Pythagorus (18 page)

Diagram showing how Archytas solved the Delian problem, evidence of how advanced Pythagorean mathematics and geometry had become in little more than one century.
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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.
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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.
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This had dramatic implications, for it was an example of something obviously present 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.
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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.
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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.
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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.'
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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.
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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.
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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.'
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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.