Editing Pythagoras (section)

== Teachings ==
=== Metempsychosis ===
Although the exact details of Pythagoras's teachings are uncertain,{{sfnp|Burkert|1972|pages=106–109}}{{sfnp|Kahn|2001|pages=5–6}} it is possible to reconstruct a general outline of his main ideas.{{sfnp|Burkert|1972|pages=106–109}}{{sfnp|Kahn|2001|pages=9–11}} Aristotle writes at length about the teachings of the Pythagoreans,{{sfnp|Copleston|2003|page=31}}{{sfnp|Burkert|1972|pages=29–30}} but without mentioning Pythagoras directly.{{sfnp|Copleston|2003|page=31}}{{sfnp|Burkert|1972|pages=29–30}} One of Pythagoras's main doctrines appears to have been ''[[metempsychosis]]'',{{sfnp|Kahn|2001|page=11}}{{sfnp|Zhmud|2012|page=232}}{{sfnp|Burkert|1985|pages=300–301}}{{sfnp|Gregory|2015|pages=24–25}}{{sfnp|Copleston|2003|pages=30–31}} the belief that all [[soul]]s are immortal and that, after death, a soul is transferred into a new body.{{sfnp|Kahn|2001|page=11}}{{sfnp|Gregory|2015|pages=24–25}} This teaching is referenced by Xenophanes, Ion of Chios, and Herodotus.{{sfnp|Kahn|2001|page=11}}<ref>{{harvp|Diog VIII|loc=§1.36}}, comp. {{harvp|Aristot. De Anima|loc=I. 2–3}}; {{harvp|Hdt. 2|loc=§123|p=425}}</ref> The earliest source on Pythagoras's metempsychosis is a satirical poem probably written after his death by the Greek philosopher [[Xenophanes of Colophon]] ({{circa|570|478|lk=no}}&nbsp;BC), who had been one of his contemporaries,{{sfnp|Joost-Gaugier|2006|page=12}}{{sfnp|Riedweg|2005|page=62}} in which Xenophanes describes Pythagoras interceding on behalf of a [[dog]] that is being beaten, professing to recognize in its cries the voice of a departed friend.{{efn|name=Xenophanes}}{{sfnp|Burkert|1985|page=299}}{{sfnp|Joost-Gaugier|2006|page=12}}{{sfnp|Copleston|2003|page=31}} Nothing whatsoever, however, is known about the nature or mechanism by which Pythagoras believed metempsychosis to occur.{{sfnp|Gregory|2015|page=25}}

[[Empedocles]] alludes in one of his poems that Pythagoras may have claimed to possess the ability to recall his former incarnations.{{sfnp|Kahn|2001|page=12}} Diogenes Laërtius reports an account from [[Heraclides Ponticus]] that Pythagoras told people that he had lived four previous lives that he could remember in detail.{{sfnp|Diog VIII|loc=§1.3–§1.5}}{{sfnp|Cornelli|McKirahan|2013|pages=164–167}}<ref>{{harvp|Porphyry, Vit. Pyth|loc=§26}}; Pausanias, ii. 17; Horace, ''Od.'' i. 28,1. 10</ref> The first of these lives was as [[Aethalides]] the son of [[Hermes]], who granted him the ability to remember all his past incarnations.{{sfnp|Cornelli|McKirahan|2013|pages=164–165}} Next, he was incarnated as [[Euphorbus]], a minor hero from the [[Trojan War]] briefly mentioned in the ''[[Iliad]]''.{{sfnp|Cornelli|McKirahan|2013|pages=165–166}} He then became the philosopher [[Hermotimus of Clazomenae|Hermotimus]],{{sfnp|Cornelli|McKirahan|2013|page=167}} who recognized the shield of Euphorbus in the temple of Apollo.{{sfnp|Cornelli|McKirahan|2013|page=167}} His final incarnation was as Pyrrhus, a fisherman from [[Delos]].{{sfnp|Cornelli|McKirahan|2013|page=167}} One of his past lives, as reported by [[Dicaearchus]], was as a beautiful courtesan.{{sfnp|Zhmud|2012|page=232}}<ref>Aulus Gellius, iv. 11</ref>

=== Numerology ===
[[File:Tetractys.svg|thumb|alt=Diagram showing the tetractys, an equilateral triangle made up of ten dots, with one dot in the top row, two in the second, three in the third, and four in the bottom.|Pythagoras is credited with having devised the [[tetractys]],{{sfnp|Gregory|2015|pages=28–29}} an important [[Religious symbol|sacred symbol]] in later Pythagoreanism.{{sfnp|Riedweg|2005|page=29}}{{sfnp|Kahn|2001|pages=1–2}}]]
Another belief attributed to Pythagoras was that of the "[[Musica universalis|harmony of the spheres]]",{{sfnp|Riedweg|2005|pages=29–30}}{{sfnp|Gregory|2015|pages=38–39}} which maintained that the planets and stars move according to mathematical equations, which correspond to musical notes and thus produce an inaudible symphony.{{sfnp|Riedweg|2005|pages=29–30}}{{sfnp|Gregory|2015|pages=38–39}} According to Porphyry, Pythagoras taught that the seven [[Muse]]s were actually the [[Classical planet|seven planets]] singing together.{{sfnp|Riedweg|2005|page=30}}

{{cquote|The so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.|author={{harvp|Aristot. Met. 1|loc=985b}} }}

Modern scholars typically ascribe these discoveries to the later Pythagorean philosopher [[Philolaus of Croton]] ({{circa|470|385|lk=no}}&nbsp;BC), whose extant fragments are the earliest texts to describe the numerological and musical theories that were later ascribed to Pythagoras.{{sfnp|Joost-Gaugier|2006|pages=87–88}} In his landmark study ''Lore and Science in Ancient Pythagoreanism'', Walter Burkert argues that Pythagoras was a charismatic political and religious teacher,{{sfnp|Kahn|2001|pages=2–3}} but that the number philosophy attributed to him was really an innovation by Philolaus.{{sfnp|Kahn|2001|page=3}} According to Burkert, Pythagoras never dealt with numbers at all, let alone made any noteworthy contribution to mathematics.{{sfnp|Kahn|2001|pages=2–3}} Burkert argues that the only mathematics the Pythagoreans ever actually engaged in was simple, [[Mathematical proof|proofless]] [[arithmetic]],{{sfnp|Burkert|1972|pages=428–433}} but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics.{{sfnp|Burkert|1972|page=465}} For the later Pythagoreans, Pythagoras was credited with devising the [[tetractys]], the triangular figure of four rows which add up to the "perfect" number, ten.{{sfnp|Gregory|2015|pages=28–29}}{{sfnp|Burkert|1972|pages=467–468}} The Pythagoreans regarded the tetractys as a symbol of utmost mystical importance.{{sfnp|Riedweg|2005|page=29}} Iamblichus, in his ''Life of Pythagoras'', states that the tetractys was "so admirable, and so divinised by those who understood [it]," that Pythagoras's students would swear oaths by it.{{sfnp|Riedweg|2005|page=29}}{{sfnp|Iamblichus, Vit. Pyth|loc=§29}}

This shouldn't be confused with a simplified version known today as "[[Numerology#Pythagorean method|Pythagorean numerology]]", involving a variant of an isopsephic technique known – among other names – as {{lang|grc|pythmenes}} {{gloss|roots}}<ref name="pythmenes1">{{harvp|Gregory|2015|pp=32–34}}</ref> or {{gloss|base numbers}},<ref>{{harvp|Zhmud|2012|p=277}}</ref> by means of which the base values of letters in a word were mathematically reduced by addition or division, in order to obtain a single value from one to nine for the whole name or word.<ref name="pythmenes1" />