Pythagorus

by Kitty Ferguson

  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 regarded such ideas as the following, from Archytas, as the teaching of the Pythagoreans:

  Those concerned with the sciences seem to me to make distinctions well, and it is not at all surprising that they have correct understanding of individual things as they are. For having made good distinctions concerning the nature of the whole, they were likely also to see well how things are in their parts. Indeed, concerning the speed of the stars and their risings and settings, as well as concerning geometry and numbers, and not least concerning music, they have handed down to us a clear set of distinctions. For these sciences seem to be akin.6

  When Archytas wrote about such matters as smoothness and nonsmoothness of the sea he was reflecting another Pythagorean traditional favourite – opposites (smoothness/lack of smoothness; motion/lack of motion) – and for him that line of thought inevitably led back to thinking about infinity. Can something be infinitely calm? Or infinitely uncalm? Or infinitely smooth; infinitely rough?

  As a politician and general, Archytas was convinced of what he was sure his Pythagorean forebears had demonstrated: The unity of all things had to include ethics and politics. The value of mathematics extended to the political arena. In the following fragment, ‘reason’ could also be translated as ‘calculation’. To a Pythagorean like Archytas, the two meanings were probably synonymous.

  When reason/calculation is discovered, it puts an end to civil strife and reinforces concord. Where this is present, greed disappears and is replaced by fairness. It is by reason/calculation that we are able to come to terms in dealings with one another. By this means do the poor receive from the affluent and the rich give to the needy, both parties convinced that by this they have what is fair.

  Plato, of course, could not have agreed more. The ability to use ‘reason’ or ‘calculation’ would make a philosopher king a superbly able ruler.

  For Archytas, the concept of unity meant he should also apply a Pythagorean search for deeper levels of mathematical understanding to optics, physical acoustics, and mechanics. His is the earliest surviving explanation of sound by ‘impact’, with stronger impacts giving higher pitches, but he nodded to his Pythagorean forebears by insisting this was a theory that had been handed down to him. By ‘impact’, Archytas meant impact on the air – whipping a stick through the air, playing a high note on a pipe by making the pipe as short as possible (making a stronger pressure on the air, he thought), and the sound of the wind whistling higher pitches as its speed increased, or a ‘bull-roarer’. That last was an instrument used in the mystery religions, a flat piece of wood on the end of a rope. Whirling it around in the air like a giant slingshot produced a fearsome howling sound; the faster the whirling the higher the pitch.7

  One of the most widely known, influential, and enduring Pythagorean ideas passed down through Archytas to Plato was the concept of the ‘music of the spheres’, the music Archytas and his Pythagorean forebears thought the planets made as they rushed through the heavens. Here is Archytas’ explanation for why humans never hear it:

  Many sounds cannot be recognised by our nature, some because of the weakness of the blow (impact), some because of the great distance from us, and some because their magnitude exceeds what can fit into our hearing, as when one pours too much into narrow mouthed vessels and nothing goes in.

  According to Pythagorean tradition, only Pythagoras could hear this music.

  Archytas was a generous man, kind to slaves and children. He invented toys and gadgets, including a wooden bird (a duck or a dove) that could fly. Aristotle was impressed by ‘Archytas’ rattle’, ‘which they give to children so that by using it they may refrain from breaking things about the house; for young things cannot keep still.’8

  This, then, was the science, mathematics, music theory, and political philosophy that Plato, from Archytas, learned to think of as Pythagorean. Through Plato, much of the image of Pythagoras and Pythagorean thought in Western civilisation is traceable to Archytas’ window on Pythagoras.

  How unclouded was this window? Archytas regarded himself as an authentic Pythagorean, true to the earliest traditions and teachings. In his era, oral accounts could still be accurate, especially in a continuing community that considered it vitally important to keep an ancient memory alive and clear. In many ways, Archytas was probably a good reflection of what it had meant to be Pythagorean when Pythagoras himself walked the paths of Megale Hellas. However, he was one of the mathematici, the school of Pythagoreanism that believed following in Pythagoras’ footsteps meant diligently seeking and increasing knowledge. The Pythagorean ideals that underlay Archytas’ thought and work led him to newer discoveries. He was among the great scholars and mathematicians of his era, by reputation the teacher of the mathematician Eudoxus. If Archytas had focused only on the knowledge of the first Pythagoreans, this would have been impossible.

  Plato himself provided a window through which we view Archytas. No matter how accurately Archytas reflected Pythagoras and Pythagorean thinking, we see him through Plato’s eyes and with Plato’s mind, the eyes and mind of one of the most creative thinkers in all history. It is in the nature of such a man, if he is impressed with an idea, to take the ball and run with it – to say, ‘This is, of course, what you mean’, and restate someone else’s good idea with a spin that makes it absolutely brilliant – and his own, not the original. Assuming Archytas was an exemplary Pythagorean, when Plato got the ball to the other end of the field, was it anything like the same ball he had caught in the pass from Archytas? That is one of the most debated questions in all the long history of those who have yearned to know what Pythagoras himself, and Pythagoreans before Plato, really discovered and thought.

  On one significant issue, Plato disagreed with Archytas, and that disagreement is a welcome clue, a clear indication of something in pre-Platonic Pythagorean thinking, undiluted by Plato, that differed from Plato. Archytas, Plato complained, was too concerned with what one could see and hear and touch, and with searching for mathematics and numbers to explain it. For Plato, the goal of studying mathematics was to turn away from experience that humans have through their five senses to a search for abstract ‘form’, out of reach of sensory perception. Numbers and mathematical understanding were a venture into abstract form, but not the same, he thought, as his own concept of the ultimate understanding of ‘the beautiful and the good’. This difference, in Plato’s view, made Archytas an inadequate philosopher and himself a better one.

  Plato’s knowledge about Pythagoras and Pythagoreans was not confined to what he learned through Archytas. There is evidence in his dialogues that he heard about them from Socrates; also, Plato and Archytas both knew of Philolaus. If the characters in Plato’s dialogue Phaedo are not entirely fictional, he was acquainted with contemporaries who were ‘disciples
of Philolaus and Eurytus’ in Phlius, a community west of Corinth, as well as with Echecrates, who speaks for them in the dialogue, and Simmias and Cebes. Plato also knew about Lysis of Tarentum who, like Philolaus, had emigrated to Thebes. The Pythagorean community there was apparently still in existence in Plato’s time.

  Plato could not have avoided also knowing about acusmatici Pythagoreans who did not agree that scholars like Archytas were Pythagoreans.9 The Greek public in the fourth century B.C. generally failed to recognise a distinction between mathematici and acusmatici and lumped all ‘Pythagorists’ together as an eccentric lot. Athenian comic dramatists lampooned them as unwashed, secretive, arrogant characters who abstained from meat and wine and went about ragged and barefoot. Plato’s pupils, educated at his Academy in the ‘Pythagorean sister sciences’ – the quadrivium of arithmetic, geometry, astronomy, and music – spoke of their philosophy and that of Pythagoras as one and the same and featured him in their books. They were certainly more in the mathematici tradition than in the acusmatici, but they nevertheless were the targets of the same jibes.

  The unshakable conviction of the men who inspired the caricature – that they were following in the authentic footsteps of Pythagoras and preserving a precious tradition – caused some of their contemporaries to feel, a bit uncomfortably, that even the most eccentric were favoured by the gods and privy to mystical secrets. Antiphanes, in his play Tarentini (the title connects it with Tarentum) spoke of Pythagoras himself as ‘thrice blessed’, and Aristophon had one of his characters report:

  He said that he had gone down to visit those below in their daily life, and he had seen all of them and that the Pythagoreans had far the best lot among the dead. For Pluto dined with them alone, because of their piety.

  Lest anyone conclude that Aristophon approved of Pythagoreans, another character commented that Pluto had to be a very easygoing god, to dine with such filthy riffraff.10

  Diodorus of Aspendus, who was not fictional, was described as a vegetarian with long hair, a beard, and a ‘crazy garment of skins’ who with ‘arrogant presumption’ drew followers about him, although ‘Pythagoreans before him wore shining bright clothes, bathed and anointed themselves, and had their hair cut according to the fashion.’11

  Aristoxenus – who interviewed the tyrant Dionysius in his Corinthian ‘retirement’ – would have none of this. He was effectively a propagandist for the mathematici, taking pleasure in contradicting the acusmatici by insisting that Pythagoras ate meat and that the aphorisms were ridiculous, and he tried to disassociate ‘true Pythagoreans’ from what he saw as this unsavoury, superstitious group who were giving the movement a bad name. He listed the pupils of Philolaus and Eurytus and called them ‘the last of the Pythagoreans’ who ‘held to their original way of life, and their science, until, not ignobly, they died out’. Because these men died a few decades before the comic allusions in the plays, dubbing them ‘the last of the Pythagoreans’ was making the point that the butts of the jokes were only pretending to be Pythagoreans.

  Aristoxenus’ public relations efforts did not succeed well. Through the fourth century B.C., the popular image of Pythagoreans continued to resemble the acusmatici more than the mathematici. But after the fourth century, the acusmatici, with a few exceptions, dwindled and vanished from notice. Had there been no other Pythagorean tradition than theirs, and if they did indeed represent the truer image of Pythagoras and his earliest followers, it would be almost impossible to explain how such an odd cult figure, not far different from others in antiquity, became so dramatically and rapidly transformed in the minds of intelligent men and women as to inspire deep and effective scientific thinking and seize the imagination of centuries of people to come.

  Was it all due to Plato? Did he get so excited about something that was mainly legend that he elaborated on it himself until he had made it hugely significant? At a minimum there had to have been the discovery in music of pattern and rationality underlying nature, and the accessibility of that rationality through numbers – and that was of no small significance. It seems much more reasonable to conclude that Pythagoras, responding to different types of interest and intelligence among his followers, encouraged both kinds of thinking – acusmatici for those who needed something naive and more regimented and conservative, and mathematici for those with minds eager to grasp difficult, nuanced concepts and explore their implications. He was personally, perhaps, not entirely unlike either group.

  However that may be, from the time of Plato, what survived as ‘Pythagorean’ and ‘Pythagoras’ was largely mathematici, and that included the conviction that right from the time of Pythagoras himself, and attributable to him, there had been a truly remarkable new approach to numbers, mathematics, philosophy, and nature.

  [1]Tarentum was the only colony established by Sparta, and Plato greatly admired the Spartan system of government. However, the people who had colonised Tarentum in 706 B.C. had come there under unusual circumstances and might not have shared Plato’s enthusiasm for Sparta. They were sons of officially arranged marriages uniting Spartan women with men who were not previously citizens. The purpose was to increase the number of male citizens who could fight in the Messenian wars. When the husbands were no longer needed as warriors, the marriages were nullified and the offspring forced to leave Sparta.

  [2]For an example of the use of movement in geometry, take a straight line, fasten down one end of it, and swing the other end about. The result is an arc. Take a right triangle and stand it upright with one of the sides serving as its base; swivel it around the upright leg and the result is a cone. (The ancient scholar Eudemus used this explanation in his description of Archytas’ solution.)

  [3]A lengthy text is needed to understand it and is available in S. Cuomo, Ancient Mathematics, Routledge, 2001, pp. 58 and 59, and on the Internet at http://mathforum.org/dr.math/faq/davies/cu/bedbl.htm

  [4]More generally, ratios such as 5:4, or 9:8, in which the larger number is one unit larger than the smaller (mathematicians call these superparticular or epimeric ratios), cannot be divided into two equal parts.

  CHAPTER 9

  ‘The ancients, our superiors who dwelt nearer to the gods, have passed this word on to us’

  Fourth Century B.C.

  Socrates was not Plato’s fictional creation. Born about thirty years after the death of Pythagoras, near the time Philolaus was born, he fought in the Peloponnesian Wars and then lived a life of intentional poverty as a teacher in Athens. He wrote nothing, and information about what he taught comes only through Plato’s dialogues and similar conversations recorded by Xenophon, another of Socrates’ pupils. Socrates’ teaching method consisted of asking questions. Plato’s dialogues are not word-for-word accounts of real question-and-answer lessons, but are almost certainly faithful to the philosophy as it emerged in conversations like these. When Socrates was seventy, he was accused of impiety and corrupting the youth of Athens. A jury of his fellow citizens sentenced him to death, probably through a dose of hemlock. He died surrounded by his friends and pupils.

  In the dialogues Plato wrote, the character Socrates usually directed the discussions, but in Plato’s Timaeus he relinquished centre stage for many pages to a fictional character named Timaeus of Locri. Timaeus was supposed to be a statesman and scientist from southern Italy, and the ideas Plato put in his mouth were heavily indebted to Archytas, with whom he had apparently spent long hours in Tarentum deep in conversation and bent over mathematical diagrams. Perhaps what Timaeus tells Socrates and his friends in the dialogue is close to what Archytas laid out for Plato. Or maybe Archytas’ ideas were only a springboard for Plato. Though many opinions have been expressed about those possibilities, no one knows for certain, and the truth likely lies somewhere in between.

  Plato carried forward two great Pythagorean themes: (1) the underlying mathematical structure of the world and the p
ower of mathematics for unlocking its secrets; and (2) the soul’s immortality.1 The stage is set for discussion of the first when Socrates asks Timaeus, an expert on such matters, to ‘tell the story of the universe till the creation of man’.2 Timaeus’ response to this daunting request is a number-haunted, Pythagorean creation story: The mathematical order of the universe was the work of a creative god, whom Plato called the demiurge – not the chief god or the only god, but a figure loosely comparable to Ptah, the Egyptian god at Memphis, or to Jesus acting in the role of the logos in the opening of the Gospel of John – ‘through him all things were created’. This craftsman god, says Timaeus, decided that the universe should be a ‘living being’, spherical and moving in ‘a uniform circular motion on the same spot; unique and alone’. Timeaus sets forth a numerical construction of the ‘world soul’:

  First the creator god took his material and ‘marked off a section of the whole’.

  Then he marked off another section ‘twice the size of the first’.

  Next he marked off a third section, ‘half again the size of the second section and three times the size of the first’.

  Next he marked off a fourth section, ‘twice the size of the second’.

  Next, a section ‘three times the third’.

  Next, a section ‘eight times the first’.

  Last, a section ‘twenty-seven times the first’.