SOURCE: B. A. G. Fuller, "Pythagoras and the Pythagoreans," in History of Greek Philosophy: Thales to Democritus, Henry Holt and Company, 1923, pp. 103-17.

[In the following essay, Fuller summarizes the contributions of Pythagoras to the fields of music, mathematics, medicine, and astronomy. He notes the influence of the Pythagorean ideas of duality and their distinction between the concepts of "form" and "matter" on later philosophical thought.]

There is no figure in the history of philosophy so mysteriously shrouded in the phosphorescent mists of legend as the person of Pythagoras. Revered by his more immediate followers as a superior being, he acquired among later disciples the majesty of a demigod. He was variously reputed to be the son of Apollo in his present existence, and to have been the child of Hermes in a prior incarnation.

Like the Bodhisattvas on the threshold of Nirvana and Buddhahood, he was said to possess through the grace of his parent, Hermes, the memory of all his past existences. As for his teachings, they were derived straight from his other father, Apollo, through the lips of the Delphic oracle. And it was reported that in the flesh he had descended into Hades. He was also credited with other scarcely less distant but more mundane journeyings which had acquainted him at first hand with all the lore of the Phoenicians, the Chaldeans, the Magi, the Hindoos, the Arabians, and the Egyptians.

It is highly improbable that, even in an age when supernatural fathers were plentiful and direct confabulation with the divine was frequent, a commonplace personality could have attracted to itself so many and so flattering legends. It is true that Pythagoras' own adherence to mystical beliefs and rites like those of the Orphics and the votaries of the Delian Apollo, made his exaltation by his followers almost a bit of routine. In them transmigration, the partial incarnation of the divine in human nature, and ecstatic union with the deity through mystical sacrament and ritual, were a matter of course. Still, the myth suggests that Pythagoras was a remarkable and distinguished man, and such historic fact as we can gather goes to bear out this assumption.

He was born at Samos, probably in the last half of the Sixth Century. His dislike of the rule of Polycrates, who became tyrant of Samos in 532 B.C., caused him to emigrate to southern Italy, where at Croton he gathered about him a company of disciples and formed them into a religious order. This Order soon became powerful and influential and, like many later religious foundations of which history readily reminds us, tried to take a hand in politics and interfere in the government of the state. The School became a target for political abuse and disorder, so much so that the Master himself had to leave Croton, and take refuge in the city of Metapontium, where he died. By the middle of the Fifth Century the political activities of the Pythagoreans had become so obnoxious that the opposition rose, burned their lodge or monastery, killed many of them, and drove out the rest. Thus the Order became diffused through Magna Graecia and Greece proper.

As we have said, the Pythagoreans were a religious community, drawing their inspiration and doctrine from the mystical side of Greek religion. Their interest centered in the purification and the redemption of the soul from the taint of the physical and the prison of the body, in her final release from the wheel of transmigration and rebirth, and her reunion with the Divine. To effect these ends the Pythagoreans offered the old mystical means of ceremonial purgations and abstinences, the avoidance of certain food and clothing, and the performance of certain ritual acts. Among them, as among the adherents of all religions, there were many doubtless who stopped with the dead letter of observance, but there were doubtless many also who reached the spirit which gave them a new and deeper life.

The exact source of the Pythagorean religious mysticism is somewhat in dispute. It has been argued that just as the Orphics reformed and purified the older cult of Dionysus out of which they sprang, so the Pythagorean Society might be regarded as essentially a reformed branch of Orphism, which sought to correct and supplement the tendency towards mere ritualism and formalism in the parent body by emphasizing the need of a real rule of life, not only in outward observances, but in thought and meditation.¹ But it has also been suggested that the fountainhead of Pythagoras' religious beliefs lay not so much in Orphism as in the worship of the Delian Apollo, who had become, like Dionysus, the center of a "religion of redemption" with mystical rites of purification dating back it may be to Minoan Crete.²

However that may be, the Order, in addition to enjoying the practice of a formal ritual of purification, like the Orphics and the Apollonian worship, also divided the sheep from the goats on broader and more spiritual lines than that of mere church membership. They distinguished three types of men in general; the lovers of pleasure and gain, the lovers of practical activity, competition, worldly success, and honor, and, best of all, the lovers of contemplation and wisdom, who were devoted to the knowledge of the highest and deepest things of life. In fact, the term "philosophy" or "love of wisdom" is reported to have been first used by Pythagoras. And it was perhaps this third way of life, inspired and devoted to the philosophic interest, rather than any mere routine of ceremonial abstinences and participation in sacraments, which he regarded as religion pure and undefiled, making clean the heart within and preparing the spirit for mystical salvation and reunion with the Godhead. Such a view receives support from the fact that for Plato, as we shall soon see, who was much influenced by Pythagorean ideas, philosophy had precisely this high and solemn office.

But whether the interest in scientific and philosophical investigation and speculation had in the eyes of the Pythagorean this religious value and function, or was simply, like the modern Jesuit's occupation with similar interests, additional and subsidiary to a central and separate religious life and experience, the fact of that interest is undoubted. Like the most distinguished and learned of modern religious orders, the Pythagoreans, also, were preëminent in their application to the problems of science—particularly of music, mathematics, medicine, and astronomy—and of philosophy.

The philosophy of the Pythagoreans is largely indebted to the Milesian School. Pythagoras is said to have been in his youth, before he left Samos, a pupil of Anaximander, and he was a contemporary of Anaximenes. Like Anaximander he believed that the substratum of all things was the Unlimited, but he seems to have characterized this Unlimited as Air, and like Anaximenes to have conceived the world as supported and animated by the inbreathing of this Air in the midst of which it floated.

Pythagorean science was also in part sprung from Milesian sources. The so-called "Pythagorean" proposition in geometry had already been debated in Miletus, and some of the astronomical ideas of the Order suggest the influence of Anaximander's theories.

The important and original contribution to philosophy comes, however, in connection with the Pythagorean answer to what we have called the Second Commandment of philosophy to show by what means and process all the richness and variety of the manifold and parti-colored world arises from the simple and undifferentiated World-Substance. Anaximander had suggested a process of separation of pairs of opposites, Anaximenes, one of condensation and rarefaction. The Pythagoreans hold in a way to Anaximander's notion of opposites, but these opposites are not conceived by them as eventually and secondarily produced within and by the Unlimited, like successive or even simultaneous births of dissimilar and quarrelsome twins. On the contrary, the opposition is a fundamental and eternal one, of one primary World-Principle with another. From all eternity, so to speak, the Unlimited finds itself confronted and conjoined with another Principle, that of Limit and Determination, which exists outside and beside it. It is only through the action of this Principle upon the Unlimited that the interminable vacancy and monotony of the latter can be broken up, and mapped, and plotted, and specified out into a world of separate, distinct, individual things, each fenced within the bounds of its particular and specific self. The world, then, is the result of the interaction of these two factors. In a word, the Universe is a measuring out or off of the Unlimited by the Limited.

But this is very vague. It leaves two questions pressing for an answer. In the first place, is there any rule for determining how much of the Unlimited must be measured out in order to make definite objects? And secondly, how can different kinds of objects all be composed of one and the same indeterminate stuff? What is the difference, for instance, between a receipt for a cat and that for a dog?

An answer to the first question was suggested to the Pythagoreans by their studies in music and medicine. They knew when they played the lyre that musical notes were vibrations imparted to the air by the quivering strings; and they were also familiar by experiment with the fact that those intervals in the scale which struck their ears as melodious and concordant were always associated with invariable arithmetical proportions in the length of one string to another. Further scrutiny showed them that the four perfectly concordant notes of which the lyre was capable were in such proportion that the two middle notes stood in the relation, also, of arithmetical means to the two extremes. The means, then, might be regarded as mixtures, according to an invariable arithmetical formula, of the extremes.

This notion of the mean as a balanced and harmonious mixture of opposites was reinforced by the Pythagoreans' medical theories. The body was obviously a combination of opposite qualities of dry here, and moist there, of heat in one place, and coolness in another. When these qualities were harmoniously balanced, when there was a "happy" mean of hot and cold, etc., the body was healthy and perfect and in a state of well-being. If there was a disturbance of the balance and one opposite upset the proportion by excess or deficiency, then there was disease.

The application of these studies in music and medicine to the philosophical situation was obvious enough. It was but a short, easy step to say that all sound, solid, clear-cut things owe, like healthy bodies, their definite and articulate nature to a harmony and balance of the factors which compose and sustain them. Every object was a correct and shapely mean between extremes of possible lopsidedness and deformity in one direction or another. The Universe, then, was the stable and well ordered, the neatly mapped and plotted and fenced affair it was, because to each of its component parts, to each cat, and dog, and tree, and blade of grass, "just enough" for that particular kind of thing, and not "too much" or "too little" had been allotted.

We should by this time be almost prepared for the answer which the Pythagoreans gave to the second question—the question which asks by what principle that balanced and happy "middle term" which means the lithe and purring cat, is differentiated from that golden mean which manifests itself in the faithful, barking watch-dog. The reply might almost rise to our lips unprompted. The measure of the one is different from the measure of the other. Each thing is a specific number, a specific amount, of the Indeterminate. Every kind of object has its own particular, mathematically expressible receipt or formula. The differences between things are then essentially differences in amount and number. In a word, the Limited, the Principle of Determination which divides up and lays out the Unlimited as an ordered and definite Universe is Number, and different things are, if one looks beyond their faces into their hearts, really nothing but different Numbers.

Before relegating this doctrine to the realm of fantastic and incomprehensible theories, let us stop and ponder it a moment. When we think of a number we generally have in mind simply the Roman or Arabic numeral by which we sum it up and symbolize it. For instance, if we think of the number "eight" there comes before our eyes simply the figure 8, or VIII. But this figure is nothing in itself. It stands for something, for eight something, or at least for eight anything. It means among other things the ability to point your finger at a line of objects and count them out. This line may be a line of all sorts of things, or, if one is abstractedly minded, it may be simply a line of plots or positions in empty space which things might occupy. But in any case the plot of space occupied by the series is the sum, so to speak, of the plots occupied by each member of the line.

Or, again, any one of the plots occupied by any member of the series may be subdivided into smaller plots that shrink eventually into points. And we may finally come to the conclusion that it would take such or such a number of little plots or points to fill in the outlines of any given object. It might take, for instance, thirtyseven of them to fill in, or rather out, the contours of a pug-dog, and two hundred and fifty to complete the measure of a man. In that case the nature of the pug would be the number 37; the essence of man, the number 250. And the difference between the "just enough" which makes that perfect and harmonious balance which we call a pug, and that which constitutes the proportion and equilibrium known as man, would be simply the arithmetical difference between the numbers 37 and 250.

Now this seems to be very much the way by which Pythagoreans came to the conclusion that things were really Numbers. They appear to have thought of numbers geometrically and to have used arrangements of dots in different forms as numerical symbols, along with, or instead of, the letters of the alphabet which the Greek commonly used. Aristotle tells us that Eurytus, a Pythagorean of the early Fourth Century, used to illustrate the "nature of horse, man and plant 'by means of pebbles' or counters,"³ in very much the same way, it would seem, as œsthetic station-masters to-day adorn the railway line at intervals with the names of their stations, or hospitable sentiments, or even symbolical figures, compounded of variegated plants or of the chaster and less frail material of whitewashed stones. Indeed, the commentator Alexander reports of Eurytus' method that "smearing the wall with plaster and sketching on it the figures of a man and a plant, he proceeded to fix some of the counters in the outline of his face, some in that of the hands, and some in that of the other parts, and so he completed the outline of the man he had imaged by a number of counters equal in number to the units which he said defined man."⁴

Later, after the impossibility of constructing the line, and hence the plane or the solid, out of pure mathematical points had been recognized, the Pythagoreans seem to have modified their views. Things were still ultimately composed of geometrical figures to be sure, but those figures and the objects which they constituted were no longer regarded as actually numbers (that is, aggregates of points) but only as like numbers. This new twist in their teaching made possible all sorts of fantastic applications. Many things, such as moral qualities, social institutions, etc., which could scarcely be regarded as composed of mathematical points might be fancied to have some picturesque or mysterious affinity with this number rather than that. Thus we find a square number assigned to justice, and the number five, which is the union of the first odd with the first even combination of units, applied to marriage.

The religious beliefs and the philosophical speculations of the Pythagoreans could not exist side by side in the same individual minds and the same school of thought without infecting each other to some extent. Just as the confluent waters of the Rhone and the Saône, so distinct from one another in color at their conjunction, soon mingle and merge, so the two elements in Pythagoreanism, drawn apparently from quite different sources, rapidly blended. The process was doubtless facilitated by the essential congeniality of the two lines of thought. Both were dualistic and emphasized the conflict of opposites in the world. The Orphic strain set the soul in violent opposition to the body, as a principle of good at war with one of evil, while philosophic speculation found Limit and Determination forever at variance with the Indeterminate.

The result was that the moral dualism was reinforced with a philosophic foundation, and the philosophic doctrine of the conflict of opposites took on a moral and religious significance. As the world split asunder into pairs of arithmetical qualities (which in the end were reduced to the fundamental opposition of the Limited and the Unlimited), all those on one side of the line of cleavage tended to be regarded as better than those on the other. The fundamental contrast of good and bad was already patent; so, also, that of the soul to the body was obviously one of the better to the worse. And the dry, the bright, and the warm were pleasanter, so far at any rate as weather and many other important considerations were concerned, than the wet, the dark, and the cold. So, too, the straight was easier and neater and honester than the crooked, and activity was to be preferred to inertia, or again the square is a trig, the oblong a lopsided, figure. Whatever the logical character of any antithesis, a reason could always be found for regarding one of its members as more respectable, the other as more disreputable, than its opposite.

Again, this pervasive contrast of good and evil in the world found in the opposition of Limit and the Indeterminate a ready and appropriate foundation. The idea of the Infinite, it will be remembered, did not strike the Hellenic as it has struck the modern mind. It suggested not perfection but formlessness and confusion. It was the limited and the definite in things which gave them character and perfection. Only as they were distinct and defined did they have a nature to realize and express. The logical, then, already suggested a moral, opposition between the principle of Limit and that of Indetermination. The Indeterminate was the natural and inevitable candidate for the principle of evil and imperfection, Limit and Number for the principle of good.

Now everything, whether good or bad, actually is, or at least is like, some number. But how is it possible that the things which are undesirable and evil should be composed of or modeled after that which is the principle of goodness and perfection? A scrutiny of the number series gives the answer. In that series a most convenient and suggestive opposition immediately reveals itself. We make a distinction between odd and even numbers. This distinction arises out of the quite different attitudes assumed by the two series towards the process of bisection. Every even number is naturally addicted to the habit of bisection. All may be bisected at least once without resistance or protest, and will go on taking "just one more" till some odd number appears on the scene and arrests them. If left, however, to the dictates of their even nature they go on bisecting and are soon increased to Falstaffian proportions below the line. Infinity alone can bound their capacity and slake their thirst. They display, that is, a propensity and sometimes a passion for the Unlimited and Indeterminate.

The even numbers, then, will always "go the limit" set by the odd, and are often literally beyond it. But the odd have principles which forbid indulgence in bisection and dalliance with the Unlimited. If they are forced to it by some overzealous admirer of arithmetic, the result is apt to be an improper, vulgar fraction and a "hang-over." Essentially limited in nature, they make it their chief mission in mathematical life to prohibit and suppress in others the bisection which they are incapable of enjoying themselves. Many a pleasant process of division is stopped, many an even number is kept from going entirely to pieces, by their interference. Nay more, each even number whose bisection they interrupt is straightway converted by them into two odd ones. All in all, if the even numbers are the Cavaliers, the odd are the Puritans, of arithmetic.

So it is that the even numbers by their addiction to bisection betray the cloven hoof. They are "limbs of Satan," in close alliance with the Indeterminate. And being already in themselves so full of the devil, they make fit material and filling for his works. It is with them that the worse member in each pair of opposites is indubitably stuffed. Scratch the crooked, or the damp, or the oblong, or the feminine, and you will surely find their Tartar "even" nature. The better member of the pair, on the other hand—a true straight line, or a dry, bright day, or a "well-rounded" square, or a real man—is some respectable odd number, the product pure and undefiled of Limit and Determination and integrity and good form.

The general development of Pythagoreanism and the modifications and embellishments which it received at the hands of its various adherents and exponents, it is very difficult, and for our purposes perhaps unprofitable, to follow. We may, however, note the view that the immediate successors of Pythagoras tended to minimize and eliminate the religious and mystical elements in his teaching, and to emphasize and develop philosophical and scientific speculation.⁵ The Pythagorean doctrine, for example, current in the time of Plato, that the soul was the harmony of the various parts and functions of the body, follows naturally enough from the views about the Mean and Number which we have just been discussing. But it is quite out of keeping with the religious belief that the soul is antagonistic to and imprisoned in the body, and is reincarnated in one new life after another. This way of looking at the matter may help explain the unembarrassed way in which during the Fourth Century Pythagoreanism as a philosophy disappeared within the portals of the School of Plato never to reëmerge.

However, the religious side of Pythagoreanism, or as we might say, the Pythagorean version of Orphism, was kept alive by some members of the School. And if it was discredited and scorned by the philosophically and scientifically minded branch of the Society, it lived to taste a splendid revenge. It was to come into its own again, riding on the crest of the swelling wave of mysticism which inundated the Grasco-Roman world in the First Century B.C. To it, rather than to the scientific and philosophical attainments of the School, were due the revival of the name and the mysterious prestige of the Master, in the so-called Neo-Pythagorean movement.

Besides their work in music and medicine and mathematics, the Pythagoreans made a distinguished contribution to astronomy. Pythagoras himself seems to have held to Anaximander's three-ring theory of hoops of fire, lunar, solar, and stellar, revolving about the earth. And apparently he thought of these rings as revolving in a musical relation to one another like that of the intervals between the concordant notes of the lyre. This is the origin of our well-known phrase "the music of the spheres."

To later members of the School, however, it occurred that the earth was not the center of the universe, but itself revolved along with the other planets about some central point. This point, in their estimation, was not the sun (which they reckoned among the planets) but what they called the central fire. We cannot see this fire because the face of the earth on which we live is always turned away from it, just as the other side of the moon is always invisible to us. They also assumed a body called the "counter-earth," which like the central fire, and for the same reasons, was always beyond our ken. But its shadow is visible in the eclipses of the moon, to explain which, apparently, its existence was assumed.

Let us now conclude our discussion of the Pythagoreans by noting such new turns as they may seem to have given to the development of philosophical speculation. The most obvious novelty perhaps is their dualistic suggestion that the world is not all of one piece, but must be referred to two equally fundamental and opposed Principles. To the Milesian School it did not occur that there could be more than one World-Substance or Principle. The great step forward of the earlier philosophers, that indeed which conferred upon their speculations the title of philosophy, lay in their sudden realization and grasp of the surface unity of the Universe. The World at any rate is one Fact; the sum total of existence is a single thing. And the singleness of its face they transferred quite naively to its heart, accepting as a matter of course that there was only one Principle at the basis of all things, and asking only whether that Principle was Water or Air or what not. Even Xenophanes, who had suggested that the World was made of two kinds of stuff, earth and water, seems to have been kept by his pantheistic musings and agnostic doubts from ever pressing the question to a point where he would have been forced to choose between a one substance theory like that of the Milesians, or some sort of dualism. He, too, was engrossed with the vision of the World as one sum and aggregate of things.

But the attention of the Pythagoreans was drawn from the very start to the diversities and contradictions on the face of things. They saw their world riven asunder on every side by oppositions, logical, moral, and perceptual, of ideas, values, and natural phenomena. When they looked more closely at these antitheses, they saw, not, like Anaximander, some sort of indefinite Unity behind them in which all were merged and swallowed up, but rather more and deeper opposition, till finally Reality was cleft to its very roots, and fell apart into the two antithetical World-Principles which we have been discussing.

Henceforth the question whether Reality was at heart One or Many was to be a philosophic problem of the first importance. And the dualistic reply which the Pythagoreans gave when they raised the question was to exert great influence in the succeeding centuries of ancient philosophy.

Furthermore, the Pythagoreans not only mothered—if indeed they did not father—the idea that the World was to be explained by the interaction of two opposite principles, but they also made a suggestion, accepted and developed by subsequent philosophers, as to what those two principles were. And in so doing they added two new and indispensable concepts to the philosophic tool-shop, notably the concept of "Form" as distinguished from that of "Matter."

The Milesians … were apparently unconscious that this distinction made any philosophic difference. Prac tically, of course, they used it every day of their lives, as any intelligence must which is human and whose experience is not a chaos. They recognized the similarities of things, and grouped them under the same or diverse species. They made different materials into the same form, as when they fashioned bronze or gold or ivory or marble into a statue. And they molded the same material into different forms, gold into a coin, or a ring, or a cup, marble into a bath or a column. But they did not think of the distinction between the "Form" and the "Matter" of a thing as something which perhaps bit deep into the inmost reality of the object and might even reach that core of things to which their speculations were seeking to penetrate. Form and Matter were for them superficial phases which were not essentially contrasted, but were soon lost and dissolved in the unity of the one living, moving, developing Thing which lay so close beneath the surface variety of the world.

The Pythagoreans, however, evoked the two concepts into separate and distinct life and made them philosophically significant. The Number which defined the space and quantity of any given thing was, as we have seen, literally its Form—was in fact an aggregate of points or dots taking on the shape of the object in question. The difference between things was a difference of Form only. It was the enumerating Number, the enclosing Figure, which made this plot of space a man, and that a tree. In the same way, the Unlimited simply provided the wherewithal for enclosure and enumeration. It was the medium on which the points or dots were laid out or impressed. In other words, it was the material which was measured out according to this or that number to take on this or that shape. Without it there would have been nothing to arrange the dots in, or to impress them on. Numbers and figures would have had to remain purely abstract and purely imaginary. The lion would have been no more fearsome than the unicorn, and there would have been no land but fairyland. In a word, if Limit and Number gave its shape to the world and determined the forms of all that dwell therein, it was the Unlimited which turned these shapes, otherwise fantastic and "unreal," into a tangible, concrete, solid, energetic, "real" Universe. The Unlimited was the matter of which things are made, just as Numbers were the forms in which they were molded.

We can see how easily this notion of the shape or figure of an object as something distinct from the matter which takes the shape in question, might develop into a wider and profounder view. After all, things do not differ from one another in point of shape alone, but in many other respects. A tree is different from a man not only in size and figure, but also in the uses to which it can be put. Its total value and significance in the world is quite different. It is defined differently. It has its own character or nature of "tree-ness." And it is this character—that is, all which is summed up in the logical concept of tree—that makes the concrete plots of existence in which it embodies itself a forest of particular trees rather than a crowd of individual men. Form has now passed from a merely physical to a logical significance. It is no longer simply the outer shape, but the total definable character of an object. This further development of the meaning of Form, and the great importance of the Pythagorean distinction between Form and Matter we shall realize when we come to discuss the Platonic and the Aristotelian systems.

We take leave, then, of the Pythagoreans, feeling that philosophy has made a notable advance. It has begun to dissect the World, and it has made a startling discovery about the anatomy of the Universe. Reality is not all one inside, but it has an internal apparatus which presents rather a complicated appearance. There, for instance, is Matter, and there, rather closely adhering to Matter but still obviously a distinct organ with a distinct function, is Form. There, too, is the Soul hitherto not clearly differentiated and localized, but now discovered to be a separate part of Reality with distinctive marks of its own and indeed displaying a marked antipathy for the physical aspect of the Universe.

Again, we find an interesting process going on by which two or three sciences are being "budded off," like young polyps or potatoes from a parent organism. Music is bulging out here, medicine and astronomy and mathematics there. They have not yet broken through the skin of the all-inclusive philosophic interest, but it is obvious that nuclei have formed about which more specialized and detached interests have begun to gather and grow. That they will eventually break out and away, and the form which they will take, can now be predicted.

Notes

¹ Cornford, From Religion to Philosophy, p. 198.

² Burnet, Greek Philosophy, Thales to Plato, pp. 40-41, Early Greek Philosophy, 3rd ed., p. 90.

³ Burnet, Greek Philosophy, Thales to Plato, p. 90.

⁴ Quoted by Burnet, op. cit., p. 91.

⁵ Burnet, Early Greek Philosophy, 3rd ed., p. 86.