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 (;  BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of P ...
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 classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...
colony of Kroton, in modern
Calabria Calabria is a Regions of Italy, region in Southern Italy. It is a peninsula bordered by the region Basilicata to the north, the Ionian Sea to the east, the Strait of Messina to the southwest, which separates it from Sicily, and the Tyrrhenian S ...
(Italy) circa 530 BC. Early Pythagorean communities spread throughout
Magna Graecia Magna Graecia refers to the Greek-speaking areas of southern Italy, encompassing the modern Regions of Italy, Italian regions of Calabria, Apulia, Basilicata, Campania, and Sicily. These regions were Greek colonisation, extensively settled by G ...
. Already during Pythagoras' life it is likely that the distinction between the ''akousmatikoi'' ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the ''mathematikoi'' ("those who learn") existed. The ancient biographers of Pythagoras,
Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
() and his master Porphyry ( ) seem to make the distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, ''akousmatikoi'', following initiation became ''mathematikoi''. It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinction mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community; subsequently did the Greek philosophical traditions become more diverse. The
Platonic Academy The Academy (), variously known as Plato's Academy, or the Platonic Academy, was founded in Classical Athens, Athens by Plato ''wikt:circa, circa'' 387 BC. The academy is regarded as the first institution of higher education in the west, where ...
was arguably a Pythagorean
cenobitic Cenobitic (or coenobitic) monasticism is a monastery, monastic tradition that stresses community life. Often in the West the community belongs to a religious order, and the life of the cenobitic monk is regulated by a Monastic rule, religious ru ...
institution, outside the city walls of Athens in the 4th century BC. As a sacred grove dedicated to Athena, and Hecademos (Academos). The academy, the sacred grove of Academos, may have existed, as the contemporaries seem to have believed, since the Bronze Age, even pre-existing the
Trojan War The Trojan War was a legendary conflict in Greek mythology that took place around the twelfth or thirteenth century BC. The war was waged by the Achaeans (Homer), Achaeans (Ancient Greece, Greeks) against the city of Troy after Paris (mytho ...
. Yet according to Plutarch it was the Athenian strategos (general) Kimon Milkiadou () who converted this, "waterless and arid spot into a well watered grove, which he provided with clear running-tracks and shady walks".
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
(less known as Aristocles) lived almost a hundred years later, circa 427 to 348 BC. On the other hand, it seems likely that this was a part of the re-building of Athens led by Kimon Milkiadou and Themistocles, following the Achaemenid destruction of Athens in 480–479 BC during the war with Persia. Kimon is at least associated with the building of the southern Wall of Themistocles, the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos. Following political instability in
Magna Graecia Magna Graecia refers to the Greek-speaking areas of southern Italy, encompassing the modern Regions of Italy, Italian regions of Calabria, Apulia, Basilicata, Campania, and Sicily. These regions were Greek colonisation, extensively settled by G ...
, some Pythagorean philosophers moved to mainland Greece while others regrouped in
Rhegium Reggio di Calabria (; ), commonly and officially referred to as Reggio Calabria, or simply Reggio by its inhabitants, is the List of cities in Italy, largest city in Calabria as well as the seat of the Metropolitan City of Reggio Calabria. As ...
. By about the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
and through him, on all of
Western philosophy Western philosophy refers to the Philosophy, philosophical thought, traditions and works of the Western world. Historically, the term refers to the philosophical thinking of Western culture, beginning with the ancient Greek philosophy of the Pre ...
. Many of the surviving sources on Pythagoras originate with
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
and the philosophers of the
Peripatetic school The Peripatetic school ( ) was a philosophical school founded in 335 BC by Aristotle in the Lyceum in ancient Athens. It was an informal institution whose members conducted philosophical and scientific inquiries. The school fell into decline afte ...
. As a philosophic tradition, Pythagoreanism was revived in the giving rise to
Neopythagoreanism Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic and Roman philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st c ...
. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism.


History

Pythagoras Pythagoras of Samos (;  BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of P ...
was already well known in ancient times for his supposed 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 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 (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
and the philosophers of the
Peripatetic school The Peripatetic school ( ) was a philosophical school founded in 335 BC by Aristotle in the Lyceum in ancient Athens. It was an informal institution whose members conducted philosophical and scientific inquiries. The school fell into decline afte ...
, 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 basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or curri ...
, doxography and the
history of science The history of science covers the development of science from ancient history, ancient times to the present. It encompasses all three major branches of science: natural science, natural, social science, social, and formal science, formal. Pr ...
. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoras' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as
Anaximander Anaximander ( ; ''Anaximandros''; ) was a Pre-Socratic philosophy, pre-Socratic Ancient Greek philosophy, Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes Ltd, George Newnes, 1961, Vol. ...
, 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 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 exposin ...
by
Xenophanes Xenophanes of Colophon ( ; ; – c. 478 BC) was a Greek philosopher, theologian, poet, and critic of Homer. He was born in Ionia and travelled throughout the Greek-speaking world in early classical antiquity. As a poet, Xenophanes was known f ...
, on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that: In a surviving fragment from
Heraclitus Heraclitus (; ; ) was an Ancient Greece, ancient Greek Pre-Socratic philosophy, pre-Socratic philosopher from the city of Ephesus, which was then part of the Achaemenid Empire, Persian Empire. He exerts a wide influence on Western philosophy, ...
, Pythagoras and his followers are described as follows: Two other surviving fragments of ancient sources on Pythagoras are by Ion of Chios and
Empedocles Empedocles (; ; , 444–443 BC) was a Ancient Greece, Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is known best for originating the Cosmogony, cosmogonic theory of the four cla ...
. 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 () was a teacher in ancient Greece in the fifth and fourth centuries BCE. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics and mathematics. They taught ''arete'', "virtue" or "excellen ...
Alcidamas 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 colony of
Taranto Taranto (; ; previously called Tarent in English) is a coastal city in Apulia, Southern Italy. It is the capital of the province of Taranto, serving as an important commercial port as well as the main Italian naval base. Founded by Spartans ...
in Italy became the home for many practitioners of Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in
Crotone Crotone (; ; or ) is a city and ''comune'' in Calabria, Italy. Founded as the Achaean colony of Kroton ( or ; ), it became a great Greek city, home of the renowned mathematician-philosopher Pythagoras amongst other famous citizens, and one ...
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&nbs ...
, both of which were Achaean colonies. Early-Pythagorean sects lived in Croton and throughout
Magna Graecia Magna Graecia refers to the Greek-speaking areas of southern Italy, encompassing the modern Regions of Italy, Italian regions of Calabria, Apulia, Basilicata, Campania, and Sicily. These regions were Greek colonisation, extensively settled by G ...
. 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 ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
listed 235 Pythagoreans by name, among them 17 women whom he described as the "most famous" women practitioners of Pythagoreanism. It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic tradition that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras was known as the site of mysteries. Pythagoras had been born on the island of
Samos Samos (, also ; , ) is a Greek island in the eastern Aegean Sea, south of Chios, north of Patmos and the Dodecanese archipelago, and off the coast of western Turkey, from which it is separated by the Mycale Strait. It is also a separate reg ...
at around 570 BC and left his homeland at around 530 BC in opposition to the policies of
Polycrates Polycrates (; ), son of Aeaces (father of Polycrates), 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 activi ...
. Before settling in Croton, Pythagoras had traveled throughout
Egypt Egypt ( , ), officially the Arab Republic of Egypt, is a country spanning the Northeast Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via the Sinai Peninsula. It is bordered by the Mediterranean Sea to northe ...
and
Babylonia Babylonia (; , ) was an Ancient history, ancient Akkadian language, Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Kuwait, Syria and Iran). It emerged as a ...
. In Croton, Pythagoras established the first Pythagorean community, described as a secret society, and attained political influence. In the early 5th century BC Croton acquired great military and economic importance. Pythagoras emphasised moderation, piety, respect for elders and of the state, and advocated a
monogamous Monogamy ( ) is a relationship of two individuals in which they form a mutual and exclusive intimate partnership. Having only one partner at any one time, whether for life or serial monogamy, contrasts with various forms of non-monogamy (e.g. ...
family structure. The Croton Council appointed him to official positions. Among others Pythagoras was in charge of education in the city. His influence as political reformer reputedly extended to other Greek colonies in southern Italy and in Sicily. Pythagoras died shortly after an arson attack on the Pythagorean meeting place in Croton.. The anti-Pythagorean attacks in were headed by Cylon of Croton. 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 refers to the Greek-speaking areas of southern Italy, encompassing the modern Regions of Italy, Italian regions of Calabria, Apulia, Basilicata, Campania, and Sicily. These regions were Greek colonisation, extensively settled by G ...
. 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 (; ), commonly and officially referred to as Reggio Calabria, or simply Reggio by its inhabitants, is the List of cities in Italy, largest city in Calabria as well as the seat of the Metropolitan City of Reggio Calabria. As ...
. 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 ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
in the early 4th century BC. The Pythagorean schools and societies died out from the 4th century BC. Pythagorean philosophers continued to practice, albeit no organised communities were established. According to surviving sources by the Neopythagorean philosopher Nicomachus, Philolaus 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, orator, writer and Academic skeptic, who tried to uphold optimate principles during the political crises tha ...
( de Orat. III 34.139), Philolaus was teacher of Archytas.. According to the
Neoplatonist Neoplatonism is a version 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 series of thinkers. Among the common id ...
philosopher
Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
, Archytas in turn became the head of the Pythagorean school about a century after the Pythagoras' death. Philolaus, Eurytus and Xenophilus are identified by
Aristoxenus Aristoxenus of Tarentum (; born 375, fl. 335 BC) was a Ancient Greece, Greek Peripatetic school, Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musi ...
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, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
: ''akousmatikoi'' and ''mathēmatikoi''. The ''mathēmatikoi'' recognised the ''akousmatikoi'' as fellow Pythagoreans, but because the ''mathēmatikoi'' allegedly followed the teachings of Hippasus, the ''akousmatikoi'' philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism. The ''akousmatikoi'' were superseded in the 4th century BC as a significant
mendicant A mendicant (from , "begging") is one who practices mendicancy, relying chiefly or exclusively on alms to survive. In principle, Mendicant orders, mendicant religious orders own little property, either individually or collectively, and in many i ...
school of philosophy by the Cynics. ''Mathēmatikoi'' philosophers were in the 4th century BC absorbed into the Platonic school of
Speusippus Speusippus (; ; c. 408 – 339/8 BC) was an ancient Greece, ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Platonic Academy, Academy, near age 60, and remai ...
,
Xenocrates Xenocrates (; ; 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 define more closely, of ...
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 and Roman philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st c ...
. 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.


''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 recognise that the continuous development of mathematical and scientific research conducted by the ''mathēmatikoi'' was in line with Pythagoras's intention. Until the demise of Pythagoreanism in the 4th century BC, the ''akousmatikoi'' continued to engage in a pious life by practicing silence, dressing simply and avoiding meat, for the purpose of attaining a privileged
afterlife The afterlife or life after death is a purported existence in which the essential part of an individual's Stream of consciousness (psychology), stream of consciousness or Personal identity, identity continues to exist after the death of their ...
. The ''akousmatikoi'' engaged deeply in questions of Pythagoras' moral teachings, concerning matters such as
harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...
,
justice In its broadest sense, justice is the idea that individuals should be treated fairly. According to the ''Stanford Encyclopedia of Philosophy'', the most plausible candidate for a core definition comes from the ''Institutes (Justinian), Inst ...
,. ritual purity and moral behavior.


''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 most basic 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 and mathematical methods such as
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
and
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
. The ''mathēmatikoi'' philosophers claimed that numbers were at the heart of everything and constructed a new view of the
cosmos The cosmos (, ; ) is an alternative name for the universe or its nature or order. Usage of the word ''cosmos'' implies viewing the universe as a complex and orderly system or entity. The cosmos is studied in cosmologya broad discipline covering ...
. In the ''mathēmatikoi'' tradition of Pythagoreanism the
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
was removed from the center of the
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
. 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, the oracular 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 Philosophy, philosophical or Religion, religious concept that the non-physical essence of a living being begins a new lifespan (disambiguation), lifespan in a different physical ...
in another human body. Like the practitioners of Orphism, a religious tradition that developed in parallel to Pythagorean religious practice, Pythagoreanism held 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 Pythagoreans also engaged in rituals to attain purity. The 4th century Greek historian and sceptic philosopher Hecataeus of Abdera asserted that Pythagoras had been inspired by
ancient Egyptian Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
philosophy in his use of ritual regulations and his belief in
reincarnation Reincarnation, also known as rebirth or transmigration, is the Philosophy, philosophical or Religion, religious concept that the non-physical essence of a living being begins a new lifespan (disambiguation), lifespan in a different physical ...
.


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 Anaximander ( ; ''Anaximandros''; ) was a Pre-Socratic philosophy, pre-Socratic Ancient Greek philosophy, Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes Ltd, George Newnes, 1961, Vol. ...
,
Anaximenes of Miletus Anaximenes of Miletus (; ; ) was an Ancient Greece, Ancient Greek, Pre-Socratic philosophy, Pre-Socratic philosopher from Miletus in Anatolia (modern-day Turkey). He was the last of the three philosophers of the Ionian School (philosophy), Milesi ...
and Pherecydes of Syros. Of the Pythagorean philosophers, Hippasus, Alcmaeon, Hippon, Archytas and Theodorus, written sources have survived.


Arithmetic and numbers

Pythagoras, in his teachings focused on the significance of
numerology Numerology (known before the 20th century 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, ...
, 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s'' – that is positive
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
(there was no
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
). 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 extensively. 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 branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
. 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 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.. Pythagoreans thought numbers existed "outside of
uman Uman (, , ) is a city in Cherkasy Oblast, central Ukraine. It is located to the east of Vinnytsia. Located in the east of the historical region of Podolia, the city rests on the banks of the Umanka River. Uman serves as the administrative c ...
minds" and separate from the world. They had many
mystical Mysticism is popularly known as becoming one with God or the Absolute, but may refer to any kind of ecstasy or altered state of consciousness which is given a religious or spiritual meaning. It may also refer to the attainment of insight ...
and magical interpretations of the roles of numbers in governing existence.


Geometry

The Pythagoreans engaged with
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
as a liberal philosophy which served to establish principles and allowed theorems to be explored abstractly and rationally. 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
platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s: the
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, the
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
and the
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
. The sides of a regular dodecahedron are regular
pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s, which for Pythagoreans symbolised 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 around ...
, 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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
. 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 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 String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
and discovered quantitative mathematical relationships of music through arithmetic ratios. Pythagoras attempted to explain subjective psychological and aesthetic feelings, such as the enjoyment of musical harmony. Pythagoras and his students experimented systematically with strings of varying length and tension, with wind instruments, with brass discs of the same diameter but different thickness, and with identical vases filled with different levels of water. Early Pythagoreans established quantitative ratios between the length of a string or pipe and the pitch of notes and the frequency of string vibration. 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, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...
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 is a system of
musical tuning In music, there are two common meanings for tuning: * #Tuning practice, Tuning practice, the act of tuning an instrument or voice. * #Tuning systems, Tuning systems, the various systems of Pitch (music), pitches used to tune an instrument, and ...
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: McGraw-Hill). Vol. I: p. 56. . This ratio, also known as the " pure"
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 f ...
, 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, except for the h sound, which is pronounced without any stricture in the vocal tract. Examples are and pronou ...
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 nobility, German aristocrat and polymath, who was a poet, novelist, philosopher and Mysticism, mystic. He is regarded as an inf ...
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 things 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 (; born 375, fl. 335 BC) was a Ancient Greece, Greek Peripatetic school, Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musi ...
, music for the purification of the soul. Pythagoreans used different types of music to arouse or calm their souls, and certain stirring songs could have notes that existed in the same ratio as the "distances of the heavenly bodies from the centre of" Earth.


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. Unity and harmony is extended to all the opposites, which descent from the so-called Pythagorean "Table of ten Opposites", mentioned by
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
. Supreme opposites are the following ten couples: limit-unlimited, odd-even, one-many, right-left, male-female, rest-motion, straight-curved, light-darkness, good-evil, and square-oblong.


Cosmology

The philosopher Philolaus, one of the most prominent figures in Pythagoreanism,. was the precursor of
Copernicus Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance polymath who formulated a mathematical model, model of Celestial spheres#Renaissance, the universe that placed heliocentrism, the Sun rather than Earth at its cen ...
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 Anaximander ( ; ''Anaximandros''; ) was a Pre-Socratic philosophy, pre-Socratic Ancient Greek philosophy, Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes Ltd, George Newnes, 1961, Vol. ...
, 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 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 () to a concept, such as to 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., in Neoplatonis ...
. Philolaus called the number
one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...
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 divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...
it produced an even number, and when added to an
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...
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 was not based on
empirical observation Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how the ...
. 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 form ...
''. They reasoned that
stars A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...
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 (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
, which contained the fifth and fourth harmonic intervals.


Justice

Pythagoreans equated
justice In its broadest sense, justice is the idea that individuals should be treated fairly. According to the ''Stanford Encyclopedia of Philosophy'', the most plausible candidate for a core definition comes from the ''Institutes (Justinian), Inst ...
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 was later referenced by
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
. 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 Heraclides Ponticus ( ''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 is best remembered for proposing that the Earth ...
, 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 indicate that while early Pythagoreans did not believe that the soul contained all psychological faculties, the soul was life and a harmony of physical elements. As such the soul died when certain arrangements of these elements ceased to exist. However, the teaching most securely identified with Pythagoras is '' metempsychosis'', or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. Pythagorean metempsychosis resembles the teachings of the Orphics, although its version contains substantial differences. Unlike the Orphics, who considered metempsychosis a cycle of grief that could be escaped by attaining liberation from it, Pythagoras seems to postulate an eternal, endless reincarnation where subsequent lives would not be conditioned by any action done in the previous.


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 () is a concept in ancient Greek thought that refers to "excellence" of any kind—especially a person or thing's "full realization of potential or inherent function." The term may also refer to excellence in "Virtue, moral virtue." The conce ...
. The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conducive to the best in a human being. The prohibition on beans may be related to older Athenian ideas associating beans with
Hades Hades (; , , later ), in the ancient Greek religion and Greek mythology, mythology, is the god of the dead and the king of the Greek underworld, underworld, with which his name became synonymous. Hades was the eldest son of Cronus and Rhea ...
, as in the cult of Cyamites. Pythagoreans advanced a grounded theory on the treatment of animals. They believed that any being that experienced pain or suffering should not have pain inflicted on it unnecessarily. Because it was not necessary to inflict pain on animals for humans to enjoy a healthy diet, they believed that animals should not be killed for the purpose of eating them. The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill 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 (; ; , 444–443 BC) was a Ancient Greece, Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is known best for originating the Cosmogony, cosmogonic theory of the four cla ...
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 BC the head of the
Platonic Academy The Academy (), variously known as Plato's Academy, or the Platonic Academy, was founded in Classical Athens, Athens by Plato ''wikt:circa, circa'' 387 BC. The academy is regarded as the first institution of higher education in the west, where ...
Polemon included vegetarianism in his concept of living according to nature. In the 1st century AD
Ovid Publius Ovidius Naso (; 20 March 43 BC – AD 17/18), known in English as Ovid ( ), was a Augustan literature (ancient Rome), Roman poet who lived during the reign of Augustus. He was a younger contemporary of Virgil and Horace, with whom he i ...
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. 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, 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. Scholars believe that Perictione I was an Athenian and contemporary of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
, because in ''On the Harmony of Woman'' she wrote in Ionic and used the same terms of virtues as Plato had done in his ''
Republic A republic, based on the Latin phrase ''res publica'' ('public affair' or 'people's affair'), is a State (polity), state in which Power (social and political), political power rests with the public (people), typically through their Representat ...
'': ''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 and is believed to have been the daughter of a Spartan admiral killed in the battle of Arginusae 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 philosophise.


Influence on Plato and Aristotle

Pythagoras' teachings and Pythagoreanism influenced
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
'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 Gorgias ( ; ; – ) was an ancient Greek sophist, pre-Socratic philosopher, and rhetorician who was a native of Leontinoi in Sicily. Several doxographers report that he was a pupil of Empedocles, although he would only have been a few years ...
'', ''
Phaedo ''Phaedo'' (; , ''Phaidōn'') is a dialogue written by Plato, in which Socrates discusses the immortality of the soul and the nature of the afterlife with his friends in the hours leading up to his death. Socrates explores various arguments fo ...
'', and ''
Republic A republic, based on the Latin phrase ''res publica'' ('public affair' or 'people's affair'), is a State (polity), state in which Power (social and political), political power rests with the public (people), typically through their Representat ...
'', while the Pythagorean cosmology was discussed in Plato's '' Timaeus''. 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 edge ...
has been discussed extensively. Plato's dialogues have become an important surviving source of Pythagorean philosophic arguments.. Plato referenced Philolaus in ''Phaedo'' and wrote a Platonic adaptation of Philolaus' metaphysical system of limiters and unlimiteds. Plato also quoted from one of the surviving Archytas fragments in the ''Republic''. However, Plato's views that the primary role of mathematics was to turn the soul towards the world of forms, as expressed in ''Timaeus'', is regarded as Platonic philosophy, rather than Pythagorean.
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
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 Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
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 and Arius Didymus. In the 1st century AD Moderatus of Gades and
Nicomachus of Gerasa Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his ...
emerged as leading teachers of Neopythagoreanism. The most significant Neopythagorean teacher was
Apollonius of Tyana Apollonius of Tyana (; ; ) was a Greek philosopher and religious leader from the town of Tyana, Cappadocia in Roman Anatolia, who spent his life travelling and teaching in the Middle East, North Africa and India. He is a central figure in Ne ...
in the 1st century AD, who was regarded as a sage and lived as an ascetic. The last Neopythagorean philosopher was Numenius of Apamea in the 2nd century. Neopythagoreanism remained an elite movement which in the 3rd century merged into
Neoplatonism Neoplatonism is a version 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 series of thinkers. Among the common id ...
. Neopythagoreans combined Pythagorean teachings with Platonic, Peripatetic, Aristotelian and Stoic philosophic traditions. Two tendencies within Neopythagorean philosophy emerged, one that owed much to Stoic
monism Monism attributes oneness or singleness () to a concept, such as to 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., in Neoplatonis ...
and another that relied on Platonic dualism. Neopythagoreans refined the idea of
God In monotheistic belief systems, God is usually viewed as the supreme being, creator, and principal object of faith. In polytheistic belief systems, a god is "a spirit or being believed to have created, or for controlling some part of the un ...
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 Abstinence is the practice of self-enforced restraint from indulging in bodily activities that are widely experienced as giving pleasure. Most frequently, the term refers to sexual abstinence, but it can also mean abstinence from alcohol (drug), ...
. 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, which states that Jesus in Christianity, Jesus is the Son of God (Christianity), Son of God and Resurrection of Jesus, rose from the dead after his Crucifixion of Jesus, crucifixion, whose ...
was influenced by a Christianised form of
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...
, 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'' ...
, 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 most basic 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 cosmology, religious cosmological or supernatural place where beings such as deity, deities, angels, souls, saints, or Veneration of the dead, venerated ancestors are said to originate, be throne, ...
, earth and
hell In religion and folklore, hell is a location or state in the afterlife in which souls are subjected to punishment after death. Religions with a linear divine history sometimes depict hells as eternal destinations, such as Christianity and I ...
. 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 Photius I of Constantinople (, ''Phōtios''; 815 – 6 February 893), also spelled ''Photius''Fr. Justin Taylor, essay "Canon Law in the Age of the Fathers" (published in Jordan Hite, T.O.R., and Daniel J. Ward, O.S.B., "Readings, Cases, Mate ...
and John of Sacrobosco. The ''Corpus Areopagiticum or Corpus Dionysiacum'' was to be referenced in the late Middle Ages by
Dante Dante Alighieri (; most likely baptized Durante di Alighiero degli Alighieri; – September 14, 1321), widely known mononymously as Dante, was an Italian Italian poetry, poet, writer, and philosopher. His ''Divine Comedy'', originally called ...
and in the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
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 Neo ...
. Early Christian theologians, such as
Clement of Alexandria Titus Flavius Clemens, also known as Clement of Alexandria (; – ), was a Christian theology, Christian theologian and philosopher who taught at the Catechetical School of Alexandria. Among his pupils were Origen and Alexander of Jerusalem. A ...
, adopted the ascetic doctrines of the neopythagoreans. The moral and ethical teachings of Pythagorean influenced early
Christianity Christianity is an Abrahamic monotheistic religion, which states that Jesus in Christianity, Jesus is the Son of God (Christianity), Son of God and Resurrection of Jesus, rose from the dead after his Crucifixion of Jesus, crucifixion, whose ...
and assimilated into early Christian texts. The ''Sextou gnomai'' ('' Sentences of Sextus''), a
Hellenistic In classical antiquity, the Hellenistic period covers the time in Greek history after Classical Greece, between the death of Alexander the Great in 323 BC and the death of Cleopatra VII in 30 BC, which was followed by the ascendancy of the R ...
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 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
. 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'' was attributed by
Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
, the 1st century biographer of Pythagoras, to ''Sextus Pythagoricus''. The assertion was repeated subsequently by
Saint Jerome Jerome (; ; ; – 30 September 420), also known as Jerome of Stridon, was an early Christian priest, confessor, theologian, translator, and historian; he is commonly known as Saint Jerome. He is best known for his translation of the Bible ...
. In the 2nd century many of the ''Sentences of Sextus'' were cited by
Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...
as Pythagorean aphorisms. The ''Sentences of Sextus'' were translated into Syriac,
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
and
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
, 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 (; ; ; ), also called , was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt. The only event in Philo's life that can be decisively dated is his representation of the Alexandrian J ...
and Nicomachus popularised the mystical and cosmological symbolism Pythagoreans attributed to
number A number is a mathematical object used to count, measure, and label. The most basic 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 ...
s. This interest in Pythagorean views on the importance of numbers was sustained by mathematicians such as Theon of Smyrna, Anatolius and
Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
. These mathematicians relied on
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
's '' Timaeus'' as their source for Pythagorean philosophy.. In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
studies and adaptations of ''Timaeus'' solidified the view that there was a numerical explanation for proportion and
harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...
among learned men. Pythagoreanism, as mediated in Plato's ''Timaeus'', spurred increasingly detailed studies of
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
and harmony. Intellectuals pondered how knowledge of the
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
in which God had arranged the
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
could be applied to life. By the 12th century Pythagorean numerological concepts had become a universal language in medieval Europe and were no longer recognised as Pythagorean. Writers such as Thierry of Chartres, William of Conches and Alexander Neckham 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, orator, writer and Academic skeptic, who tried to uphold optimate principles during the political crises tha ...
,
Ovid Publius Ovidius Naso (; 20 March 43 BC – AD 17/18), known in English as Ovid ( ), was a Augustan literature (ancient Rome), Roman poet who lived during the reign of Augustus. He was a younger contemporary of Virgil and Horace, with whom he i ...
and Pliny, leading them to believe that mathematics was the key to understanding
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
and
nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
. Another important text on Pythagorean numerology was
Boethius Anicius Manlius Severinus Boethius, commonly known simply as Boethius (; Latin: ''Boetius''; 480–524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', polymath, historian, and philosopher of the Early Middl ...
's ''De arithmetica'', which was widely reproduced in the West. Boethius had relied on Nicomachus's writings as a source of Pythagoreanism.. The 11th-century Byzantine professor of philosophy
Michael Psellus Michael Psellos or Psellus (, ) 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 have died in 1078, although it has also been maintained tha ...
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 () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
to Pythagoras. Psellus thought to reconstruct
Iamblichus Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
' 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 by modern con ...
, a set of texts thought to be genuinely antique and which would be prolifically reproduced in the late Middle Ages. Manuel Bryennios introduced Pythagorean numerology to Byzantine music with his treatise ''Harmonics''. He argued that the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
was essential in attaining perfect harmony. In the Jewish communities the development of the
Kabbalah Kabbalah or Qabalah ( ; , ; ) is an esoteric method, discipline and school of thought in Jewish mysticism. It forms the foundation of Mysticism, mystical religious interpretations within Judaism. A traditional Kabbalist is called a Mekubbal ...
as esoteric doctrine became associated with numerology. It was only in the 1st century that Philo of Alexandria, developed a Jewish Pythagoreanism. In the 3rd century Hermippus popularised the belief that Pythagoras had been the basis for establishing key dates in Judaism. In the 4th century this assertion was further developed by Aristobulus. The Jewish Pythagorean numerology developed by Philo held that God as the unique One was the creator of all numbers, of which seven was the most divine and ten the most perfect. The medieval edition of the Kabbalah focused largely on a cosmological scheme of creation, in reference to early Pythagorean philosophers Philolaus and
Empedocles Empedocles (; ; , 444–443 BC) was a Ancient Greece, Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is known best for originating the Cosmogony, cosmogonic theory of the four cla ...
and helped to disseminate Jewish Pythagorean numerology.


On mathematics

Nicomachus' treatises were well known in Greek, Latin and the Arabic worlds. In the 9th century an Arabic translation of Nicomachus' ''Introduction to Arithmetic'' was published. 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 Italians, Italian translator of scientific books from Arabic into Latin. He worked in Toledo, Spain, Toledo, Kingdom of Castile and obtained the Arabic books in the libr ...
, 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 discussed
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
and division 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 mandatory financial charge or levy imposed on an individual or legal person, legal entity by a governmental organization to support government spending and public expenditures collectively or to Pigouvian tax, regulate and reduce nega ...
,
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 to ...
, 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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. In the 13th century Leonardo of Pisa, better known as Fibonacci, published the ''Libre quadratorum'' ('' The Book of Squares''). Fibonacci had studied scripts from Egypt, Syria, Greece and Sicily, and was learned in Hindu, Arabic and Greek methodologies. Using the
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
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 us ...
always arise through the addition of consecutive odd numbers starting with unity.
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
put forward a method of generating sets of three square numbers that satisfied the relationship first attributed to Pythagoras by
Vitruvius Vitruvius ( ; ; –70 BC – after ) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work titled . As the only treatise on architecture to survive from antiquity, it has been regarded since the Renaissan ...
, 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 , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
.


In the Middle Ages

In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
, from the 5th until the 15th century, Pythagorean texts remained popular. Late antique writers had produced adaptions of the '' Sentences of Sextus'' as '' The golden verses of Pythagoras''. The ''Golden Verses'' gained popularity and Christian adaptations of it appeared. These Christian adaptations were adopted by monastic orders, such as
Saint Benedict Benedict of Nursia (; ; 2 March 480 – 21 March 547), often known as Saint Benedict, was a Great Church, Christian monk. He is famed in the Catholic Church, the Eastern Orthodox Church, the Lutheran Churches, the Anglican Communion, and Old ...
, as authoritative Christian doctrine. In the Latin medieval western world, the ''Golden Verses'' became a widely reproduced text. Although the concept of the quadrivium 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 (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
. According to Proclus, Pythagoreanism divided all mathematical sciences into four categories:
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
,
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
,
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
.
Boethius Anicius Manlius Severinus Boethius, commonly known simply as Boethius (; Latin: ''Boetius''; 480–524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', polymath, historian, and philosopher of the Early Middl ...
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 (; : curriculums or curricula ) is the totality of student experiences that occur in an educational process. The term often refers specifically to a planned sequence of instruction, or to a view of the student's experi ...
s in medieval
school A school is the educational institution (and, in the case of in-person learning, the Educational architecture, building) designed to provide learning environments for the teaching of students, usually under the direction of teachers. Most co ...
s and
universities A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ...
. In the 12th century Pythagoras was credited by Hugh of Saint Victor 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, grammar is the set of rules for how a natural language is structured, as demonstrated by its speakers or writers. Grammar rules may concern the use of clauses, phrases, and words. The term may also refer to the study of such rul ...
,
rhetoric Rhetoric is the art of persuasion. It is one of the three ancient arts of discourse ( trivium) along with grammar and logic/ dialectic. As an academic discipline within the humanities, rhetoric aims to study the techniques that speakers or w ...
and
dialectic Dialectic (; ), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argument. Dialectic resembles debate, but the ...
. 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. In the early 6th century the Roman philosopher
Boethius Anicius Manlius Severinus Boethius, commonly known simply as Boethius (; Latin: ''Boetius''; 480–524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', polymath, historian, and philosopher of the Early Middl ...
popularised Pythagorean and Platonic conceptions of the universe and expounded the supreme importance of numerical ratios. The 7th century Bishop
Isidore of Seville Isidore of Seville (; 4 April 636) was a Spania, Hispano-Roman scholar, theologian and Roman Catholic Archdiocese of Seville, archbishop of Seville. He is widely regarded, in the words of the 19th-century historian Charles Forbes René de Montal ...
expressed his preference for the Pythagorean vision of a universe governed by the mystical properties of certain numbers, over the newly emerging
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
ean notion that knowledge could be built through deductive proofs. Isidore relied on the
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
of Nicomachus, who had fashioned himself as heir of Pythagoras, and took things further by studying the
etymology Etymology ( ) is the study of the origin and evolution of words—including their constituent units of sound and meaning—across time. In the 21st century a subfield within linguistics, etymology has become a more rigorously scientific study. ...
of the name of each number. The 12th century theologian Hugh of Saint Victor 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 rebuked the preoccupation with Pythagorean numerology, arguing that nature could not only be explained in terms of numbers.
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
's '' Timaeus'' 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 most basic 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 and
harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...
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 and al-Kindi, who pointed to the similarity between the harmony of music and the harmony of the soul. But Islamic philosophers such as
al-Farabi file:A21-133 grande.webp, thumbnail, 200px, Postage stamp of the USSR, issued on the 1100th anniversary of the birth of Al-Farabi (1975) Abu Nasr Muhammad al-Farabi (; – 14 December 950–12 January 951), known in the Greek East and Latin West ...
and
Ibn Sina Ibn Sina ( – 22 June 1037), commonly known in the West as Avicenna ( ), was a preeminent philosopher and physician of the Muslim world, flourishing during the Islamic Golden Age, serving in the courts of various Iranian peoples, Iranian ...
vehemently rejected this Pythagorean doctrine. in '' Kitab al-Musiqa al-Kabir'' Al-Farabi rejected the notion of celestial harmony on the grounds that it was "plainly wrong" and that it was not possible for the heavens, orbs and stars to emit sounds through their motions. The four books of the ''Corpus Areopagiticum or Corpus Dionysiacum'' (''The Celestrial Hierarchy'', ''The Ecclesiastical Hierarchy'', ''On Divine Names'' and ''The Mystical Theology'') 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'' ...
became enormously popular during the Middle Ages in the
Byzantine The Byzantine Empire, also known as the Eastern Roman Empire, was the continuation of the Roman Empire centred on Constantinople during late antiquity and the Middle Ages. Having survived the events that caused the fall of the Western Roman E ...
world, where 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 cosmology, religious cosmological or supernatural place where beings such as deity, deities, angels, souls, saints, or Veneration of the dead, venerated ancestors are said to originate, be throne, ...
, earth and
hell In religion and folklore, hell is a location or state in the afterlife in which souls are subjected to punishment after death. Religions with a linear divine history sometimes depict hells as eternal destinations, such as Christianity and I ...
, 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 Photius I of Constantinople (, ''Phōtios''; 815 – 6 February 893), also spelled ''Photius''Fr. Justin Taylor, essay "Canon Law in the Age of the Fathers" (published in Jordan Hite, T.O.R., and Daniel J. Ward, O.S.B., "Readings, Cases, Mate ...
. The 13th century astronomer and mathematician John of Sacrobosco in turn credited Pseudo-Dionysius when discussing the twelve signs of the zodiac. In the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
various classical texts that discussed Pythagorean ideas were reproduced and translated. Plato's '' Timaeus'' 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, William of Conches and Alexander Neckham referred not only to Plato but also to 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, orator, writer and Academic skeptic, who tried to uphold optimate principles during the political crises tha ...
,
Ovid Publius Ovidius Naso (; 20 March 43 BC – AD 17/18), known in English as Ovid ( ), was a Augustan literature (ancient Rome), Roman poet who lived during the reign of Augustus. He was a younger contemporary of Virgil and Horace, with whom he i ...
and Pliny. 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 preface of '' De revolutionibus'',
Copernicus Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance polymath who formulated a mathematical model, model of Celestial spheres#Renaissance, the universe that placed heliocentrism, the Sun rather than Earth at its cen ...
cites three pythagorean philosophers as precursors of the Heliocentric Theory: Hicetas, Philolaus and Ecphantus. :At first I found in
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, orator, writer and Academic skeptic, who tried to uphold optimate principles during the political crises tha ...
that Hicetas supposed the earth to move. Later I also discovered in
Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...
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 ( ''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 is best remembered for proposing that the Earth ...
and Ecphantus the Pythagorean make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center." In the 16th century
Vincenzo Galilei Vincenzo Galilei (3 April 1520 – 2 July 1591) was an Italian lutenist, composer, and music theory, music theorist. His children included the astronomer and physicist Galileo Galilei and the lute virtuoso and composer Michelagnolo Galilei. Vinc ...
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), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...
, engaged in an extended public exchange with his former teacher Zarlino. Zarlino supported the theory that if two weights in the ratio of 2 to 1 were attached to two strings, the pitches generated by the two strings would produce the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
. 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) was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England under King James I. Bacon argued for the importance of nat ...
, 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 of p ...
, Mersenne, 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 elasticity (solid mechanics), elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in a ...
were carried out by philosophers attached to the
French Academy of Sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
, the Accademia del Cimento 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 History of science, modern science during the early modern period, when developments in History of mathematics#Mathematics during the Scientific Revolution, mathemati ...
, as
Aristotelianism Aristotelianism ( ) is a philosophical tradition inspired by the work of Aristotle, usually characterized by Prior Analytics, deductive logic and an Posterior Analytics, analytic inductive method in the study of natural philosophy and metaphysics ...
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 of p ...
's '' harmonices mundi'' and
Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
's '' pre-established harmony''.
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
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 discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
. This mathematization of the physical reality climaxed in the 20th century. The pioneer of
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...
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) * Esoteric cosmology * Hippasus *
Ionian School (philosophy) The Ionian school of pre-Socratic philosophy refers to Ancient Greek philosophers, or a school of thought, in Ionia in the 6th century B.C, the first in the Western tradition. The Ionian school included such thinkers as Thales, Anaximander, ...
* 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''. Vol. 4. Century. pp. 379–380. The fal ...
'' *
Mathematical beauty Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as ...
* Mathematicism *
Pyrrhonism Pyrrhonism is an Ancient Greek school of philosophical skepticism which rejects dogma and advocates the suspension of judgement over the truth of all beliefs. It was founded by Aenesidemus in the first century BCE, and said to have been inspired ...
* Sacred geometry *
Tetractys The tetractys (), 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 representation of the fourt ...
* Unit-point atomism


References


Bibliography

* Reprinted by Dover Publications, Mineola, NY, 1960. * * * * * * * * * * ** * * * * * *


External links

* * {{Authority control Asceticism Presocratic philosophy Western esotericism Intentional communities Religions that require vegetarianism