Pythagoreanism began in the 6th century BC, based on the teachings and beliefs of Pythagoras and his followers, the Pythagoreans. Pythagoras created the first Pythagorean community in the ancient Greek city of Croton, located in modern-day Calabria, Italy, around 530 BC. Early Pythagorean groups spread across Magna Graecia, a region in southern Italy.
During Pythagoras's lifetime, it is likely that the Pythagorean community was divided into two groups: the akousmatikoi, who focused on religious and ritual practices, and the mathematikoi, who studied mathematics and philosophy. Ancient writers, such as Iamblichus and Porphyry, described these groups as "beginners" and "advanced" members. As Pythagoreans followed a secret path similar to ancient mystery schools, members of the akousmatikoi became mathematikoi after completing initiation rituals. It is incorrect to say that Pythagoreans were replaced by the Cynics in the 4th century BC. However, the Cynics are known for rejecting the rules and traditions important to the Pythagorean community. Over time, Greek philosophy became more varied. The Platonic Academy, founded in the 4th century BC outside Athens, may have been inspired by Pythagorean traditions. This area, dedicated to Athena and a figure named Academos, was described by Plutarch as a place transformed by the Athenian general Kimon, who lived around 510–450 BC. Kimon and the leader Themistocles likely helped rebuild Athens after the Persian destruction of the city in 480–479 BC.
Political changes in Magna Graecia led some Pythagorean philosophers to move to mainland Greece, while others gathered in Rhegium. By about 400 BC, most Pythagorean philosophers had left Italy. Their ideas greatly influenced Plato, who in turn shaped much of Western philosophy. Many records about Pythagoras come from Aristotle and his followers, the Peripatetic school.
As a philosophy, Pythagoreanism was revived in the 1st century BC, leading to Neopythagoreanism. Worship of Pythagoras continued in Italy, and Pythagoreans may have survived as part of or influenced religious groups like the Bacchic cults and Orphism.
History
Pythagoras was well known in ancient times for his mathematical discovery called the Pythagorean theorem. This theorem states 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 discovering that music has mathematical foundations. Ancient sources credit him with inventing the monochord, a tool used to demonstrate the relationship between musical intervals.
Most surviving information about Pythagoras comes from Aristotle and the philosophers of the Peripatetic school, who helped develop traditions like writing biographies and histories of science. Sources from the 5th century BC about Pythagoras and early Pythagoreanism do not include supernatural elements. However, sources from the 4th century BC added legends and stories about his teachings. Philosophers like Anaximander, Andron of Ephesus, Heraclides, and Neanthes studied Pythagoreanism using both written records and oral traditions. By the 4th century BC, these traditions included more legends. Neopythagorean philosophers, who wrote many surviving sources on Pythagoreanism, continued this tradition of adding stories and fantasy.
The earliest surviving ancient source about Pythagoras is a satire by Xenophanes, which describes Pythagorean beliefs about the transmigration of souls. Xenophanes wrote:
"Once they say that he was passing by when a puppy was being whipped,
And he took pity and said:
‘Stop! Do not beat it! For it is the soul of a friend
That I recognised when I heard it giving tongue.’"
A surviving fragment from Heraclitus describes Pythagoras and his followers as:
"Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selected of these writings made for himself a wisdom or made a wisdom of his own: a polymathy, an imposture."
Other surviving fragments about Pythagoras are from Ion of Chios and Empedocles, both born after his death. By that time, he was known as a wise man, and his fame spread throughout Greece. Ion described Pythagoras as:
"…distinguished for his manly virtue and modesty, even in death has a life which is pleasing to his soul, if Pythagoras the wise truly achieved knowledge and understanding beyond that of all men."
Empedocles described Pythagoras as "a man of surpassing knowledge, master especially of all kinds of wise works, who had acquired the utmost wealth of understanding." In the 4th century BC, the Sophist Alcidamas wrote that Pythagoras was widely honored by Italians.
Today, scholars divide Pythagoreanism into two periods: early-Pythagoreanism (6th to 5th century BC) and late-Pythagoreanism (4th to 3rd century BC). The Spartan colony of Taranto in Italy became a center for Pythagoreanism and later for Neopythagorean philosophers. Pythagoras had also lived in Crotone and Metaponto, which were Achaean colonies. Early-Pythagorean groups lived in Croton and other parts of Magna Graecia. They followed strict rules about diet, clothing, and behavior and believed in the immortality of the soul.
Early-Pythagorean groups were closed societies, and new members were chosen based on merit and discipline. Ancient sources say that early-Pythagoreans listened to teachings (akousmata) in silence for five years. Initiates could join the inner circle after passing a test. Pythagoreans could also leave the community if they wished. Iamblichus listed 235 Pythagoreans by name, including 17 women he called the "most famous" women practitioners of Pythagoreanism. Family members often became Pythagoreans, as Pythagoreanism became a tradition with rules for daily life and secrets. The home of Pythagoras was known as a place of mysteries.
Pythagoras was born on the island of Samos around 570 BC and left his homeland around 530 BC in opposition to the policies of Polycrates. Before settling in Croton, he traveled to Egypt and Babylonia. In Croton, he established the first Pythagorean community, described as a secret society, and gained political influence. In the early 5th century BC, Croton became an important city. Pythagoras emphasized moderation, piety, respect for elders and the state, and supported monogamy. The Croton Council gave him official roles, including overseeing education. His influence as a political reformer reportedly reached other Greek colonies in southern Italy and Sicily. Pythagoras died shortly after an arson attack on a Pythagorean meeting place in Croton.
In about 508 BC, anti-Pythagorean attacks led by Cylon of Croton forced Pythagoras to flee to Metapontium. After these attacks and Pythagoras’s death, Pythagorean communities in Croton and elsewhere continued to grow. Around 450 BC, attacks on Pythagorean groups occurred across Magna Graecia. In Croton, a house where Pythagoreans gathered was set on fire, and most of the philosophers inside died. Attacks also occurred in other cities, killing leaders. These events happened during widespread violence in Magna Graecia. Later, some Pythagoreans fled to mainland Greece, while others regrouped in Rhegium. By about 400 BC, most Pythagoreans had left Italy. Archytas stayed in Italy, and ancient sources say that young Plato visited him in the early 4th century BC. Pythagorean schools and societies declined by the 4th century BC. Though some Pythagoreans continued to practice, no organized
Philosophic traditions
After Pythagoras died, disagreements about his teachings led to the formation of two groups within Pythagoreanism in Italy: the akousmatikoi and the mathēmatikoi. The mathēmatikoi considered the akousmatikoi to be fellow Pythagoreans, but the akousmatikoi did not recognize the mathēmatikoi because the mathēmatikoi were believed to follow the teachings of Hippasus. Despite this, both groups were seen by others at the time as part of the Pythagorean tradition.
The akousmatikoi became less influential in the 4th century BC as the Cynics grew in importance as a school of philosophy. The mathēmatikoi, however, were absorbed into the Platonic school led by Speusippus, Xenocrates, and Polemon during the same period. Pythagoreanism was revived in the 1st century BC, leading to the development of Neopythagoreanism. Worship of Pythagoras continued in Italy for two centuries after his death. As a religious group, Pythagoreans may have survived or greatly influenced the Bacchic cults and Orphism.
The akousmatikoi believed humans should act in specific, proper ways. They collected all of Pythagoras’s sayings in a text called the Akousmata, which they treated as divine teachings. They refused to change or reinterpret Pythagoras’s ideas. People who followed most of the Akousmata were considered wise. The akousmatikoi rejected the mathēmatikoi’s scientific and mathematical work, claiming it did not align with Pythagoras’s goals. Until the 4th century BC, the akousmatikoi lived simply, practiced silence, avoided meat, and focused on moral teachings about harmony, justice, ritual purity, and behavior to achieve a better afterlife.
The mathēmatikoi accepted the religious aspects of Pythagoreanism and focused on learning and studying, especially mathematics, as part of their practice. While their work was mostly mathematical, they also explored other scientific fields Pythagoras had studied. A conflict developed between the strict akousmatikoi and the more progressive mathēmatikoi. This tension continued until the 4th century BC, when the philosopher Archytas used advanced mathematics to honor Pythagoras’s teachings.
Today, Pythagoras is best known for his mathematical ideas and the work of early Pythagoreans in developing theories about harmonic music, numbers, proportions, and methods like arithmetic and geometry. The mathēmatikoi believed numbers were the foundation of the universe and proposed a new view of the cosmos. They taught that the Earth, along with other celestial bodies, orbited a central fire, creating a cosmic harmony.
Pythagoreanism was both a philosophy and a religious practice. As a religious group, they followed oral teachings and worshipped Pythian Apollo, the god of the Delphic Oracle. They lived strictly, believing the soul was trapped in the body, which acted as a prison in this life. The highest goal was for the soul to join the gods and escape reincarnation. Like followers of Orphism, they believed the soul was punished for past wrongs and could be purified through rituals and disciplined living. The 4th-century historian Hecataeus of Abdera claimed Pythagoras was influenced by ancient Egyptian philosophy in his use of rituals and belief in reincarnation.
Philosophy
Early Pythagoreanism was based on studying and collecting knowledge from books written by other philosophers. Pythagoras' teachings directly referenced the ideas of Anaximander, Anaximenes of Miletus, and Pherecydes of Syros. Among the Pythagorean philosophers, Hippasus, Alcmaeon, Hippon, Archytas, and Theodorus are known because written records of their work have survived.
Pythagoras focused on the importance of numbers in explaining the universe. At his time, numbers were what we now call natural numbers—positive integers (without zero). Unlike other Greek thinkers, Pythagoreans used dots, called psiphi (pebbles), to represent numbers in shapes like triangles, squares, and pentagons. This helped them visualize math and explore how numbers relate to geometry. They studied numbers extensively, defining perfect numbers as those equal to the sum of their divisors, such as 28 = 1 + 2 + 4 + 7 + 14. The difference between odd and even numbers was central to their math. They arranged dots to show that even and odd numbers alternate, like 2, 4, 6… and 3, 5, 7…
Early Pythagorean philosophers like Philolaus and Archytas believed math could help solve important philosophical questions. For them, numbers were connected to abstract ideas. The number one represented intellect, two represented thought, and four symbolized justice because 2 × 2 = 4. The number three was especially important, as it represented the whole world, combining beginning, middle, and end. Pythagoreans also believed a person’s goodness had three parts: wisdom, drive, and fortune.
Pythagoreans thought numbers existed independently of human minds and were separate from the physical world. They gave numbers mystical meanings, believing they governed existence.
Pythagoreans studied geometry as a way to explore principles and theorems logically. They saw a close connection between numbers and shapes. They proved simple geometric rules, such as the sum of a triangle’s angles equaling two right angles. They also identified three of the five platonic solids: the tetrahedron, cube, and dodecahedron. The dodecahedron’s sides were regular pentagons, which Pythagoreans linked to health. They revered the pentagram because its diagonals divided each other in the golden ratio. Combining Babylonian algebra with their arithmetic helped Greek mathematicians develop geometric algebra. Pythagoreans helped create strict rules for solving math problems, laying the foundation for axiomatic methods.
Pythagoras studied music using math. He measured physical properties like string length and discovered relationships between musical notes and ratios. He explored how math could explain emotional responses to music. He and his students tested strings, wind instruments, brass discs, and water-filled vases to find patterns. They found that the most harmonious musical intervals came from simple ratios of the first four natural numbers: the octave (1/2), the fifth (2/3), and the fourth (3/4). The sum of these numbers, 10, was considered a perfect number because it represented the "whole essential nature of numbers." Werner Heisenberg called this discovery a major scientific advance because it allowed sound to be measured in space.
Pythagorean tuning uses the ratio 3:2, known as the perfect fifth, to create musical intervals. This ratio was chosen because it is easy to tune and linked to the importance of the number 3. Novalis called these musical proportions "particularly correct natural proportions."
Pythagoreans believed math could explain human emotions and the nature of reality. They saw numbers as the foundation of all things and the universe as a whole. Music was central to their lives, used to purify the soul, as noted by Aristoxenus. They used music to calm or excite emotions, and some songs had notes matching the distances of celestial bodies from Earth.
For Pythagoreans, harmony meant combining different elements into balance. They applied numeric harmony to math, medicine, psychology, art, and the study of the cosmos. They believed numbers governed all things and were the causes of existence. Numbers were the building blocks of all beings, and the universe was made of harmony and numbers.
Pythagoreans believed harmony connected all opposites, as listed in the "Table of ten Opposites" mentioned by Aristotle. These pairs included limit-unlimited, odd-even, one-many, right-left, male-female, rest-motion, straight-curved, light-darkness, good-evil, and square-oblong.
Philolaus, a key Pythagorean philosopher, influenced later thinkers like Copernicus by suggesting Earth was a planet, not the center of the cosmos. Anaximander, a teacher of Pythagoras, is credited with being the first to measure the size and distance of planets in the 6th century BC. Pythagoreans were among the first to try to explain the order of planets. Philolaus believed the cosmos was made of limited and unlimited elements.
Female philosophers
The stories about Pythagoras's life mention that his mother, wife, and daughters were important members of his group. Women who were part of the Pythagorean community had the same chances to study as men and learned both philosophy and practical skills for home life.
Many writings by female Pythagorean philosophers remain today. These writings are part of a collection called pseudoepigrapha Pythagorica, which was created by later followers of Pythagoras in the 1st or 2nd century. Some of the oldest writings in this collection are from early-Pythagorean women, while most of the surviving texts are from women who lived in the 4th and 3rd centuries BC. These women are among the earliest known female philosophers whose writings have survived.
Theano of Croton, the wife of Pythagoras, was an important leader in early-Pythagoreanism. She was known as a respected philosopher, and stories say she led the school after Pythagoras died. Texts from other female philosophers from the late-Pythagorean period also remain. These include Perictione I, Perictione II, Aesara of Lucania, and Phintys of Sparta.
Scholars think Perictione I lived in Athens and was alive at the same time as Plato. In her work On the Harmony of Woman, she used the same terms for virtues as Plato did in The Republic: andreia (courage), sophrosyne (self-control), dikaiosyne (justice), and sophia (wisdom). Perictione I wrote that these virtues help women develop wisdom and self-control, which can benefit their families and even their cities if women govern. She also said a wife should stay loyal to her husband, no matter his actions, which scholars believe reflects the limited legal rights of women in Athens.
Phyntis of Sparta was a Spartan woman and the daughter of an admiral who died in the battle of Arginusae in 406 BC. She wrote a work called Moderation of Women, in which she taught that moderation is important for women but also said that courage, justice, and wisdom belong to both men and women. Phyntis supported the idea that women should be allowed to study philosophy.
Influence on Plato and Aristotle
Pythagoras's teachings and Pythagoreanism affected Plato's writings about the universe, the mind, ethics, and government in the 5th century BC. However, Plato followed the main Greek philosophy of his time, and his ideas did not support the mix of experiments and math that was important in Pythagoreanism. The influence of Pythagoreanism lasted for many years because the belief in reincarnation was mentioned in Plato's works like Gorgias, Phaedo, and Republic. Also, Pythagorean ideas about the structure of the universe were discussed in Timaeus. Some scholars have studied how Pythagoreanism might have influenced Plato's ideas about harmony and geometric shapes called Platonic solids. The belief in reincarnation also appeared in Plato's dialogues. Plato's dialogues are now an important source of Pythagorean philosophical ideas. Plato mentioned Philolaus in Phaedo and adapted Philolaus's system of "limiters and unlimiteds" in his own way. He also quoted a part of Archytas's work in Republic. However, Plato's belief that math helps the soul understand the world of perfect forms, as described in Timaeus, is seen as part of his own philosophy, not Pythagoreanism.
In the 4th century BC, Aristotle disagreed with using math to study the world. He thought numbers were only a way to measure things and had no real value in explaining existence. It is hard to understand Aristotle's views on Pythagoreanism because he disliked their arguments and their ideas did not match his own. In On the Heavens, Aristotle rejected the Pythagorean idea of "the harmony of the spheres." However, he wrote a treatise about the Pythagoreans, though only parts of it remain. In that work, he described Pythagoras as a religious teacher with magical powers.
Neopythagoreanism
The Neopythagoreans were a school and a religious group. Their revival of Pythagorean ideas is linked to figures like Publius Nigidius Figulus, Eudorus of Alexandria, and Arius Didymus. In the 1st century AD, Moderatus of Gades and Nicomachus of Gerasa became important teachers of Neopythagoreanism. The most influential Neopythagorean teacher was Apollonius of Tyana in the 1st century AD, who was seen as a wise person and lived a simple, self-disciplined life. The last major Neopythagorean philosopher was Numenius of Apamea in the 2nd century. Neopythagoreanism remained a movement for educated people and later blended with Neoplatonism in the 3rd century.
Neopythagoreans combined Pythagorean teachings with ideas from Plato, Aristotle, and the Stoics. Two main ideas developed within Neopythagorean philosophy: one influenced by Stoic beliefs that everything is part of a single whole, and another shaped by Platonic ideas that separate the spiritual and physical worlds. Neopythagoreans improved the concept of God, placing Him beyond the physical world so He could not interact with physical things. They emphasized worshiping God in a spiritual way and believed life needed purification through self-restraint.
Neopythagoreans were deeply interested in numerology and the mystical aspects of Pythagoreanism. They connected these ideas with teachings from Plato’s followers. Neopythagorean philosophers often claimed their ideas were based on the teachings of Pythagoras himself, hoping to gain authority for their views.
Later influence
Christianity was influenced by a version of Platonism that had been adapted for Christianity. This version was explained in four books called the Corpus Areopagiticum or Corpus Dionysiacum: The Celestial Hierarchy, The Ecclesiastical Hierarchy, On Divine Names, and The Mystical Theology. These books were attributed to a figure named Pseudo-Dionysius the Areopagite. They described the relationships among celestial beings, humans, God, and the universe. Numbers were central to this explanation. According to The Celestial Hierarchy, the universe was divided into three parts: heaven, earth, and hell. Sunlight was seen as evidence of God’s presence. In the Middle Ages, this idea of dividing the universe into three parts was linked to the Pythagoreans. Earlier, it was considered an important source of Christian teaching by figures like Photius and John of Sacrobosco. The Corpus Areopagiticum was later referenced by Dante in the late Middle Ages and translated again by Marsilio Ficino during the Renaissance.
Early Christian theologians, such as Clement of Alexandria, adopted ideas from the neopythagoreans about self-discipline and morality. The teachings of the Pythagoreans influenced early Christianity and were included in Christian writings. A text called The Sentences of Sextus, which was originally a Hellenistic Pythagorean work, was modified to reflect Christian ideas. This text, which existed at least as early as the 2nd century, remained popular among Christians until the Middle Ages. It contained 451 sayings or principles, such as instructions to love truth, avoid pleasure, avoid flatterers, and control one’s speech. The text was attributed to a figure named Sextus Pythagoricus by Iamblichus, a 1st-century writer about Pythagoras. This claim was later repeated by Saint Jerome. In the 2nd century, Plutarch cited many of these sayings as Pythagorean aphorisms. The text was translated into Syriac, Latin, and Arabic, but only in the Latin-speaking world did it become widely used as a guide for daily life.
In the 1st century, writings by Philo and Nicomachus helped spread the Pythagorean belief that numbers had symbolic and cosmic meanings. This interest in Pythagorean ideas about numbers was continued by mathematicians such as Theon of Smyrna, Anatolius, and Iamblichus. These scholars used Plato’s Timaeus as a source for Pythagorean philosophy.
During the Middle Ages, studies of Timaeus reinforced the belief that numbers explained balance and harmony among learned people. Pythagorean ideas, as presented in Timaeus, led to more detailed studies of symmetry and harmony. Scholars wondered how the geometry of the universe, as arranged by God, could be applied to life. By the 12th century, Pythagorean numerology had become a common language in medieval Europe and was no longer seen as specifically Pythagorean. Thinkers like Thierry of Chartres, William of Conches, and Alexander Neckham referenced classical writers such as Cicero, Ovid, and Pliny, leading them to believe that mathematics was key to understanding astronomy and nature. Another important text was Boethius’s De arithmetica, which was widely copied in the West. Boethius relied on Nicomachus’s writings as a source for Pythagorean ideas.
In the 11th century, the Byzantine scholar Michael Psellus promoted Pythagorean numerology in his theological writings. He argued that Plato inherited Pythagorean knowledge and claimed that Diophantus’s mathematical inventions were linked to Pythagoras. Psellus also tried to reconstruct Iamblichus’s 10-book encyclopedia on Pythagoreanism from surviving fragments, spreading Iamblichus’s descriptions of Pythagorean physics, ethics, and theology at the Byzantine court. Psellus was said to have possessed the Hermetica, a set of ancient texts that were widely copied in the late Middle Ages. Manuel Bryennios introduced Pythagorean numerology to Byzantine music through his treatise Harmonics, arguing that the octave was essential for perfect harmony.
In Jewish communities, the development of the Kabbalah, a secret teaching system, became linked to numerology. Philo of Alexandria, in the 1st century, created a Jewish version of Pythagorean ideas. In the 3rd century, Hermippus claimed Pythagoras influenced key dates in Judaism. This idea was expanded by Aristobulus in the 4th century. Philo’s Jewish Pythagorean numerology taught that God, as the singular One, created all numbers, with seven being the most divine and ten the most perfect. The medieval Kabbalah focused on a creation plan inspired by early Pythagorean thinkers like Philolaus and Empedocles, helping spread Jewish Pythagorean numerology.
Nicomachus’s writings were well known in Greek, Latin, and Arabic cultures. In the 9th century, an Arabic translation of his Introduction to Arithmetic was published. These Arabic translations were later translated into Latin by Gerard of Cremona, making them part of the Latin tradition of numerology. The Pythagorean theorem appeared in Arabic manuscripts. Scholars in the Arabic world showed strong interest in Pythagorean ideas. In the 10th century, Abu al-Wafa’ Buzjani discussed multiplication and division in a treatise on arithmetic for business administrators, referencing Nicomachus. However, Islamic mathematicians focused more on practical problems like taxation, measurement, and trade rather than the numerology developed in the Latin world. Their arithmetic system was based on Hindu methods, which did not see numbers and geometry as symbolic.
In the Middle Ages, from the 5th to the 15th century, Pythagorean texts remained widely read. Late antique writers adapted The Sentences of Sextus into a text called The Golden Verses of Pythagoras. These verses became popular, and Christian versions of them were adopted by monastic groups, such as Saint Benedict, as official Christian teachings. In medieval Western Europe, The Golden Verses became a commonly copied text.
Although the idea of the quadrivium, a group of four subjects, originated with Archytas in the 4th century BC, it was