Plato (428–348 BC)
Plato was a leading Greek philosopher and pupil of Socrates, who founded the Academy (see below) where Aristotle studied. His early dialogues present a portrait of Socrates as a destructive arguer, but in the great middle dialogues he develops his own doctrines such as the Theory of Forms (Republic), the immortality of the soul (Phaedo), knowledge as recollection of the Forms by the soul (Meno), virtue as knowledge (Protagoras), and attacks hedonism and the idea that "might is right" (Gorgias). The Symposium and Phaedrus sublimate love into a beatific vision of the Forms of the Good and the Beautiful. The late dialogues (Sophist, Theaetetus, Politicus, Philebus, Parmenides) are difficult and technical; the Timaeus contains cosmological speculation.
Plato was convinced that ultimate reality lay in ideas and what he called Forms, perfect abstractions of things, which were knowable only through the trained mind. Plato believed that what we see around us are no more than distortions of the truth – twisted reflections of some Platonic ideal. For example, a particular tree, which might have a branch or two missing or a gnarled trunk, have been struck by lightning, or bear a carving with lovers' initials, is merely a flawed embodiment of the ideal Form of a tree from which its existence derives. Outside of space and time, outside of materiality, is the one pure, transcendent Tree that allows us to identify the imperfect reflections of all particular trees around us. Only reason, guided by the proper use of logic, Plato insisted, makes the perception of such ideal Forms possible. In mathematics, Platonism is the belief that mathematical objects exist independent of physical models.
The highest function of the human soul, according to Plato, is to achieve the vision of the Form of the Good. Drawing an analogy between the soul and the state, he presents his famous ideal state ruled by philosophers, who correspond to the rational part of the soul. In the late Laws Plato develops in detail his ideas of the state. His idealist philosophy, his insistence on order and harmony, his moral fervor and asceticism, and his literary genius have made Plato a dominant figure in Western thought.
Plato's beliefs led him to oppose the claims of atomism put forward by Leucippus, Democritus, and others, that there are other inhabited worlds. In the Timaeus, he writes: "There is and ever will be one only-begotten and created heaven." This statement derived partly from Plato's belief that a unique Creator implies a unique creation. His student Aristotle also argued vigorously in favor of a single kosmos and a single seat of life throughout all of space and time.
In 4th century BC Greece, all the major thinkers and seats of learning were congregated in a few key city-states of which Athens was preeminent. Just outside the city walls of the capital lay the Academy, the Harvard of the ancient world. And at the head of the Academy was Plato, foremost thinker of his age.
The Academy supposedly took its name from Hekademos, a mythical Attic hero at the time of the Trojan War who, legend has it, planted twelve olive groves on land he owned about a mile from the center of Athens using shoots from the sacred tree of Athena on the Acropolis. He then bequeathed the place for use as a public gymnasium (an athletic training ground) and shrine to the chief goddess of Greece and other deities. Several hundred years later, in the 6th century BC, Hippias built a wall around the site and put up some statues and temples, while the statesman Kimon went so far as to have the course of the river Cephisus changed so that it would make the dry land of this popular park more fertile. Festivals were held there, as were athletic events in which runners would race between the various altars. Then, in about 387 BC, Plato inherited a house nearby, together with a garden inside the grounds of the park. And here, within this pleasant, leafy retreat, he founded his Academy.
It's often been described as the first university in the West – a fair enough description in the sense that it became a focus of intellectual energy, a place set aside from the workaday world to which keen minds came to learn and discuss lofty ideas across a range of disciplines. There were seminars and informal talks and meetings. Yet the only university-style lectures in the Academy were in mathematics. Above the door, Plato is said to have had inscribed "Let no one who is not a geometer enter." And while that may be myth, given that the first reference to the inscription appears in a document written more than 700 years after Plato died, mathematics unquestionably loomed large in Plato's cosmic master-plan. He was drawn to the subject because of its idealized abstractions, its transcendent purity, the fact that it stood aloof from the material world, somehow above and beyond it. Natural philosophy – science, as we now call it – was anathema to him, an inferior and unworthy sort of knowledge. Mathematics in its most unadulterated form, Plato believed, could have nothing to do with the gross and imperfect goings-on in everyday life. Where it interfaced with reality at all, it must be well outside the flawed human realm – working at the most fundamental level, underpinning the very nature of things, and also, on the grandest of scales, encapsulating the structure of the universe as a whole. In such musings there's more than a whiff of intellectual snobbery: the aristocrat of knowledge, from a privileged family – his father's side claimed descent from the sea god Poseidon – waited upon by slaves, not wanting to deal with the sordid reality of commonplace data. But we can also glimpse an early attempt to devise a "theory of everything," a way of accounting for all the most basic ingredients of nature within a unified mathematical framework.
At the heart of Plato's cosmological scheme lie the simplest and most perfect of three-dimensional geometric shapes, a point he drives home most emphatically in one of his later and best known dialogues, Timaeus. (The bulk of Plato's major writings take the form of contrived two-way conversations, often involving his teacher, the great Socrates.) In Timaeus, Plato talks about the five, and only five, possible regular solids – those with equivalent faces and with all lines and angles, formed by those faces, equal. They are the four-sided tetrahedron, the six-sided hexahedron or cube, the eight-sided octahedron, the twelve-sided dodecahedron, and the twenty-sided icosahedron. Today, we call these shapes the Platonic solids because they first became widely known in medieval Europe through their exposure in Timaeus. But Plato didn't discover them. Almost certainly, he learned of their existence during the ten years or so he spent in Sicily and Southern Italy before setting up the Academy, probably from his close friend Archytas, a senior member of the Pythagorean school of thought (see Pythagoras). In fact, the bulk of Plato's knowledge and philosophy of math was culled directly from the extraordinary Pythagorean sect.
Pythagoras, born in about 570 BC on the Ionian island of Samos, and his followers practiced a weird blend of mysticism and mathematics under the rubric "All is number." They lived by a litany of madcap rules, such as never look in a mirror by lamp light and don't eat beans or put your shoe on the right foot first, and held some eccentric beliefs (Pythagoras himself thought he was semi-divine) as well as a few enlightened ones, including that men and women are equal – something virtually unheard of at the time. Crucially, they were the world's first pure mathematicians. And like a good many pure mathematicians and theoretical physicists today, they started from the premise that thought is a surer guide than the senses, that intuition ranks above observation. From the Pythagoreans, Plato inherited his most unshakable conviction – that behind the world we see lies a more fundamental, eternal realm accessible only via the intellect. From them too he gained knowledge of the five regular solids. And although "Platonic solids" is a misnomer, Plato was genuinely original in how he interpreted the significance of these shapes. He linked them with the classical elements of earth, water, air, and fire, and, in so doing, formed a bridge between the mathematical and the material. In Timaeus he writes:
To earth, then, let us assign the cubic form, for earth is the most immovable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature.
Noting that the tetrahedron has the smallest volume for its surface area and the icosahedron the largest, Plato saw in them the properties of dryness and wetness respectively, and hence a correspondence with the elements fire and water. The octahedron, which rotates freely when held by two opposite corners, he regarded as a natural partner for air, the most mobile of the elemental quartet.
But there are five regular solids. To Plato, utterly convinced of the truth of his geometric worldview and of the unassailable power of intuition, this discrepancy between theory and observation could mean only one thing: there must be another element in addition to the four already known. There must be, in other words, a quinta essentia or quintessence, a "fifth essence," not familiar on Earth. Surely, he reasoned, this quintessence was the stuff of the heavens and its form the remaining regular solid – the dodecahedron. In support of his claim he noted that there are 12 sides on the dodecahedron and 12 signs of the zodiac – the constellations that the Sun passes through in the course of a year. "God used this solid for the whole universe," he declared, "embroidering figures on it."
An element of truth
A twelve-sided cosmos? Dreamed up long before humanity fathomed the true nature of stars and galaxies and the immensity of space and time? It seems, on the face of it, just another quaint idea, surely long overtaken by events. But in Oct 2003, Jean-Pierre Luminet and his colleagues at the Paris Observatory published a paper in the journal Nature1 arguing, on the basis of data collected by the orbiting Wilkinson Microwave Anisotropy Probe (WMAP), that the universe does indeed take the shape of a dodecahedron.
WMAP, launched in 2001, is designed to survey very precisely the so-called cosmic microwave background, the much-cooled glow of the vast explosion in which the universe began. The wavelength of this radiation is remarkably pure, but like a musical note it has harmonics associated with it. These harmonics reflect the shape of the object in which the waves were produced. In the case of a note, that object is the musical instrument upon which the note is played. In the case of the microwave background, the object is the universe itself. WMAP's measurements revealed that the second and third harmonics of the microwave radiation – the quadrupole and octupole – are weaker than expected. This weakness can be explained, according to the French team, if the universe is assumed to be finite and dodecahedron-shaped. Unfortunately their model doesn't involve anything quite so simple as a giant Platonic solid, because an ordinary dodecahedron has a definite inside and an outside and exists in "flat" space – the kind of space we're familiar with in everyday life and to which Euclid's geometry applies. What Luminet and his coworkers proposed is something called dodecahedral space, first described by their fellow countryman Henri Poincaré in the 19th century. Also known as a Poincaré manifold, this is a weird type of mathematical space that doesn't lend itself to being easily visualized. But the key point is it has the same kind of symmetry as the dodecahedral cosmos that Plato had in mind.
Plato may have struck lucky on another point, too. It's easy to look at the classical elements – earth, water, air and fire – and conclude that they've very little in common with the elements known to modern science: hydrogen, helium, carbon, iron and the rest. But that's not really a fair comparison. The classical elements don't seem much like the elements of today's periodic table, it's true; however, they do correspond very closely with what we now call the states of matter. Read earth for solid, water for liquid, air for gas, and fire for plasma (an ionized gas, often described as the fourth state of matter) and the ancients no longer seem so far off the mark. That leaves Plato's quintessence without a modern-day partner. Nothing in 20th-century science seemed to correspond to this esoteric, celestial stuff. But then, without any warning, along came dark matter. At least four-fifths of all the mass in the universe, it turns out, consists of this invisible ingredient whose nature remains a subject of intense debate. Even more recently, astronomers have found evidence for another mysterious cosmic component quite unlike anything ever seen on Earth. This is dark energy. Both dark matter and dark energy have, for the purpose of various theories, been tagged "quintessence" by modern physicists mindful of Plato's seminal ideas.
Allegory of the cave
The allegory of the cave appears in Plato's Socratic dialogue, The Republic (c. 360 BC). It begins with an underground cave that is inhabited by prisoners who have been chained there since childhood. The prisoners can only look toward the back of the wall, where flickering shadows stimulate their imaginations and cause them to think that all they imagine is real. However, if a prisoner were to get free and see the cause of the shadows – figures walking in the vicinity of a flickering fire – he would begin to reassess what he thought was real. Moreover, if this prisoner were to escape from the cave, he would then be able to see the sun itself, which illuminates everything in the world in the most real way. However, if this free man were to return to the cave to explain his findings to the other prisoners, he no longer would be accustomed to the darkness that they share, and to those ignorant people he would sound like a fool or worse.
This allegory has been highly influential in the history of philosophy for its succinct depiction of Plato's epistemological, ethical, metaphysical, and educational thought. The cave represents our world; we humans are prisoners who imagine such things as sex, power, and money to be the overpowering real and important things of life when , in fact, they are shadows of greater goods that we have the capacity to know. The fire is the inspiration that helps us ascend until we finally come face to face with reality. The liberated prisoner's decent back into the cave represents the ethical duty of the philosopher, who having discovered the truth, tries to help others seek enlightenment.
1. Luminet, J.-P., J. R. Weeks, A. Riazuelo, R. Lehoucq, and J.-P. Uzan. "Dodecahedral Space Topology as an Explanation for Weak Wide-Angle Temperature Correlations in the Cosmic Microwave Background." Nature 425 (2003): 593-95.