The other day, speaking on the subject of space and time, I mentioned symmetry as a fundamental concept, and I also mentioned that symmetry involves the feeling of permanence in time. This is because symmetry means the invariance of an object under the action of a group of motions: we see the object as remaining identical with itself in spite of change all around. There's a very hard question lurking here: why do we seek that feeling of invariance? Why is it so necessary for us, no less than air and food? Here we can only speculate, so let me speculate a little. Let's go back to the main subject of our course: our identity. Often this is taken as a question: who am I? And as often this is answered linguistically and numerically: my name is such and such, I was born on this date, in that place, I weigh so much, my social security number is 1-2-3, I went to school at Slippery Swamp State College, and so on. This is an insufficient answer, and it is made only more opaque by adding that I belong to some ill-defined categories: My profession is university professor; I am Hispanic, say, or Irish-American, or Black, or some combination thereof. The real question is: what is my identity? Or in other words, since identity means sameness: what makes my sameness? You who are sitting here in this classroom were brushing your teeth a few hours ago and will be playing or sleeping a few hours hence: what makes you the same you, all through these quite different activities? The usual reply here is: memory. You remember being you, through all these changes. But, again, this answer is insufficient. For memory in itself could lead you to the opposite conclusion; if it were possible for you to remember every single detail of what you experienced two hours ago, and compare them to every single detail of what you are experiencing now, the logical conclusion would seem to be that these two infinite, vastly different experiences have nothing in common, nothing in them is the same. This is what Heraclitus meant when he said that no one can step into the same waters, not merely because the river is not the same, but because we are not the same. And the Buddha meant something similar when he said that our own identity is nothing but an illusion. But it so happens that we have a feeling that in the midst of all these changes something has remained invariant; whether this feeling is true, that is, justified by science, is beyond the point—indeed, science may well side with Heraclitus and conclude that identity is only an illusion; the fact is, we feel it, and without this feeling there would be no identity, no ME, no YOU. When we perceive symmetry, that feeling of invariance through change which constitutes our identity is evoked, and this evocation, this recalling of our feeling, makes us feel one and the same being again, makes me feel me. What I said in my last lecture about spatial symmetry is also true of repetition, which is symmetry in time.
At the risk of boring you, I must repeat what I said in my previous lecture, in relation to architecture: extreme symmetry and extreme repetition are boring. The precisely symmetric features of the models on magazine covers are not beautiful; to acquire beauty that symmetry should be broken by some non-symmetric feature, smile or wince. To listen always to the same note, the same rhythm, may be hypnotic but is not spiritual. Incidentally, hypnotic phenomena are a good example of loss of identity; under hypnosis my I gets lost and I become someone else, the hypnotist, or the crowd. I could go into the loss of identity caused by music with driving, constant rhythms, and I could go on showing how extreme symmetry and repetition in our societies forces the individual to melt into the crowd and fall into goosestep—but I won't for now—perhaps in the second semester. On the other hand, when we listen to J.S. Bach's Goldberg Variations, for example, we notice that the theme at the beginning is exactly repeated, note by note, at the end; but in between, what a fantastic wealth of variations! The struggle between change—variation and sameness makes this piece of music spiritual, for it affirms our own identity in an ocean of change (remember my definition of spirituality as an ongoing engagement with the double nature of time). A similar example: I mentioned last time Proust's novel, Remembrance of Things Past. There you'll find the most quoted example of spiritual repetition or symmetry. The narrator, Marcel, happens to eat a cookie, a madeleine, and the taste takes him, unconsciously and surprisingly, back to his childhood; thus his whole life appears, for one brief moment, as a unity. Had he been eating the same kind of cookie every day, his spiritual experience of repetition and oneness could not have occurred; it is the feeling of sameness after a long period of difference that makes it interesting and moving. Another great writer I mentioned previously, Kierkegaard, wrote long before Proust, in the 1840's, a book titled Repetition, in which he stated that repetition is the fundamental concept of philosophy, and noted that this kind of spiritual experience of oneness brought about by repetition cannot be willed by us—such willed repetitions are either impossible or a disappointment. Marcel did not find those cookies and thought, "Hey, I ate these as a child, so let's see if by eating one now I can recapture..." No, such experiences must take our brain by surprise. On the other hand, we read in the Katha Upanishad,
"Who sees variety and not the unity wanders on from death to death."
I may add that I know of no more intense joy than the fleeting feeling that all my previous life, all the manifold experiences I've had, suddenly appear as a unity, thus endowing them with sense. We call this an illumination. Those of you who are interested in poetic expressions of this feeling of oneness may also read Wordsworth's poem "The Prelude."
What does all this have to do with the birth of modern science? Today I will confine myself to astronomy and what's called the astronomical revolution of the 16th and 17th centuries. Before we start with astronomy, you should remember that the simplest and most completely symmetrical finite object one can think of is a circle in two dimensions and a sphere in three. This is why the Greeks considered them the most perfect forms. The Greek obsession with symmetry has been of extreme importance—I would even say fateful—for us. Remember also that at the very beginning of Greek thought, Anaximander thought the earth was suspended at the center of a spherical universe, and a century later Parmenides thought the whole cosmos was an eternal, unchanging sphere.
Of all symmetries and repetitions we experience, the most obvious and constant is the rising and setting of celestial bodies—the sun, the moon, the planets, the stars. Not only do these phenomena repeat: they are periodic, that is, they repeat at roughly equal intervals of time. Day follows night and night follows day; the seasons and the phases of the moon come and go only to return; nothing in human experience can be foretold with more certainty. This may be why celestial bodies were since very early times thought of as immortal gods. The interruption of their appointed paths was deemed the most powerful miracle. In the Bible, the sun stood still for Joshua, allowing him a longer day to annihilate his enemies; when in the 11th century CE, during the First Crusade, hundreds of Jews were slaughtered in the Rhineland, the Jewish chronicler asked reproachfully, "Why didn't the sun darken?" Eclipses and miracles apart, nothing appeared more regular to our remote ancestors than the motions of the stars. Egyptians and Babylonians, Mayans and Chinese, many civilizations kept careful watch of those motions, drew calendars, built huge buildings oriented toward notable celestial points. What's a calendar? It's a method for predicting celestial phenomena. For example, a calendar may be able to tell you on which day will fall the first full moon after the spring equinox: this happens to be important because it determines the date of Passover and Easter, in the Jewish and Christian traditions. Here we cannot go into the intricate history of early astronomy; I will only say that its main practical application was the measurement of time. Our 24 hour division of day and night had its origin in the Egyptian system, in which the rising in the horizon of certain stars signaled the beginning of another period, another hour of the night. Similarly, our own division of the hour in sixty minutes and the minute in sixty seconds had its origin in the Babylonian system of numeration, based on 60—our system is based on 10. The rhythms of human life and of ritual—perhaps the rhythms of all life on earth—depend on the rhythms of celestial bodies, and so it is not surprising that in some cultures the belief became prevalent that those celestial phenomena governed also the destinies of states and individuals. This belief is called astrology, which survives today as a superstition (the word can be interpreted literally as the survival of old beliefs when they do not correspond to reality anymore). In any case, because of the regularity of celestial phenomena, they became the object of the earliest exact science, that is, they were the first to be described by mathematics.
Having said only this much about the astronomy of ancient civilizations, we are going to focus on Greek astronomy. This is not because I am Hellenocentric, convinced that the Greeks were the absolute peak of humanity. That was the intellectual fashion between the Renaissance and roughly the 1960's; now it has become fashionable to pretend that all cultures are equally interesting and we should not spend too much time on the Greeks; so I have to justify my focus. In Greek astronomy we already see the process that's at work in modern science. When it comes to careful mathematical measurements of the motions of the stars, long before the Greeks the Babylonians were second to none; what the Greeks contributed was a mathematical model (actually several different ones), and this means a theory involving geometric diagrams, and then the logical results we extract from this diagrams are compared to the perceived facts, which either confirm the model or cause it to be revised. As Ortega y Gasset says in his book, this is the essence of modern science. The Greeks were the first modern scientists. Plato expressed the role of scientific theories thus: they must "save" the phenomena. This is a wonderful phrase: if we just take our perceptions—the phenomena—as given, without adding any theory, any hierarchy to them, our world would be pure chaos; so in this sense theory "saves" the phenomena. Theory also "saves" the phenomena in the same sense as Christianity called Christ "the Savior," with the same Greek word, sotér: for Christianity Jesus' sacrifice abolishes our death and makes us eternal; for Plato, as for modern science, theory transforms the fleeting, transient appearances into manifestations of eternal laws.
Eudoxus of Cnidus (5th-4th century BC), a colleague of Plato in the famous Academy, is credited with being the originator of scientific astronomy. He was not only an astronomer, but a great pure mathematician: Euclid's Fifth Book contains his celebrated theory of proportions. None of Eudoxus' works has survived. His astronomical model consisted in a system of spheres (over 20 of them), all having the center at the center of the earth, and each revolving on an axis which was fixed to the next larger sphere. Aristotle, whose surviving works were to exert a tremendous influence during the Middle Ages, was also a student at Plato's Academy and took over Eudoxus system, but since Aristotle was not a mathematician and so didn't care much about details, it came to happen that Aristotle's simplified system of concentric spheres, of which the outermost was called "the prime mover," the God who puts the whole cosmos in motion, became Christian dogma. For challenging this dogma, Giordano Bruno was burnt at the stake in Rome in 1600, and Galileo had to do public repentance and penance in Rome in 1633. Another early model in Greek astronomy was that of Aristarchus of Samos (3rd century BC). He put the sun at the center of the world, and the planets, including the earth, revolving around it in circles, with constant speed. Note two things: first, Greek astronomical models were based either on spheres or circles—how could the most perfect motions, those of the divine celestial bodies, be based on anything other than the most perfectly symmetric forms? Thus reasoned Plato and his friends. And secondly, in all these models the earth itself was spherical. This last point runs contrary to what many of us were taught in high school, that before Columbus and Magellan people believed the earth was flat. No doubt there must have been people who did believe that, as there are people today who believe in astrology or voodoo; but few scientifically educated people entertained such notions.
The reality of celestial motions presents difficult problems when we try to measure them by number. Remember that the Pythagoreans discovered the correspondence between musical intervals and simple fractions, or ratios (for ex, a fifth, C-G, corresponds to 3:2). Now, there is no such simple ratio when it comes to comparing the time it takes for the earth to complete its orbit around the sun with the time it takes for the earth to revolve around itself; in other words, a year is not a simple, precise number of days: it is—but only approximately—365 days, 5 hours, 49 minutes and 12 seconds. Similarly, a lunar month is not exactly 29 or 30 days, but some complicated number in between. This fact is of practical importance: errors accumulate. Suppose your watch gains one second every day: this may seem like a very tiny deviation, but over a century that watch will have gained more than ten hours. Another difficulty which plagued ancient astronomers appears when we consider the motions of the planets. Sometimes Jupiter, for example, proceeds in one direction from our point of view, while at other times it proceeds in the contrary direction (retrogade motion). This is why the Greeks called them "planets," a word which means literally the ones wandering about. To account for these and other facts, mathematical models based either on spheres or circles had to become more and more complex, if they were to be of any use to people like navigators, who needed good accuracy. The Eudoxian system of concentric spheres, although it survived for millenia under its Aristotelian guise, was abandoned very early by mathematicians, who favored another system, due to Apollonius (3rd ctry BC) and Hipparchus (2nd ctry BC), based on circles (or cycles) and epicycles. These latter are curves described by a point on a rotating circle whose center itself rotates on a larger circle. Finally, Ptolemy (2nd century AD) put together the system which was going to dominate Western and Arabic astronomy essentially up to the 17th century. In his work titled Mathematical Syntaxis, which means mathematical putting together, later called the Almagest (a funny word, mixture of Arabic and Greek, meaning "the greatest"), and one of the great scientific works of all ages, he used circles and epicycles, and he introduced a very original modification: the speeds of rotation along these circles were not constant, something which went very much against the Platonic spirit, but which was necessary to account for the observed positions of the planets. Ptolemy, too, was a compiler of good trigonometric tables, which are necessary for astronomical computations. For more details about the Ptolemaic system, I refer you to Ekeland's book. With Ptolemy, we reach the end and culmination of Greek astronomy.
The next hero in our story is Nicolaus Copernicus, whose main work, De revolutionibus orbium coelestium, On the revolutions of celestial spheres, was published in 1543, the year its author died. That was quite a year, for not only did Copernicus change forever our ideas of the world out there, but also in 1543 a book on anatomy was published by Andrea Vesalius, De humani corporis fabrica, On the structure of the human body, which changed the prevalent ideas about the world inside ourselves. The medieval macrocosm and microcosm: both were revolutionized. Let me summarize what Copernicus did. Firstly, the observations he played with were the same or no better than those of Ptolemy; his mathematics was not, save in minor respects, much of an improvement on that of the Greeks. Secondly, his theory does not fit the observed facts any better than the old Ptolemaic theory but just about as well. So what made Copernicus' work appealing? In a sense, he was more conservative than Ptolemy. What Copernicus objected to in Ptolemy's system was the use of non-uniform motion (i.e. circular motion with variable speed): like Plato many centuries before, he found it "contrary to reason." To avoid this lack of uniformity, Copernicus kept using circles and epicycles, but put the sun at the center of those circles or spheres, rather than the earth. As we said before, already in the 3rd century BC Aristarchus had a heliocentric system; Copernicus', though, had the extra sophistication and predicting power of the Ptolemaic.
The simplification in calculation brought about by Copernicus had to be paid for by swallowing something extremely hard: that the earth moved around the sun and around its own axis at great speeds. For many centuries, scientists had good and strong objections against such an idea. For one thing, if the earth moves around the sun, the so-called fixed stars would appear at different inclinations through the year, which was not found to happen (this phenomenon is called paralax). Copernicus answered that objection by postulating that the distance between earth and sun is negligibly small compared to the distance between earth and the fixed stars. Of course, this is so in modern astronomy too. Also, and even more damaging to the idea of a fast-moving earth: objects standing on the earth would tend to fly off it, and if you throw a stone straight up it wouldn't fall down where you are because the motion of the earth would leave the stone behind. The solution to this objection could not be given within the framework of Aristotelian mechanics that Copernicus used; it had to wait until Galileo, of whom we'll talk next time, changed our notions of how bodies move. Therefore, the Copernican system was generally taken as an aid to computation of celestial events, but not as a description of how things stand in reality. The Roman Catholic Church did not object to the Copernican system until later, when Bruno and Galileo claimed that it actually described reality. On the other hand, Luther was objecting to it as early as 1539 (four years before the publication of Copernicus' book!): pretending that the sun is fixed and the earth moves goes against Scripture, Luther claimed, for Joshua—the Joshua we mentioned at the beginning of this lecture—asked the sun to stand still, see, not the earth! Luther's position is the same as that of some Christians who nowadays object to modern evolutionary biology. The demotion of our home from the center of the universe to the status of just another planet had enormous psychological implications. It was the first step in what is now called "the de-centralization of man." From then on, we became tiny specks living upon a tiny speck lost somewhere in the immensity of the universe. Quite a revolution in our sense of our human identity. But to that we'll return in the second semester.
The astronomer who finally broke away with the symmetry of circles and spheres which so obsessed the Greeks was not Copernicus but Johannes Kepler (1571-1630). Kepler was of the same generation as Galileo, born in 1564; it was this generation and the following one, that of Descartes, which inaugurated science as we know it. Kepler possessed a wealth of astronomical data, very accurate, which had been collected by a brilliant observer of the heavens, the Dane Tycho Brahe (1546-1601), the first man in the West to record the observation of a nova, an exploding star, on November 11 1572 (the Chinese had known of novas long before). The importance of this nova is that it showed that, contrary to what all philosophers thought, the heavens are not unchangeable and incorruptible, since a star could suddenly appear, shine for a while then disappear. Coming back to Kepler, his life and personality are interesting enough to have inspired more than one novel, but we have enough time only to mention his celebrated laws of planetary motion, to which he arrived after a long, laborious analysis of Brahe's observations. Kepler's first law says that the planetary orbits are elliptical, with the sun at one focus. Ellipses were very well known to the Greeks, but they would have never dreamt that celestial objects would travel on them. Around 1080, Arzaquen, a Muslim astronomer in Toledo, in Spain, declared that planetary orbits are elliptical, but the time was not ripe, and no one paid attention to him. As a matter of fact, the Greek mathematicians of the school of Plato discovered ellipses as plane sections of a circular cone, and those curves (ellipses, hyperbolas and parabolas, called conic sections) were used by them to solve a famous problem which had nothing to do with astronomy or physics: given a cube with side a, its volume is equal to aaa = a3; now, how can we construct a cube with twice that volume? Here, then, we have one of the first examples of a phenomenon well known to scientists: something invented or discovered with one problem in mind, turns out, maybe 2,000 years later, to be precisely what one needs to solve a completely different problem.
Kepler's second law says that the line joining the sun with a planet sweeps equal areas in equal times no matter where the planet is located—the "area velocity" is constant. And, finally, the third law tells exactly how the orbital period of each planet depends on its distance from the sun. Ancient astronomers were aware that the farther the planet is from the sun, the longer its solar year, its period; Kepler gave the precise mathematical law: the square of a planet's period is proportional to the cube of its mean distance from the sun. This third law, which may seem complicated at first blush, is historically of enormous importance, because two generations later Newton started from it to obtain his law of gravitation. Please remember this for later, when we deal with Newton in the second semester.
Finally, to end this discussion on the astronomical revolution, let's go back to our concept of symmetry. What was gained by the breaking of the perfect symmetry of circles and spheres, so dear to the Greeks, and the adoption of less symmetric curves? More accuracy, to be sure, and accuracy was important for calendar making and for navigation. But this is not the end of the story, nor is it the most important thing. If you have read Ekeland's book, you know by now that the absolute accuracy of Kepler's and Newton's model, of which the 18th and 19th centuries were so proud and which vastly influenced their ideas, is just an illusion. That illusion was shattered by Henri Poincaré. In fact, we cannot hope, even with the fastest computers imaginable, achieve absolute accuracy or anything close to it. There is something else, immensely important, that was gained by the breaking of the staid symmetry of circles and spheres, and that is unity. For the Aristotelian thought which dominated science up to the 17th century, the nature of the celestial bodies—their matter, their motions, the laws of those motions—was entirely different from the nature of things here on earth. The earth was heavy, the celestial objects were weightless. Down here there was corruption, decomposition, friction, collisions, etc. Up there, none of these things. With Kepler, and later with Newtonian gravitation, the same laws apply everywhere, here on earth as well as in the heavens. One science applies to the whole cosmos. This breaking of symmetry for the sake of unity is a hallmark of all fundamental breakthroughs in science. And, I submit, not only in science, but in art, in human communication, in our own lives. But why break with the cozy symmetries of life, with the structures that have become routine, with the safe assumptions which serve us so well? Why this crazy search for unity, why not rest content with the fragmentary, the disconnected, the ad-hoc? This is far from being an idle question, but I don't know the answer. The search for unity, the need, as Ortega says, to agree with oneself, for the cosmos to agree with itself and us with it, seems to be one of the deepest characteristics of our human identity and of our spirituality.
For a very brief summary of 2,000 years of astronomical developments, the first two chapters of Ekeland's book, Mathematics and the Unexpected, Chicago. For an account of the crises at the beginning of modern science, Ortega y Gasset, Man and Crisis, Norton (whole book).
For a more extended treatment of Copernicus and Kepler, see Alexandre Koyré, The Astronomical Revolution, Dover Publications.
The best general history of astronomy up to the 1600's is J.L.E. Dreyer, A History of Astronomy from Thales to Kepler, Dover Publications.
A history of cosmology up to the 1600's: Michael J. Crowe, Theories of the World from Antiquity to the Copernican Revolution, Dover Publications.