by Adam Frank
It was Thales’ students who carried the revolution forward, articulating foundational ideas of natural philosophy that, in some cases, would wait two millennia to be further articulated by modern science. Thales’ pupil Anaximander proposed the world was built of “intermingled opposites: hot and cold, dry and wet, light and dark”.36 The tension between these opposites produced a dynamic, evolving world. In Anaximander’s account, all animals and humans evolved from lesser ocean creatures—a prototype of Charles Darwin’s vision.
What made Thales and those who followed him so utterly different than what came before was their assertion that the universe was comprehensible to the human mind. Nature was rationally constructed; there was no need to turn back to the supernatural for explanation, counsel or divination. The Greek philosophers picked up on Thales’ approach and added new tools by describing nature with an expanding language of mathematics instead of the language of priestly divination.
Pythagoras, another native Ionian, forged many of the mathematical tools Greek philosophers would use to build their new universe. It was Pythagoras who began formulating theorems with economy and rigour, developing geometry to a level that later thinkers such as Euclid would inherit. Born on the Ionian island of Samos sometime in the early 500s BCE, Pythagoras is reputed to have travelled widely across the civilizations of the Mediterranean, metabolizing what he learned into a new mode of thought.37
The philosophical school Pythagoras established around the mid-400s BCE exerted enormous influence over the rest of classical history. No strangers to deeply felt mysticism, the Pythagoreans were a secret fraternity. Unlike the cult religions that had preceded them, however, the basis of Pythagorean religious rapture was the contemplation and exploration of mathematical beauty. “All is number” was their creed. For the Pythagoreans, reality was mathematics. The concept of “Kosmos”—a universe that could be studied and contemplated mathematically—is believed to be a Pythagorean invention. Those who followed Pythagoras’ tradition found enlightenment in mathematical explorations of this cosmos. For Pythagoreans, both ancient and modern, the universe is “suffused with arithmetic divinity”.38
Central to the Pythagorean vision were the five geometric constructs that would later be called the Platonic solids. The Platonic solids constitute five highly symmetrical three-dimensional forms constructed using the rules of geometry. The simplest is the perfectly symmetrical sphere. Next is the four-sided pyramid called the tetrahedron. Above it on the ladder of complexity is the cube with its six sides, followed by the octahedron (eight sides), the dodecahedron (twelve sides) and the icosahedrons (twenty faces).
Each Platonic solid embodies a wealth of elegant geometrical relationships. For example, with a proper choice of sizing, all five solids can be nested into each other with the vertices (tips) of each inner figure gently touching the interior surface of the next, outer figure. Better yet, each solid can be nested into that most perfect of all forms—the sphere—with the vertexes just glancing the sphere’s inner surface. For the Pythagoreans, the Platonic solids were the embodiment of beauty and harmony. This was an essential point, for in their view of the cosmos, beauty and truth were one and the same. The elegant symmetries of Platonic solids led these Greek mathematician-philosophers to associate each figure with an elemental property of the physical world. The cube was the element earth, fire was bound into the tetrahedron, air was embodied within the octahedron and water’s essence lay in the icosahedrons; the dodecahedron was associated with the whole of the cosmos, which Pythagoreans believed was bound into the perfect sphere.39
The association of elemental mathematical form with elementary physical essence was a profoundly new development for cosmological thinking. With it came an association of underlying mathematical realities with the forms and behaviour of the observable world. The logic this association set in motion still drives scientific attempts to understand the world, as seen in the emphasis on finding ever more abstract mathematical models to describe subatomic particles or space-time itself. With the cosmology of today now standing on the edge of its own transformation, beauty and elegance in competing mathematical models plays an important, and perhaps exaggerated, role in capturing scientists’ attention. Such is the legacy of Pythagoras and his followers.
“Let no one enter here who is ignorant of geometry.” This admonition crowned the gates of the Academy, Plato’s famous school of philosophy. Plato, an Athenian who lived during the city’s golden age of the fourth and third centuries BCE, continued the Pythagorean enchantment with numbers and mathematics. While he never developed detailed models of the cosmos like other Greek thinkers, Plato contributed an all-encompassing idea that would be critical in shaping Western science.
Building on theories of the Pythagoreans, Plato argued that behind the appearances of reality lay a purely perfect, fully mathematical world. This flawless realm of mathematical forms acted as a blueprint for all that we see. For Plato, the time-bound world we experience so vividly is a corrupted version of the ideal and timeless world of mathematical forms. This concept was sometimes called the Doctrine of Forms (or Doctrine of Ideals), and it would lodge in the minds of philosophers like a burr, beginning a millennia-spanning search for the world’s pure, underlying architecture. After Plato, mathematics was seen as a skeleton on which the flesh of the world was to be hung. Even today a strong, if tacit, background of Platonic idealism characterizes the efforts of modern physicists and cosmologists.
One of the most enduring consequences of Plato’s Doctrine of Ideals was his exhortation of Greek astronomers to “save the appearances”, or find mathematical models of the heavens that fit the Greek ideal of order and beauty. Rather than seek the nature of the cosmos through observation alone, these models must begin with the assumption that nature was ordered along mathematical, or geometrical, form. Using geometry, the models must account for, and predict, all celestial motions—including the looping retrograde motions of planets against the stars.
The Greek geometric image of perfection was the circle. Thus the true motion of a planet should be a constant, stately march along a circular orbit. The platonic demand that any true model of the solar system would invoke all planets on circular orbits, moving with constant velocities. The apparent planetary disorder seen on the sky—the speeding up, slowing down and puzzling loop-the-loops—must be shown as an Earth-bound illusion. It would be Plato’s greatest student who fully answered his challenge and built the foundations for an Earth-centred, or geocentric, universe that would last more than a millennia.
This student was Aristotle. After time spent tutoring Alexander the Great, Aristotle returned to Athens to found his own school, the Lyceum. There he developed a philosophy that differed strongly from his teacher’s, and those differences would lay their own brand on Western civilization. Aristotle considered all aspects of the natural world—biology, physics, astronomy—but he was not a scientist in the modern sense of the word. He did not validate his theories through rigorous experimentation but instead combined a set of core beliefs he held to be self-evident with select observations about the world’s behaviour.40 In this way Aristotle deduced an elaborate account for the cosmos built on an unequal balance of reason and evidence.
Ruling minds for more than fifty generations, Aristotle’s cosmos was a divided kingdom. The spherical Earth was the centre of creation, but it was a polluted realm—according to Aristotle, the sublunar domain was home to decay and imperfection. Aristotle’s vision for physics (as important to history as his astronomy) depended on this division of sub- and superlunar cosmic domains. Five basic elements existed in Aristotle’s cosmos: earth, water, fire, air and aether. Each element had a “natural” motion associated with it. Earth and water “naturally” sought movement towards the Earth’s centre. Air and fire naturally rose towards the celestial domain. The aether was a divine substance constituting the heavenly spheres. These “natural” inclinations seemed self-evident to Aristotle and did not require separate
tests. Only many centuries later would a new breed of scientists such as Galileo (in the late sixteenth and early seventeenth centuries) demand that a hypothesis such as natural motion be validated through experiments.
To reclaim order from the chaos of apparent celestial motions (retrograde loops most of all), Aristotle built his cosmos on a model proposed by Eudoxus, another of Plato’s students.41 It was a fully geometrical, geocentric vision of the cosmos with the Earth surrounded by a concentric set of rotating spherical shells. The sun and each of the planets revolved around the Earth attached to a different crystalline shell, each of which guided its heavenly body through mathematically perfect circular motions at a constant speed. By tuning each shell’s rotation to observations, Eudoxus could use his “universe mechanism” to recover many of the appearances Plato had demanded be saved.
Aristotle’s model of the solar system captured the Greek imagination with its mix of geocentric egotism and geometric harmony. It was, however, ill-suited for detailed comparisons with the ever more accurate observations of astronomers such as Hipparchus of Rhodes. The final crowning achievement in the creation of the Greek cosmos would come not from Greece but from the great library of Alexandria in Egypt—the intellectual glory of the classical world and its version of a fully funded research institute. It was in Alexandria that astronomer Claudius Ptolemy solved Plato’s challenge once and for all.
Ptolemy created a truly accurate geometric, geocentric model of celestial motion that could stand up to Hipparchus’ data. The Ptolemaic universe was built using only uniform circular motions (though Ptolemy was forced to stretch the meaning of “uniform” somewhat). Any astronomer of average competency could use Ptolemy’s work to predict the motions of the sun, moon and planets with an accuracy that matched the best naked-eye observations (telescopes were still fifteen hundred years in the future).
So powerful was Ptolemy’s work that his book Syntaxis Mathematica, written in 140 CE, became the standard astronomy textbook for more than a millennium. The Arabic astronomers, who carried science forward while western Europe huddled in the Dark Ages, called Ptolemy’s book Almagest, or simply “The Greatest”. For the next 1,300 years, the study of astronomy was the study of Ptolemy.
Claudius Ptolemy achieved his accuracy at a price, however, and that price was simplicity. The appearances could be saved in his geocentric model only by adding an impressive array of geometric bells and whistles. Capturing retrograde motion, for example, demanded that each planet not move directly on its circular orbit about the Earth. Instead, planets tracked along on a smaller circle called an epicycle, and it was the centre of the epicycle that moved with uniform speed around the Earth. When the planet’s motion on the epicycle was in the same direction as the epicycles’ orbital motion, it would appear to an Earth-bound observer to move fairly steadily across the sky. Recall that this motion is always against the fixed stars. As the planet steadily marched against the stars towards the eastern horizon, its direction on the epicycle would match the epicycle’s direction along the orbit. But things changed when the planet looped around to the other side of its epicycle; its motion now moved in opposition to the epicycle as a whole. Thus the planet would appear to change direction in the sky and move backwards towards the western horizon. It was the combination of motions—the planet on its epicycle and the epicycle orbiting the Earth—that allowed Ptolemy’s model to get Plato’s homework problem right.42
Only in hindsight does Ptolemy’s vision of the solar system look like a convoluted Rube Goldberg contraption. For astronomers of the classical world (and for the next fifteen centuries), the sway of Greek philosophical preferences for circles and circular motion made Ptolemy’s work seem a triumph of reason. Using geometry alone, Ptolemy had mapped out the perfect mechanics of the perfect heavens.
FIGURE 2.6. Retrograde motion and epicycles in Ptolemaic astronomy. Ptolemy explained the periodic loops in planetary motion by imagining each planet riding an “epicycle” whose centre orbited the Earth on its deferent. For an observer on the Earth the combination of circular motion on the epicycle and circular motion on the deferent created the appearance of retrograde motion on the sky.
TIME, CHANGE AND THE FIVE COSMOLOGICAL QUESTIONS
While the later Greek thinkers, with their emphasis on maths and geometry, created sophisticated models of the cosmos—which for them meant the solar system—there were deeper questions that needed to be addressed. Lurking behind the philosophical and astronomical achievements of Plato, Aristotle and their descendants was an essential question touching the very nature of time itself: is change, and therefore time, real, or is it an illusion?
The polarities arising in response to this over-arching question still echo down to our day as cosmology searches for its next step beyond the Big Bang. The conflict is cleanly embodied in the teachings of two famous pre-Socratic philosophers: Parmenides of Elea (in Italy)43 and Heraclitus of Ephesus (now the Turkish coast).44
Parmenides looked at the world of change and saw it as nothing more than an illusion of the senses. At the deepest levels of reality there was no change, no transformation and hence no time. He reached this conclusion not by considering what is but by contemplating what is not. In The Way of Truth, a section of his sole surviving work, Parmenides considers the relationship between existence and nonexistence. He concluded: “It is necessary to speak and to think what is; for being is, but nothing is not.”
With the phrase “Being is, but nothing is not” Parmenides rejected the very idea of nonexistence and was forced to conclude that things cannot come into being from nothing; neither can they disappear. “Nothing can come from nothing” was his claim. But movement, which must involve moving into a void—the place where nothing existed a moment before—cannot be possible either. Thus, behind the appearance of change and the appearance of movement must be a single unified timeless reality. Change and time are illusions.
Heraclitus drew the opposite conclusion. “All is flux” was his doctrine, and the famous dictum “One cannot step in the same river twice” is attributed to his works. In Heraclitan philosophy the world was composed of ceaseless movement and transformation. Every object was considered new from moment to moment. The basis of this endless flux was a constant conflict between different elements of the world. Fire was primary among the four elements and its transformations were the basis for the apprehended world: “All things are an equal exchange for fire, and fire for all things, as goods are for gold and gold for goods.”
Thus the dichotomy was set early. Either an eternal, timeless world existed behind appearance, or time and change were the only essential reality. Plato, with his Doctrine of Ideals, was naturally an admirer of Parmenides. Heraclitus had his devotees as well; the Stoics, a school of philosophy that subsequently gained influence in both Greece and the Roman Empire, would claim Heraclitus’ rule of change as their own. In their cosmology, the world emerged through ekpyrosis (literally, “out of fire”) and would return to fire again.45
One far-reaching attempt to reconcile the different views of reality and change emerged in the work of the atomists.46 In a display of remarkable prescience, Greek philosophers such as Democritus argued in 400 BCE that everything visible was caused by the movement of tiny, indivisible specks of matter called atoms. The eternal nature of these atoms satisfied Parmenides’ stand against change. But while Democritus accepted most of Parmenides’ conclusions, he and the other atomists rejected the idea that change is an illusion. Atoms did move, and they moved through an infinite void.47
The themes of Heraclitus and Parmenides would reverberate across millennia to our own time, as would the ideas of the atomists. One view would gain favour over the other only to be discarded again later. Thus, echoing the debate between Heraclitus and Parmenides, cosmic time in its human context shows both constancy and change. There is constancy in the questions humans ask about the universe and there is change in the answers that dominate one culture or another as the centuries pass
. Understanding this balance is critical. Just as important is seeing how the questions and answers cosmology offers predate even the Greeks.
Humanity’s first thematic encounter with time and the cosmos began with the mythological imagination of the urban revolution. And, in a sense, it ended there as well. Looking broadly across creation myths from the Egyptians to the Hittites to the early Chinese civilizations and on to the cultures of Mesoamerica (Aztec, Inca and so on), we find almost all potential explanations for time and cosmos clearly mapped out. Thus, there are only so many ways to think about the origin of time.
This is a point cosmologist Marcelo Gleiser eloquently explored in his book The Dancing Universe. Myth forms a storehouse of possible solutions to questions of time and the universe’s origins. As Gleiser wrote, “In their variety these myths encompass all the logical answers we can give to the question of the origin of the Universe, including those found in modern theories of cosmology.”48
Humanity’s first rational encounter with time and the cosmos began with the Greeks, producing results that have stayed with us until the present day. The dichotomies of Parmenides and Heraclitus, the prescience of the atomists, the arguments about infinite space and time, the nature of the vacuum and the mathematical models of the solar system as cosmos—these all set the stage for what would follow. While it would be foolish to argue that no novelty emerged in the next two thousand years of cosmological thinking, it would be equally detrimental to ignore the way the Greeks established the palette of colours that later thinkers would use to paint their universes.
