Pythagoras and Pythagoreanism

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Pythagoras was an Ionian Greek born on the island of Samos, probably about 570 BCE. His dislike of the policies of the Samian tyrant Polycrates caused him to immigrate to Crotona in southern Italy. There he founded a society with religious and political, as well as philosophical, aims that gained power in the city and considerably extended its influence over the surrounding area. A certain Cylon, however, stirred up a revolt against the society in which a number of its leading members were killed, and Pythagoras retired to Metapontum. The community recovered its influence until a more serious persecution took place in the middle of the fifth century, from which the survivors scattered to various parts of the Greek worldnotably Thebes, Phleius, and Tarentum. In these places "they preserved their original ways and their science, although the sect was dwindling, until, not ignobly, they died out" (in the late fourth century), to quote the epitaph written by a contemporary.

Nature of the Evidence

The obstacles to an appraisal of classical Pythagoreanism are formidable. There exists no Pythagorean literature before Plato, and it was said that little had been written, owing to a rule of secrecy. Information from the Christian era is abundant but highly suspect. Pythagoras himself, though a fully historical figure, underwent a kind of canonization. His life was quickly obscured by legend, and piety attributed all the school's teaching to him personally. Moreover, the dispersion of the school inevitably led to divergences of doctrine in the various groups. Aristotle makes it clear that by the late fifth century some Pythagoreans were teaching one thing and some another. A further reason for division was that the universal genius of Pythagoras, for whom religion and science were two aspects of the same integrated worldview, was beyond the scope of lesser men. Some naturally inclined more to the religious and superstitious; others, to the intellectual and scientific side, as is confirmed by later references to the division between acusmatici and mathematici.

As early evidence there are several references to Pythagoras in works of his contemporaries or near contemporaries (for instance, Xenophanes satirized his belief in the transmigration of souls), a valuable reference in Plato to the relationship between astronomy and harmonics in the Pythagorean system, a quantity of information from Aristotle (who at least would not confuse the Pythagoreans with Plato, as later writers excusably did), and some quotations from pupils of Aristotle who were personally acquainted with the last generation of the school.

Given the nature of the sources, the following is a fairly conservative summary of Pythagoreanism before Plato.

Man and the Cosmos

In contrast with the Milesians, the Pythagoreans were not motivated by disinterested scientific curiosity. For Pythagoras, philosophy was the basis of a way of life, leading to salvation of the soul. "Their whole life," said a fourth-century writer, "is ordered with a view to following God, and it is the governing principle of their philosophy." At philosophy's center, therefore, were man and his relation to other forms of life and to the cosmos. Purity was to be sought by silence, self-examination, abstention from flesh and beans, and the observance of other primitive taboos that the Pythagoreans interpreted symbolically. Of the recognized gods they worshiped Apollo, guardian of the typically Greek ideal of moderation ("nothing too much"), of whom Pythagoras was believed to be an incarnation.

Behind both the superstition and the science was the notion of kinship or sympathy. The kinship and essential unity of all life made possible the belief in the transmigration of souls and accounted for the prohibition of meat: a sheep might house the soul of an ancestor. Not only animate nature in our sense but the whole world was akin, for the cosmos itself was a living, breathing creature. The cosmos was one, eternal, and divine; men were divided and mortal. But the essential part of man, his soul, was not mortal; it was a fragment of the divine, universal soul that was cut off and imprisoned in a mortal body. Men should therefore cultivate and purify the soul, preparing it for a return to the universal soul of which it was a part. Until then, since it was still contaminated by the body, it must tread the wheel of reincarnation, entering a new body of man or animal after the death of its previous tenement.

These tenets were also taught by the religious movement known as Orphism, from which the religious side of Pythagoreanism can hardly be separated. (Pythagoras himself was said in the fifth century to have written books under the name of Orpheus.) But whereas the Orphics sought salvation by purely religious meanssacramental ceremonies and the observance of ritual prohibitionsPythagoras added a new way, the way of philosophy.

Philosophy, for Pythagoras as for others, meant the use of reason and observation to gain understanding of the universe. The link between this procedure and his overriding aim of salvation seems to have been the principle that like is known by like, a widespread tenet of pre-Socratic thought, common to such diverse systems as the philosophicoreligious synthesis of Empedocles and the scientific atomism of Democritus. Hence, an understanding of the divine universe would bring man's nature closer to its own. In this conception we meet the typically Pythagorean conception of kosmos, a word that combines in an untranslatable way the notion of orderly arrangement or structural perfection with that of beauty. Closely linked with it is peras, meaning limit. An organic whole, particularly one that, like the universe, lives forever, must of necessity exhibit limit and order in the highest degree. What is unlimited has no telos (end) and is a-teles, which means both "endless" and "incomplete." But the world is a perfect whole, a model of order and regularity, supremely exemplified in Greek eyes by the ceaseless wheeling of the heavenly bodies in (as they believed) perfect circles, bringing about the unvarying succession of day and night and seasons. It was said of Pythagoras that he was the first to call the world kosmos, "from its inherent order." By studying this order, we reproduce it in our own souls, and philosophy becomes an assimilation to the divine, as far as that is possible within the limitations imposed by our mortal bodies.

The Doctrine that Things are Numbers

The Pythagoreans studied mathematics in a cosmic context, and for them numbers always retained a mystical significance as the key to the divine cosmos. "They supposed the whole heaven to be a harmonia and a number," said Aristotle. Harmonia, though specially applied to music, could signify any well-organized structure of parts fitted together in due proportion. Its effect in music seems to have burst on Pythagoras as a revelation of the whole cosmic system. We may accept the many later statements that he discovered the numerical ratios underlying the intervals that the Greeks called consonant and used as the basis of their scale. They involve only the numbers 1 to 41:2, octave; 3:2, fifth; 4:3, fourth. These numbers add up to 10, a sacred number for the Pythagoreans, which was symbolized by the dotted triangle (tetractys ), "source and root of everlasting nature." From the discovery that the sounds they recognized as beautiful depended on inherent, objective, mathematical order, they leaped to the conclusion that number was the key to the element of order in nature as a whole.

With this innovation the Pythagoreans would seem to have taken the momentous step from explanation in terms of matter (as the Milesians had sought it) to explanation in terms of form. Yet philosophy was not quite ready for that step, nor could the distinction between matter and form be clearly grasped. They saw simply the ultimate, single nature (physis ) of things in their mathematical structure. There seems little doubt that probably until well on in the fifth century they thought it possible to speak of things as actually made up of "numbers" that were regarded simultaneously as units, geometrical points, and physical atoms. Lines are made of points; surfaces, of lines; solids, of surfaces; and physical bodies, of solids. In this scheme two points made a line; three, the minimum surface (triangle); four, the minimum solid (tetrahedron). A later theory spoke of the "fluxion" of point into line, line into surface, and so on, which gave a geometrical progression (1, 2, 4, 8) instead of the arithmetical (1, 2, 3, 4), and the sequence of point, line, square, cube. Based on continuity, it seems designed to avoid the problem of incommensurable magnitudes or irrational numbers.

Whenever they were discovered (probably not much later than 450), incommensurables had dealt a blow to the original "things are numbers" doctrine, the idea that geometrical figuresand thus ultimately the physical worldare based on a series of integers. No ratio between integers can either describe the relation between the diagonal of a square and its side or serve as the basis of construction of a right triangle. If, however, magnitudes are regarded as continuous and hence infinitely divisible, the existence of incommensurable or irrational magnitudes (those which cannot be expressed as a ratio of natural numbers) could be explained and the difficulty overcome.

The Ultimate Principles

The analysis went further than that outlined above, for numbers themselves have their elements. The ultimate principles were limit and the unlimited, which were equated with good and bad respectively; moral concepts went side by side with physical concepts in this extraordinary system. Abstractions as well as physical phenomena were equated with numbers; for instance, justice was 4, the first square number, symbolizing equality or requital. After limit and the unlimited came odd and even instances, respectively, of these two. They generated the unit (considered to be outside the number series, and both odd and even), from the unit sprang numbers, and from numbers came the world. There seems no doubt that the scheme goes back to an ultimate duality that corresponds to the moral dualism of Pythagoreanism, but one can also see how monistically minded Neoplatonic commentators could speak of the cosmos as originating from the One. In general terms, kosmos was achieved by the imposition of limit on the unlimited in order to make the limited, just as the imposition of definite ratios on the indefinite range of musical pitch produced the harmonia of the scale.

Cosmogny and Cosmology

Cosmogony starts with the planting of a unit in the infinite. Aristotle called it, among other things, a seed; and since limit was associated with male and unlimited with female, the Pythagoreans probably thought of the generation of the living cosmos as taking place as did that of other animals. It grows by drawing in and assimilating the unlimited outside, that is, by conforming it to limit and giving it numerical structure. Physically the process resembles inspiration, and the unlimited is also called breath.

The unit seed had the nature of fire and in the completed cosmos (which evidently grew from the center outward) became a fire at its center. There are traces of two different cosmological schemes, a geocentric one that spoke of a fire at the center of Earth, and a more remarkable one attributed, in later sources at least, to the fifth-century Pythagorean Philolaus, which made Earth a planet. (Nicolas Copernicus in De Revolutionibus says that reading of this Pythagorean doctrine gave him courage to consider explaining the heavenly motions on the basis of a moving Earth.) According to this latter scheme, Earth, planets, sun, and moonand an extra body called the counterearthall revolved about the center of the universe, which was occupied by a fire invisible to man because he lived on the opposite side of Earth. It was known that the moon's light is borrowed, and the idea was extended to the sun, whose heat and light were said to be reflected from the central fire. The moon was eclipsed by the interposition of both Earth and the counterearth and, according to some, of further, otherwise unknown, planetary bodies. These caused the comparatively frequent lunar eclipses.

In this system, the mixture of religion and science in Pythagoreanism is well brought out. Fire was given the central position, not for any scientific reason but because it was regarded with religious aweand the center is the most "honorable" place. It was lauded with such titles as Hearth of the Universe, Tower of Zeus, and Throne of Zeus. Yet the same thinkers were aware that with Earth in orbit "the phenomena would not be the same" as in a geocentric scheme (presumably they were thinking of the lack of stellar parallax and variations in the apparent size of the sun and moon). They pointed out that even with a central Earth, an observer would be separated from the center by the distance of its radius, and they argued that the visible effect would be as negligible in one case as in the other. This assumes that the heavenly bodies are at vast distances from Earth; and it is not known how, if at all, this system was related to the theory later known as the harmony of the spheres.

In any case, there are many divergent versions of this doctrine. In outline, the idea was that large bodies in motion must inevitably produce a sound; that the speeds of the heavenly bodies, judged by their distances, are in the ratios of the musical consonances; and that therefore the sound made by their simultaneous revolution is concordant. We do not hear it because it has been with us from birth, and sound is perceptible only by contrast with silence. It has been plausibly argued that in the original version Pythagoras, like Anaximander, took only three orbits into account (sun, moon, and all the stars); this would relate it to his original musical discovery about the fourth, fifth, and octave. Later versions speak of seven, eight (Plato), and ten orbits. In any form, the doctrine emphasizes the universal importance, in Pythagorean eyes, of mathematical and musical laws and their intimate relation to astronomy.


The influence of Pythagorean thought on the history of philosophy and religion has been exercised largely through the medium of Plato, who enthusiastically adopted its main doctrines of the immortality of the soul, philosophy as an assimilation to the divine, and the mathematical basis of the cosmos. Later antiquity regarded him as a Pythagorean source, so that post-Platonic writings are of little help in distinguishing Pythagorean from original Platonic material in the dialogues. The Neo-Pythagorean movement, which started in the first century BCE, was an amalgam of early Pythagorean material with the teachings of Plato, the Peripatetics, and the Stoics. All of this material was credited to Pythagoras, who was revered as the revealer of esoteric religious truths. The interests of Neo-Pythagoreanism were religious and, in accordance with the prevailing tendencies of the time, it emphasized the mystical and superstitious sides of the earlier doctrine, its astral theology and number-mysticism, to the detriment of philosophical thinking. It cannot be called a system, but rather is a trend that in different forms continued until the rise of Neoplatonism in the third century CE, when it lost its identity in that broader and more powerful current. Besides contributing to Neoplatonism, it influenced Jewish thought through Philo of Alexandria and Christian thought through Clement of Alexandria. Prominent Neo-Pythagoreans were Cicero's acquaintance, Nigidius Figulus, and Apollonius of Tyana, a wandering mystic and ascetic of the first century CE, credited with miraculous and prophetic powers. Numenius of Apamea in the late second century was called both Pythagorean and Platonist, and was the immediate precursor of Neoplatonism.

See also Apeiron/Peras; Aristotle; Atomism; Cicero, Marcus Tullius; Clement of Alexandria; Continuity; Copernicus, Nicolas; Cosmology; Empedocles; Geometry; Leucippus and Democritus; Neoplatonism; Numenius of Apamea; Philo Judaeus; Philolaus of Croton; Plato; Platonism and the Platonic Tradition; Pre-Socratic Philosophy; Reason; Xenophanes of Colophon.


Delatte, A. Études sur la littérature pythagoricienne. Paris: E. Champion, 1915. For specialists.

Fritz, K. von. Pythagorean Politics in South Italy: An Analysis of the Sources. New York, 1940.

Guthrie, W. K. C. "Pythagoras and the Pythagoreans." In his History of Greek Philosophy. Vol. I, 146340. Cambridge, U.K.: Cambridge University Press, 1962. References to much of the literature of the subject will be found in this general account.

Minar, E. L., Jr. Early Pythagorean Politics. Baltimore: Waverly Press, 1942.

Morrison, J. S. "Pythagoras of Samos." Classical Quarterly, n.s., 6 (1956): 135156.

Thesleff, H. An Introduction to the Pythagorean Writings of the Hellenistic Period. Turku, Finland, 1961. For specialists.

Timpanaro-Cardini, Maria. Pitagorici: testimonianze e frammenti. 2 vols. Florence, 1958 and 1962. Texts from Fragmente der Vorsokratiker, edited by H. Diels and W. Kranz, translated into Italian, with introduction and commentary.

Van der Waerden, B. L. "Die Arithmetik der Pythagoreer." Mathematische Annalen 120 (1948): 127153 and 676700. For specialists.

Van der Waerden, B. L. Die Astronomie der Pythagoreer. Amsterdam: North-Holland, 1951. For specialists. Both this and the other van der Waerden work do not require a knowledge of Greek; texts are given in German.

W. K. C. Guthrie (1967)