Pythagoreanism

The NEN Lue. Brit generated from something unlimited; but they added the idea of the imposition of limit upon the unlimited and the sense of a musical harmony in the universe. Again, IS edi kon like the Ionians, they devoted themselves to astronomical and geometrical speculation. Combining, as it does, a rationalistic theory of number with a mystic numerology Macragpedia 1/57; 322-326 and a speculative cosmology with a theory of the deeper, more enigmatic reaches of the soul, Pythagoreanism interweaves Rationalism and irrationalism more inseparably than does any other movement in ancient Greek Holger ThestefÆ thought (see PHILOSOPHY, HISTORY OF WESTERN). AND TEACHINGS CONCERNS MAJOR The problem of describing Pythagoreanism is complicat- Motasser Thiosophy Hehkisk ed by the fact that the surviving picture is far from complete, being based chiefly on a small number of fragments from the time before Plato and on various discussions in authors who wrote much later—most of whom were either Aristotelians or Neoplatonisis (see below History of Pythagoreanism). In spite of the historical uncertainties. however, that have plagued searching scholars, the contribution of Pythagoreanism to Western culture has been significant and therefore justifies the effort, however inadequate, to depict what its teachings may have been. Moreover, the heterogeneousness of Pythagorean doctrines has been well documented ever since Heracleitus, a classic early-Sth-century Greek philosopher who, scoffing at Pythagoras’ wide-ranging knowledge, said that it “does not teach one to have intelligence.” There probably never existed a strictly uniform system of Pythagorean philosophy and religious beliefs, even if the school did have a certain internal organization. Pythagoras appears to have taught by pregnant, cryptic akousmata (“something heard”) or symbola. His pupils handed these on, formed them partly into Hieroi Logoi (“Sacred a Pythagoreanism The philosophical schoo] and religious Discourses"), of which different versions were current brotherhood from the 4th century on, and interpreted them according known as Pythagoreanism is believed to have been founded by Pythagoras of Samos, who settled in Croton in southern Italy about 525 nc. GENERAL FEATURES OF PYTHAGOREANISM The character of the original Pythagoreanism is controversial, and the conglomeration of disparate features that +S e q it displayed is intrinsically confusing. Its fame rests, however, on some very influential ideas, not always correctly understood, that have been ascribed to it since antiquity. These ideas include those of (1) the metaphysic of number and the conception that reality, including music and astronomy, is, at its deepest level, mathematical in nature; (2) the use of philosophy as a means of spiritual purification; (3) the heavenly destiny of the soul and the possibility of its rising to union with the divine: (4) the appeal to certain symbols, sometimes mystical. such as the fetraktys, the golden section, and the harmony of the spheres (to be discussed below); (5) the Pythagorean theorem; and (6) the demand that members of the order shall observe a strict loyalty and secrecy. By laying stress on certain inner experiences and intuitive truths revealed only to the initiated, Pythagoreanism seems to have represented a soul-directed subjectivism alien to the mainstream: of Pre-Socratic Greek thought centring on the lonian coast of Asia Minor (Thales, Anaximander, Anaxagoras, and others), which was preoccupied with determining what the basic cosmic substance is. In contrast with such lonian naturalism, Pythagorcanism was akin to trends seen in mystery religions a emotional movements. such as Orphism, which often to their convictions. Religion and ethics. sit otf . qe [17 “ur "plate The belief in the transmigration of souls provided a basis for the Pythagorean way of life. Some Pythagoreans deduced from this belief the principle of “the kinship of all beings,” the ethical implications of which were later stressed in 4th-century speculation. Pythagoras himself seems to have claimed a semidivine status in close association with the superior god Apollo; he believed that he was able to remember his earlier incarnations and, hence, to know more than others knew. Recent research has emphasized shamanistic traits deriving from the ecstatic cult practices of Thracian medicine men in the early Pythagorean outlook. The rules for the religious life that Pythagoras taught were largely ritualistic: refrain from speaking about the holy, wear white clothes, observe sexual purity, do not touch beans, ‘ind so forth. He seems also to have taught purification Of the soul by means of music and mental activity (later called philosophy) in order to reach higher incarnations. “To be like your Master” and so “to come nearer to the Fods" was the challenge that he imposed on his pupils. Sulvation, and perhaps ultimate union with the divine cosmos through the study of the cosmic order, became une of the leading ideas in his school. The advanced ethics and political theories sometimes ascribed to Pythagoreanism may to some extent reflect ideas later developed in the circle of Archytas, the lead- Ing 4th-century Pythagorean. But a picture current among the Peripatetics (the school founded by Aristotle} of Pythagoras as the educator of the Greeks, who publicy preached a gospel of humanity, is clearly anachronistic. Several of the Peripatetic writers, Aristoxen us, Dicclaimed to achieve through intoxication a spiritual insight “earchus, and Timaeus, seem to have interpreted some into the divine origin and nature of the soul. Yet there are also aspects of it that appear to have owed much to Principles—properly laid down only for esoteric use in the brotherhood—as though these applied 10 all manthe more sober. “Homeric” philosophy of the Tonians. ind: the internal loyalty, modesty, self-discipline, piety, The Pythagoreans, for example, displayed an interest in metaphysics (the nature of Being), as did their naturalislic predecessors, though they claimed to find its key in pr abstinence required by the secret doctrinal system; € higher view of womanhood reflected in the admission es omen to the school; a certain community of propermathematical form rather than in any substance. They accepted the essentially Ionian doctrines that the world is composed of opposites (wet-dry, hot-cold, etc.) and y: and perhaps the drawing of a parallelism between the pr rocosm (the universe) and the microcosm (man), in ich (for instance) the Pythagorean idea that the yLon

Pythagoreanism 323 324 Pythagoreanism lowing is, therefore, a compromise between the widely cosmos is an organism was applied to the state, which should thus mix monarchy, oligarchy, and democracy into a hurmonic whole—these were all universalized. Metaphysics and number theory. According to Aristodivergent views of modern scholars. Use of the . gnómones tle, number speculation is the most characteristic feature of Pythagoreaniam. Things “are” number, or “resemble” number. To many Pythagoreans this concept meant that things are measurable and commensurable or proportional in terms of number—an idea of considerable significance for Western civilization. But there were also attempts to arrange a certain minimum number of pebbles so as to represent the shape of a thing—as, for instance, stars in a constellation that seem to represent an animal. For the Pythagoreans even abstracted things “have” their number: “justice” is associated with the number four and Figure 2: Gnomons of Pythagorean numbartheory{see text). with @ square, “marriage” with the number five, and so on. The psychological associations at work here have not been clarified. The harmony of the cosmos. numbers is put around the unit as gnomons, they always produce squares; thus, the members of the series 4, 9, 16, 25, . . . are “square” numbers. If even numbers are The sacred decad in particular has a cosmic significance in Pythagoreanism: its mystical name, tetraktys (meaning approximately “fourness"), implies 1 + 2 + 3 + 4 = 10; but it can also be thought of as a “perfect triangle,” as in Figure 1. Speculation on number and proportion led to an intuidepicted in a similar way, the resulting figures (which offer infinite variations) represent “oblong” numbers, The such as those of the series 2, 6, 12, 20. . .. On the other hand, a triangle represented by three dots (as in the upper part of the fetrakiys) can be extended by a series of natural numbers to form the “triangular” numbers 6, 10 (the fefraktys), 15. 21. . . . This procedure, which was, so far, Pythagorean, led later, perhaps in the Platonic Academy, to a speculation on “polygonal” numbers. Probably the square numbers of the gnomons were early tetrakiys tive feeling of the harmonia (“fitting together") of the kosmos (“the beautiful order of things”); and the application of the tetrakıys to the theory of music (see below Music) revealed a hidden order in the range of sound. Pythagoras may have referred, vaguely, to the “music of the heavens," which he alone seemed able to hear; and later Pythagoreans seem to have assumed that the distances of the heavenly bodies from the earth somehow correspond to musical intervals—a theory that, under the influence of Platonic conceptions, resulted in the famous idea of the “harmony of the spheres.” Though number to the early Pythagoreans was still a kind of cosmic matter, like the water or air proposed by the lonians, their stress upon numerical proportions, harmony, and order comprised a decisive step toward a metaphysic in which form is the basic reality. The doctrine of opposites. From the Ionians, the PyIhagoreans adopted the idea of cosmic opposites, which they—perhaps secondarily—applied to their number speculation. The principal pair of opposites is the limit and the unlimited; the limit (or limiting), represented by the odd (3,5,7, . . .), is an active force effecting order, harmony, “cosmos,” in the unlimited, represented by the even. All kinds of opposites somehow “fit together” within the cosmos, as they do, microcosmically, in an individual man and in the Pythagorean society. There was also a Pythagorean “table of ten opposites,” to which Aristotle has referred—limit-unlimited, odd-even, onemany, right-left, male-female, rest-motion, straightcurved, light-darkness, good-evil, and square-oblong. The arrangement of this table reflects a dualistic conception, which was apparently not original with the school, however, or accepted by all of its members. The Pythagorean number metaphysic was also reflected in its cosmology. The unit (1), being the starting point of the number series and its principle of construction, is not itself strictly a number; for, to be a number is to be even or odd, whereas, in the Pythagorean view, “one” is seen as both even and odd. This ambivalence applies, similarly, to the total universe, conceived as the One. There was also a cosmogonical theory (of cosmic origins) that explained the generation of numbers and number-things from the limiting-odd and the unlimited-even—a theory that, by stages unknown to scholars, was ultimately incorporated into Plato’s philosophy in his doctrine of the derivation of sensed realities from mathematical principles (sce PLATONISM AND NEOPLATONISM: Greek Platonism from Aristotle through Middle Platonism, its nature and history). Mathematics and science. Pythagorean thought was scientific as well as metaphysical and included specific developments in arithmetic and geometry, in the science of musical tones and harmonies, and in astronomy. Arithmetic. Early Pythagorean achievements in mathematics are unclear and largely disputable, and the fol- In the speculation on odd and even numbers, the early Pythagoreans used so-called gnômones (Greek: “carpenter's squares”). Judging from Aristotle's account, gnomon numbers, represented by dots or pebbles, were arranged in the manner shown in Figure 2. If a series of odd associated with the Pythagorean theorem (likely to have been used in practice in Greece, however, before Pythagoras), which holds that for a right triangle a Square drawn on the hypotenuse is equal in area to the sum of the squares drawn on its sides; in the gnomons it can easily be seen, in the case of a 3,4,5-triangle for example (see Figure 3), that the addition of a square Figure 3: Gnomon for Pythagorean thecrem. The matked off “carpenter's square” Ss’ —comprising 3 groups of 3 dots each (3 X 3)—thus represents 32, which when added to 42 yields 52 {ihe totat gnomon). gnomon number to a square makes a new square: 3* + 4 = $, and this gives a method for finding two square numbers the sum of which is also a square. Some Sth-century Pythagoreans seem to have been puz- The problem of irrational numbers zled by apparent arithmetical anomalies: the mutual relationships of triangular and square numbers; the anomalous properties of the regular pentagon; the fact that the length of the diagonal of a square is incommensurable with its sides—i.c., that no fraction composed of integers can express this ratio exactly (the resulting decimal is thus defined as irrational); and the irrationality of the mathematical proportions in musical scales. The discovery of such irrationality was disquieting because it had fatal consequences for the naive view that the universe is expressible in whole numbers; the Pythagorean Mippasus is said to have been expelled from the brotherhood, according to some sources even drowned, because he made a point of the irrationality. In the 4th century, Pythagorizing mathematicians made a significant advance in the theory of irrational numbers, such as the-square-root-of-n (1/#), n being any rational number, when they developed a method for finding progressive approximations to /2 by forming sets of socalled diagonal numbers. Geometry. In geometry. the Pythagoreans cannot be credited with any proofs in the Euclidean sense. They were evidently concerned. however, with some speculation on geometrical figures, as in the case of the Pythagorean theorem. and the concept that the point, line, triangle, and tetrahedron correspond to the elements of the sczraktys, since they are determined by one, two, three, and four points. respectively. They possibly knew practical methods of constructing the five regular solids, but the theoretical basis for such constructions was given by non-Pythagoreans in the 4th century.

It is notable that the properties of the circle seem not to Most of these literary sources hark back ultimately to have interested the early Pythagoreans. But perhaps the tradition that Pythagoras himself discovered that the sum of the three angles of any triangle is equal to two right the environment of Plato and Aristotle; and here the importance of one of Aristotle’s students has become obvious, viz., the musicologist and philosopher Aristoxangles may be trusted. The idea of geometric proportions is probably Pythagorean in origin; but the so-called golden section—which divides a line at a point such that the smaller part is to the greater as the greater is to the whole—is hardly an early Pythagorean contribution. Some advance in geometry was made later, by 4th-centuenus, who in spite of his bias possessed firsthand information independent of the point of view of Plato’s Academy. The role played by Dicaearchus, another of Aristotle’s pupils, and by the Sicilian historian Timaeus, of the early 3rd century mc, is less clear. Recently, the reliability of Aristotle’s account of Pythagoreanism has also been emphasized against the doubts that had been expressed by ry Pythagoreans; e.g., Archytas offered an interesting solution to the problem of the duplication of the cube— in which a cube twice the volume of a given cube is constructed—by an essentially geometrical construction in three dimensions; and the conception of geometry as a “flaw” of points into lines, of lines into surfaces, and so on, may have been contributed by Archytas; but on the whole the achievements of non-Pythagorean mathematicians were in fact more conspicuous than those of some modern scholars; but Aristotle’s sources, in turn, hardly lead farther back than to the late 5th century (perhaps to Philolaus; see below Two Pythagorean sects). In addition, there ate scattered hints in various early authors and in some not very substantial remains of 4thcentury Pythagorean literature. The mosaic of reconthe Pythagoreans. Music. The achievements of the early Pythagoreans in musical theory are somewhat less controversial. The scientific approach to music, in which musical intervals are expressed as numerical proportions, originated with them, as did also the more specific idea of harmonic “means.” At an early date they discovered empirically that the basic intervals of Greek music include the elemy? Matre LORS aule mito ral. lone: struction thus has to be to some extent subjective. Early Pythagoreanism. Within the ancient Pythagorean movement four chief periods can be distinguished: carly Pythagoreanism, dating from the late 6th century nc and extending to about 400 Bc; 4th-century Pythagoreanism; the Hellenistic trends; and Neo-Pythagoreanism, a revival that occurred in the mid-1st century AD and lasted for two and a half centuries. Background, The background of Pythagoreanism is complex, but two main groups of sources can be distinments of the tetraktys, since they have the proportions guished. The Ionian philosophers (Thales, Anaximander, 1:2 (octave). 3:2 (fifth), and 4:3 (fourth). The discovery could have been made, for instance, in pipes or flutes or stringed instruments: the tone of a plucked string held at its middle is an octave higher than that of the whole string; the tone of a string held at the 34 point is a fifth higher; and that of one held at the % point is a Anaximenes, and others) provided Pythagoras with the problem of a single cosmic principle, the doctrine of nSwoe°Msole fourth higher. Moreover, they noticed that the subtraction of intervals is accomplished by dividing these ratios by one another. In the course of the Sth century they calculated the intervals for the usual diatonic scale, the tone being represented by 9:8 (fifth minus fourth); ie. 3/2 + 4/3, and the semitone by 256:243 (fourth minus two tones); i.e., 4/3 - (9/8 x 9/8). Archytas made some modification to this doctrine and also worked out the relationships of the notes in the chromatic (12-tone) scale and the enharmonic scale (involving such minute upposites, and whatever reflections of Oriental mathematics there are in Pythagoreanism; and from the technicians of his birthplace, the Isle of Samos, he learned to understand the importance of number, measuring, and proportions. Popular cults and beliefs current in the 6th century and reflected in Orphism introduced him to occultism, ritualism, and the doctrine of individual immortality. In view of the shamanistic traits of Pythagoreanism, reminiscent of Thracian cults, it is interesting to note that Pythagoras seems to have had a Thracian slave. E differences as that between A flat and G sharp, which on a piano are played by the same key). Pythagorean communities. The school apparently founded by Pythagoras at Croton in southern Italy seems to have been primarily a religious brotherhood centred around Pythagoras and the cults of Apollo and of the Muses, ancient patron goddesses of poetry and culture. It became ‘Perhaps successively institutionalized and re- Astronomy. In their cosmological views the earliest Pythagoreans probably differed little from their Ionian predecessors. They made a point of studying the stellar heavens; but—with the possible exception of the theory of musical intervals in the cosmos—no new contributions ceived different classes of esoteric members and exoteric sympathizers. The rigorism of the ritual and ethical observances demanded of the members is unparalleled in carly Greece; in addition to the rules of life mentioned above, it is fairly well-attested that secrecy and a long to astronomy can be ascribed to them with any degree of silence during the novitiate were required. The exoteric probability. Late in the 5th century, or possibly in the 4th ME, however, were politically active and estabcentury, a Pythagorean boldly abandoned the geocentric en a Crotonian hegemony in southern Italy. About view and posited a cosmological model in which the RC a coup by a rival party caused Pythagoras to take Earth, Sun, and stars circle about an (unseen) central fire—a view traditionally attributed to the Sth-century refuge in Metapontum, where he died. During the early 5th century, Pythagorean communities ENCES, HISTORY OF). pr doctrinal differentiation and diffusion. In the course Pythagorean Philolaus of Croton (sec PHYSICAL SCIexisted in many southern Italian cities, a fact i time the politics of the Pythagorean parties became seal antidemocratic. About the middle of the cen- HISTORY OF PYTHAGOREANISM The life of Pythagoras and the origins of Pythagorcanism à a violent democratic revolution swept over southern appear only dimly through a thick veil of Jegend and = Y, many Pythagoreans were killed, and only a few semihistorical tradition, The literary sources for the teachings of the Pythagoreans present extremely complicated problems. Special difficulties arise from the oral and esoteric transmission of the early doctrines, the profuse accumulation of tendentious legends, and confusion caused by the split in the school in the 5ih century RC. In the 4th century, Plato's inclination toward Pythagoreanism created a tendency—manifest already in the middle of the century in the works of his pupils—t0 interpret Platonic concepts as originally Pythagorean. But the radical skepticism as 10 the reliability of the sources shown by some modern scholars has on the whole been abandoned in recent rescarch. It now seents possible to extract bits of reliable evidence from a wide range of ancient authors, such as Porphyry and Jamblichus (see below Neo-Pythagorcanism). that led to of u among them Lysis of Tarentum and Philolaus en \ tha roton, who went to Greece and formed small Pyprea circles in Thebes and Phlious. tiene Pythagorean sects. Little is known about Pythag3 ‘se! activity during the latter part of the 5th century. si differentiation of the school into two main sects, later Da , akousmatikoi Td,” (Greek: akousma, “something viz., the esoteric teachings) and mathématikoi bs ek: mathematikos, “scientific”), may have occurred “ i time. The acousmatics devoted themselves to the of ee of rituals and rules and to the interpretation concer Sayings of the master; the “mathematics” were Philope cd with the scientific aspects of Pythagoreanism. lished = Who was rather a mathematic, probably pubin the & summary of Pythagorean philosophy and science late Sth century.

Pythagoreanism Fourth-century Pythagoreanism. In the first half of the 4th century, Tarentum, in southern Italy, rose into considerable significance. Under the political and spiritual leadership of the mathematic Archytas, a friend of Plato, Tarentum became a new centre of Pythagoreanism, from Archytas and his school which acousmatics—so-called Pythagorists who did not sympathize with Archytas—went out travelling as mendicant ascetics all around the Greek-speaking world. a pupil of Plotinus), thought of himself as a Pythagorean sage and about ap 300 wrote the last great synthesis of Pythagoreanism, in which most of the disparate postclassical traditions are reflected. It is characteristic of the Neo-Pythagoreans that they were chiefly interested in the Pythagorean way of life and in the pseudoscience of number mysticism. On a more popular level, Pythagoras and Archytas were remembered as magicians. Moreover, it has been suggested that Pythagorean legends also influenced the Christian monastic tradition. Medieval and modern trends. In the Middle Ages the popular conception of Pythagoras the magician was combined with that of Pythagoras “the father of the quadrition, and harmonic proportions, became applied to aesthetics. To many Humanists, moreover, Pythagoras was the father of the exact sciences. In the early 16th Pythagorean tradition. . It is doubtful whether advanced modern philosophy has ever drawn from sources thought to be distinctly Pythagorean. Yet Platonic-Neoplatonic notions, such as the mathematical of reality or the philosopher's union with the universe and various mystical beliefs are still likely to be stamped as Pythagorean. Even today uncritical admiration of Pythagoreanism is common. EVALUATION The history of the projection of Pythagoreanism into subsequent thought indicates how fertile some of its core concepts were. Plato is here the great catalyst; but it is possible to perceive behind him, however dimly, a series of Pythagorean ideas of paramount potential significance: the combination of religious esoterism (or exclusivism) with the germs of a new philosophy of mind, present in the belief in the progress of the soul toward the actualization of its divine nature and toward knowlplied contemporary ethical notions to them. In the Hellenistic Age, the Academic and Peripatetic views gave rise to a rather fanciful antiquarian literature on Pythagoreanism. There also appeared a large and yet more lamblichus, a pupil of Porphyry (who in turn had been vium”; Le., of the more specialized liberal arts of the curriculum. From the Italian Renaissance onward, some “Pythagorean” ideas, such as the tetrad, the golden seccentury, Nicolaus Copernicus, who developed the view that the Earth revolves around the Sun, considered his system to be essentially Pythagorean or “Philolaic,” and Galileo was called a Pythagorean. The 17th-century Rationalist G.W. Leibniz appears to have been ths last great i r and scientist who felt himself to be in the philosophe The acousmatics seem to have preserved some early Pythagorean Hieroi Logoi and ritual practices, Archytas himself, on the other hand, concentrated on scientific problems, and the organization of his Pythagorean brotherhood was evidently less rigorous than that of the early school. After the 380s there was a give-and-take between the school of Archytas and the Academy of Plato, a relationship that makes it almost impossible to disentangle the original achievements of Archytas from joint involvements (but see above, Geometry and Music). The Hellenistic Age. Whereas the school of Archytas apparently sank into inactivity after the death of its founder (probably after 350 BC), the Academics of the next generation continued “Pythagorizing" Platonic doctrines, such as that of the supreme One, the indefinite dyad (a metaphysical principle), and the tripartite soul (see PLATONISM AND NEOPLATONISM). At the same time, various Peripatetics of the school of Aristotle, including Aristoxenus, collected Pythagorean legends and apheterogeneous mass of apocryphal writings falsely attributed to different Pythagoreans, as if attempts were being made to revive the school. The texts fathered on Archytas display Academic and Peripatetic philosophies mixed with some notions that were originally Pythagorean. Other texts were fathered on Pythagoras himself or on his immediate pupils, imagined or real. Some show, for instance, that Pythagoreanism had become confused with Orphism; others suggest that Pythagoras was considered a magician and an astrologist; there are also indications of Pythagoras “the athlete" and “the Dorian nationalist.” But the anonymous authors of this pseudo-Pythagorean literature did not succeed in re-establishing the school, and the “Pythagorean” congregations formed in early imperial Rome seem to have had little in common with original Pythagoreanism; they were ritualistic sects that adopted, eclectically, various occult practices. Neo-Pythagoreanism. With the ascetic sage Apollonius of Tyana, about the middle of the Ist century AD, a distinct Neo-Pythagorean trend appeared. Apollonius studied the Pythagorean legends of the previous centuries, created and propagated the ideal of a Pythagorean life— of occult wisdom, purity, universal tolerance, and approximation to the divine—and felt himself to be a reincarnation of Pythagoras. Through the activities of Neo-Pythagorean Platonists, such as Moderatus of Gades, a pagan trinitarian, and the arithmetician Nicomachus of Gerasa, both of the 1st century AD, and, in the 2nd or 3rd century, Numenius of Apamea, forerunner of Plotinus (an epoch-making elaborator of Platonism), Neo-Pythagoreanism gradually became a part of the expression of Platonism known as Neoplatonism; and it did so without having achieved a scholastic system of its own. The founder of a Syrian school of Neoplatonism, Pythagoreanism edge; stress upon harmony and order, and upon limit as the good; the primacy of form, proportion, and numerical expression; and in ethics, an emphasis upon such virtues as friendship and modesty. The fact that Pythagoras, to later ages, also became alternatively a Dorian nationalist, a sportsman, an educator of the people, or a great magician is a more curious consequence of the productivity of his teaching. i BIBLIOGRAPHY Fragments and texts: The collection of the fragments in HERMANN DIELS and WALTHER KRANZ, Die Fragmente der Vorsokratiker, 6th ed., vol. 1 (1951), is insufficient; additions are given in MARIA TIMPANARO CARDINI (ed.), Piragorici: Testimonianze e frammenti, 3 vol. (1958-64); and in CORNELIA 3. DE VOGEL, Pythagoras and Early Pythagoreanism (1966). For the pseudo-Pythagoreans, se¢ HOLGER THESLEFF (ed.), The Pythagorean Texts of the Hellenistic Period (1965). Early Pythagoreanism: The best comprehensive introduction to Pythagoreanism is the long chapter “Pythagoras and the Pythagoreans,” in w.x.c. GUTHRIE, A History of Greek Philosophy, vol. 1, pp. 146-340 (1962). Somewhat different Amalgamation with Neoplatonism approaches have been taken by DE VOGEL (op. eit.); and JAMES A. PHILIP, Pythagoras and Early Pythagoreanism (1966), works that demand more active criticism by the reader, Fairly full references to the discussion of Pythagoreanism up to 1960 are in WALTER BURKERT, Weishelt und Wissenschaft: Studien zu Pythagoras, Philolaos, und Platon (1962; Eng. trans., Lore and Science in Ancient Pythagoreantem, 1972), a highly technical and at times rather avercritical work. Among later technical discussions, likely to become influential, are the articles “Pythagoras” and "Pythagoreer”” in Pauly-Wissowa Realencyclopädie, vol. 47, (1963), and suppl. vol. 10 (1965)—of the contributors, KURT VON FRITZ and H. DORRIE arrive at less controversial conclusions than B.L. VAN DER WAERDEN. Hellenistic Pythagoreanism: HOLGER THESLEFP, An Introduction to the Pythagorean Writings of the Hellenistic Period (1961); additions and corrections in Entretiens Fondation Hardt, vol. 18 (1972). Neo-Pythagoreanism: Neoplatonism (1953). PHILIP MERLAN, From Platonism to For up-to-date bibliographical accession, see L'Année phil. ologique (annual), under the subject heading “Pythagorica” and the various Pythagoreans, Heben Thea tt

Philosophy, History of Western art Fr Me sic of number. All of the philosophies men- Fair =far are in various ways historically akin to one another. Toward the end of the 6th century, however, there arose quite independently another kind of philosophy, which only later entered into interrelation with the Pythagoras of Samos developments just mentioned: the philosophy of Pythagoras of Samos. Pythagoras travelled extensively in the East and in Egypt and, after his return to Samos, emigrated to southern Italy because of his dislike of the tyranny of Polycrates. At Croton and „Metapontum he founded a philosophical society with strict rules and soon gained considerable political influence. He appears to have brought his doctrine of the transmigration of souls from the East. Much more important for the history of philosophy and science, however, was his doctrine that “all things are numbers,” which means that the essences and structures of all things can be determined by finding the numerical relations contained in them. Originally, this, too, was a very broad generalization made on the basis of comparatively few observations: for instance, that the same harmonies can be produced with different instruments—strings, pipes, disks, etc.—by means of the same numerical ratios—1 :2, 2:3, 3:4—in one-dimensional extensions; the observation that certain regularities exist in the movements of the celestial bodies: and the discovery that the form of a triangle is determined by the ratio of the lengths of its sides. But, because the followers of Pythagoras tried to apply their principle everywhere with the greatest of accuracy, one of them—Hippasus of Metapontum—around 450 ac made one of the most fundamental discoveries of the entire history of science, that of incommensurability, viz., that the quantitative relation between the side and diagonal of such simple figures as the square and the regular pentagon cannot be expressed as a ratio of integers, however great. At first sight this discovery seemed to destroy the very basis of the Pythagorean philosophy, and the school thus split into two sections, one of which engaged in rather abstruse numerical speculations while the other succeeded in overcoming the difficulty by ingenious mathematical inventions and laid the foundations of all quantitative science. Pythagorean philosophy also exerted a great influence on the development of Plato's thought in his later years. The speculations described so far comprise in many ways the most important part of the history of Greek philosophy because all of the most fundamental problems of Western philosophy turned up here for the first time and one finds here the formation of a great many concepts that have continued to dominate Western philosonhv and science down to the present day.

Pythagorean brotherhood, an singy or cult founded in the 6th centuryArt dedicated to the Muses and to Sa get. lo. The exact purposes of the€ ily Be; ! known, but the association originall qe ! strong religious motivation; also, yy # - thagoras' society eventually died out towar the end of the 4th century sc, largely beca of the violent democratic opposition to y, 15edition Mreropedie. Setting houses had become centres of politiical/power. Some carly Pythagoreans a his writings have survived, and “E «RS. or ing houses, too “meals together, had a distinctive dress, underis teachings t to disting from those of his disciples. Non“, invariably supported their doctrines by indy, criminately citing their master’s authority. Py thagoras, however, is generally credi va the theory of the functional significance yy numbers in the objective world and in Mure Other mathematical principles and discoveria often attributed to him (e.8., the Pythagore theorem for right triangles) were only later by the Pythagorean school at a tm when mat! tical concepts had evolved to a -were doubiless‘: maintained vihroughout-the troubled history of the order, swith relative importance. given at different ;places and times to mathematics, music, fi Zoran Pythagorean centres were located ‘at Thebes, the home of ithe Pythagoreans »Rhegium and Philius. About 454 ec meeting cal enemies and by about , 390 most of the imen Pythagoreans (except Archytas) had forced into exile, The brotherhood was, scribed certain secret cultic practices. f Pythagoreanism deeply influenced the deve opment of classical Greek philosophy s medieval European thought (especially the» ; a wi rean -ruler, Archytas, was a friend of Plato; and at E : houses of the order were déstroyed by politi- «for all practical purposes, inactive by the end ‘of the 4th century. gr. 4" sei dl» -Pythagoreanism 15:322, :e --philosophical school and religious brotherhood, believed to ‘have been founded by late in the -6th century ac, holding: (1) that at its deepest ‘level, reality is mathematical in nature; (2) ‘that philosophy can be used for spiritual purification; (3) that the soul can rise to union “with the divine; (4) that certain symbols have ‘a mystical significance; and (5) that all brothers of the order should observe strict loyalty : The text article covers the general features of -Pythagoreanism; its major concems and chings in religion and ethics, metaphysics and number theory, and mathematics and Science; and its history, including literary different orbital radii of the planetary so sources, early, 4th-century, Hellenistic, and Neo-Pythagoreanism, and medieval and mod“em trends, e REFERENCES in other-texi articles: .... . ascetic practices 2:136€ .... ... >»: causality in relation to ...... re. + Aristolelianism 1:1169b "77 97 7 system. content and influence in Archaic.. . “art of western Greeks 8:353h ‘celibate communal life 3:1042a dualistic basis of numerical “cosmological significance of man 12:799f * -cosmology based on mathematical . i trological belief that the number harmon Y € the universe decidedly affects all human deavour). In the scientific realm Pyt mathematics aided the discovery of Keple* Law, which philosophy. -adaptations sophical and ethical teachings can be a cribed: the famed dictum “all is numbg* meaning that all existing things can be yy. mately reduced to number relationships; tx the brain as the locus of the soul; and pr that, although religious in nature, formulated principles that influenced the thought of Plato and Aristotle, and contributed to the development of mathematics and Western rational went periods of initiation and testing, and - were admitted to different of membermore advanced state. More probably the buy of the intellectual tradition originating wig Pythagoras himself belongs to mystical na dom rather than to scientific scholarshi Accordingly, it is only to the legacy of Pi i oe tradition that the following phiy dependence of the dynamics of world stnx ture on the interaction of contraries, or par of opposites, the first of which is the even-cg relationship essential to numbers; the viewing of the soul as a self-moving number expenex ing a form of mctempsychosis, or succes reincarnation in different species until its ever tual purification (particularly through the e tellectual life of the ethically rigorous P: thagoreans), and liberation from the repeuts “wheel of life”; the understanding, as in pr Socratic tradition, that all existing objec: were fundamentally composed of form an, not of material substance. Additional A thagorean doctrine applied number relatire mo to music theory, acoustics, geomeln and astronomy (in which the Earth sx termed a moving spheroid, but without ı clear expression of heliocentrism); Pythagoras (b. c. 580 sc, Samos, Greece— d. c. 500 sc, Metapontum, near present Metaponto, Italy), philosopher, mathematician, and founder of the Pythagorean brotherhood yo” “Tt iydificult. to distinguish Pytha t The new Enc Brill engaged in political activity creating _resentment in many cities where Py! posited the relation between & 8’ Greece 8:333d . . 2 CU. £ relations 18:1007b --u: +Kepler's mathematical astronomy :14:386f speculation 5:1067a «early geographic theory 7:1036b Earth sphericity theory 6:1g aa «evolution of Pythagorean doctrines 15:}°° «harmony and divided string 8:647d -hereditary origins of human life 7:994b Pa «history of calculatory device and table 1 ‘Indian influence on doctrines 8:913e *mathematical model of universe 14:384f «nrusic's mathematical basis 12:663b .mystery cult beliefs and practices -12:779g - «number as ultimate reality 12:15e . = “numbers and development of . :.:-— E slight properties theory 10:928h -music’s mathematical basis 12:663b «music study as philosophy 2:97g “number as ultimate reality 12:15e “Philo of Alexandria influenced M:246e. x -origins of number metaphysic 14:25, _ «pitch and tuning in Western music 12: “Rationalist idea of mathematical reality 15:528g -recollection of carlier lives 18:411b -teachings and influence in Archaic Greece 8:333d -theosophic brotherhood and teachings 18:276h «Western music's physical basis 12: 704 Pythagoras. contormate engraved belween AD 395 and 410; ın the Bibliothèque Nationale, Paris Bon veto, at me Bbinihenne Nanonate. Pars Pythagoras migrated to southern Italy c. 532, apparently to escape Samos’ tyrranical rule, and established his ethico-political academy at Croton (now Crotona). Flourishing throughout southern Italy and Greece, Py- . A and tuning in Western music 12:747b APiaio's rejectian of number views 14:538e ‘polygonal and figurate numbers, and + Pythagorean triples 13:348g: illus.J49 “principles and influence 8:35de - --. 5... -. science history-and trathition :36:3661, soul es attunement of body -22:18g 27.2. : -. |” zodiacal applications 2:221f . .. RELATED ENTRIES in the Ready Reference and Index: number metapbysic, Pythagorean; opposites, table of; Pythagorean Brotherhood