The writing of this book has been a great adventure, reflecting a lifelong engagement with the nature of time. As any traveler does, I owe an enormous debt to the many people who assisted, encouraged, guided, and sometimes led me on this journey.
The adventure began in 1980, when I was studying for the summer in Oxford as a guest of Roger Penrose. Roger told me that if I really wanted to think about the nature of time I had to talk with a fellow named Julian Barbour, who lived and worked in a village near Oxford. Arrangements were made, and along with the philosopher of science Amelia Rechel-Cohn, I paid him a visit. It was during those discussions that Julian began his philosophical mentorship of me, introducing me to Leibniz’s writings and the ideas of relational space and time. I was one of the first, but far from the last, young physicists to have his head turned round and his thinking set in the right direction by Julian’s tutelage.
The journey took an unexpected turn in 1986, when Andrew Strominger told me about his discovery of a great number of string theories and his worries that their abundance would defeat any attempt to derive the Standard Model of Particle Physics from pure principles. Meditating on this, I imagined a landscape of string theories similar to fitness landscapes in biology, in which a mechanism analogous to natural selection would govern the evolution of laws. Encouraged, shortly before her untimely death, by my dear friend the physician and playwright Laura Kuckes, I developed the idea of cosmological natural selection, which I published in 1992 and described in my first book, The Life of the Cosmos.
While I was finishing that book, another friend, Drucilla Cornell, told me to read the Brazilian philosopher Roberto Mangabeira Unger, who in a book on social theory had also argued for the evolution of laws in cosmology. She put us in touch, and after a thrilling conversation in his office at Harvard, he proposed that we collaborate on a rigorous academic book about the implications of the reality of time and the evolution of laws. That project, under way for the last five years, has been the main impetus and vehicle for the development of the ideas in this book. Thanks mainly to Roberto’s clear and provocative thinking, I eventually came to appreciate the radicality of the proposal that time is real. The crux of the Epilogue — that the openness of the future should prompt human beings to reach for novel solutions to problems on every scale — was inspired largely by his writings. Those readers wanting a more rigorous formulation of the arguments sketched here are encouraged to turn to the forthcoming book, tentatively titled The Singular Universe and the Reality of Time.
Also in 1986, I began work on the quantization of the new formulation of general relativity invented the previous year by Abhay Ashtekar. This led to the discovery of loop quantum gravity, which I realized in joint work with Carlo Rovelli. Our technical work was motivated and framed by constant discussions on the nature of time with Abhay and Carlo along with Louis Crane, Ted Jacobson, Chris Isham, Laurent Freidel, João Magueijo, Fotini Markopoulou, Giovanni Amelino-Camelia, Jerzy Kowalski-Glikman, and Renate Loll, among many others. Indeed, I came later than several of my friends to the idea that the relativity of simultaneity would have to be given up on a cosmological scale. Antony Valentini understood many years ago that this was the cost of adopting a hidden-variables theory, and João Magueijo was already being provocative about violating relativity when we met in London in 1999. Fotini Markopoulou was the first to emphasize the importance of the inverse problem and to forcefully advocate an approach to quantum gravity in which time is fundamental and space emerges, and the main ideas of Chapter 15, including disordered locality and geometrogenesis, are due to her.
Although I’m a physicist, I have been fortunate to have been a frequent guest in the house of the philosophy of science, where I’ve made many friends who, over the years, have listened patiently to and critically read my attempts to think clearly about the nature of time. These include Simon Saunders, Steve Weinstein, Harvey Brown, Patricia Marino, Jim Brown, Jenan Ismael, Cheryl Misak, Ian Hacking, Joseph Berkovitz and Jeremy Butterfield, as well as my original teacher in the philosophy of physics, Abner Shimony. Julian Barbour, Jim Brown, Drucilla Cornell, Jenan Ismael, Roberto Mangabeira Unger, and Simon Saunders were kind enough to read the whole draft and give me crucial feedback.
I am grateful to Sean Carroll, Matt Johnson, Paul Steinhardt, Neil Turok, and Alex Vilenkin, whose conversations and comments on draft chapters straightened me out on cosmological issues. My work on quantum foundations described here has benefited greatly from interactions with the foundations community at Perimeter Institute, particularly Chris Fuchs, Lucien Hardy, Adrian Kent, Markus Müller, Rob Spekkens, and Antony Valentini.
Crucial encouragement and critical insight on early drafts, and much besides, was, as always, given by Saint Clair Cemin, Jaron Lanier, and Donna Moylan.
The Epilogue’s concern with climate change was inspired and informed by a seminar on threshold behavior in political science, organized jointly with Thomas Homer-Dixon, of the Balsillie School of International Affairs. I am grateful to Tad and other participants, particularly Manjana Milkoreit and Tatiana Barlyaeva, for discussions and collaborations on this issue.
For an education in economics, reflected also in the Epilogue, I thank those who worked with me to organize a conference on the Economic Crisis and its Implications for the Science of Economics, held at Perimeter Institute in May 2009; and others I met at the conference and its aftermath, particularly Brian Arthur, Mike Brown, Emanuel Derman, Doyne Farmer, Richard Freeman, Pia Malaney, Nassim Taleb and Eric Weinstein.
For their friendship, collaboration, and views on self-organization, I owe an enormous debt of thanks to Stu Kauffman and Per Bak.
I am forever grateful to Howard Burton and Mike Lazaridis for the honor and unique opportunity of helping to establish Perimeter Institute for Theoretical Physics, and I am grateful also to Neil Turok for his continual encouragement of our efforts at making breakthroughs and discoveries. Every scientist and scholar should have the good fortune I have had to find an intellectual home as stimulating, diverse, and supportive of big ambitions as Perimeter Institute.
My work in physics has been generously funded by the NSF, NSERC, the Jesse Philips Foundation, Fqxi and the Templeton Foundation, to all of whom I am very thankful for the opportunity to do my work as well as to support and mentor promising young scientists.
This book is much better than it might have been thanks to comments from early readers, who gave me feedback on all or part of the manuscript. In addition to those mentioned above, these include Jan Ambjørn, Brian Arthur, Krista Blake, Howard Burton, Marina Cortes, Emanuel Derman, Michael Duschenes, Laurent Freidel, James George, Dina Graser, Thomas Homer-Dixon, Sabine Hossenfelder, Tim Koslowski, Renate Loll, Fotini Markopoulou, Catherine Paleczny, Nathalie Quagliotto, Henry Reich, Carlo Rovelli, Pauline Smolin, Michael Smolin, Rita Tourkova, Antony Valentini, Natasha Waxman, and Ric Young. Many thanks to Davide Topper for pointing out some historical errors in the story I tell in the first chapter.
I’m one of those writers who loves being edited — because he is painfully aware that he benefits from it. Due to the vicissitudes of publishing, no book of mine has had a more dedicated set of editors. The book owes its conception to the efforts of Amanda Cook, now at Crown Publishing, to whom I am indebted for her belief in the project as well as her conviction that it deserved the time to be sharpened and focused. Courtney Young, of Houghton Mifflin Harcourt, and Sara Lippincott, with their many comments and suggestions, have been the best editors an author could want. The book has also greatly benefited from the wise insights of Louise Dennys, of Knopf Canada. Thomas Penn’s encouragement at crucial moments is also well appreciated. I am also very grateful to Henry Reich, who made many of the figures. As with all my books, I owe a huge debt of gratitude to John Brockman, Katinka Matson, and Max Brockman; without their faith in it, this book would never have come to be.
My thanks to Rodila Gregorio for continual lessons in patience, grace, and loving responsibility — and to Kai, from whom I learned everything I know about time that is not discussed here. Thanks to Pauline, Mike, and Lorna for their love and their confidence in me. Finally, words are insufficient to express my thanks to Dina, who with infinite love and patience held me together through the pressures of trying to finish a book on time on time.
Here is a selection of mostly popular books
on the topic of time in physics or cosmology (and related issues), many of them
presenting alternative or conflicting ideas to those I have proposed in these
pages.
Guido Bacciagaluppi & Antony Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference (New York: Cambridge University Press, 2009).
Per Bak, How Nature Works: The Science of Self-Organized Criticality (New York: Copernicus, 1996).
Julian B. Barbour, The End of Time: The Next Revolution in Physics (New York: Oxford University Press, 2000).
—, The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories (New York: Oxford University Press, 2001).
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed. (New York: Cambridge University Press, 2004).
James Robert Brown, Platonism, Naturalism, and Mathematical Knowledge (Oxford, U.K.: Routledge, 2011).
Bernard Carr, ed., Universe or Multiverse? (New York: Cambridge University Press, 2007).
Sean Carroll, From Eternity to Here: The Quest for the Ultimate Arrow of Time (New York: Dutton, 2010).
P.C.W. Davies, The Physics of Time Asymmetry (San Francisco: University of California Press, 1974).
David Deutsch, The Fabric of Reality: The Science of Parallel Universes — and Its Implications (New York: Allen Lane/Penguin Press, 1997).
Dan Falk, In Search of Time: The History, Physics and Philosophy of Time (New York: St. Martin’s, 2010).
Adam Frank, About Time: Cosmology and Culture at the Twilight of the Big Bang (New York: Free Press, 2011).
Rodolfo Gambini & Jorge Pullin, A First Course in Loop Quantum Gravity (New York: Oxford University Press, 2011).
Marcelo Gleiser, A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Universe (New York: Free Press, 2010).
Brian Greene, The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos (New York: Knopf, 2011).
Stephen W. Hawking & Leonard Mlodinow, The Grand Design (New York: Bantam, 2010).
Stuart A. Kauffman, At Home in the Universe: The Search for the Laws of Self-Organization and Complexity (New York: Oxford University Press, 1995).
—, The Origins of Order: Self-Organization and Selection in Evolution (New York: Oxford University Press, 1993).
Helge Kragh, Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology (New York: Oxford University Press, 2011).
Janna Levin, How the Universe Got Its Spots: Diary of a Finite Time in a Finite Space (Princeton, NJ: Princeton University Press, 2002).
João Magueijo, Faster than the Speed of Light: The Story of a Scientific Speculation (Cambridge, MA: Perseus, 2003).
Roberto Mangabeira Unger, The Self Awakened: Pragmatism Unbound (Cambridge, MA: Harvard University Press, 2007).
Harold Morowitz, Energy Flow in Biology, (New York: Academic Press, 1968).
Richard Panek, The 4-Percent Universe: Dark Matter, Dark Energy, and the Race to Discover the Rest of Reality (Boston, MA: Houghton Mifflin Harcourt, 2011).
Roger Penrose, Cycles of Time: An Extraordinary New View of the Universe (New York: Knopf, 2011).
—, The Road to Reality: A Complete Guide to the Laws of the Universe (New York: Knopf, 2005).
—, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (New York: Oxford University Press, 1989).
Huw Price, Time’s Arrow and Archimedes’ Point: New Directions for the Physics of Time (New York: Oxford University Press, 1996).
Lisa Randall, Warped Passages: Unraveling the Mysteries of the Universe’s Hidden Dimensions (New York: Ecco/HarperCollins, 2005).
Carlo Rovelli, The First Scientist: Anaximander and His Legacy (Yardley, PA: Westholme Publishing, 2011).
Simon Saunders et al., eds., Many Worlds? Everett, Quantum Theory, and Reality (New York: Oxford University Press, 2010).
Lee Smolin, The Life of the Cosmos (New York: Oxford University Press, 1997).
—, Three Roads to Quantum Gravity (New York: Basic Books, 2001).
—, The Trouble with Physics (Boston, MA: Houghton Mifflin Harcourt, 2006).
Paul J. Steinhardt & Neil Turok, Endless Universe: Beyond the Big Bang (New York: Doubleday, 2007).
Leonard Susskind, The Cosmic Landscape: String Theory and the Illusion of Intelligent Design (New York: Little, Brown, 2005).
Alex Vilenkin, Many Worlds in One: The Search for Other Universes (New York: Hill & Wang, 2006).
Before starting this or any other journey of discovery, we should heed the advice of the Greek philosopher Heraclitus, who, barely a few steps into the epic story that is science, had the wisdom to warn us that “Nature loves to hide.” And indeed she does; consider that most of the forces and particles that science now considers fundamental lay hidden within the atom until the last century. Some of Heraclitus’s contemporaries spoke of atoms, but without really knowing whether or not they existed. And their concept was wrong, for they imagined atoms as indivisible. It took until Einstein’s papers of 1905 for science to catch up and form the consensus that matter is made of atoms. And six years later the atom itself was broken into pieces. Thus began the unraveling of the interior of atoms and the discoveries of the worlds hidden within.
The largest exception to the modesty of nature is gravity. It is the only one of the fundamental forces whose effects everyone observes with no need for special instruments. Our very first experiences of struggle and failure are against gravity. Consequently, gravity must have been among the first natural phenomena to be named by our species.
Nonetheless, key aspects of the common experience of falling remained hidden in plain sight until the dawn of science, and much remains hidden still. As we shall see in later chapters, one thing that remains hidden about gravity is its relation to time. So we start our journey toward the discovery of time with falling.
“Why can’t I fly, Daddy?”
We were on the top deck, looking down three floors to the back garden.
“I’ll just jump off and fly down to Mommy in the garden, like those birds.”
“Bird” had been his first word, uttered at the sparrows fluttering in the tree outside his nursery window. Here is the elemental conflict of parenthood: We want our children to feel free to soar beyond us, but we also fear for their safety in an uncertain world.
I told him sternly that people can’t fly and he was absolutely never to try, and he burst into tears. To distract him, I took the opportunity to tell him about gravity. Gravity is what holds us down to Earth. It is why we fall, and why everything else falls.
The next word out of his mouth was, unsurprisingly, “Why?” Even a three-year-old knows that to name a phenomenon is not to explain it.
But we could play a game to see how things fall. Soon we were throwing all kinds of toys down into the garden, doing “speriments” to see whether they all fell the same way or not. I quickly found myself thinking of a question that transcends the powers of a three-year-old mind. When we throw an object and it falls as it moves away from us, it traces a curve in space. What sort of curve is it?
It’s not surprising that this question doesn’t occur to a three-year-old. It doesn’t seem to have been an important question for thousands of years after we regarded ourselves as highly civilized. It seems that Plato and some of the other great philosophers of the ancient world were content to watch things fall around them without wondering whether falling bodies travel along a specific kind of curve. The one ancient philosopher who did speculate about the paths taken by falling bodies – Aristotle – proposed an answer that was easy to disprove but nonetheless was blindly believed for more than a thousand years
The first person to understand correctly the paths traced by falling bodies was the Italian Galileo Galilei, early in the 17th century. He presented his results in Dialogue Concerning Two New Sciences which he wrote during his seventies, when he was under house arrest by the Inquisition. In this book, he reported that falling bodies always travel along the same sort of curve, which is a parabola.
Galileo not only discovered how objects fall but also explained his discovery. The fact that falling bodies trace parabolas is a direct consequence of another fact he was the first to observe, which is that all objects, whether thrown or dropped, fall with a constant acceleration.
Galileo’s observation that all falling objects trace a parabola is one of the most wonderful discoveries in all of science. Falling is universal, and so is the kind of curve that falling bodies trace. It doesn’t matter what the object is made of, how it is put together, or what its function is. Nor does it matter how many times, from what height, or with what forward speed we drop or throw the object. We can repeat the experiment over and over, and each time it’s a parabola. The parabola is one of the simplest curves to describe. It is the set of points equidistant from a point and a line. So one of the most universal phenomena is also one of the simplest.
Figure 1: Definition of a parabola: the points equidistant from a point and a line.
A parabola is a concept from mathematics — an example of what we call a mathematical object — that was known to mathematicians well before Galileo’s time. Galileo’s observation that bodies fall along parabolas is one of the first examples we have of a law of nature — that is, a regularity in the behavior of some small subsystem of the universe. In this case, the subsystem is an object falling near the surface of a planet. This has happened a great number of times and in a great number of places since the universe began; hence there are many instances to which the law applies.
Here’s a question children may ask when they’re a bit older: What does it say about the world that falling objects trace such a simple curve? Why should a mathematical concept like a parabola, an invention of pure thought, have anything to do with nature? And why should such a universal phenomenon as falling have a mathematical counterpart that is one of the simplest and most beautiful curves in all of geometry?
Since Galileo’s discovery, physicists have profitably used mathematics in the description of physical phenomena. It may seem obvious to us now that a law must be mathematical, but it took almost 2,000 years after Euclid codified his axioms of geometry for someone to propose a correct mathematical law applying to the motion of objects on Earth. From the time of the ancient Greeks to the 17th century, educated people knew what a parabola was, but not a single one of them seems to have wondered whether the balls, arrows, and other objects they dropped, flung, or shot fell along any particular sort of curve.1 Any one of them could have made Galileo’s discovery; the tools he used were available in the Athens of Plato and the Alexandria of Hypatia. But nobody did. What changed to make Galileo think that mathematics had a role in describing something as simple as how things fall?
This question takes us into the heart of some questions easy to state but hard to answer: What is mathematics about? Why does it come into science?
Mathematical objects are constituted out of pure thought. We don’t discover parabolas in the world, we invent them. A parabola or a circle or a straight line is an idea. It must be formulated and then captured in a definition. “A circle is a set of points equidistant from a single point. … A parabola is a set of points equidistant from a point and a line.” Once we have the concept, we can reason directly from the definition of a curve to its properties. As we learned in high school geometry class, this reasoning can be formalized in a proof, each argument of which follows from earlier arguments by simple rules of reasoning. At no stage in this formal process of reasoning is there a role for observation or measurement.2
A drawing can approximate the properties demonstrated by a proof, but always imperfectly. The same is true of curves we find in the world: the curve of a cat’s back when she stretches or the sweep of the cables of a suspension bridge. They will only approximately trace a mathematical curve; when we look closer, there’s always some imperfection in the realization. Thus the basic paradox of mathematics: The things it studies are unreal, yet they somehow illuminate reality. But how? The relationship between reality and mathematics is far from evident, even in this simple case.
You may wonder what an exploration of mathematics has to do with an exploration of gravity. But this is a necessary digression, because mathematics is as much at the heart of the mystery of time as gravity is, and we need to sort out how mathematics relates to nature in a simple case, such as bodies falling along curves. Otherwise when we get to the present era and encounter statements like “The universe is a four-dimensional spacetime manifold,” we will be rudderless. Without having navigated waters shallow enough for us to see bottom, we’ll be easy prey to mystifiers who want to sell us radical metaphysical fantasies in the guise of science.
Although perfect circles and parabolas are never to be found in nature, they share one feature with natural objects: a resistance to manipulation by our fantasy and our will. The number pi — the ratio of a circle’s circumference to its diameter — is an idea. But once the concept was invented, its value became an objective property, one that must be discovered by further reasoning. There have been attempts to legislate the value of pi, and they have revealed a profound misunderstanding. No amount of wishing will make the value of pi anything other than it is. The same is true for all the other properties of curves and other objects in mathematics; these objects are what they are, and we can be right or wrong about their properties but we can’t change them.
Most of us get over our inability to fly. We eventually concede that we have no influence on many of the aspects of nature. But isn’t it a bit unsettling that there are concepts existing only in our minds whose properties are as objective and immune to our will as things in nature? We invent the curves and numbers of mathematics, but once we have invented them we cannot alter them.
But even if curves and numbers resemble objects in the natural world in the stability of their properties and their resistance to our will, they are not the same as natural objects. They lack one basic property shared by every single thing in nature. Here in the real world, it is always some moment of time. Everything we know of in the world participates in the flow of time. Every observation we make of the world can be dated. Each of us, and everything we know of in nature, exists for an interval of time; before and after that interval, we and they do not exist.
Curves and other mathematical objects do not live in time. The value of pi does not come with a date before which it was different or undefined and after which it will change. If it’s true that two parallel lines never meet in the plane as defined by Euclid, it always was and always will be true. Statements about mathematical objects like curves and numbers are true in a way that doesn’t need any qualification with regard to time. Mathematical objects transcend time. But how can anything exist without existing in time?3
People have been arguing about these issues for millennia, and philosophers have yet to reach agreement about them. But one proposal has been on the table ever since these questions were first debated. It holds that curves, numbers, and other mathematical objects exist just as solidly as what we see in nature — except that they are not in our world but in another realm, a realm without time. So there are not two kinds of things in our world, time-bound things and timeless things. There are, rather, two worlds: a world bound in time and a timeless world.
The idea that mathematical objects exist in a separate, timeless world is often associated with Plato. He taught that when mathematics speaks of a triangle, it is not any triangle in the world but an ideal triangle, which is just as real (and even more so) but exists in another realm, one outside time. The theorem that the angles of a triangle add up to 180 degrees is not precisely true of any real triangle in our physical world, but it is absolutely and precisely true of that ideal mathematical triangle existing in the mathematical world. So when we prove the theorem, we are gaining knowledge of something that exists outside time and demonstrating a truth that, likewise, is not bounded by present, past, or future.
If Plato is right, then simply by reasoning we human beings can transcend time and learn timeless truths about a timeless realm of existence. Some mathematicians claim to have deduced certain knowledge about the Platonic realm. This claim, if true, gives them a trace of divinity. How do they imagine they pulled this off? Is their claim credible?
When I want a dose of Platonism, I ask my friend Jim Brown for lunch. Both of us enjoy a good meal, during which he will patiently, and not for the first time, explain the case for belief in the timeless reality of the mathematical world. Jim is unusual among philosophers in coupling a razor-sharp mind with a sunny disposition. You sense that he’s happy in life, and it makes you happy to know him. He’s a good philosopher; he knows all the arguments on each side, and he has no trouble discussing those he can’t refute. But I haven’t found a way to challenge his confidence in the existence of a timeless realm of mathematical objects. I sometimes wonder if his belief in truths beyond the ken of humans contributes to his happiness at being human.
One question that Jim and other Platonists admit is hard for them to answer is how we human beings, who live bounded in time, in contact only with other things similarly bounded, can have definite knowledge of the timeless realm of mathematics. We get to the truths of mathematics by reasoning, but can we really be sure our reasoning is correct? Indeed, we cannot. Occasionally errors are discovered in the proofs published in textbooks, so it’s likely that errors remain. You can try to get out of the difficulty by asserting that mathematical objects don’t exist at all, even outside time. But what sense does it make to assert that we have reliable knowledge about a domain of nonexistent objects?
Another friend I discuss Platonism with is the English mathematical physicist Roger Penrose. He holds that the truths of the mathematical world have a reality not captured by any system of axioms. He follows the great logician Kurt Gödel in arguing that we can reason directly to truths about the mathematical realm — truths that are beyond formal axiomatic proof. Once, he said something like the following to me: “You’re certainly sure that one plus one equals two. That’s a fact about the mathematical world that you can grasp in your intuition and be sure of. So one-plus-one-equals-two is, by itself, evidence enough that reason can transcend time. How about two plus two equals four? You’re sure of that, too! Now, how about five plus five equals ten? You have no doubts, do you? So there are a very large number of facts about the timeless realm of mathematics that you’re confident you know.” Penrose believes that our minds can transcend the ever changing flow of experience and reach a timeless eternal reality behind it.4
We discovered the phenomenon of gravity when we realized that our experience of falling is an encounter with a universal natural occurrence. In our attempts to comprehend this phenomenon, we discerned an amazing regularity: All objects fall along a simple curve the ancients invented called parabolas. Thus we can relate a universal phenomenon affecting time-bound things in the world with an invented concept that, in its perfection, suggests the possibility of truths — and of existence — outside time. If you’re a Platonist, like Brown and Pen-rose, the discovery that bodies universally fall along parabolas is no less than the perception of a relationship between our earthly time-bound world and another, timeless world of eternal truth and beauty. Galileo’s simple discovery then takes on a transcendental or religious significance: It is the discovery of a reflection of timeless divinity acting universally in our world. The falling of a body in time in our imperfect world reveals a timeless essence of perfection at nature’s heart.
This vision of transcendence to the timeless via science has drawn many into science, including myself, but now I’m sure it’s wrong. The dream of transcendence has a fatal flaw at its core, related to its claim to explain the time-bound by the timeless. Because we have no physical access to the imagined timeless world, sooner or later we’ll find ourselves just making stuff up (I’ll present you with examples of this failing in chapters to come). There’s a cheapness at the core of any claim that our universe is ultimately explained by another, more perfect world standing apart from everything we perceive. If we succumb to that claim, we render the boundary between science and mysticism porous.
Our desire for transcendence is at root a religious aspiration. The yearning to be liberated from death and from the pains and limitations of our lives is the fuel of religions and of mysticism. Does the seeking of mathematical knowledge make one a kind of priest, with special access to an extraordinary form of knowledge? Should we simply recognize mathematics for the religious activity it is? Or should we be concerned when the most rational of our thinkers, the mathematicians, speak of what they do as if it were the route to transcendence from the bounds of human life?
It is far more challenging to accept the discipline of having to explain the universe we perceive and experience only in terms of itself — to explain the real only by the real, and the time-bound only by the time-bound. But although it’s more challenging, this restricted, less romantic route will ultimately be the more successful. The prize that awaits us is to understand, finally, the meaning of time on its own terms.
Galileo was not the first to associate motion with curves. He was just the first to do it for motion on Earth. One reason it may never have occurred to anyone before Galileo that bodies fall on parabolas is that no one had perceived those parabolas directly. The paths of falling bodies were simply too fast to see.1 But long before Galileo, people did have examples of motion slow enough to easily record. These were the motions of the sun, moon, and planets in the sky. Plato and his students had records of their positions, which the Egyptians and Babylonians had been keeping for thousands of years.
Such records amazed and delighted those who studied them, because they contained patterns — some obvious, like the annual motion of the sun, and others far from obvious, like the cycle of eighteen years and eleven days found in records of solar eclipses. These patterns were clues to the true constitution of the universe the ancients found themselves in. Over many centuries, scholars worked to decipher them, and it is by these efforts that mathematics first entered science.
But this isn’t the whole answer. Galileo used no tool not available to the Greeks, so there must have been some conceptual reason for the lack of progress on earthly motion. Did Galileo’s predecessors have some blind spot about motion on Earth that Galileo lacked? What did they believe that he didn’t?
Let’s consider the discovery of one of the simplest and most profound patterns found by ancient astronomers. The word “planet” comes from the Greek word for wanderer, but the planets don’t wander all over the sky. They all move along a great circle called the ecliptic, which is fixed with respect to the stars. The discovery of the ecliptic must have been the first step in decoding the records of planetary positions.
A circle is a mathematical object, defined by a simple rule. What does it mean if a circle is seen in the motions in the sky? Is this the visitation of a timeless phenomenon into the ephemeral, time-bound world? This might be how we would see it, but this is not how the ancients understood it. The universe, for the ancients, was split into two realms: the earthly realm, which was the arena of birth and death, of change and decay, and the heavenly realm above, which was a place of timeless perfection. For them the sky was already a transcendental realm; it was populated by divine objects that neither grew nor decayed. This was, after all, what they observed. Aristotle himself noted that “in the whole range of time past, so far as our inherited records reach, no change appears to have taken place either in the whole scheme of the outermost Heaven or in any of its proper parts.”2
If the objects in this divine realm were to move, these movements could only be perfect and thus eternal. To the ancients, it was evident that the planets move along a circle because, being divine and perfect, they could move only on the curve that was the most perfect. But the earthly realm is not perfect, so it might have seemed bizarre to them to describe motion on Earth in terms of perfect mathematical curves.
The division of the world into an earthly realm and heavenly spheres was codified in Aristotelian physics. Everything in the earthly realm was composed of mixtures of four elements: earth, air, fire, and water. Each had a natural motion: The natural motion of earth, for example, was to seek the center of the universe. Change followed from the mixing of these four essences. Aether was a fifth element, the quintessence, which made up the heavenly realm and the objects that moved across it.
This division was the origin of the connection of elevation with transcendence. God, the heavens, perfection — these are above us, while we are trapped here below. From this perspective, the discovery that mathematical shapes are traced by motions in the sky makes sense, because both the mathematical and the heavenly are realms that transcend time and change. To know each of them is to transcend the earthly realm.
Mathematics, then, entered science as an expression of a belief in the timeless perfection of the heavens. Useful as mathematics has turned out to be, the postulation of timeless mathematical laws is never completely innocent, for it always carries a trace of the metaphysical fantasy of transcendence from our earthly world to one of perfect forms.
Long after science has moved on from the cosmos of the ancients, its basic shape influences everyday speech and metaphor. We speak of rising to the occasion. We look upward for inspiration. Whereas to fall (as in “falling in love,” for instance) means to surrender to loss of control. More than that, the opposition of “ascending” and “falling” symbolizes the conflict between the corporeal and the spiritual. Heaven is above us, Hell is below. When we degrade ourselves, we sink downward into the earth. God, and everything we ultimately seek, is above us.
Music was another way the ancients experienced transcendence. Listening to music, we often experience a profound beauty that takes us “out of the moment.” It’s not surprising that behind the beauty of music the ancients sensed mathematical mysteries waiting to be decoded. Among the great discoveries of the school of Pythagoras was the association of musical harmonies with simple ratios of numbers. For the ancients, this was a second clue that mathematics captures the patterns in the divine. We know few personal details about Pythagoras and his followers, but we can imagine that they noticed that an affinity for mathematics often accompanies a talent for music. We would say that mathematicians and musicians share an ability to recognize, create, and manipulate abstract patterns. The ancients might have talked instead of a shared ability to perceive the divine.
Galileo was exposed to music as a child, before he was a scientist.3 His father, Vincenzo Galilei, was a composer and an influential music theorist, who is said to have stretched violin strings across the attic of their house in Pisa so his young son could experience the relationship between harmony and ratio. Bored during a service in the Pisa cathedral, Galileo noticed that the time it took a hanging lamp to sway from side to side was independent of how wide its swing was. This independence of the period (meaning the time it takes to complete one swing or orbit) from the amplitude of a pendulum was one of his first discoveries. How did he manage it? We would use a stopwatch or a clock, but Galileo didn’t have those available. We can imagine that he simply sang to himself as he watched the lamps sway over his head, since he later claimed to be able to measure time to within a tenth of a pulse-beat.
Galileo evinced a musician’s showmanship as well, when he took the case for Copernicanism to the people. He wrote his ideas down in Italian instead of Latin, the language of scholars, vividly conveying them through dialogues in which imagined characters converse about science as they share a meal or a walk. For this he is praised as a democrat who disdained the hierarchy of church and university to appeal directly to the intelligence of the common person.
But as brilliant a polemicist and experimenter as he was, what’s stunning about Galileo’s work are the new questions he asked — thanks in part to the liberation from ancient dogma that was the legacy of the Italian Renaissance. The ancient distinction between the earthly and divine realms that had long kept people from thinking seems to have left Galileo unimpressed. Leonardo had discovered proportion and harmony in static form, but Galileo looked for mathematical harmony in everyday motions, such as those of pendulums and balls rolling down inclined planes. Before he was democratic in his strategy of communication, he was a democrat about the universe.
Galileo destroyed the divinity of the sky when he discovered that heavenly perfection was a lie. He did not invent the telescope, and he may not have been the only one who used the new invention to look at the heavens. But his unique perspective and talents led him to make a fuss about what he saw there, which was imperfection. The sun has spots. The moon is not a perfect sphere of quintessence; it has mountains, just like Earth. Saturn has a strange threefold shape. Jupiter has moons, and there are vastly more stars than those seen with the naked eye.
This decline of divinity had been anticipated a few years earlier, in 1577, when the Danish astronomer Tycho Brahe watched a comet penetrate the perfect spheres of Heaven. Tycho was the last and greatest of the naked-eye astronomers, and he and his assistants accumulated over his lifetime the best measurements of planetary motions that had ever been made. These sat in his record books undecoded until 1600, when he employed an irascible young assistant, Johannes Kepler.
The planets move along the ecliptic, but they are not seen to move consistently. They all move in the same direction, but occasionally pause and reverse themselves, moving backward for a while. This retrograde motion was a great mystery to the ancients. Its real meaning is that the Earth is a planet, too, which moves around the sun as the other planets do. The planets appear to stop and start only from Earth’s perspective. Mars moves eastward in our sky when it’s ahead of us and reverses direction when Earth catches up. Its retrograde motion is simply an effect of Earth’s motion, but the ancients couldn’t see it that way, because they were stuck with the false idea that the Earth is at rest at the center of the universe. Since Earth is still, the perceived motion of the planets must be their real motion; hence the ancient astronomers had to explain the retrograde motions as if they were caused by the planets’ intrinsic motion. To do so, they imagined an awkward arrangement involving two kinds of circles, in which each planet was attached to a small circle rotating around a point that itself moved on a bigger circle around the Earth.
The epicycles, as these mini-circles were called, rotated with a period of one Earth year, because they were nothing but the shadow of Earth’s motion.4 Other adjustments required still more circles; it took fifty-five circles to get it all to work. By assigning the right periods to each of the big circles, the Alexandrian astronomer Ptolemy calibrated the model to a remarkable degree of accuracy. A few centuries later, Islamic astronomers fine-tuned the Ptolemaic model, and in Tycho’s time it predicted the positions of the planets, the sun, and the moon to an accuracy of 1 part in 1,000 — good enough to agree with most of Tycho’s observations. Ptolemy’s model was beautiful mathematically, and its success convinced astronomers and theologians for more than a millennium that its premises were correct. And how could they be wrong? After all, the model had been confirmed by observation.
Figure 2: A schematic of the Ptolemaic vision of the universe.5
There’s a lesson here, which is that neither mathematical beauty nor agreement with experiment can guarantee that the ideas a theory is based on bear the slightest relation to reality. Sometimes a decoding of the patterns in nature takes us in the wrong direction. Sometimes we fool ourselves badly, as individuals and as a society. Ptolemy and Aristotle were no less scientific than today’s scientists. They were just unlucky, in that several false hypotheses conspired to work well together. There is no antidote for our ability to fool ourselves except to keep the process of science moving so that errors are eventually forced into the light.
It fell to Copernicus to decipher the meaning of the fact that all the epicycles have the same period and move in phase with the sun’s orbit. He put Earth in its rightful place, as a planet, and the sun near the center of the universe. This simplified the model but introduced a tension the ancient cosmology couldn’t survive. Why should Earth’s sphere be any different from those of the heavens, if Earth is just another planet traveling through the heavens?
However, Copernicus was a reluctant revolutionary who missed other clues. A big one was that even after Earth’s motion was accounted for, the planets’ orbits were not precisely circles. Unable to escape the idea that motions on the sky must be compounded from circles, he solved this problem just as Ptolemy had fourteen centuries earlier. He introduced epicyles as needed to get the theory to fit the data.
The least circular orbit is that of Mars. It was Kepler’s great luck — and science’s too — that Tycho assigned to him the problem of deciphering the orbit of Mars, and, after working for many years after he left Tycho’s service, Kepler found that Mars traces an ellipse, not a circle, in space.6
This was revolutionary in ways that may not be apparent to a modern reader. In an Earth-centered cosmology, the planets don’t trace a closed path of any sort, because their paths relative to Earth each combine two circular motions with different periods. It is only when the orbits are plotted with respect to the sun that they make closed paths. Only then does it become possible to ask what the shape of an orbit is. So putting the sun at the center deepens the harmony of the world.
Once the planetary orbits were understood to be ellipses, the explanatory power of Ptolemy’s theory was shattered. A slew of new questions arose: Why do the planets move in elliptical orbits? And what keeps them from wandering off? What compels them to move at all, rather than just sitting still in space? Kepler’s answer was a wild guess that turned out to be half right: What moves the planets around in their orbits is a force from the sun. Imagine the sun as a rotating octopus, its arms sweeping the planets around as it turns. This was the first time anyone had suggested the sun as the source of a force that affects the planets. He just got the direction of the force wrong.
Tycho and Kepler smashed the heavenly spheres and in so doing unified the world. This unification had grave implications for the understanding of time. In the cosmology of Aristotle and Ptolemy, a timeless realm of eternal perfection surrounds the earthly realm. Growth, decay, change, all the evidence of a time-bound world is restricted to the small domain below the sphere of the moon. Above it is perfect circular motion, unchanging and eternal. Now that the sphere separating the time-bound and the timeless was smashed, there could be only one notion of time. Would this new world be time-bound throughout, with the whole universe subject to growth and decay? Or would timeless perfection be extended to all of creation, so that change, birth, and death would be seen as mere illusions? We still struggle with this question.
Kepler and Galileo did not solve the mystery of the relationship between the divine, timeless realm of mathematics and the real world we live in. They deepened it. They breached the barrier between sky and Earth, putting Earth in the sky as one of the divine planets. They found mathematical curves in the motions of bodies on Earth and the planets around the sun. But they could not heal the fundamental rift between time-bound reality and timeless mathematics.
By the middle of the 17th century, scientists and philosophers confronted a stark choice. Either the world is in essence mathematical or it lives in time. Two clues to the nature of reality hung in the air, expectant and unresolved. Kepler had discovered that the planets move along ellipses. Galileo had discovered that falling objects move along parabolas. Each was expressed by a simple mathematical curve and each was a partial decoding of the secret of motion. Separately they were profound discoveries; together they were the seeds of the Scientific Revolution, which was about to flower.