The Pythagorean Tradition and its Development
TO SURVEY THE INFLUENCE of Pythagorean thought would expand this introductory essay beyond reasonable boundaries, insofar as the basic conceptions of the Pythagoreans have influenced a long line of thinkers from antiquity reaching up until the present day. Nonetheless, something should be said about the thought of the early Pythagoreans and the Neopythagoreans of the first centuries B.C.E. and C.E., thus helping to place some of the writings here assembled into their proper context.
As the biographical accounts in this volume show, the downfall of the original Pythagorean school had much to do with anti-aristocratic sentiments amongst the populace of south Italy. A revolt was led against the Pythagoreans by Cylon in 500 B.C.E. -- some say because he was rejected admission into the school -- and a period of unrest followed. During the revolt led by Cylon, or during another revolt which followed, various meeting houses were attacked and a good number of Pythagoreans may have perished in the flames. This final attack seems to have been rather successful, and those Pythagoreans that remained alive seem to have migrated to mainland Greece with the exception of Archytas at Tarentum.
Unfortunately, the details concerning the attack on the school are sketchy and little more can be said than the above. The Pythagoreans did, however, carry on in mainland Greece where centers were established at Phlious and Thebes. Echecrates went to Phlious, Xenophilus went to Athens, and the names of Lysis and Philolaus are associated with Thebes, and it was there that Philolaus taught Simmias and Cebes who appear as characters in Plato's Phaedo. Philolaus, who was born around 474 B.C.E., was the first Pythagorean to actually record the teachings of the school in writing; hence his fragments, which are collected in this volume, possess an exceptional value.
Archytas (first half of the fourth century B.C.E.), who was the general of Tarentum and one of the Pythagorean mathematikoi like Philolaus, made contributions to mathematics, geometry, and harmonic theory. He was visited by Plato in 388 B.C.E. and it is possible that Archytas was Plato's model for the so-called "philosopher king."
This brings us to a discussion of Plato himself (428-348 B.C.E.) which is not a topic of minor significance for, as W.K.C. Guthrie has observed, "In general the separation of early Pythagoreanism from the teaching of Plato is one of the historian's most difficult tasks, to which he can scarcely avoid bringing a subjective bias of his own. If later Pythagoreanism was coloured by Platonic influences, it is equally undeniable that Plato himself was deeply affected by earlier Pythagorean belief." [49]
Many important Pythagorean influences have already been noted on the thought of Plato and perhaps it would be fair to view Plato as the most important Pythagorean thinker in the history of the West. There was quite a bit of interest in Pythagorean thought in the early Academy as well, and it has been suggested that the idea of the Academy was in part due to the inspiration of the earlier Pythagorean school. Whatever the case, some points of contact include the tripartite division of the soul; [50] Plato's usage of the One and the Indefinite Dyad; [51] the theory of education in the Republic; [52] the identification of the One and the Good in "the unwritten doctrine" referred to by Aristotle; [53] the Pythagorean character of Plato's lecture "On the Good" reported on by Aristotle;[54] the idea that the soul of the philosopher attains order by contemplating those things which possess order in nature; [55] the idea that "the goodness of anything is due to order and arrangement"; [56] the idea that various beings are linked together through geometrical equality; [57] a doctrine of idea-numbers in the dogmata agrapha reported by Aristotle; [58] and various examples of Pythagorean musical symbolism. [59]
Following Plato in the leadership of the Academy was his nephew Speusippus (407-339 B.C.E.) who was also quite interested in Pythagorean thought: he suggested that there exists a One above being (an important teaching of later Neopythagorean and Neoplatonic thought), and also wrote a work On Pythagorean Numbers about the Tetraktys and numbers comprising the Decad. [60] This treatise was based on the writings of Philolaus and an interesting fragment of it survives.
Aristotle showed an interest in the Pythagorean school and even wrote an essay On the Pythagoreans which does not survive. One of his students, Aristoxenus of Tarentum, was a music theorist and was in touch with the last surviving generation of Pythagoreans at Phlious. Aristoxenus, who might have been one of the Pythagorean mathematikoi, possessed an antiquarian interest in the school and wrote a biography of Pythagoras which is quoted from by Porphyry, Iamblichus and Diogenes Laertius.
After the time of Aristotle we are left with an uncomfortable gap in the history of Pythagorean thought until the Neopythagorean revival commencing in the first century B.CE. Yet it is precisely during this period that most of the Pythagorean ethical and political tractates contained in the second section of this volume were probably composed. But the question remains, by whom were they written? And why?
Unfortunately, no one is certain even about the date or location of their composition. There now exists a tendency to see these writings as being somewhat earlier than previously thought, and Holger Thesleff suggests that the bulk of them were composed around the third century B.C.E. It will be noted that these writings are attributed to original members of the Pythagorean school, which in fact is actually not the case. This does not mean that these writings are "forgeries" in the modern sense of the word, for it was a fairly common practice in antiquity to publish writings as pseudepigrapha, attributing them to earlier, more-renowned individuals. It was probably out of reverence for their master -- and also perhaps because they were discussing authoritative school traditions -- that certain Pythagoreans who published writings attributed them directly to Pythagoras himself. Even Pythagoras is said to have attributed some poems of his to Orpheus. Other examples which might be cited include the many Jewish pseudepigrapha of the time, Orphic fragments, the Hermetic writings, and even a number of Pauline epistles from the New Testament.
A careful study of these writings will show that they are deeply imbued with many Pythagorean ideas - what, in fact, could be more Pythagorean than comparing the structure of the family or society to a well-tuned lyre? -- a particularly beautiful and useful simile which appears more than once in the Pythagorica here collected. Yet, alongside the central Pythagorean core of these writings are found strong Academic and Peripatetic influences as well. Thesleff is probably correct in suggesting that these writings were composed as philosophical textbooks for laymen, [61] but it is unlikely that the exact date or locale of their composition will ever be decisively settled. However, as has been suggested, a careful study of these texts might well provide for some valuable insights into the thought of the early Academy.
The next phase of Pythagorean thought involves the so-called Neopythagorean revival at the beginning of the common era. Due to the Hellenization of the ancient world stemming from the conquests of Alexander the Great, interest in Greek philosophy was no longer limited to one small part of the world. This was especially true of the interest in Pythagorean and Platonic thought, and the names of many Pythagorean philosophers are known from the Hellenistic age. Unfortunately, for some of the most important thinkers the information concerning them is quite fragmentary, which is perhaps one reason why no one has attempted the kind of full scale study that the topic deserves: the study of the Neopythagorean thought of this period is not only significant for its own sake, but also for understanding the thought of Plotinus and the later Neoplatonists who were influenced by a range of Neopythagorean ideas. Actually, as John Dillon has succinctly observed, during this period "Middle Platonism" and "Neopythagoreanism" existed as something of a continuous tradition, with Neopythagoreanism representing "an attitude that might be taken up within Platonism." [62] Keeping this in mind, it might be useful to mention some of the thinkers during this period who were influenced by Pythagorean thought, for it is only through such a listing that one can get a true feeling for the creative ferment of this period.
The first word that we have concerning a renewed interest in Pythagorean thought comes from Cicero, regarding his friend Nigidius Figulus (98-45 B.C.E.), who was attempting to revive Neopythagoreanism in Rome. It appears however that Nigidius was less interested in abstract philosophy than in integrating astrological, ritual and a variety of occultist beliefs.
Eudorus of Alexandria (fl. 30 B.C.E.) appears to have been influenced by Pythagorean thought, and attempted to show that the Pythagoreans held that a Supreme Principle, the One, existed above the Monad and the Dyad. Whether or not the earlier Pythagoreans actually held such a belief is another question altogether, but the notion of a transcendent One surpassing the principles of Limited and Unlimited is important for the history of later philosophy.
Philo of Alexandria (20 B.C.E.-40 C.E.), a Hellenized Jew who interpreted Jewish scripture in light of Greek philosophy, shows a deep interest in Pythagorean thought, especially arithmology, in his voluminous writings. Philo was primarily a Platonist who subscribed to a emanationist cosmology which he tried to reconcile with Jewish thought, but it is interesting to note that he was referred to by Clement of Alexandria, an early church father, simply as "Philo the Pythagorean." [63]
We should not fail to mention Apollonius of Tyana, a colorful figure who flourished during the first half of the first century C.E. Apollonius was perhaps more of a Pythagorean wonderworking ascetic than a philosopher, who travelled through the ancient world as something of a pagan missionary, meeting with priests, performing marvels, and restituting cults of worship to their former purity. His life and exploits are chronicled in an entertaining historical novel by Philostratus. Whether or not this biography gives a well-rounded picture of Apollonius remains uncertain. It does seem, however, that Apollonius saw himself as a reincarnation of Pythagoras and also possessed much information on the life of the sage, which he used in compiling a biography, subsequently used by Porphyry and Iamblichus. In addition to the Life of Philostratus, several letters attributed to Apollonius are extant. [64]
Often overlooked as an important witness to Pythagorean thought is Plutarch of Chaeronea (45-125 C.E.), well known for his famous Lives. What is not so generally well known is that Plutarch was a Platonist with Neopythagorean leanings and was also a priest of Apollo at Delphi. In certain writings of his Moralia he displays a keen interest in the interpretation of myth, the religio-philosophical esoterism of the time, and various bits of Neopythagorean lore, including arithmology. His writings remain a vital resource for understanding the profound, inner dimensions of the contemporary spiritual universe and, like Plotinus, he refers to the Pythagorean interpretation of the name Apollo, which equates Apollo with the One (a = not; pollon = of many). [65]
About the same time as Plutarch we have Moderatus of Gades (fl. second half of the first century) who has been termed an "aggressive Pythagorean" for the severe criticism he applied to Plato, accusing him of using ideas of Pythagoras without giving proper credit where credit was due. In his cosmology Moderatus taught the existence of three unities: the first and highest, the One above being which he identified with the Good; secondly, a unified, active logos, identified with the intelligible realm; and thirdly, the realm of soul. Needless to say, the resemblance between these ideas and those of Plotinus are quite striking.
Theon of Smyrna brings us into the second century (fl. circa 125 C.E.). He was a Platonist and wrote a work Mathematics Useful for Understanding Plato, of which a good English translation exists, and which is equally useful for understanding aspects of Pythagorean thought. [66] In addition to discussing the principles of arithmetic, harmonics and astronomy, Theon also treats the symbolism of the first ten integers and various forms of the Tetraktys (see Appendix I). The work also dealt with the principles of geometry, but this section no longer survives.
Working in a similar vein, and not much later, was Nicomachus of Gerasa (active 140-150 C.E.), whose Introduction to Arithmetic, [67] translated into Latin by Apuleius and Boethius, remained a definitive handbook up until the Renaissance. Also surviving is Nichomachus' Manual of Harmonics, and fragments of his Theology of Arithmetic, a work on Pythagorean arithmology. He also wrote a Life of Pythagoras which was used by the later biographers and an Introduction to Geometry which did not survive. He is known to have been familiar with the practice of gematria, [68] and it has been suggested that Iamblichus' "Pythagorean encyclopedia" found its inspiration in the wide-ranging works of this scholar.
Numenius of Apamea in Syria (fl. 160 C.E.) was also a Platonist with Neopythagorean leanings. Some fragments of his works remain, but the majority have perished. He wrote On the Good; On the Indestructibility of the Soul; On the Secret Doctrines of Plato; a work called Hoopoe, after the bird of the same name; On Numbers (perhaps a work on arithmology); On Place; and On the Divergence of the Academics from Plato. He had an associate, Cronius, who flourished about the same date and wrote a work On Reincarnation.
With Numenius, the mixture of Middle Platonic and Neopythagorean thought begins to transform itself into Neoplatonism, of which philosophy the most brilliant and beautiful expositor was Plotinus. Plotinus seems to have been influenced to a certain extent by the thought of Numenius, and John Dillon sees Plotinus' direct teacher Ammonius Saccas (fl. circa 230 C.E.) as a being a Platonist of a strongly Neopythagorean cast.
The main feature which differentiates Neoplatonism from Middle Platonism is the Neoplatonic doctrine of the transcendent absolute, the One, which exists above the realm of Being. However, as we have seen, Plato's nephew Speusippus, Eudorus of Alexandria, and Maderatus of Gades all posited the existence of such a transcendent principle; even Plato himself, in the Republic, suggests that the Good. which he identified with the One, exists above being. [69] As is typical, the "clear cut" distinctions between various schools and periods are not always so sharp as one has been led to believe. This is especially true of the distinctions between Middle Platonism and Neopythagoreanism in the period outlined above.
In addition to the doctrine of the One above being, Plotinus (204-269 CE.) also held that the intelligible realm, which he identified with nous or Mind, exists as a unity-diversity, as a differentiated "image" of the One. Hence, this world of Forms, which contains all the laws and principles of the universe, can be seen as the living union of the Monad and the Indefinite Dyad, with the Monad acting as the limiting and form-giving principle in the realm of nous, while the Indefinite Dyad acts as the "intelligible matter" upon which the Monad acts. The Indefinite Dyad also provides for the element of Infinity which allows for the existence of an unlimited number of forms and souls in the realm of Mind.
Henceforward, Pythagorean ideas played an important role in subsequent Neoplatonic thought. Porphyry (233-305 C.E.), Plotinus' successor, wrote a biography of Pythagoras which was part of his History of Philosophy in Ten Books. Iamblichus (d. circa 330), who was a student of Porphyry and thought of himself as something of a full-fledged Pythagorean, undertook the task of writing a multivolume "Pythagorean encyclopedia" which included The life of Pythagoras here reproduced and a number of other works: The Exhortation to Philosophy, On the Common Mathematical Science, Commentary on Nichomachus' Introduction to Arithmetic, three books On the Natural, Ethical and Divine Conceptions which are Perceived in the Science of Numbers (of which the anonymous Theology of Arithmetic is based on the third book); and three lost works on Pythagorean harmonies, geometry and astronomy, bringing the total number of volumes up to 10, the Pythagorean Perfect Number.
The Neopythagorean component of Neoplatonism did not end with Iamblichus but rather continued through to the closing of the school in Athens, only to resurface in Renaissance Florence with the thinkers associated with the Cosimo de' Medici's Platonic Academy, of which Marsilio Ficino was the head.
This is not to imply that only pagan thinkers were followers of Pythagorean thought. On the contrary, many of the early church fathers held Pythagoras and his teachings in high esteem. Not only that, it became quite fashionable, after the manner of Philo, to enlist the help of Pythagorean number symbolism in the interpretation of scripture. Justin Martyr (100-164 C.E.) was rejected by a Pythagorean teacher on account of his inadequate mathematical knowledge (recalling the words engraved above the Academy door); he turned to Platonism, and then to Christianity, but never gave up his admiration for the Greek sages. Clement of Alexandria (fl. circa 200 CE.) was also an admirer of Hellenistic thought and even applied the ratios of the Harmonic Proportion to the exegesis of the holy writ. [70] Augustine (354-430), heavily influenced by Neoplatonism, also loved to indulge in numerical exegesis, and he was instrumental in helping to transmit an interest in number symbolism to the Middle Ages.
Also important in the transmission of the Pythagorean ideas to the following age were the pagan encyclopediasts of the late antique world. Macrobius (first part of the fifth century) discussed Pythagorean thought in his Commentary on the Dream of Scipio, and Martianus Capella (fl. 410-429) in his allegorical work on the seven liberal arts, The Marriage of Philology and Mercury, discussed arithmology in Book VII. Another important source for the medievals was Boethius (480-525), especially his works On Arithmetic and On Music.
Having arrived at the close of the ancient world we have also arrived at the end of our survey. Pythagorean ideas continued to be transmitted in the work of Christian thinkers and applied in the realm of sacred architecture by groups of medieval masons. Insofar as Pythagorean thought had been Christianized, it had been changed, yet nonetheless many important conceptions -- such as the ideas of celestial harmony and the significance of Number as a cosmic paradigm -- remained unaltered. There was a brief and beautiful Renaissance of Pythagorean thought at the cathedral school of Chartres in France during the 12th century, due in part to a Latin translation of Plato's Timaeus, and of course there was a renewed interest in Pythagorean thought with the rediscovery of classical writings during the rebirth of learning in Renaissance Italy.
A Philosophy of Whole Systems
IT SHOULD COME AS NO SURPRISE that the figure of Pythagoras appealed strongly to those scholars of Renaissance Italy, as he has invariably appealed to the more universal thinkers of every age. The appeal lies in his important and influential conceptions, which in many ways seem to directly reveal important principles existing at the sacred "root of Nature's fount," but also in the man's character, for Pythagoras himself seems to well represent the possibility of an integrated approach to the study of Nature as a philosophical way of life.
As we have seen, the central focus of Pythagorean thought is in many respects placed on the principle of harmonia. The Universe is One, but the phenomenal realm is a differentiated image of this unity -- the world is a unity in multiplicity. What maintains the unity of the whole, even though it consists of many parts, is the hierarchical principle of harmony, the logos of relation, which enables every part to have its place in the fabric of the all.
Because of Pythagoras' approach, integrating mathematics, psychology, ethics, and political philosophy into one comprehensive whole, it would be quite inappropriate to end this essay without devoting a few words to the contemporary significance of the Pythagorean approach. Pythagoras would have never wished that his insights remain the focus of a merely antiquarian interest, and so we shall honor his intentions by inquiring into what value Pythagorean thought might possess in the contemporary world, addressing these matters in quite general terms.
Pythagoras, no doubt, would have disapproved of the radical split which occurred between the sciences and philosophy during the 17th century "enlightenment" and which haunts the intellectual and social fabric of Western civilization to this day. In retrospect perhaps we can see that man is most happily at home in the universe as long as he can relate his experiences to both the universal and the particular, the eternal and the temporal levels of being.
Natural science takes an Aristotelian approach to the universe, delighting in the multiplicity of the phenomenal web. It is concerned with the individual parts as opposed to the whole, and its method is one of particularizing the universal. Natural science attempts to quantify the universal, through the reduction of living form and qualitative relations to mathematical and statistical formulations based on the classification of material artifacts.
By contrast, natural philosophy is primarily Platonic in that it is concerned with the whole as opposed to the part. Realizing that all things are essentially related to certain eternal forms and principles, the approach of the natural philosopher strives to understand the relation that the particular has with the universal. Through the language of natural philosophy, and through the Pythagorean approach to whole systems, it is possible to relate the temporal with the eternal and to know the organic relation between multiplicity and unity.
If the scientific spirit is seen as a desire to study the universe in its totality, it will be seen that both approaches are complementary and necessary in scientific inquiry, for an inclusive cosmology must be equally at home in dealing with the part or the whole. The great scientists of Western civilization -- Kepler, Copernicus, Newton, Einstein, and those before and after -- were able to combine both approaches in a valuable and fruitful way.
It is interesting that the split between science and philosophy coincides roughly with the industrial revolution -- for once freed from the philosophical element, which anchors scientific inquiry to the whole of life and human values, science ceases to be science in a traditional sense, and is transformed into a servile nursemaid of technology, the development and employment of mechanization. Now machines are quite useful as long as they are subservient to human good, in all the ramifications of that word -- but as it turned out, the industrial revolution also coincided with a mechanistic conceptualization of the natural order, which sought to increase material profit at the expense of the human spirit. This era, which gave rise to the nightmare of the modern factory -- William Blake's "dark satanic mills" -- gained its strength through the naive premise that the human spirit might be elevated and perfected through the agency of the machine.
Today, in many circles, to a large part fueled by the desire for economic reward, science has nearly become confused with and subservient to technology, and from this perspective it might be said that the ideal of a universal or inclusive science has been lost. This is because the ideal scientist is also a natural philosopher who is interested in relating his discoveries to a larger universal framework, whereas the dull-minded technologist, if he has any interest in universal principles at all, limits that interest to their specific mechanistic applications rather than their intrinsic worth of study. Yet, those who study universal principles as principles-in-themselves, often find that these principles have many applications in a wide variety of fields.
While Pythagoras would have taken a dim view of this artificial and dangerous split between science and philosophy, the negative consequences of this rupture have not gone unnoticed. Yet with his emphasis on the unity of all life, Pythagoras would have been in an excellent position to foresee the negative consequences: ecological imbalance, materialism, the varied effects of personal greed, the disintegration of human values, the decline of the arts, a lack of interest in personal excellence and achievement. In a sense these problems, not necessarily unique to this age, result from a lack of balance and an ability to see the parts in relation to the whole. As the poet Francis Thompson said, "You cannot move a flower without troubling a star," and so it is with every individual and collective action. Pythagoras correctly observed that all things are linked together proportionately, by justice, harmony -- call it what you will. By cultivating an awareness of harmonic forming principles and working within the bounds set by necessity, mankind possesses the potential to become a sacred steward of the earth and co-creator with Nature; but the inevitable corollary is that humanity also has every power to create and inhabit a hell of its own making. The simple fact remains that the scales of justice are inexorable -- it is a principle of Nature, and not merely of human morals, that each should receive his due. If we poison our rivers, we poison ourselves; if we act in stupidity, it is only appropriate that we suffer the consequences. If there is a moral to the story it is simply that individuals and societies are far less likely to run into trouble should they possess an awareness of these principles and relationships. And if one would like to cultivate the innate human ability to see things as they are, in whole-part relations, there is scarcely a better guide than the Pythagorean sciences. There has been much talk among the avant garde of "whole systems," the "philosophy of holism," etc., but few have realized that it is actually Pythagoras who is the tutelary genius and founder of the philosophy of whole systems.
We have mentioned the split between science and philosophy because it is an easy and self-evident example. Yet Pythagoras would also have something to say about the structure of our educational system as well. It has become fashionable to create ever more specialized disciplines -- a Ph.D. thesis is considered proportionately better the fewer the number of people that can understand it. This is not to imply that specialized knowledge lacks value, but rather to say that a danger exists in the self- inflicted alienation of academia and the sciences. Great things cannot fail to happen when minds get together and one mind fertilizes another -- when disciplines inspire one another. Pythagoras would say that, from the standpoint of natural philosophy, a superfluous multiplicity of facts and compartmentalized data is useless in a higher sense unless one can determine their relation to the whole, or the universal patterns which underlie all creation.
Interestingly, it is the modern-day physicists who have come most closely to approximating Pythagorean conceptions. Hell-bent on proving the mechanistic notions of 18th-century materialism, physicists have discovered that the deeper they push into matter the more it looks like the cosmos of the Pythagoreans and Platonists. Each atom is a Pythagorean universe, the sight of eternity in a grain of sand, consisting of an arithmetic number of particles, geometrically distributed in space, dancing and vibrating like a miniature solar system to the music of the spheres. A modern physicist would have little difficulty comprehending the teaching of the Orphic theologians that "the essence of the Gods" -- that is to say the formative principles -- "is defined by Number." [71]
Matter and energy are but different aspects of one, underlying continuum. Advancing to the subatomic level, quantity becomes quality, energy becomes information. Many physicists, proceeding from particulars to universals, are now on the verge of recognizing the essential truth of the statement, common to all spiritual traditions, that "Through consciousness the universe is but one single thing; all is interdependent with all." [72] The science of physics, proceeding from matter to energy, from energy to intelligence (i.e., pattern, logos), and from intelligence to Nous, has all but discovered the deus absconditus of the alchemists, the God hidden in matter. If the alchemical poeticism be allowed, even matter, if properly tortured, slain and resurrected, contains the innate potentiality of revealing the Hermetic mercury of eternal being. With the atomic accelerator at their disposal, modern physicists indeed have the capability to change lead into gold. They have taught the world that flesh, coal and diamond are made of the same basic stuff (carbon), driving home the reality that soul and Form is the essential component of all things.
One important Pythagorean insight which possesses ramifications for both the sciences and human behavior is the observation that the phenomenal universe is a mixture, a synthesis of Limited and Unlimited elements. Plato, drawing upon this notion in the Timaeus, compares the limited world of the Forms to a father, and the unlimited Receptacle of Space to a mother, the "nurse of becoming" as he puts it. [73] From their conjunction is born an offspring: the visible, phenomenal universe, the world of eternal principles manifesting in time and space.
The significance of this observation lies in the fact that it paints a picture of the phenomenal realm existing as a manifestation of what might be called "ordered chaos" -- we exist in an intermediate realm. Platonism, which posits the existence of an extratemporal and extraspatial world of perfect form, recognizes that the universe in which we live mirrors this perfection in an imperfect way. Hence Plato notes in the Timaeus that the receptacle of becoming, which we inhabit, was initially "filled with powers that were neither alike nor evenly balanced." [74] The receptacle might be compared to a sea, in which various currents provide for an ambient randomness. Stated in the terms of contemporary physics, one might observe that in the intelligible realm light, as principle, travels in a perfectly straight line, while in the realm of manifestation its path -- and the fabric of space itself -- is warped to a degree by gravitational mass.
While the Pythagoreans identified the principle of Limit with the Good, it should also be observed that without the principle of the Unlimited all manifestation would be impossible. [75] Moreover, working in conjunction with its partner, the principle of Unlimitedness is equally responsible for the organic beauty of the phenomenal realm. All trees of the same species more-or-less follow the same laws of growth, but at each juncture of growth there exists an indefinite number of possibilities. It is precisely the unlimited element which makes for the beauty of a forest, which would be much less beautiful if each tree were exactly the same. A musical composition also relies on order and randomness (change); should either element come to predominate it ceases to be beautiful.
Whereas the Platonism before the Renaissance possessed a tendency to focus on the transcendent world of forms, the first Pythagoreans seemed to concentrate more on the incarnate manifestations of universal principles. After all, not only did they study harmonia as a universal principle, seeing it reflected on all levels of the beautiful cosmos, they incorporated the principle into the fabric of their daily lives as well. But already with Aristotle we find a lack of insight into the Pythagorean view -- he cannot understand how the Pythagoreans view Number as possessing "magnitude." [16] This could well represent a misunderstanding of the incarnationalist dimension of Pythagorean thought, and perhaps reveals an overly literalist interpretation on Aristotle's part as well.
The Pythagorean view of the universe as a living, harmonic mixture is not only indispensable as a scientific concept, but it beautifully articulates the position of man in the cosmos as well. If, along with Plato, we view time as a moving image of eternity, [77] then each generation of humanity stands poised between the present moment and the timeless immensity of the eternal. Rather than being a worthless speck meaninglessly situated in the infinite expanse of space, each person, according to the Pythagorean view, is a microcosm, a complete image of the entire cosmos, with one foot located in the realm of eternal principles and the other foot rooted in a particular world of manifestation. Poised as he is between time and eternity, matter and spirit, man possesses an incredible freedom to learn, create and know, limited only by those principles on which creation is based. From this vantage point, humanity is engaged in a never-ceasing dialectic between time and eternity, possessing the ability to incarnate eternal principles in time (and in this sense mirror the creative work of Nature), yet also possessing the ability to elevate the particular to the universal through conscious understanding.
In relation to this theme, one final observation is in order: the creative endeavors of humanity seem to attain their peak of excellence precisely at that point when the intermediate nature of humanity is actively recognized. For with this recognition comes the realization that one must actively integrate the particular and universal aspects of being. Hence, the best science will once again embrace its sister, philosophia: both deal with universal principles and particular phenomena -- and together they will not attempt to build up a system of thought either from "the top down," deduced from purely a priori formulations, nor will they dare to start exclusively from "the bottom up" abstracting observations only from particular phenomena while ignoring universal principles. This approach I believe is fundamentally Pythagorean: the harmonic proportion, according to legend discovered by Pythagoras, exists as a purely universal principle, but it would have never been discovered without empirical experimentation on the monochord. The value of the harmonic proportion lies in both its universal nature as well as the significance and usefulness of its particular applications. Through the creative dialectic between the temporal and the eternal, there necessarily occurs a form of integration between otherwise purely theoretic and pragmatic approaches. Another benefit of this realization -- the realization that all things are composed of constants and variables -- is that, if seriously embraced, it actively encourages honest inquiry, rendering the twin dangers of Fundamentalism and Relativism equally impotent, for universal justice has its own means of dealing with individuals who mistakenly believe that they possess the Absolute Truth, or, conversely, think that "everything is relative."
One final point needs to be made about the Pythagorean approach, and that concerns the topic of value. There are many occasions where it is useful to take a divisive approach to Nature for the purposes of abstract analysis, yet there are also times when it becomes expedient to stress the unity of all being. For Aristotle Number was merely an abstraction as opposed to an innately existing a priori principle, so it is easy to see how he might become confused. by the notion that abstract number possesses "magnitude." Yet, if Number acts as a geometrical forming principle in the sphere of natural phenomena, as some of the studies cited in the bibliography abundantly demonstrate, it seems unwise to deny its immanent efficacy. Likewise, Aristotle was equally bewildered by the Pythagorean symbolism which equated certain archetypal number forms with principles such as "justice." Yet the truth of the matter is that it is precisely through the Pythagorean approach that quantity (number) and quality are discovered to be integrally related. As Ernst Levy has pointed out in an important article, "The Pythagorean Concept of Measure," this is shown to be especially true in the realm of music where each tone is actually a number, yet also a qualitative phenomenon possessing value. [78] It is particularly true in the realms of music and what has been called "sacred geometry" that one can gain insight into the Pythagorean conception of Number as both creative paradigm and qualitative relation. Levy suggests that in order to once again benefit from a unified scientific and philosophical synthesis that "a new mental attitude is required which many among us will be reluctant to assume, because it is contrary to the scientifically determined mind. The definition of that attitude is simple enough. It consists in this, that we must be willing to ascribe equal reality and equal importance to quality and quantity." [79] It is worth observing that the Pythagorean approach, while realizing the necessity of employing antithetical pairs of opposites in a conceptual sense, always arrives at a position which emphasizes the unity of the all.
To conclude that Number, in the most Pythagorean sense of the term, and the cosmos itself possesses a dimension of meaning is, within the context of mechanistic "science" or modern reductionistic "philosophy," perhaps the ultimate heresy; yet, for the traditional scientist and philosopher such a realization is only the starting point. If Pythagoras had but one imperative for the present age -- or any age -- it would be, as F.M. Cornford has suggested, this:
Seek truth and beauty together; you will never find them apart. [80]
-- DAVID R. FlDELER
NOTES TO THE INTRODUCTION
References to writings appearing in this volume follow Guthrie's chapter divisions.
1) It will not be my intention in this introductory essay to attempt to pierce through the various mysteries surrounding the figure of Pythagoras or to embark on the overwhelming task of textual criticism. Nor do I desire to write much about the life of Pythagoras, seeing that all of the primary source material is presented in this volume. Rather, I will attempt to briefly sketch out the history of the Pythagorean school and its influence, discussing those doctrines which are generally agreed upon, and to provide some type of context into which the writings of this sourcebook may be placed by the general reader. Indeed, I have tried to keep the general reader in mind throughout the introduction, and have limited more specialized comments to these notes, which also include references for further reading on topics which can only be touched upon here.
2) The source for this is Ion of Chios, quoted by Diogenes Laertius, Life of Pythagoras, chapter 5. For a discussion of Pythagorean Orphica see West, The Orphic Poems, Oxford University Press, 1984, 7- 15.
3) For Plato's views on writing about matters of ultimate concern see his Seventh Letter.
4) Since Pythagoras left no writings, this presents the historian with some difficulties. What, in fact, can safely be attributed to Pythagoras? Moreover, what do we know about his life? At a very early date a body of legends grew up around Pythagoras; many of these beautiful and amusing stories are recorded in the biographies, which constitute the first part of this book. For a long time, due to the nearly miraculous accounts, certain scholars dismissed the biographies as "late" and "unreliable." However, Aristotle in his lost monograph On the Pythagoreans emphasized how Pythagoras was seen at two places at once, how he showed his golden thigh, how he was thought to be the Hyperborean Apollo, and how he was addressed by a certain river. Obviously these stories are not "late Neopythagorean inventions" but go back to the time of Plato or before. Another source of these accounts was The Tripod of Andron of Ephesus, who was roughly contemporary with Aristotle. While the interpretive dimension of Iamblichus' biography is certainly colored by later Neopythagorean and Neoplatonic influence, it is now taken for granted that the biographies contain a great deal of early information about Pythagoras and his school, and much of the information is taken from older authorities whose work has since perished. Some of the ealiest authorities include Timaeus of Tauromenium (circa 352-256 B.C.E.) who wrote a History of Sicily which contained information of the Pythagoreans and the speeches of Pythagoras, and Dicaearchus of Messina (fourth century B.C.E.), a pupil of Aristotle who wrote a comprehensive study of Greek history which also treated the Pythagoreans. Another student of Aristotle, Aristoxenus of Tarentum, wrote several works on the Pythagoreans, used by the later biographers, which drew on early sources and his first-hand contact with members of the Pythagorean school.
5) Porphyry, Life of Pythagoras, chapter 12.
6) Iamblichus, Life of Pythagoras, chapter 5. For all we know, Pythagoras may have been invited to go to Croton. While this, to my knowledge, has not been previously suggested, it seems unlikely that he would have moved his teaching activities to a distant city without having some contact with and knowledge of the inhabitants. If this is correct, it would help explain his rapid acceptance by the populace.
7) See Vogel, Pythagoras and Early Pythagoreanism, for an analysis of these speeches. Iamblichus' source for these is Timaeus of Tauromenium.
8) Theon of Smyrna, Mathematics Useful for Understanding Plato, 12.
9) This concept -- that the One or principle of unity is the source of all numbers -- is easily grasped if one envisions "the One" as a circle in which various polygons are inscribed; the polygons, or numbers contained within the circle, may then be seen as various manifest aspects of the underlying unity. Another analogy is found in Pythagorean harmonics: the monochord, symbolizing unity (1/1), innately contains the entire overtone series (2/1, 3/1, 4/1, 5/1, etc.), which is manifested when the string is plucked.
10) Theon of Smyrna, Mathematics Useful for Understanding Plato, 66.
11) Ibid, 66.
12) Cornford, "Science and Mysticism in the Pythagorean Tradition," part 2, 3.
13) Ibid.
14) Aristotle, Physics 203 a 10.
15) Plato, Timaeus 35b f.
16) See Levin, The Harmonics of Nichomachus and the Pythagorean Tradition, chapter 6.
17) Different musical instruments emphasize different overtones. For example, the clarinet emphasizes the odd numbered overtones, thus accounting for its peculiar timbre.
18) This relation has been defined by Flora R. Levin in her Harmonics of Nicomachus and the Pythagorean Tradition, 1, as the "metaphysical octave," the characteristic feature of which is "a perfect fusion of parts (3:4 and 2:3) into a whole (2: 1)." Richard L. Crocker in his article "Pythagorean Mathematics and Music," 330, observes: "This construction, dividing as it does the first consonance by the second and third in a curious, interlocking way, has every right to be called the harmony. Here the inner affinity of whole-number arithmetic and music finds its most congenial expression."
19) The leimma is the excess of the fourth over the double tone: 4/3 : 9/8 x 9/8 = 4/3 x 64/81 = 256/243.
20) Needless to say, the "Music of the Spheres" is one of the most influential conceptions of Pythagoras. For a complete discussion which deals with its long and interesting history, as well as the significance of the concept as a poetic fact, see Joscelyn Godwin's The Harmonies of Heaven and Earth.
21) Alcmaeon of Croton in Freeman, Ancilla to the Pre-Socratic Philosophers, 41 (24 DK 4).
22) For more on figured numbers and their properties, see Heath, A History of Greek Mathematics, vol. 1, chapter 3, "Pythagorean Arithmetic." Also see the first volume of Ivor Thomas' Greek Mathematical Fragments.
23) Relating to the musical ratios of the Tetraktys is the Pythagorean saying "What is the Oracle at Delphi?" The answer is: "The Tetraktys, the very thing which is the Harmony of the Sirens" (Iamblichus, Life of Pythagoras, chapter 18). Nicomachus of Gerasa also identifies the harmonic ratio of 6:8: :9: 12 as a version of the Tetraktys in his Manual of Harmonics, vii, 10.
24) One modern proponent of this approach was Buckrninster Fuller. Some important sources for the study of this approach and geometrical forming principles include Keith Critchlow's Order in Space and Islamic Patterns, Ghyka's The Geometry of Art and Life, and The Geometrical Basis of Natural Structure by Robert Williams.
25) Iamblichus, The Life of Pythagoras, chapter 12.
26) Pythagoras was thought to be associated in some way with the God Apollo. This is only natural since Apollo is related to the celestial principles of harmonic order and logos, these being also the principles with which Pythagoras was most concerned. This connection is even made plain by the name of the philosopher -- for Pythios is the name of Apollo at Delphi, his most sacred shrine from which his oracles were delivered. This obvious etymological connection led Diogenes Laertius to interpret the name as meaning that he spoke (agoreuein) the truth no less than Apollo (Pythios) at Delphi.
27) This period of silence may have only been a ritual matter required during the religious ceremonies of the society, not during everyday life. "The ceremonies are conducted by Pythagoras behind a veil or curtain. Those who have passed this five-year test may pass behind the curtain and see him face to face during the ceremonies; the others must merely listen." Minar, "Pythagorean Communism," 39.
28) This is Minar's analysis in "Pythagorean Communism."
29) From an Orphic gold funerary plate, translated in Freeman, Ancilla to the Pre-Socratic Philosophers, 5. (l DK 18)
30) A fragment quoted by Iamblichus maintains that the nature of the Gods is related to Number (Iamblichus, Life of Pythagoras, chapter 28), and there was even an Orphic "Hymn to Number," portions of which are found in Kern, Orphicorum Fragmenta, Berlin, Weidemann, 1922.
Cameron, in his important study of Pythagorean thought observes that harmonia in Pythagorean thought inevitably possesses a religious dimension. He goes on to note that both harmonia -- there is no "h" in the Greek spelling -- and arithmos appear to be descended from the single root ar. This seems to "indicate that somewhere in the unrecorded past, the Number religion, which dealt in concepts of harmony or attunement, made itself felt in Greek lands. And it is probable that the religious element belonged to the arithmos-harmonia combination in prehistoric times, for we find that ritus in Latin comes from the same Indo-European root." (Alister Cameron, The Pythagorean Background of Recollection, 26.)
31) Burnet, Early Greek Philosophy, 83.
32) Ibid.
33) The Life of Pythagoras preserved by Photius, chapter 15.
34) Stocks, "Plato and the Tripartite Soul," 210-11.
35) For a good discussion of the Pythagorean view of justice as proportion see John Robinson, An Introduction to Early Greek Philosophy, 81-83.
36) Plato, Republic 444d (Comford translation, 143).
37) Plato, Republic 443d f. (Cornford translation, 141-2).
38) For sources and a discussion of the soul as a harmonia see Guthrie, A History of Early Greek Philosophy, vol. 1, 307 f.
39) Quoted by Clement of Alexandria, Stromateis ii, 84.
40) Certain scholars, appealing primarily to the "Table of Opposites," have argued that the orientation of Pythagoreanism was essentially dualistic. Such a simplistic view overlooks the fact that every philosophical system employs dualistic typologies and that "as a religious philosophy, Pythagoreanism unquestionably attached central importance to the idea of unity, in particular the unity of all life, divine, human, and animal, implied in the scheme of transmigration." (F.M. Cornford, Plato and Parmenides, 4.)
41) After writing this section I came across the observation of Vilctor Goldschmidt: "Our capacity to apprehend the outside world may be explained thus, that there are processes in our mind (microcosm) which are analogous to those in nature. These psychological processes we call natural laws. " (Quoted by Ernst Levy, "The Pythagorean Concept of Measure," 53.)
42) Heath, A History of Greek Mathematics, vol. 1, 10.
43) Plato, Republic 527b (Cornford translation, p. 244).
44) Tzetzes, Chiliad, viii. 972.
45) Plato, Republic 527b (Cornford translation, p. 244).
46) Vogel, Pythagoras and Early Pythagoreanism, 197.
47) Iamblichus, Life of Pythagoras, chapter 21.
48) Within this context it might be noted that Pythagorean metaphysics has, from ancient Greece to the present day, had an influence on the arts. The lesser can lead to the greater, and natural beauty, which is relative, can lead to the apprehension of transcendent Beauty which is absolute. This is, of course, a Platonic sentiment, but it is foreshadowed in the structure of Pythagorean thought. Perhaps, however, it would be more accurate in Pythagorean thought, with its immanent metaphysics, to suggest that the universal is realized through the particular. The use of mathematical and geometrical harmonies in sacred architecture, for example, can lead to the perception of, and resonance with, universal harmony. For example, through the medium of "Pythagorean geometry," a sacred edifice has the potential to become a celestial mediator: through the harmonic nature of its structure, the heavenly principles of harmonic form are reflected on earth; yet through the effect that this harmony has on those that are receptive to its beauty, the particular may be exalted to perceive the universal. This two- fold principle is applicable not only in architecture, however, but indeed in all the arts.
49) Guthrie, A History of Early Greek Philosophy, vol. 1, 170.
50) Plato, Republic 434d -44lc; also see J.L. Stocks, "Plato's Tripartite Soul."
51) Aristotle, Metaphysics i 6, 987 a 29 f.
52) Plato, Republic, particularly 398c -403c and 52lc -531c.
53) This was stated in Plato's lecture "On the Good." See the note below.
54) Aristotle enjoyed telling a story about Plato's lecture "On the Good": "Everyone went there with the idea that he would be put in the way of getting one or other of the things in human life which are usually accounted good, such as Riches, Health, Strength, or, generally, any extraordinary gift of fortune. But when they found that Plato discoursed about mathematics, arithmetic, geometry, and astronomy, and finally declared the One to be the Good, no wonder they were altogether taken by surprise; insomuch that in the end some of the audience were inclined to scoff at the whole thing, while others objected to it altogether." (Aristoxenus, Harmonica ii ad init., quoted by Heath, A History of Greek Mathematics, vol. 1, 24.)
55) Republic 500c.
56) Gorgias 506e.
57) Gorgias 508a.
58) Aristotle, Metaphysics xiii 7; see also the discussion in Dillon and Vogel (below).
59) For Pythagorean musical symbolism in Plato see Ernest McClain, The Pythagorean Plato. For more on similarities and differences between Plato and the Pythagoreans, and the characteristics of post-Platonic Pythagoreanism, see the valuable discussion in chapter 8 of Vogel's Pythagoras and Early Pythagoreanism. For a brief discussion of the dogmata agrapha and the early Academy see Dillon, The Middle Platonists, chapter 1.
60) A translation of the fragment appears in volume I of Ivor Thomas' Greek Mathematical Fragments. Dillon gives a succinct overview of the thought of Speusippus in chapter 1 of The Middle Platonists; Taran's Speusippus of Athens is a comprehensive study of the remaining Greek fragments and the thought of Speusippus.
61) Holger Thesleff, An Introduction to the Pythagorean Writings of the Hellenistic Period, 72.
62) Dillon in A.H. Armstrong, ed., Classical Mediterranean Spirituality, NY. Crossroad, 1987. 226.
63) Clement of Alexandria, Stromateis, i, 15.
64) Philostratus' Life of Apollonius of Tyana, which includes the letters, appears in the Loeb Classical Library.
65) For the equation of Apollo with the One see Plutarch, Moralia 270f, 393c and 436b. For his Pythagorizing tendencies see "On the Mysteries of Isis and Osiris" and his Delphic essays, collected in volume 5 of Plutarch's Moralia, Harvard. 1936.
66) Theon of Smyrna, Mathematics Useful for Understanding Plato, translated by Robert and Deborah Lawlor, San Diego, Wizards Bookshelf, 1979.
67) Nicomachus of Gerasa, Introduction to Arithmetic, translated by M.L. D'Ooge. New York, MacMillan, 1926.
68) Dillon, The Middle Platonists, 359. Each letter of the Greek alphabet possesses a numerical value and hence each word may also be represented as a number. The science of gematria involves the conscious use of this numerical symbolism and is not to be confused with either arithmology or numerology. A well known example is found in the name of the Gnostic divinity Abraxas, the numerical value of which is 365, the number of days in a solar year. The Babylonian divinities were represented by whole numbers and a Babylonian clay tablet indicates that Sargon II (fl. 720 B.C.E.) ordered that the wall of Khorsabad be constructed to have a length of 16,283 cubits, the numerical value of his name. The present writer has researched the history of gematria in some depth and has discovered an extremely strong body of evidence that gematria was utilized in Greece prior to the Hellenistic period.
69) Plato, Republic 509b.
70) In book ix, chapter II of the Stromateis, which deals with "The Mystical Meanings in the Proportions of Number, Geometrical Ratios, and Music." Clement also refers to the sage from Samos as "Pythagoras the great." (Stromateis i, 21.)
71) Iamblichus, The Life of Pythagoras, chapter 28.
72) R.A. Schwaller de Lubicz, Nature Word, West Stockbridge, MA, Lindisfarne Press, 1982, 99.
73) Plato, Timaeus 49.
74) Plato, Timaeus 52e.
75) Plato's nephew Speusippus thought that it was inappropriate to apply ethical values to the principles of peras and apeiron.
76) Aristotle, Metaphysics 1080 b 16; 1080 b 31; 1083 b 9; and 1090 a 20.
77) Plato, Timaeus 37d. Plato adds that time is a moving image of eternity "according to number."
78) Ernest Levy, "The Pythagorean Concept of Measure," 53.
79) Ibid.
80) Francis M. Cornford, "The Harmony of the Spheres." 27.