Perfect Rigour

by Masha Gessen

  No one knew what was occupying Perelman’s mind. Even Gromov heard nothing from him and assumed he was still stuck on Alexandrov spaces—in other words, that he had joined the sizable ranks of talented mathematicians who did brilliant early work and then disappeared into the black hole of some impossible problem.

  In February 2000, Mike Anderson at Stony Brook suddenly received an e-mail message from Perelman. “Dear Mike,” it began. “I’ve just read your paper20 on generalised Lichnerovicz thm, and there is one point in your paper that disturbs me.” Perelman went on to describe the nature of his doubts in one long, perfectly constructed sentence and finished with: “Am I missing something? Best regards, Grisha.” There were no unnecessary niceties one might expect in a letter like this—nothing along the lines of “I hope this finds you well” or “It has been a long time.” But the letter was perfectly polite, and Perelman’s English—presumably disused for more than five years—all but impeccable.

  Anderson responded the next day with a letter that by the standards of the mathematical world was downright effusive:

  Dear Grisha,21

  It was a surprise to hear from you again—a pleasant surprise. I often ask people who I see from St. Petersburg if they know how you are and what you are thinking about these days.

  I just returned from a short trip, and so haven’t been able to think yet in detail about your remarks on my stationary paper. But I see your points, and agree I have made an error here. I don’t think these two errors effect [sic] the results, and that the proofs require only minor modifications. I will think this through in the next couple of days and report back to you.

  I’d also like to hear how you are, and what kind of mathematical or other issues you are concerned about these days.

  Best regards to you,

  Mike

  Three days later, Anderson sent Perelman a more detailed e-mail message, outlining a fix for the mistakes Perelman had found. Again, he inserted a note of personal and professional interest: “I thank you very much for spotting these errors. Are you becoming interesting [sic] in these areas yourself?” Anderson also complained that so few people were working in his area—geometrization—that he had no one to double-check his ideas. He asked if Perelman had looked at his other two papers on related topics.22

  Perelman replied the next day. He thanked Anderson for his prompt response but ignored every single one of his questions. He wrote only that Anderson’s paper had drawn his attention because it was “tangentially related” to Perelman’s own current interests — and also, he noted, because it was short. He did not invite further communication. Nor did he promise he would look at Anderson’s other papers—he wrote that he had them but had not read them. In fact, it seems likely that he subsequently read the papers but, finding no errors, saw no reason to write to Anderson again.

  Anderson still tried to pursue the dialogue. He sent Perelman a file containing a more detailed fix for his paper. Perelman responded by saying he could not open the file23 without somebody’s help (“I do not know computers at all,” he claimed) and explained that his sister had helped print out the original Anderson papers when he visited her in Rehovot, where she was a graduate student. He proceeded to write that sending the file to a Steklov computer to open there might make it accessible to other people, so in the end he would rather wait until Anderson published the paper. In other words, he had gotten all he needed from this exchange with his colleague.

  The message was a curious document in other respects. It seems that in the five years since leaving the United States, Perelman had drifted far from the practical aspects even of mathematics: he didn’t seem to know how to use his office computer to log onto the SUNY e-mail account he used to correspond with Anderson or how to forward the file to a Web-based address no one else could access. At the same time, Perelman was using his lack of technical expertise to close the conversation, which had evidently outworn its usefulness to him. After all, when he actually needed Anderson’s preprints, he had been resourceful enough to ask his sister for help. It is remarkable too how casually Perelman shared the details of his life and his sister’s. It was never his intention to hide his family life or refuse to discuss himself or his relatives; it was just very rarely relevant to any conversation he found worth having.

  It would be two and a half years before Mike Anderson heard from Perelman again.

  PERFECT RIGOR

  THE PROBLEM

  8

  The Problem

  THE VERY POSSIBILITY of mathematical science seems1 an insoluble contradiction.” So, more than a century ago, wrote Henri Poincaré, known among mathematicians as the last universalist, for he excelled in all areas of mathematics. If the objects of study are confined to the imagination, “from whence is derived that perfect rigor which is challenged by none?” And when rules of formal logic have replaced the experiment, “how is it that mathematics is not reduced to a gigantic tautology?” Finally, “are we then to admit that . . . all the theorems with which so many volumes are filled are only indirect ways of saying that A is A?”

  Poincaré went on to explain that mathematics was a science because its reasoning traveled from the particular to the general. A mathematician who conducted his mental experiments with sufficient rigor could derive the rules that governed the rest of the imaginary terrain he shared with other mathematicians. In other words, he not only proved that A was A but also explained what made A quintessentially an A and where other A’s might be found or how they might be constructed. “We know what it is to be in love or to feel pain,2 and we don’t need precise definitions to communicate,” wrote an American mathematics professor who, after authoring many academic books, undertook to explain topology to a general audience. “The objects of mathematics lie outside common experience, however. If one doesn’t define these objects carefully, one cannot manipulate them meaningfully or talk to others about them.” This may or may not be so. Most of us are, in fact, perfectly satisfied with our casual understandings of distances long and short, of slopes smooth and steep, and of lines and circles and spheres. We’re satisfied with a gut feeling that puncturing a hole can sometimes but not always change the nature of an object—that is, a punctured balloon is entirely different from an intact one, while, say, a jelly-filled doughnut without a hole is, to us, essentially similar to a doughnut with a hole in the middle, with or without jelly. All of these things are in their simplest forms parts of our common experience. But in the disjointed world of the mathematician, shifting understandings and imprecise coordinates muddle the picture intolerably. In his world, nothing is like anything else unless proven similar; nothing is familiar until thoroughly defined; nothing—or very nearly nothing—is self-evident.

  At the dawn of mathematics, Euclid attempted to start with things that were self-evident. He began his Elements with thirty-five definitions, five postulates, and five common notions, or axioms. Definitions ranged from that of a point (“that which has no parts, or which has no magnitude”3) to that of parallel straight lines (“such as are in the same plane, and which being produced ever so far do not meet”4). Then he made a series of statements such as “things that are equal to the same thing are equal to one another.”5 And the five postulates were:

  “A straight line may be drawn from any one point to any other point” (interpreted to mean that only one straight line may be drawn from any point to any other).6

  “A terminated straight line may be produced at any length in a straight line” (in other words, a segment may be extended indefinitely into a straight line).

  “A circle may be described at any center, at any distance from that center.”7

  “All right angles equal one another.”

  “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on
that side on which are the angles less than the two right angles.”8

  To a true classifier, even these five statements take too much for granted. “I had been told that Euclid proved things,9 and was much disappointed that he started with axioms,” wrote Bertrand Russell of his first childhood encounter with the Elements. “At first, I refused to accept them unless my brother could offer me some reason for doing so, but he said, ‘If you don’t accept them, we cannot go on,’ and so, as I wished to go on, I reluctantly admitted them.”

  As a place to start, the first four postulates struck Euclid, his contemporaries, and the generations of mathematicians to follow as indeed self-evident. Since they are confined to a space we can not only visualize but actually see, the postulates could be checked empirically by drawing with a straightedge or a compass or by stretching a piece of string. As a segment grew in length or a circle in radius, even past the point where a human eye might be able to grasp it, it would not change essentially, and this was as close as anything could get to being obvious and not requiring further proof. But the fifth postulate made claims on the imagination. It said that if two lines were not parallel, they had to cross eventually. Conversely, it said that two parallel lines would never cross, no matter how far they traveled. It was also interpreted to mean that for any straight line, only one parallel could be drawn through any given point not on the original line. This was not obvious; it could not be verified. And because it could not be verified, it had to be proved. For centuries mathematicians struggled to find proof of this claim, and found none.

  The eighteenth century saw two mathematicians’ attempts to prove the fifth postulate by first assuming that it was not correct. The objective of such an exercise is to build on an assumption until it grows evidently absurd, thereby debunking the original premise. But the examples failed to show themselves wrong; the exercises produced internally consistent pictures that settled in the imagination quite comfortably and quite separately from Euclid’s fifth. Both mathematicians deemed this ridiculous and abandoned their efforts. After another century, three different mathematicians—the Russian Nikolai Lobachevski, the Hungarian János Bolyai, and his teacher the German Johann Karl Friedrich Gauss—decided that other, non-Euclidean geometries could exist where four of the postulates obtained but the fifth did not. But what does it mean that they could exist? Do they exist? They do, as long as mathematicians can find no holes or, rather, internal contradictions in them. Can we see them the same way we can see a line segment and a circle? Sure, no less and no more than we can see a strictly Euclidean geometry. So how do we know which is right? The great American mathematician Richard Courant (for whom the Courant Institute of Mathematical Sciences at New York University is named) and his coauthor Herbert Robbins, then a professor at Rutgers University, wrote that for our purposes it did not matter and we might as well choose Euclid: “Since the Euclidean system is rather simpler to deal with,10 we are justified in using it exclusively as long as fairly small distances (of a few million miles!) are under consideration. But we should not necessarily expect it to be suitable for describing the universe as a whole.”

  But how about describing a small piece of the universe? Say, the planet Earth. Or an apple. Remember this for future reference: the Earth and an apple are essentially the same. Let us think about the surface of the Earth, or of an apple, as the plane we are studying. Take an apple and draw a triangle on it. Now, if Euclidean geometry obtained for the surface of the apple, the sum of the angles of this triangle would equal 180 degrees. But because the surface of the apple is curved, the sum of the angles of the triangle is greater. This would mean that the fifth postulate is not true for this surface. Indeed, it is easy to see that on this surface, any two straight lines—a straight line being the extension of a segment that connects two points in the shortest possible way—will cross. All straight lines on the apple, or on the Earth, are “great circles” with their centers at the center of the sphere.

  It was the nineteenth-century German mathematician Bernhard Riemann who developed a geometry of curved spaces, where straight lines are called geodesics and any two of them will cross. The geometry is called elliptic, or simply Riemannian, geometry,11 and it is the geometry used in Einstein’s general theory of relativity.

  Euclid’s world, limited to his immediate surroundings, was, for all intents and purposes, flat. Our world is curved. Humans now routinely travel distances great enough to make the curvature of the Earth part of our lived experience. Not all of us travel so far all the time, but in the imagination—the very place where mathematics resides—the shortest distance between two points is the trajectory described by an airplane, which generally lies along a geodesic, even if we have never heard the word. These straight lines do not go on forever but, being circles, inevitably close in on themselves. And, of course, they cross, any two of them. What seemed absurd in the eighteenth century is now an accurate reflection of the way we experience the world.

  In other words, our world has grown bigger. But that raises two questions: How much bigger can it get? and What does bigger mean? Here, allow me formally to introduce topology,12 an area of mathematics born in St. Petersburg in 1736, when the Swiss mathematician Leonhard Euler, who was teaching there, freed geometry of the burden of measuring distances. He published a paper on the solution to the Königsberg bridge problem, which had been posed by the mayor of the eponymous city, who had wanted Euler to devise a walking tour that would have an individual pass through each of Königsberg’s seven bridges exactly once. Euler concluded that this could not be done. He also showed, first, that in any city with bridges, such a walking tour could be designed if and only if an odd number of bridges led to two areas of the town or to no areas, and, second, that it could not be designed if an odd number of bridges led to one area or to more than two. The third thing that Euler did while solving a problem where locations, not distances, were important was herald a new area of mathematics, which he termed “geometry of position.”

  In this new discipline, size—distance—in the familiar sense of the word did not matter. The number of steps that made up the walking tour made no difference; it was the way these steps were taken. What made an object lesser or greater in this new field was the amount of information required to locate it; to be precise, it was the number of coordinates needed to describe it. A single point has dimension zero; a line segment has one dimension; the surface of something such as a triangle or a square or a sphere has two dimensions. That is correct: the surface of something that we envision as flat and the surface of something that we envision as solid are, for the purposes of topology, the same. This is because when topologists talk about the surface of a sphere, or, say, an apple, they mean just the surface, with no regard for the solid internal space of the apple. Put another way, a topologist is like a tiny bug crawling on the apple, or like Euclid walking on the Earth: neither the bug nor Euclid has much reason to suspect that a triangle he describes will have angles amounting in sum to more than 180 degrees or that the straight line he is walking will not go on forever but will eventually close in on itself, describing a great circle. The curved nature of the surface is a function of the third dimension, of which neither of them has any experience.

  We modern humans, who know firsthand that the Earth is a sphere and that its surface is therefore curved, live in three dimensions. But there is a fourth dimension—we know there is—and it is called time. We cannot move ourselves back and forth in time, so we cannot observe our three-dimensional habitat the way we can observe, by being lifted up into the air, the two-dimensional habitat of lesser animals. We are reduced to exploring the space that surrounds us and making guesses as to what it would look like from a vantage point we can suggest but cannot experience or, really, imagine. This is the nature of the Poincaré Conjecture: the last universalist supposed that the universe was shaped like a sphere—a three-dimensional sphere.

  The young mathematician who gave me topology
lessons for this book—who watched as I painfully tried to wrap my mind, like so many tight rubber bands, around the basic concepts of topology—cringed whenever he encountered references to the Poincaré Conjecture describing the shape of the universe. It would be more accurate to state that the proof of the Poincaré Conjecture will probably aid science greatly in learning the shape and properties of the universe, but this was not the issue Grigory Perelman tackled: he attacked a simply stated, much discussed mathematical problem that had gone unsolved for more than a century. Just like my young tutor and many other mathematicians I met along the way, he emphatically did not care about the physical shape of the universe or the experience of people who inhabited it; mathematics had given him the liberty to live among abstract objects in his own imagination, which was exactly where this problem had to be solved.

  In 1904 Henri Poincaré published a paper on three-dimensional manifolds. What is a manifold? It is an object, or a space, existing in the mathematician’s imagination—whether or not something like it can actually be observed in reality—that can be divided into many neighborhoods. Each neighborhood, taken separately, has a basic Euclidean geometry or can be explained through it, but all the neighborhoods together may add up to something much more complicated. The best example of a manifold is the Earth as portrayed through a series of maps, each showing only a small part of its surface. Imagine a map of Manhattan, for example: its Euclidean nature is obvious. When maps are put together in an atlas, their parallel lines continue not to cross and their triangles maintain their 180-degree nature. But if we used the maps to try to replicate the actual surface of the Earth, we would start with something that looked like a many-many-faceted disco ball, and then we would smooth out the edges and ultimately get a globe that reflected the Earth’s curved complexity—and if we extended Manhattan’s First Avenue and Second Avenue, they would cross. These concepts—maps, atlases, and manifolds—are basic to topology.