# Pythagoras and the Pythagoreans

# PYTHAGORAS AND THE PYTHAGOREANS

Pythagoras (fl. *c.* 530 b.c.), Greek philosopher and mathematician, was one of the most influential and creative thinkers of the pre-Socratic period. A native of the island of Samos off the western coast of Asia Minor, he is reported to have traveled extensively, but this cannot be verified. At about the age of 40 he fled from Samos to avoid the tyranny of Polycrates and migrated to Croton in southern Italy, where he lived for 20 years. It was here that he founded his famous society. His last years were spent in Metapontum, farther north from Croton, where he died on an unknown date. The Pythagorean society continued in existence for well over a century, disappearing finally in the fourth century b.c.

Pythagoras himself apparently wrote nothing, preferring to transmit his teachings orally to the members of the school. Moreover, the members of the society were supposedly sworn to secrecy on their doctrines which, if true, would militate further against the development of a written tradition. As a result, most knowledge of the Pythagoreans has come indirectly through later classical writers. Furthermore, Pythagoras was held in such high esteem by members of the school that they regularly attributed their ideas to him. Thus, although there must have been some development of the tradition, it is difficult to distinguish the teachings of the various Pythagoreans. Even Aristotle usually refers to the "Pythagoreans" as a common group rather than to Pythagoras himself. The later members of the school included Philolaus, Eurytus, Lysis, Xenophilos, Phanton, Echekrates, Diokles, Polymnastos, Timaeus of Locri, and Archytas of Tarentum, a friend of Plato. The authenticity of the fragments attributed to Philolaus has been questioned.

**Basic Principles.** The Pythagorean society was a quasi-religious community devoted to the purification of the soul and its liberation from the "wheel of birth." The emphasis on communal life as a road to self-perfection, along with the doctrine of the transmigration of souls, may have derived from the earlier Orphic religious tradition. The members of the society were bound by numerous dietary and other practical proscriptions to purify the soul in accordance with their belief in metempsychosis. Over and above these religious and ascetic aspects, they were also actively involved in political life, which led to the eventual expulsion of the society from Croton. But above all, they placed value on the discovery and cultivation of knowledge, giving the highest place to mathematics. They held that devotion to science was the noblest form of purification.

Unlike the Homeric tradition in which the soul is merely the shadow of the man, that of the Pythagoreans held that the soul is the man and is imprisoned in the body, thus introducing the soul-body dichotomy so prominent later in Plato's thought. It is difficult to reconcile this with the doctrine, also attributed to the Pythagoreans, that the soul is the harmony or attunement of the four elements constituting the body. They also looked upon virtue as a harmonious mathematical proportion in human action.

The Pythagorean system was based on a duality of first principles: the limited, as the source of definiteness, and the unlimited, as the source of divisibility. The entire universe was a harmony of these two principles. Aristotle reports (*Meta.* 986a 23–27) that some members of the school expanded the basic duality into a table of ten opposites: (1) limited and unlimited, (2) odd and even, (3) one and plurality, (4) right and left, (5) male and female,(6) rest and moving, (7) straight and curved, (8) light and darkness, (9) good and bad, (10) square and oblong.

**Mathematical Teachings.** The first product of the limited and unlimited was the number one, which in turn is the principle of all the cardinal numbers. According to the testimony of Aristotle, the Pythagoreans identified number with the physical universe. This latter point is difficult to see and has been the subject of much discussion.

*Number and the Universe.* The theory seems to have been based on an identification, or at least an association, of arithmetical numbers, geometrical entities, and physical bodies. Aristotle suggests (*Meta.* 985b 31–986a 3) that Pythagoras's discovery that musical harmony is expressible in terms of the ratios of small whole numbers prompted him to conclude that all other phenomena in the universe are also mathematical in nature. Moreover, the Pythagoreans customarily represented numbers by placing a pebble or dot for each unit in the number. In this way the number one became a point; two, a line; three, a plane triangle; and four, a tetrahedron. Possibly because the first four numbers culminate in a solid figure, which is identifiable with an extended body, these numbers arranged in a plane triangular configuration became a sacred figure, the "tetractys," and the sum of the first four numbers, that is, 10, was a sacred number. In this way the Pythagoreans could say that things are numbers: numbers were geometrized, and geometrical entities were incarnated. Though this theory may seem strange and difficult for the modern mind, it embodies a hope and an ideal of perennial recurrence, that is, the reduction of the physical universe to mathematical structure. Many of the founders of modern physical science, especially J. kepler and G. galilei, were motivated by this same ideal.

*Types of Numbers.* The Pythagoreans were characterized by several further mathematical teachings. Their distinction between triangular, square, and oblong numbers was based on their practice of representing a number by a corresponding group of dots. Triangular numbers (3, 6, 10, 15, etc.) were the sums of the unbroken natural number series beginning with one and arranged in a triangular pattern. The "tetractys" was, of course, the chief triangular number pattern. Square numbers (4, 9, 16, 25, etc.) were the sums of the odd numbers beginning with one and arranged in the form of a square. Such numbers have a rational square root. Oblong numbers (2, 6, 12, 20, etc.) were the series of even numbers beginning with two and arranged in a rectangle. Each successive addition to the latter two geometrical patterns was called a "gnomon."

*Pythagorean Theorem.* The use of the lengths 3, 4, and 5 for the purpose of constructing a right triangle dates back to the very earliest times. To Pythagoras belongs the credit of universalizing this relation into the famous theorem that still bears his name: the square on the hypotenuse of a right triangle is equal to the sum of the squares on its sides. But this discovery was a mixed blessing for the Pythagoreans. For the use of this theorem results in many cases in the uncovering of incommensurable quantities, that is, the hypotenuse and sides of the triangle are so related that no unit of measurement however small can be divided into them without a fractional remainder. This in effect was the discovery of irrational numbers, a scandal for Pythagorean cosmology. For if numbers are irrational,

then so are things. Hippasos of Metapontum was reportedly drowned at sea for revealing this embarrassment. The solution to this problem of incommensurability escaped the Pythagoreans and had to wait for the theory of proportions developed by Eudoxus in the Academy. The Pythagoreans appear to have known only three of the five regular solids: the cube, tetrahedron, and dodecahedron. The discovery of the octahedron and icosahedron, along with the use of the five regular solids as cosmological principles, were later Platonic developments.

**Astronomical Views.** Pythagorean astronomical theory was both ingenious and highly influential. The universe as a whole is surrounded by a "boundless breath" that is inhaled by the universe after the manner of a huge animal. As inhaled, this breath plays the role of empty space separating the celestial bodies. At the middle of the universe is the central fire, variously called the hearth of the world, the watchtower of Zeus, and the mother of the gods, which is never seen by us because it is always on the other side of the earth. The celestial bodies from the central fire outward are the counterearth (antichthon), earth, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, and the fixed stars. The counterearth, also never seen from our side of the earth, was posited according to Aristotle either to bring the celestial bodies up to the sacred number 10 or to explain why there are more lunar than solar eclipses. The heavenly bodies are visible as orifices in wheels of fire that revolve at a methodic pace producing the musical harmony of the heavens, which is not noticed because it is always heard. The celestial bodies periodically return to their original orientation in a time interval called the Pythagorean Great Year. Other notable elements in the theory were the doctrine of the sphericity of the earth and the distinction between the diurnal westward motion of the heavens and the slower eastward motion of planets. These astronomical views are very prominent in Plato's *Timaeus.* The early Pythagoreans clearly did not introduce the heliocentric hypothesis, an honor that must be reserved for Aristarchus of Samos (fl. 281 b.c.).

**Neo-Pythagoreanism.** The first century b.c. saw a Neo-Pythagorean revival that continued into the second century of the Christian Era. The emphasis in this movement was on the religious and ascetic traditions of the old school, but intellectual pursuits were not overlooked. The result was a rather eclectic combination of Pythagoreanism with Stoic, Aristotelian, and especially Platonic elements. The chief figures in Neo-Pythagoreanism were Nigidius Figulus, Plutarch of Chaeronea, Apollonius of Tyana, Nicomachus of Gerasa, Numenius of Apamea, and Philostratus of Lemnos.

See Also:greek philosophy; science (in antiquity).

**Bibliography:** w. k. guthrie, *A History of Greek Philosophy* (Cambridge, Eng. 1962–) v. 1. j. burnet, *Early Greek Philosophy* (4th ed. London 1930). j. e. raven, *Pythagoreans and Eleatics* (Cambridge, Eng. 1948). e. l. minar, *Early Pythagorean Politics in Practice and Theory* (Baltimore, Md. 1942). t. l. heath, *A History of Greek Mathematics,* 2 v. (Oxford 1921) 1:65–117, 141–169; *Aristarchus of Samos: The Ancient Copernicus* (Oxford 1913).

[r. j. blackwell]

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