch it was the Athenian strategos (general) Kimon ({{circa|510|450 BC}}) who converted this, "waterless and arid spot into a well-watered grove, which he provided with clear running-tracks and shady walks". [[Plato]] lived almost a hundred years later, circa 427 to 348 BC. On the other hand, it seems likely that this was a part of the rebuilding of Athens led by Kimon and [[Themistocles]], following the [[Achaemenid destruction of Athens]] in 480–479 BC during the [[Second Persian invasion of Greece|war with Persia]]. Kimon is at least associated with the building of the southern [[Themistoclean Wall|Wall of Themistocles]], the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos.
Following political instability in [[Magna Graecia]], some Pythagorean philosophers moved to mainland Greece while others regrouped in [[Rhegium]]. By about {{nobr|400 BC}} the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on [[Plato]] and through him,[{{cite book |last=George |first=Calian Florin |year=2021 |chapter=Numbers, ontologically speaking: Plato on numerosity |title=Numbers and Numeracy in the Greek Polis |publisher=Brill |page=219 ff }}] on all of [[Western philosophy]]. Many of the surviving sources on Pythagoras originate with [[Aristotle]] and the philosophers of the [[Peripatetic school]].
As a philosophic tradition, Pythagoreanism was revived in the {{nobr|1st century BC,}} giving rise to [[Neopythagoreanism]]. The worship of Pythagoras continued in Italy and as a [[Intentional community|religious community]] Pythagoreans appear to have survived as part of, or deeply influenced, the [[Bacchic|Bacchic cults]] and [[Orphism (religion)|Orphism]].
==History==
[[File:Plimpton 322.jpg|thumb|The [[Plimpton 322|Plimpton 322 tablet]] records [[Pythagorean triples]] from Babylonian times.[{{cite book |url=https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA36 |first=Otto |last=Neugebauer |year=1969 |orig-year=1957 |title=The Exact Sciences in Antiquity |page=36 |edition=reprint of 2nd |publisher=Courier Dover Publications |isbn=0-486-22332-9 }}]]]
[[File:Pythagorean theorem - Ani.gif|thumb|Animation demonstrating the smallest whole number Pythagorean triple, {{nobr| 3² + 4² {{=}} 5² }}]]
[[Image:Kapitolinischer Pythagoras adjusted.jpg|thumb|Bust of [[Pythagoras]], [[Musei Capitolini]], [[Rome]].]]
[[Pythagoras]] was already well known in ancient times for his supposed mathematical achievement of the [[Pythagorean theorem]].{{sfn|Riedweg|2008|p=26}} Pythagoras had been credited with discovering that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered [[Interval (music)|music intervals]] also credit him as the inventor of the [[monochord]], a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals.{{sfn|Riedweg|2008|p=27}}
Much of the surviving sources on Pythagoras originated with [[Aristotle]] and the philosophers of the [[Peripatetic school]], which founded historiographical academic traditions such as [[biography]], [[doxography]] and the [[history of science]]. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoras's teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as [[Anaximander]], Andron of Ephesus, [[Heraclides Ponticus|Heraclides]] and [[Neanthes of Cyzicus|Neanthes]] had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline. [[Neopythagoreanism|Neopythagorean]] philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy.{{sfn|Zhmud|2012|p=29}}
The earliest surviving ancient source on Pythagoras and his followers is a [[satire]] by [[Xenophanes]], on the Pythagorean beliefs on the transmigration of souls.{{sfn|Zhmud|2012|p=30}} Xenophanes wrote of Pythagoras that:
{{blockquote|Once they say that he was passing by when a puppy was being whipped,
And he took pity and said:
"Stop! Do not beat it! For it is the soul of a friend
That I recognised when I heard it giving tongue."{{sfn|Zhmud|2012|p=30}} }}
In a surviving fragment from [[Heraclitus]], Pythagoras and his followers are described as follows:
{{blockquote|
Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selecting of these writings made for himself a wisdom or made a wisdom of his own: a polymathy, an imposture.{{sfn|Zhmud|2012|p=33}}
}}
Two other surviving fragments of ancient sources on Pythagoras are by [[Ion of Chios]] and [[Empedocles]]. Both were born in the 490s, after Pythagoras's death. By that time, he was known as a sage and his fame had spread throughout Greece.{{sfn|Zhmud|2012|p=38}} According to Ion, Pythagoras was:
{{blockquote|
... distinguished for his manly virtue and modesty, even in death has a life which is pleasing to his soul, if Pythagoras the wise truly achieved knowledge and understanding beyond that of all men.{{sfn|Zhmud|2012|p=38}}
}}
Empedocles described Pythagoras as "a man of surpassing knowledge, master especially of all kinds of wise works, who had acquired the upmost wealth of understanding".{{sfn|Zhmud|2012|p=39}} In the 4th century BC the [[Sophist]] [[Alcidamas]] wrote that Pythagoras was widely honored by Italians.{{sfn|Zhmud|2012|p=44}}
Today scholars typically distinguish two periods of Pythagoreanism: early-Pythagoreanism, from the 6th until the 5th century BC, and late-Pythagoreanism, from the 4th until the 3rd century BC.{{sfn|Ballif|Moran|2005|p=315}} The [[Spartan]] colony of [[Taranto]] in Italy became the home for many practitioners of Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in [[Crotone]] and [[Metaponto]], both of which were [[Achaeans (tribe)|Achaean]] colonies.{{sfn|Pomeroy|2013|p=xvi}} Early-Pythagorean sects lived in Croton and throughout [[Magna Graecia]]. They espoused to a rigorous life of the intellect and strict rules on diet, clothing and behavior. Their burial rites were tied to their belief in the immortality of the soul.{{sfn|Ballif|Moran|2005|p=315}}
Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five-year initiation period of listening to the teachings (''akousmata'') in silence. Initiates could through a test become members of the inner circle. However, Pythagoreans could also leave the community if they wished.[{{Cite book|title=Classical Rhetorics and Rhetoricians: Critical Studies and Sources|author1=Michelle Ballif |author2=Michael G. Moran |publisher=Greenwood Publishing Group|year=2005|isbn=978-0-313-32178-8|page=316}}] [[Iamblichus]] listed 235 Pythagoreans by name, among them 17 women whom he described as the "most famous" women practitioners of Pythagoreanism. It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic tradition that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras was known as the site of mysteries.{{sfn|Pomeroy|2013|p=1}}
Pythagoras had been born on the island of [[Samos]] at around 570 BC and left his homeland at around 530 BC in opposition to the policies of [[Polycrates]]. Before settling in Croton, Pythagoras had traveled throughout [[Ancient Egypt|Egypt]] and [[Babylonia]]. In Croton, Pythagoras established the first Pythagorean community, described as a secret society, and attained political influence. In the early 5th century BC Croton acquired great military and economic importance. Pythagoras emphasised moderation, piety, respect for elders and of the state, and advocated a [[monogamous]] family structure. The Croton Council appointed him to official positions. Among others Pythagoras was in charge of education in the city. His influence as political reformer reputedly extended to other Greek colonies in southern Italy and in Sicily. Pythagoras died shortly after an arson attack on the Pythagorean meeting place in Croton.[{{harvnb|McKirahan|2011|p=79}}.]
The anti-Pythagorean attacks in {{Circa|508 BC}} were headed by [[Cylon of Croton]].[{{cite book |last1=Willers |first1=Michael |title=Algebra: The x and y of Everyday Math |year=2009 |publisher=Quid Publishing |isbn=978-1-4351-1400-5 |page=43 |edition=2009}}] Pythagoras escaped to Metapontium. After these initial attacks and the death of Pythagoras, Pythagorean communities in Croton and elsewhere continued to flourish. At around 450 BC attacks on Pythagorean communities were carried out across [[Magna Graecia]]. In Croton, a house where Pythagoreans gathered was set on fire and all but two of the Pythagorean philosophers burned alive. Pythagorean meeting places in other cities were also attacked and philosophic leaders killed. These attacks occurred in the context of widespread violence and destruction in Magna Graecia. Following the political instability in the region, some Pythagorean philosophers fled to mainland Greece while others regrouped in [[Rhegium]]. By about 400 BC the majority of Pythagorean philosophers had left Italy. [[Archytas]] remained in Italy and ancient sources record that he was visited there by young [[Plato]] in the early 4th century BC. The Pythagorean schools and societies died out from the 4th century BC. Pythagorean philosophers continued to practice, albeit no organised communities were established.[{{cite book|url=https://archive.org/details/philolaosdespyt01philgoog |quote=Pythagoras Lehren nebst den Bruchstücken seines Werkes. |page=[https://archive.org/details/philolaosdespyt01philgoog/page/n19 14]|year=1819|title=Philolaos des Pythagoreers Lehren nebst den Bruchstücken seines Werkes|publisher=In der Vossischen Buchhandlung |author=August Böckh}}] According to [[Cicero]] ([[De Oratore|de Orat.]] III 34.139), Philolaus was teacher of [[Archytas]].[{{harvnb|Huffman|2024}}.] According to the [[Neoplatonist]] philosopher [[Iamblichus]], [[Archytas]] in turn became the head of the Pythagorean school about a century after the Pythagoras's death.{{sfn|Ball|1908|p=28}} Philolaus, [[Eurytus (Pythagorean)|Eurytus]] and [[Xenophilus (philosopher)|Xenophilus]] are identified by [[Aristoxenus]] as the teachers of the last generation of Pythagoreans.{{sfn|Huffman|2024}}
== Philosophic traditions ==
Following Pythagoras's death, disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism in [[Italy]]: ''akousmatikoi'' and ''mathēmatikoi''. The ''mathēmatikoi'' recognised the ''akousmatikoi'' as fellow Pythagoreans, but because the ''mathēmatikoi'' allegedly followed the teachings of [[Hippasus]], the ''akousmatikoi'' philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism.{{sfn|McKirahan|2011|pp=88–89}}
The ''akousmatikoi'' were superseded in the 4th century BC as a significant [[mendicant]] school of philosophy by the [[Cynicism (philosophy)|Cynics]]. ''Mathēmatikoi'' philosophers were in the 4th century BC absorbed into the [[Platonic Academy|Platonic school]] of [[Speusippus]], [[Xenocrates]] and [[Polemon (scholarch)|Polemon]]. As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to [[Neopythagoreanism]].{{sfn|Kahn|2001|p=72}} The worship of Pythagoras continued in Italy in the two intervening centuries. As a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the [[Bacchic|Bacchic cults]] and [[Orphism (religion)|Orphism]].{{sfn|Kahn|2001|pp=73–74}}
=== ''Akousmatikoi'' ===
[[Image:Bronnikov gimnpifagoreizev.jpg|thumb|''Pythagoreans celebrate sunrise'', 1869 painting by [[Fyodor Bronnikov]]. ]]
The ''akousmatikoi'' believed that humans had to act in appropriate ways. The ''Akousmata'' (translated as "oral saying") was the collection of all the sayings of Pythagoras as divine dogma. The tradition of the ''akousmatikoi'' resisted any reinterpretation or philosophical evolution of Pythagoras's teachings. Individuals who strictly followed most ''akousmata'' were regarded as wise. The ''akousmatikoi'' philosophers refused to recognise that the continuous development of mathematical and scientific research conducted by the ''mathēmatikoi'' was in line with Pythagoras's intention. Until the demise of Pythagoreanism in the 4th century BC, the ''akousmatikoi'' continued to engage in a pious life by practicing silence, dressing simply and avoiding meat, for the purpose of attaining a privileged [[afterlife]]. The ''akousmatikoi'' engaged deeply in questions of Pythagoras's moral teachings, concerning matters such as [[harmony]], [[justice]],[{{harvnb|McKirahan|2011|p=89}}] ritual purity, and moral behavior.[{{harvnb|McKirahan|2011|p=90}}]
=== ''Mathēmatikoi'' ===
[[File:Archytas curve.png|thumb|The Archytas curve]]
The ''mathēmatikoi'' acknowledged the religious underpinning of Pythagoreanism and engaged in ''mathēma'' (translated as "learning" or "studying") as part of their practice. While their scientific pursuits were largely mathematical, they also promoted other fields of scientific study in which Pythagoras had engaged during his lifetime. A sectarianism developed between the dogmatic ''akousmatikoi'' and the ''mathēmatikoi'', who in their intellectual activism became regarded as increasingly progressive. This tension persisted until the 4th century BC, when the philosopher [[Archytas]] engaged in advanced mathematics as part of his devotion to Pythagoras's teachings.[{{harvnb|Vamvacas|2009|p=65}}.]
Early-Pythagorean philosophers such as [[Philolaus]] and [[Archytas]] held the conviction that mathematics could help in addressing important philosophical problems.[{{harvnb|Long|1999|p=84}}.] In Pythagoreanism numbers became related to intangible concepts. The ''one'' was related to the intellect and being, the ''two'' to thought, the number ''four'' was related to justice because 2 * 2 = 4 and equally even. A dominant symbolism was awarded to the number ''three'', Pythagoreans believed that the whole world and all things in it are summed up in this number, because end, middle and beginning give the number of the whole. The [[Trichotomy (philosophy)|triad]] had for Pythagoreans an ethical dimension, as the goodness of each person was believed to be threefold: prudence, drive and good fortune.[{{harvnb|Vamvacas|2009|p=72}}.]
Pythagoreans thought numbers existed "outside of [human] minds" and separate from the world.[{{Cite book |last=Singer |first=Charles |title=A Short History of Science to the Nineteenth Century |publisher=[[Oxford University Press]] |year=1949 |isbn=1-306-39093-1 |location=[[Oxford]] |pages=18–19 |oclc=869524436}}] They had many [[Mysticism|mystical]] and [[Magic (supernatural)|magical]] interpretations of the roles of numbers in governing existence.[{{harvnb|Vamvacas|2009|p=68}}.]
===Music===
{{anchor|Music and harmony}}
[[Image:Pythagoras and Philolaus.png|right|thumb|Medieval woodcut by [[Franchino Gaffurio]], depicting [[Pythagoras]] and [[Philolaus]] conducting musical investigations.]]
Pythagoras pioneered the mathematical and experimental study of music. He objectively measured physical quantities, such as the length of a [[String (music)|string]] and discovered quantitative mathematical relationships of music through arithmetic ratios. Pythagoras attempted to explain subjective psychological and aesthetic feelings, such as the enjoyment of musical harmony. Pythagoras and his students experimented systematically with strings of varying length and tension, with [[wind instruments]], with brass discs of the same diameter but different thickness, and with identical vases filled with different levels of water. Early Pythagoreans established quantitative ratios between the length of a string or pipe and the pitch of notes and the frequency of string vibration.[{{harvnb|Vamvacas|2009|p=69}}.]
[[Pythagorean tuning]] is a system of [[musical tuning]] in which the [[frequency ratio]]s of all [[interval (music)|intervals]] are based on the ratio [[Perfect fifth|3:2]].[{{cite book |first1=Bruce |last1=Benward |first2=Marilyn Nadine |last2=Saker |year=2003 |title=Music: In Theory and Practice |edition=seventh |location=Boston |publisher=McGraw-Hill |volume=I |page=56 |isbn=978-0-07-294262-0}}] This ratio, also known as the "[[Five-limit tuning#The justest ratios|pure]]" [[perfect fifth]], is chosen because it is one of the most [[consonance and dissonance|consonant]] and easiest to tune by ear and because of importance attributed to the integer 3. As [[Novalis]] put it, "The musical proportions seem to me to be particularly correct natural proportions."[{{cite book |first1=Kenneth Sylvan |last1=Guthrie |first2=David R. |last2=Fideler |year=1987 |title=The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy |page=24 |publisher=Red Wheel/Weiser |isbn=9780933999510}}]
The fact that mathematics could explain the human sentimental world had a profound impact on the Pythagorean philosophy. Pythagoreanism became the quest for establishing the fundamental essences of reality. Pythagorean philosophers advanced the unshakable belief that the essence of all things are numbers and that the universe was sustained by harmony.[{{harvnb|Vamvacas|2009|p=70}}.] Pythagoras had in his teachings named numbers and the symmetries of them as the first principle and called these numeric symmetries harmony.{{sfn|Vamvacas|2009|p=71}} This numeric harmony could be discovered in rules throughout nature. Numbers governed the properties and conditions of all beings and were regarded the causes of being in everything else. Pythagorean philosophers believed that numbers were the elements of all beings and the universe as a whole was composed of harmony and numbers.[{{cite web |url=https://www.britannica.com/topic/table-of-opposites|title=Table of opposites |archive-url=https://archive.today/20250324075923/https://www.britannica.com/topic/table-of-opposites|archive-date=March 24, 2025 |url-status=live}}]
===Cosmology===
[[Image:Antichthon.svg|thumb|According to a collection of ancient philosophical texts by [[Stobaeus]] in the 5th century AD, [[Philolaus]] believed there was a "Counter-Earth" (''Antichthon'') orbiting a "central fire" but not visible from [[Earth]].{{sfn|Orr|1913}}{{page needed |date=February 2025}} ]]
{{Main|Pythagorean astronomical system}}
The philosopher [[Philolaus]], one of the most prominent figures in Pythagoreanism,[{{cite book|last=Aristotle|title=On the Heavens|url=http://classics.mit.edu/Aristotle/heavens.2.ii.html|access-date=Apr 17, 2016|volume=II|chapter=13}}]
It is not known whether Philolaus believed Earth to be round or flat,[{{harvnb|Huffman|2024}}, citing {{cite journal |last=Burch |first=G B |year=1954 |title=The counter-Earth |journal=Osiris |volume=11 |pages=272–73|doi=10.1086/368583 }}] but he did not believe the earth rotated, so that the Counter-Earth and the Central Fire were both not visible from Earth's surface, or at least not from the hemisphere where Greece was located.[{{Cite book|title=The Pythagorean Sourcebook and Library|author=David R. Fideler|publisher=Red Wheel/Weiser|year=1987|isbn=978-1-60925-394-3|page=33}}] Early-Pythagoreans believed that after the death of the body, the soul would be punished or rewarded. Humans could, through their conduct, ensure that their soul was admitted to another world. The reincarnation in this world equated to a punishment. In Pythagoreanism life in this world is social[{{Cite book|title=What is Justice?: Justice, Law, and Politics in the Mirror of Science: Collected Essays|author=Hans Kelsen|publisher=The Lawbook Exchange|year=2000|isbn=978-1-58477-101-2|page=99}}] and in the realm of society justice existed when each part of society received its due. The Pythagorean tradition of [[universal justice]] was later referenced by [[Plato]]. For Pythagorean philosophers the soul was the source of justice and through the harmony of the soul, divinity could be achieved. Injustice inverted the natural order. According to the 4th century BC philosopher [[Heraclides Ponticus]], Pythagoras taught that "happiness consists in knowledge of the perfection of the numbers of the soul.[{{Cite book|title=Women Writers of Ancient Greece and Rome: An Anthology|author=Ian Michael Plant|publisher=Equinox Publishing|year=2004|isbn=978-1-904768-02-9|page=81}}]}}
===Body and soul===
{{See also|Pythagoras#Metempsychosis}}
Pythagoreans believed that body and soul functioned together, and a healthy body required a healthy psyche.[{{harvnb|Campbell|2014|p=539}}.] Early Pythagoreans conceived of the soul as the seat of sensation and emotion. They regarded the soul as distinct from the intellect.[{{Cite book|title=The Concept of Woman: The Early Humanist Reformation, 1250-1500, Part 1|author=Prudence Allen|publisher=Wm. B. Eerdmans Publishing|year=2005|isbn=978-0-8028-3346-4|page=5}}] However, only fragments of the early Pythagorean texts have survived, and it is not certain whether they believed the soul was immortal. The surviving texts of the Pythagorean philosopher [[Philolaus]] indicate that while early Pythagoreans did not believe that the soul contained all psychological faculties, the soul was life and a harmony of physical elements. As such the soul died when certain arrangements of these elements ceased to exist.[{{Cite book|title=The Concept of Woman: The Early Humanist Reformation, 1250-1500, Part 1|author=Prudence Allen|publisher=Wm. B. Eerdmans Publishing|year=2005|isbn=978-0-8028-3346-4|page=22}}]
However, the teaching most securely identified with Pythagoras is ''[[metempsychosis]]'', or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body.{{sfnm|Kahn|2001|1pp=4, 11–12, 14|Zhmud|2012|2p=221}} Pythagorean metempsychosis resembles the teachings of the [[Orphism (religion)|Orphics]], although its version contains substantial differences. Unlike the Orphics, who considered metempsychosis a cycle of grief that could be escaped by attaining liberation from it, Pythagoras seems to postulate an eternal, endless reincarnation where subsequent lives would not be conditioned by any action done in the previous.{{sfn|Zhmud|2012|p=232–33}}
===Vegetarianism===
[[File:Do Not Eat Beans.jpg|thumb|right|Pythagoras and [[Vicia faba|faba beans]], French, 1512/1514.{{citation needed|date=June 2018}} Pythagoreans refused to eat beans. Already in antique times there was much speculation about the reason for this custom.{{sfn|McKirahan|2011|p=88}}]]
Some medieval authors refer to a "Pythagorean diet", entailing the abstention from eating meat, beans or fish.[See for instance the popular treatise by Antonio Cocchi, ''Del vitto pitagorico per uso della medicina'', Firenze 1743, which initiated a debate on the "Pythagorean diet".] Pythagoreans believed that a vegetarian diet fostered a healthy body and enhanced the search for [[Arete]]. The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conducive to the best in a human being. The prohibition on beans may be related to older Athenian ideas associating beans with [[Hades]], as in the cult of [[Cyamites]]. Pythagoreans advanced a grounded theory on the treatment of animals. They believed that any being that experienced pain or suffering should not have pain inflicted on it unnecessarily. Because it was not necessary to inflict pain on animals for humans to enjoy a healthy diet, they believed that animals should not be killed for the purpose of eating them. The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill it and that doing so would diminish the moral status of a human. By failing to show justice to the animal, humans diminish themselves.[{{harvnb|Campbell|2014|p=540}}.] The arguments advanced by Pythagoreans convinced numerous of their philosopher contemporaries to adopt a vegetarian diet.[{{Cite book |last=Glenn |first=Cheryl |title=Rhetoric retold: regendering the tradition from antiquity through the Renaissance |date=2010 |publisher=Southern Illinois University Press |isbn=978-0-8093-2137-7 |location=Carbondale |pages=30–31 |url={{google books |id=zgTKrbMSsiQC |plainurl=y }} }}]
[[File:The story of the greatest nations; a comprehensive history, extending from the earliest times to the present, founded on the most modern authorities, and including chronological summaries and (14783288925).jpg|thumb|upright=1.4|Illustration from 1913 showing Pythagoras teaching a class of women]]
Many of the surviving texts of women Pythagorean philosophers are part of a collection, known as ''pseudoepigrapha Pythagorica'', which was compiled by Neopythagoreans in the 1st or 2nd century. Some surviving fragments of this collection are by early-Pythagorean women philosophers, while the bulk of surviving writings are from late-Pythagorean women philosophers who wrote in the 4th and 3rd century BC.{{sfn|Ballif|Moran|2005|p=315}} Female Pythagoreans are some of the first female philosophers from which texts have survived.
[[Theano of Croton]], the wife of Pythagoras, is considered a major figure in early-Pythagoreanism. She was noted as distinguished philosopher and in the lore that surrounds her, is said to have taken over the leadership of the school after his death. Text fragments have also survived from women philosophers of the late-Pythagorean period. These include [[Perictione I]], [[Perictione II]], [[Aesara of Lucania]] and [[Phintys of Sparta]].[{{Cite book|title=Classical Rhetorics and Rhetoricians: Critical Studies and Sources|author1=Michelle Ballif |author2=Michael G. Moran |publisher=Greenwood Publishing Group|year=2005|isbn=978-0-313-32178-8|page=317}}] The woman Pythagorean philosopher Phyntis was [[Spartan]] and is believed to have been the daughter of a Spartan admiral killed in the battle of [[Arginusae]] in 406 BC.[{{harvnb|Vamvacas|2009|p=76}}.] The influence of Pythagoreanism extended throughout and beyond antiquity because the Pythagorean doctrine of reincarnation was recounted in Plato's ''[[Gorgias]]'', ''[[Phaedo]]'', and ''[[Republic (Plato)|Republic]]'', while the Pythagorean cosmology was discussed in Plato's ''[[Timaeus (dialogue)|Timaeus]]''. The possible influence of Pythagoreanism on Plato's concept of harmony and the [[Platonic solids]] has been discussed extensively. The belief in transmigration or reincarnation found its way into Plato's dialogue teaching as well. Plato's dialogues have become an important surviving source of Pythagorean philosophic arguments.[{{harvnb|McKirahan|2011|p=80}}.] Plato referenced [[Philolaus]] in ''Phaedo'' and wrote a Platonic adaptation of Philolaus' metaphysical system of limiters and unlimiteds. Plato also quoted from one of the surviving [[Archytas]] fragments in the ''Republic''. However, Plato's views that the primary role of mathematics was to turn the soul towards the world of forms, as expressed in ''Timaeus'', is regarded as Platonic philosophy, rather than Pythagorean.[{{Cite book |last=Janos |first=Damien |title=Method, Structure, and Development in al-Fārābī's Cosmology |date=2012 |publisher=Brill |isbn=978-90-04-21732-4 |location=Leiden |url=https://brill.com/display/title/18309 |page=118 }}] Nevertheless, he wrote a treatise on the Pythagoreans of which only fragments survive, in which he treats Pythagoras as a wonder-working religious teacher.[{{Cite book|title=Greek Thought: A Guide to Classical Knowledge|author=Pierre Pellegrin|publisher=Harvard University Press|year=2000|isbn= 978-0-674-00261-6|page=918}}]
==Neopythagoreanism==
{{Main|Neopythagoreanism}}
The Neopythagoreans were a school and a religious community. The revival of Pythagoreanism has been attributed to [[Publius Nigidius Figulus]], [[Eudorus of Alexandria]] and [[Arius Didymus]]. In the 1st century AD [[Moderatus of Gades]] and [[Nicomachus of Gerasa]] emerged as leading teachers of Neopythagoreanism.[{{Cite book|title=Encyclopedia of Early Christianity, Second Edition|author=Everett Ferguson|publisher=Routledge|year=2013|isbn=978-1-136-61158-2|page=804}}]{{sfn|Riedweg|2008|p=73}} The most significant Neopythagorean teacher was [[Apollonius of Tyana]] in the 1st century AD, who was regarded as a sage and lived as an [[Asceticism|ascetic]]. The last Neopythagorean philosopher was [[Numenius of Apamea]] in the 2nd century. Neopythagoreanism remained an elite movement which in the 3rd century merged into [[Neoplatonism]].[{{harvnb|Joost-Gaugier|2007|p=118}}.] In the Middle Ages this numerological division of the universe was credited to the Pythagoreans, while early on it was regarded as an authoritative source of Christian doctrine by [[Photius]] and [[John of Sacrobosco]]. The ''Corpus Areopagiticum or Corpus Dionysiacum'' was to be referenced in the late Middle Ages by [[Dante]] and in the [[Renaissance]] a new translation of it was produced by [[Marsilio Ficino]].[{{harvnb|Joost-Gaugier|2007|page=119}}]
Early Christian theologians, such as [[Clement of Alexandria]], adopted the [[ascetic]] doctrines of the neopythagoreans.[{{harvnb|Joost-Gaugier|2007|page=117}}.]
===On numerology===
[[File:Tetractys.svg|thumb|Pythagoras is credited with having devised the [[tetractys]],[{{cite book |last1=Bruhn |first1=Siglind |title=The musical order of the world: Kepler, Hesse, Hindemith |date=2005 |publisher=Pendragon Press |location=Hillsdale, NY |isbn=978-1-57647-117-3 |pages=65–66}}] an important [[Religious symbol|sacred symbol]] in later Pythagoreanism.{{sfnm|Riedweg|2008|1p=29|Kahn|2001|2pp=31–36}}]]
First century treatises of [[Philo]] and [[Nicomachus]] popularised the mystical and cosmological symbolism Pythagoreans attributed to [[number]]s. This interest in Pythagorean views on the importance of numbers was sustained by mathematicians such as [[Theon of Smyrna]], [[Anatolius]] and [[Iamblichus]]. These mathematicians relied on [[Plato]]'s ''[[Timaeus (dialogue)|Timaeus]]'' as their source for Pythagorean philosophy.[{{harvnb|Joost-Gaugier|2007|pages=117–118}}.]
In the [[Middle Ages]] studies and adaptations of ''Timaeus'' solidified the view that there was a numerical explanation for [[Proportionality (mathematics)|proportion]] and [[harmony]] among learned men. Pythagoreanism, as mediated in Plato's ''Timaeus'', spurred increasingly detailed studies of [[symmetry]] and harmony. Intellectuals pondered how knowledge of the [[geometry]] in which God had arranged the [[universe]] could be applied to life. By the 12th century Pythagorean numerological concepts had become a universal language in medieval Europe and were no longer recognised as Pythagorean.[{{harvnb|Joost-Gaugier|2007|page=120}}.]
The 11th-century Byzantine professor of philosophy [[Michael Psellus]] popularised Pythagorean numerology in his treatise on theology, arguing that Plato was the inheritor of the Pythagorean secret. Psellus also attributed arithmetical inventions by [[Diophantus]] to Pythagoras. Psellus thought to reconstruct [[Iamblichus]]' 10 book encyclopedia on Pythagoreanism from surviving fragments, leading to the popularisation of Iamblichus' description of Pythagorean physics, ethics and theology at the Byzantine court. Psellus was reputably in the possession of the [[Hermetica]], a set of texts thought to be genuinely antique and which would be prolifically reproduced in the late Middle Ages. [[Manuel Bryennios]] introduced Pythagorean numerology to Byzantine music with his treatise ''Harmonics''. He argued that the [[octave]] was essential in attaining perfect harmony.[{{harvnb|Joost-Gaugier|2007|page=123}}.]
In the Jewish communities the development of the [[Kabbalah]] as esoteric doctrine became associated with numerology. It was only in the 1st century that [[Philo of Alexandria]], developed a Jewish Pythagoreanism. In the 3rd century [[Hermippus]] popularised the belief that Pythagoras had been the basis for establishing key dates in Judaism. In the 4th century this assertion was further developed by [[Aristobulus of Alexandria|Aristobulus]]. The Jewish Pythagorean numerology developed by Philo held that God as the unique One was the creator of all numbers, of which seven was the most divine and ten the most perfect. The medieval edition of the Kabbalah focused largely on a cosmological scheme of creation, in reference to early Pythagorean philosophers [[Philolaus]] and [[Empedocles]] and helped to disseminate Jewish Pythagorean numerology.{{sfn|Joost-Gaugier|2007|page=125}}
===On mathematics===
[[Image:Liber abbaci magliab f124r.jpg|thumb|A page of [[Fibonacci]]'s ''[[Liber Abaci]]'' from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) the [[Fibonacci sequence]] with the position in the sequence labeled in Roman numerals and the value in [[Eastern Arabic numerals]].]]
[[Nicomachus]]' treatises were well known in Greek, Latin and the Arabic worlds. In the 9th century an Arabic translation of Nicomachus' ''Introduction to Arithmetic'' was published.[{{cite encyclopedia |last=Zarepour |first=Mohammad Saleh |title=Arabic and Islamic Philosophy of Mathematics |date=2022 |encyclopedia=Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/arabic-islamic-phil-math/ |access-date=2025-02-03 |edition=Summer 2022 |publisher=Metaphysics Research Lab, Stanford University}}] The Arabic translations of Nicomachus' treatises were in turn translated into Latin by [[Gerard of Cremona]], making them part of the Latin tradition of numerology. The [[Pythagorean theorem]] was referenced in Arabic manuscripts.[{{harvnb|Joost-Gaugier|2007|page=124}}.]
Besides the enthusiasm that developed in the Latin and Byzantine worlds in the Middle Ages for Pythagorean numerology, the Pythagorean tradition of perfect numbers inspired profound scholarship in [[mathematics]]. In the 13th century [[Leonardo of Pisa]], better known as Fibonacci, published the ''Libre quadratorum'' (''[[The Book of Squares]]''). Fibonacci had studied scripts from Egypt, Syria, Greece and Sicily, and was learned in Hindu, Arabic and Greek methodologies. Using the [[Hindu–Arabic numeral system]] instead of the [[Roman numerals]], he explored numerology as it had been set forth by Nicomachus. Fibonacci observed that [[square numbers]] always arise through the addition of consecutive odd numbers starting with unity. [[Fibonacci]] put forward a method of generating sets of three square numbers that satisfied the relationship first attributed to Pythagoras by [[Vitruvius]], that {{math|1=''a''2 + ''b''2 = ''c''2}}. This equation is now known as the [[Pythagorean triple]].[{{harvnb|Joost-Gaugier|2007|page=126}}.]
===In the Middle Ages===
In the [[Middle Ages]], from the 5th until the 15th century, Pythagorean texts remained popular. Late antique writers had produced adaptions of the ''[[Sentences of Sextus]]'' as ''[[The golden verses of Pythagoras]]''. The ''Golden Verses'' gained popularity and Christian adaptations of it appeared. These Christian adaptations were adopted by [[monastic orders]], such as [[Saint Benedict]], as authoritative Christian doctrine. In the Latin medieval western world, the ''Golden Verses'' became a widely reproduced text.[{{harvnb|Joost-Gaugier|2007|pages=121–122}}]
In the early 6th century the Roman philosopher [[Boethius]] popularised Pythagorean and [[Platonism|Platonic]] conceptions of the universe and expounded the supreme importance of [[ratio|numerical ratios]].{{sfn|Mancosu|2003|p=599}} The 7th century Bishop [[Isidore of Seville]] expressed his preference for the Pythagorean vision of a universe governed by the mystical properties of certain numbers, over the newly emerging [[Euclid]]ean notion that knowledge could be built through deductive proofs. Isidore relied on the [[arithmetic]] of [[Nicomachus]], who had fashioned himself as heir of Pythagoras, and took things further by studying the [[etymology]] of the name of each number.[{{Cite book|title=Jawanmardi|author=Lloyd Ridgeon|publisher=Edinburgh University Press|year=2011|isbn=978-0-7486-8815-9|page=158}}] in ''[[Kitab al-Musiqa al-Kabir]]'' Al-Farabi rejected the notion of celestial harmony on the grounds that it was "plainly wrong" and that it was not possible for the heavens, orbs and stars to emit sounds through their motions.[{{cite book |last1=Cicero |first1=Marcus Tullius |title=The Academics of Cicero |date=1885 |publisher=Macmillan |page=81 |url=https://books.google.com/books?id=phdIbuiWzYUC&pg=PA81 |language=en}}] Later I also discovered in [[Aetius (philosopher)|Plutarch]] that others were of this opinion. I have decided to set his words down here, so that they may be available to everybody: "Some think that the earth remains at rest. But [[Philolaus|Philolaus the Pythagorean]] believes that, like the sun and moon, it revolves around the fire in an oblique circle. [[Heraclides of Pontus]] and [[Ecphantus the Pythagorean]] make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center."[{{cite web |title=Pseudo-Plutarch, Placita Philosophorum, Book 3., chapter 13 |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0404%3Abook%3D3%3Achapter%3D13 |website=www.perseus.tufts.edu}}][{{cite book |last1=Copernicus |first1=Nicholas |last2=Dobrzycki |first2=Jerzy |last3=Rosen |first3=Edward |title=On the Revolutions: Volume 2 |date=2016 |publisher=Springer |isbn=978-1-349-01776-8 |pages=4–5 |url=https://books.google.com/books?id=2zSFCwAAQBAJ&pg=PA4 |language=en}}]{{sfn|Rosen|1978}}
In the 16th century [[Vincenzo Galilei]] challenged the prevailing Pythagorean wisdom about the relationship between pitches and weights attached to strings. Vincenzo Galilei, the father of [[Galileo Galilei]], engaged in an extended public exchange with his former teacher [[Zarlino]]. Zarlino supported the theory that if two weights in the ratio of 2 to 1 were attached to two strings, the pitches generated by the two strings would produce the [[octave]]. Vincenzo Galilei proclaimed that he had been a committed Pythagorean, until he "ascertained the truth by means of experiment, the teacher of all things". He devised an experiment which showed that the weights attached to the two strings needed to increase as the square of the string length.{{sfn|Mancosu|2003|p=603}} This public challenge to prevailing numerology in musical theory triggered an experimental and physical approach to [[acoustics]] in the 17th century. Acoustics emerged as a mathematical field of music theory and later an independent branch of physics. In the experimental investigation of sound phenomena, numbers had no symbolic meaning and were merely used to measure physical phenomena and relationships such as frequency and vibration of a string.{{sfn|Mancosu|2003|pp=597–98}}
Many of the most eminent 17th century natural philosophers in Europe, including [[Francis Bacon]], [[Descartes]], [[Isaac Beeckman|Beeckman]], [[Kepler]], [[Mersenne]], [[Stevin]] and Galileo, had a keen interest in music and acoustics.{{sfn|Mancosu|2003|p=604}} By the late 17th century it was accepted that sound travels like a wave in the air at a finite speed and experiments to establish the [[speed of sound]] were carried out by philosophers attached to the [[French Academy of Sciences]], the [[Accademia del Cimento]] and the [[Royal Society]].{{sfn|Mancosu|2003|p=609}}
At the height of the [[Scientific Revolution]], as [[Aristotelianism]] declined in Europe, the ideas of early-Pythagoreanism were revived. Mathematics regained importance and influenced philosophy as well as science. Mathematics was used by Kepler, Galileo, Descartes, [[Christiaan Huygens|Huygens]] and [[Isaac Newton|Newton]] to advance physical laws that reflected the inherent order of the universe. Twenty-one centuries after Pythagoreas had taught his disciples in Italy, Galileo announced to the world that "the great book of nature" could only be read by those who understood the language of mathematics. He set out to measure whatever is measurable, and to render everything measurable that is not.[{{harvnb|Vamvacas|2009|p=77}}.] The Pythagorean concept of cosmic harmony deeply influenced western science. It served as the basis for [[Kepler]]'s ''[[Harmonices Mundi|harmonices mundi]]'' and [[Leibniz]]'s ''[[pre-established harmony]]''.