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.* 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 in-flated 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 center 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 short-changed 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 Dyad” (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, organizing 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 Pythagore
an 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.* 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.
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.* 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 toward 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.
Forms, in Plato.
Plato spoke of two levels of reality:
(1) the divine realm of immutable Forms, which is the model for
(2) the realm in which humans live and where everything is continually changing, ruled by the passions, subject to opinion.
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 judgment—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 skepticism 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.
Plato probably wrote his Meno after his first visit with Archytas. In it, Socrates speaks of “wise priests and priestesses” whose authority is reliable, who teach about immortality and reincarnation—a bow to the Pythagoreans. Plato had Socrates attribute to those “priests and priestesses” the idea that because of what we have experienced earlier, much of what we know in this present life is “recollection.” This does not seem out of line with what Pythagoras claimed for himself, but Plato had something different in mind that he and his pupils thought was compatible with Pythagorean teaching.
As Plato interpreted the Pythagorean concept of the transmigrating soul, the possibility of escape from the parade of reincarnations lay in “becoming just and pious with wisdom,” freeing the soul from fear and the passions and pains of the body. The highest goal was “becoming like god,” as Plato phrased it in his Theaetetus. Several centuries later, the pagan neo-Platonist Porphyry (Pythagoras’ biographer) listed Hercules, Pythagoras, and Jesus among those who had succeeded in this ultimate achievement o
f “becoming like god.”
“Recollection,” for Plato, did not however mean memories of past lives. Instead, it was the mysterious, innate, a priori knowledge that humans seem to possess, that cannot be explained by what one has learned in one’s present life. Plato did not imply that anyone could recall acquiring this knowledge. As a demonstration, he used the geometric exercise that some believe reflected Pythagoras’ proof of the Pythagorean theorem.
In this scene from Plato’s Meno, Socrates and Meno are discussing a figure Socrates has drawn in the sand, a four-foot square. The task is to double the size of the square. Socrates intends to demonstrate that innate knowledge—not of the correct answer but of the underlying geometry that will lead to the correct answer—lies hidden in Meno’s slave boy, waiting to be reawakened. Socrates is acting as a sort of midwife.
SOCRATES (to Meno): Now notice what he will discover by seeking the truth in company with me,
though I simply ask him questions without teaching him. Be ready to catch me if I give him any instruction or explanation instead of simply interrogating him on his own opinions. (Socrates rubs out previous figures in the sand and starts again with a four-foot square.)
Tell me, boy, is not this our square of four feet? You understand?
The Music of Pythagoras Page 15
