Pythagoreans
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Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
colony of Kroton, in modern
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(Italy). Early Pythagorean communities spread throughout
Magna Graecia Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these re ...
. Pythagoras' death and disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism. The ''akousmatikoi'' were superseded in the 4th century BC as a significant
mendicant A mendicant (from la, mendicans, "begging") is one who practices mendicancy, relying chiefly or exclusively on alms to survive. In principle, mendicant religious orders own little property, either individually or collectively, and in many inst ...
school of philosophy by the Cynics. The ''mathēmatikoi'' philosophers were absorbed into the Platonic school in the 4th century BC. Following political instability in Magna Graecia, some Pythagorean philosophers fled to mainland Greece while others regrouped in
Rhegium Reggio di Calabria ( scn, label= Southern Calabrian, Riggiu; el, label=Calabrian Greek, Ρήγι, Rìji), usually referred to as Reggio Calabria, or simply Reggio by its inhabitants, is the largest city in Calabria. It has an estimated popula ...
. By about 400 BC the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
and through him, on all of
Western philosophy Western philosophy encompasses the philosophical thought and work of the Western world. Historically, the term refers to the philosophical thinking of Western culture, beginning with the ancient Greek philosophy of the pre-Socratics. The word ' ...
. Many of the surviving sources on Pythagoras originate with
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
and the philosophers of the
Peripatetic school The Peripatetic school was a school of philosophy in Ancient Greece. Its teachings derived from its founder, Aristotle (384–322 BC), and ''peripatetic'' is an adjective ascribed to his followers. The school dates from around 335 BC when Aristo ...
. As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to
Neopythagoreanism Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC ...
. The worship of Pythagoras continued in Italy and as a
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Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism.


History

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
was already well known in ancient times for the mathematical achievement of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
. 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 music intervals also credit him as the inventor of the
monochord A monochord, also known as sonometer (see below), is an ancient musical and scientific laboratory instrument, involving one (mono-) string ( chord). The term ''monochord'' is sometimes used as the class-name for any musical stringed instrument h ...
, a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals. Much of the surviving sources on Pythagoras originated with
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
and the philosophers of the
Peripatetic school The Peripatetic school was a school of philosophy in Ancient Greece. Its teachings derived from its founder, Aristotle (384–322 BC), and ''peripatetic'' is an adjective ascribed to his followers. The school dates from around 335 BC when Aristo ...
, which founded historiographical academic traditions such as
biography A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or ...
,
doxography Doxography ( el, δόξα – "an opinion", "a point of view" +  – "to write", "to describe") is a term used especially for the works of classical historians, describing the points of view of past philosophers and scientists. The term ...
and the
history of science The history of science covers the development of science from ancient times to the present. It encompasses all three major branches of science: natural, social, and formal. Science's earliest roots can be traced to Ancient Egypt and Meso ...
. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoreas' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander, Andron of Ephesus, Heraclides and Neanthes had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline.
Neopythagorean Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC ...
philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy. The earliest surviving ancient source on Pythagoras and his followers is a
satire Satire is a genre of the visual, literary, and performing arts, usually in the form of fiction and less frequently non-fiction, in which vices, follies, abuses, and shortcomings are held up to ridicule, often with the intent of shaming ...
by
Xenophanes Xenophanes of Colophon (; grc, Ξενοφάνης ὁ Κολοφώνιος ; c. 570 – c. 478 BC) was a Greek philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φ ...
, on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that: In a surviving fragment from
Heraclitus Heraclitus of Ephesus (; grc-gre, Ἡράκλειτος , "Glory of Hera"; ) was an ancient Greek pre-Socratic philosopher from the city of Ephesus, which was then part of the Persian Empire. Little is known of Heraclitus's life. He wrote ...
, Pythagoras and his followers are described as follows: Two other surviving fragments of ancient sources on Pythagoras are by
Ion of Chios Ion of Chios (; grc-gre, Ἴων ὁ Χῖος; c. 490/480 – c. 420 BC) was a Greek writer, dramatist, lyric poet and philosopher. He was a contemporary of Aeschylus, Euripides and Sophocles. Of his many plays and poems only a few titles and fr ...
and
Empedocles Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
. Both were born in the 490s, after Pythagoras' death. By that time he was known as a sage and his fame had spread throughout Greece. According to Ion, Pythagoras was: 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." In the 4th century BC the
Sophist A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ' ...
Alcidamas Alcidamas ( grc-gre, Ἀλκιδάμας), of Elaea, in Aeolis, was a Greek sophist and rhetorician, who flourished in the 4th century BC. Life He was the pupil and successor of Gorgias and taught at Athens at the same time as Isocrates, to wh ...
wrote that Pythagoras was widely honored by Italians. 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. The
Spartan Sparta ( Doric Greek: Σπάρτα, ''Spártā''; Attic Greek: Σπάρτη, ''Spártē'') was a prominent city-state in Laconia, in ancient Greece. In antiquity, the city-state was known as Lacedaemon (, ), while the name Sparta refe ...
colony of
Taranto Taranto (, also ; ; nap, label= Tarantino, Tarde; Latin: Tarentum; Old Italian: ''Tarento''; Ancient Greek: Τάρᾱς) is a coastal city in Apulia, Southern Italy. It is the capital of the Province of Taranto, serving as an important com ...
in Italy became the home for many practitioners of Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in Crotone and
Metaponto Metaponto is a small town of about 1,000 people in the province of Matera, Basilicata, Italy. Administratively it is a frazione of Bernalda. History The town was built by the ancient Greeks to defend Sybaris from the growth of Taranto. A 1  ...
, both were Achaean colonies. Early-Pythagorean sects lived in Croton and throughout
Magna Graecia Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these re ...
. 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. 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.
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
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 traditions that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras was known as the site of mysteries. Pythagoras had been born on the island of
Samos Samos (, also ; el, Σάμος ) is a Greek island in the eastern Aegean Sea, south of Chios, north of Patmos and the Dodecanese, and off the coast of western Turkey, from which it is separated by the -wide Mycale Strait. It is also a separate ...
at around 570 BC and left his homeland at around 530 BC in opposition to the policies of
Polycrates Polycrates (; grc-gre, Πολυκράτης), son of Aeaces, was the tyrant of Samos from the 540s BC to 522 BC. He had a reputation as both a fierce warrior and an enlightened tyrant. Sources The main source for Polycrates' life and activit ...
. Before settling in Croton, Pythagoras had traveled throughout
Egypt Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
and
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
. 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 emphasized 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 reputably 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. The anti-Pythagorean attacks in c. 508 BC were headed by
Cylon of Croton Cylon of Croton was a leading citizen of Croton, who led a revolt against the Pythagoreans, probably around 509 BC. According to Iamblichus' ''De Vita Pythagorae'', Cylon had previously tried and failed to be accepted into the Pythagorean order ( ...
. 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 Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these re ...
. 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 Reggio di Calabria ( scn, label= Southern Calabrian, Riggiu; el, label=Calabrian Greek, Ρήγι, Rìji), usually referred to as Reggio Calabria, or simply Reggio by its inhabitants, is the largest city in Calabria. It has an estimated popula ...
. 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 Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
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 organized communities were established. According to surviving sources by the
Neopythagorean Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC ...
philosopher
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
,
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
was the successor of Pythagoras. According to
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
( de Orat. III 34.139), Philolaus was teacher of Archytas. According to the
Neoplatonist Neoplatonism is a strand of Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as a chain of thinkers. But there are some id ...
philosopher
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
, Archytas in turn became the head of the Pythagorean school about a century after the Pythagoras' death. Philolaus,
Eurytus Eurytus, Eurytos (; Ancient Greek: Εὔρυτος) or Erytus (Ἔρυτος) is the name of several characters in Greek mythology, and of at least one historical figure. Mythological *Eurytus, one of the Giants, sons of Gaia, killed by Dionysus ...
and
Xenophilus Xenophilus ( el, Ξενόφιλος; 4th century BC), of Chalcidice, was a Pythagorean philosopher and musician who lived in the first half of the 4th century BC. Aulus Gellius relates that Xenophilus was the intimate friend and teacher of Aristo ...
are identified by
Aristoxenus Aristoxenus of Tarentum ( el, Ἀριστόξενος ὁ Ταραντῖνος; born 375, fl. 335 BC) was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been ...
as the teachers of the last generation of Pythagoreans.


Philosophic traditions

Following Pythagoras’ death, disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism in
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...
: ''akousmatikoi'' and ''mathēmatikoi''. The ''mathēmatikoi'' recognized the ''akousmatikoi'' as fellow Pythagoreans, but because the ''mathēmatikoi'' allegedly followed the teachings of
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
, the ''akousmatikoi'' philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism. The ''akousmatikoi'' were superseded in the 4th century BC as significant
mendicant A mendicant (from la, mendicans, "begging") is one who practices mendicancy, relying chiefly or exclusively on alms to survive. In principle, mendicant religious orders own little property, either individually or collectively, and in many inst ...
school of philosophy by the Cynics. ''Mathēmatikoi'' philosophers were in the 4th century BC absorbed into the Platonic school of
Speusippus Speusippus (; grc-gre, Σπεύσιππος; c. 408 – 339/8 BC) was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Academy, near age 60, and remained ...
,
Xenocrates Xenocrates (; el, Ξενοκράτης; c. 396/5314/3 BC) of Chalcedon was a Greek philosopher, mathematician, and leader (scholarch) of the Platonic Academy from 339/8 to 314/3 BC. His teachings followed those of Plato, which he attempted to d ...
and Polemon. As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to
Neopythagoreanism Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC ...
. 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 cults and Orphism.


The ''akousmatikoi''

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' teachings. Individuals who strictly followed most ''akousmata'' were regarded as wise. The ''akousmatikoi'' philosophers refused to recognize 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 afterlife (also referred to as life after death) is a purported existence in which the essential part of an individual's identity or their stream of consciousness continues to live after the death of their physical body. The surviving ess ...
. The ''akousmatikoi'' engaged deeply in questions of Pythagoras' moral teachings, concerning matters such as
harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
,
justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and perspective ...
, ritual purity and moral behavior.


The ''mathēmatikoi''

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' teachings. Today, Pythagoras is mostly remembered for his mathematical ideas, and by association with the work early Pythagoreans did in advancing mathematical concepts and theories on harmonic musical intervals, the definition of
numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
,
proportion Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
and mathematical methods such as
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. The ''mathēmatikoi'' philosophers claimed that numbers were at the heart of everything and constructed a new view of the
cosmos The cosmos (, ) is another name for the Universe. Using the word ''cosmos'' implies viewing the universe as a complex and orderly system or entity. The cosmos, and understandings of the reasons for its existence and significance, are studied in ...
. In the ''mathēmatikoi'' tradition of Pythagoreanism the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
was removed from the center of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. The ''mathēmatikoi'' believed that the Earth, along with other celestial bodies, orbited around a central fire. This, they believed, constituted a celestial harmony.


Rituals

Pythagoreanism was a philosophic tradition as well as a religious practice. As a religious community they relied on oral teachings and worshiped the
Pythian Apollo Apollo, grc, Ἀπόλλωνος, Apóllōnos, label=genitive , ; , grc-dor, Ἀπέλλων, Apéllōn, ; grc, Ἀπείλων, Apeílōn, label=Arcadocypriot Greek, ; grc-aeo, Ἄπλουν, Áploun, la, Apollō, la, Apollinis, label= ...
, the
oracular An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word ''or ...
god of Delphic Oracle. Pythagoreans preached an austere life. They believed that the soul was buried in the body, which acted as a tomb for the soul in this life. The highest reward a human could attain was for the soul to join in the life of the gods and thus escaped the cycle of
reincarnation Reincarnation, also known as rebirth or transmigration, is the philosophical or religious concept that the non-physical essence of a living being begins a new life in a different physical form or body after biological death. Resurrection is a ...
in another human body. Like the practitioners of Orphism, a religious tradition that developed in parallel to Pythagorean religious practice, Pythagoreanism believed that the soul was buried in the body as a punishment for a committed offense and that the soul could be purified. Aside from conducting their daily lives according to strict rules Pythagorean also engaged in rituals to attain purity. The 4th century Greek historian and
sceptic Skepticism, also spelled scepticism, is a questioning attitude or doubt toward knowledge claims that are seen as mere belief or dogma. For example, if a person is skeptical about claims made by their government about an ongoing war then the p ...
philosopher
Hecataeus of Abdera :''See Hecataeus of Miletus for the earlier historian.'' Hecataeus of Abdera or of Teos ( el, Ἑκαταῖος ὁ Ἀβδηρίτης), was a Greek historian and Pyrrhonist philosopher who flourished in the 4th century BC. Life Diogenes La ...
asserted that Pythagoras had been inspired by ancient Egyptian philosophy in his use of ritual regulations and his belief in
reincarnation Reincarnation, also known as rebirth or transmigration, is the philosophical or religious concept that the non-physical essence of a living being begins a new life in a different physical form or body after biological death. Resurrection is a ...
.


Philosophy

Early Pythagoreanism was based on research and the accumulation of knowledge from the books written by other philosophers. Pythagoras' philosophic teachings made direct reference to the philosophy of Anaximander,
Anaximenes of Miletus Anaximenes of Miletus (; grc-gre, Ἀναξιμένης ὁ Μιλήσιος, translit=Anaximenēs ho Milēsios; ) was an Ancient Greek, Ionian Pre-Socratic philosopher from Miletus in Asia Minor (modern-day Turkey), active in the latter half of ...
and
Pherecydes of Syros Pherecydes of Syros (; grc-gre, Φερεκύδης ὁ Σύριος; fl. 6th century BCE) was an Ancient Greek mythographer and proto- philosopher from the island of Syros. Little is known about his life and death. Some ancient testimonies cou ...
. Of the Pythagorean philosophers,
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
, Alcmaeon, Hippon, Archytas and Theodorus, written sources have survived.


Arithmetic and numbers

Pythagoras, in his teachings focused on the significance of
numerology Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in ...
, he believed that numbers themselves explained the true nature of the Universe. ''Numbers'' were in the Greek world of Pythagoras' days ''
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s'' – that is positive
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
(there was no
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
). But unlike their Greek contemporaries, the Pythagorean philosophers represented numbers graphically, not symbolically through letters. Pythagoreans used dots, also known as ''psiphi'' (pebbles), to represent numbers in triangles, squares, rectangles and pentagons. This enabled a visual comprehension of mathematics and allowed for a geometrical exploration of numerical relationships. Pythagorean philosophers investigated the relationship of numbers exhaustively. They defined ''perfect numbers'' as those that were equal to the sum of all their divisors. For example: 28 = 1 + 2 + 4 + 7 + 14. The theory of odd and even numbers was central to Pythagorean
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
. This distinction was for the Pythagorean philosophers direct and visual, as they arranged triangular dots so that the even and odd numbers successively alternate: 2, 4, 6, ... 3, 5, 7, ... Early-Pythagorean philosophers such as
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
and Archytas held the conviction that mathematics could help in addressing important philosophical problems. 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 triad had for Pythagoreans an ethical dimension, as the goodness of each person was believed to be threefold: prudence, drive and good fortune.


Geometry

The Pythagoreans engaged with
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
as a liberal philosophy which served to establish principles and allowed theorems to be explored abstractly and mentally. Pythagorean philosophers believed that there was a close relationship between numbers and geometrical forms. Early-Pythagorean philosophers proved simple geometrical theorems, including "the sum of the angles of a triangle equals two right angles". Pythagoreans also came up with three of the five regular
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
: the
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
and the
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
. The sides of a regular dodecahedron are regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
s, which for Pythagoreans symbolized health. They also revered the
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
, as each diagonal divides the two others at the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. When linear geometrical figures replaced the dots, the combination of Babylonian algebra and Pythagorean arithmetic provided the basis for Greek geometric algebra. By attempting to establish a system of concrete and permanent rules, Pythagoreans helped to establish strict
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
procedures of solving mathematical problems.


Music

Pythagoras pioneered the mathematical and experimental study of music. He objectively measured physical quantities, such as the length of a 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 A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitc ...
, 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. Pythagoras is credited with discovering that the most harmonious musical intervals are created by the simple numerical ratio of the first four natural numbers which derive respectively from the relations of string length: the octave (1/2), the fifth (2/3) and the fourth (3/4). The sum of those numbers 1 + 2 + 3 + 4 = 10 was for Pythagoreans the perfect number, because it contained in itself "the whole essential nature of numbers".
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
has called this formulation of musical arithmetic as "among the most powerful advances of human science" because it enables the measurement of sound in space.
Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
is a system of musical tuning in which the
frequency ratio In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (), 1.5, and may be approximated by an equal tempered perfect fifth () which is 27/ ...
s of all
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: McGraw-Hill). Vol. I: p. 56. . This ratio, also known as the "
pure Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, F ...
"
perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...
, is chosen because it is one of the most
consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
and easiest to tune by ear and because of importance attributed to the integer 3. As
Novalis Georg Philipp Friedrich Freiherr von Hardenberg (2 May 1772 – 25 March 1801), pen name Novalis (), was a German polymath who was a writer, philosopher, poet, aristocrat and mystic. He is regarded as an idiosyncratic and influential figure of ...
put it, "The musical proportions seem to me to be particularly correct natural proportions." 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 thing are numbers and that the universe was sustained by harmony. According to ancient sources music was central to the lives of those practicing Pythagoreanism. They used medicines for the purification ('' katharsis'') of the body and, according to
Aristoxenus Aristoxenus of Tarentum ( el, Ἀριστόξενος ὁ Ταραντῖνος; born 375, fl. 335 BC) was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been ...
, music for the purification of the soul. Pythagoreans used different types of music to arouse or calm their souls.


Harmony

For Pythagoreans, harmony signified the "unification of a multifarious composition and the agreement of unlike spirits". In Pythagoreanism, numeric harmony was applied in mathematical, medical, psychological, aesthetic, metaphysical and cosmological problems. For Pythagorean philosophers, the basic property of numbers was expressed in the harmonious interplay of opposite pairs. Harmony assured the balance of opposite forces. Pythagoras had in his teachings named numbers and the symmetries of them as the first principle, and called these numeric symmetries harmony. 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.


Cosmology

The philosopher
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
, one of the most prominent figures in Pythagoreanism,Philolaus
Stanford Encyclopedia of Philosophy.
was the precursor of
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulated ...
in moving the earth from the center of the cosmos and making it a planet. According to Aristotle's student Eudemus of Cyprus, the first philosopher to determine quantitatively the size of the known planets and the distance between them was Anaximander, a teacher to Pythagoras, in the 6th century BC. Historic sources credit the Pythagorean philosophers with being the first to attempt a clarification of the planet sequence. The early-Pythagorean philosopher
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
believed that limited and unlimited things were the components of the cosmos and these had existed ever since. The center of the universe, according to Philolaus, was the number one (''hēn''), which equated to the unity of
Monism Monism attributes oneness or singleness (Greek: μόνος) to a concept e.g., existence. Various kinds of monism can be distinguished: * Priority monism states that all existing things go back to a source that is distinct from them; e.g., i ...
. Philolaus called the number one an "even-odd" because it was able to generate both even and odd numbers. When one was added to an
odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
it produced an even number, and when added to an even number it produced an odd number. Philolaus further reasoned that the fitting together of the earth and the universe corresponded to the construction of the number one out of the even and the odd. Pythagorean philosophers believed that the even was unlimited and the odd was limited. Aristotle recorded in the 4th century BC on the Pythagorean astronomical system: :It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position, there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the center. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the center. They further construct another earth in opposition to ours to which they give the name counterearth. It is not known whether Philolaus believed Earth to be round or flat, 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. But the conclusion of Pythagorean philosophers that the universe is not
geocentric In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, an ...
was not based on
empirical observation Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
. Instead, as Aristotle noted, the Pythagorean view of the astronomical system was grounded in a fundamental reflection on the value of individual things and the hierarchical order of the universe. Pythagoreans believed in a ''
musica universalis The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a ...
''. They reasoned that
stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
must produce a sound because they were large swiftly moving bodies. Pythagoreans also determined that stars revolved at distances and speeds that were proportional to each other. They reasoned that because of this numerical proportion the revolution of the stars produced a harmonic sound. The early-Pythagorean philosopher Philolaus argued that the structure of the cosmos was determined by the musical numerical proportions of the diatonic
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
, which contained the fifth and fourth harmonic intervals.


Justice

Pythagoreans equated
justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and perspective ...
with geometrical proportion, because proportion ensured that each part receives what it is due. 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 and in the realm of society justice existed when each part of society received its due. The Pythagorean tradition of
universal justice Universal jurisdiction is a legal principle that allows states or international organizations to claim criminal jurisdiction over an accused person regardless of where the alleged crime was committed, and regardless of the accused's nationality, ...
was later referenced by
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
. 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. A surviving fragment from the 3rd century BC by the late-Pythagorean philosopher Aesara reasoned that:


Body and soul

Pythagoreans believed that body and soul functioned together, and a healthy body required a healthy psyche. Early Pythagoreans conceived of the soul as the seat of sensation and emotion. They regarded the soul as distinct from the intellect. 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 Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
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 passed away when certain arrangements of these elements ceased to exist. However, the teaching most securely identified with Pythagoras is ''
metempsychosis Metempsychosis ( grc-gre, μετεμψύχωσις), in philosophy, is the Reincarnation#Conceptual definitions, transmigration of the soul, especially its reincarnation after death. The term is derived from ancient Greek philosophy, and has be ...
'', or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. Pythagorean metempsychosis resembles the teachings of the
Orphics Orphism (more rarely Orphicism; grc, Ὀρφικά, Orphiká) is the name given to a set of religious beliefs and practices originating in the ancient Greek and Hellenistic world, associated with literature ascribed to the mythical poet Orpheus ...
, 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.


Vegetarianism

Some Medieval authors refer to a "Pythagorean diet", entailing the abstention from eating meat, beans or fish. Pythagoreans believed that a vegetarian diet fostered a healthy body and enhanced the search for
Arete ''Arete'' (Greek: ) is a concept in ancient Greek thought that, in its most basic sense, refers to 'excellence' of any kind Liddell, H.G. & Scott, R. ''A Greek–English Lexicon'', 9th ed. (Oxford, 1940), s.v.br>—especially a person or thi ...
. The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conductive to the best in a human being. 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 an animal and that doing so would diminish the moral status of a human. By failing to show justice to the animal, humans diminish themselves. Pythagoreans believed that human beings were animals, but with an advanced intellect and therefore humans had to purify themselves through training. Through purification humans could join the psychic force that pervaded the cosmos. Pythagoreans reasoned that the logic of this argument could not be avoided by killing an animal painlessly. The Pythagoreans also thought that animals were sentient and minimally rational. The arguments advanced by Pythagoreans convinced numerous of their philosopher contemporaries to adopt a vegetarian diet. The Pythagorean sense of kinship with non-humans positioned them as a counterculture in the dominant meat-eating culture. The philosopher
Empedocles Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
is said to have refused the customary blood sacrifice by offering a substitute sacrifice after his victory in a horse race in Olympia. Late-Pythagorean philosophers were absorbed into the Platonic school of philosophy and in the 4th century AD the head of the
Platonic Academy The Academy (Ancient Greek: Ἀκαδημία) was founded by Plato in c. 387 BC in Classical Athens, Athens. Aristotle studied there for twenty years (367–347 BC) before founding his own school, the Lyceum (classical), Lyceum. The Academy ...
Polemon included vegetarianism in his concept of living according to nature. In the 1st century AD
Ovid Pūblius Ovidius Nāsō (; 20 March 43 BC – 17/18 AD), known in English as Ovid ( ), was a Roman poet who lived during the reign of Augustus. He was a contemporary of the older Virgil and Horace, with whom he is often ranked as one of the th ...
identified Pythagoras as the first opponent to meat-eating. But the fuller argument Pythagoreans advanced against the maltreatment of animals did not sustain. Pythagoreans had argued that certain types of food arouse the passions and hindered spiritual ascent. Thus Porphyry would rely on the teachings of the Pythagoreans when arguing that abstinence from eating meat for the purpose of spiritual purification should be practiced only by philosophers, whose aim was to reach a divine state.


Female philosophers

The biographical tradition on Pythagoras holds that his mother, wife and daughters were part of his inner circle. Women were given equal opportunity to study as Pythagoreans and learned practical domestic skills in addition to philosophy.Glenn, Cheryl, ''Rhetoric Retold: Regendering the Tradition from Antiquity Through the Renaissance''. Southern Illinois University, 1997. 30–31. 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. Female Pythagoreans are some of the first female philosophers from which texts have survived.
Theano of Croton Theano of Crotone (; el, Θεανώ) was a 6th-century BC Pythagoreanism, Pythagorean philosopher. She has been called the wife or student of Pythagoras, although others see her as the wife of Brontinus. Her place of birth and the identity of her ...
, 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 Phintys was a Pythagorean philosopher, probably from the third century BC. She wrote a work on the correct behaviour of women, two extracts of which are preserved by Stobaeus. According to Stobaeus, Phintys was the daughter of Callicrates, who is ...
. Scholars believe that Perictione I was an Athenian and contemporary of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, because in ''On the Harmony of Woman'' she wrote in Ionic and used the same terms of
virtues Virtue ( la, virtus) is moral excellence. A virtue is a trait or quality that is deemed to be morally good and thus is valued as a foundation of principle and good moral being. In other words, it is a behavior that shows high moral standard ...
as Plato had done in his ''
Republic A republic () is a "state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th c ...
'': ''andreia'', ''sophrosyne'', ''dikaiosyne'' and ''sophia''. In ''On the Harmony of Woman'' Perictione I outlines the condition that enable women to nurture wisdom and self-control. These virtues will, according to Perictione I, bring "worthwhile things" for a woman, her husband, her children, the household and even the city "if, at any rate, such a woman should govern cities and tribes". Her assertion that a wife should remain devoted to her husband, regardless of his behavior, has been interpreted by scholars as a pragmatic response to the legal rights of women in Athens. The woman Pythagorean philosopher Phyntis was
Spartan Sparta ( Doric Greek: Σπάρτα, ''Spártā''; Attic Greek: Σπάρτη, ''Spártē'') was a prominent city-state in Laconia, in ancient Greece. In antiquity, the city-state was known as Lacedaemon (, ), while the name Sparta refe ...
and is believed to have been the daughter of a Spartan admiral killed in the battle of
Arginusae In classical antiquity, the Arginusae ( grc, Ἀργινοῦσαι ''Arginousai'') were three islands off the Dikili Peninsula on the coast of modern-day Turkey, famous as the site of the Battle of Arginusae during the Peloponnesian War. They were ...
in 406 BC. Phyntis authored the treatise '' Moderation of Women'', in which she assigned the virtue of moderation to women, but asserted that "courage and justice and wisdom are common to both" men and women. Phyntis defended the right of women to philosophize.


Influence on Plato and Aristotle

Pythagoras' teachings and Pythagoreanism influenced
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's writings on physical cosmology, psychology, ethics and political philosophy in the 5th century BC. However, Plato adhered to the dominant Greek philosophy, and the Platonic philosophy suppressed the combination of experimental method and mathematics which was an inherent part of Pythagoreanism. The influence of Pythagoreanism extended throughout and beyond antiquity because the Pythagorean doctrine of reincarnation was recounted in Plato's '' Gorgias'', ''
Phaedo ''Phædo'' or ''Phaedo'' (; el, Φαίδων, ''Phaidōn'' ), also known to ancient readers as ''On The Soul'', is one of the best-known dialogues of Plato's middle period, along with the ''Republic'' and the ''Symposium.'' The philosophical ...
'', and ''
Republic A republic () is a "state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th c ...
'', while the Pythagorean cosmology was discussed in Plato's ''
Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...
''. The possible influence of Pythagoreanism on Plato's concept of harmony and the
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
has been discussed extensively. Plato's dialogues have become an important surviving source of Pythagorean philosophic arguments. Plato referenced
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
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.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
in the 4th century BC rejected mathematics as a tool for investigation and understanding of the world. He believed that numbers constituted simply a quantitative determinant and had no
ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
value. Aristotle's discussion of Pythagorean philosophy is difficult to interpret, because he had little patience for Pythagorean philosophic arguments, and Pythagoreanism does not fit with his philosophic doctrine. In ''On the Heavens'', Aristotle refuted the Pythagorean doctrine on the harmony of the spheres. Nevertheless, he wrote a treatise on the Pythagoreans of which only fragments survive, in which he treats Pythagoras as a wonder-working religious teacher.


Neopythagoreanism

The Neopythagoreans were a school and a religious community. The revival of Pythagoreanism has been attributed to Publius Nigidius Figulus,
Eudorus of Alexandria Eudorus of Alexandria ( el, Εὔδωρος ὁ Ἀλεξανδρεύς; 1st century BC) was an ancient Greek philosopher, and a representative of Middle Platonism. He attempted to reconstruct Plato's philosophy in terms of Pythagoreanism. He form ...
and Arius Didymus. In the 1st century AD
Moderatus of Gades Moderatus of Gades ( el, Μοδερᾶτος) was a Greek philosopher of the Neopythagorean school, who lived in the 1st century AD (contemporary with Apollonius of Tyana). He wrote a great work on the doctrines of the Pythagoreans, and tried to s ...
and
Nicomachus of Gerasa Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
emerged as leading teachers of Neopythagoreanism. The most significant Neopythagorean teacher was
Apollonius of Tyana Apollonius of Tyana ( grc, Ἀπολλώνιος ὁ Τυανεύς; c. 3 BC – c. 97 AD) was a Greek Neopythagorean philosopher from the town of Tyana in the Roman province of Cappadocia in Anatolia. He is the subject of ''L ...
in the 1st century AD, who was regarded as a sage and lived as ascet. The last Neopythagorean philosopher was
Numenius of Apamea Numenius of Apamea ( grc-gre, Νουμήνιος ὁ ἐξ Ἀπαμείας, ''Noumēnios ho ex Apameias''; la, Numenius Apamensis) was a Greek philosopher, who lived in Apamea in Syria and Rome, and flourished during the latter half of the 2 ...
in the 2nd century. Neopythagoreanism remained an elite movement which in the 3rd century merged into
Neoplatonism Neoplatonism is a strand of Platonism, Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and Hellenistic religion, religion. The term does not encapsulate a set of ideas as much as a chain of ...
. Neopythagoreans combined Pythagorean teachings with
Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...
,
Peripatetic Peripatetic may refer to: *Peripatetic school, a school of philosophy in Ancient Greece *Peripatetic axiom * Peripatetic minority, a mobile population moving among settled populations offering a craft or trade. *Peripatetic Jats There are several ...
, Aristotelian and
Stoic Stoic may refer to: * An adherent of Stoicism; one whose moral quality is associated with that school of philosophy * STOIC, a programming language * ''Stoic'' (film), a 2009 film by Uwe Boll * ''Stoic'' (mixtape), a 2012 mixtape by rapper T-Pain * ...
philosophic traditions. Two tendencies within Neopythagorean philosophy emerged, one that owed much to Stoic
monism Monism attributes oneness or singleness (Greek: μόνος) to a concept e.g., existence. Various kinds of monism can be distinguished: * Priority monism states that all existing things go back to a source that is distinct from them; e.g., i ...
and another that relied on Platonic dualism. Neopythagoreans refined the idea of
God In monotheism, monotheistic thought, God is usually viewed as the supreme being, creator deity, creator, and principal object of Faith#Religious views, faith.Richard Swinburne, Swinburne, R.G. "God" in Ted Honderich, Honderich, Ted. (ed)''The Ox ...
and located him beyond the finite so that God could not come into contact with anything corporeal. Neopythagoreans insisted on a spiritual worship of God and that life had to be purified through abstinence. Neopythagoreans manifested a strong interest in numerology and the superstitious aspects of Pythagoreanism. They combined this with the teachings of Plato's philosophic successors. Neopythagorean philosophers engaged in the common ancient practice of ascribing their doctrines to the designated ''founder'' of their philosophy and by crediting their doctrines to Pythagoras himself, they hoped to gain authority for their views.


Later influence


On early Christianity

Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...
was influenced by a Christianized form of
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...
, which had been set out in the four books of the ''Corpus Areopagiticum or Corpus Dionysiacum'': ''The Celestrial Hierarchy'', ''The Ecclesiastical Hierarchy'', ''On Divine Names'' and ''The Mystical Theology''. Having been attributed to
Pseudo-Dionysius the Areopagite Pseudo-Dionysius the Areopagite (or Dionysius the Pseudo-Areopagite) was a Greek author, Christian theologian and Neoplatonic philosopher of the late 5th to early 6th century, who wrote a set of works known as the ''Corpus Areopagiticum'' or ...
, the books explained the relationship among celestial beings, humans, God and the universe. At the heart of the explanation were
numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
. According to ''The Celestrial Hierarchy'', the universe consisted of a threefold division:
heaven Heaven or the heavens, is a common religious cosmological or transcendent supernatural place where beings such as deities, angels, souls, saints, or venerated ancestors are said to originate, be enthroned, or reside. According to the belie ...
, earth and
hell In religion and folklore, hell is a location in the afterlife in which evil souls are subjected to punitive suffering, most often through torture, as eternal punishment after death. Religions with a linear divine history often depict hell ...
. Sunlight lit up the universe and was proof of God's presence. 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 Johannes de Sacrobosco, also written Ioannes de Sacro Bosco, later called John of Holywood or John of Holybush ( 1195 – 1256), was a scholar, monk, and astronomer who taught at the University of Paris. He wrote a short introduction to the Hi ...
. The ''Corpus Areopagiticum or Corpus Dionysiacum'' was to be referenced in the late Middle Ages by
Dante Dante Alighieri (; – 14 September 1321), probably baptized Durante di Alighiero degli Alighieri and often referred to as Dante (, ), was an Italian poet, writer and philosopher. His ''Divine Comedy'', originally called (modern Italian: '' ...
and in the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
a new translation of it was produced by
Marsilio Ficino Marsilio Ficino (; Latin name: ; 19 October 1433 – 1 October 1499) was an Italian scholar and Catholic priest who was one of the most influential humanist philosophers of the early Italian Renaissance. He was an astrologer, a reviver of ...
. Early Christian theologians, such as
Clement of Alexandria Titus Flavius Clemens, also known as Clement of Alexandria ( grc , Κλήμης ὁ Ἀλεξανδρεύς; – ), was a Christian theologian and philosopher who taught at the Catechetical School of Alexandria. Among his pupils were Origen and ...
, adopted the
ascetic Asceticism (; from the el, ἄσκησις, áskesis, exercise', 'training) is a lifestyle characterized by abstinence from sensual pleasures, often for the purpose of pursuing spiritual goals. Ascetics may withdraw from the world for their p ...
doctrines of the neopythagoreans. The moral and ethical teachings of Pythagorean influenced early
Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...
and assimilated into early Christian texts. The ''Sextou gnomai'' (''
Sentences of Sextus The ''Sentences of Sextus'', also called the ''Sayings of Sextus'', is a Hellenistic Pythagorean collection of maxims which was popular among Christians and translated into several languages. The identity of the Sextus who originated the collect ...
''), a
Hellenistic In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
Pythagorean text modified to reflect a Christian viewpoint, existed from at least the 2nd century and remained popular among Christians well into the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
. The ''Sentences of Sextus'' consisted of 451 sayings or principles, such as injunctions to love the truth, to avoid the pollution of the body with pleasure, to shun flatterers and to let one's tongue be harnessed by one's mind. The contents of the ''
Sentences of Sextus The ''Sentences of Sextus'', also called the ''Sayings of Sextus'', is a Hellenistic Pythagorean collection of maxims which was popular among Christians and translated into several languages. The identity of the Sextus who originated the collect ...
'' was attributed by
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
, the 1st century biographer of Pythagoras, to ''Sextus Pythagoricus''. The assertion was repeated subsequently by
Saint Jerome Jerome (; la, Eusebius Sophronius Hieronymus; grc-gre, Εὐσέβιος Σωφρόνιος Ἱερώνυμος; – 30 September 420), also known as Jerome of Stridon, was a Christian priest, confessor, theologian, and historian; he is comm ...
. In the 2nd century many of the ''Sentences of Sextus'' were cited by
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
as Pythagorean aphorisms. The ''Sentences of Sextus'' were translated into
Syriac Syriac may refer to: *Syriac language, an ancient dialect of Middle Aramaic *Sureth, one of the modern dialects of Syriac spoken in the Nineveh Plains region * Syriac alphabet ** Syriac (Unicode block) ** Syriac Supplement * Neo-Aramaic languages a ...
,
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
and
Arabic Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
, then the written language of both Muslims and Jews, but only in the Latin world did they become a guide to daily life that was widely circulated.


On numerology

1st century treatises of
Philo Philo of Alexandria (; grc, Φίλων, Phílōn; he, יְדִידְיָה, Yəḏīḏyāh (Jedediah); ), also called Philo Judaeus, was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt. Philo's de ...
and
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
popularised the mystical and cosmological symbolism Pythagoreans attributed to
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s. This interest in Pythagorean views on the importance of numbers was sustained by mathematicians such as
Theon of Smyrna Theon of Smyrna ( el, Θέων ὁ Σμυρναῖος ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. Hi ...
,
Anatolius Anatolius is both a given name and a surname. Notable people with the name include: Given name * Anatolius of Laodicea (died 283), Bishop of Laodicea in Syria, also known as Anatolius of Alexandria * Anatolius, Vicarius of the Diocese of Asia in 35 ...
and
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
. These mathematicians relied on
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's ''
Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...
'' as source for Pythagorean philosophy. In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
studies and adaptations of ''Timaeus'' solidified the view that there was a numerical explanation for
proportion Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
and
harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
among learned men. Pythagoreanism, as mediated in Plato's ''Timaeus'', spurred increasingly detailed studies of
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
and harmony. Intellectuals pondered how knowledge of the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
in which God had arranged the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
could be applied to life. By the 12th century Pythagorean numerological concepts had become a universal language in Medieval Europe and were no longer recognized as Pythagorean. Writers such as
Thierry of Chartres Thierry of Chartres (''Theodoricus Chartrensis'') or Theodoric the Breton (''Theodericus Brito'') (died before 1155, probably 1150) was a twelfth-century philosopher working at Chartres and Paris, France. The cathedral school at Chartres promoted ...
,
William of Conches William of Conches (c. 1090/1091 – c. 1155/1170s) was a French scholastic philosopher who sought to expand the bounds of Christian humanism by studying secular works of the classics and fostering empirical science. He was a prominent member ...
and
Alexander Neckham Alexander Neckam (8 September 115731 March 1217) was an English magnetician, poet, theologian, and writer. He was an abbot of Cirencester Abbey from 1213 until his death. Early life Born on 8 September 1157 in St Albans, Alexander shared his bi ...
referenced classical writers that had discussed Pythagoreanism, including
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
,
Ovid Pūblius Ovidius Nāsō (; 20 March 43 BC – 17/18 AD), known in English as Ovid ( ), was a Roman poet who lived during the reign of Augustus. He was a contemporary of the older Virgil and Horace, with whom he is often ranked as one of the th ...
and
Pliny Pliny may refer to: People * Pliny the Elder (23–79 CE), ancient Roman nobleman, scientist, historian, and author of ''Naturalis Historia'' (''Pliny's Natural History'') * Pliny the Younger (died 113), ancient Roman statesman, orator, w ...
, leading them to believe that mathematics was the key to understanding
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
. Another important text on Pythagorean numerology was
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
's ''De arithmetica'', which was widely reproduced in the West. Boethius had relied on
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
's writings as a source of Pythagoreanism. In the Byzantine world the influential professor of philosophy
Michael Psellus Michael Psellos or Psellus ( grc-gre, Μιχαὴλ Ψελλός, Michaḗl Psellós, ) was a Byzantine Greek monk, savant, writer, philosopher, imperial courtier, historian and music theorist. He was born in 1017 or 1018, and is believed to hav ...
in the 11th century 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 Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
to Pythagoras. Psellus thought to reconstruct
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
’ 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 The ''Hermetica'' are texts attributed to the legendary Hellenistic figure Hermes Trismegistus, a syncretic combination of the Greek god Hermes and the Egyptian god Thoth. These texts may vary widely in content and purpose, but are usually subd ...
, a set of texts thought to be genuinely antique and which would be prolifically reproduced in the late Middle Ages.
Manuel Bryennios Manuel Bryennios or Bryennius ( el, Μανουήλ Βρυέννιος; c. 1275 – c. 1340). was a Byzantine scholar who flourished in Constantinople about 1300 teaching astronomy, mathematics and musical theory.. His only surviving work is the ''H ...
introduced Pythagorean numerology to Byzantine music with his treatise ''Harmonics''. He argued that the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
was essential in attaining perfect harmony. In the Jewish communities the development of the
Kabbalah Kabbalah ( he, קַבָּלָה ''Qabbālā'', literally "reception, tradition") is an esoteric method, discipline and Jewish theology, school of thought in Jewish mysticism. A traditional Kabbalist is called a Mekubbal ( ''Məqūbbāl'' "rece ...
as esoteric doctrine became associated with numerology. It was only in the 1st century that
Philo of Alexandria Philo of Alexandria (; grc, Φίλων, Phílōn; he, יְדִידְיָה, Yəḏīḏyāh (Jedediah); ), also called Philo Judaeus, was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt. Philo's de ...
, developed a Jewish Pythagoreanism. In the 3rd century
Hermippus Hermippus ( grc-gre, Ἕρμιππος; fl. 5th century BC) was the one-eyed Athenian writer of the Old Comedy, who flourished during the Peloponnesian War. Life He was the son of Lysis, and the brother of the comic poet Myrtilus. He was younger t ...
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 Aristobulus or Aristoboulos may refer to: *Aristobulus I (died 103 BC), king of the Hebrew Hasmonean Dynasty, 104–103 BC *Aristobulus II (died 49 BC), king of Judea from the Hasmonean Dynasty, 67–63 BC *Aristobulus III of Judea (53 BC–36 BC), ...
. 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 Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
and
Empedocles Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
, and helped to disseminate Jewish Pythagorean numerology.


On mathematics

Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
' treatises were well known in Greek, Latin and the Arabic worlds. In the 1st century an Arabic translation of Nicomachus’ ''Introduction to Arithmetic'' was published. The Arabic translations of Nicomachus' treatises were in turn translated into Latin by
Gerard of Cremona Gerard of Cremona (Latin: ''Gerardus Cremonensis''; c. 1114 – 1187) was an Italian translator of scientific books from Arabic into Latin. He worked in Toledo, Kingdom of Castile and obtained the Arabic books in the libraries at Toledo. Some of ...
, making them part of the Latin tradition of numerology. The
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
was referenced in Arabic manuscripts. Scholars in the Arabic world displayed a strong interest in Pythagorean concepts. In the 10th century
Abu al-Wafa' Buzjani Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian mathematician ...
discussed
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
in a treatise on arithmetic for business administrators in reference to Nicomachus. However, the primary interest of Islamic arithmeticians was in solving practical problems, such as
taxation A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer (an individual or legal person, legal entity) by a governmental organization in order to fund government spending and various public expenditures (regiona ...
,
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
, the estimation of agricultural values and business applications for the buying and selling of goods. There was little interest for the Pythagorean numerology that developed in the Latin world. The primary arithmetical system used by Islamic mathematicians was based on Hindu arithmetic, which rejected the notion that the relations between numbers and geometrical forms were symbolic. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. In the 13th century
Leonardo of Pisa Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
, better known as Fibonacci, published the ''Libre quadratorum'' (''
The Book of Squares ''The Book of Squares'', ''(Liber Quadratorum'' in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor Frederick II (German: ''Friedrich''; Italian: ''Federi ...
''). 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 The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
instead of the
Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
, he explored numerology as it had been set forth by Nicomachus. Fibonacci observed that
square numbers In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usua ...
always arise through the addition of consecutive odd numbers starting with unity.
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
put forward a method of generating sets of three square numbers that satisfied the relationship first attributed to Pythagoras by
Vitruvius Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...
, that . This equation is now known as the
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
.


In the Middle Ages

In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
, from the 5th until the 15th century, Pythagorean texts remained popular. Late antique writers had produced adaptions of the ''
Sentences of Sextus The ''Sentences of Sextus'', also called the ''Sayings of Sextus'', is a Hellenistic Pythagorean collection of maxims which was popular among Christians and translated into several languages. The identity of the Sextus who originated the collect ...
'' 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 Monasticism (from Ancient Greek , , from , , 'alone'), also referred to as monachism, or monkhood, is a religious way of life in which one renounces worldly pursuits to devote oneself fully to spiritual work. Monastic life plays an important role ...
, such as
Saint Benedict Benedict of Nursia ( la, Benedictus Nursiae; it, Benedetto da Norcia; 2 March AD 480 – 21 March AD 548) was an Christianity in Italy, Italian Christian monk, writer, and theologian who is venerated in the Catholic Church, the Eastern Ortho ...
, as authoritative Christian doctrine. In the Latin medieval western world, the ''Golden Verses'' became a widely reproduced text. Although the concept of the
quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...
originated with Archytas in the 4th century BC and was a familiar concept among academics in the antiquity, it was attributed as Pythagorean in the 5th century by
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
. According to Proclus, Pythagoreanism divided all mathematical sciences into four categories:
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
,
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
.
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
developed this theory further, arguing that a fourfold path led to the attainment of knowledge. Arithmetic, music, geometry and astronomy went on to become the essential parts of
curriculum In education, a curriculum (; : curricula or curriculums) is broadly defined as the totality of student experiences that occur in the educational process. The term often refers specifically to a planned sequence of instruction, or to a view ...
s in medieval
school A school is an educational institution designed to provide learning spaces and learning environments for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is sometimes compuls ...
s and
universities A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, t ...
. In the 12th century Pythagoras was credited by
Hugh of Saint Victor Hugh of Saint Victor ( 1096 – 11 February 1141), was a Saxon canon regular and a leading theologian and writer on mystical theology. Life As with many medieval figures, little is known about Hugh's early life. He was probably born in the 1090s. ...
with having written a book on quadrivium. The role of harmony had its roots in the triadic thinking of Plato and Aristotle and included the trivium of
grammar In linguistics, the grammar of a natural language is its set of structure, structural constraints on speakers' or writers' composition of clause (linguistics), clauses, phrases, and words. The term can also refer to the study of such constraint ...
,
rhetoric Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate parti ...
and
dialectic Dialectic ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing ...
. From the 9th century onwards, both the quadrivium and the trivium were commonly taught in schools and the newly emerging universities. They came to be known as the
Seven liberal arts Liberal arts education (from Latin "free" and "art or principled practice") is the traditional academic course in Western higher education. ''Liberal arts'' takes the term '' art'' in the sense of a learned skill rather than specifically th ...
. In the early 6th century the Roman philosopher
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
popularized Pythagorean and
Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...
conceptions of the universe and expounded the supreme importance of numerical ratios. The 7th century Bishop
Isidore of Seville Isidore of Seville ( la, Isidorus Hispalensis; c. 560 – 4 April 636) was a Spanish scholar, theologian, and archbishop of Seville. He is widely regarded, in the words of 19th-century historian Montalembert, as "the last scholar of ...
expressed his preference for the Pythagorean vision of a universe governed by the mystical properties of certain numbers, over the newly emerging
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
ean notion that knowledge could be built through deductive proofs. Isidore relied on the
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
of
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
, who had fashioned himself as heir of Pythagoras, and took things further by studying the
etymology Etymology ()The New Oxford Dictionary of English (1998) – p. 633 "Etymology /ˌɛtɪˈmɒlədʒi/ the study of the class in words and the way their meanings have changed throughout time". is the study of the history of the Phonological chan ...
of the name of each number. The 12th century theologian
Hugh of Saint Victor Hugh of Saint Victor ( 1096 – 11 February 1141), was a Saxon canon regular and a leading theologian and writer on mystical theology. Life As with many medieval figures, little is known about Hugh's early life. He was probably born in the 1090s. ...
found Pythagorean numerology so alluring that he set out to explain the human body entirely in numbers. In the 13th century the fashion for numerology dwindled. The Christian scholar
Albertus Magnus Albertus Magnus (c. 1200 – 15 November 1280), also known as Saint Albert the Great or Albert of Cologne, was a German Dominican friar, philosopher, scientist, and bishop. Later canonised as a Catholic saint, he was known during his li ...
rebuked the preoccupation with Pythagorean numerology, arguing that nature could not only be explained in terms of numbers.
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's ''
Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...
'' became a popular source on the mystical and cosmological symbolism Pythagoreans attributed to
numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
. The preoccupation for finding a numerical explanation for
proportion Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
and
harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
culminated in the French cathedrals of the 11th, 12th and 13th century. Arabic translations of the ''Golden Verses'' were produced in the 11th and 12th centuries. In the Medieval Islamic world a Pythagorean tradition took hold, whereby spheres or stars produced music. This doctrine was further developed by
Ikhwan al-Safa The Brethren of Purity ( ar, إخوان‌ الصفا, Ikhwān Al-Ṣafā; also The Brethren of Sincerity) were a secret society of Muslim philosophers in Basra, Iraq, in the 9th or 10th century CE. The structure of the organization and the id ...
and
al-Kindi Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (; ar, أبو يوسف يعقوب بن إسحاق الصبّاح الكندي; la, Alkindus; c. 801–873 AD) was an Arab Muslim philosopher, polymath, mathematician, physician ...
, who pointed to the similarity between the harmony of music and the harmony of the soul. But Islamic philosophers such as
al-Farabi Abu Nasr Muhammad Al-Farabi ( fa, ابونصر محمد فارابی), ( ar, أبو نصر محمد الفارابي), known in the West as Alpharabius; (c. 872 – between 14 December, 950 and 12 January, 951)PDF version was a renowned early Isl ...
and Ibn Sina vehemently rejected this Pythagorean doctrine. in ''
Kitab al-Musiqa al-Kabir ''Kitab al-Musiqa al-Kabir'' ( ar, كتاب الموسيقى الكبير, en, the Great Book of Music) is a treatise on music in east by the medieval philosopher al-Farabi (872-950/951). The work prescribes different aspects of music such as ma ...
'' 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. The four books of the ''Corpus Areopagiticum or Corpus Dionysiacum'' (''The Celestrial Hierarchy'', ''
The Ecclesiastical Hierarchy ''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the m ...
'', ''
On Divine Names On, on, or ON may refer to: Arts and entertainment Music * On (band), a solo project of Ken Andrews * ''On'' (EP), a 1993 EP by Aphex Twin * ''On'' (Echobelly album), 1995 * ''On'' (Gary Glitter album), 2001 * ''On'' (Imperial Teen album), 200 ...
'' and ''
The Mystical Theology ''The'' () is a grammatical Article (grammar), article in English language, English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite ...
'') by
Pseudo-Dionysius the Areopagite Pseudo-Dionysius the Areopagite (or Dionysius the Pseudo-Areopagite) was a Greek author, Christian theologian and Neoplatonic philosopher of the late 5th to early 6th century, who wrote a set of works known as the ''Corpus Areopagiticum'' or ...
became enormously popular during the Middle Ages in the
Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...
world, were they had first been published in the 1st century, but also the Latin world when they were translated in the 9th century. The division of the universe into
heaven Heaven or the heavens, is a common religious cosmological or transcendent supernatural place where beings such as deities, angels, souls, saints, or venerated ancestors are said to originate, be enthroned, or reside. According to the belie ...
, earth and
hell In religion and folklore, hell is a location in the afterlife in which evil souls are subjected to punitive suffering, most often through torture, as eternal punishment after death. Religions with a linear divine history often depict hell ...
, and the 12 orders of heaven were credited as Pythagoras’ teachings by an anonymous biographer, who was quoted in the treatise of the 9th century Byzantine patriarch Photius. The 13th century astronomer and mathematician
John of Sacrobosco Johannes de Sacrobosco, also written Ioannes de Sacro Bosco, later called John of Holywood or John of Holybush ( 1195 – 1256), was a scholar, monk, and astronomer who taught at the University of Paris. He wrote a short introduction to the Hi ...
in turn credited Pseudo-Dionysius when discussing the twelve
signs of the zodiac The zodiac is a belt-shaped region of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The path ...
. In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
various classical texts that discussed Pythagorean ideas were reproduced and translated. Plato's ''
Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...
'' was translated and republished with commentary in the Arab and Jewish worlds. In the 12th century the study of Plato gave rise to a vast body of literature explicating the glory of God as it reflected in the orderliness of the universe. Writers such as
Thierry of Chartres Thierry of Chartres (''Theodoricus Chartrensis'') or Theodoric the Breton (''Theodericus Brito'') (died before 1155, probably 1150) was a twelfth-century philosopher working at Chartres and Paris, France. The cathedral school at Chartres promoted ...
,
William of Conches William of Conches (c. 1090/1091 – c. 1155/1170s) was a French scholastic philosopher who sought to expand the bounds of Christian humanism by studying secular works of the classics and fostering empirical science. He was a prominent member ...
and
Alexander Neckham Alexander Neckam (8 September 115731 March 1217) was an English magnetician, poet, theologian, and writer. He was an abbot of Cirencester Abbey from 1213 until his death. Early life Born on 8 September 1157 in St Albans, Alexander shared his bi ...
referenced not only Plato but also other classical authors that had discussed Pythagoreanism, including
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
,
Ovid Pūblius Ovidius Nāsō (; 20 March 43 BC – 17/18 AD), known in English as Ovid ( ), was a Roman poet who lived during the reign of Augustus. He was a contemporary of the older Virgil and Horace, with whom he is often ranked as one of the th ...
and
Pliny Pliny may refer to: People * Pliny the Elder (23–79 CE), ancient Roman nobleman, scientist, historian, and author of ''Naturalis Historia'' (''Pliny's Natural History'') * Pliny the Younger (died 113), ancient Roman statesman, orator, w ...
. William of Conches argued that Plato was an important Pythagorean. In this medieval Pythagorean understanding of Plato, God was a craftsman when he designed the universe.


On Western science

In the ''
De revolutionibus ''De revolutionibus orbium coelestium'' (English translation: ''On the Revolutions of the Heavenly Spheres'') is the seminal work on the heliocentric theory of the astronomer Nicolaus Copernicus (1473–1543) of the Polish Renaissance. The book ...
'',
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulated ...
cites three pythagorean philosophers as precursors of the
Heliocentric Theory Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth a ...
: :At first I found in
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the estab ...
that
Hicetas Hicetas ( grc, Ἱκέτας or ; c. 400 – c. 335 BC) was a Greek philosopher of the Pythagorean School. He was born in Syracuse. Like his fellow Pythagorean Ecphantus and the Academic Heraclides Ponticus, he believed that the daily movemen ...
supposed the earth to move. Later I also discovered in
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
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 the Pythagorean believes that, like the sun and moon, it revolves around the fire in an oblique circle.
Heraclides of Pontus Heraclides Ponticus ( grc-gre, Ἡρακλείδης ὁ Ποντικός ''Herakleides''; c. 390 BC – c. 310 BC) was a Greek philosopher and astronomer who was born in Heraclea Pontica, now Karadeniz Ereğli, Turkey, and migrated to Athens. He ...
and
Ecphantus the Pythagorean Ecphantus or Ecphantos ( grc, Ἔκφαντος) or Ephantus () is a shadowy Greek pre-Socratic philosopher. He may not have actually existed. He is identified as a Pythagorean of the 4th century BCE, and as a supporter of the heliocentric theory. ...
make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center. In the 16th century
Vincenzo Galilei Vincenzo Galilei (born 3 April 1520, Santa Maria a Monte, Italy died 2 July 1591, Florence, Italy) was an Italian lutenist, composer, and music theorist. His children included the astronomer and physicist Galileo Galilei and the lute virtuoso and ...
challenged the prevailing Pythagorean wisdom about the relationship between pitches and weights attached to strings. Vincenzo Galilei, the father of
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
, engaged in an extended public exchange with his former teacher
Zarlino Gioseffo Zarlino (31 January or 22 March 1517 – 4 February 1590) was an Italian music theorist and composer of the Renaissance. He made a large contribution to the theory of counterpoint as well as to musical tuning. Life and career Zarlino w ...
. 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 In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
. 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. This public challenge to prevailing numerology in musical theory triggered an experimental and physical approach to
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
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. Many of the most eminent 17th century natural philosophers in Europe, including
Francis Bacon Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England. Bacon led the advancement of both ...
, Descartes, Beeckman,
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
,
Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
, Stevin and Galileo, had a keen interest in music and acoustics. 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 The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
were carried out by philosophers attached to the
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
, the
Accademia del Cimento The Accademia del Cimento (Academy of Experiment), an early scientific society, was founded in Florence in 1657 by students of Galileo, Giovanni Alfonso Borelli and Vincenzo Viviani and ceased to exist about a decade later. The foundation of Acade ...
and the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. At the height of the
Scientific Revolution The Scientific Revolution was a series of events that marked the emergence of modern science during the early modern period, when developments in mathematics, physics, astronomy, biology (including human anatomy) and chemistry transfo ...
, as
Aristotelianism Aristotelianism ( ) is a philosophical tradition inspired by the work of Aristotle, usually characterized by deductive logic and an analytic inductive method in the study of natural philosophy and metaphysics. It covers the treatment of the socia ...
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, Huygens and 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. The Pythagorean concept of cosmic harmony deeply influenced western science. It served as the basis for
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
's ''
harmonices mundi ''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...
'' and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's ''
pre-established harmony Gottfried Leibniz's theory of pre-established harmony (french: harmonie préétablie) is a philosophical theory about causation under which every " substance" affects only itself, but all the substances (both bodies and minds) in the world never ...
''.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
believed that through this ''pre-established harmony'', the productive unison between the spiritual and material world was possible. The Pythagorean belief that all bodies are composed of numbers and that all properties and causes could be expressed in numbers, served as the basis for a mathematization of
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
. This mathematization of the physical reality climaxed in the 20th century. The pioneer of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
argued that "this mode of observing nature, which led in part to a true dominion over natural forces and thus contributes decisively to the development of humanity, in an unforeseen manner vindicated the Pythagorean faith".


See also

*
Dyad (Greek philosophy) The Dyad is a title used by the Pythagoreans for the number two, representing the principle of "twoness" or "otherness". Numenius of Apamea, a Neopythagorean philosopher in the latter 2nd century CE, said that Pythagoras gave the name of Monad ...
* Esoteric cosmology *
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
*
Ionian School (philosophy) The Ionian school of Pre-Socratic philosophy was centred in Miletus, Ionia in the 6th century BC. Miletus and its environment was a thriving mercantile melting pot of current ideas of the time. The Ionian School included such thinkers as Thales, ...
* Incommensurable magnitudes * ''
Ipse dixit ''Ipse dixit'' (Latin for "he said it himself") is an assertion without proof, or a dogmatic expression of opinion.Whitney, William Dwight. (1906)"''Ipse dixit''" ''The Century dictionary and cyclopedia,'' pp. 379–380; Westbrook, Robert B"John ...
'' * Mathematical beauty *
Mathematicism Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy'. or else it is the epistemological view that reality is fundamentally mathematical. The term has been appli ...
* Pyrrhonism *
Sacred geometry Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions. It is associated with the belief that a god or goddess is the creator of the universal geometer. The geometry used in the des ...
*
Tetractys The tetractys ( el, τετρακτύς), or tetrad, or the tetractys of the decad is a triangular number, triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical repr ...
*
Unit-point atomism According to some 20th century philosophy, twentieth-century philosophers,Paul Tannery (1887), ''Pour l'histoire de la science Hellène'' (Paris), and John Raven, J. E. Raven (1948), ''Pythagoreans and Eleatics'' (Cambridge: Cambridge University Pr ...


References


External links

* {{Authority control Asceticism Presocratic philosophy Western esotericism