Read an Excerpt
CHAPTER 1
Falling
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.
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. 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.
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?
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.
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 Penrose, 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.
(Continues…)
Excerpted from "Time Reborn"
by .
Copyright © 2013 Spin Networks, Ltd..
Excerpted by permission of Houghton Mifflin Harcourt Publishing Company.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.