Fundamentals Read online

  The seventeenth century saw dramatic theoretical and technological progress on many fronts, including in the design of mechanical machines and ships, of optical instruments (including, notably, microscopes and telescopes), of clocks, and of calendars. As a direct result, people could wield more power, see more things, and regulate their affairs more reliably. But what makes the so-called Scientific Revolution unique, and fully deserving of the name, is something less tangible. It was a change in outlook: a new ambition, a new confidence.

  The method of Kepler, Galileo, and Newton combines the humble discipline of respecting the facts and learning from Nature with the systematic chutzpah of using what you think you’ve learned aggressively, applying it everywhere you can, even in situations that go beyond your original evidence. If it works, then you’ve discovered something useful; if it doesn’t, then you’ve learned something important. I’ve called that attitude Radical Conservatism, and to me it’s the essential innovation of the Scientific Revolution.

  Radical Conservatism is conservative because it asks us to learn from Nature and to respect facts—key aspects of what is called the scientific method. But it is radical, too, because it pushes what you’ve learned for all it’s worth. This is no less essential to how science actually works. It provides science with its cutting edge.

  IV

  This new outlook was inspired, above all, by developments in a subject that even in the seventeenth century was already ancient and well developed: celestial mechanics, the description of how objects in the sky appear to move.

  Since long before the beginning of written history, people have recognized such regularities as the alternation of night and day, the cycle of seasons, the phases of the Moon, and the orderly procession of stars. With the rise of agriculture, it became crucial to keep track of seasons, in order to plant and harvest at the most appropriate times. Another powerful, if misguided, motivation for accurate observations was the belief that human life was directly connected to cosmic rhythms: astrology. In any case, for a mixture of reasons—including simple curiosity—people studied the sky carefully.

  It emerged that the vast majority of stars move in a reasonably simple, predictable way. Today, we interpret their apparent motion as resulting from Earth rotating around its axis. The “fixed stars” are so far away that relatively small changes in their distance, whether due to their own proper motion or to the motion of Earth around the Sun, are invisible to the naked eye. But a few exceptional objects—the Sun, the Moon, and a few “wanderers,” including the naked-eye planets Mercury, Venus, Mars, Jupiter, and Saturn—do not follow that pattern.

  Ancient astronomers, over many generations, recorded the positions of those special objects, and eventually learned how to predict their changes with fair accuracy. That task required calculations in geometry and trigonometry, following complicated, but perfectly definite, recipes. Ptolemy (c. 100–170) brought this material together in a mathematical text that became known as Almagest. (Magest is a Greek superlative meaning “greatest.” It has the same root as “majestic.” Al is simply Arabic for “the.”)

  Ptolemy’s synthesis was a magnificent achievement, but it had two shortcomings. One was its complexity and, related to this, its ugliness. In particular, the recipes it used to calculate planetary motions brought in many numbers that were determined purely by fitting the calculations to observations, without deeper guiding principles connecting them. Copernicus (1473–1543) noticed that the values of some of those numbers were related to one another in surprisingly simple ways. These otherwise mysterious, “coincidental” relationships could be explained geometrically, if one assumed that Earth together with Venus, Mars, Jupiter, and Saturn all revolve around the Sun as center (and the Moon further revolves around Earth).

  The second shortcoming of Ptolemy’s synthesis is more straightforward: It simply isn’t accurate. Tycho Brahe (1546–1601), in an anticipation of today’s “Big Science,” designed elaborate instruments and spent a lot of money building an observatory that enabled much more precise observations of planetary positions. The new observations showed unmistakable deviations from Ptolemy’s predictions.

  Johannes Kepler (1571–1630) set out to make a geometric model of planetary motion that was both simple and accurate. He incorporated Copernicus’s ideas and made other important technical changes to Ptolemy’s model. Specifically, he allowed the planetary orbits around the Sun to deviate from simple circles, substituting ellipses, with the Sun at one focus. He also allowed the rate at which the planets orbit the Sun to vary with their distance from it, in such a way that they sweep out equal areas in equal times. After those reforms, the system was considerably simpler, and it also worked better.

  Meanwhile, back on the surface of Earth, Galileo Galilei (1564–1642) made careful studies of simple forms of motion, such as the way balls roll down inclined planes and how pendulums oscillate. Those humble studies, putting numbers to positions and times, might seem pitifully inadequate to addressing big questions about how the world works. Certainly, to most of Galileo’s academic contemporaries, concerned with grand questions of philosophy, they seemed trivial. But Galileo aspired to a different kind of understanding. He wanted to understand something precisely, rather than everything vaguely. He sought—and found—definite mathematical formulas that described his humble observations fully.

  Isaac Newton (1643–1727) weaved together Kepler’s geometry of planetary motion and Galileo’s dynamical description of motion on Earth. He demonstrated that both Kepler’s theory of planetary motions and Galileo’s theory of special motions were best understood as special cases of general laws, laws that apply to all bodies everywhere and for all time. Newton’s theory, which we now call classical mechanics, went from triumph to triumph, accounting for the tides on Earth, predicting the paths of comets, and empowering new feats of engineering.

  Newton’s work showed, by convincing example, that one could address grand questions by building up from a detailed understanding of simple cases. Newton called this method analysis and synthesis. It is the archetype of scientific Radical Conservatism.

  Here is what Newton himself had to say about that method:

  As in mathematics, so in natural philosophy the investigation of difficult things by the method of analysis ought ever to precede the method of composition. This analysis consists of making experiments and observations, and in drawing general conclusions from them by induction. . . . By this way of analysis we may proceed from compounds to ingredients, and from motions to the forces producing them; and in general from effects to their causes, and from particular causes to more general ones till the argument end in the most general. This is the method of analysis: and the synthesis consists in assuming the causes discovered and established as principles, and by them explaining the phenomena preceding from them, and proving the explanations.

  V

  Before leaving Newton, it seems appropriate to add another quotation, which reflects his kinship with his predecessors Galileo and Kepler, and with all of us who follow in their footsteps:

  To explain all nature is too difficult a task for any one man or even for any one age. ’Tis much better to do a little with certainty & leave the rest for others that come after you.

  A more recent quotation from John R. Pierce, a pioneer of modern information science, beautifully captures the contrast between the modern concept of scientific understanding and all other approaches:

  We require that our theories harmonize in detail with the very wide range of phenomena they seek to explain. And we insist that they provide us with useful guidance rather than with rationalizations.

  As Pierce was acutely aware, this heightened standard comes at a painful price. It involves a loss of innocence. “We will never again understand nature as well as Greek philosophers did. . . . We know too much.” That price, I think, is not too high. In any case, there’s no going back.

  I

  W
hat There Is

  1

  THERE’S PLENTY OF SPACE

  PLENTY OUTSIDE AND PLENTY WITHIN

  When we say that the something is big—be it the visible universe or a human brain—we have to ask: Compared with what? The natural point of reference is the scope of everyday human life. This is the context of our first world-models, which we construct as children. The scope of the physical world, as revealed by science, is something we discover when we allow ourselves to be born again.

  By the standards of everyday life, the world “out there” is truly gigantic. That outer plenty is what we sense intuitively when, on a clear night, we look up at a starry sky. We feel, with no need for careful analysis, that the universe has distances vastly larger than our human bodies, and larger than any distance we are ever likely to travel. Scientific understanding not only supports but greatly expands that sense of vastness.

  The world’s scale can make people feel overwhelmed. The French mathematician, physicist, and religious philosopher Blaise Pascal (1623–1662) felt that way, and it gnawed at him. He wrote that “the universe grasps me and swallows me up like a speck.”

  Sentiments like Pascal’s—roughly, “I’m very small, I make no difference in the universe”—are a common theme in literature, philosophy, and theology. They appear in many prayers and psalms. Such sentiments are a natural reaction to the human condition of cosmic insignificance, when measured by size.

  The good news is that raw size isn’t everything. Our inner plenty is subtler, but at least equally profound. We come to see this when we consider things from the other end, bottom up. There’s plenty of room at the bottom. In all the ways that really matter, we’re abundantly large.

  In grade school, we learn that the basic structural units of matter are atoms and molecules. In terms of those units, a human body is huge. The number of atoms in a single human body is roughly 1028−1 followed by 28 zeros: 10,000,000,000, 000,000,000,000,000,000.

  That is a number far beyond what we can visualize. We can name it—ten octillion—and, after some instruction and practice, we can learn to calculate with it. But it overwhelms ordinary intuition, which is built on everyday experience, when we never have occasion to count that high. Visualizing that many individual dots far exceeds the holding capacity of our brains.

  The number of stars visible to unaided human vision, in clear air on a moonless night, is at best a few thousand. Ten octillion, on the other hand, the number of atoms within us, is about a million times the number of stars in the entire visible universe. In that very concrete sense, a universe dwells within us.

  Walt Whitman (1819–1892), the big-spirited American poet, felt our inner largeness instinctively. In his “Song of Myself” he wrote, “I am large, I contain multitudes.” Whitman’s joyful celebration of abundance is just as grounded in objective facts as Pascal’s cosmic envy, and it is much more relevant to our actual experience.

  The world is large, but we are not small. It is truer to say that there’s plenty of space, whether we scale up or down. One shouldn’t envy the universe just because it’s big. We’re big, too. We’re big enough, specifically, to contain the outer universe within our minds. Pascal himself took comfort from that insight, as he followed his lament that “the universe grasps me and swallows me up like a speck” with the consolation “but through thought I grasp it.”

  The abundance of space—both its outer and its inner plenty—is the main topic of this chapter. We’ll look deeper into the hard facts, and then venture a bit beyond.

  OUTER PLENTY: WHAT WE KNOW AND HOW WE KNOW IT

  Prelude: Geometry and Reality

  Scientific discussion of cosmic distances is built on the foundation of our understanding of physical space and how to measure distance: the science of geometry. Let us begin, therefore, with the relationship between geometry and reality.

  Direct, everyday experience teaches us that objects can move from place to place without changing their properties. This leads us to the idea of “space” as a kind of receptacle, wherein nature deposits objects.

  Practical applications in surveying, architecture, and navigation led people to measure distances and angles among nearby objects. Through such work, they discovered the regularities on display in Euclidean geometry.

  As practical applications got more extensive and demanding, that framework held up impressively. So successful was Euclid’s geometry, and so majestic is its logical structure, that critical tests of its validity as a description of physical reality were rarely undertaken. In the early nineteenth century, Carl Friedrich Gauss (1777–1855), one of the all-time great mathematicians, thought it was worth doing a reality check. He measured the angles in a triangle formed by three distant mountain stations in Germany and found that they added up to 180°, as Euclid predicts, within measurement uncertainties. Today’s Global Positioning System (GPS) is based on Euclidean geometry. It performs millions of experiments like Gauss’s every day, but on larger scales and with much greater precision. Let’s take a quick look at its workings.

  To get your position using GPS, you tap into broadcasts from a collection of artificial satellites high above Earth, which know where they are. (We’ll come back to how that happens.) Currently there are more than thirty of these satellites strategically arranged around the globe. Their radio broadcasts don’t translate into talk or music. Instead, they send out simple announcements of where they are, in a digital format tailored to computers. The announcements include time stamps, which specify when they were sent. Each satellite carries a superb atomic clock onboard. That clock ensures that the time stamps are accurate. Then:

  Your GPS unit’s receiver picks up some of the satellite signals. The unit, which also has access to signals from an extensive network of ground-based clocks, computes how long the different satellite signals took to arrive. Since those signals travel at a known speed—the speed of light—the transit times can be used to determine the satellites’ distances.

  Using those distances, the positions of the satellites, and Euclidean geometry, the computer determines a unique position of the source—that is, you—by triangulation.

  The computer reports that result, and you learn where you are.

  The fully implemented GPS adds many clever refinements, but that is its central idea. This system bears an uncanny resemblance to Albert Einstein’s “thought design” of reference frames in his original paper on special relativity. In 1905, he anticipated using light beams and transit times to map out spatial positions. Einstein liked that idea because it uses a technique rooted in basic physics—the fixed speed of light—to map out space. Technology has a way of catching up with thought experiments.

  As an exercise in visual imagination, you might try to convince yourself that your distances to four satellites—each in a known position—provide more than enough information to reconstruct your position.

  (Here’s a hint: Points at a given distance from a satellite lie on a sphere centered on that satellite. If we take two spheres, centered on two different satellites, they will either intersect in a circle or not at all. Since your location is somewhere in the intersection, they’d better intersect! Now consider how a third sphere, corresponding to a third satellite, intersects that circle. Generally, they will intersect at two points. Finally, the fourth satellite’s sphere will single out one of those two points.)

  Now let’s return to the question of how the GPS’s satellites know where they are. The technical details get complicated, but the underlying idea is simple: They start from known positions, and then they track their own motion. By putting those two pieces of information together, they calculate where they are.

  In more detail: The satellites monitor their motion using onboard gyroscopes and accelerometers, like the ones in your iPhone. From the observed response of those instruments, the satellite’s computer can read out the satellite’s acceleration, using the phy
sics of Newtonian mechanics. From that input, using calculus, it calculates how much the satellite has moved. Indeed, Newton invented calculus to solve problems like this.

  If you review all the steps, you’ll see that the engineers who designed the Global Positioning System built on many non-obvious assumptions. The system relies on the idea that the speed of light is constant. It uses atomic clocks, whose design and interpretation relies on advanced principles of quantum theory, to do accurate timing. It uses the tools of classical mechanics to calculate the position of the satellites it deploys. It also makes corrections for the effect, predicted by general relativity, that the rate of clocks depends slightly on their elevation above Earth. Clocks run slower near Earth’s surface, where its gravitational field is stronger.

  Since the Global Positioning System relies on so many other assumptions in addition to the validity of Euclidean geometry, we cannot claim that it provides a clean, pure test of that geometry. Indeed, the success of GPS is not a clean, pure test of any single principle. It is a complicated system, whose design relies on a tangled web of assumptions.

  Any of those assumptions might be wrong or, to put it more diplomatically, only approximately true. If any of the assumptions that engineers assumed to be “approximately true” were significantly wrong, GPS would give inconsistent results. For instance, you might derive different positions from triangulating on different sets of satellites. Hard use can reveal hidden weaknesses.