The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

Chapter 13. TRUTH, BEAUTY, AND MATHEMATICS

How far can investigators, trying to survey the universe’s hidden dimensions, proceed in the absence of any physical proof? The same question, of course, can also be put to string theorists trying to concoct a complete theory of nature without the benefit of empirical feedback. It’s like exploring a vast, dark cavern—whose contours are largely unknown—with just a flickering candle, if that, to illuminate the path. Although proceeding under such circumstances may seem like sheer folly to some, the situation is by no means unprecedented in the history of science. In the early stages of theory building, periods of fumbling in the dark are rather common, especially when it comes to developing, and pushing through, ideas of great scope. At various junctures like this, when there is no experimental data to lean on, mathematical beauty may be all we have to guide us.

The British physicist Paul Dirac “cited mathematical beauty as the ultimate criterion for selecting the way forward in theoretical physics,” writes the physicist Peter Goddard.1 Sometimes this approach has paid off handsomely, as when Dirac predicted the existence of the positron (like an electron with positive charge), solely because mathematical reasoning led him to believe that such particles must exist. Sure enough, the positron was discovered a few years later, thereby affirming his faith in mathematics.

Indeed, one of the things we’ve found over and over again is that the ideas that hold up mathematically, and meet the criteria of simplicity and beauty, tend to be the ones that we eventually see realized in nature. Why that is the case is surely baffling. The physicist Eugene Wigner, for one, was perplexed by “the unreasonable effectiveness of mathematics in the natural sciences”—the mystery being how purely mathematical constructs, with no apparent connection to the natural world, can nevertheless describe that world so accurately.2

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world.” When he asked the geometer S. S. Chern how it was possible that “mathematicians dreamed up these concepts out of nowhere,” Chern protested, “No, no. These concepts were not dreamed up. They were natural and real.”3

There certainly is no shortage of abstract ideas that came to mathematicians seemingly out of thin air and that were later found to describe natural phenomena. Not all of these, by the way, were the products of modern mathematics. Conic sections, which are the curves—the circle, ellipse, parabola, and hyperbola—made by slicing a cone with a plane, were reportedly discovered by the Greek geometer Menaechmus around 300 B.C. and systematically explored a century later by Apollonius of Perga in his treatise Conics. These forms, however, did not find a major scientific application until the early 1600s, when Kepler discovered the elliptical orbits of planets in our solar system.

Similarly, “buckyballs,” or buckminsterfullerenes—a novel form of carbon consisting of sixty carbon atoms arranged on a spherelike structure composed of pentagonal and hexagonal faces—were discovered by chemists in the 1980s. Yet the shape of these molecules had been described by Archimedes some two thousand years earlier.4 Knot theory, a branch of pure mathematics that has evolved since the late 1800s, found applications more than a century later in string theory and in studies of DNA.

It’s hard to say why ideas from mathematics keep popping up in nature. Richard Feynman found it equally hard to explain why “every one of our physical laws is a purely mathematical statement.” The key to these puzzles, he felt, may lie somewhere in the connection between math, nature, and beauty. “To those who do not know mathematics,” Feynman said, “it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature.”5

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13.1—Conic sections are the three basic curves you get when you intersect a plane with a cone (or, actually, a kind of double cone attached at the pointy ends). Those curves are the parabola, ellipse (of which a circle is a special case), and hyperbola.

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13.2—Whereas the regular icosahedron consists of twenty triangular faces, the truncated icosahedron (shown here) consists instead of twenty hexagonal faces and twelve pentagonal faces (in which no two pentagons have an edge in common). Unlike the regular icosahedron, which is classified as a Platonic solid, the truncated icosahedron is an Archimedean solid, named after the Greek mathematician who explored these shapes more than two thousand years ago. This shape resembles a soccer ball and one version of a so-called buckyball—the molecular structure of a form of carbon, consisting of sixty atoms, that was discovered in 1985 by the chemists Harold Kroto and Richard Smalley. The term buckyball is short for buckminsterfullerene—the name given to this class of molecules in honor of R. Buckminster Fuller, the inventor of the similarly shaped geodesic dome.

Of course, if beauty is to guide us in any way—even temporarily, until more tangible clues come in—that leaves the problem of trying to define it, a task that some feel might be best left to poets. While mathematicians and physicists may view this concept somewhat differently, in both disciplines the ideas we call beautiful tend to be those that can be stated clearly and concisely, yet have great power and broad reach. Even so, for a notion as subjective as beauty, personal taste inevitably comes into play as well. I’m reminded of a toast made at the wedding of a longtime bachelor who settled down relatively late in life after many years of playing the field. What kind of woman, people had wondered, would it take to get this guy to tie the knot? The bachelor himself was curious about that, too. “You’ll know it when you see it,” a friend repeatedly advised him prior to his finding “the one.”

I know what he means. I felt that way when I met my wife in the Berkeley math library many decades ago, although I’d be hard-pressed to capture the exact feeling in words. And, with no offense intended to my wife, I felt something similar—a vague, tingly sense of euphoria—after proving the Calabi conjecture in the mid-1970s. With the proof complete after months of exertion and exhaustion—stretched out, of course, over the span of years—I was finally able to relax and admire the complex, multidimensional spaces I had found. You might say it was love at first sight, although after all that labor, I felt that I already knew these objects well, even upon first “viewing” them. Perhaps my confidence was misplaced, but I sensed then (as I still do) that somehow these spaces would play a role, and possibly an important one, in the physical world. Now it’s up to string theorists—or perhaps researchers from some unrelated branch of science—to find out if that hunch is correct.

It ought to be reassuring to string theorists, mathematician Michael Atiyah argues, “that what they’re playing with, even if we can’t measure it experimentally, appears to have a very rich . . . mathematical structure, which not only is consistent but actually opens up new doors and gives new results and so on. They’re onto something, obviously. Whether that something is what God’s created for the universe remains to be seen. But if He didn’t do it for the universe, it must have been for something.”6

I don’t know what that something is, but it strikes me as being far too much to be nothing. Yet I’m also fully aware—as is Atiyah—of the risk of being lulled by elegance onto shaky ground. “Beauty can be a slippery thing,” cautions Jim Holt, a string theory skeptic writing in the New Yorker.7 Or as Atiyah puts it, “the mathematical take-over of physics has its dangers, as it could tempt us into realms of thought which embody mathematical perfection but might be far removed, or even alien to, physical reality.”8

There’s no doubt that a blind adherence to mathematical beauty could lead us astray, and even when it does point us in the right direction, beauty alone can never carry us all the way to the goal line. Eventually, it has to be backed up by something else, something more substantial, or our theories will never go beyond the level of informed speculation, no matter how well motivated and plausible that speculation may be.

“Beauty cannot guarantee truth,” asserted the physicist Robert Mills, the second half of the Yang-Mills duo. “Nor is there any logical reason why the truth must be beautiful, but our experience has repeatedly led us to expect beauty at the heart of things and to use this expectation as a guide in seeking deeper theoretical understanding of the fundamental structures of nature.” Conversely, Mills added, “if a proposed theory is inelegant, we have learned to be dubious.”9

So where does that leave string theory and the mathematics behind it? Cornell physicist Henry Tye believes “string theory is too beautiful, rich, creative, and subtle not to be used by nature. That would be such a waste.”10 That in itself is not enough to make string theory right, and critical treatments of the subject, such as The Trouble with Physics and Not Even Wrong, have raised doubts in the public consciousness at a time when the theory is in a kind of doldrums, not having seen a major breakthrough in years. Even an enthusiast like Brian Greene, author of The Elegant Universe, acknowledges that a physical theory cannot be judged solely on the basis of elegance: “You judge it on the basis of whether it makes predictions that are going to be confirmed by experiment.”11

While writing this book, I’ve had occasion to discuss its contents with a number of laypeople—exactly the educated types who I hope might eventually want to read this sort of thing. When they heard it related to the mathematical foundations of string theory, the response often went like this: “Wait a minute. Isn’t string theory supposed to be wrong?” Their questions suggested that writing a book about the mathematics of string theory was like writing a book about the fantastic blueprints that went into the making of the Titanic. A mathematics colleague of mine, who should have known better, even went on record as saying that because “the jury is still out on string theory,” the jury is still out on the mathematics associated with string theory as well.

Such a claim implies a fundamental misconception about the nature of mathematics and its relation to the empirical sciences. Whereas the final proof in physics is an experiment, that is not the case in math. You can have a billion bits of evidence that something is true, yet on the billionth-and-first time, it fails. Until something is proven by pure logic, it remains a conjecture.

In physics and other empirical sciences, something thought to be true is always subject to revision. Newton’s theory of gravitation held up well for more than two centuries until its limitations were finally appreciated and it was replaced by Einstein’s version, whose own limitations may someday be addressed by a theory of quantum gravity like string theory. Nevertheless, the math that goes into Newtonian mechanics is 100 percent correct, and that will never change.

In fact, to formulate his theory of gravitation, Newton had to invent (or co-invent) calculus; when Newton’s theory of gravity broke down at the limits that general relativity was designed to address, we didn’t throw out calculus. We kept the math—which is not only sound but indispensable—realizing that Newtonian mechanics is a perfectly good tool in most situations, though it is not applicable in the most extreme cases.

Now for something a little more contemporary—and closer to my heart. Thirty-some years ago, I proved the existence of spaces we now call Calabi-Yau. Their existence, moreover, is not at all contingent on whether string theory turns out to be “the theory of nature.” Admittedly, weak points can be uncovered in a proof, after the fact, and topple the argument like a house of cards. But in the case of the Calabi conjecture, the proof has been gone over so many times that the chances of finding a mistake are essentially nil. Not only are Calabi-Yau spaces here to stay, but the techniques I used to attack the problem have been applied with great success to many other mathematical problems, including those in algebraic geometry that have no obvious ties to the original conjecture.

Indeed, the utility of Calabi-Yau spaces in physics is, in some sense, irrelevant to the question of whether the mathematics is important. At the risk of sounding immodest, I might add that I was awarded the Fields Medal in 1982—one of the highest honors in mathematics—largely because of my proof of the Calabi conjecture. The award, you’ll notice, was granted a couple of years before physicists knew about Calabi-Yau manifolds and before string theory itself was really on the map.

As for string theory, the mathematics underlying it or inspired by it can be absolutely correct, no matter what the jury finally decides regarding the theory itself. I’ll go farther: If the mathematics associated with string theory is solid and has been rigorously proven, then it will stand regardless of whether we live in a ten-dimensional universe made of strings or branes.

So, what, if anything, can this tell us about the physics? As I said before, because I am a mathematician, it’s not for me to judge the validity of string theory, but I will offer some opinions and observations. Granted, string theory remains not only unproven but untested. Nevertheless, one major tool for checking the work of physicists has been mathematical consistency, and so far, the theory has passed those exams with flying colors. Consistency in this case means there’s no contradiction. It means that if what you put into the string theory equations is correct, what you get out of the equations is correct, too. It means that when you do a calculation, the numbers don’t blow up and go to infinity. The functions remain well-spoken rather than spouting off gibberish. While that’s not nearly enough to satisfy the strictures of science, it is an important starting point. And to me it suggests there must be some truth to this idea, even if nature doesn’t follow the same script.

Edward Witten seems to share this opinion. Mathematical consistency has been, he claims, “one of the most reliable guides to physicists in the last century.”12

Given how hard it is to devise an experiment that could access Planck-scale physics and how expensive such an experiment might be if we ever manage to come up with one, all we’ll have in many cases will be these consistency checks, which, nevertheless, “can be very powerful,” according to Berkeley mathematician Nicolai Reshetikhin. “That’s why the high end of theoretical physics is becoming more and more mathematical. If your ideas are not mathematically consistent, you can rule them out right away.”13

Beyond mathematical consistency, string theory also seems to be consistent with everything we’ve learned about particle physics, while offering new perspectives for grappling with issues of space and time—gravity, black holes, and various other conundrums. Not only does string theory agree with the established, well-tested physics of quantum field theories, but it appears to be inextricably tied to those theories as well. No one doubts, for example, that gauge theories—such as the Yang-Mills equations for describing the strong interaction—are a fundamental description of nature, argues Robbert Dijkgraaf, a physicist at the University of Amsterdam. “But gauge theories are fundamentally connected with strings.” That’s true because of all the dualities, which establish an equivalence between field theories and string theories, showing them to be different ways of looking at the same thing. “It isn’t possible to argue whether string theory belongs in physics, since it’s continuously connected to all the things we hold dear,” Dijkgraaf adds. “So we can’t get rid of string theory, regardless of whether our universe is described by it. It’s just one more tool for thinking about the fundamental properties of physics.”14

String theory was also the first theory to quantize gravity in a consistent way, which was the point all along. But it goes even further than that. “String theory has the remarkable property of predicting gravity,” Witten proclaims. By that he means that string theory does more than just describe gravity. The phenomenon is embedded within the theory’s very framework, and someone who knew nothing about gravity could discover it as a natural consequence of the theory itself.15 In addition to quantizing gravity, string theory has gone far toward solving problems like the black hole entropy puzzle that had resisted solutions by other means. Viewed in that sense, string theory can already be considered a successful theory on some level, even if it doesn’t turn out to be the ultimate theory of physics.

While that matter is being adjudicated, there’s no denying that string theory has led to a treasure trove of new ideas, new tools, and new directions in mathematics. The discovery of mirror symmetry, for instance, has created a cottage industry in the fields of algebraic and enumerative geometry. Mirror symmetry—the notion that most Calabi-Yau spaces have a topologically distinct mirror partner (or partners) that gives rise to identical physics—was discovered in the context of string theory, and its validity was later confirmed by mathematicians. (That, as we have seen, is a typical pattern: Whereas string theory may provide concepts, hints, and indications, in most cases mathematics delivers the proof.)

One reason mirror symmetry has been so valuable for math is that a difficult calculation for one Calabi-Yau can be a much simpler calculation in its mirror partner. As a result, researchers were soon able to solve centuries-old problems in mathematics. Homological mirror symmetry and the theory of Strominger-Yau-Zaslow (SYZ), which have developed since the mid-1990s, have forged unexpected though fruitful connections between symplectic geometry and algebraic geometry—two branches of mathematics that had previously been considered separate. Although mirror symmetry was uncovered through string theory research, its mathematical basis does not depend on string theory for its truth. The phenomenon, notes Andrew Strominger, “can be described in a way that doesn’t involve string theory at all, [but] it would have been a long time before we figured it out had it not been for string theory.”16

To cite another example, a 1996 paper by my former postdoc Eric Zaslow and I used an idea from string theory to solve a classic problem in algebraic geometry related to counting the number of so-called rational curves on a four-dimensional K3 surface. (Please keep in mind that the term K3 refers to a whole class of surfaces—not one but an infinite number of them.) The “curves” in this case are two-dimensional Riemann surfaces defined by algebraic equations that are the topological equivalent of spheres embedded on that surface. The counting of these curves, it turns out, depends only on the number of nodes a curve possesses—nodes being points where a curve crosses itself. A figure eight, for example, has just one node, whereas a circle has zero nodes.

Here’s another way of thinking about nodes that relates to our previous discussion of conifold transitions (in Chapter 10): If you take a two-dimensional donut and shrink one of the circles that run through the hole down to a point, you’ll get something that looks like a croissant whose two ends are attached. If you separate those two ends and blow up the surface, you’ll have the topological equivalent of a sphere. So you can consider this pinched donut or “attached croissant” a sphere with one node (or crossing). Similarly, we could go to a higher genus and look at a double-holed donut: First we’re going to pinch a circle down to a point in the “inner wall” between the two holes and do the same somewhere on the donut’s “outer wall.” The object with these two pinch points is actually a sphere with two nodes because if we were to separate those two points and inflate the surface, we’d have a sphere. The point is that if you start with a surface of higher genus (say two, three, or more holes), you can end up with a curve, or sphere, with more nodes, too.

Let me restate the problem in algebraic geometry that we were originally trying to solve: For a K3 surface, we’d like to know the number of rational curves with g nodes that can fit on that surface for any (positive integer) value of g. Using conventional techniques, mathematicians had devised a formula that worked for curves with six or fewer nodes but not beyond. Zaslow and I set out to tackle the more general situation of curves with an arbitrary number of nodes. Instead of the usual approach, we took a string theory perspective, viewing the problem in terms of branes inside a Calabi-Yau.

String theory tells us there are branes associated with a K3 surface that consist of curves (or two-dimensional surfaces, as previously defined) plus a so-called flat line bundle attached to each curve. To get a sense of what this line bundle is all about, suppose a person walks around the equator holding a stick of any length—even an infinitely long one—that is perpendicular to the equator and tangent to the surface of the sphere. Eventually, the stick will trace out a cylinder, which is called a trivial line bundle. If the person rotated the stick 180 degrees once during the walk, it would trace a Möbius strip. Both of these line bundles, by the way, are “flat,” meaning that they have zero curvature.

Zaslow and I observed that if you take the space of all the branes containing curves of a fixed genus g that are associated with a given K3 and then compute the Euler characteristic of that space, the number you get will be exactly equal to the number of rational curves with g nodes that fit in that K3 surface.

In this way, my colleague and I had converted the original problem into a different form, showing that it all came down to getting the Euler characteristic of the space of branes in question. We then used a string theory duality, developed by Cumrun Vafa and Witten, to calculate the Euler characteristic. String theory had thus provided a new tool for attacking this problem, as well as hints about a new way of framing the problem. Algebraic geometers hadn’t been able to solve the problem before, because they weren’t thinking of branes: It never occurred to them to frame this problem in terms of the moduli space, which encompasses the totality of all possible branes of this type.

Although Zaslow and I sketched out the general approach, the actual proof of this idea was completed by others—Jim Bryan of the University of British Columbia and Naichung Conan Leung of the University of Minnesota—a few years later. As a result, we now have a theorem in mathematics that should be true regardless of whether string theory is right or wrong.

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13.3—If you walked around the equator, all the while holding a pole parallel to the ground (and essentially tangent to the surface), you would sweep out a cylinder. If instead you gradually rotated the pole 180 degrees as you circumnavigated the globe, you would sweep out a more complicated surface—which has just one side rather than two—known as a Möbius strip.

In addition, the formula that Zaslow and I derived for counting rational curves on K3 surfaces gives you a function for generating all the numbers you get for rational curves with an arbitrary number of nodes. It turns out that this function essentially reproduces the famous tau function, which was introduced in 1916 by the Indian mathematician and self-taught genius Srinivasa Ramanujan. 17 This function and the conjectures that Ramanujan raised in conjunction with it have since led to many important advances in number theory. Our work, as far as I know, established the first solid link between enumerative geometry—the subject of counting curves—and the tau function.

This link has been strengthened by the recent contributions of Yu-Jong Tzeng, a young mathematician recently hired by Harvard and trained by my former student, Jun Li. Tzeng showed that not only are the rational curves on a K3 connected to the tau function but the counting of any curves of arbitrary genus on any algebraic surfaces is connected to the tau function. And Tzeng did this by proving a conjecture made by the German mathematician Lothar Goettsche, which generalized the so-called Yau-Zaslow formula for rational curves on K3 surfaces.18 The new, generalized formula, whose validity Tzeng upheld, bears the name Goettsche-Yau-Zaslow. (A few years earlier, a former graduate student of mine, A. K. Liu, published a proof of the Goettsche-Yau-Zaslow formula.19 But his proof, rooted in a highly technical, analytic approach, did not provide an explanation in the form that algebraic geometers were looking for. As such, Liu’s paper was not regarded as the final confirmation of that formula. Tzeng’s proof, which was based on algebraic geometry arguments, is more widely accepted.)

The broader point is that through a finding that originally stemmed from string theory, we’ve learned that the ties between enumerative geometry and Ramanujan’s tau function are probably deeper than anyone imagined. We’re always looking for connections like this between different branches of mathematics because these unexpected links can often lead us to new insights in both subjects. I suspect that, in time, more ties between enumerative geometry and the tau will be uncovered.

In the 1990s, in yet another, more celebrated example of how string theory has enriched mathematics, Witten and Nathan Seiberg of Rutgers University developed a set of equations called the Seiberg-Witten equations, which (as discussed in Chapter 3) have accelerated the study of four-dimensional spaces. The equations were much easier to use than existing methods, leading to an explosion in the things we can do in four dimensions—the main one being the attempt to categorize and classify all possible shapes. Although the Seiberg-Witten equations were initially discovered as a statement in field theory, it was soon shown that they could be derived from string theory as well. Putting this idea in the context of string theory, moreover, greatly enhanced our understanding of it. “On various occasions,” says a publicity-shy colleague of mine, “Witten has basically said to mathematicians, ‘Here, take these equations; they might be useful.’ And, by golly, they were useful.”

“String theory has been such a boon to mathematics, such a tremendous source of new ideas, that even if it turns out to be completely wrong as a theory of nature, it has done more for mathematics than just about any other human endeavor I can think of,” claims my longtime collaborator Bong Lian of Brandeis University.20 Although I’d phrase it more moderately than Lian, I agree that the payoff has been unexpectedly huge. On this point, we seem to be in accord with Atiyah, as well. String theory, he said, “has transformed and revitalized and revolutionized large parts of mathematics . . . in areas that seem far removed from physics.” Many fields, “geometry, topology, and algebraic geometry and group theory, almost anything you want, seem to be thrown into the mixture—and in a way that seems to be very deeply connected with their central content, not just tangential contact, but into the heart of mathematics.”21

While other areas of physics have informed math in the past, the influence of string theory has penetrated much deeper into the internal structure of mathematics, leading us to new conceptual breakthroughs. The advent of string theory, ironically, has led to harmonious collaborations within mathematics itself, because string theory has demanded a lot from mathematicians in areas that include differential geometry, algebraic geometry, Lie group theory, number theory, and others. In a funny way, our best hope so far for a unified theory of physics has helped promote the unification of mathematics as well.

Despite the beauty of string theory and its deeply felt impacts on mathematics, the question remains: How long must we wait for outside corroboration—for some connection, any connection, with the real world? Brian Greene believes that patience is in order, given that “we are trying to answer the most difficult, the most profound questions in the history of science. [Even] if we haven’t gotten there in 50 or 100 years, that’s a pursuit we should keep on with.”22 Sean Carroll, a physicist at the California Institute of Technology, agrees: “Profound ideas don’t come with expiration dates.”23 Or, putting it in other terms, what’s the big rush, anyway?

A historical precedent might be useful here. “In the nineteenth century, the question of why water boils at 100 degrees centigrade was hopelessly inaccessible,” Witten notes. “If you told a 19th-century physicist that by the 20th century you would be able to calculate this, it would have seemed like a fairy tale.”24

Neutron stars, black holes, and gravitational lenses—dense concentrations of matter that act like magnifying glasses in the sky—were similarly dismissed as sheer fantasy until they were actually seen by astronomers. “The history of science is littered with predictions that such and such an idea wasn’t practical and would never be tested,” Witten adds. But the history of physics also shows that “good ideas get tested.”25 New technology that could not even be guessed at a generation before can make ideas that seemed beyond the pale become science fact rather than science fiction.

“The more important the question is, the more patient one should be in the testing game,” maintains MIT physicist Alan Guth, one of the architects of inflationary theory, which holds that our universe underwent a brief, explosive, expansionary burst during the earliest moments of the Big Bang. “When we were working on inflation in the early days, I never thought for a moment that it would be tested in my lifetime,” Guth says. “We had to be lucky for inflation to be tested, and we were. Though it wasn’t luck so much as the tremendous skill of the observers. The same thing could happen with string theory. And maybe we won’t have to wait a hundred years.”26

While string theory must be regarded as speculation, there’s nothing necessarily wrong with that. The conjectures of mathematics, like Calabi’s, are nothing more than speculation rooted in mathematical theory. They are absolutely essential to progress in my field. Nor could we get anywhere in physics and advance our understanding were it not for speculation of the learned, rather than idle, kind. Nevertheless, the word does imply some degree of doubt, and how you react to that is a matter of your temperament, as well as your personal investment in a problem. When it comes to string theory, some are in it for the long haul, hopeful that it will eventually pan out. Others, who can’t get beyond the lingering questions, put the uncertainties front and center, waving metaphorical placards that read: “Stop! You’re making a big mistake.”

There was a time, not too many centuries ago, when people warned about sailing too far from one’s shores, lest the ship and its passengers fall off the edge of the earth. But some intrepid travelers did set sail, nevertheless, and rather than falling off the edge of the world, they discovered the New World instead.

Perhaps that’s where we are today. I’m of the camp of pushing forward because that’s what mathematicians do. We carry on. And we can do so with or without any input from the external world—or the realm of experiment—while keeping productive as well.

Though, personally, I find it helpful to keep tabs with the physicists. Indeed, I’ve spent the bulk of my career working in the interstices between math and physics, partly owing to my conviction that interactions between the two fields are crucial for furthering our grasp of the universe. All told, these interactions have been mostly harmonious over the decades. Sometimes, the mathematics has developed before applications in physics were found—as occurred with the great works of Michael Atiyah, Elie Cartan, S. S. Chern, I. M. Singer, Hermann Weyl, and others—and sometimes the physics has outpaced the math, as occurred with the discovery of mirror symmetry. But perhaps I shouldn’t characterize the current arrangement between mathematicians and physicists as entirely cozy. “There’s a good deal of healthy and generally good-natured competition” between the two fields, according to Brian Greene, and I think that’s a fair assessment.27 Competition, however, is not always bad, as it usually helps move things along.

In different times in history, the divisions between the fields—or lack thereof—have changed significantly. People like Newton and Gauss were certainly comfortable moving freely between math, physics, and astronomy. Indeed, Gauss, who was one of the greatest mathematicians of all time, served as professor of astronomy at the Göttingen Observatory for almost fifty years, right up to the moment of his death.

The introduction of the Maxwell equations of electromagnetism, and subsequent developments in quantum mechanics, created a wedge between math and physics that persisted for the better part of a century. In the 1940s, 1950s, and 1960s, many mathematicians didn’t think much of physicists and didn’t interact with them. Many physicists, on the other hand, were arrogant as well, having little use for mathematicians. When the time came for math, they figured they could work it out for themselves.

MIT physicist Max Tegmark supports this interpretation, citing a “cultural gap” between the two fields. “Some mathematicians look down their nose at physicists for being sloppy—for doing calculations that lack rigor,” he says. “Quantum electrodynamics is an example of an extremely successful theory that is not mathematically well defined.” Some physicists, he adds, are scornful of mathematicians, thinking that “you guys take forever to derive things that we can get in minutes. And if you had our intuition, you’d see it’s all unnecessary.”28

Ever since string theory entered the picture—with theoretical physics relying to an increasing degree on advanced mathematics—that cultural gap has started to narrow. The mathematics that comes up in string theory is so complicated—and so integral to everything in the theory—that physicists not only needed help but also welcomed it. While mathematicians became interested in Calabi-Yau spaces before the physicists did, to pick one example, the physicists eventually got there, and when they did, they showed us a few tricks as well. We’re now in a period of “reconvergence,” as Atiyah puts it, and that’s a good thing.

I can’t say whether string theory will ever get past its most serious hurdle—coming up with a testable prediction and then showing that the theory actually gives us the right answer. (The math part of things, as I’ve said, is already on much firmer ground.) Nevertheless, I do believe the best chance for arriving at a successful theory lies in pooling the resources of mathematicians and physicists, combining the strengths of the two disciplines and their different ways of approaching the world. We can work on complementary tracks, sometimes crossing over to the other side for the benefit of both.

Cliff Taubes, a math colleague of mine at Harvard, summed up the differences between the fields well. Though the tools of math and physics may be the same, Taubes said, the aims are different. “Physics is the study of the world, while mathematics is the study of all possible worlds.”29

That’s one of the reasons I love mathematics. Physicists get to speculate about other worlds and other universes, just as we do. But at the end of the day, they eventually have to bring it back to our world and think about what’s real. I get to think about what’s possible—not only “all possible worlds,” as Cliff put it, but the even broader category of all possible spaces. As I see it, that’s our job. While physicists, by and large, tend to look at one space and see what it can tell us about nature, we mathematicians need to look at the totality of all spaces in order to find some general rules and guiding principles that apply to the cases of greatest concern.

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13.4—This cartoon by the physicist Robbert Dijkgraaf shows the interplay between mathematicians and physicists. (Image courtesy of Robbert H. Dijkgraaf)

Still, all spaces are not created equal, and some command my attention more than others, especially those spaces wherein the extra dimensions of nature are thought to reside. A critical challenge before us is to figure out the shape of that hidden realm, which, theory holds, dictates both the kinds of matter we see in the cosmos and the kinds of physics we see. That problem has been on my plate for a while, and it’s not likely to be disposed of soon.

Although I take on a variety of projects from time to time, I keep coming back to this one. And, despite my forays into other areas of math and physics, I keep coming back to geometry. If peace comes through understanding, geometry is my way of trying to achieve some semblance of inner calm. Putting it more broadly, it’s my way of trying to make sense of our universe and to fathom the mysterious spaces—named in part after me—that may lurk within.