Pythagoreanism emerged in the 6th century BC, founded upon the teachings and beliefs of Pythagoras and his followers. Pythagoras established the first Pythagorean community in the ancient Greek colony of Croton, located in modern Calabria (Italy), around 530 BC. These initial Pythagorean communities subsequently expanded across Magna Graecia.
During Pythagoras's lifetime, a probable distinction existed between the akousmatikoi ("those who listen"), traditionally associated with religious and ritual practices and oral tradition, and the mathematikoi ("those who learn"). Ancient biographers of Pythagoras, including Iamblichus (c. 245 – c. AD 325) and his mentor Porphyry (c. 234 – c. AD 305), appear to have differentiated these groups as 'beginners' and 'advanced' adherents. Given that Pythagorean cenobites pursued an esoteric path, akin to ancient mystery schools, adherents initially categorized as akousmatikoi would progress to become mathematikoi after initiation. While it is inaccurate to assert that Cynics superseded Pythagoreans in the 4th century BC, the Cynic disregard for hierarchy, protocol, and initiatory procedures—elements crucial to the Pythagorean community—marked a significant distinction. Consequently, Greek philosophical traditions diversified. The Platonic Academy, established in the 4th century BC outside Athens' city walls, was arguably a Pythagorean cenobitic institution, situated within a sacred grove dedicated to Athena and Hecademos (Academos). Contemporaries seemingly believed that the sacred grove of Academos, where the Academy was located, might have existed since the Bronze Age, potentially predating the Trojan War. However, Plutarch attributes the transformation of this "waterless and arid spot" into a "well-watered grove, which he provided with clear running-tracks and shady walks" to the Athenian strategos (general) Kimon (c. 510 – c. 450 BC). Plato, who lived approximately a century later (circa 427 to 348 BC), would have encountered this transformed space. Conversely, this development was likely part of Athens' reconstruction efforts, spearheaded by Kimon and Themistocles, subsequent to the Achaemenid destruction of the city in 480–479 BC during the Persian Wars. Kimon is, at minimum, associated with the construction of the southern section of Themistocles' Wall, part of ancient Athens' fortifications. It is plausible that Athenians perceived this as a revitalization of the sacred grove of Academos.
Political instability in Magna Graecia prompted some Pythagorean philosophers to relocate to mainland Greece, while others reconvened in Rhegium. By approximately 400 BC, most Pythagorean philosophers had departed from Italy. Pythagorean concepts significantly influenced Plato, and through his work, profoundly impacted the entirety of Western philosophy. A substantial portion of extant sources concerning Pythagoras derive from Aristotle and the Peripatetic school of philosophers.
Pythagoreanism experienced a resurgence as a philosophical tradition in the 1st century BC, leading to the emergence of Neopythagoreanism. The veneration of Pythagoras persisted in Italy, and as a religious community, Pythagoreans seemingly endured either as constituents of, or significant influences upon, Bacchic cults and Orphism.
History
In antiquity, Pythagoras was renowned for his purported mathematical accomplishment, the Pythagorean theorem. He was credited with the discovery that, in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Furthermore, ancient accounts acknowledge Pythagoras for his revelation of the mathematical underpinnings of music. Historical sources attribute to Pythagoras not only the initial discovery of musical intervals but also the invention of the monochord, a device comprising a straight rod, a string, and a movable bridge, utilized to illustrate the relationships between these intervals.
The majority of extant sources concerning Pythagoras derive from Aristotle and the Peripatetic school, which established academic historiographical traditions including biography, doxography, and the history of science. Fifth-century BCE sources on Pythagoras and early Pythagoreanism lack supernatural elements, whereas fourth-century BCE accounts of his teachings incorporated legendary and mythical narratives. Philosophers like Anaximander, Andron of Ephesus, Heraclides, and Neanthes, who discussed Pythagoreanism, utilized both historical written records and the oral tradition, which was diminishing by the fourth century BCE. Neopythagorean philosophers, responsible for many of the surviving texts on Pythagoreanism, perpetuated this tradition of legend and fantasy.
The earliest extant ancient reference to Pythagoras and his adherents is a satirical work by Xenophanes, addressing Pythagorean doctrines concerning the transmigration of souls. Xenophanes recounted of Pythagoras:
It is recounted that, upon passing by a whipped puppy,
He expressed compassion, stating:
"Cease! Do not strike it! For it embodies the soul of a friend,
Which I recognized upon hearing its cry."
An extant fragment from Heraclitus characterizes Pythagoras and his disciples thus:
Pythagoras, son of Mnesarchus, pursued inquiry beyond all others, and by selecting from these writings, he fashioned a wisdom for himself, or created his own wisdom: a polymathy, an imposture.
Additional extant ancient fragments concerning Pythagoras originate from Ion of Chios and Empedocles. Both individuals were born in the 490s BCE, subsequent to Pythagoras's demise. By this era, he was recognized as a sage, and his renown had disseminated across Greece. Ion described Pythagoras as:
... distinguished by his masculine virtue and modesty, possessing even in death a life pleasing to his soul, provided that Pythagoras the wise genuinely attained knowledge and understanding surpassing all others.
Empedocles characterized Pythagoras as "a man of exceptional knowledge, particularly adept in all forms of sagacious endeavors, who had amassed the greatest wealth of understanding." During the fourth century BCE, the Sophist Alcidamas recorded that Pythagoras received widespread veneration from Italians.
Contemporary scholarship generally differentiates two distinct periods of Pythagoreanism: early Pythagoreanism, spanning the sixth to fifth centuries BCE, and late Pythagoreanism, from the fourth to third centuries BCE. The Spartan colony of Taranto in Italy served as a significant center for numerous Pythagorean practitioners and subsequently for Neopythagorean philosophers. Pythagoras himself had resided in Crotone and Metaponto, both Achaean colonies. Early Pythagorean communities flourished in Croton and across Magna Graecia, advocating a rigorous intellectual life and stringent regulations concerning diet, attire, and conduct. Their funerary practices were intrinsically linked to their conviction in the soul's immortality.
Early Pythagorean communities functioned as exclusive societies, with new members selected based on merit and discipline. Ancient accounts indicate that early Pythagoreans underwent a five-year initiation phase, involving silent reception of the teachings (akousmata). Successful initiates could, through an examination, gain admission to the inner circle. Nevertheless, Pythagoreans retained the option to depart from the community. Iamblichus enumerated 235 Pythagoreans by name, including 17 women whom he identified as the "most famous" female adherents of Pythagoreanism. It was common for family members to join the Pythagoreans, as the philosophy evolved into a tradition encompassing daily life regulations, and its members were bound by secrecy. Pythagoras's residence was recognized as a location of mysteries.
Born on the island of Samos around 570 BC, Pythagoras departed his homeland around 530 BC, opposing the policies of Polycrates. Before settling in Croton, he traveled extensively throughout Egypt and Babylonia. In Croton, Pythagoras established the first Pythagorean community, described as a secret society, and achieved political influence. Croton itself gained significant military and economic importance in the early 5th century BC. Pythagoras advocated for moderation, piety, respect for elders and the state, and promoted a monogamous family structure. The Croton Council appointed him to official positions, including responsibility for education within the city. His influence as a political reformer reportedly extended to other Greek colonies in southern Italy and Sicily. Pythagoras died shortly after an arson attack on the Pythagorean meeting place in Croton.
Anti-Pythagorean attacks, spearheaded by Cylon of Croton, took place around c. 508 BC, prompting Pythagoras to seek refuge in Metapontium. Despite these initial assaults and Pythagoras's death, Pythagorean communities in Croton and elsewhere continued to flourish. However, around 450 BC, a wave of attacks targeted Pythagorean communities across Magna Graecia. In Croton, a house where Pythagoreans convened was set on fire, resulting in the deaths of all but two of the philosophers. Pythagorean meeting places in other cities also suffered attacks, leading to the killing of philosophical leaders. These violent events occurred within a broader context of widespread destruction and instability in Magna Graecia. Following this political turmoil, some Pythagorean philosophers fled to mainland Greece, while others regrouped in Rhegium. By approximately 400 BC, the majority of Pythagorean philosophers had left Italy. Archytas, however, remained, and ancient sources record that he was visited there by a young Plato in the early 4th century BC. Pythagorean schools and societies ceased to exist as organized entities from the 4th century BC, though individual Pythagorean philosophers continued their practices without establishing formal communities.
Extant sources from the Neopythagorean philosopher Nicomachus identify Philolaus as Pythagoras's successor. Cicero (de Orat. III 34.139) further indicates that Philolaus taught Archytas. According to the Neoplatonist philosopher Iamblichus, Archytas, in turn, assumed leadership of the Pythagorean school approximately a century after Pythagoras's death. Aristoxenus names Philolaus, Eurytus, and Xenophilus as the teachers of the last generation of Pythagoreans.
Philosophical Traditions
Following Pythagoras's death, disagreements concerning his teachings led to the emergence of two distinct philosophical traditions within Pythagoreanism in Italy: the akousmatikoi and the mathēmatikoi. Although the mathēmatikoi recognized the akousmatikoi as fellow Pythagoreans, the akousmatikoi philosophers did not extend this recognition, reportedly because the mathēmatikoi adhered to the teachings of Hippasus. Despite this internal division, both groups were regarded by their contemporaries as practitioners of Pythagoreanism.
In the 4th century BC, the akousmatikoi were superseded as a significant mendicant school of philosophy by the Cynics. Concurrently, mathēmatikoi philosophers were absorbed into the Platonic school, which included Speusippus, Xenocrates, and Polemon. Pythagoreanism, as a philosophical tradition, experienced a revival in the 1st century BC, giving rise to Neopythagoreanism. The veneration of Pythagoras continued in Italy during the two intervening centuries. As a religious community, Pythagoreans appear to have persisted either as part of, or significantly influenced by, the Bacchic cults and Orphism.
Akousmatikoi
The akousmatikoi adhered to the principle that human conduct must be appropriate. The Akousmata (translated as "oral saying") comprised the complete collection of Pythagoras's pronouncements, revered as divine dogma. The tradition of the akousmatikoi opposed any reinterpretation or philosophical development of Pythagoras's doctrines. Adherents who rigorously observed the majority of the akousmata were esteemed for their wisdom. The akousmatikoi philosophers declined to acknowledge that the ongoing mathematical and scientific advancements pursued by the mathēmatikoi aligned with Pythagoras's original intent. Until the 4th century BC, when Pythagoreanism declined, the akousmatikoi maintained a devout lifestyle, characterized by silence, simple attire, and vegetarianism, with the aim of securing a favored afterlife. The akousmatikoi focused extensively on Pythagoras's ethical doctrines, encompassing concepts like harmony, justice, ritual purity, and virtuous conduct.
Mathēmatikoi
The mathēmatikoi recognized the religious foundations of Pythagoreanism and incorporated mathēma (meaning "learning" or "studying") into their practices. Although their scientific endeavors were predominantly mathematical, they also championed other scientific disciplines that Pythagoras had explored during his lifetime. A sectarian division emerged between the dogmatic akousmatikoi and the mathēmatikoi, the latter of whom were perceived as increasingly progressive due to their intellectual pursuits. This divergence continued until the 4th century BC, when the philosopher Archytas integrated advanced mathematics into his commitment to Pythagorean doctrines.
Presently, Pythagoras is primarily recognized for his mathematical contributions, and for the advancements made by early Pythagoreans in mathematical concepts and theories, including harmonic musical intervals, the definition of numbers, proportion, and mathematical methodologies such as arithmetic and geometry. The mathēmatikoi philosophers asserted the fundamental importance of numbers in all phenomena and developed a novel cosmological perspective. Within the mathēmatikoi branch of Pythagoreanism, the Earth was dislodged from its traditional position at the universe's center. The mathēmatikoi postulated that the Earth, alongside other celestial entities, revolved around a central fire. This arrangement, they contended, established a celestial harmony.
Rituals
Pythagoreanism encompassed both a philosophical tradition and a religious practice. As a religious fraternity, they adhered to oral doctrines and venerated Pythian Apollo, the oracular deity of the Delphic Oracle. Pythagoreans advocated for an ascetic lifestyle. They posited that the soul was interred within the body, which served as its sepulchre during earthly existence. The ultimate human aspiration was for the soul to assimilate into the divine realm, thereby transcending the cycle of reincarnation into another mortal form. Similar to the adherents of Orphism, a religious tradition concurrent with Pythagorean practice, Pythagoreanism maintained that the soul's embodiment was a punitive consequence for transgressions, and that purification of the soul was achievable. Beyond adhering to stringent daily regulations, Pythagoreans also performed rituals aimed at achieving spiritual purity. Hecataeus of Abdera, a 4th-century Greek historian and skeptical philosopher, contended that Pythagoras's adoption of ritualistic precepts and his belief in metempsychosis were influenced by ancient Egyptian philosophy.
Philosophy
Initial Pythagoreanism was founded upon scholarly inquiry and the synthesis of knowledge derived from the works of other philosophers. Pythagoras's philosophical doctrines directly referenced the philosophies of Anaximander, Anaximenes of Miletus, and Pherecydes of Syros. Written accounts pertaining to the Pythagorean philosophers Hippasus, Alcmaeon, Hippon, Archytas, and Theodorus are extant.
Arithmetic and numbers
Pythagoras's teachings emphasized the profound significance of numerology, positing that numbers inherently elucidated the universe's true nature. In the Hellenic context of Pythagoras's era, numbers were understood as natural numbers, specifically positive integers, as the concept of zero had not yet emerged. Distinct from their Greek contemporaries, Pythagorean philosophers depicted numbers graphically rather than through symbolic letters. They employed dots, also referred to as psiphi (pebbles), to illustrate numerical concepts within geometric shapes such as triangles, squares, rectangles, and pentagons. This approach facilitated a visual understanding of mathematics and enabled a geometric investigation of numerical relationships. Pythagorean philosophers conducted extensive research into numerical relationships, defining perfect numbers as those equivalent to the sum of all their proper divisors (e.g., 28 = 1 + 2 + 4 + 7 + 14). The theory of odd and even numbers constituted a fundamental aspect of Pythagorean arithmetic. This classification was perceived by Pythagorean philosophers as both direct and visual, as they arranged triangular dots to demonstrate the successive alternation of even and odd numbers (e.g., 2, 4, 6, ... and 3, 5, 7, ...).
Early Pythagorean philosophers, including Philolaus and Archytas, maintained the belief that mathematics offered a means to resolve significant philosophical challenges. Within Pythagoreanism, numbers acquired associations with abstract concepts. For instance, one was linked to intellect and being, two to thought, and the number four to justice, owing to its derivation (2 × 2 = 4) and its even parity. The number three held prominent symbolic significance, as Pythagoreans contended that the entire cosmos and its constituents were encapsulated within this number, representing the synthesis of beginning, middle, and end. Furthermore, the triad possessed an ethical dimension for Pythagoreans, who considered individual goodness to be tripartite: prudence, drive, and good fortune.
Pythagoreans posited that numbers existed independently of human cognition and distinct from the physical world. They attributed numerous mystical and magical interpretations to the roles numbers played in governing existence.
Geometry
The Pythagoreans approached geometry as a liberal philosophy, utilizing it to establish fundamental principles and to facilitate the abstract and rational exploration of theorems. Pythagorean philosophers asserted a profound connection between numbers and geometric forms. Early Pythagorean thinkers demonstrated elementary geometric theorems, such as the principle that the sum of a triangle's angles equals two right angles. Furthermore, Pythagoreans identified three of the five Platonic solids: the tetrahedron, the cube, and the dodecahedron. The regular pentagonal faces of the dodecahedron held symbolic significance for Pythagoreans, representing health. They also venerated the pentagram, noting that each of its diagonals bisects the other two at the golden ratio. The transition from dots to linear geometric figures, combined with Babylonian algebra and Pythagorean arithmetic, laid the groundwork for Greek geometric algebra. Through their efforts to establish a system of concrete and immutable rules, Pythagoreans contributed significantly to the development of rigorous axiomatic procedures for mathematical problem-solving.
Music
Pythagoras initiated the mathematical and experimental investigation of music. He objectively quantified physical attributes, such as string length, thereby uncovering quantitative mathematical relationships in music through arithmetic ratios. Pythagoras also endeavored to elucidate subjective psychological and aesthetic experiences, including the appreciation of musical harmony. Collaborating with his students, Pythagoras systematically experimented with strings of diverse lengths and tensions, various wind instruments, brass discs of identical diameter but differing thickness, and identical vases containing varying water levels. Early Pythagoreans successfully established quantitative ratios correlating the length of a string or pipe with the pitch of notes and the frequency of string vibration.
Pythagoras is attributed the discovery that the most harmonious musical intervals result from the simple numerical ratios of the first four natural numbers, specifically derived from string length relationships: the octave (1/2), the fifth (2/3), and the fourth (3/4). For Pythagoreans, the sum of these numbers (1 + 2 + 3 + 4 = 10) constituted the perfect number, as it was believed to encapsulate "the whole essential nature of numbers." Werner Heisenberg characterized this mathematical formulation of musical principles as "among the most powerful advances of human science," noting its capacity to facilitate the spatial measurement of sound.
Pythagorean tuning represents a musical temperament system where all interval frequency ratios are fundamentally derived from the 3:2 ratio. This specific ratio, recognized as the "pure" perfect fifth, was selected due to its high consonance, ease of aural tuning, and the symbolic significance ascribed to the integer three. Novalis articulated this concept by stating, "The musical proportions seem to me to be particularly correct natural proportions."
The realization that mathematical principles could elucidate the human emotional realm profoundly influenced Pythagorean philosophy. Consequently, Pythagoreanism evolved into an inquiry focused on discerning the fundamental essences of reality. Pythagorean thinkers firmly asserted that numbers constituted the essence of all phenomena and that the cosmos was maintained by an inherent harmony. Ancient accounts indicate that music held a pivotal role in the lives of adherents to Pythagoreanism. They employed medicinal remedies for bodily purification (katharsis) and, as noted by Aristoxenus, utilized music for the purification of the soul. Pythagoreans strategically applied various musical forms to either stimulate or tranquilize their psyches, with specific evocative melodies featuring notes whose ratios mirrored the "distances of the heavenly bodies from the centre of" Earth.
Harmony
Within Pythagorean thought, harmony denoted the "unification of a multifarious composition and the agreement of unlike spirits." Numeric harmony, within Pythagoreanism, found application across mathematical, medical, psychological, aesthetic, metaphysical, and cosmological domains. Pythagorean philosophers posited that the fundamental characteristic of numbers manifested through the harmonious interaction of opposing pairs. This harmony was understood to ensure the equilibrium of antithetical forces. Pythagoras, in his doctrines, identified numbers and their inherent symmetries as the primary principle, designating these numeric symmetries as harmony. Such numeric harmony was believed to be discernible in the governing principles across all of nature. Numbers were considered to dictate the properties and conditions of all entities and were regarded as the foundational causes of existence in all other phenomena. Ultimately, Pythagorean philosophers contended that numbers constituted the elemental components of all beings, and that the universe itself was fundamentally composed of harmony and numbers.
The concepts of unity and harmony were extended to encompass all opposing principles, which originated from the Pythagorean "Table of ten Opposites," as documented by Aristotle. These supreme opposing pairs include: limit-unlimited, odd-even, one-many, right-left, male-female, rest-motion, straight-curved, light-darkness, good-evil, and square-oblong.
Cosmology
Philolaus, a distinguished Pythagorean philosopher, anticipated Copernicus by positing Earth as a planet, rather than the cosmic center. Eudemus of Cyprus, a student of Aristotle, attributed to Anaximander—a teacher of Pythagoras in the 6th century BC—the pioneering effort to quantitatively determine the sizes of known planets and their inter-planetary distances. Historical accounts often credit Pythagorean philosophers with the initial attempts to systematically order the planets. Philolaus, an early Pythagorean, posited that the cosmos was composed of both limited and unlimited elements, which had perpetually existed. For Philolaus, the cosmic center was represented by the number one (hēn), which he equated with the unity inherent in Monism. He termed the number one "even-odd" due to its capacity to generate both even and odd numbers. Specifically, adding one to an odd number yielded an even number, while adding it to an even number produced an odd number. Philolaus further theorized that the structural integration of Earth and the universe mirrored the formation of the number one from the interaction of even and odd. Pythagorean philosophers generally held that the even represented the unlimited, whereas the odd symbolized the limited.
In the 4th century BC, Aristotle documented the Pythagorean astronomical system, stating:
- Aristotle noted the ongoing discourse regarding Earth's position, motion, and morphology. While the majority, particularly those who considered the cosmos finite, posited Earth at the center, the Italian philosophers known as Pythagoreans held an opposing perspective. They theorized that a central fire occupied the universe's core, with Earth functioning as a celestial body that generated day and night through its orbital movement around this central point. Furthermore, they conceived of an additional celestial body, termed the "counter-earth," positioned opposite to Earth.
The precise shape of Earth, whether spherical or flat, according to Philolaus, remains undetermined; however, he did not subscribe to the idea of Earth's rotation. Consequently, neither the Counter-Earth nor the Central Fire would have been observable from Earth's surface, or at least not from the Greek hemisphere. Importantly, the Pythagorean philosophers' rejection of a geocentric universe was not derived from empirical data. Instead, as Aristotle observed, their astronomical model stemmed from a profound philosophical contemplation concerning the intrinsic worth of individual entities and the cosmos's hierarchical structure.
Adherents of Pythagoreanism posited the existence of a musica universalis. Their rationale suggested that celestial bodies, being massive and rapidly moving, inherently generated sound. Furthermore, Pythagoreans concluded that stars orbited at mutually proportional distances and velocities. This numerical proportionality, they argued, resulted in the production of a harmonic sound during stellar revolutions. Philolaus, an early Pythagorean philosopher, contended that the cosmic structure was governed by the musical numerical ratios inherent in the diatonic octave, which encompasses the fifth and fourth harmonic intervals.
Justice
Pythagoreans conceptualized justice as analogous to geometrical proportion, asserting that proportionality guaranteed each component received its rightful share. Early Pythagoreans held that post-mortem, the soul would undergo either punishment or reward. Human actions, they believed, could determine the soul's admission to an alternative realm, with reincarnation into this world signifying a form of retribution. Within Pythagorean philosophy, earthly life was inherently social, and societal justice was realized when every societal segment obtained its due. Plato subsequently cited this Pythagorean concept of universal justice. For Pythagorean thinkers, the soul represented the fount of justice, and through its harmony, divine realization was attainable. Conversely, injustice was perceived as a subversion of the natural order. Heraclides Ponticus, a 4th-century BC philosopher, attributed to Pythagoras the teaching that "happiness consists in knowledge of the perfection of the numbers of the soul." A fragment from the 3rd century BC by the later Pythagorean philosopher Aesara articulated:
I believe that human nature establishes a universal benchmark for law and justice, applicable to both the familial unit and the polis. Those who introspectively explore and seek will uncover this truth, for within lies law and justice, which constitutes the soul's appropriate configuration.
Body and Soul
Pythagorean philosophy posited an integrated function of body and soul, asserting that physical well-being necessitated a sound psyche. Early Pythagoreans conceptualized the soul as the locus of sensation and emotion, distinguishing it from the intellect. Nevertheless, due to the fragmentary nature of surviving early Pythagorean texts, the certainty of their belief in soul immortality remains elusive. Extant writings by the Pythagorean philosopher Philolaus suggest that while early Pythagoreans did not attribute all psychological faculties to the soul, they considered it the essence of life and a harmonious arrangement of physical constituents. Consequently, the soul was believed to perish when specific configurations of these elements dissolved.
Nevertheless, the doctrine most definitively associated with Pythagoras is metempsychosis, or the "transmigration of souls," which asserts the immortality of every soul and its subsequent entry into a new physical form upon death. While Pythagorean metempsychosis bears similarities to Orphic teachings, its interpretation presents significant distinctions. In contrast to the Orphics, who viewed metempsychosis as a sorrowful cycle from which liberation could be achieved, Pythagoras appears to have proposed an perpetual, unending reincarnation where successive existences were not contingent upon prior actions.
Vegetarianism
Medieval scholars documented a "Pythagorean diet," characterized by the avoidance of meat, beans, and fish. Adherents of Pythagoreanism posited that a vegetarian regimen promoted physical well-being and facilitated the pursuit of Arete. Vegetarianism within Pythagorean philosophy was not conceived as an act of self-abnegation but rather as a practice fostering human excellence. The proscription against beans might stem from ancient Athenian beliefs linking them to Hades, as observed in the cult of Cyamites. Pythagoreans developed a comprehensive theoretical framework concerning the ethical treatment of animals. They maintained that no sentient being should be subjected to gratuitous pain or suffering. Given that a healthy human diet did not necessitate inflicting pain upon animals, Pythagoreans concluded that animals should not be slaughtered for consumption. Furthermore, Pythagoreans argued that killing an animal was unjustifiable unless it presented a direct threat to a human, asserting that such an act would degrade human moral standing. Consequently, a failure to extend justice to animals was seen as a self-diminishing act for humanity.
Pythagoreans conceptualized humans as animals endowed with superior intellect, necessitating self-purification through rigorous training. This purification process was believed to enable humans to integrate with the pervasive psychic force of the cosmos. They contended that the ethical implications of this argument could not be circumvented by merely ensuring a painless death for an animal. Moreover, Pythagoreans considered animals to be sentient and possessing a rudimentary form of rationality. The compelling arguments put forth by Pythagoreans persuaded many contemporary philosophers to embrace a vegetarian lifestyle. Their profound sense of kinship with non-human life established them as a countercultural movement within a predominantly meat-eating society. The philosopher Empedocles, for instance, reportedly declined the traditional blood sacrifice following his victory in an Olympic horse race, offering a substitute instead.
Later Pythagorean philosophers were assimilated into the Platonic school, and by the 4th century BC, Polemon, the head of the Platonic Academy, incorporated vegetarianism into his philosophy of living in harmony with nature. In the 1st century AD, Ovid recognized Pythagoras as the pioneering advocate against meat consumption. However, the comprehensive Pythagorean argument against animal mistreatment did not endure universally. Pythagoreans had previously asserted that specific foods stimulated passions and impeded spiritual progress. Consequently, Porphyry drew upon Pythagorean doctrines to contend that abstaining from meat for spiritual purification should be exclusively practiced by philosophers striving for a divine state.
Female Philosophers
Biographical accounts of Pythagoras indicate that his mother, wife, and daughters were integral members of his inner circle. Women were afforded equitable opportunities for Pythagorean study, acquiring both philosophical knowledge and practical domestic competencies.
A significant portion of extant texts by female Pythagorean philosophers belongs to a compilation known as pseudoepigrapha Pythagorica, assembled by Neopythagoreans during the 1st or 2nd century. While some surviving fragments from this collection are attributed to early-Pythagorean female philosophers, the majority of the extant writings originate from late-Pythagorean women philosophers active in the 4th and 3rd centuries BC. Female Pythagoreans represent some of the earliest documented female philosophers whose writings have been preserved.
Theano of Croton, Pythagoras's wife, is recognized as a prominent figure in early Pythagoreanism. She was esteemed as a distinguished philosopher, and according to prevailing lore, she assumed leadership of the school following his demise. Furthermore, textual fragments from female philosophers of the late-Pythagorean era have been preserved. Notable among these are Perictione I, Perictione II, Aesara of Lucania, and Phintys of Sparta.
Academic consensus suggests that Perictione I, an Athenian, was a contemporary of Plato. This conclusion is drawn from her work, On the Harmony of Woman, written in Ionic, which employs the same virtue terminology—andreia, sophrosyne, dikaiosyne, and sophia—as found in Plato's Republic. Within On the Harmony of Woman, Perictione I delineates the prerequisites for women to cultivate wisdom and self-control. She posits that these virtues would yield substantial benefits for a woman, her spouse, her offspring, the household, and even the polis, particularly "if, at any rate, such a woman should govern cities and tribes". Scholars interpret her advocacy for a wife's unwavering devotion to her husband, irrespective of his conduct, as a pragmatic adaptation to the prevailing legal framework for women in Athenian society. The Pythagorean philosopher Phyntis, a Spartan, is traditionally identified as the daughter of a Spartan admiral who perished in the Battle of Arginusae in 406 BC. Phyntis authored the treatise Moderation of Women, wherein she ascribed the virtue of moderation specifically to women, yet concurrently affirmed that "courage and justice and wisdom are common to both" genders. Furthermore, Phyntis championed the right of women to engage in philosophical inquiry.
The Impact on Plato and Aristotle
The doctrines of Pythagoras and the broader tenets of Pythagoreanism significantly shaped Plato's philosophical discourse on physical cosmology, psychology, ethics, and political philosophy during the 5th century BC. Nevertheless, Plato's adherence to prevailing Greek philosophical traditions led Platonic philosophy to de-emphasize the integration of experimental methodology and mathematics, a combination intrinsic to Pythagorean thought. Pythagoreanism's influence persisted throughout and beyond antiquity, notably through Plato's works; its doctrine of reincarnation is articulated in his Gorgias, Phaedo, and Republic, while Pythagorean cosmology is explored in his Timaeus. The potential impact of Pythagoreanism on Plato's theories of harmony and the Platonic solids has been a subject of extensive scholarly examination. Furthermore, the concept of transmigration, or reincarnation, was incorporated into Plato's dialogic pedagogy. Consequently, Plato's dialogues serve as a crucial extant repository for Pythagorean philosophical arguments. Plato explicitly referenced Philolaus in Phaedo and subsequently developed a Platonic interpretation of Philolaus' metaphysical framework concerning limiters and unlimiteds. Additionally, he cited a surviving fragment from Archytas within the Republic. Nonetheless, Plato's perspective, articulated in Timaeus, that mathematics primarily functions to orient the soul toward the realm of Forms, is generally categorized as distinctly Platonic rather than Pythagorean.
During the 4th century BC, Aristotle dismissed mathematics as a valid instrument for investigating and comprehending the empirical world. His conviction was that numbers merely represented quantitative descriptors, devoid of inherent ontological significance. Interpreting Aristotle's engagement with Pythagorean philosophy presents challenges, primarily due to his apparent lack of affinity for Pythagorean arguments and the incongruence of Pythagoreanism with his own philosophical framework. Specifically, in On the Heavens, Aristotle directly challenged the Pythagorean concept of the harmony of the spheres. Despite this, he authored a treatise concerning the Pythagoreans, of which only fragments remain, portraying Pythagoras as a miraculous religious instructor.
Neopythagoreanism
Neopythagoreanism constituted both a philosophical school and a religious fellowship. The resurgence of Pythagorean thought is often credited to figures such as Publius Nigidius Figulus, Eudorus of Alexandria, and Arius Didymus. During the 1st century AD, Moderatus of Gades and Nicomachus of Gerasa rose to prominence as influential exponents of Neopythagoreanism. Apollonius of Tyana, also active in the 1st century AD, is considered the most notable Neopythagorean teacher, revered as a sage and known for his ascetic lifestyle. Numenius of Apamea, active in the 2nd century, is recognized as the final significant Neopythagorean philosopher. Ultimately, Neopythagoreanism persisted as an exclusive intellectual current, eventually integrating into Neoplatonism by the 3rd century.
Neopythagoreans synthesized Pythagorean doctrines with elements drawn from Platonic, Peripatetic, Aristotelian, and Stoic philosophical traditions. Within Neopythagorean philosophy, two distinct currents developed: one significantly influenced by Stoic monism, and another grounded in Platonic dualism. They further refined the concept of the divine, positing God as transcendent beyond the finite realm, thereby precluding any direct interaction with corporeal existence. Neopythagoreans advocated for a spiritual form of divine worship and emphasized the necessity of purifying one's life through ascetic abstinence.
Neopythagoreans exhibited a profound interest in numerology and the superstitious dimensions of Pythagorean thought, integrating these with the philosophical teachings of Plato's successors. Following a common ancient practice, Neopythagorean philosophers frequently attributed their doctrines to the designated founder of their tradition, specifically Pythagoras himself, to enhance the authoritative standing of their perspectives.
Subsequent Influence
Influence on Early Christianity
Early Christianity was significantly shaped by a Christianized form of Platonism, articulated within the four books of the Corpus Areopagiticum or Corpus Dionysiacum: The Celestial Hierarchy, The Ecclesiastical Hierarchy, On Divine Names, and The Mystical Theology. These texts, attributed to Pseudo-Dionysius the Areopagite, elucidated the intricate relationships among celestial beings, humanity, God, and the cosmos. Central to this exposition was the role of numbers. Specifically, The Celestial Hierarchy posited a threefold cosmic division comprising heaven, earth, and hell. Sunlight, illuminating the universe, was presented as evidence of God's omnipresence. During the Middle Ages, this numerological cosmic division was ascribed to Pythagorean influence, although earlier it had been considered an authoritative source of Christian doctrine by figures such as Photius and John of Sacrobosco. The Corpus Areopagiticum or Corpus Dionysiacum was later referenced by Dante in the late Middle Ages, and a new translation was produced by Marsilio Ficino during the Renaissance.
Prominent early Christian theologians, including Clement of Alexandria, incorporated ascetic doctrines derived from Neopythagoreanism. Pythagorean moral and ethical teachings influenced early Christianity, becoming assimilated into foundational Christian texts. The Sextou gnomai (Sentences of Sextus), a Hellenistic Pythagorean text adapted to a Christian perspective, was extant from at least the 2nd century and maintained considerable popularity among Christians throughout the Middle Ages. Comprising 451 sayings or principles, the Sentences of Sextus included precepts such as loving truth, avoiding bodily indulgence, shunning flatterers, and exercising mental control over speech. Iamblichus, a 1st-century biographer of Pythagoras, attributed the content of the Sentences of Sextus to Sextus Pythagoricus, an attribution later reiterated by Saint Jerome. In the 2nd century, Plutarch cited numerous passages from the Sentences of Sextus as Pythagorean aphorisms. While the Sentences of Sextus were translated into Syriac, Latin, and Arabic—the common written language for both Muslims and Jews at the time—their widespread circulation as a guide for daily life was primarily confined to the Latin-speaking world.
Influence on Numerology
First-century treatises by Philo and Nicomachus significantly popularized the mystical and cosmological symbolism that Pythagoreans ascribed to numbers. This scholarly interest in Pythagorean perspectives on numerical significance was perpetuated by mathematicians including Theon of Smyrna, Anatolius, and Iamblichus. These mathematicians consistently referenced Plato's Timaeus as a primary source for Pythagorean philosophy.
During the Middle Ages, scholarly examinations and adaptations of Plato's Timaeus reinforced the prevailing belief among scholars that numerical principles underpinned proportion and harmony. Pythagoreanism, as interpreted through Plato's Timaeus, stimulated progressively intricate investigations into symmetry and harmony. Intellectuals contemplated the practical application of understanding the divine geometry structuring the universe. By the 12th century, Pythagorean numerological concepts had become so pervasive in medieval Europe that their Pythagorean origins were often no longer recognized. Authors such as Thierry of Chartres, William of Conches, and Alexander Neckham consulted classical writers who had discussed Pythagoreanism, including Cicero, Ovid, and Pliny, which led them to conclude that mathematics was fundamental to comprehending astronomy and nature. Boethius's De arithmetica, another significant text on Pythagorean numerology, was extensively disseminated throughout the Western world. Boethius himself drew upon Nicomachus's works as a foundational source for Pythagoreanism.
The 11th-century Byzantine professor of philosophy Michael Psellus significantly popularized Pythagorean numerology through his theological treatise, asserting that Plato had inherited the Pythagorean secret. Psellus also erroneously ascribed Diophantus's arithmetical innovations to Pythagoras. He further endeavored to reconstruct Iamblichus's ten-volume encyclopedia on Pythagoreanism from extant fragments, thereby disseminating Iamblichus's interpretations of Pythagorean physics, ethics, and theology within the Byzantine court. Psellus was reportedly in possession of the Hermetica, a collection of texts believed to be genuinely ancient, which subsequently saw extensive reproduction during the late Middle Ages. Manuel Bryennios subsequently introduced Pythagorean numerology into Byzantine music via his treatise, Harmonics, contending that the octave was fundamental for achieving perfect harmony.
Within Jewish communities, the evolution of Kabbalah as an esoteric doctrine became intertwined with numerology. Philo of Alexandria initiated a distinct Jewish Pythagoreanism during the 1st century. By the 3rd century, Hermippus propagated the notion that Pythagoras had provided the foundational principles for determining significant dates in Judaism. This assertion was subsequently elaborated upon by Aristobulus in the 4th century. Philo's Jewish Pythagorean numerology posited that God, as the singular One, was the progenitor of all numbers, with seven being considered the most divine and ten the most perfect. The medieval rendition of the Kabbalah primarily concentrated on a cosmological framework of creation, referencing early Pythagorean philosophers such as Philolaus and Empedocles, which facilitated the broader dissemination of Jewish Pythagorean numerology.
On Mathematics
Nicomachus's treatises achieved widespread recognition across the Greek, Latin, and Arabic intellectual spheres. An Arabic translation of Nicomachus's Introduction to Arithmetic appeared in the 9th century. These Arabic renditions of Nicomachus's works were subsequently translated into Latin by Gerard of Cremona, thereby integrating them into the Latin numerological tradition. The Pythagorean theorem was also cited in Arabic manuscripts, indicating a significant scholarly engagement with Pythagorean concepts within the Arabic world. For instance, in the 10th century, Abu al-Wafa' Buzjani addressed multiplication and division in an arithmetical treatise intended for business administrators, referencing Nicomachus. Nevertheless, the principal focus of Islamic arithmeticians lay in resolving pragmatic issues, including taxation, measurement, agricultural valuation, and commercial applications for trade. Consequently, there was minimal interest in the Pythagorean numerology that had evolved in the Latin world. The predominant arithmetical system employed by Islamic mathematicians derived from Hindu arithmetic, which fundamentally rejected the symbolic interpretation of relationships between numbers and geometrical forms.
Beyond the considerable interest in Pythagorean numerology that emerged in the Latin and Byzantine regions during the Middle Ages, the Pythagorean legacy concerning perfect numbers stimulated significant mathematical scholarship. In the 13th century, Leonardo of Pisa, more commonly known as Fibonacci, authored the Libre quadratorum (The Book of Squares). Fibonacci's extensive studies encompassed texts from Egypt, Syria, Greece, and Sicily, leading to his proficiency in Hindu, Arabic, and Greek mathematical methodologies. He investigated numerology, as articulated by Nicomachus, utilizing the Hindu–Arabic numeral system rather than Roman numerals. Fibonacci noted that square numbers invariably result from the summation of consecutive odd numbers commencing with unity. Furthermore, Fibonacci proposed a method for generating sets of three square numbers that conform to the relationship initially ascribed to Pythagoras by Vitruvius: a§78§ + b§1112§ = c§1516§. This specific equation is presently recognized as a Pythagorean triple.
In the Middle Ages
Throughout the Middle Ages, spanning from the 5th to the 15th century, Pythagorean texts maintained their prominence. Late antique authors created adaptations of the Sentences of Sextus, titling them The golden verses of Pythagoras. The Golden Verses subsequently achieved widespread popularity, leading to the emergence of Christian adaptations. These Christianized versions were embraced by monastic orders, including that of Saint Benedict, as authoritative Christian doctrine. Within the Latin medieval Western world, the Golden Verses became a frequently reproduced text.
While the concept of the quadrivium originated with Archytas in the fourth century BCE and was a familiar notion among ancient academics, Proclus attributed it to Pythagoreanism in the fifth century. Proclus posited that Pythagorean philosophy categorized all mathematical sciences into four distinct areas: arithmetic, music, geometry, and astronomy. Boethius subsequently expanded upon this theory, asserting that a four-part intellectual pathway facilitated the acquisition of knowledge. Consequently, arithmetic, music, geometry, and astronomy became fundamental components of curricula in medieval educational institutions. In the twelfth century, Hugh of Saint Victor credited Pythagoras with authoring a treatise on the quadrivium. The concept of harmony, rooted in the triadic philosophical frameworks of Plato and Aristotle, also encompassed the trivium, comprising grammar, rhetoric, and dialectic. From the ninth century onward, both the quadrivium and the trivium were routinely incorporated into the educational programs of schools and nascent universities, collectively becoming known as the Seven Liberal Arts.
In the early sixth century, the Roman philosopher Boethius significantly popularized Pythagorean and Platonic cosmological perspectives, emphasizing the paramount importance of numerical ratios. The seventh-century Bishop Isidore of Seville favored the Pythagorean concept of a universe governed by the mystical properties of specific numbers, contrasting it with the emerging Euclidean paradigm that posited knowledge could be constructed through deductive proofs. Isidore's approach drew upon the arithmetic of Nicomachus, who self-identified as an heir to Pythagoras, and extended this by investigating the etymology of each number's name. The twelfth-century theologian Hugh of Saint Victor found Pythagorean numerology so compelling that he endeavored to explicate the human body entirely through numerical principles. However, the prominence of numerology diminished in the thirteenth century, with the Christian scholar Albertus Magnus critiquing the excessive focus on Pythagorean numerology and contending that nature could not be exclusively explained by numbers. Plato's Timaeus emerged as a significant resource for understanding the mystical and cosmological symbolism that Pythagoreans ascribed to numbers. The intense pursuit of numerical explanations for proportion and harmony ultimately found its architectural expression in the French cathedrals of the eleventh, twelfth, and thirteenth centuries.
Arabic translations of the Golden Verses were produced during the eleventh and twelfth centuries. Within the Medieval Islamic world, a Pythagorean tradition developed, positing that celestial spheres or stars generated music. This doctrine was further elaborated by Ikhwan al-Safa and al-Kindi, who highlighted the congruence between musical harmony and the harmony of the soul. Nevertheless, prominent Islamic philosophers such as al-Farabi and Ibn Sina vehemently rejected this Pythagorean doctrine. In Kitab al-Musiqa al-Kabir, Al-Farabi refuted the notion of celestial harmony, asserting it was "plainly wrong" and that the heavens, orbs, and stars were incapable of emitting sounds through their motions.
The four treatises comprising the Corpus Areopagiticum or Corpus Dionysiacum (The Celestrial Hierarchy, The Ecclesiastical Hierarchy, On Divine Names, and The Mystical Theology) by Pseudo-Dionysius the Areopagite achieved immense popularity during the Middle Ages, initially in the Byzantine world where they were first published in the first century, and subsequently in the Latin world following their translation in the ninth century. The cosmological division of the universe into heaven, earth, and hell, along with the twelve orders of heaven, were attributed to Pythagoras's teachings by an anonymous biographer, whose work was cited in the ninth-century treatise of the Byzantine patriarch Photius. In the thirteenth century, the astronomer and mathematician John of Sacrobosco, in turn, credited Pseudo-Dionysius when discussing the twelve signs of the zodiac.
During the Middle Ages, numerous classical texts discussing Pythagorean concepts were reproduced and translated. Plato's Timaeus, for instance, was translated and reissued with extensive commentary in both the Arab and Jewish intellectual spheres. By the 12th century, the study of Plato had stimulated a substantial body of literature that elucidated the divine glory as manifested in the universe's inherent orderliness. Scholars such as Thierry of Chartres, William of Conches, and Alexander Neckham referenced not only Plato but also other classical authors who had explored Pythagoreanism, including Cicero, Ovid, and Pliny. William of Conches specifically posited Plato as a significant Pythagorean. Within this medieval Pythagorean interpretation of Plato, God was conceptualized as a craftsman in the design of the cosmos.
Influence on Western Science
In the preface to De revolutionibus, Copernicus identifies three Pythagorean philosophers—Hicetas, Philolaus, and Ecphantus—as antecedents to the Heliocentric Theory.
- Copernicus noted, "At first I found in Cicero that Hicetas supposed the earth to move. Later I also discovered in Plutarch that others were of this opinion. I have decided to set his words down here, so that they may be available to everybody: 'Some think that the earth remains at rest. But Philolaus the Pythagorean believes that, like the sun and moon, it revolves around the fire in an oblique circle. Heraclides of Pontus and Ecphantus the Pythagorean make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center.'"
In the 16th century, Vincenzo Galilei challenged the prevailing Pythagorean understanding concerning the relationship between musical pitches and the weights attached to strings. Vincenzo Galilei, father of Galileo Galilei, engaged in a protracted public debate with his former instructor, Zarlino. Zarlino advocated the theory that if two weights in a 2:1 ratio were affixed to two strings, the resulting pitches would produce an octave. Vincenzo Galilei, however, declared his prior adherence to Pythagorean principles, stating he remained so "until he ascertained the truth by means of experiment, the teacher of all things." He devised an experiment demonstrating that the weights attached to the two strings needed to increase proportionally to the square of the string's length. This public refutation of established numerology in musical theory catalyzed an experimental and physical approach to acoustics in the 17th century. Acoustics subsequently emerged as a mathematical subfield of music theory and later evolved into an autonomous branch of physics. In the empirical investigation of sound phenomena, numerical values lost their symbolic significance, serving instead merely to quantify physical phenomena and relationships, such as frequency and string vibration.
Many prominent 17th-century European natural philosophers, including Francis Bacon, Descartes, Beeckman, Kepler, Mersenne, Stevin, and Galileo, demonstrated a profound interest in music and acoustics. By the close of the 17th century, the understanding that sound propagates as a wave through air at a finite speed was widely accepted, leading to experiments conducted by scholars affiliated with institutions such as the French Academy of Sciences, the Accademia del Cimento, and the Royal Society to determine the speed of sound.
During the zenith of the Scientific Revolution, as Aristotelianism waned across Europe, the tenets of early-Pythagoreanism experienced a resurgence. Mathematics regained its prominence, influencing both philosophy and scientific inquiry. Key figures such as Kepler, Galileo, Descartes, Huygens, and Newton employed mathematics to formulate physical laws that elucidated the universe's inherent order. Twenty-one centuries after Pythagoras instructed his disciples in Italy, Galileo famously asserted that "the great book of nature" could only be deciphered by those fluent in the language of mathematics. He committed to quantifying all measurable aspects and rendering immeasurable phenomena quantifiable. The Pythagorean concept of cosmic harmony profoundly shaped Western science, forming the foundation for Kepler's Harmonices Mundi and Leibniz's pre-established harmony. Albert Einstein posited that through this pre-established harmony, a productive synthesis between the spiritual and material realms was achievable.
The Pythagorean conviction that all physical entities are fundamentally numerical and that their attributes and causal relationships are quantifiable provided the foundational framework for the mathematization of scientific inquiry. This mathematical approach to physical reality reached its zenith in the 20th century. Werner Heisenberg, a pioneering physicist, asserted that "this method of observing nature, which partially resulted in genuine control over natural forces and thereby significantly advanced human development, unexpectedly affirmed the Pythagorean tenet."
Dyad (Greek philosophy)
- Dyad (Greek philosophy)
- Esoteric cosmology
- Hippasus
- Ionian School (philosophy)
- Incommensurable magnitudes
- Ipse dixit
- Mathematical beauty
- Mathematicism
- Pyrrhonism
- Sacred geometry
- Tetractys
- Unit-point atomism
References
Bibliography
Media pertaining to Pythagoreanism.
- Media related to Pythagoreanism at Wikimedia Commons
- Huffman, Carl. "Pythagoreanism." In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. ISSN 1095-5054. OCLC 429049174.Source: TORIma Academy Archive