The Music of Pythagoras (15 page)

A
WORSE POSSIBILITY
for Pythagoras’ image is that he took the theorem from the Babylonians and claimed it as his own. According to Heraclitus, he “practiced inquiry more than any other man, and selecting from these writings he made a wisdom of his own—much learning, mere fraudulence.”
*
It would certainly not have surprised Heraclitus if Pythagoras had stolen the Pythagorean theorem lock, stock, and barrel from the Babylonians. However, the fragments in which Heraclitus dismissed him as an imposter also placed Pythagoras high in the echelon of thinkers. Two of Heraclitus’ other targets, Xenophanes and Hecataeus, were renowned polymaths. “Inquiry” meant not study in general but Milesian science. Most scholars think that Heraclitus had no basis for his attacks. He had an aversion to polymaths, and he was simply an ornery and contentious man being ornery and contentious. On another occasion he commented that “Homer should be turned out and whipped!”

If the Pythagoreans did come up with the theorem independently, the question remains whether credit should go to Pythagoras and his contemporaries or to later generations of Pythagoreans. Intemperate Heraclitus would not have been pleased to know that evidence coming from his own work places the appearance of the Pythagorean mathematical achievements in Pythagoras’ lifetime: Heraclitus followed up on Pythagorean ideas about the soul and immortality and continued to develop the idea of harmony. For him, the lyre and the bow—Apollo’s musical instrument and weapon—symbolized the order of nature. The bow was “strife,” the lyre
harmonia
. The significance of the bow
(“strife”) was original with Heraclitus, but the role of the lyre and
harmonia
were developments from Pythagorean thought, which suggests that the idea of connections between numerical proportions, musical consonances, and the Pythagorean numerical arrangement of the cosmos dated from the time of Pythagoras himself. Heraclitus was only one generation younger than Pythagoras.

In the first century
B.C
., the theorem seems to have been widely attributed to Pythagoras. A case in point: The great Roman architect Marcus Vitruvius Pollio, better known as Vitruvius, knew it well, attributed it without question to Pythagoras, and, in Book 9 of his ten-volume
De architectura
, mentioned the sacrifice to celebrate it. Apparently Vitruvius could write about Pythagoras as the discoverer of the theorem and assume that no one would gainsay him. He knew other methods of forming a right triangle, but found Pythagoras’ much the easiest:

Pythagoras demonstrated the method of forming a right triangle without the aid of the instruments of artificers: and that which they scarcely, even with great trouble, exactly obtain, may be performed by his rules with great facility.

Let three rods be procured, one three feet, one four feet, and the other five feet long; and let them be so joined as to touch each other at their extremities; they will then form a triangle, one of whose angles will be a right angle. For if, on the length of each of the rods, squares be described, that whose length is three feet will have an area of nine feet; that of four, of sixteen feet; and that of five, of twenty-five feet: so that the number of feet contained in the two areas of the square of three and four feet added together, are equal to those contained in the square, whose side is five feet.
¹⁷

W
HERE, THEN, DOES
this discussion end? In spite of the certainty that Vitruvius and his contemporaries shared, the most skeptical modern scholars think Pythagoras had nothing to do with the theorem at all. Others do not close the door to the possibilities that Pythagoras and/or his early followers may have made the discovery independently, unaware of previous knowledge of the theorem, or that they learned it elsewhere but were the first to introduce it to the Greeks.

My own conclusion is that there is no good reason to decide that
Pythagoras and the Pythagoreans had nothing to do with the theorem, and several meaningful hints that they did, including the fact that Plato chose to assume that right triangles were the basic building blocks of the universe when he wrote his
Timaeus
, the dialogue most influenced by Pythagorean thinking.
*
If earlier knowledge of the theorem had indeed been lost, then someone had rediscovered it at about the time of Pythagoras. Of all those who were aware of right angles and triangles and used them in practical and artistic ways, the Pythagoreans were unique in their approach to the world, apparently having the motivation and leisure to give top priority to ideas and study. Their intellectual elitism kept them focused beyond the nitty-gritty of “what works” on the artisan level, and their musical discovery led them to think beyond number problem solving for its own sake—causing them to turn their eyes beneath the surface and view nature in an iconic way. For the Pythagoreans (as for no others among their contemporaries), the theorem would have represented an example of the wondrous underlying number structure of the universe, reinforcing their view of nature and numbers and the unity of all being, as well as the conviction that their inquiry was worthwhile, and that their secretive elitism was something to be treasured and maintained. Has any other ruling class—and the Pythagoreans seem also to have been busy ruling—had that same set of priorities? Regarding the possibility that they began with the triple, I like the fact that their having it would not imply a continuum with the Old Babylonian mathematical tradition—a continuum that scholars like Robson have convincingly argued did not exist. And in this scenario, the Babylonian evidence, instead of pulling Pythagoras off the pedestal, actually suggests a way that he and his followers could have rediscovered the theorem in the time and place that tradition has always said they did without being disillusioned too soon by the discovery of incommensurability.

Bronowski credited Pythagoras with discovering the link between the geometry of the right triangle and the truth of primordial human experience. He echoed Plato’s reverence for right triangles as the basics of creation when he wrote, “What Pythagoras established is a fundamental characterization of the space in which we move. It was the first time that was
translated
into numbers. And the exact fit of the numbers
describes the exact laws that bind the universe.” Bronowski for that reason thought it not extravagant to call the “theorem of Pythagoras” “the most important single theorem in the whole of mathematics.”
¹⁸

But what of the ox? Did Pythagoras sacrifice it, or perhaps forty of them (some stories say), in thanksgiving for the discovery of the theorem? That Apollodorus referred to this “famous” story does not necessarily mean he believed it. Many dismiss the tale as impossible on the grounds that Pythagoras, who ate no meat, would not have sacrificed an ox. However, there is plenty of evidence that he had no objection to the slaughter of animals for ritual purposes. If vegetarianism is a clue, it may point in a different direction: Later Pythagoreans were more ready than early ones to believe that Pythagoras was a strict vegetarian. Burkert thought the existence of the sacrifice story “ought rather to be considered an indication of antiquity,” weighing in on behalf of the argument that Pythagoras or his earliest followers made the discovery that spawned the tale. A later generation would not have made up this story about their hero.

Fifth Century
B.C
.

A
FTER
P
YTHAGORAS
’
DEATH
, the demise of the Pythagorean brotherhood in southern Italy did not take place overnight or in a few short years. Many Pythagoreans survived the violence that ushered in the fifth century
B.C
., and the Pythagorean drama, minus its lead character, continued in the colonial cities. Only one Pythagorean is known for certain by name from that period: Hippasus of Metapontum. He was a scholar and perhaps a brilliant one, apparently part of the Pythagorean inner circle, who worked in music theory, mathematics, and natural philosophy and considered fire a first principle. One report credits Hippasus with constructing the dodecahedron, the twelve-sided solid—he “first drew the sphere constructed out of twelve pentagons.” He may have taught the cantankerous Heraclitus. However, after Pythagoras’ death, Hippasus fell from grace, and he is chiefly remembered as an ill-fated, perhaps ill-intentioned figure.

When the Pythagoreans discovered that mathematical relationships underlie nature, they did not announce this to the world. Secrecy was their custom. Hippasus, however, had to have been privy to the discovery, because he performed the successful experiment with bronze disks. Some stories connected him with the discovery of incommensurability, or even made him the unlucky discoverer. Accounts differed about how
he erred, but somehow, while all good things were attributed to Pythagoras, all bad things seemed to get hung on Hippasus. His transgression was revealing a secret of geometry, or discovering incommensurability, or effrontery to the gods by making a discovery in geometry (that could have been the dodecahedron), or taking credit for a discovery instead of attributing it to Pythagoras.

The most nuanced and authentic-sounding material about Hippasus comes from Aristotle and Aristoxenus, and it links Hippasus with a fault line that developed in the Pythagorean brotherhood between two factions calling themselves the
acusmatici
and the
mathematici
.
¹
The antagonism, which may have had its roots in a two-level hierarchy initiated by Pythagoras, the better to organize his brotherhood according to interests and abilities, split the community into opposing camps.

The
acusmatici
were devoted to rote learning. Their philosophy (quoting Iamblichus) “consisted of unproven and unargued aphorisms, and they attempted to preserve the things Pythagoras said as though they were divine doctrines.” Scholars surmise that these aphorisms were relics of the most elementary, easily remembered part of Pythagoras’ teaching. Some were folk maxims with added interpretations, with knowledge of the interpretations possibly serving as passwords or signifying rank in the community. There were three kinds of aphorisms (this according to Iamblichus): Some asked what something was: “What are the Isles of the Blessed? The sun and the moon.” A second kind indicated superlatives: “What is most wise? Number.” “What is most truly said? That men are wretched.” A third concerned minutiae about “what one must do or not do.” Many sounded pointless to anyone unaware of the secret interpretations. “Do not turn aside into a temple” meant “Do not treat God as a digression.” “Do not help anyone put down a burden; rather, help him take it up” meant “Do not encourage idleness.” “Do not break a loaf of bread” because “it is disadvantageous with regard to the judgment in Hades.” Iamblichus threw up his hands at that and called it “far-fetched.”
Acusmatici
evidently understood the connection. They claimed the title “Pythagorean” for themselves exclusively.

The
mathematici
, on the other hand, were willing to admit the
acusmatici
under the banner of the brotherhood, but they preserved and extended a different kind of Pythagorean knowledge. Though not always agreeing among themselves, they shared a conviction that the
acusmatici
’s refusal to allow knowledge to develop further was contrary to
the spirit in which Pythagoreanism had been practiced when Pythagoras was alive.

According to Aristotle and Aristoxenus, Hippasus was one of the
mathematici
, or one of those who would be labeled
mathematici
when the groups became fully polarized. The opposing camp—those who would be known as
acusmatici
—frowned on his work as new and subversive. The
mathematici
might have been expected to defend Hippasus, but they were engaged in delicate maneuvers, insisting they were not introducing new doctrines, merely working on explication of the doctrines of Pythagoras. They disassociated themselves from Hippasus, to no avail since the
acusmatici
continued to accuse them of following him rather than Pythagoras. Hippasus was caught in the crossfire. His punishment, from the gods or the Pythagoreans, depending on which story to believe, was drowning at sea, expulsion from the community, and/or the construction of a tomb to him as though he were dead.

Hippasus’ story provides a clue for dating some of the Pythagorean discoveries. Modern historians are skeptical about the claim that Hippasus taught Heraclitus, but they think the fact that many believed he did dates Hippasus reliably. He was supposed to have taught Heraclitus, not the other way around, and since Heraclitus’ lifetime overlapped Pythagoras’, Hippasus must have been an even earlier contemporary of Pythagoras. Furthermore, Hippasus’ disk experiment had to have happened after the discovery of the ratios and before Hippasus’ disgrace, which occurred (dated by the split in the brotherhood) shortly after Pythagoras’ death. This chronology makes it impossible for the discovery of the musical ratios to have been made later, in the next generation. They were an authentically early Pythagorean discovery.

Hippasus’ disgrace and the
mathematici/acusmatici
conflict are not the only evidence that some Pythagoreans who survived the turn-of-thecentury upheavals stayed in Magna Graecia. Remnants of the brotherhood persisted throughout the region. Iamblichus had information that Pythagoras’ “successor” was Aristaeus, who married his widow Theano, “carried on the school,” and educated Pythagoras’ children. A son of Pythagoras named Mnesarchus reputedly took over the school when Aristaeus became too old. If folk memory in Metapontum had it right, Pythagoras survived for a while in exile and established a school there.