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Although there are no extant fragments of the writings of Pythagoras, his views were influential in the ancient world and have been referred to by a number of philosophical writers, among them Plato, Aristotle, Porphyry, and Diogenes Laërtius. As one might expect, the accounts are not entirely consistent, and it is often difficult to determine precisely or even approximately what view Pythagoras held on a question under discussion, but there is a body of beliefs that critics generally attribute to Pythagoras or to his followers. The followers are generally assumed either to have inherited the master’s views or to have been inspired by his philosophy and practice to develop their ideas along lines that have a distinctive inherited character.
Pythagoras (like many ancient Greek philosophers) did not distinguish his metaphysical convictions from his beliefs about the physical world: His ontology (theory of being), cosmology (theory of cosmic origin and development), epistemology (theory of knowledge), theology, and ethics appear to be grounded in certain abstract mathematical ideas and beliefs and to be interrelated.
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According to Aristotle, the Pythagoreans believed that all things are numbers in the sense that the principles of numbers are the principles of all things. “There is but one number, the mathematical,” is a view attributed to Pythagoras by Aristotle, together with the related propositions that all objects of sense are numbers and that numbers are prior, both in power and existence, as well as logically, to physical objects. Accordingly, the Pythagoreans differed from the Milesian philosophers, who found in water, fire, or earth the fundamental substance and cause of things; for the Pythagoreans, the primary cause and substance of all things is number. Not only physical things but also justice and the other virtues, as well as the soul and reason, are in principle and composition numbers. The early writers attributed this philosophical tendency of Pythagoras and his followers—the tendency to take number as primary, creative essence and substance—to the Pythagoreans’ having noticed similarities between numbers and objects of sense (although it is not clear what sorts of relationships counted as similarities). Thus, Aristotle writes, “They see many qualities of numbers in bodies perceived by sense” and “in numbers, . . . they thought they saw many likenesses to things that are and that are coming to be.”
No doubt part of the belief in the creative power of numbers stemmed from the discovery of the numerical ratios involved in musical harmony. If numbers can so order sound as to achieve harmony, then it is credible that numbers so order the unlimited as to achieve a harmonious universe and that numbers so order the things within the universe as to endow them with distinctive numerical natures and to make physical harmony possible. Thus, since the Pythagoreans regarded ten as “the very nature of number” (Aetius) and as perfect (Aristotle), they declared that the number of heavenly bodies must be ten (although they had observed only nine and thereby presumed the tenth to be an unobservable body between the earth and the sun, a “counter-earth”).
One is tempted to suppose that the Pythagoreans regarded number as the nature of things because they discovered constant and harmonizing arithmetical ratios in nature; it is as if they fastened on the abstract relationships that contemporary physics attempts to fix by mathematical equations. According to Aristotle, however, number for the Pythagoreans was not only the “first principle” of all things—the formal aspect or essence—but also “as it were the matter in things and in their conditions and states.” Hence, number was not only form but also matter; the numbers themselves were quantities or magnitudes, not simply the formal aspects of that which has quantity. Pythagoras (or the Pythagoreans) believed that the odd and the even are the elements of numbers; the even is unlimited and is identified with the infinite; the odd is limited. Unity (the number 1) is the product of the odd and even, and all number arises from this original unity.
According to Aristotle, some Pythagoreans—proceeding from a dedication to ten as the perfect number—maintained that there are ten fundamental principles of all being, each principle consisting of a pair of opposites: the limited and the unlimited, the odd and the even, the one and the many, the right and the left, the male and the female, the resting and the moving, the straight and the crooked, the light and the dark, the good and the bad, the square and the oblong. Here again, Aristotle surmises, the principles appear to have been ranged “under the category of matter, for they say that being is compounded and formed from them, and that they inhere in it.”
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The Pythagorean account of the origin of the universe as an ordered system consistently accords to numbers the power of generation and the essential determination of the direction and quality of world order. According to Aristotle in Physica (second Athenian period, 335-323 b.c.e.; Physics, 1812), the Pythagoreans argued that void entered into heaven, which breathed it in from the Unlimited. Somehow “void defines the nature of things,” and first of all defined numbers. Void is described as “a kind of separating and distinguishing factor between terms in a series.” (It is not clear from Aristotle’s account—and perhaps it was not clear to Aristotle—whether the Pythagoreans believed that number was somehow in void and then drawn out of void by a movement, like breathing, stemming from a resolution of tension between the limited and the unlimited, or that void somehow actually gave rise to number. In any case, the universe results from the forming power of number, according to the Pythagoreans.)
Aristotle remarks that although the common belief is that the earth is at the center of the universe, the Pythagoreans (who were dedicated astronomers) believed that a central fire is the center and that the earth creates day and night by circling about this fire. Fire as the center of space, matter, and nature also was regarded as the authoritative guard of all being, “the guard of Zeus.”
Aristotle also comments on the Pythagorean view that there is a music of the spheres, a harmony of sound produced by the movement of the heavenly bodies in accord with the intervals determined by numbers. The belief in this heavenly music followed from their assumptions about the effect of the determination of all things by numbers (just as the belief in the tenth planet, the “counter-earth,” was required by their belief in ten as the perfect number). The Pythagoreans accounted for the fact that human beings are not aware of the heavenly sounds by pointing out that because this sound is part of the nature of things and has been with us from birth, it is part of our lives, a constant background, and hence not noticeable.
Even the virtues can be understood mathematically, the Pythagoreans believed, according to the sources. Aristotle remarks in the Ethica Nicomachea (second Athenian period work, 335-323 b.c.e.; Nicomachean Ethics, 1797) that the Pythagoreans regarded the good as the limited and the evil as unlimited. (The idea of moral virtue as involving constraint within limits and even as exhibiting a kind of harmony was characteristic of the Greeks, including Plato and Aristotle.) Aristotle also mentions that the Pythagoreans defined the just as that which is reciprocal. Aristotle declares that Pythagoras was mistaken in attempting to discuss goodness by reference to numbers. Such a reference is inappropriate, Aristotle asserts; after all, he insists, “justice is not a square number.”
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Hippolytus speaks of the Pythagoreans as combining astromony, geometry, and music in their study of nature. He reports that Pythagoras claimed that God is a monad, and he mentions the Pythagorean belief that the universe is melodic and that the stars move rhythmically and hence melodiously. For the Pythagoreans, Hippolytus continues, number is the first principle, and this first principle is a male monad in substance, “begetting as a father all other numbers.” The dyad is female, the triad male; that is, even is female; odd is male. All numbers are fours, and four generates ten, the perfect number. (If one adds the numbers 1, 2, 3, and 4, the total is 10.) The four parts of the decad—number, monad, power, and cube—by combining, account for all growth.
Hippolytus also calls attention to the Pythagorean belief in the immortality of the soul and in the soul’s moving from one body to another. (He mentions the Pythagorean prohibition against the eating of beans because “at the beginning and composition of all things when the earth was still a whole, the bean arose.”)
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The Neoplatonic philosopher Proclus alludes to the Pythagorean discovery that the square of the hypotenuse of a right-angle triangle is equal to the sum of the squares of the other two sides. (The Pythagorean practice of arranging units or “dots” in squares may have contributed to some of their mathematical discoveries as well as to their metaphysical conviction that all things are numbers. As the Greek philosopher Speusippus points out, for the Pythagoreans 1 is the point, 2 is the line, 3 is the triangle, and 4 is the pyramid. The tetraktys, a triangle with a four dot base, then a line of three, then two, then one dot—making ten in all—was a key figure.)
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The Pythagoreans were subject to a number of rules of considerable moral and religious importance but hardly of philosophical significance (such as “Stir not the fire with iron,” “Speak not of Pythagorean matters without light,” and “Let not a swallow nest under your roof”). These rules, together with others—such as the prohibitions against the eating of flesh and beans and against the sacrifice of animals—stem from certain beliefs involved in the religion of the Pythagoreans as influenced by Orphism (the cult of Orpheus). The strictures against the eating of flesh and the sacrifice of animals, for example, are required by the belief in the transmigration of souls.
The moral emphasis in religious beliefs and practices of the Pythagoreans was on the purification of the soul. Porphyry writes of their beliefs that the soul is immortal and changes into other kinds of living things, that events recur in cycles, and that all living things be regarded as kin. Herodotus also speaks of the cyclical theory and applies it specifically to the transmigration of souls: From a human body, the soul enters the body of an animal born at the time of death of the human organism; the soul then makes the rounds of land and sea creatures; finally, after three thousand years, it enters a human body again. Diogenes Laërtius tells a tale in which Pythagoras calls upon someone to stop whipping a puppy because Pythagoras had recognized in the yelping of the dog the voice of a departed friend.
Through philosophy, the use of reason, music, religious observances, and the inculcation and development of a spirit of universal sympathy, the Pythagoreans sought the purification of the soul. (Because of the fundamental metaphysical belief in the ultimate reality and power of numbers, these various routes to purification were unified; the reliance on music, for example, was due at least in part to the discovery of the arithmetical proportions exhibited in musical harmony.) The soul, then, was to be educated, trained, ordered, and harmonized. Through the restraint of desire, the soul was to find its proper limits and balance; it could thereupon fit into the universal scheme of things, the universe itself exhibiting the beauty of harmony resulting from its essence in sacred numbers. As Aristotle wrote, for the Pythagoreans “the whole heavens were harmony and number.”
Although the Pythagoreans apparently meant literally to claim that all things are numbers and that harmony is achieved through the proper arithmetical relationships, their philosophy probably could not have elicited the kind of dedication it did had not the emphasis on numbers been made “mystically”—that is, in such a way as to transform a mathematical metaphysics into a Greek ethics. Number as first principle was regarded as indefinable (according to Hippolytus); hence, it lent itself to symbolic extension as the possibility of order in life and to moral application in the form of injunctions calling for the attainment of inner harmony and the recognition of a universal harmony that provides an ideal, a ten. Although the discovery of the theorem that bears the master’s name was a magnificent intellectual accomplishment, the moral use of metaphysics by the Pythagoreans contributed to the distinctive Greek emphasis on the use of reason, the recognition of opposites, the attainment of the mean, the setting of proper limits, and the harmonizing of the self and the world.
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Bamford, Christopher, ed. Homage to Pythagoras: Rediscovering Sacred Science. Hudson, N.Y.: Lindisfarne Press, 1994. This collection of essays touches on Pythagoras’s ideas as they affect architecture and religion, among other topics. Includes bibliography.
Boudouris, K. I., ed. Pythagorean Philosophy. Athens: International Center for Greek Philosophy and Culture, 1992. This volume examines Pythagoras and the Pythagorean school. Includes bibliography.
Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Translated by Edwin L. Minar, Jr. Cambridge, Mass.: Harvard University Press, 1972. This study, translated from the German, attempts to disentangle Pythagoreanism from Platonism and to describe the various aspects of Pythagoreanism, from music theory to what is called shamanistic religion. Includes extensive bibliography.
Godwin, Joscelyn, ed. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993. This volume examines the effect that the philosophy and aesthetics of Pythagoras, particularly the concept of the harmony of the spheres, had on music. Includes bibliography and indexes.
Guthrie, W. K. C. The Earlier Presocratics and the Pythagoreans. Vol. 1 in A History of Greek Philosophy. Cambridge, England: Cambridge University Press, 1962. Contains an excellent, nearly two-hundred-page chapter on Pythagoras and a half dozen Pythagoreans.
Kingsley, Peter. Ancient Philosophy, Mystery, and Magic: Empedocles and Pythagorean Tradition. Oxford: Clarendon Press, 1995. This book illuminates Pythagorean philosophy by showing how it influenced Empedocles. It demonstrates the Pythagorean origin of Plato’s myths. It examines connections between ancient magic, science, and religion, tracing a line of transmission from Empedocles and the Pythagoreans into the world of Islam.
Kirk, Geoffrey S., John E. Raven, and M. Schofield. The Presocratic Philosophers. 2d ed. Cambridge, England: Cambridge University Press, 1983. One chapter contains a scholarly account of Pythagorean philosophy; includes Greek text of testimony (no fragments).
Mourelatos, Alexander P. D. The Pre-Socratics: A Collection of Critical Essays. Princeton, N.J.: Princeton University Press, 1993. This volume includes two essays on Pythagoreanism. F. M. Cornford argues that the early Pythagorean school exhibited two radically opposed systems of thought, the mystical and the scientific, which have been mistakenly conflated. Charles H. Kahn addresses the question of how much of the Pythagorean doctrine can be traced back to some earlier period of the school and specifically to Pythagoras.