Greeks II: The Cult of Numbers
On the Pythagoreans, their fear of irrational numbers, and the search for universal music.
The so-called Pythagoreans applied themselves to mathematics,
and were the first to develop this science;
and through studying it they came to believe
that its principles are the principles of everything.
~ Metaphysics, Aristotle
Hippasus, they say, was drowned for revealing the secrets of the Pythagoreans. Exactly which of their secrets he divulged is unknown. Perhaps it was the art of constructing the dodecahedron, a shape they might have regarded as mystical. Or perhaps it was the existence of irrational numbers, a thing the Pythagoreans saw as blasphemous.
Neither is the method of drowning known. Some say the Pythagoreans threw Hippasus overboard in a fit of rage; others say the gods punished him, and caused his ship to sink in a sudden tempest. Whether you choose to believe the story or not, the legend at least tells us the Pythagoreans took their numbers seriously.
And why not? After all, according to those same legends, Pythagoras was a mathematical genius. He divined the secrets of geometry, revealed the hidden ratios of music, and used the perfection of numbers to study the mysteries of space. Even today we hold him in high regard: his theorem on triangles, Johannes Kepler once wrote, is one of the great treasures of geometry.
And that is not all. Both Copernicus and Newton believed they were restoring the long-ignored work of Pythagoras to its rightful place. It was he and his followers, Copernicus wrote, that had been the first to conceive of a moving Earth; and likewise, Newton later said, they had first laid down the essence of the law of gravitation.
But was Pythagoras really the great man these legends speak of? Were the Pythagoreans really tormented by the discovery of irrational numbers, and sufficiently afraid of them to murder a man? And did he really write down the great discoveries of science, only for them to be lost or covered up for two millennia?
I. Pythagoras, Son of a God
There is an odd tale about the origin of Pythagoras. According to Diogenes Laertius and Heraclides of Pontus, the soul of Pythagoras had once resided in the body of Aethalides. He was the son of the god Hermes, and this god offered him the choice of any gift bar immortality. Aethalides thus asked for a perfect memory: one so good that it would persist even after death.
Many years later, after Aethalides died, his soul passed into the body of Euphorbus, who recalled the migrations of that same soul through the underworld and through the mortal beings of plants and animals until it returned to a human form. After, it passed to Hermotimus, and then to a fisherman named Pyrrhus, and finally into Pythagoras, who still remembered all.
In his lifetime, Pythagoras made no secret of his past lives or his belief in reincarnation. We know little for sure of his teachings, but the most certain is that of metempsychosis, or the immortality of souls. According to Pythagoras, souls could not die but could be transformed, and so enter a new body after death. This, he said, had happened to him.
Of the rest, things are very uncertain. Pythagoras may have written books, though if he did they were kept secret and later lost. We have no direct accounts of his life, and the texts that do describe him in detail were written centuries after he died. In one later period the Greeks developed something of an obsession with Pythagoras, and at that time recast him as an almost divine prophet.
That makes it very hard to work out what is real and what is myth. But generally the story runs like this. Pythagoras was born on the island of Samos around 570 BC. He spent some time travelling, and studied under the priests of Egypt and Babylon. Upon his return to Samos he found the island ruled by a tyrant, and so left for Croton in the south of Italy. There he founded a school or cult and attracted a wide following.
But in time that brought trouble, and eventually he was the victim of an uprising. In some accounts his school was burned and he was killed. In others he fled, but the rioters caught up with him and put him to death. His followers, the Pythagoreans, remained in Italy for some time, but eventually they too were driven out and scattered across the Greek world.
II. The Cult of Numbers
It is said Pythagoras was a strikingly charismatic and handsome man, that he dressed all in white, wore trousers – an unusual thing in ancient Greece – and went around with a wreath of gold on his head. Various miracles are attributed to him. One story tells of a time he was bitten by a snake, only for Pythagoras to bite the snake back and kill it.
In another, foreshadowing the later miracles of Jesus, Pythagoras came across a group of fishermen. He made a bet with them: if, he said, he could tell them precisely how many fish were in their nets then they would grant him any wish. They agreed, and then were astonished when Pythagoras gave them the correct sum. His wish was granted: he asked the fishermen to return all the fish to the sea, upon which they miraculously returned to life.
It seems clear, then, that Pythagoras was no scholar, but instead some kind of holy man who founded a cult or semi-religious order. His teachings, as best we know them, focused on a particular way of life. It appears he prescribed certain rituals for funerals and deaths, and considered the eating of animals immoral. Famously he is said to have banned the eating of beans, for he believed they contained souls waiting to be incarnated.
But it is for mathematics and geometry he is best known today. Here the historical record is surprisingly weak. There is no real evidence to say he did any geometry himself, though he may have studied the subject in Egypt and Babylonia and brought the knowledge back to Greece. Certainly his famous theorem was already well known to both those civilizations, and predates Pythagoras by at least a thousand years.
Yet the Pythagoreans did regard numbers as mystical. Where Thales thought the world was ultimately made of water, and Anaximenes of air, the Pythagoreans taught that it was made of numbers. They searched for patterns in those numbers and shapes, and regarded some numbers as special, or perfect, and believed that nature must arrange itself in accordance.
Take, for example, the planets. The Pythagoreans saw ten as one of the most perfect numbers. In the heavens, however, they only counted nine bodies: the Earth, the Sun, and the Moon, the five visible planets, and the celestial sphere holding the stars. Where was the tenth body, they asked?
Philolaus, a student of Pythagoras, found the answer. The Earth, Sun, and Moon, he said, all revolved around an invisible fire that marked the centre of the universe. Behind this fire lay another planet: the Counter-Earth, a world perpetually hidden from view. Philolaus was perhaps the first person to dislodge the Earth from the centre of the cosmos, but the Counter-Earth, sadly, does not exist.
III. Musica Universalis
The Counter-Earth was not the only Pythagorean contribution to cosmology. They also developed the concept of musica universalis, arguing that the planets created music through their movements. No human other than Pythagoras could hear this sound, but it was said to be harmonious.
The Pythagoreans were no doubt inspired by the real relationships between musical notes. These follow strict ratios, and Pythagoras and his followers put much effort into working them out. The perfection of the number ten was ultimately derived from these ratios, and so too, it seems, were many of their other beliefs about the mystical nature of mathematics.
But this belief also led them into trouble. The study of geometry inevitably results in the discovery of irrational numbers; that is, numbers that cannot be expressed as a ratio or a fraction. For the Pythagoreans their existence was troubling, and a rude challenge to their ideas of mathematical perfection.
Whether they really killed Hippasus over this realisation, or punished him for some other reason, it does seem that the existence of irrational numbers posed a threat to their way of life. After all, ultimately, they were not really scientists, open to challenging ideas and shifting conceptions of the world, but more like a cult that believed in the mystical powers of numbers and geometry.
Yet the ideas they developed persisted. Their belief in an inherent perfection in nature influenced Plato. The concept of musica universalis, the harmony of the planets, was taken up centuries later by Kepler and inspired his discovery of the patterns behind the orbits in the solar system.
They have persisted, too, in modern beliefs about the elegance of physics and in the power of equations and algebra to resolve the secrets of nature. The planets, we still say, follow musical resonances in their movements. So too do atoms, electrons, and other fundamental particles. Nature, in some senses of modern theory, is a grand symphony of waves and vibrations performing together in a way that would have delighted the Pythagoreans.
In that, the ideas and philosophy of Pythagoras are about more than just a few rules of geometry. It is a belief in the way numbers underlie nature, and a conviction that patterns in mathematics can help divine the way the natural world works.
There is still a great mystery here, and indeed something mystical about why this works at all. Modern scientists, after all, still sometimes wonder at the unreasonable effectiveness of mathematics. They owe that, ultimately, to Pythagoras.
"Behind this fire lay another planet: the Counter-Earth, a world perpetually hidden from view. Philolaus was perhaps the first person to dislodge the Earth from the centre of the cosmos, but the Counter-Earth, sadly, does not exist."
No doubt. But, as we cannot see through the center of the galaxy, we cannot discount the possibility that there is a Counter-Earth lurking there.