INTRODUCTION
IT HAS BEEN SUGGESTED, by Alfred North Whitehead, that "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. " If such be the case, what might then be said of Pythagoras, to whose philosophy Plato was so greatly indebted? While no definitive answer will be attempted here, it might do well to note that not only did Pythagoras first employ the term philosophy, and define the discipline thereof in the classic sense, but that he bequeathed to his followers, and to the whole of Western civilization, many important studies and sciences which he was instrumental in either formulating or systematizing.
True as this may be, much mystery surrounds the figure of Pythagoras, despite the significant influence of Pythagorean thought in antiquity. Of course, many things can be precisely stated. He was both a natural philosopher and a spiritual philosopher, a scientist and a religious thinker. He was a political theorist, and was even involved in local government. While he may not have been the first to discover the ratios of the musical scale, with which he is credited, there can be no doubt that he did conduct extensive research into musical harmonics and tuning systems. Pythagoras is well known as a mathematician, but few realize that he was also a music therapist having, in fact, founded the discipline. Pythagoras taught the kinship of all living things; hence, he and his followers were vegetarians. Yet, while all these things may be safely stated, quite a bit of mystery still remains. [1] This is due in large part to the fact that Pythagoras left no writings, although it is said that he wrote some poems under the name of Orpheus [2]. Pythagoras' teaching was of an oral nature. While he seems to have made some speeches upon his arrival in southern Italy to the populace, the true fruits of his philosophic inquiries were presented only to those students who were equipped to assimilate them. Pythagoras no doubt felt, like his later admirer Plato, that philosophic doctrines of ultimate concern should never be published, seeing that philosophy is a process, and that books can never answer questions, nor engage in philosophical enquiry. [3]
Yet, despite the lack of first-hand writings by Pythagoras himself, we need not be deterred. There is an immense amount of material in the biographies of Pythagoras which goes back to a very early date, and it is certainly possible to sketch an accurate if not complete picture of early Pythagorean philosophy, even being quite specific on many points. [4] Let us then begin with Pythagoras himself.
Pythagoras
THE PRIMARY SOURCES of information about the life of Pythagoras are to be found in this anthology. In addition to containing quite a bit of information about Pythagoras which goes back to an early date, these four biographies also demonstrate in an admirable fashion the high esteem in which the philosopher was held.
Pythagoras was born around 570 B.C.E. to Mnesarchus of Samos, a gemengraver, and his wife, Pythais.
The biographies of Pythagoras are unanimous that at an early age he travelled widely to assimilate the wisdom of the ancients wherever it might be found. He is said by Iamblichus to have spent some 22 years in Egypt studying there with the priests, and is also said to have studied the wisdom of the Chaldeans firsthand. These accounts are generally accepted by most scholars -- as indeed they should be, owing to the high degree of contact between Asia Minor and other cultures -- although it is doubtful, while not impossible, that he travelled to Persia to study the teachings of Zoroaster. In these distant lands Pythagoras not only studied the sciences there cultivated, including mathematical sciences we may safely presume, but was also initiated into the religious mysteries of the "barbarians." As Porphyry succinctly observes, "It was from his stay among these foreigners that Pythagoras acquired the greater part of his wisdom." [5]
After his studies abroad, Pythagoras returned home to the island of Samos, where he continued his philosophical researches. It is said that he outfitted a cave especially designed for the study of philosophy, and it was there that he made his home. About this time Pythagoras opened his first school, as we are told by Porphyry, yet it probably was not long-lived, as Pythagoras decided to leave Samos at the age of 40, owing to the tyranny of Poly crates which was then flourishing.
From Samos Pythagoras journeyed to South Italy, arriving at Croton, "conceiving that his real fatherland must be the country containing the greatest number of the most scholarly men." [6]
It would seem that his reputation preceded the philosopher, for he was shortly asked to speak to the populace of Croton -- men, women, and children -- on the proper conduct of life. The essence of these speeches is to be found in the biography by Iamblichus. While obviously not recorded verbatim, it seems quite likely that the content of these talks is genuinely Pythagorean and goes back to Pythagoras himself. [7] According to the biographies, the populace was enthralled by the wisdom of this man, owing to which he was invited to become involved in local government.
Number, Kosmos, Harmonia
... and the ancients, who were superior to us and dwelt nearer to the Gods, have handed down a tradition that all things that are said to exist consist of a One and a Many and contain in themselves the connate principles of Limit and Unlimitedness.
-- Plato, Philebus 16c
WHERE PYTHAGORAS DEVELOPED his interest in Number we do not know, although it is likely that he was not the first to be concerned with its sacred or metaphysical dimension. What we do know is that a metaphysical philosophy of Number lay at the heart of his thought and teaching, permeating, as we shall see, even the domains of psychology, ethics and political philosophy.
The Pythagorean understanding of Number is quite different from the predominately quantitative understanding of today. For the Pythagoreans, Number is a living, qualitative reality which must be approached in an experiential manner. Whereas the typical modern usage of number is as a sign, to denote a specific quantity or amount, the Pythagorean usage is not, in a sense, even a usage at all: Number is not something to be used; rather, its nature is to be discovered. In other words, we use numbers as tokens to represent things, but for Pythagoreans Number is a universal principle, as real as light (electromagnetism) or sound. As modern physics has demonstrated, it is precisely the numeric, vibrational frequency of electromagnetic energy -- the "wavelength" -- which determines its particular manifestation. Pythagoras, of course, had already determined this in the case of sound.
Because Pythagorean science possessed a sacred dimension, Number is seen not only as a universal principle, it is a divine principle as well. The two, in fact, are synonymous: because Number is universal it is divine; but one could as easily say that because it is divine, it is universal. Hence, the aim of Pythagorean and later Platonic science is different from that of modern "Aristotelian" science: it is not so much involved with the investigation of things, as the investigation of principles. It should be very firmly emphasized, however, that for Pythagoras the scientific and religious dimensions of number were never at odds with each other. Moreover, the Pythagorean approach to Number, for the first time in Greece, elevated mathematics to a study worth pursuing above any purely utilitarian ends for which it had previously been employed.
The Pythagoreans believed that Number is "the principle, the source and the root of all things." [8] But to make things more explicit: the Monad, or Unity, is the principle of Number. In other words, the Pythagoreans did not see One as a number at all, but as the principle underlying number, which is to say that numbers -- especially the first ten -- may be seen as manifestations of diversity in a unified continuum [9] To quote Theon of Smyrna:
Unity is the principle of all things and the most dominant of all that is: all things emanate from it and it emanates from nothing. It is indivisible and it is everything in power. It is immutable and never departs from its own nature through multiplication (lxl=1). Everything that is intelligible and not yet created exists in it; the nature of ideas, God himself, the soul, the beautiful and the good, and every intelligible essence, such as beauty itself, justice itself, equality itself, for we conceive each of these things as being one and as existing in itself. [10]
If One represents the principle of Unity from which all things arise, then Two, the Dyad, represents Duality, the beginning of multiplicity, the beginning of strife, yet also the possibility of logos, the relation of one thing to another:
The first increase. the first change from unity is made by the doubling of unity which becomes 2, in which are seen matter and all this is perceptible, the generation of motion, multiplication and addition, composition and the relationship of one thing to another. [11]
With the Dyad arises the duality of subject and object, the knower and the known. With the advent of the Triad, however, the gulf of dualism is bridged, for it is through the third term that a Relation or Harmonia ("joining together") is obtained between the two extremes. While Two represents the first possibility of logos, the relation of one thing to another, the Triad achieves that relation in actuality. If this process of emergence is represented graphically as in figure I, we can see that the Triad not only binds together the Two, but also, in the process, centrally reflects the nature of the One in a "microcosmic" and balanced fashion. (See figure 1.)
FIGURE 1. UNlTY, DUALITY AND HARMONY
What we have seen in this example of Pythagorean paradigm, based on the universal principles of pure Number and Form, is the emergence of Duality out of Unity, and the subsequent unification of duality, which in turn results in a dynamic, differentiated image of the One in three parts-a continuum of beginning, middle and end, or of two extremes bound together with a mean term. This process, in fact, is the archaic and archetypal paradigm of cosmogenesis, the pattern of creation which results in the world. As F.M. Corn ford has observed:
The abstract formula which is common to the early cosmogonies is as follows: (1) There is an undifferentiated unity. (2) From this unity two opposite powers are separated out to form the world order. (3) The two opposites unite again to generate life. [12]
Cornford goes on to demonstrate how this universal pattern underlies not only the cosmogonies of Greek myth, but also those of the early Ionian scientific tradition. [13] It also underlies, as one may suspect, the Pythagorean view of the kosmos, literally "world-order" or "ordered-world," a term that Pythagoras is credited with first applying to the universe. The word kosmos, in addition to its primary meaning of order, also means ornament. The world, according to Pythagoras, is ornamented with order. This is another way of saying that the universe is beautifully ordered.
The idea of order is intimately connected with Limit (peras), the opposite of which is the Unlimited (apeiron), and these are the two most basic, and hence most universal, principles of Pythagorean cosmology. According to the Pythagoreans, the world or cosmos is compounded of these elements, summarized in the famous "Table of Opposites" which has been preserved by Aristotle in his Metaphysics (i. 5 986 a 23):
FIGURE 2. THE PYTHAGOREAN TABLE OF OPPOSITES
Limit is a definite boundary; the Unlimited is indefinite and is therefore in need of Limit. Apeiron also may be translated as Infinite, but it is infinite in a negative sense: that is, it is infinitely or indefinitely divisible, and hence weak, rather than the modern "positive" usage of the term, which is often synonymous with "powerful." To avoid any confusion between the ancient and modern meanings, Apeiron has been translated as either Indefinite or Unlimited in the writings which appear in this book, unless the context suggests otherwise.
Aristotle stated that the Pythagoreans made everything out to be created of numbers; what he means to say is that everything is created out of the elements of number, which include the Limited and the Indefinite, the Odd and the Even.
The Pythagoreans were in the habit of representing arithmetical numbers as geometrical forms, through which they arrived at some interesting insights. In fact, Aristotle makes reference to this very practice:
The Pythagoreans identify the Unlimited with the Even. For this. they say, when it is enclosed and limited by the Odd provides things with the element of unlimitedness. An indication of this is what happens in numbers: if gnomons are placed around the unit and apart from the unit, in the latter case the resulting figure is always other, in the former it is always one. [14]
Aristotle is referring to the following figures:
FIGURE 3. SQUARE NUMBER
FIGURE 4. OBLONG NUMBER
The Greek word gnomon signifies a "carpenter's square." In figure 3 gnomons have been placed around the One, in figure 4 around the Dyad. From this arrangement several patterns arise. In the case of figure 3, each gnomon or band of points is odd, in figure 4 each gnomon is even. From the above diagrams, we can easily see why the Pythagoreans, in the Table of Opposites, identified the Odd and Even with the Square and Oblong respectively. Moreover, the principles of Limit and the Unlimited are also most manifest in these representations, for figure 3 is limited by the stable form of the square, while figure 4 is infinitely variable: with each successive gnomon, the shape and its corresponding lateral to horizontal ratio changes each time, for it is the nature of the Unlimited to be eternally variable and multifarious.
According to the paradigms of ancient cosmology, Matter (the Indefinite) receives and is shaped by Form (Limit); hence, these two principles of peras and apeiron may be seen at the two most universal and essential elements which are absolutely necessary for the manifestation of phenomenal reality. From this perspective it becomes easy to see the logic behind the Pythagorean sentiment that the cosmos is created out of the elements of Number, namely the Limited and the Indefinite. Plato, in fact, takes over this Pythagorean cosmology to the letter. His only change, and a minor one at that, is that he referred to Limit as the One, the Unlimited as the Indefinite Dyad, terms which have even more Pythagorean implications than the originals.
In the Pythagorean and Platonic cosmology, Limit and the Indefinite, Form and Matter, are woven together through numerical harmony: their offspring, existing in the indefinite receptacle of space, is the phenomenal universe, in which every being is composed of universal constants and local variables. Hence, in his Pythagorean cosmogony of the Timaeus, Plato shows how the fabricator of the cosmos parcels out the stuff of the World Soul according to the numerical proportions of the musical scale. [15]
The Monochord: The Mathematics of Harmonic Mediation
The musical proportions seem to me to be particularly correct natural proportions.
- Novalis
PYTHAGORAS IS SAID to have discovered the musical intervals. While the story of the musical smithy is probably a Middle Eastern folk tale, [16] there can be no doubt that Pythagoras experimented with the monochord (figure 5), a one-stringed musical instrument with a moveable bridge, used to investigate the principles and problems of tuning theory.
FIGURE 5. THE MONOCHORD. String, sounding box and moveable bridge.
The monochord affords an excellent example of how the primary principles of peras and apeiron underlie the realm of acoustic phenomena. Of course, the fact that numerical proportions underlie musical harmony has become a commonplace since the days of Pythagoras; yet there is something about the perfect beauty of these proportions, and their manifestations in the realm of sound, which will exercise a curious fascination over anyone who chooses to actually investigate them on the monochord.
The problem which the monochord presents is that the string can be divided at any point. The string represents an Indefinite continuum of tonal flux which may be infinitely divided. How, then, is it possible to "create" a musical scale at all? The solution, of course, resides in the limiting power of Number.
A curious phenomenon occurs when a string is plucked. First, the string vibrates as a unit. Then, in two parts, then in three parts, four, and so on. As the string vibrates in smaller parts higher tones are produced, this being the so-called harmonic overtone series. [17] While they are not as loud as the fundamental tone of the entire string vibrating, with practice the overtones can nonetheless be heard.
Through the power of Limit, the most formal manifestation of which is Number, harmonic nodal points naturally and innately exist on the string, dividing its length in halves, thirds, fourths, and so on, as shown in figure 6. Plucking the string at one end, and simultaneously touching one of the nodal points without the bridge, will produce the corresponding overtone vibration. In this fashion, one can play out the overtone series, as far as is practical. However, dampening the string at any other point will just deaden out the string. (See figure 6.)
The overtone series provides, as it were, the architectural foundation of the musical scale, the basic "field" of which is the octave, 1:2, or the doubling of the vibrational frequency, which inversely correlates with a halving of the string. Returning again to the basic question of how one bridges the tonal flux, we know the answer to be Number, but now we can see more clearly that the problem itself is one of mediation or harmonia, through the medium of numerical proportion or logos. The solution, in fact, can be seen as performing a marriage of opposites, linking together the upper and the lower (I :2), in a truly cosmic fashion, which is to say in a manner partaking of both order and beauty.
While the complete ratios of the scale are set out in Appendix IV, "The Ratios and Formation of the Pythagorean Scale," we shall here note the essentials.
In order to arrive at whole number solutions, we will use the octave of 6:12.
1) The first step is one of arithmetic mediation. To find the arithmetic mean we take the two extremes, add them together, and divide by 2. The result is a vibration of 9, which, in relation to 6, is in the ratio of 2: 3. This is the perfect fifth, the most powerful musical relationship.
2) The second form of mediation is harmonic. It is arrived at by multiplying together the two extremes, doubling the sum, and dividing that result by the sum of the two extremes (i.e., 2AB / A+B). The harmonic mean linking together 6 and 12 then is 8. This proportion, 6:8 or 3:4, is the perfect fourth, which is actually the inverse of the perfect fifth.
3) Through only two operations we have arrived at the foundation of the musical scale, the so-called "musical" or "harmonic" proportion, 6:8 :: 9:12, the discovery of which was attributed to Pythagoras. (See figure 7.)
FIGURE 6. THE HARMONIC NODAL POINTS AND OVERTONE SERIES ON THE MONOCHORD. The above figure illustrates the reciprocal relation which exists between string length and vibrational frequency. By stopping the string at the geometrical nodal points the harmonic overtones may be individually emphasized.
FIGURE 7. THE HARMONIC PROPORTION
This arrangement of the perfect consonances of the octave, fifth and fourth needs to be played out, preferably on the monochord, in order to fully appreciate its significance. While we have not "created" a complete musical scale, we have arrived at the architectural foundation on which it is based. By carefully observing the above arrangement, however, we shall discover enough information to complete the scale.
First of all, it would be well to notice the peculiar form of musical and mathematical "dialectic" which is occurring. That is to say, not only is 6:9 a perfect fifth, but 8: 12 is as well; i.e., 6:9 :: 8: 12. Nor is that all, for while 6:8 is a fourth, so too is 9:12; or, 6:8 :: 9:12. Again, the significance of this harmonic symmetry will be fully realized by playing these relations out. [18] However, not only are the fourth and fifth manifested in these multiple ways, but the ratio of 8:9 defines the whole tone as well.
The tone having been defined, the final creation of the scale is quite simple. The vibration of the tonic C is increased by the ratio 8:9 to arrive at D. D is increased by 8:9 to arrive at E. Now, if E were increased by that ratio, it would overshoot F; hence there we must stop. The ratio between E and F ends up being 243:256, called in Greek the leimma, or "left over," corresponding to our semi-tone. [19] Ascending from G, the same 8:9 ratio is used to fill up the remaining intervals. Likewise, the interval between B and C is the leimma.
While the fourth and fifth mediate between the two extremes via harmonic and arithmetic proportion, the scale is filled up through the continued geometrical proportion of 8:9; hence, the geometric mean between C and E would be D. All these forms of proportion interpenetrate, cooperate and harmonize with one another to produce the musical scale.
In summary, we can see the paramount importance of the musical scale and its formation in Pythagorean thought. First of all, the experiments conducted by the Pythagoreans on the monochord confirmed the importance of numerical peras as the limiting factor in the otherwise indefinite realm of manifestation. It also suggested for the first time that if a mathematical harmony underlies the realm of tone and music, that Number may account for other phenomena in the cosmic order -- for example, planetary motion, which was also thought of being related to the mathematical harmonia of the scale, this being the famous "Music of the Spheres." [20] Moreover, the world is full of beings and phenomena which reflect the harmonic principle of dynamic symmetry present in the musical proportion as well. Through their investigation of musical harmony, the Pythagoreans shifted philosophic inquiry away from the materialistic cosmologies of the earlier Ionic tradition to the consideration of Form, which was now to be seen as constituting the world of First Principles. In addition to shifting emphasis from Matter to Form, the Pythagoreans also discovered the principle of harmonia, the fitting together of the high and the low, the hot and the cold, the moist and the dry. From then on, Health was seen as the perfect harmony of the elements comprising the body, disease as that state in which one of the elements becomes too weak or strong, destroying the proper symmetry of the arrangement. Indeed, it was Alcmaeon of Croton, a young man when Pythagoras was old, who first defined health as "the harmonious mixture of the qualities." [21] This had an inestimable effect on Hippocratic medicine. As is so apparent in their various cultural achievements, the ancient Greeks had a very special affinity with the principles of Form, Symmetry, and Harmony. The Pythagoreans were the inheritors of this affinity, and helped to articulate these principles in new, important ways which have profoundly influenced the arts and sciences of Western civilization.
The Tetraktys: Number as Paradigm
I swear by the discoverer of the Tetraktys
Which is the spring of all our wisdom
The perennial fount and root of Nature.
-- Pythagorean Oath
THE PYTHAGOREANS PERCEIVED another principle of Number, in addition to seeing it as a formative agent active in nature. This is perhaps best exemplified in the figure of the Tetraktys which, as we might say in the present century, stood as a numerical paradigm of whole systems.
As we have observed, the Pythagoreans were accustomed to arranging numbers in geometrical shapes, and there are a variety of descriptions which have come down to us from antiquity of triangular, square, pentagonal, and other figured numbers and their properties. [22] This way of representing numbers may have well resulted in the discovery of geometrical theorems. Moreover, the observation that the relations between different types of "geometrical numbers" follow certain definite patterns surely furthered the Pythagorean contention that mathematical study is an important route leading, to the perception of universal laws.
The most well known example of such a "figured number" is the famous Pythagorean Tetraktys ("Quaternary"), consisting of the first four integers arranged in a triangle of ten points:
FIGURE 8. THE TETRAKTYS.
For the Pythagoreans the Tetraktys symbolized the perfection of Number and the elements which comprise it. In one sense it would be proper to say that the Tetraktys symbolize, like the musical scale, a differentiated image of Unity; in the case of the Tetraktys, it is an image of unity starting at One, proceeding through four levels of manifestation, and returning to unity, i.e., Ten. In the sphere of geometry, One represents the point , Two represents the line , Three represents the surface , and Four the tetrahedron , the first three-dimensional form. Hence, in the realm of space the Tetraktys represent the continuity linking the dimensionless point with the manifestation of the first body; the figure of the Tetraktys itself also represents the vertical hierarchy of relation between Unity and emerging Multiplicity. In the realm of music, it will be seen that the Tetraktys also contains the symphonic ratios which underlie the mathematical harmony of the musical scale: 1:2, the octave; 2:3, the perfect fifth; and 3:4, the perfect fourth. [23]
We might further note that the Tetraktys, being a Triangular number, is composed of consecutive integers, incorporating both the Odd and Even, whereas Square number (Limited) is composed of consecutive odd integers, and Oblong number of consecutive even integers (Indefinite). Since the universe is comprised of peras and apeiron woven together through mathematical harmonia, it is easy to see from these considerations why the Tetraktys, or the Decad, was called Kosmos (world-order), Ouranos (heaven), and Pan (the All). In Pythagorean thought the Tetraktys came to represent an inclusive paradigm of the four-fold pattern which underlies different classes of phenomena, as exemplified by Theon of Smyrna in Appendix 1. Not only does a four-fold pattern underlie each class, but each level is in a certain fashion analogous or proportionately similar with that same level in every other class of phenomena. In many respects Pythagorean philosophy is a philosophy of analogia.
The Pythagoreans, then, were the first to use numerical and geometrical diagrams as models of cosmic wholeness and the celestial order. This use of arithmetic and geometrical paradigms of whole systems has a long and interesting history, extending from antiquity through Medieval times, through the Renaissance, up until the modern era. [24] If geometrical principles actually shape the phenomena of nature, why not use those same geometrical forms to illustrate the harmonies and symmetries which exist between natural phenomena? This is no doubt the reasoning behind this symbolic usage of number and geometry, and its appeal seems firmly rooted in the human imagination. In fact, it might be argued that such paradigms possess greater merit than more arbitrary typologies insofar that, being based on the principles of natural order, "Pythagorean" models have more intrinsically in common with the phenomena they seek to classify than other typologies which are of merely human invention. Whereas other models sometimes fail, Pythagorean cosmological symbolism seems particularly well suited in showing how parts relate to a larger whole, thus illustrating the principle of unity underlying diversity.
The Way of Philosophy and the Three Lives
ACCORDING TO A BEAUTIFUL and well known account, Pythagoras likened the entrance of men into the present life to the progression of a crowd to some public spectacle. There assemble men of all descriptions and views. One hastens to sell his wares for money and gain; another exhibits his bodily strength for renown; but the most liberal assemble to observe the landscape, the beautiful works of art, the specimens of valor, and the customary literary productions. So also in the present life men of manifold pursuits are assembled. Some are influenced by the desire of riches and luxury; others, by the love of power and dominion, or by insane ambition for glory. But the purest and most genuine character is that of the man who devotes himself to the contemplation of the most beautiful things, and he may properly be called a philosopher. [25]
Likewise, the story is told of how Pythagoras was indeed the first man to call himself a philosopher. Others before had called themselves wise (sophos), but Pythagoras was the first to call himself a philosopher, literally a lover of wisdom.
More importantly, for Pythagoras and his followers philosophy was not merely an intellectual pursuit, but a way of life, the aim of which was the assimilation to God. Even in the days of Plato the surviving Pythagoreans were noted for their distinctive bios Pythagorikos, or Pythagorean way of life, as Plato puts it in the Republic (600a-b).
The school of Pythagoras in Croton appears to have been a religious society centered around the Muses, the goddesses of learning and culture, and their leader Apollo. [26] Iamblichus' description of the school gives it something of a monastic flavor, and there was indeed a "rule" of life, but while the Pythagoreans gathered together at certain times of the day, most of them did not live together.
Apparently there were different levels within the school. One group, the akousmatikoi or "auditors" (from the verb akouo, to hear), went through a three year probationary period and were limited mainly to hearing lectures. A more advanced group, the mathematikoi or "students," went through a five year period of "silence," [27] and held their property in common whereas the akousmatikoi did not; there is, however, nothing to indicate that the mathematikoi took anything like a vow of poverty. Rather, their property was managed by certain members of the society -- the politikoi -- and they received an adequate subsistence in return for its use. [28]
Pythagoras himself was heavily influenced by Orphism, an esoteric, private religion of ancient Greece, named after the legendary musician Orpheus, "the founder of initiations," which also featured a distinctive way of life. According to Orphism, the soul, a divine spark of Dionysus, is bound to the body (soma) as to a tomb (sema). Mankind is in a state of forgetfulness of its true, spiritual nature. The soul is immortal, but descends into the realm of generation, being bound to the "hard and deeply- grievous circle" of incarnations, [29] until it is released through a series of purifications and rites, regaining its true nature as a divine being.
Pythagoras fully accepted the Orphic belief in transmigration or "reincarnation" -- in fact, he is said to have possessed the power to remember his previous lives, and the ability to remind his associates of theirs as well. Yet while Pythagoreanism remains closely related to the Orphic thought of the period, [30] the clearly distinguishing factor between the two is that for the Pythagoreans liberation from the wheel is obtained not through religious rite, but through philosophy, the contemplation of first principles. Hence, philosophia is a form of purification, a way to immortality. As others have observed, whereas the Eleusinian mysteries offered a single revelation, and Orphism a religious way of life, Pythagoras offered a way of life based on philosophy. Burnett notes that this conception lies at the heart of Plato's Phaedo. itself "dedicated, as it were, to a Pythagorean community at Phlious"; [31] moreover, "This way of regarding philosophy is henceforth characteristic of the best Greek thought." [32]
One may well ask how assimilation to God is possible through philosophy. The answer is to be found in the nature of man:
Pythagoras said that man is a microcosm, which means a compendium of the universe; not because, like other animals, even the least, he is constituted by the four elements, but because he contains all the powers of the cosmos. For the universe contains Gods, the four elements, animals and plants. All of these powers are contained in man. He has reason. which is a divine power; he has the nature of the elements, and the powers of moving, growing, and reproduction. [33]
Man, by comprising a world-order in miniature, contains all of those principles constituting the greater cosmos, of which he is a reflection, including the powers of divinity. The problem is not so much of becoming divine as becoming aware of the divine, universal principles within. It is this end, primarily, toward which the Pythagorean curriculum was focused. Plato alludes to the Pythagorean theory of philosophy in the Republic (500c) when he observes:
a man should come to resemble that with which it delights him to associate... Hence the philosopher through the association with what is divine and orderly (kosmios) becomes divine and orderly (kosmios) insofar as a man may.
Man realizes the divine by knowing the universal and divine principles which constitute the cosmos -- i.e., for the Pythagoreans, Number. To know the cosmos is to seek and know the divine element within, and one must become divine and harmonized since only like can know like. From this perspective it also becomes obvious that philosophy is nothing other, at least in one respect, than the care of the soul.
The Soul, its Nature and Care
ACCORDING TO SEVERAL ancient sources, it was from the Pythagoreans that Plato received his doctrine of the tripartite soul, a doctrine which underlies Pythagoras' parable of the three lives: one group of humanity is covetous, another ambitious, and the other curious. As J.L. Stocks has pointed out, "What the division specifies is the three typical motives of human action, and all three motives will be found in operation at different times in every normal human soul." [34] These motives are the desire for profit, honor, and knowledge.
Plato, apparently in line with the Pythagorean tradition, divides the soul into three parts: one part is reasoning, another part is "spirited," and the last desires the pleasures of nutrition and generation. Unlike certain schools of modern psychology, the Platonic division of the soul is hierarchical: the reasoning part is superior to the other two, and deserves more attention, for it is this dimension of the soul which makes us uniquely human. We might summarize the relation between the levels of the soul and their attendant virtues, or forms of excellence, as shown in figure 9.
FIGURE 9. THE THREE LIVES
Seen in this perspective, it becomes plain that psychic health must result when the three "parts" of the soul are brought into a state of harmony, which is not to say a state of equality. Rather, this state of balance could be seen as a state of attunement, where each part receives what it is due. Psychic disturbance results when each part of the soul tries to go its own separate way; the psyche then becomes a house divided, resulting in dissociation and fragmentation, as opposed to the realization of psychic wholeness.
The grand project behind Plato's Republic is to define the nature of justice. We know that the Pythagoreans identified justice with proportion, especially geometrical proportion, because it is through proportion that "each part receives what it is due." [35] Following the Pythagorean tradition, Plato observes that in the realm of society justice exists when each part of society receives its due, and is able to achieve the function for which it is truly best suited. Justice, as a universal principle, operates in exactly the same fashion in the realm of the soul. There, "justice is produced in the soul, like health in the body, by establishing the elements concerned in their natural relations of control and subordination, whereas injustice is like disease and means that this natural order is inverted." [36] As Plato notes, in a magnificently Pythagorean passage:
... The just man does not allow the several elements in his soul to usurp one another's functions; he is indeed one who sets his house in order, by self-mastery and discipline coming to be at peace with himself, and bringing into tune those three parts, like the terms in the proportion of a musical scale, the highest and lowest notes and the mean between them, with all the intermediate intervals. Only when he has linked these parts together in well-tempered harmony and has made himself one man instead of many, will he be ready to go about whatever he may have to do, whether it be making money and satisfying bodily wants, or business transactions, or the affairs of state. In all these fields when he speaks of just and honorable conduct, he will mean the behavior that helps to produce and preserve this habit of mind; and by wisdom he will mean the knowledge which presides over such conduct. Any action which tends to break down this habit will be for him unjust; and the notions governing it he will call ignorance and folly. (My emphasis.) [37]
If Pythagorean philosophy, then, constitutes a care of the soul, of what precisely is that care comprised? The answer to this is to be found in the ethical and educational conceptions of the Pythagoreans, as well as in those special pursuits and studies for which they were renowned.
Pythagorean Educational Theory
WE HAVE SEEN that for Pythagoras philosophy represents a "purification," the aim of which is the assimilation to God. The universe is divine because of its order (kosmos), and the harmonies and symmetries which it contains and reflects. These principles make the universe divine for they are the characteristics of divinity, and they also innately subsist within the human soul. The Pythagoreans taught that the soul is a harmony. [38] If we are to become like God, then according to Pythagorean philosophy the soul must become aware of its harmonic origin, structure and content. Since the source of all harmony and order is the divine principle of Number, we can perhaps come to understand the initially enigmatic statement of Heracleides that, according to Pythagoras, true "happiness consists in knowledge of the perfection of the numbers of the soul." [39]
In the realm of epistemology the presence of Number is most evident: progress in rational thought depends on a fundamentally dyadic relationship between knower and known, subject and object. Moreover, as certainly as the principle of polarity underlies the world of phenomenal manifestation, so too does the mind depend on dualistic typologies, such as the Table of Opposites, in order to make intellectual progress. [40] Knowledge itself is the third, harmonic element which conjoins the two poles of subject and object. Knowledge then is unifying, much like the harmonic ratios of the musical scale or the central circle in figure 1. Moreover, as we shall see, to the Pythagoreans the knowledge of divine harmony can be either abstract or experiential or, indeed, both.
Even more immediately evident is the undeniable influence of Number on our psychic state through the medium of music, depending as it does on numerical proportion. Certain musical proportions express a sense of cheerfulness; others, such as the minor third, possess a bittersweet quality that can make us sad. The fact that Number can influence a person's emotional state is indeed mysterious and points toward a dimension of qualitative Number which transcends the merely quantitative.
Related to the question of music and harmony is the principle of resonance: two strings, tuned to the same frequency, will both vibrate if only one is plucked, the unplucked string resonating in sympathy with the first. This, of course, is accomplished through the medium of the vibrating air, but the principle underlying the phenomenon is one of harmonic attunement. If, as the Pythagoreans held, man is a microcosm, and the soul is a harmony, perhaps it is through a form of resonance that we relate so intensely to the archetypal ratios of musical proportion. [41] Moreover, by experientially investigating and employing the principles of harmony in the external world, one comes to understand and activate those same principles within. This idea in fact underlies the Pythagorean approach to mathematical study.
The Pythagoreans divided the study of Number into four branches which may be analyzed in the following fashion:
Arithmetic = Number in itself.
Geometry = Number in space.
Music or Harmonics = Number in time.
Astronomy = Number in space and time.
Plato, of course, was heavily influenced by the Pythagorean study of number and incorporates the above quadrivium into his own educational curriculum set out in the Republic, adding another branch of study, Stereometry, the investigation of Number in three-dimensional space, which probably relates to the regular "Platonic" solids and other polyhedra. For Plato -- who believed that God geometrizes always [42] -- "geometry is the knowledge of the eternally existent," [43] and the emphasis that he placed on the study is well known from the legendary inscription above the Academy door, "Let no one ignorant of geometry enter here. " [44]
Within the Platonic curriculum, the purpose of mathematical studies is to purify the eye of the intellect, for mathematical studies have the propensity "to draw the soul towards truth and to direct upwards the philosophic intelligence which is now wrongly turned earthwards." [45] Number, for Plato, is a transcendent Form to which we must intellectually ascend. For the earlier Pythagoreans, however, the emphasis was clearly on the immanence of Number.
While the Pythagoreans moved the direction of philosophical inquiry from the realm of matter to that of Form and principles, Plato took this movement even further than his predecessors. For Plato mathematical studies are a preparation for the contemplation of divine principles; for the Pythagoreans, mathematical studies are the contemplation of divine principles. As Cornelia de Vogel has lucidly observed, for the Pythagoreans.
The contemplation of divine Law, which was the content of the study of mathematics, was a direct contact with a divine Reality: Divinity immanent in the cosmos.
It was different for Plato. He adopts the Pythagorean notion that number is the principle of order in the cosmos and life, but number as such to him is not yet a theion [divinity]. It points at a purely intelligible Number which is a 'Form' (eidos) -- no immanent principle of order within the objects, but a transcendent Example. This is the basic difference between the Pythagorean doctrine of number and Plato's Theory of Forms. Plato's philosophy is a metaphysic of the transcendent; the Pythagorean philosophy is a metaphysic of the immanent order. [46]
This particular difference between the earlier Pythagoreans and Plato must have manifested itself in the sphere of praxis. For Plato it was, in a sense, best to pursue mathematical contemplation with as little reference to physical objects as possible: truth must be approached through intellect, and through intellect alone. For the Pythagoreans, truth manifests itself through the world of physical phenomena; for example, the Pythagoreans no doubt felt that through experimentation on the monochord one could experience the divine principles of harmony which underlie the structure of the cosmos.
The differing views between Plato and the earlier Pythagoreans can also be seen in the realm of music. Plato refers to different musical modes throughout his writings, and to the negative effects that some forms of music can have on the soul and on society. The Pythagoreans, however, actually used certain forms of music to pacify and harmonize the psychic state. In the same way that the music of Orpheus enchanted the wild beasts of the field, so too did the Pythagoreans use music to quell and harmonize the irrational passions.
While the Pythagoreans placed emphasis on the immanence of divine Number and Harmonia, they certainly did not ignore the transcendental dimension. This is made clear by their emphasis on peras and apeiron, the elements of Number, which they obviously took to be universal principles of the first order. It seems that, rather than focusing exclusively on either the immanent or transcendent levels of being, the Pythagoreans were intent on unifying all levels of human experience through the principles of harmony. The divine harmony can be grasped through the mind, yet can also be perceived through the senses. The experiential perception of harmony through the senses can lead to its intellectual apprehension. By means of theoria or contemplation the universal and abstract principles of harmony may be perceived, but through praxis they may be felt in the soul, itself a harmonic entity. Yet there is another level, that of therapeia, where harmonic principles can be used to effect changes in the psychic disposition.
Through the use of proper music, diet, and exercise, the early Pythagoreans sought to nurture and maintain the natural harmony of the psychic and somatic faculties. According to Iamblichus, "They took solitary morning walks to places which happened to be appropriately quiet, to temples or groves, or other suitable places. They thought it inadvisable to converse with anyone until they had gained inner serenity, focusing their reasoning powers. They considered it turbulent to mingle in a crowd as soon as they rose from bed, and that is the reason why these Pythagoreans always selected the most sacred spots to walk." [47] All of these practices can be seen as a form of philosophic "purification" (catharsis) or "practice"(praxis), designed to regulate the body and the emotions. On the intellectual and psychic levels, through their study of mathematics and the natural world, the Pythagoreans approached the principles of harmony experientially through the study of harmonics on the monochord and through geometrical constructions. The Pythagoreans also pursued the study of purely abstract mathematics.
Recalling that the end of all of these pursuits was to follow God, it is interesting to briefly contrast the Pythagorean approach to divinisation with the Christian mysticism of the late Hellenistic period and thereafter. The first stage of "the mystical ascent" consists of the ethical purification of the soul commonly known as praxis. The second stage is contemplation or theoria; in Christianity, however, the contemplation of Nature and universal principles, so characteristic of the Greek philosophic tradition, is replaced predominately by the contemplation of scripture. The final stage, theosis, is the union of the mystic with God. In Hellenistic Christian mysticism of the late antique world, however, the first two stages lose virtually all significance when the final stage is reached. Catharsis and theoria are merely the steps of a ladder; when the summit is reached, the ladder is oftentimes kicked away.
The ancient Pythagorean approach to divinisation would have never sanctioned kicking away the ladder. In early Pythagorean thought there nowhere appears an earnest desire to escape from the world. True, like the Orphics, the Pythagoreans believed in reincarnation, and looked upon the body as limiting the soul. But even so, there is no firm evidence that, like the Orphics, the Pythagoreans sought exemption from the Wheel of Generation. Rather than transcend the world, Pythagorean religiosity held as its goal to exist within the cosmos in a state of emotional repose and intellectual acuteness. Man, while possessing a soul which clearly transcends the limitations of the body, the realm of time and space, is nonetheless a reflection of the entire universe, a microcosm, and is linked together with nature, other living beings, and the Gods through harmony, justice, and proportion. The Pythagorean goal is not to leave the divinely beautiful cosmos behind for a realm of transcendent harmony, but rather to become aware of, and enhance the function of, transcendent harmony in the natural, psychological and social orders. [48]
Pythagorean Political and Ethical Theory
THE CENTRAL INSIGHTS of the Pythagoreans concerning the significance of harmonia were applied to political theory as well: in the same way that harmonic proportion underlies the health of the balanced soul, so too does the principle of justice underlie the living structure of a healthy state. In this regard Plato's Republic, which is a study of justice in both the psychic and social realms, appears to be firmly based on earlier Pythagorean conceptions. Plato, via analogia, identifies the three parts of the soul with three different parts of society, and shows how both the soul and society attain their peak of excellence when "each part receives its due" and when each of the three parts fulfils the particular function for which it is best adapted. It is not possible to say whether Plato's tripartite division of society corresponds precisely with an earlier Pythagorean division, although it is known that the Pythagoreans identified justice with proportion, of which they viewed geometrical proportion as being the most perfect. Like Plato the early Pythagoreans were aristocrats, in the authentic sense of the word, believing that the best government will be composed of those best qualified to govern, as opposed to other political systems in which leadership is based on wealth, on power, or on the choice of the populace. Finally, it should be noted that the Pythagoreans were the first philosophical school to concern themselves with such social and political questions, which fell outside the natural philosophy of the earlier Ionian tradition.
As for Pythagorean ethics, little needs to be said, as the entire idea of "proper action" is tied up with the ideas of philosophy as a way of life, and the nature of the soul and the cosmos. As one discovers the structure and nature of the soul, and experiences and begins to understand the principles of harmony, it seems inevitable that such insight will leave a mark on one's personal conduct and dealings with others. Nonetheless, as the writings in this volume demonstrate, the Pythagoreans did not hesitate to make use of aphorisms and other explicit ethical teachings. At all events, however, it will be seen that these teachings reflect and spring from the more universal insights of Pythagorean thought. In short, because each part is linked to the whole through harmonia, every action has its repercussions, either beneficial or not, for which the individual is supremely responsible.