The Music of Pythagoras

by Kitty Ferguson

  Most of the ancient world regarded natural phenomena as beyond human understanding or explanation, subject to the whims of capricious deities and best dealt with in imaginative stories. Philolaus referred to the central fire as the “home of Zeus,” perhaps to make his contemporaries feel comfortable with the notion. But what we learn from him is that the first Pythagoreans, led by a man who was, by some descriptions, more shaman than scientist or mathematician, were trying a new way of securing a foothold on the climb to understanding nature and the universe, through numbers. The earliest pre-Socratic philosophers—Thales, Anaximander, Anaximenes—for all their yearning to get at the roots of things did not connect or confirm their philosophical ideas with numbers or mathematics. The Mesopotamians of the First Babylonian Dynasty had found numbers useful and enjoyed using impressive mathematics in exercises that had no practical applications, but apparently did not think that numbers and mathematics were a way to reach a profounder, all-encompassing truth. Philolaus wrote that “nature itself admits of divine and not of human knowledge,” but he was convinced that number relationships underlay the origin of the universe and the soul’s relationship with the divine, making it possible for humans to figure such things out. This insight was a fresh departure, a sea change of enormous proportions, and Pythagoreans such as Philolaus regarded the relationship of rational humans to a rational universe with awe. The kinship was reflected in a doctrine of the unity of all being. A fragment in Against the Mathematicians, by the skeptic philosopher and physician Sextus Empiricus (second–third century A.D.) states: “The Pythagoreans say that reason is the criterion of truth—not reason in general, but mathematical reason, as Philolaus said, which, inasmuch as it considers the nature of the universe, has a certain affinity to it (for like is naturally apprehended by like).”

  Bust of Plato

  That “certain affinity”—the fact that human mathematical reasoning does match up with what is really happening in nature—was not something that the Pythagoreans, or Philolaus, or anyone since them could or can explain. It was enough to know that numbers were tied in a fundamental way to the origin and nature of the cosmos.

  CHAPTER 8

  Plato’s Search for Pythagoras

  Fourth Century B.C.

  IN ABOUT THE YEAR 389 B.C., Plato left his home in Athens and boarded a ship setting sail westward into the Ionian Sea. His destination was Tarentum, one of the old colonial cities of southern Italy, in the coastland known to him as Megale Hellas. He was going in search of Pythagoras.1

  In the 110 years since his death, Pythagoras had become the stuff of legend. Some believed he had been the wisest man who ever lived, almost a god. There were stories that a wealth of precious knowledge had perished with him and his followers in upheavals that had destroyed their communities in 500 and 454 B.C. Though no one alive was old enough to have known Pythagoras, Plato had heard that in Megale Hellas there were still men calling themselves Pythagoreans. So, in his thirty-eighth year, he sailed to the shores where Pythagoras at about that same age had preceded him and walked and taught and died. The stones of the promontories, the pleasant coastlines, the very dust of the roads, ought to remember him.

  Plato’s investigation began in Tarentum, on a small peninsula at the western extreme of the instep of the Italian boot, the first port of call for ships crossing from Greece.* The only story connecting Pythagoras with that city was that he had convinced a bull there not to eat beans, but Tarentum had been far enough from Croton for refugees from the fifth-century attacks to have settled, felt reasonably safe, and started their own exile Pythagorean community. It had survived, and Plato knew that its most prominent member now was Archytas of Tarentum—“Archytas the Pythagorean.”

  In Archytas, Plato found a man who embodied Pythagorean ideals both in his lifestyle and his studies. Archytas was an outstanding scholar and mathematician working in the Pythagorean mathematici tradition, and also an able civic leader. Meeting him must have confirmed for Plato that the years of Pythagorean rule in Megale Hellas had been an era of peace and stability, strengthening his conviction that men who knew philosophy and mathematics made splendid rulers. Plato and Archytas were within a year of each other in age. The visit in 389 was the first of several during which Plato conversed with him and his Pythagorean friends, absorbing knowledge and information that only a handful of men in the world could have given him. Megale Hellas would continue to draw Plato, not only because of Archytas.

  At the time of Plato’s first visit, the southern Italian cities were living under the encroaching shadow of a formidable enemy—Dionysius, tyrant of Syracuse, close across the water in Sicily. “Tyrant” did not necessarily have negative connotations then. The term meant a ruler whose claim to power was not hereditary, and, indeed, Dionysius had begun in the lowly position of clerk in a city office. However, he also fit the later, ugly definition. Tactics that made him hugely successful shocked even his contemporaries. Dionysius reigned for nearly forty years, preserving Syracuse’s independence during repeated invasions while most of the rest of Sicily fell to the Carthaginians from North Africa. Syracuse became one of the most powerful cities in the world, her fleet for a time the strongest in the Mediterranean. It was certain that if Dionysius chose to move against his Italian neighbors, no one could stop him. Plato had come to an unstable, dangerous region, but instead of heading directly back to safer Athens, he decided to experience at first hand the court of a powerful, gifted ruler. Here was no theoretical governance. It was the real thing.

  Dionysius’ capital was, or was in the process of becoming, a splendid, well-fortified city, built strategically on an island separated from the mainland of Sicily by a narrow swath of water. There was a Pythagorean community in Syracuse, begun like the one in Tarentum by fifth-century Pythagoreans who in this case had fled west across the Gulf of Messina, but Plato was more interested in the court of Dionysius. He was becoming increasingly intrigued with public affairs, and he seems to have enjoyed—perhaps too well for his own good—rubbing shoulders with powerful courtiers among whom he felt more than able to hold his own. On this first visit, Plato met one of the most influential men in Syracuse, the tyrant’s brother-in-law Dion. Plato was impressed with Dion . . . and Dion with Plato.

  Not long after Plato’s visit, Dionysius’ invading forces wreaked devastation on the south Italian cities, and the entire region fell to Syracuse. In terms of the map, the football had kicked the boot. Meanwhile, back in Athens, Plato went on to establish his Academy, adopting a “Pythagorean curriculum” that he had learned from Archytas: a “quadrivium” of arithmetic, geometry, astronomy, and music. The inclusion of music was an exceptionally Pythagorean touch.

  The ruthless Dionysius died in 367, survived by his son, Dionysius the Younger. Unfortunately for Syracuse—though perhaps to the relief of many in the region—the son was a much less able leader than the father. Plato’s acquaintance Dion, the new ruler’s uncle, was dubious about his nephew’s ability to keep Syracuse as dominant as the old tyrant had left it. For whatever well-meaning or devious reasons (history records the events but not the motivation) Dion decided to improve his nephew by seeing to his belated education. The father had been an innately brilliant leader with literary pretensions (though his writing was widely judged to be embarrassingly bad), but the son needed assistance if he was to rule effectively and continue to frustrate the Carthaginians’ desire to complete their takeover of Sicily. Dion recalled his conversations with Plato twenty years earlier and some of Plato’s dialogues that he had read since then, in which Plato had been developing the idea that men like Pythagoras and Archytas—philosophers for whom the “quadrivium” was bread and butter—should be the political rulers. To fill such shoes and be a “philosopher king,” as Plato coined the term, Dionysius the Younger needed training only Plato could provide. Dion decided to try to convince Plato, by then sixty-one and famous in Athens and far beyond, to return to Syracuse and tutor him.

  In spite of what must hav
e been a yearning to foster a philosopher king in a world power like Syracuse, Plato was not initially keen about Dion’s proposal, thinking it would be a risky undertaking and unlikely to succeed. Archytas convinced Plato to change his mind. Partly tempted by the opportunity for more conversations with Archytas, Plato sailed for Syracuse. For a while, he was on sufficiently good terms with Dionysius the Younger to do some networking on Archytas’ behalf. A friendly relationship between Dionysius and Archytas was advantageous for the city of Tarentum. However, Dionysius did not study with Plato long. Before the year 366 ended, he banished Dion; Plato, suspecting that his own best interests did not lie in this court, prudently took his leave.

  Yet five or six years later, in 361–360 B.C., Plato was back, invited by the tyrant himself. Dionysius sent an emissary named Archedemus, a friend of Archytas, on a special ship to summon Plato. The banished Dion also had a clandestine hand in his return. He asked Plato to engineer a reconciliation between him and Dionysius.

  Plato arrived and Dionysius’ lessons resumed, but any hope of transforming Dionysius into a philosopher king was, again, short-lived. It cannot have helped that Plato was at court partly at the behest of the banished Dion. Plato was soon not only out of favor but in danger for his life. He got word to Archytas, and that resourceful man, using the influence he retained with Dionysius, sent an ambassador with a ship from Tarentum and persuaded the tyrant to release Plato. Afterward Archytas was not only known as “Archytas of Tarentum” or “the Pythagorean” but also as “Archytas who saved Plato’s life.”

  Dion captured Syracuse three years later and was assassinated three years after that at the behest of another Syracusan acquaintance of Plato. Dionysius regained control for a short period, but he seems never to have had much talent or inclination for ruling, and it may have come as a relief to him in 344 when the Corinthian general Timoleon compelled him to surrender and retire to Corinth. There he became a language teacher. Perhaps Plato’s efforts had not been entirely wasted and a former tyrant was well qualified to teach.

  In Corinth, Dionysius met Aristotle’s pupil Aristoxenus, who was collecting information about Pythagoras and the Pythagoreans. Aristoxenus would turn out to be one of the earliest and most valuable sources, for Tarentum was his birthplace and he said his father knew Archytas. From Dionysius, who had been rather useless at nearly everything else, Aristoxenus was able to glean firsthand knowledge about Pythagoreans in the fourth century in Syracuse, not far from the area where the society had originated.

  As Dionysius told the story to Aristoxenus, some of his courtiers in Syracuse had spoken disparagingly of the local Pythagoreans as arrogant, pious fakes whose rumored moral strength and superiority would evaporate in a crisis. Other courtiers disagreed, and the two sides contrived a way to settle the dispute. Would one Pythagorean be willing to stake his life on the dependability and faithfulness of another? Would the other’s faithfulness and dependability—to the death—prove deserving of such trust?

  The courtiers accused a man named Phintias, a member of the local Pythagorean community, of plotting against Dionysius. When Dionysius sentenced him to death, Phintias asked for a stay of execution for the remainder of the day, long enough to settle his affairs. It was a Pythagorean custom, established by Pythagoras himself, to keep no private property but own all things in common. Phintias was the oldest among the local brotherhood and chiefly in charge of the management of finances. Dionysius and his court, following their plan, allowed him to send for another Pythagorean, Damon, to remain as hostage until his return. To the astonishment of the court, Damon willingly came to stand as personal surety for Phintias. Phintias departed, and the courtiers—sure they had seen the last of him—sneered at Damon for being such a trusting fool. But the faithful Phintias returned at sunset to face his death rather than leave his friend to be executed in his stead. “All present were astonished and subdued,” reported Dionysius, who was so impressed that he embraced the two men and asked to be allowed to join their bond of friendship. Not surprisingly, “they would by no means consent to anything of the kind.” What happened to them then is not known. Plato, so often at court in Syracuse, also likely heard about this incident, but he never wrote about it.

  Plato’s activities in Megale Hellas went beyond learning about Pythagoras and Pythagorean teachings, experiencing day-to-day reality in a tyrant’s court, and abortive attempts to tutor Dionysius. He helped Archytas strengthen his position in Tarentum as a minor philosopher king. Archytas went on to play a prominent role in political affairs among the cities of Megale Hellas and Sicily, in accordance with the Pythagorean tradition of wise and able involvement in public service.

  IN A SEARCH for the real Pythagoras and the Pythagoreans and what they believed and taught, the information about Archytas, Plato, and Dionysius the Younger provides valuable clues. Most significantly, it reveals a link between Plato and a Pythagorean community that still existed in the fourth century B.C. in the region where Pythagoras and his followers had had their golden age in the late sixth century. Plato knew, and knew of, other fourth-century Pythagoreans, but after his visits to Tarentum, when he thought “Pythagorean” he was probably mostly thinking of Archytas and his associates. When he thought “Pythagorean mathematics and learning” he was thinking of the mathematics and learning of Archytas.

  What was he like, this man who was, for Plato, the best available evidence of what it meant to be Pythagorean and what “Pythagorean knowledge” was? What could Plato have learned from him about Pythagoras and what Pythagorean teaching had been more than a century earlier?

  Archytas was known to be a mild-mannered man who ruled in Tarentum through a democratic set of laws—information deduced from the news that these were not always obeyed: Though the “laws” said a man should serve no more than one year, the city “elected” Archytas seven times to the office of strategos, or ruling general.2 Aristoxenus wrote that Archytas was never defeated as a general except once, when his political opponents forced him to resign and the enemy immediately captured his men. Archytas, said Aristoxenus—whose father, he claimed, had known the man in person—was “admired for excellence of every sort.”

  There can be no doubt that as a scholar Archytas lived by the great insight that set the Pythagoreans apart from other ancient thinkers: that numbers and number relationships were the key to vast knowledge about the universe. Archytas was a rigorous mathematician who solved an infamous problem in Greek mathematics known as the Delian problem, or doubling a cube, that is, creating a new cube twice the volume of the first. Archytas’ solution was sophisticated, requiring new geometry using three dimensions—“solid” geometry—and involving the idea of movement.3*

  Diagram showing how Archytas solved the Delian problem, evidence of how advanced Pythagorean mathematics and geometry had become in little more than one century.†

  Viewing the world through the eyes of his Pythagorean forebears, Archytas could not avoid pondering the possible hidden, underlying numbers and geometry. “Why are the parts of plants and animals (except for the organs) all round?” he asked, “of plants, the stems and branches; of animals, the legs, thighs, arms, thorax? Neither the whole animal nor any part is triangular or polygonal.” He suspected there was a “proportion of equality in natural motion, since all things move proportionately, and this is the only motion that returns back to itself, so that when it occurs it produces circles and rounded curves.”

  Later scholars, among them Euclid and Ptolemy, agreed that Archytas’ precise work in the mathematics of music was fundamentally linked with the earliest Pythagorean mathematics and music theory.4 Archytas extended the study of numerical ratios between notes of the scale and showed that if you defined a whole tone as the interval separating the fourth and fifth notes of the scale (such as F and G in a scale beginning with C), as Greek music theorists were doing, then a whole tone could not be divided into two equal halves.* This had dramatic implications, for it was an example of something obviously prese
nt in the real world that could not be measured precisely. A different example, discovered in the right triangle, had famously caused the first Pythagoreans to have a devastating crisis of faith in the rationality of the universe, but incommensurability seemed no longer to disturb Pythagoreans like Archytas in the fourth century B.C.

  In astronomy, Archytas puzzled over the question of whether the cosmos is infinitely large, and was notorious for asking: “If I come to the limit of the heavens, can I extend my arm or my staff outside, or not?” He replied that whatever the answer—yea or nay—if he were out there performing this experiment, he could not actually be at the limit of the heavens. If he could not extend his arm or staff farther, something beyond the supposed limit had to be stopping him.5

  In archaic-sounding litanies, the first Pythagoreans had asked “What are the isles of the blessed?” and answered “The Sun and the Moon.” Archytas brought this up-to-date in a more sophisticated catechism, asking “What is calm?” and answering as a parent might answer a child, with an example: “What is a man?” “Daddy is a man.” Similarly, Archytas’ reply to “What is calm?” was “Smoothness of the sea.” His catechism, however, implied more than “example answers,” for he liked to connect the specific with the general, reflecting the Pythagorean doctrine of the unity of all being, and he enjoyed thinking about the relationship between the whole and the parts or particulars. His questions and answers about the weather and the sea were particular cases of deeper questions about smoothness and motion. The problem of dividing a whole tone into equal halves was a particular case of a mathematical discovery about ratios that could not be equally divided. His observations about the roundness in trees, plants, and animals were particular manifestations of a “proportion of equality in natural motion.” Archytas was convinced of a tight connection between understanding the universe, or anything else, as a whole and understanding the details. Plato wrote in his Republic that this paragraph from Archytas was “the teaching of the Pythagoreans”: