Pitágoras

Pythagoras (c.570–c.497 bc)

Pythagoras of Samos was an early Greek sage and religious innovator. He taught the kinship of all life and the immortality and transmigration of the soul. Pythagoras founded a religious community of men and women in southern Italy that was also of considerable political influence. His followers, who became known as Pythagoreans, went beyond these essentially religious beliefs of the master to develop philosophical, mathematical, astronomical, and musical theories with which they tended to credit Pythagoras himself. The tradition established by Pythagoras weaves through much of Greek philosophy, leaving its mark particularly on the thought of Empedocles, Plato, and later Platonists.

1 Life and deeds

Pythagoras, son of Mnesarchus, was born on the island of Samos. For the first half of his life Pythagoras travelled widely, not only in Greece but supposedly also in Egypt, Phoenicia and Babylonia, where he is reputed to have acquired much of his knowledge and religious wisdom. Perhaps to escape the rule of Polycrates, the tyrant of Samos, he emigrated to Croton in southern Italy. His moral stature and eloquence gained him many adherents. With his followers, both men and women, Pythagoras practised a simple, communal life whose goal was to live in harmony with the divine. To that end he prescribed a regimen of purification that included dietary restrictions, periods of silence and contemplation, and other ascetic practices. In addition to the religious and monastic aspects of the Pythagorean society, we hear of Pythagorean political associations (hetaireiai) that played an important role in the public affairs of Croton and other southern Italian cities (it appears they initiated social reforms and supported aristocratic constitutions). After a time their dominance came to be resented and a ‘Pythagorean revolt’ ensued, in the course of which many Pythagoreans were killed or scattered abroad. Pythagoras himself, possibly as a result of this upheaval, moved to Metapontum where he died.

Already during his lifetime Pythagoras was regarded with near religious veneration. It is therefore not surprising that the stories told about him after his death should turn into hagiology and include many fantastic elements: that Pythagoras was the Hyperborean Apollo and had a golden thigh to prove it; that he was seen in two places at one time; that he could converse with animals and control natural phenomena. Pythagoras’ wonder-working clearly belongs to the realm of legend, although it reinforces the picture of him as a ‘shaman’. A more difficult matter is to establish what Pythagoras actually taught, since the oral and then the written traditions attribute to him not only miracles but also sophisticated mathematical and philosophical achievements. This habit of tracing all things back to the master, coupled with evidence of the quasi-religious avoidance of uttering his name, is typified in the expression common among Pythagoreans: ‘he himself said’ (autos epha; Latin ipse dixit). However, because Pythagoras wrote nothing and shrouded his lectures in secrecy, it is impossible to verify all that is ascribed to him. What remains certain is that he was a highly influential religious teacher whose main tenets dealt with the soul and the rites required for its purification and salvation. This made the Pythagorean movement especially popular in Magna Graecia, a fertile soil for mystery cults of all kinds. Pythagoras is also connected with certain ‘Orphic’ writings, since these share eschatological concerns similar to his oral teachings (see Orphism). For these reasons the following account will emphasize those doctrines that accord with Pythagoras’ reputation as an early Greek sage and religious innovator (for the philosophical and scientific theories traditionally associated with his name, see Pythagoreanism).

2 Teachings

Pythagoras believed that the world was animate and that the planets were gods. This view of the universe as living and divine was characteristic of early Greek thought (see Thales §2), but what appears unique with Pythagoras is the corollary he drew on an anthropological level: there is an element in human beings that is related to the universe and that, like the universe in which events recur in eternal cycles, is eternal. This divine, immortal element is the soul (see Psychē). With the death of the body the soul passes into another body, human or animal. An early witness to Pythagoras’ belief in transmigration (or metempsychōsis) is the poet-philosopher Xenophanes (§1), who satirizes him for claiming to recognize the soul of a friend when he heard the voice of a puppy that was being beaten. Pythagoras asserted of himself, as is typical of a religious figure who draws on personal experience, that he had once been the Homeric hero Euphorbus, and he exhorted his disciples to recall their own past lives.

The immortality of the soul underlies many of Pythagoras’ practical teachings, for the soul, as the most important element within a person, required nurture to ensure not only equanimity in this life but also a better incarnation in the life to come. These ends could be achieved by bringing the soul into harmony with the divine, cosmic order (according to a disputed doxography, Pythagoras was the first to pronounce the world a kosmos, a term that in Greek combines the ideas of adornment, beauty and order). In so far, however, as the soul resided in a body, it needed to be freed from the turmoils and corrupting influences of the body. Hence Pythagoras preached a strict way of life that centred on purification (see Katharsis) and asceticism. Furthermore, he practised a form of musical therapy for both body and soul. He valued friendship highly as a means of promoting equality and concord; the love of friends was a specific instance of the universal sympathy existing in the cosmos.

The rules of Pythagoras found expression in short, pithy sayings known as akousmata, a term that implies oral transmission, or, more frequently, as symbola. The latter most likely functioned as secret passwords for Pythagorean initiates but, as the name suggests and their often esoteric and oracular nature prompted, they were also subjects for ‘symbolic’ interpretation. The symbola range from primitive religious taboos to simple moral precepts and various dietary prohibitions (abstinence from certain parts of animals and the famous ban on eating beans).

3 Legacy

Pythagoras, according to Plato (Republic X 600b), handed down to his followers a distinctive way of life (bios) ‘they call Pythagorean to this day’. By Plato’s day the Pythagorean life meant, besides purifications of the soul, inquiries in philosophy, mathematics, astronomy and music. Did Pythagoras bequeath these enterprises as well? Empedocles (§2), who was greatly influenced by Pythagoras, praised him as a man of surpassing knowledge, with a vast wealth of understanding, capable of all kinds of wise works. Heraclitus (§1), while agreeing that Pythagoras ‘practised inquiry beyond all other men’, saw the result as a peculiar wisdom consisting of polymathy and evil artifice. From both witnesses, however, Pythagoras emerges as a figure active in a wide variety of fields and therefore likely at least to have dabbled in those studies for which Pythagoreans were known in Plato’s time. Still, the only sure legacy of Pythagoras is the immortality of the soul. Although this was primarily a religious belief, it carried philosophical import. By singling out the immortal soul as the essential element of life Pythagoras foreshadowed the Parmenidean/Platonic distinction between eternal being and changeable becoming and, in general, the dualism of mind and matter that informs so much of Western philosophy.

Pythagoreanism

Pythagoreanism refers to a Greek religious-philosophical movement that originated with Pythagoras in the sixth century bc. Although Pythagoreanism in its historical development embraced a wide range of interests in politics, mysticism, music, mathematics and astronomy, the common denominator remained a general adherence among Pythagoreans to the name of the founder and his religious beliefs. Pythagoras taught the immortality and transmigration of the soul (reincarnation) and recommended a way of life that through ascetic practices, dietary rules and ethical conduct promised to purify the soul and bring it into harmony with the surrounding universe. Thereby the soul would become godlike since Pythagoras believed that the cosmos, in view of its orderly and harmonious workings and structure, was divine. Pythagoreanism thus has from its beginnings a cosmological context that saw further evolution along mathematical lines in the succeeding centuries. Pythagorean philosophers, drawing on musical theories that may go back to Pythagoras, expressed the harmony of the universe in terms of numerical relations and possibly even claimed that things are numbers. Notwithstanding a certain confusion in Pythagorean number philosophy between abstract and concrete, Pythagoreanism represents a valid attempt, outstanding in early Greek philosophy, to explain the world by formal, structural principles. Overall, the combination of religious, philosophical and mathematical speculations that characterizes Pythagoreanism exercised a significant influence on Greek thinkers, notably on Plato and his immediate successors as well as those Platonic philosophers known as Neo-Pythagoreans and Neoplatonists.

1 History

In the second half of the sixth century bc, Pythagoras founded a community in the southern Italian city of Croton whose members were united by the belief in the transmigration of the soul, an ascetic way of life that centred on the purification of the soul, and a political outlook that aimed at social reform along aristocratic lines. Pythagorean associations (hetaireiai), which also formed in other cities of Magna Graecia, acquired considerable political authority, but their dominance eventually met with opposition, both during the lifetime of Pythagoras and later again about 450 bc. In the wake of these anti-Pythagorean movements the followers of Pythagoras were scattered throughout the Greek world, so that by the time of Plato there is little evidence of formal Pythagorean societies. Individual Pythagoreans, however, continued to be recognized, some by their distinctive lifestyle in matters of food, dress and purificatory practices, others by the additional pursuit of various philosophical, mathematical and musical theories with which they credited Pythagoras. Later tradition refers to these two types of Pythagoreans as ‘hearers’ (akousmatikoi) and ‘learners’ (mathēmatikoi). The distinction supposedly goes back to the original society in which some members were only fit to accept the oral teachings of Pythagoras without arguments and proofs, while those with more leisure and perhaps philosophical ability were further instructed in the rational foundations of the master’s teachings. Whatever differences there may have been between groups of Pythagoreans, and even among individual mathēmatikoi, all professed allegiance to Pythagoras.

From the time of Pythagoras to Plato there were several famous Pythagoreans – Hippasus, Philolaus and Archytas. Aristotle, in his extant works, speaks only generally of the Pythagoreans (‘some Pythagoreans say… ’); his special treatise on Pythagorean beliefs unfortunately no longer survives. In the third and second centuries bc we do not hear of philosophers who were known as or called themselves Pythagoreans, but interest in ‘Pythagoreanism’ continued, as is evidenced by the wealth of apocryphal writings in prose and verse on Pythagorean themes that mostly date from this period. The actual practice of Pythagoreanism experienced a revival in the Roman world from the first century bc to the first century ad; Latin writers such as Cicero, Ovid and Seneca testify to its popularity. In the first two centuries ad the theoretical side of Pythagoreanism marked certain philosophers to the extent that they may be called Neo-Pythagoreans (see Neo-Pythagoreanism) and these in turn influenced later Platonic philosophers (see Neoplatonism).

2 Music, mathematics, and cosmology

Plato says of the true philosopher, whose mind is on the higher realities (that is, the Platonic Forms):

he looks unto the fixed and eternally immutable realm where… all is orderly and according to reason, and he imitates this realm and, as much as possible, assimilates himself to it… and by association with the divine order becomes himself orderly and divine as far as a human being can…

(Republic VII 500c)

Plato’s philosophic ideal, as well as one of the methods he prescribed to achieve it – namely mathematics – owes much to Pythagoreanism. Pythagoras had taught that life should be in harmony with the divine cosmos (see Pythagoras §2). In Pythagoreanism harmony (harmonia) became a central tenet and was explained through numerical relations, possibly in connection with musical theory. For example, the Pythagoreans thought that the motions of the orbiting planets produced a sound which, given the belief that the intervals between the heavenly bodies corresponded to musical ratios, was harmonious. So Aristotle explains the ‘music of the spheres’ – one of several explanations offered for this famous Pythagorean image. Of seminal importance for illustrating the coherence of music and number was the discovery of musical ratios – that music should result when the first four integers of the numerical system, used as components in the harmonic ratios of the octave (2 : 1), the fifth (3 : 2) and the fourth (4 : 3), were imposed upon the continuum of sound. Whether or not Pythagoras, as tradition holds, was the ‘discoverer’ of the musical concords, the Pythagoreans fixed upon the first four numbers as the building blocks of nature. These four sufficed to give extension and shape to bodies in the sequence of point–line–surface–solid:

Moreover, the first four integers add up to ten, which the Pythagoreans considered a perfect number and represented in a figure called the tetraktys.

3 Soul and ethics

Perhaps in no other ancient Greek philosophy is the human being as intimately linked to the cosmos as in Pythagoreanism. Although the correspondence between microcosm and macrocosm, the individual and the universe, is found in much of Greek thought, it received a particularly sharp outline in the Pythagorean view that by assimilation to the divine cosmos the self would come to reflect the cosmic order and harmony. The true self of every person was the soul (see Psychē), the essential element in the partnership of body and soul. Pythagoras’ teaching that the soul survived the dissolution of the body and reappeared in other bodies was steadfastly adhered to throughout the history of Pythagoreanism, even if it was sometimes eclipsed by other interests (see Immortality in ancient philosophy §1). The doctrine of the immortality and transmigration of the soul translated into a practical and clearly defined way of life that combined ritual purity with high ethical standards and whose precepts were embodied in sayings known as akousmata and symbola (see Pythagoras §2). The belief in the kinship of all life, as a corollary to the belief in transmigration, imposed a great moral responsibility towards parents, children, friends and fellow citizens, and generally entailed a respectful attitude towards all forms of life in which soul may be embodied; hence the Pythagoreans were, to varying degrees, vegetarians. Their dietary laws were also intended to free them from bodily pollution. The body was seen as a prison, even as a grave (Plato, Gorgias 493a) from which the soul was to rise, ever achieving superior reincarnations and culminating in a state of divinity. Thus the Pythagoreans postulated three kinds of rational creatures: ‘gods, men, and such as Pythagoras’. Pythagoras was thought to have achieved semi-divine status, which in Greek could be expressed by ‘daemon’. It is in a Pythagorean vein that Empedocles (§2) proclaims human beings to be fallen ‘daemons’. (A complex ‘daemonology’ was to become an ingredient of Neo-Pythagoreanism.) The recognition of the religious and moral mandates of the Pythagorean life and of its eschatological meaning constituted wisdom (sophia) and the lover of such a life was a philosophos, a term that in this sense, as certain traditions report, was first coined in Pythagorean circles.

From whatever angle one approached the tetraktys in counting, the addition always resulted in ten, the basis of the decimal system. The tetraktys also produced an equilateral triangle, the simplest plane figure, and a pyramid, the simplest three-dimensional shape.

The Pythagoreans viewed the tetraktys as a sacred symbol and used it in the following oath: ‘By him [Pythagoras] who handed down to us the tetraktys, source and root of everlasting nature’. In short, the Pythagoreans supposed the nature of things could be understood numerically; indeed, some of them apparently went so far as to say that things are numbers: ‘…they assumed the elements of numbers to be the elements of all things that exist, and the whole universe to be harmonia and number’ (Aristotle, Metaphysics 986a2).

Since numbers functioned as the constituent elements of the cosmos, it is important to see how the Pythagoreans understood the nature and generation of number itself. Aristotle again offers the best starting-point:

The elements of number are the even and the odd, the latter limited, the former unlimited. The One is composed of both of these (for it is both even and odd) and number comes from the one; and numbers… are the whole universe.

(Metaphysics 986a17)

How the One can be both odd and even is best explained in the sense that the One, as the first number, is the principle of both odd and even numbers (zero was unknown in Greek mathematics). Thus ‘number comes from the One’, and the generation of number was simultaneously a cosmogonical process since ‘numbers… are the whole universe’. Although in its origin the Pythagoreans viewed the One as both odd and even, limited and unlimited, in its practical application, that is, in its interaction with other numbers, they treated it as odd and limiting. This can be seen in the schema by which the Pythagoreans illustrated the correspondence of odd/even to limited/unlimited. Gnomons (carpenter squares) were placed around an arrangement of points (or pebbles) as follows:

When a gnomon is placed around one point, and the process is continued in sequence, the resulting figure is always of ‘limited’ shape, that is, always a square, whereas when it is placed around two points, the result is a series of oblongs whose sides stand in an ‘unlimited’, that is, an infinite variation to each other. In this scheme the One, as a single unit or point, is equated with the odd. The One is also ranked with limited and odd in the Pythagorean table of ten opposites:

The position of limited/unlimited at the beginning is not accidental, for their opposition, embodied in the original composition of the One, was considered primal and basic to the development of number and the universe, while the opposition between odd and even can be generally subsumed under this fundamental distinction. The connection of the One with the limited reappears in another Pythagorean text (Aristotle, Metaphysics 1091a17), where it is said that the limited (tantamount to the One) ‘breathes in’ and is thus penetrated by the unlimited. In this cosmological fragment the unlimited is equated with the void, to be thought of as infinite space and serving as a dividing principle. The One now becomes a Two (reminiscent of the mythological separation of heaven and earth), which in effect marks the beginning of number, plurality and the existence of discrete physical bodies. The cosmogonic process is sometimes put the other way round: the unlimited is limited (or penetrated) by the limit. This way of stating it brings to the fore the characteristic Greek feeling that what is unlimited, without bounds, is without order and somehow evil (hence ‘bad’ is listed under the unlimited in the above table) and therefore needs to be curtailed and bounded by limit and measure (as the limits of harmonic ratios must be imposed upon the limitless range of sound to produce music). The result, in cosmological terms, is then precisely a kosmos, an orderly and structured universe. In so far as the perfected and integral cosmos represents a harmonia, a unity stemming from a reconciliation of opposites, Pythagoreanism has a monistic aspect, yet in its source theory it remains essentially dualistic: number comes from the One, and the One is composed of limited and unlimited. From these first two principles everything derives, even though the One, when identified with limit, odd, male, and so on, was felt to be the ‘good’ element in this cosmic dualism. (There can be no question of positing the One as the sole ultimate principle without imposing Platonic and Neo-Pythagorean notions upon early Pythagoreanism.)

Limited und unlimited are the principles of number. Numbers do not merely express the substance, shape and quantitative differences of things but actually appear to constitute bodies of physical dimension. The identification of things with numbers confuses physical bodies with abstractions, or, in Aristotelian terms, the material causes of things with their formal causes. Here Pythagoreanism connects with much of Presocratic thought according to which the source and informing element of the universe was thought of as some kind of matter, be it water, air, fire, earth, or a combination of these (see Archē). At the same time the numerical theories of the Pythagoreans represent a true advance in the history of Greek philosophy: the attempt to explain the nature of things by numbers is a valid philosophical striving to understand the world by its formal or structural principles, even if the Pythagoreans then equated these with the things themselves. And while the Pythagoreans’ interest in numbers was often infused with a mystical element (their solemn veneration of the tetraktys) and a primitive number symbolism that blurred distinctions between abstract and concrete (a moral concept such as justice was considered the embodiment of four, a square number of ‘just’, that is, equal reciprocity), no less an authority than Aristotle acknowledged that the Pythagoreans were pioneers and made advances in the mathematical fields (arithmetic, astronomy, harmonics and some geometry). For the ‘Pythagorean theorem’, although long known to Babylonian mathematicians, the Pythagoreans are generally considered to have found the proofs and, although in the wake of this theorem they discovered that the ratios of geometrical figures to each other cannot simply be expressed by a series of rational integers – the discovery of irrational numbers and the principle of incommensurability upset the Pythagorean notion that the world is harmony and number – their very setbacks contributed to subsequent work in Greek mathematics. Plato valued mathematics as a useful discipline to train the philosopher in the perception of eternal and transcendent truth, since in his mind it dealt essentially with invisible and eternally valid realities. By divorcing number from physical substance Plato transforms Pythagorean mathematics, yet few doubt that his interest in number, arithmetic and measure largely rests upon Pythagorean foundations (consider the mathematical model of the universe in his Timaeus (see Plato §16). Plato’s immediate successors in the Academy were steeped in Pythagorean number theory (see Speusippus §2; Xenocrates §2).