Counting the squares in each line gives 1, 2, 3, 4, 9, 8, 27. The first four of those numbers are the numbers in the tetractus and the Pythagorean musical ratios, 2:1, 3:2, 4:3, but it is an interesting challenge to discern a meaningful pattern in the rest of the numbers, and how they could work in a creation scheme. The answer is that if you square each of the first two numbers 2 and 3 (1 not being a ‘number’), you get 4 and 9. Cube the same two numbers, 2 and 3, and you get 8 and 27. For a Pythagorean it was significant that each pair was an even and an odd number. Plato stopped with the cubes because in the creation of three-dimensional, solid physical reality, only three dimensions are needed.
Next, according to the account Plato put in Timaeus’ mouth, the creator divided his material into smaller parts, filling in harmonic and arithmetical means between those numbers and connecting the ‘world soul’ with a diatonic scale in music.[1] Plato used a scale developed by Philolaus, not the one developed by Archytas.
Astronomy in his Timaeus was also worked out in numbers, with the ‘world soul’ cut into two strips bent around to form an ‘X’ at one point, making an inner and outer ring. Two such rings really exist in astronomy, the celestial equator and the ecliptic. The celestial equator is on the plane of Earth’s equator and anchors the sphere of the fixed stars that do not change their positions in the sky relative to one another and the celestial equator. This was the ring Timaeus called ‘the Same’. It stays the same and never changes. The ecliptic is the circular path that the Sun appears to follow in its daily round, with the planets appearing to orbit in a band centred on it. This ring was Timaeus’s ‘the Different’, for it changes He called the planets ‘instruments by which Time can be measured’.[2] The creator cut the Different into seven narrower strips to accommodate Sun, Moon, and five planets, with the radiuses of their orbits proportional to the numbers 1, 2, 3, 4, 8, 9, and 27. Both rings – Same and Different – were in constant motion, which, Plato thought, nothing but a living soul could be, unless something else pushed it. The rings moved in opposite directions, the Same east to west, the Different west to east, and the seven strips of the Different moved at different speeds, corresponding to the speeds of the Sun, the Moon and the planets.
Plato had Timaeus explain that the movement humans see in the sky is the result of this combination: The daily rotation of the Same with the sphere of fixed stars carries everything around with it, east to west, including Sun, Moon, and planets. But the Sun, Moon, and planets – the seven bodies of the Different – have in addition their own contrary west-to-east motion against that background. They ‘swim upstream’, so to speak, against the current of the Same, at varying speeds, and sometimes back up. This, says Timaeus, is because they are souls, and souls exercise independent choices and power of movement. It is believed to be one of the Pythagorean triumphs, showing up in Philolaus’ fragments, in more detail in Archytas’ work, and then in Plato, to have explained heavenly motion correctly as a combination of opposite movements.
Geometry, Plato had Timaeus explain, had a detailed role in creation when primordial disorder was sorted into four elements – earth, fire, air, and water – and the creator introduced four geometric figures – cube, pyramid or tetrahedron, octahedron, and icosahedron. These ‘Pythagorean’ or ‘Platonic’ solids are four of the five possible solids in which all the edges are the same length and all the faces are the same shape.[3] Each element – earth, fire, air, and water – was made up of tiny pieces in one of those shapes, too small to be visible to the eye.
Plato had Timaeus continue: The four elements and four solids were not the alphabet of the universe. The solids were constructed of something even more basic, two types of right triangles. Plato, through Timaeus, admitted there was room for argument about which triangles were most basic, but he thought he was correct to choose the isosceles triangle and scalene triangle. Both are right triangles.
The isosceles triangle is made by cutting a square into equal halves on the diagonal. Obviously, two isosceles triangles make a square, and squares make up cubes (one of the solids).
In a scalene triangle, the diagonal is twice as long as the shortest side. Two scalene triangles set back to back create an equilateral triangle – none other than the Pythagorean tetractus. The faces of the tetrahedron, octahedron, and icosahedron are equilateral triangles.
Here is Plato’s explanation.
Cube: Fasten together the edges of six squares (each made by pairing two isosceles triangles). The result is a cube, the only regular solid that uses the isosceles triangle or square for its construction.
Pyramid or tetrahedron: Fasten together the edges of four equilateral triangles (each made by pairing two right scalene triangles). The result is a pyramid or tetrahedron.
Octahedron: Fasten together the edges of eight equilateral triangles. The result is an octahedron.
Icosahedron: Fasten together the edges of twenty equilateral triangles. The result is an icosahedron.
The Pythagoreans and Plato knew the dodecahedron, the only regular solid made of pentagons (12 of them), but Plato did not use it in his scheme.
Beyond those five – cube, pyramid, octahedron, icosahedron, and dodecahedron – there are no other regular solids (polyhedrons). Try to fasten together any other number of any regular figure (polygon). You get no fit. No wonder the Pythagoreans, Plato, and later Kepler thought these solids were mysterious.
Timaeus explains to Socrates and the other characters in the dialogue that earth is made up of microscopic cubes, fire of tetrahedrons, air of octahedrons, water of icosahedrons. The pairings were based on how easily movable each solid was, how sharp, how penetrating, and on considerations of what qualities it would give an element to be made up of tiny pieces in this shape.
Timaeus pairs the fifth regular solid, the dodecahedron, with ‘the whole spherical heaven’, and in his Phaedo, Plato associated it with the spherical Earth, in spite of the fact that in his time most of the Greek world, except for the scattered Pythagorean communities, still assumed the Earth was flat. The dodecahedron comes close to actually being a sphere. In fact, the earliest mention of a dodecahedron was in sports, with twelve pentagonal pieces of cloth sewn together and the result inflated to create a ball. Each of the five solids fits into a sphere with each of its points touching the inner surface of the sphere, and a sphere can be fitted into each of the solids so as to touch the centre of each surface, which makes sense of Philolaus’ enigmatic (and controversial) fragment: ‘The bodies in the sphere are five: fire, water, earth, and air, and fifthly the hull of the sphere.’
Though the triangles making up the solids in Plato’s scheme may have been the basic ‘alphabet’ of creation, he thought they were not the fundamentals or archai. In the dialogue Philebus, Socrates says knowledge of the principles of unlimited and limiting is ‘a gift of the gods to human beings, tossed down from the gods by some Prometheus together with the most brilliant fire. And the ancients, our superiors who dwelt nearer to the gods, have passed this word on to us.’3 Plato’s contemporaries and generations of later readers thought that by ‘some Prometheus’, he meant Pythagoras, and that ‘the ancients, our superiors who dwelt nearer to the gods’, were the Pythagoreans, which contributed substantially to the image of Pythagoras as a channel for superhuman knowledge and wisdom. If Plato meant that, he shortchanged Anaximander, who had talked of ‘unlimited’ and ‘limiting’ earlier.
According to Plato, one thing that ‘some Prometheus’ tossed down concerning the unlimited and the limiting was that ‘all things that are said to be are always derived from One and from Many, having Limit and Unlimitedness inherent in their nature.’4 He explained this in unpublished lectures at his Academy that Aristotle reported firsthand.
Plato chose to transform the concepts of unlimited and limiting into something slightly easier to understand: unity and plurality. He called these ‘the One’ (unity) and the ‘Indefinite Dya
d’ (plurality). It is easy enough to grasp what is meant by One, or unity, but the Indefinite Dyad is a more difficult concept. Think of it as more than one, or everything that is not One, or – more vaguely, but closer to what Plato apparently meant – something implying the possibility of numbers or a role for numbers (there would be no role for numbers if everything were One), but not implying that numbers actually exist. The Indefinite Dyad also implied the possibility of opposites – large/small; hot/cold – for if everything were One, opposites would not exist.
To start things off, the One acted in some fashion upon the Indefinite Dyad and the result was a definite number, 2. The One went on acting on the Dyad, generating the numbers up to 10. Once they existed, the numbers 2, 3, and 4 – numbers in the Pythagorean tetractus – predictably played a special role, organising the Dyad to produce geometry. Plato introduced again the progression point–line–surface–solid, connecting the One and the Indefinite Dyad with the world as humans knew it. The meeting of the One and the Indefinite Dyad had been the flashpoint that brought everything else into existence.
On a more mundane level, Plato connected numbers with ideas about an ideal society and ideal rulers. He had probably only recently returned from his first visit to Syracuse when he wrote his Gorgias, his earliest dialogue to deal explicitly with political matters. A character named Callicles in the dialogue lusts for power and luxury, and Socrates admonishes him in words that ring with Pythagorean conviction:
Wise men say that heaven and earth and gods and men all are held together by community, friendship, orderliness, self-control and justice, which is why they call this universe a kosmos (a world order, or universe) – not disorderliness or intemperance. But, I fear, you ignore them, though you are wise yourself, and fail to see what power is wielded among both gods and men by geometrical equality. Hence your defence of selfish aggrandisement. It arises from your neglect of geometry.5
Also in Plato’s Timaeus, before Socrates relinquishes the floor, he reminds his listeners of two decidedly Pythagorean conditions of ‘equality’ needed for an ideal society: Those whose duty is to defend the community, internally and externally, should hold no private property but own all things in common. Women should share in all occupations, in war and in the rest of life. However, sharing in all occupations did not apparently indicate true equality for women, for later in the same dialogue Timaeus says that if a man fails to live a good life he may be relegated to being a woman in the next.
Archytas had introduced Plato to the Pythagorean quadrivium, the curriculum comprising arithmetic, geometry, astronomy, and music. Plato had Socrates declare, ‘I think we may say that, in the same way that our eyes are made for astronomy, so our ears are made for harmony, and that the two are, as the Pythagoreans say, sisters of one another, and we agree.’ That was Plato’s only mention of the Pythagoreans by name, but as Socrates continues he clearly is still talking about them: ‘They gave great attention to these studies, and we should allow ourselves to be taught by them.’
Plato was not, however, entirely in agreement with the Pythagorean approach: Studying the stars and their movements was useful insofar as it got one beyond surface appearances to underlying mathematical principles and laws of motion, but, though the stars and their movements illustrated these realities, they never got them precisely right. A philosopher had to go further than what they could show him and attempt to understand ‘the true realities, which reason and thought can perceive but which are not visible to the eye’.6 Plato was convinced that a new manner of education was needed.
Not long after his first visit to Syracuse, Plato had acquired property near Athens that included an olive grove, a park, and a gymnasium sacred to the legendary hero Academus. In that pleasant setting, he had founded his Academy – the name deriving from the legendary hero. For the rest of his life, except for sojourns abroad, he taught there, lectured, and set problems for his students.[4] His trainees spent ten years (between the ages of twenty and thirty) mastering the quadrivium, but this was only a preliminary step in Plato’s preparation of them to serve as civic leaders who were also philosophers. Education continued in the form of ‘dialectic’. It is not surprising that Dionysius – in the middle of running an impossibly unwieldy tyranny – balked, though this was the training Plato and Dion felt would enable him to rule effectively.
Forms, in Plato.
Plato spoke of two levels of reality:
the divine realm of immutable Forms, which is the model for
the realm in which humans live and where everything is continually changing, ruled by the passions, subject to opinion.
Plato required the dialectic, not merely the quadrivium, because he believed that the world as humans can know it is at best only an imperfect likeness to something else – only a flawed copy of a unique, perfect, eternal model that just ‘is’, ‘always is’, ‘never becomes’, and can never change or be destroyed. In the world perceptible to humans, things resembled this higher realm of the ‘Forms’ and had the same names, but they were not perfect and eternal.[5] They changed – they ‘became’. They began to exist, came to an end, moved about, and were subject to opinions and passions. They were copies or imitations of the Forms; but ‘never fully real’.7 The realm of the Forms could not be perceived by the human senses, but through reasoning and intelligence humans could come nearer to perceiving it. To stretch towards it, Plato thought, you had to use discussion and debate, hence ‘dialectic’. That was what his characters did in his dialogues – Socrates’ question-and-answer lessons – those discussions that never settled anything definitely.
Where did Plato place numbers and mathematics in this picture? Parting company with the Pythagoreans and with Archytas, he thought that although the logic of mathematics and geometry might be part of the universal, immutable truths of the Forms, there was no way humans could find out whether or not they were. Human mathematics was earthbound, deductive reasoning, capable of building only on its own previous knowledge, making the truth of human mathematics only hypothetical, not necessarily Truth with a capital T. In Plato’s house, there was no complete staircase from the human-experience level to the level of the Forms. Numbers and mathematics could take you up a few flights. By using dialectic, argument, thought, and logic, you could go higher, but those flights also fell short of reaching the top. You could never find out whether what was up there on the unreachable level was mathematical or not. The Pythagorean house, by contrast, had a complete staircase made entirely of numbers and mathematics. Humans could climb it and, reaching the top, would discover that what was up there was also mathematical. Pythagoreans were sure they knew that mathematics and numbers were the rationality of the universe and the key to complete understanding and reunion with the divine level of reality.
‘Knowing’ in a context like this was problematic for Plato, for it was not compatible with a universe in which the Forms could never be fully known. The Pythagoreans, however, had had an experience that Plato lacked. The discovery that mathematical logic and pattern underlie nature had apparently come as a shocking, intuitive impression for them. Mathematics and numbers were the rational, unconditional principles of the universe, waiting to be discovered, not deduced from things already known. Had they heard Plato speak about a search for ‘the invisible and incorporeal realm of Form’, one of them might well have raised a hand and insisted they had found it. Their experience was that the vein of Truth (call it Forms, if you are Plato), mostly buried deep beyond the reach of the senses, at some rare points lies close enough to the surface to be perceived, like a vein of gold gleaming through a thin layer of dust and rock. The realm of music was one of those thin places.
Plato’s pupils, and their pupils, continued to ponder the issues he had wrestled with, including the questions about whether the numbers are Forms. Speusippus allowed numbers and ‘mathematicals’ to take the place of the Forms, while Xenocrates said that the Forms were
the numbers. Both thought of themselves, and Plato, as Pythagoreans.
Many scientists and mathematicians today still hold to a Pythagorean faith that truth about the universe is inherently mathematical, and that it is possible to grasp at least bits of that truth by using our human level of mathematics. A few insist that mathematics is the only discipline in which some things are unarguably true and not subject to opinion, while others will not grant it that. Still others redefine ‘complete truth’ as ‘truth that human beings can discover through mathematics’, stretching the Pythagoreans beyond their own meaning and performing an end run around Plato.
The second Pythagorean theme that inspired Plato was the creation and destiny of the soul. He applied the mathematical proportions that went into the creation of the ‘world soul’ also to the human soul, and even described the soul in terms of a version of the Same and Different, reflecting two types of competing judgement – the ability and privilege of a human to say yes or no. For Plato, this free will was the essence of rational thought. But things were not easy for a soul living in a physical body on Earth, the Moon, or one of the planets. At the mercy of all the passions of its body, it inevitably got distorted and stirred up. Proper education could restore it to harmonious equilibrium by reawakening it to its link with the world soul. One way this could happen was through something heard and understood – the musical scale, the proportions of the world soul reproduced in sound.
The Pythagorean belief that a soul could ultimately escape the distorting influences of the world and be reunited with the divine level of the universe fascinated Plato. His ideas about immortality ranged from scepticism in his Apology to mystical speculation in his Gorgias, where Socrates attributes some of his thoughts about the soul to ‘some clever Sicilian or Italian’ – an allusion to the Pythagoreans and probably to the philosopher Empedocles, who was often included under the Pythagorean banner. The dialogue ended with a myth in which souls witness the horrible punishment of incurable sinners in Hades. Plato, in this dialogue, did not argue for a doctrine of reincarnation, but his myth assumed that reincarnation occurred for those witnesses.
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