What is now proved was once only imagined.

Lisa Randall knows a thing or two about gravity.

On the one hand, as an avid rock climber, she regularly defies gravity’s all-too-powerful pull, not always successfully. From first-hand experience she could testify that gravity can pose some serious problems.

On the other hand, as a physicist, she encounters a completely opposite problem with gravity. In the hierarchy of nature’s forces, gravity is the weakling—magnetism is more than 100 times as strong, and the forces binding an atomic nucleus together are much stronger still. Gravity is the feeblest force in the universe, and for decades physicists have wondered why.

Randall knows no magic way to reduce the risks that gravity brings to rock climbing. But she does have a good idea for explaining why gravity on the cosmic scale is so weak.

For not only is Randall an intrepid explorer of national parks, she explores the universe as well. And since her spaceship is strictly mathematical, she doesn’t have to worry about falling into a black hole or being blasted to smithereens by a supernova. And she doesn’t have to restrict her travels to the space that we can see. Lisa Randall journeys to other dimensions. There she learns about the secrets of gravity and the existence of other worlds.

She is, in a way, a twenty-first-century version of A. Square, the protagonist of Edwin Abbott’s nineteenth-century fantasy novel Flatland.

Abbott, a schoolteacher and theologian, described a totalitarian world whose inhabitants were like shadows on a sheet of paper, living very flat lives. Most men were polygons—triangles, squares, or hexagons, for example. High Priests were circles. Women were segments of straight lines.¹ The edges of all these figures were luminous so Flatlanders could see one another.

One day A. Square, a Flatlander mathematician, observed a small circle in his home. The circle would grow larger, expanding to a maximum width of about 13 inches, and then would contract again to a point before vanishing. This oscillating circle, A. Square eventually realized, was really a sphere from the third dimension, a realm of space previously unknown to the Flatlanders. Passing through the Flatlanders’ sheet-of-paper universe, the sphere would become visible only where it intersected the sheet. And the intersection of a sphere with a plane is a circle.

When A. Square tried to explain the discovery of other dimensions to his fellow Flatlanders, they threw him in jail. A. Square’s account of his ordeal was therefore dedicated with a plea to the outsideworlders who live in “space” to open their minds to the insights that extra dimensions had to offer.

Abbott’s allegory has long been popular among mathematicians and scientists. But I doubt that the depth of the truth he revealed has

been appreciated until very recently. For most of the past century, the common concept of an additional dimension stemmed from Einstein’s theory of relativity, in which the “fourth dimension” is defined as time.² Time was not what Abbott had in mind, though. He clearly proposed the existence of higher dimensions of space, urging the citizens of “spaceland” to aspire to discovering the secrets of four, five, or even six spatial dimensions.

But surely, space itself has only three dimensions. Up and down, left and right, back and forth—three ways to move. Any movement can be described as some combination of movements in those three directions. Latitude, longitude, altitude—three dimensions—three numbers to specify the location of any object. What could be more obvious?

Only that it takes some imagination to see beyond the obvious. And that is what great scientists do.

A. Square saw beyond his two-dimensional world to a universe beyond. Lisa Randall sees beyond three-dimensional space to a realm of multiple dimensions, spaces that cannot be seen because light itself refuses to go there.

In fact, Lisa Randall’s exploration of unknown dimensions shows how Abbott’s prescient fantasy captures the essence of discovering the undiscovered—seeing beyond the horizons of the obvious. It may seem obvious to most people that space has three dimensions. But Abbott saw then, as Randall sees now, that exploring only the known dimensions restricts the prospects for great discoveries—and deeper understanding—about the universe and all that it contains. If all truths were obvious truths, there would be no need for science, or for scientists. But much of nature hides itself from human senses, rendering it unfamiliar when eventually revealed. Consequently, exploring the universe turns up some pretty strange things.

Scientists find those strange things in two places—out in space, and in their heads. Roughly speaking, those who seek novelty in space

are called observers. Those who can work with their eyes closed are called theorists. There is an obvious symbiosis between these two species of scientist. When observers see something new in the cosmos, they call on theorists to explain it. When theorists dream up a new idea for something that the universe might contain, they expect observers to find it.

Theorists of the past have imagined many strange things in advance of their discovery by observers: antimatter, electromagnetic waves, black holes, and neutron stars; the expansion of the universe, the neutrino, quarks inside atoms, even atoms themselves. I like to call such instances of theoretical anticipation “prediscoveries.” They suggest that science is richer and more creative than often presented—not mere observation, experiment, induction and deduction, but a process flush with creativity and imagination.

As have their intellectual ancestors, theorists of today have imagined many strange things that observers haven’t yet found. Astrophysics journals, cosmology conferences, and World Wide Web pages are full of lengthy discourses on undiscovered objects and phenomena. Often the imaginations of scientists run completely unrestrained, and the resulting ideas bear no recognizable relation to standard science. Or even any thinkable future relation. But eliminating the flakes leaves plenty of exciting science on the edge— speculative yet plausible proposals about exotic objects that observers really do have a chance of finding someday.

Sometimes these proposals get a fair amount of media attention. For the most part, though, forecasts of new phenomena don’t get the same respect as the odd things that actually have been found. But it’s those ideas, those possible future discoveries, that transport us to the frontiers of the universe and point the way beyond. Strange ideas from theorists’ imaginations guide observers in their efforts to learn more about matter, space, and time—and in fact, to explore the realm of reality beyond matter, space, and time.

To be sure, categorizing reality in terms of matter, space, and time has served science well. And in fact, even the boldest suggestions for novel phenomena usually build on the space-time-matter framework. Many historical examples of prediscovery—and some of the best candidates today—involve inferring the existence of new kinds of particles of matter, for example. It has become something of a physics tradition. Murray Gell-Mann imagined quarks years before evidence for them appeared in experiments. Decades earlier Wolfgang Pauli had invented a bizarre new particle called the neutrino, seemingly impossible to detect but essential for salvaging an important law of physics. Experimental proof of the neutrino’s existence came a quarter century after Pauli’s prediction.

Nowadays, inventing new forms of matter is a favorite pastime for physicists attempting to solve what astronomers call the “dark matter” problem. Based on the way galaxies spin and congregate in space, scientists can tell that the universe contains more matter than anybody can see. Perhaps it is ordinary matter, just not shining like stars. But few experts think so. Most of the evidence favors dark matter of some exotic flavor, probably made of particles of a species never detected on Earth. Inspired by the mystery of the dark matter—and by the elegance of certain equations—physicists have conceived of entire zoos of undiscovered particles that might very well pervade the cosmos.

Other dark matter candidates emerge from the work of physicists like Vic Teplitz, who seeks evidence of known matter particles in new disguises, perhaps in the form of “strange quark nuggets” that might be silently raining onto (and perhaps zipping through) Earth. While computer programmers tweak the software for his nuggetdetecting “seismic telescope,” Teplitz wanders through a looking glass to explore a “mirror world” of particles and stars. They are particles and stars that can’t be seen but might be detected by the force of gravity—just the sorts of things that might make up the dark matter.

Further efforts to understand the cosmos provoke many of today’s most magnificent prognostications. Great thinkers of the past, such as Alexander Friedmann, imagined a universe more dynamic than any scientist had before, a universe growing larger and larger. His vision was soon confirmed by observations. Today equally imaginative ideas suggest vast new frontiers of space and time for future scientists to explore. Most dramatically, scientists like Andrei Linde and Alan Guth have conceived of countless new universes bursting into existence far beyond our view. Guth and Linde are among many advocates of the view that our universe isn’t the only one—though perhaps it’s the best one, for living things like us.

But a multiplex of universes isn’t the end of the story. More new ideas are needed to explain the universe we already know about. Another cosmotheorist, Paul Steinhardt, is a leader in the effort to explain why our universe seems to be expanding faster now than it used to be. He and others propose new versions of a ubiquitous cosmic fluid that may occupy every groove and wrinkle of all of space.

Besides identifying dark matter and understanding the cosmos, a third grand motivation inspires many attempts at prediscovery: the urge to unify science’s theories of matter and force. Such efforts celebrate a noble tradition, exemplified by James Clerk Maxwell’s nineteenth-century theory of electromagnetism. By figuring out the math that simultaneously described both magnetism and electricity, Maxwell prediscovered radio waves—as well as other forms of electromagnetic rays that transformed human lifestyles in the twentieth century. Today every physicist hopes to see the day when electromagnetism and the forces of the atomic nucleus are mathematically joined with gravity. The road to that unification leads toward a foggy horizon, but the pioneers suspect that their journey will reveal ultratiny objects called superstrings—or perhaps something even stranger.

Already, it seems, that road to unification twists and turns in some

of those new dimensions of space explored by Lisa Randall and others, like Stanford’s Savas Dimopoulos. (Always an enthusiast for new ideas, Dimopoulos promotes those extra dimensions as a place where Bill Gates might someday want to store information.) And space’s secrets may get even stranger. Other authorities proclaim that the universe may be wrapped around itself in such a way as to create ghost images of every galaxy. If so, a distant patch of light on the night sky might just be the backside of our own galaxy, the Milky Way. And if that’s not bizarre enough, free thinkers like Harvard’s Cumrun Vafa would like to tell you that space is not the only place where dimensions should be added; the world may be big enough for more than one dimension of time as well.

It should go without saying that not all of the visions at the frontiers of physics will turn out to be true. Some are mirages, propelling the pioneers forward toward disappointment. But forward nonetheless. And surely some of the airy visions will soon solidify, at least if the successes of the past are any guide.

Therein lies the central mystery, however. Most of the prediscoveries of the past have exploited the power of mathematics to represent reality, even parts of reality that have until then never been seen. How can math do that? Or as my friend Rosie Mestel says, how is it that squiggles on paper can tell us of the existence of things in the real world never before encountered?³

Usually this question is posed in terms of “the unreasonable effectiveness of mathematics,” from a famous 1960 paper by the physicist Eugene Wigner. He emphasized the ability of scientists to take the crude observations of complicated experience and extract equations capturing regularities within the complexity. Perhaps, he suggested, such success is achieved because the typical physicist is rather irresponsible.

“When he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at

the conclusion that the connection is that discussed in mathematics,” Wigner comments. “The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.”⁴

But Wigner was talking about the success of theories after the fact, so to speak. He marvels at the precision of quantum mechanics for computing the energy of electrons in a helium atom, even though Werner Heisenberg’s original quantum equations were based on properties that helium does not possess. Somehow, Wigner says, we “got something out of the equations that we did not put in.”⁵ But he does not discuss the power of mathematics to reveal previously unimagined phenomena. Apparently he found the ability of math to work at all mysterious enough.

In a recent book, though, two cognitive scientists—a psychologist and a linguist—argue that Wigner’s mystery is illusory. Mathematics, these authors say, is simply an invention of the human mind, based on common human experience. There is no idealized, “platonic” mathematics inherent in nature; math is what people make it.

“It is sometimes assumed that the effectiveness of mathematics as a scientific tool shows that mathematics itself exists in the structure of the physical universe,” these scientists write.⁶ “This, of course, is not a scientific argument with any empirical scientific basis.”

Oh? I would say the “of course” in the previous sentence lacks substantial justification. But these scientists argue vigorously in their book that math has nothing to do with the way the universe works apart from human descriptions of it.

“All the ‘fitting’ between mathematics and the regularities of the physical world is done within the minds of physicists who comprehend both,” the cognitive scientists assert. “The mathematics is in the mind of the mathematically trained observer, not in the regularities of the physical universe.”⁷

So there is no mystery, they say. We impose our math on the world in order to describe it. That’s why math works.

Frankly, I am not impressed by this argument. Although it is surely true, at least in some senses, that math is a human invention, it does not logically follow that the universe does not live by mathematical laws. The idea of math as merely a human invention may explain much of its success. But I do not see how it explains the way that math reveals unseen, even unimagined, features of reality. It’s one thing to fit equations to aspects of reality that are already known; it’s something else for that math to tell of phenomena never previously suspected. When Paul Dirac’s equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac’s math disclosed their existence. If math is a human invention, nature seems to have already known what was going to be invented.

It may well be true that humans have built mathematics out of concepts drawn from human experience. Yet somehow that realization does not resolve the mystery about why math works so well, but rather deepens it. For even with all the latest advances in brain science—revealing how humans think and reason and understand their environment—math’s power to predict the reality of strange matters remains unexplained. But perhaps exploring the prediscoveries of the past and the potential prediscoveries of today can provide some clues to that mystery.