Pythagoras

Greek philosopher and mathematician

• Born: c. 580 b.c.e.
• Birthplace: Samos, Ionia, Greece
• Died: c. 500 b.c.e.
• Place of death: Metapontum, Lucania (now in Italy)

Pythagoras set an inspiring example with his energetic search for knowledge of universal order. His specific discoveries and accomplishments in philosophy, mathematics, astronomy, and music theory make him an important figure in Western intellectual history.

Early Life

Pythagoras (pih-THAG-oh-ruhs), son of Mnesarchus, probably was born about 580 b.c.e. (various sources offer dates ranging from 597 to 560). His birthplace was the Greek island of Samos in the Mediterranean Sea. Aside from these details, information about his early life—most of it from the third and fourth centuries b.c.e., up to one hundred years after he died—is extremely sketchy. On the other hand, sources roughly contemporary with him tend to contradict one another, possibly because those who had been his students developed in many different directions after his death.

Aristotle’s Metaphysica (335-323 b.c.e.; Metaphysics, 1801), one source of information about Pythagorean philosophy, never refers to Pythagoras himself but always to “the Pythagoreans.” Furthermore, it is known that many ideas attributed to Pythagoras have been filtered through Platonism. Nevertheless, certain doctrines and biographical events can be traced with reasonable certainty to Pythagoras himself. His teachers in Greece are said to have included Creophilus and Pherecydes of Syros; the latter (who is identified as history’s first prose writer) probably encouraged Pythagoras’s belief in the transmigration of souls, which became a major tenet of Pythagorean philosophy. A less certain but more detailed tradition has him also studying under Thales of Miletus, who built a philosophy on rational, positive integers. In fact, these integers were to prove a stumbling block to Pythagoras but would lead to his discovery of irrational numbers such as the square root of two.

Following his studies in Greece, Pythagoras traveled extensively in Egypt, Babylonia, and other Mediterranean lands, learning the rules of thumb that, collectively, passed for geometry at that time. He was to raise geometry to the level of a true science through his pioneering work on geometric proofs and the axioms, or postulates, from which these are derived.

A bust now housed at Rome’s Capitoline Museum (the sculptor is not known) portrays the philosopher as having close-cropped, wavy Greek hair and beard, his features expressing the relentlessly inquiring Ionian mind—a mind that insisted on knowing for metaphysical reasons the exact ratio of the side of a square to its diagonal. Pythagoras’s eyes suggest an inward focus even as they gaze intently at the viewer. The furrowed forehead conveys solemnity and powerful concentration, yet deeply etched lines around the mouth and the hint of a crinkle about the eyes reveal that this great man was fully capable of laughter.

Life’s Work

When Pythagoras returned to Samos from his studies abroad, he found his native land in the grip of the tyrant Polycrates, who had come to power about 538 b.c.e. In the meantime, the Greek mainland had been partially overrun by the Persians. Probably because of these developments, in 529 Pythagoras migrated to Croton, a Dorian colony in southern Italy, and entered into what became the historically important period of his life.

At Croton he founded a school of philosophy that in some ways resembled a monastic order. Its members were pledged to a pure and devout life, close friendship, and political harmony. In the immediately preceding years, southern Italy had been nearly destroyed by the strife of political factions. Modern historians speculate that Pythagoras thought that political power would give his organization an opportunity to lead others to salvation through the disciplines of nonviolence, vegetarianism, personal alignment with the mathematical laws that govern the universe, and the practice of ethics in order to earn a superior reincarnation. (Pythagoras believed in metempsychosis, the transmigration of souls from one body to another, possibly from humans to animals. Indeed, Pythagoras claimed that he could remember four previous human lifetimes in detail.)

His adherents he divided into two hierarchical groups. The first was the akousmatikoi, or listeners, who were enjoined to remain silent, listen to and absorb Pythagoras’s spoken precepts, and practice the special way of life taught by him. The second group was the mathematikoi (students of theoretical subjects, or simply “those who know”), who pursued the subjects of arithmetic, the theory of music, astronomy, and cosmology. (Though mathematikoi later came to mean “scientists” or “mathematicians,” originally it meant those who had attained advanced knowledge in a broader sense.) The mathematikoi, after a long period of training, could ask questions and express opinions of their own.

Despite the later divergences among his students—fostered perhaps by his having divided them into two classes—Pythagoras himself drew a close connection between his metaphysical and scientific teachings. In his time, hardly anyone conceived of a split between science and religion or metaphysics. Nevertheless, some modern historians deny any real relation between the scientific doctrines of the Pythagorean society and its spiritualism and personal disciplines. In the twentieth century, Pythagoras’s findings in astronomy, mathematics, and music theory are much more widely appreciated than the metaphysical philosophy that, to him, was the logical outcome of those findings.

Pythagoras developed a philosophy of number to account for the essence of all things. This concept rested on three basic observations: the mathematical relationships of musical harmonies, the fact that any triangle whose sides are in a ratio of 3:4:5 is always a right triangle, and the fixed numerical relations among the movement of stars and planets. It was the consistency of ratios among musical harmonies and geometrical shapes in different sizes and materials that impressed Pythagoras.

His first perception (which some historians consider his greatest) was that musical intervals depend on arithmetical ratios among lengths of string on the lyre (the most widely played instrument of Pythagoras’s time), provided that these strings are at the same tension. For example, a ratio of 2:1 produces an octave; that is, a string twice as long as another string, at the same tension, produces the same note an octave below the shorter string. Similarly, 3:2 produces a fifth and 4:3 produces a fourth. Using these ratios, one could assign numbers to the four fixed strings of the lyre: 6, 8, 9, and 12. Moreover, if these ratios are transferred to another instrument—such as the flute, also highly popular in that era—the same harmonies will result. Hippasus of Metapontum, a mathematikos living a generation after Pythagoras, extended this music theory through experiments to produce the same harmonies with empty and partly filled glass containers and metal disks of varying thicknesses.

Pythagoras determined that the most important musical intervals can be expressed in ratios among the numbers 1, 2, 3, and 4, and he concluded that the number 10—the sum of these first four integers—comprehends the entire nature of number. Tradition has it that the later Pythagoreans, rather than swear by the gods as most other people did, swore by the “Tetrachtys of the Decad” (the sum of 1, 2, 3, and 4). The Pythagoreans also sought the special character of each number. The tetrachtys was called a “triangular number” because its components can readily be arranged as a triangle.

By extension, the number 1 is reason because it never changes; 2 is opinion; 4 is justice (a concept surviving in the term “a square deal”). Odd numbers are masculine and even numbers are feminine; therefore, 5, the first number representing the sum of an odd and an even number (1, “unity,” not being considered for this purpose), symbolizes marriage. Seven is parthenos, or virgin, because among the first ten integers it has neither factors nor products. Other surviving Pythagorean concepts include unlucky 13 and “the seventh son of a seventh son.”

To some people in the twentieth century, these number concepts seem merely superstitious. Nevertheless, Pythagoras and his followers did important work in several branches of mathematics and exerted a lasting influence on the field. The best-known example is the Pythagorean theorem, the statement that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Special applications of the theorem were known in Mesopotamia as early as the eighteenth century b.c.e., but Pythagoras sought to generalize it for a characteristically Greek reason: This theorem measures the ratio of the side of a square to its diagonal, and he was determined to know the precise ratio. It cannot be expressed as a whole number, however, so Pythagoras found a common denominator by showing a relationship among the squares of the sides of a right triangle. The Pythagorean theorem is set forth in book 1 of Euclid’s Stoicheia (Elements), Euclid being one of several later Greek thinkers whom Pythagoras strongly influenced and who transmitted his ideas in much-modified form to posterity.

Pythagoras also is said to have discovered the theory of proportion and the arithmetic, geometric, and harmonic means. The terms of certain arithmetic and harmonic means yield the three musical intervals. In addition, the ancient historian Proclus credited Pythagoras with discovering the construction of the five regular geometrical solids, though modern scholars think it more likely that he discovered three—the pyramid, the tetrahedron, and the dodecahedron—and that Theaetetus (after whom a Platonic dialogue is named) later discovered the construction of the remaining two, the octahedron and the icosahedron.

The field of astronomy, too, is indebted to Pythagoras. He was among the first to contend that the earth and the universe are spherical. He understood that the sun, the moon, and the planets rotate on their own axes and also orbit a central point outside themselves, though he believed that this central point was the earth. Later Pythagoreans deposed the earth as the center of the universe and substituted a “central fire,” which, however, they did not identify as the sun—this they saw as another planet. Nearest the central fire was the “counter-earth,” which always accompanied the earth in its orbit. The Pythagoreans assumed that the earth’s rotation and its revolution around the central fire took the same amount of time—twenty-four hours. According to Aristotle, the idea of a counter-earth—besides bringing the number of revolving bodies up to the mystical number of ten—helped to explain lunar eclipses, which were thought to be caused by the counter-earth’s interposition between sun and moon. Two thousand years later, Nicolaus Copernicus saw the Pythagorean system as anticipating his own; he had in mind both the Pythagoreans’ concept of the day-and-night cycle and their explanation of eclipses.

Like Copernicus in his time, Pythagoras and his followers in their time were highly controversial. For many years, the Pythagoreans did exert a strong political and philosophical influence throughout southern Italy. The closing years of the sixth century b.c.e., however, saw the rise of democratic sentiments, and a reaction set in against the Pythagoreans, whom the democrats regarded as elitist.

Indeed, this political reaction led either to Pythagoras’s exile or to his death—there are two traditions surrounding it. One is that a democrat named Cylon led a revolt against the power of the Pythagorean brotherhood and forced Pythagoras to retire to Metapontum, where he died peacefully about the end of the sixth century b.c.e. According to the other tradition, Pythagoras perished when his adversaries set fire to his school in Croton in 504 b.c.e. The story is that of his vast library of scrolls, only one was brought out of the fire; it contained his most esoteric secrets, which were passed on to succeeding generations of Pythagoreans.

Whichever account is true, Pythagoras’s followers continued to be powerful throughout Magna Graecia until at least the middle of the fifth century b.c.e., when another reaction set in against them, and their meetinghouses were sacked and burned. The survivors scattered in exile and did not return to Italy until the end of the fifth century. During the ensuing decades, the leading Pythagorean was Philolaus, who wrote the first systematic exposition of Pythagorean philosophy. Philolaus’s influence can be traced to Plato through their mutual friend Archytas, who ruled Taras (Tarentum) in Italy for many years. The Platonic dialogue Timaeus (360-347 b.c.e.), named for its main character, a young Pythagorean astronomer, describes Pythagorean ideas in detail.

Significance

“Of all men,” said Heraclitus, “Pythagoras, the son of Mnesarchus, was the most assiduous inquirer.” Pythagoras is said to have been the first person to call himself a philosopher, or lover of wisdom. He believed that the universe is a logical, symmetrical whole, which can be understood in simple terms. For Pythagoras and his students, there was no gap between the scientific or mathematical ideal and the aesthetic. The beauty of his concepts and of the universe they described lies in their simplicity and consistency.

Quite aside from any of Pythagoras’s specific intellectual accomplishments, his belief in universal order, and the energy he displayed in seeking it out, provided a galvanizing example for others. Sketchy as are the details of his personal life, his ideals left their mark on later poets, artists, scientists, and philosophers from Plato and Aristotle through the Renaissance and down to the twentieth century. Indirectly, through Pythagoras’s disciple Philolaus, his ideas were transmitted to Plato and Aristotle, and, through these better-known thinkers, to the entire Western world.

Among Pythagoras’s specific accomplishments, his systematic exposition of mathematical principles alone would have been enough to make him an important figure in Western intellectual history, but the spiritual beliefs he espoused make him also one of the great religious teachers of ancient Greek times. Even those ideas of his that are seen as intellectually disreputable have inspired generations of poets and artists. For example, the Pythagorean concept of the harmony of the spheres, suggested by the analogy between musical ratios and those of planetary orbits, became a central metaphor of Renaissance literature.

Bibliography

Guthrie, Kenneth Sylvan, ed. The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy. Grand Rapids, Mich.: Phanes Press, 1988. This anthology of Pythagorean writings contains the four ancient biographies of Pythagoras as well as later Pythagorean and Neopythagorean writings.

Kahn, Charles H. Pythagoras and the Pythagoreans. Indianapolis: Hackett, 2001. Surveys Pythagorean tradition from Pythagoras’s time to early modern times, including his influence on early modern math, music, and astronomy. Indexed by ancient and early modern name and by modern name.

Kirk, Geoffrey S., and John E. Raven. The Presocratic Philosophers. New York: Cambridge University Press, 1983. Provides a good account of Pythagoras and his followers, in their historical context, from a philosopher’s point of view.

Muir, Jane. Of Men and Numbers. New York: Dover, 1996. Written for lay readers. Contains a chapter on Pythagoras’s mathematical work and its influence on later scientists, especially Euclid.

Philip, J. A. Pythagoras and Early Pythagoreanism. Toronto: University of Toronto Press, 1968. Attempts to separate the valid information from the legends surrounding Pythagoras and his teachings. Includes notes and a selected bibliography.

Strohmeier, John, and Peter Westbrook. Divine Harmony: The Life and Teachings of Pythagoras. Berkeley, Calif.: Berkeley Hills Books, 2003. Describes Pythagoras’s travels in Egypt, Phoenicia, Babylonia, and Greece and examines Pythagorean ideas as taught at his scholarly community in southern Italy. Includes illustrations, map, introduction, and bibliography.