Pythagoras of Samos

Mathematics Pythagoras and his school made fundamental contributions to the theory of numbers, geometry, and the concept of mathematical proof. The Pythagorean theorem (a² + b² = c²), though known empirically in earlier civilizations, received its first deductive proof in the Greek tradition through the Pythagorean school. The Pythagoreans studied figurate numbers (triangular, square, pentagonal), perfect numbers, and the properties of odd and even integers. The discovery of irrational numbers—specifically the incommensurability of the diagonal of a square with its side—emerged within the school and posed a fundamental challenge to their doctrine that all things are numbers.

Music Theory Pythagoras discovered the mathematical basis of musical harmony by demonstrating that consonant musical intervals correspond to simple numerical ratios. Using a monochord, he showed that an octave corresponds to a 2:1 ratio, a perfect fifth to 3:2, and a perfect fourth to 4:3. This insight formed the foundation of the Pythagorean tuning system and led to the broader philosophical claim that the cosmos itself is organized according to harmonic proportions (the 'Harmony of the Spheres').

Philosophy and Cosmology Pythagoras pioneered the idea that the universe is fundamentally mathematical in structure. His doctrine that 'all is number' represents one of the earliest formulations of mathematical realism. The Pythagorean cosmology, later developed by Philolaus, proposed a non-geocentric model of the universe with a central fire, anticipating aspects of heliocentric thought.

Ethics and Way of Life Pythagoras established a comprehensive ethical framework centered on the purification and elevation of the soul through intellectual discipline, ascetic practice, and communal living. The doctrine of metempsychosis (transmigration of souls) provided a moral cosmology linking human conduct to the soul's ultimate fate.

Pythagoras was born around 570 BCE on the island of Samos, off the coast of Asia Minor (modern Turkey), to Mnesarchus, a prosperous gem-engraver or merchant, and Pythais. Ancient sources agree that he received an excellent education, learning to play the lyre, studying poetry, and reciting Homer from an early age.

Three philosophers exercised formative influence on the young Pythagoras. Pherecydes of Syros, widely regarded as his most important teacher, introduced him to ideas about the immortality of the soul. Thales of Miletus, whom Pythagoras reportedly visited between the ages of 18 and 20, sparked his interest in mathematics and astronomy and advised him to travel to Egypt for further study. Anaximander, Thales's pupil, exposed him to cosmological speculation.

According to later tradition, Pythagoras undertook extensive travels lasting perhaps two decades, studying with Egyptian priests in the Nile Delta, Babylonian scholars (where he may have encountered advanced arithmetic and astronomical tables), and possibly Phoenician and Persian sages. While the exact duration and scope of these travels remain debated among scholars, the breadth of Pythagoras's intellectual synthesis suggests genuine exposure to multiple knowledge traditions.

Around 530 BCE, Pythagoras emigrated to Croton (modern Crotone), a Greek colony in southern Italy. His departure from Samos is commonly attributed to disagreement with the tyranny of Polycrates, though some scholars suggest the political climate of Persian expansion into Ionia was the primary motivating factor.

At Croton, Pythagoras founded a philosophical school that functioned simultaneously as a religious brotherhood, an intellectual academy, and a political association. The community followed strict rules of conduct, including dietary restrictions (abstinence from meat and beans), ritual practices, periods of silence, and communal ownership of property.

Members were organized into two classes: the mathematikoi (learners), who constituted the inner circle and engaged in advanced study of arithmetic, geometry, music theory, and astronomy; and the akousmatikoi (listeners), who adhered to Pythagorean ethical precepts and oral teachings without necessarily pursuing theoretical studies. New initiates reportedly underwent a five-year period of silence before being admitted to the inner circle.

The school's intellectual program was structured around four disciplines that later became known as the quadrivium: arithmetic (number in itself), geometry (number in space), music/harmonics (number in time), and astronomy (number in space and time). This fourfold division of mathematical education would dominate Western curricula for nearly two millennia.

Pythagoras rapidly acquired significant political influence in Croton, serving as an adviser to the city's elite and delivering public speeches on virtue and civic duty. The Pythagorean community's political involvement, however, generated increasing resentment.

Around 509 BCE, a violent anti-Pythagorean revolt erupted in Croton, traditionally attributed to Cylon of Croton, a wealthy aristocrat who had been rejected for admission to the brotherhood. The revolt resulted in the burning of Pythagorean meeting houses and the persecution of members. Pythagoras fled to Metapontum, another Greek city in southern Italy, where he reportedly spent his final years.

He died around 495 BCE at Metapontum. The circumstances of his death are uncertain: some accounts describe him dying of grief over the destruction of his community, while others place him among the victims of anti-Pythagorean violence. Despite this political catastrophe, the Pythagorean movement survived and continued to develop, producing significant thinkers such as Philolaus and Archytas in the following century.