I am a little piece of nature.

When the young journalist Walter Lippmann wrote a book in 1922 called Public Opinion, he titled the first chapter “The World Outside and the Pictures in Our Heads.” There was, he observed, a big difference between the two.

“What each man does is based not on direct and certain knowledge, but on pictures made by himself or given to him,”¹ Lippmann noted. “The world that we have to deal with politically is out of reach, out of sight, out of mind. It has to be explored, reported, and imagined.”²

Such truths about the political world, Lippmann would no doubt have agreed, apply to the natural world as well. For the pictures of the natural world that science provides are not snapshots of a naked reality, but artistic renditions that attempt to capture reality’s essence. Reality always appears distorted by the imperfection of human senses

and the filters of the human mind. Scientists interpret reality’s shadows, helping to guide life’s journey through nature’s jungles.

But if nature is separate from science’s pictures of it, on what foundation does science build its claim to reflect some “truth” about reality? As its critics often point out, science’s version of the “truth” is always changing. From the ancient Greeks, to Newton, to Einstein, to today, science’s comprehension of the universe has evolved, mutated, matured, and been reborn. New nuances constantly emerge, and radical departures from textbook dogma occasionally appear.

Yet while scientists agree that their knowledge is tentative, almost all nevertheless insist that it captures some truth about an objective reality. Many of science’s critics, on the other hand, dispute the very idea of a truth to be captured. They argue that “reality” is beyond the reach of the human mind. The laws of physics are not really deep truths about nature, the critics claim, but mere agreements among a community of scientists about how to talk about nature.

In responding to such claims, many scientists are content to side with the seventeenth-century German philosopher Gottfried Wilhelm von Leibniz. If we can apply reason to the world and not be deceived, Leibniz said, the world is “real enough.” And from a practical viewpoint, science’s success in applying reason to reality is spectacular.

But I think a more compelling case can be made for science’s ability to grasp an independent reality. That case hinges on the success of mathematics in describing the physical world and on math’s ability to enable prediscoveries.

Somehow, humans have been able to discover the laws governing nature, in the form of symbols and the rules for combining them. Those symbols, imaginatively manipulated, have foretold the existence of strange objects and phenomena—antimatter and quarks, neutrinos and black holes, radio waves and vacuum energy, the

expansion of the universe and the curvature of space. Most scientists believe that math’s success in this regard signifies something deep and true about the universe, disclosing an inherent mathematical structure that rules the cosmos, or at least makes it comprehensible.

Nevertheless, scientists have a tough time explaining how it is that math works so well. As Eugene Wigner expressed it so eloquently four decades ago, “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”³ And Wigner’s essay did not even address the power of mathematics to discover new things in advance of any physical clues to their existence. It’s those prediscoveries, I believe, that provide science’s best clue to the existence of a reality that science can perceive. Is there something real out there? Math’s ability to divine the presence of strange matters in the universe argues strongly that yes, there is.

Yet a recent popular book seems to contradict that idea completely. As I mentioned in the Introduction, two cognitive scientists argue that math is merely a human invention that has nothing inherent to do with any external reality.

George Lakoff, of the University of California, Berkeley, and Rafael Núñez, of Berkeley and the University of Freiburg, in Germany, see a world ruled not by math, but by the human brain. Whatever might be “real,” they write, human knowledge depends solely on the brain and its own ways of finding things out. Math is not a discovery about the external world, but an invention rooted in metaphors linked to human thoughts, sensations, and actions.

“Where does mathematics come from?” Lakoff and Núñez ask. “It comes from us!” they answer in their book, Where Mathematics Comes From. “We create it, but it is not arbitrary,” they write. “It uses the

basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history.”⁴

All mathematical ideas, Lakoff and Núñez contend, are elaborate metaphors. Those metaphors are drawn from real-world experience and then linked and mixed to guide mathematical practice. Basic principles of arithmetic, algebra, trigonometry, mathematical logic, and other math fields all rely on metaphorical reasoning.

Such metaphors underlie rigorous systems of deduction and calculation. Arithmetic, for example, can be envisioned as movement along a path with markers placed at equal intervals, the metaphorical basis for the concept of a number line. Other metaphors can be identified to illustrate the ideas underlying more advanced math. Therefore, Lakoff and Núñez conclude, math is a mere human invention, a systematic way of capturing the way the brain sees the world.

In their book, Lakoff and Núñez declare that math succeeds in science only because scientists force it to. The fit between mathematics and the regularities in the world is all in the mind of the mathematician-scientist, not in the physical universe outside.

At a meeting in San Francisco, in February 2001, I heard Lakoff defend this view. He hedged a little on the issue of whether math really does exist in the world apart from in the human brain. “The only mathematics that we know is the mathematics that our brain allows us to know,” he said.⁵ So any question of math’s being inherent in physical reality is moot, since there is no way to know whether it is or not. “Mathematics may or may not be out there in the world, but there’s no way that we scientifically could possibly tell,” he argued.

Well, I suppose he might be right. But I doubt it. That view surely does not correspond to what many great scientists think about the issue. I appreciate much more the view expressed by Heinrich Hertz,

who discovered electromagnetic waves after James Clerk Maxwell prediscovered them, in commenting on Maxwell’s theory.

“It is impossible to study this wonderful theory without feeling as if the mathematical equations had an independent life and an intelligence of their own, as if they were wiser than ourselves, indeed wiser than their discoverer, as if they gave forth more than he had put into them,” Hertz said.⁶

Murray Gell-Mann, prediscoverer of quarks, makes a similar point. He views efforts to find the ultimate theory of nature’s particles and forces not as a construction job, but as an exploration. “It seems that this whole theory is lurking there in some mathematical space,” Gell-Mann said during a talk at Caltech in 2000. “It is there to be found. . . . The search for it appears to be a process of discovery, not invention. You are not adding bells and whistles in an effort to fit some empirical facts. You are gradually finding out what that preexisting self-consistent structure is.”⁷

I don’t think Gell-Mann would like Lakoff and Núñez’s book.

Still, even if math resides in the physical world, that fact doesn’t solve the big mystery. Sure, humans can observe the universe and then find equations that capture patterns in what goes on out there. You can watch planets move through the sky, for example, and detect regular features that allow you to infer equations that describe the path of any of them. That’s what physics is all about—finding the formula that applies in general to everything within a class of phenomena. Humans, in other words, can translate reality into squiggles on paper.

But how is it that those squiggles then reveal aspects of reality that had never been observed? It’s like translating Euripides from Greek into English and getting the works of Sophocles as a bonus. How do you get more out of the equations than what you put in? How is prediscovery possible?

During the months I spent writing this book, I posed that last question to many physicists. The quickest answer came from Rocky Kolb.

“A lot more things are proposed than ever turn out to be true,” he said. “If enough people are proposing enough things, some of them are bound to be right. A lot of it is luck.”

No doubt that is part of the answer. If that were the whole story, though, I could end the book now. But Rocky acknowledged that there is more to it than that. In the game of prediscovery, some people are suspiciously lucky.

“When they do it more than once, it’s like winning the lottery twice in a row,” Rocky says. “You start to suspect there’s something funny about the balls in there. Obviously they have some insight or imagination that I don’t see, that I don’t have.”⁸

Somehow the great prediscoverers have an intimate relationship with nature, or have an enhanced intuition about how the universe operates. Hertz said something like that about Maxwell and his equations. “Such comprehensive and accurate equations only reveal themselves to those who with keen insight pick out every indication of the truth which is faintly visible in nature,” Hertz said.⁹

Maxwell developed his equations through contemplating actual physical devices. His appreciation of the way the physical world worked guided the search for the right math. In a similar way, Alexander Friedmann’s intimacy with the atmosphere guided his interpretation of Einstein’s equations applied to the universe. Friedmann knew the weather; he knew how to translate the squiggles on paper into wind speeds and air pressures. His experience relating equations to the atmosphere surely prepared him to better comprehend the relationship of Einstein’s equations to all of space.

Einstein himself, the greatest prediscoverer of them all, snared secrets from nature as though he somehow had illegal access to

inside information. Gerald Holton summed it up this way, using Einstein as the archetype for scientific genius: “There is a mutual mapping of the mind and lifestyle of this scientist, and of the laws of nature.”¹⁰

In other words, Einstein’s mind somehow embodied a map of the natural world. But when Einstein said he was “a little piece of nature,” he was speaking not only for himself. Everybody possesses a mental model of reality. Your brain consults that model when directing your actions, using it as a guide for navigating in the world. Why does this work? Because people aren’t mere observers of a real world outside. People and their brains are part of the world that science tries to discover and explain.

From this point of view, Lakoff and Núñez may not be quite as far off base as they seem. In fact, they may have latched on to part of the solution, even while denying the existence of the mystery. They insist, you’ll recall, that math is merely a human invention and that any usefulness for describing nature arises simply from the fact that this is the way our brains work. But if math is merely a human invention, then how can it tell us about things in the real world that have not yet been seen? Perhaps because our brains are part of the physical world, too. Maybe the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life).

Einstein was able to exploit this intimate relationship with nature more successfully than most humans. But how did he do it—let’s say, how did he get in touch with his inner self? It looks to me that he accomplished what he did by recognizing the power of ideas, especially of ideas in which he could find the principles that nature puts on a pedestal. Einstein showed that by understanding some one important thing about reality, and then by elevating that insight to the status of a principle, you could discern many previously unknown things about reality.

At this point, I’m struck with how opposite this picture is to the one I drew in my previous book, The Bit and the Pendulum. In that book I identified great technologies as the inspirations for progress in science. The mechanical clock provided the metaphor exploited by Newton to generate the mechanical “clockwork” view of a universe governed by force. The steam engine inspired Sadi Carnot to lay the foundations for thermodynamics, a science picturing a universe ruled by energy. In the twentieth century the computer infiltrated scientific culture, driving new investigations of natural processes that can be described in terms of information storage and processing.

Now it looks as though I’m saying the opposite. Progress in science—and especially in prediscovering unknown things— is driven not by grand technologies, but by grand ideas. In fact, I see three main ideas that have taken turns inspiring the last two centuries of progress in physics and cosmology. In the nineteenth century (and before), the guiding idea was “geometry.” In the twentieth century, the most fruitful principle was “symmetry.” And the idea of the twenty-first century, I believe, will be “duality.” Examining those ideas goes a long way to helping understand the mystery of prediscovery.

The idea behind geometry is old and simple: insight into nature can be gained by building an edifice of logical deductions based on “selfevident” axioms. The tricky thing about geometry, of course, is the apparently innocent idea of self-evident. It seems to me that self-evident is just a way of saying that that’s the way things look like in the world. Geometry, in other words, did not originate from pure thought but from human experiences with nature. Long before Euclid got around to codifying its logical foundations, geometry developed from solving the practical problems encountered by Egyptian surveyors.

Euclid compiled a lot of existing geometrical knowledge and showed how its insights could be derived from a small number of

definitions and propositions. To the extent that those propositions merely reflected common experience, geometry was not really an independent, strictly logical system divorced from any certain connection to reality. It was rooted in reality, or at least in reality as it appeared to the early geometers. In particular, Euclid’s famous fifth postulate could only have reflected physical considerations. Parallel lines never meet, Euclid said. Or the shortest line connecting two parallel lines meets them at right angles. It’s all the same thing. But there’s no way to prove it—there’s no strictly logical reason why it has to be true, even given the other axioms. The only way you would know it’s “true” is by drawing lines and measuring angles in the real world.

Nevertheless, nineteenth-century thinkers thought Euclid’s geometry revealed truths about reality. Here’s the odd thing, though. Euclidean geometry—the geometry that incorporates the lessons from real life—turns out to be the wrong geometry for describing real life. The non-Euclidean geometries of the nineteenth century did away with Euclid’s fifth postulate, sticking to a more rigorously logical-mathematical approach, uncontaminated by physical appearances. Gauss, Lobachevsky, Bolyai, and especially Riemann recognized the limitations of experience in grasping reality. It was the divorce of geometry from experience that led to the great prediscovery of space (or spacetime) curvature. Pure mathematical reasoning gave back something that wasn’t put into it.

Geometry, of course, is intimately connected with the notion of symmetry. Moving and rotating and reflecting geometrical figures leave the deductions about them unchanged—a good thing, or otherwise Euclid would have been wasting his time. Einstein exploited the symmetry in geometry to explain how the laws of nature remained unchanged from different points of view.

In his general theory of relativity, Einstein reaped the benefits of Riemann’s insights, showing how the new geometry could describe gravity in such a way that everybody observes the same law of gravitation. That accomplishment was rooted in Einstein’s success with symmetry in special relativity. Maxwell’s equations, in Einstein’s eyes, captured something essential about reality. It would be heresy to alter those equations merely to accommodate the motion of some observer at a constant rate of speed along a straight line. There must be a symmetry that keeps the laws the same even when motion changes. Einstein built special relativity on that notion.

The success of special relativity, and then of general relativity a decade later, revealed symmetry’s power to twentieth-century science. Later physicists, building on Einstein’s inspiration and the work of others like Emmy Noether, Hermann Weyl, and Eugene Wigner, constructed a whole outline of existence from the blueprints contained in mathematical symmetries. The symmetry approach brought science to an incredibly deep understanding of nature’s laws, in the form of the Standard Model of particles and forces, well before the end of the twentieth century.

Just as the symmetries recorded their greatest victories, though, progress stalled. Then, in the twentieth century’s last years, a new idea for the new millennium arose, with the promise of sustaining the quest for the ultimate understanding of the universe. That idea encompasses a special kind of symmetry that goes by the name of duality.

Duality at first glance seems to be an utterly simple idea. The back of a house looks different from the front, but it’s the same house. So what? Such a duality seems to offer a rather trivial insight. As superstring theorist Brian Greene points out, many dualities are trivial—say, Einstein’s theory presented in Chinese or in English.

True, they are different-looking (“dual”) views of the same theory. But if you learned Einstein’s theory in English, you will discover nothing more about it by reading a Chinese translation.

In other situations, duality runs deeper—like the duality between ice and steam. Suppose you are a scientist in a primitive culture, trying to figure out what steam is made of. It won’t be easy—it’s hard to keep steam contained, and you’re likely to scald yourself. But then you discover that ice and steam are just two very different appearances of the same substance. Ice is easier to study. You can experiment with it, measure things about it, and ultimately figure out that it’s made of molecules containing hydrogen and oxygen. You can learn things you never knew about steam by studying ice, thanks to the ice-steam duality.

In the same way, theories of the universe can look very different yet be identical on some deeper level. And one of those theories might be much easier to use. This is precisely the situation in certain string theories. When strings interact strongly, the math describing them is very hard to do. But when strings interact weakly, the math is much easier. To solve hard problems with strongly interacting strings, you can use the dual theory, with weakly interacting strings.

Recently physicists have explored another profound duality, discovered by Juan Maldacena at Harvard. (He is now at the Institute for Advanced Study in Princeton). He found that the physical description of a volume of space (let’s say, for example, the interior of a black hole) can be equally well represented by the boundary of that space (the surface, or “horizon” of the black hole). The two descriptions are dual. It’s like saying you could describe all the three-dimensional objects in a room merely by looking at a wall—a two-dimensional surface containing three-dimensional information. (Think about it: the universe might be like a room with walls covered by mirrors.)

Physicists are still trying to figure out what all of this means. But at least it’s clear that the search for an ultimate physical theory will

not produce one and only one picture. Different points of view reveal different pictures.

“The main lesson of recent progress in string dualities has been the recognition of the existence of different viewpoints on a physical theory all of which are good for answering some question,” says Harvard’s Cumrun Vafa. “It is this democracy of physical descriptions (not the superiority of one over the other) which is the lesson of string duality.”¹¹

Duality therefore expresses a special kind of symmetry, a symmetry between theories. But duality symmetries still share much with the symmetries of geometry and the symmetry of the twentieth century embodied in gauge theories. Geometry, symmetry, and duality are themselves just different aspects of the same thing. They all point to an underlying sameness, an underlying identity that scientists struggle to discern. Geometry describes symmetries of space. In Einstein’s hands, symmetry transformed space into spacetime, an arena where gauge symmetry could describe particles and forces. Duality symmetries show now that spacetime is not the final answer. But nobody yet knows what to replace it with.

At least the idea of duality has helped me immensely in reconciling my previous book with this one. Obviously, the two books are dual to each other.

Geometry, symmetry, and duality have taken science a long way. Taken together, they also explain a lot about why an ultimate picture of reality has been so elusive. It’s because things are not always what they seem. Euclidean geometry looks right, but only if there’s no matter around to warp space. Newton’s laws of motion seem fine, unless you happen to be moving really fast. J. J. Thomson won a Nobel Prize for proving the electron is a particle; then his son George

won a “dual” Nobel 31 years later for showing that the electron is a wave.

At the same time, though, geometry, symmetry, and duality show how to find an underlying sameness that makes the world consistent with itself. Riemannian geometry makes general relativity work, so that you can describe the universe with whatever system of coordinates you choose. The symmetries of special relativity translate the description of slow-moving or fast-moving bodies into the same language, keeping nature’s laws the same for everybody. Duality permits string theory to appear in many disguises, but also allowed Edward Witten to realize that the five superstring theories were just five doors to the same M-theory house. All this progress points to an underlying unity in a hard-to-see reality. And that unity is our strongest hint of objective reality, however dimly we perceive it.

This picture is clearly in harmony with Einstein’s longtime quest to find the unified theory of gravity and electromagnetism, a quest that has mutated into today’s search for a “theory of everything.” And it’s a picture in harmony with the writings of the seventeenth-century Dutch philosopher Baruch Spinoza, who deeply influenced Einstein’s attitudes on these issues. Spinoza’s God, Einstein wrote, “revealed himself in the harmony of all being.” Understanding that harmony was Einstein’s lifelong goal.

In Spinoza’s view, all the variety in the perceived world ultimately stemmed from an essential underlying harmony.¹² But science can’t grasp that reality as a whole. Science must deal with the fragments of nature accessible to human perception. To Spinoza (and, apparently, to Einstein), all the fragments are pieces of an infinite puzzle. The whole puzzle is a unified “substance” at reality’s foundation. Spinoza’s substance is only imperfectly perceptible, as humans have access only to its manifestations, not its inherent unity.

I think this point of view explains prediscovery. At least I think such a case can be made—and has, in fact, been made pretty well by

the French mathematician Henri Poincaré. During the first years of the twentieth century, Poincaré was one of France’s great scientific popularizers. He thought deeply and wrote clearly about the relationship of science and mathematics to reality. And while he never (as far as I’ve been able to find) addressed the idea of prediscovery directly, things he said in different places offer a clue to what he would have said about it had he been asked. My reconstruction of what he would have said goes something like this:

1. You can’t add apples and oranges.

2. If it gets you where you want to go, the map must be right.

3. If you used math to make the map, the world is not a fruit salad.

Of course, Poincaré said these things a little more eloquently.

“It might be asked, why in physical science generalization so readily takes the mathematical form,” he wrote in Science and Hypothesis. “It is because the observable phenomenon is due to the superposition of a large number of elementary phenomena which are all similar to each other.”¹³ (Spinoza might say the elementary phenomena are all expressions of his harmonious substance at the root of reality.) In other words, math works because it deals with similar things.

“Mathematics teaches us, in fact, to combine like with like,” Poincaré wrote. “Its object is to divine the result of a combination without having to reconstruct that combination element by element.”¹⁴ You don’t have to count to 100 five times, for instance, to learn that 5 times 100 equals 500. But for math’s shortcut to work, you have to combine like with like. Multiplying 5 apples times 100 oranges would be senseless. And so math’s success in science implies that science deals with “likes.”

The second point, about maps getting you where you want to go, refers to the use of equations to represent reality. We can’t see the

“true” underlying reality directly, so we substitute images for the “true” objects. We assign symbols that can be manipulated mathematically to represent these images. The success of the math signifies that the relations between the images conform to the relations between the underlying “real” objects.

In other words, equations represent relationships. Using equations you can construct a mathematical map that helps you find your way through nature.

For example, Poincaré suggested, consider physical concepts such as motion or electric current. You can assign symbols to stand for these concepts. Then you can deduce physical consequences by manipulating the symbols according to the formal rules of math. By definition, those formal rules have nothing to do with reality. But those rules do succeed in making accurate physical deductions. The rules of math must therefore contain something true about the relations between the physical things in the real world. “If the equations remain true, it is because the relations preserve their reality,” Poincaré wrote.¹⁵

To sum it up, Poincaré says math works because it deals with like things. Since math works for science, science must be dealing with like things, not a fruit salad of disconnected realities. Math’s success suggests a deep underlying “likeness” in the universe, a simplicity, or unity, which reveals a connectedness, or universal set of relationships, connecting absolutely everything. It is math’s ability to express those relationships that allows science to identify truths about reality before they are observed. Since everything is harmoniously connected, observing part of the whole can tell you about other parts you haven’t seen yet.

Some theories are, of course, better than others at facilitating prediscovery. Observations are always approximate, and theories based solely on observations may therefore not capture precisely the right relationships. Those theories might explain observations over a restricted range of conditions but fail under other circumstances.

This is true even of some very good theories, such as Newton’s explanation of gravity. It fails when gravity is very strong.

Other theories, though, capture more of the whole of the universe and preserve more nearly all of the underlying connections and relations. These theories, like Einstein’s, provide a vastly greater range of insights into the universe, and lead to prediscoveries.

Poincaré understood how this process works. Imagine, he says, a chart depicting all “the variations of the world.”¹⁶ At each point in time everything in the universe is in a particular arrangement. At the next instant the arrangement will be slightly different. (The differences from one instant to the next would be the result of the combined operations of all the laws of physics.) A graph of those changes over time would take the shape of a curve. A good mathematician could figure out an equation to describe that curve. With that equation, then, we could extend the curve to predict the future of the universe or ascertain the past.

But earthbound mathematicians can never see the whole curve. Human theories are always based only on one arc, one piece of the universal curve. Two theories based on different arcs might deduce different equations to describe the whole curve. (Quantum mechanics and general relativity describe different aspects of reality exquisitely well, for example, while appearing to be incompatible.)

However, Poincaré notes, a greater intellect, or a similar intellect with a wider field of view, could perceive the region between these two arcs and construct a better equation. That equation could describe not only both arcs but also the part of the curve in between. And sometimes human scientists can figure out that better equation before they see the whole curve. If they get the right equation, it will then tell them things about regions of the curve that have not yet been measured. In that way the math can reveal things about the physical world that haven’t yet been seen.

Historically, this is almost exactly what happened a century ago when Max Planck introduced quantum theory to the world. In 1900

there were two formulas for predicting how bright different colors of light would be when emitted from a hot oven (or “black body” cavity). One formula worked well if wavelengths of the light were short, toward the blue end of the spectrum; the other worked well when wavelengths were long, toward the red end of the spectrum. Planck found the equation that worked for all wavelengths.

And then he figured out what his equation meant. His equation could be true, he found, only if energy could not be divided into smaller and smaller amounts. Energy had to come in packets, or quanta, the way money comes in units no smaller than pennies. This realization launched the quantum revolution. Today scientists know that the world at its foundations is very, very strange, obeying laws that strike many people as bizarre. But it’s the way the world is.

Planck’s formula explained the spectrum of black body radiation. But that formula told him much more than just the intensity of light at different colors. It told him, once he thought about it, something deep and true about the nature of reality at the most fundamental of levels, something that applied to much more than light coming out of an oven. Somehow the math told Planck about something real. Planck’s math revealed something that nature had been hiding. He got more out of his equation than what he put in.

In this regard I think we should remember something else that Planck once noted. “Great caution must be exercised,” he said, “in using the word, real.”¹⁷ Poincaré expressed similar caution. “What this world consists of, we cannot say or conjecture; we can only conjecture what it seems, or might seem to be to minds not too different from ours,” he wrote.¹⁸ What we learn about reality are relationships, relationships expressed in mathematics. The objects of reality we think we have discerned are merely images that allow us to visualize the relationships that math reveals.

“The true relations between these real objects,” Poincaré commented, “are the only reality we can attain.”¹⁹ Equations may show a relationship between motion and electric current, for example. “But

these are merely names of the images we substituted for the real objects which Nature will hide for ever from our eyes,” Poincaré wrote.

Nevertheless, math’s success tells us that the universe is real. Science evolves, though, because our imperfect perceptions of the underlying reality can never capture the entire true picture. One approximate picture after another emerges as new ideas inspire new images. To describe what that reality is like, we have no recourse but to say what it seems to be like, and we naturally choose for comparison those tangible mechanisms and processes that capture and shape our cultural imaginations.

To ancient thinkers, with minds different from ours, the world seemed to be a different place—a series of concentric spheres. In the late Middle Ages, people began to view the universe as a clockwork, inspired in this belief by the cultural importance of the mechanical clock. By the end of the nineteenth century the universe seemed more like a big steam engine (one that was running out of steam), the prime mover of the Industrial Revolution. Nowadays, to many scientists, the world seems a lot like a computer—a point of view clearly inspired by the computer’s role as society’s dominant machine.²⁰

These metaphors are like the images that Poincaré spoke of, capturing essential relations stemming from that unseen reality beyond our senses. This may strike some as unsatisfying, for it seems that what we usually think of as real isn’t really real at all. But in the end the situation is just as Lippmann perceived it in politics. The world outside is different from the pictures in our heads, and the “real” world is not exactly the same thing as the pictures in the scientists’ heads. As Lippmann wrote in Public Opinion, “the real environment is altogether too big, too complex, and too fleeting for direct acquaintance.”²¹ And scientists are only human.

“Man is no Aristotelian god contemplating all existence at one glance,” Lippmann wrote. “He is the creature of an evolution who can just about span a sufficient portion of reality to manage his survival.”²²