A seemingly endless variety of food was sprawled over several tables at the home of Judith L. Baxter and her husband, mathematician Stephen D. Smith, in Oak Park, Ill., on a cool Friday evening in September 2011. Canapés, homemade meatballs, cheese plates and grilled shrimp on skewers crowded against pastries, pâtés, olives, salmon with dill sprigs and feta wrapped in eggplant. Dessert choices included—but were not limited to—a lemon mascarpone cake and an African pumpkin cake. The sun set, and champagne flowed, as the 60 guests, about half of them mathematicians, ate and drank and ate some more.
The colossal spread was fitting for a party celebrating a mammoth achievement. Four mathematicians at the dinner—Smith, Michael Aschbacher, Richard Lyons and Ronald Solomon—had just published a book, more than 180 years in the making, that gave a broad overview of the biggest division problem in mathematics history.
Their treatise did not land on any best-seller lists, which was understandable, given its title: The Classification of Finite Simple Groups. But for algebraists, the 350-page tome was a milestone. It was the short version, the CliffsNotes, of this universal classification. The full proof reaches some 15,000 pages—some say it is closer to 10,000—that are scattered across hundreds of journal articles by more than 100 authors. The assertion that it supports is known, appropriately, as the Enormous Theorem. (The theorem itself is quite simple. It is the proof that gets gigantic.) The cornucopia at Smith's house seemed an appropriate way to honor this behemoth. The proof is the largest in the history of mathematics.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
And now it is in peril. The 2011 work sketches only an outline of the proof. The unmatched heft of the actual documentation places it on the teetering edge of human unmanageability. “I don't know that anyone has read everything,” says Solomon, age 66, who studied the proof his entire career. (He retired from Ohio State University two years ago.) Solomon and the other three mathematicians honored at the party may be the only people alive today who understand the proof, and their advancing years have everyone worried. Smith is 67, Aschbacher is 71 and Lyons is 70. “We're all getting old now, and we want to get these ideas down before it's too late,” Smith says. “We could die, or we could retire, or we could forget.”
That loss would be, well, enormous. In a nutshell, the work brings order to group theory, which is the mathematical study of symmetry. Research on symmetry, in turn, is critical to scientific areas such as modern particle physics. The Standard Model—the cornerstone theory that lays out all known particles in existence, found and yet to be found—depends on the tools of symmetry provided by group theory. Big ideas about symmetry at the smallest scales helped physicists figure out the equations used in experiments that would reveal exotic fundamental particles, such as the quarks that combine to make the more familiar protons and neutrons.
Group theory also led physicists to the unsettling idea that mass itself—the amount of matter in an object such as this magazine, you, everything you can hold and see—formed because symmetry broke down at some fundamental level. Moreover, that idea pointed the way to the discovery of the most celebrated particle in recent years, the Higgs boson, which can exist only if symmetry falters at the quantum scale. The notion of the Higgs popped out of group theory in the 1960s but was not discovered until 2012, after experiments at CERN's Large Hadron Collider near Geneva.
Symmetry is the concept that something can undergo a series of transformations—spinning, folding, reflecting, moving through time—and, at the end of all those changes, appear unchanged. It lurks everywhere in the universe, from the configuration of quarks to the arrangement of galaxies in the cosmos.
The Enormous Theorem demonstrates with mathematical precision that any kind of symmetry can be broken down and grouped into one of four families, according to shared features. For mathematicians devoted to the rigorous study of symmetry, or group theorists, the theorem is an accomplishment no less sweeping, important or fundamental than the periodic table of the elements was for chemists. In the future, it could lead to other profound discoveries about the fabric of the universe and the nature of reality.
Except, of course, that it is a mess: the equations, corollaries and conjectures of the proof have been tossed amid more than 500 journal articles, some buried in thick volumes, filled with the mixture of Greek, Latin and other characters used in the dense language of mathematics. Add to that chaos the fact that each contributor wrote in his or her idiosyncratic style.
That mess is a problem because without every piece of the proof in position, the entirety trembles. For comparison, imagine the two million-plus stones of the Great Pyramid of Giza strewn haphazardly across the Sahara, with only a few people who know how they fit together. Without an accessible proof of the Enormous Theorem, future mathematicians would have two perilous choices: simply trust the proof without knowing much about how it works or reinvent the wheel. (No mathematician would ever be comfortable with the first option, and the second option would be nearly impossible.)
The 2011 outline put together by Smith, Solomon, Aschbacher and Lyons was part of an ambitious survival plan to make the theorem accessible to the next generation of mathematicians. “To some extent, most people these days treat the theorem like a black box,” Solomon laments. The bulk of that plan calls for a streamlined proof that brings all the disparate pieces of the theorem together. The plan was conceived more than 30 years ago and is now only half-finished.
If a theorem is important, its proof is doubly so. A proof establishes the honest dependability of a theorem and allows one mathematician to convince another—even when separated by continents or centuries—of the truth of a statement. Then these statements beget new conjectures and proofs, such that the collaborative heart of mathematics stretches back millennia.
Inna Capdeboscq of the University of Warwick in England is one of the few younger researchers to have delved into the theorem. At age 44, soft-spoken and confident, she lights up when she describes the importance of truly understanding how the Enormous Theorem works. “What is classification? What does it mean to give you a list?” she ponders. “Do we know what every object on this list is? Otherwise, it's just a bunch of symbols.”
Reality’s deepest secrets
Mathematicians first began dreaming of the proof at least as early as the 1890s, as a new field called group theory took hold. In math, the word “group” refers to a set of objects connected to one another by some mathematical operation. If you apply that operation to any member of the group, the result is yet another member.
Symmetries, or movements that do not change the look of an object, fit this bill. Consider, as an example, that you have a cube with every side painted the same color. Spin the cube 90 degrees—or 180 or 270—and the cube will look exactly as it did when you started. Flip it over, top to bottom, and it will appear unchanged. Leave the room and let a friend spin or flip the cube—or execute some combination of spins and flips—and when you return, you will not know what he or she has done. In all, there are 24 distinct rotations that leave a cube appearing unchanged. Those 24 rotations make a finite group.
Simple finite groups are analogous to atoms. They are the basic units of construction for other, larger things. Simple finite groups combine to form larger, more complicated finite groups. The Enormous Theorem organizes these groups the way the periodic table organizes the elements. It says that every simple finite group belongs to one of three families—or to a fourth family of wild outliers. The largest of these rogues, called the Monster, has more than 1053 elements and exists in 196,883 dimensions.* (There is even a whole field of investigation called monsterology in which researchers search for signs of the beast in other areas of math and science.) The first finite simple groups were identified by 1830, and by the 1890s mathematicians had made new inroads into finding more of those building blocks. Theorists also began to suspect the groups could all be put together in a big list.
Mathematicians in the early 20th century laid the foundation for the Enormous Theorem, but the guts of the proof did not materialize until midcentury. Between 1950 and 1980—a period which mathematician Daniel Gorenstein of Rutgers University called the “Thirty Years' War”—heavyweights pushed the field of group theory further than ever before, finding finite simple groups and grouping them together into families. These mathematicians wielded 200-page manuscripts like algebraic machetes, cutting away abstract weeds to reveal the deepest foundations of symmetry. (Freeman Dyson of the Institute for Advanced Study in Princeton, N.J., referred to the onslaught of discovery of strange, beautiful groups as a “magnificent zoo.”)
Those were heady times: Richard Foote, then a graduate student at the University of Cambridge and now a professor at the University of Vermont, once sat in a dank office and witnessed two famous theorists—John Thompson, now at the University of Florida, and John Conway, now at Princeton University—hashing out the details of a particularly unwieldy group. “It was amazing, like two Titans with lightning going between their brains,” Foote says. “They never seemed to be at a loss for some absolutely wonderful and totally off-the-wall techniques for doing something. It was breathtaking.”
It was during these decades that two of the proof's biggest milestones occurred. In 1963 a theorem by mathematicians Walter Feit and John Thompson laid out a recipe for finding more simple finite groups. After that breakthrough, in 1972 Gorenstein laid out a 16-step plan for proving the Enormous Theorem—a project that would, once and for all, put all the finite simple groups in their place. It involved bringing together all the known finite simple groups, finding the missing ones, putting all the pieces into appropriate categories and proving there could not be any others. It was big, ambitious, unruly and, some said, implausible.
The man with the plan
Yet Gorenstein was a charismatic algebraist, and his vision energized a new group of mathematicians—with ambitions neither simple nor finite—who were eager to make their mark. “He was a larger than life personality,” says Lyons, who is at Rutgers. “He was tremendously aggressive in the way he conceived of problems and conceived of solutions. And he was very persuasive in convincing other people to help him.”
Solomon, who describes his first encounter with group theory as “love at first sight,” met Gorenstein in 1970. The National Science Foundation was hosting a summer institute on group theory at Bowdoin College, and every week mathematical celebrities were invited to the campus to give a lecture. Solomon, who was then a graduate student, remembers Gorenstein's visit vividly. The mathematical celebrity, just arrived from his summer home on Martha's Vineyard, was electrifying in both appearance and message.
“I'd never seen a mathematician in hot-pink pants before,” Solomon recalls.
In 1972, Solomon says, most mathematicians thought that the proof would not be done by the end of the 20th century. But within four years the end was in sight. Gorenstein largely credited the inspired methods and feverish pace of Aschbacher, who is a professor at the California Institute of Technology, for hastening the proof's completion.
One reason the proof is so huge is that it stipulates that its list of finite simple groups is complete. That means the list includes every building block, and there are not any more. Oftentimes proving something does not exist—such as proving there cannot be any more groups—is more work than proving it does.
In 1981 Gorenstein declared the first version of the proof finished, but his celebration was premature. A problem emerged with a particularly thorny 800-page chunk, and it took some debate to resolve it successfully. Mathematicians occasionally claimed to find other flaws in the proof or to have found new groups that broke the rules. To date, those claims have failed to topple the proof, and Solomon says he is fairly confident that it will stand.
Gorenstein soon saw the theorem's documentation for the sprawling, disorganized tangle that it had become. It was the product of a haphazard evolution. So he persuaded Lyons—and in 1982 the two of them ambushed Solomon—to help forge a revision, a more accessible and organized presentation, which would become the so-called second-generation proof. Their goals were to lay out its logic and keep future generations from having to reinvent the arguments, Lyons says. In addition, the effort would whittle the proof's 15,000 pages down, reducing it to a mere 3,000 or 4,000.
Gorenstein envisioned a series of books that would neatly collect all the disparate pieces and streamline the logic to iron over idiosyncrasies and eliminate redundancies. In the 1980s the proof was inaccessible to all but the seasoned veterans of its forging. Mathematicians had labored on it for decades, after all, and wanted to be able to share their work with future generations. A second-generation proof would give Gorenstein a way to assuage his worries that their efforts would be lost amid heavy books in dusty libraries.
Gorenstein did not live to see the last piece put in place, much less raise a glass at the Smith and Baxter house. He died of lung cancer on Martha's Vineyard in 1992. “He never stopped working,” Lyons recalls. “We had three conversations the day before he died, all about the proof. There were no good-byes or anything; it was all business.”
Proving it again
The first volume of the second-generation proof appeared in 1994. It was more expository than a standard math text and included only two of 30 proposed sections that could entirely span the Enormous Theorem. The second volume was published in 1996, and subsequent ones have continued to the present—the sixth appeared in 2005.
Foote says the second-generation pieces fit together better than the original chunks. “The parts that have appeared are more coherently written and much better organized,” he says. “From a historical perspective, it's important to have the proof in one place. Otherwise, it becomes sort of folklore, in a sense. Even if you believe it's been done, it becomes impossible to check.”
Solomon and Lyons are finishing the seventh book this summer, and a small band of mathematicians have already made inroads into the eighth and ninth. Solomon estimates that the streamlined proof will eventually take up 10 or 11 volumes, which means that just more than half of the revised proof has been published.
Solomon notes that the 10 or 11 volumes still will not entirely cover the second-generation proof. Even the new, streamlined version includes references to supplementary volumes and previous theorems, proved elsewhere. In some ways, that reach speaks to the cumulative nature of mathematics: every proof is a product not only of its time but of all the thousands of years of thought that came before.
In a 2005 article in the Notices of the American Mathematical Society, mathematician E. Brian Davies of King's College London pointed out that the “proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual.” His article brought up the uncomfortable idea that some mathematical efforts may be too complex to be understood by mere mortals. Davies's words drove Smith and his three co-authors to put together the comparatively concise book that was celebrated at the party in Oak Park.
The Enormous Theorem's proof may be beyond the scope of most mathematicians—to say nothing of curious amateurs—but its organizing principle provides a valuable tool for the future. Mathematicians have a long-standing habit of proving abstract truths decades, if not centuries, before they become useful outside the field.
“One thing that makes the future exciting is that it is difficult to predict,” Solomon observes. “Geniuses come along with ideas that nobody of our generation has had. There is this temptation, this wish and dream, that there is some deeper understanding still out there.”
The next generation
These decades of deep thinking did not only move the proof forward; they built a community. Judith Baxter—who trained as a mathematician—says group theorists form an unusually social group. “The people in group theory are often lifelong friends,” she observes. “You see them at meetings, travel with them, go to parties with them, and it is really is a wonderful community.”
Not surprisingly, these mathematicians who lived through the excitement of finishing the first iteration of the proof are eager to preserve its ideas. Accordingly, Solomon and Lyons have recruited other mathematicians to help them finish the new version and preserve it for the future. That is not easy: many younger mathematicians see the proof as something that has already been done, and they are eager for something different.
In addition, working on rewriting a proof that has already been established takes a kind of reckless enthusiasm for group theory. Solomon found a familiar devotee to the field in Capdeboscq, one of a handful of younger mathematicians carrying the torch for the completion of the second-generation proof. She became enamored of group theory after taking a class from Solomon.
“To my surprise, I remember reading and doing the exercises and thinking that I loved it. It was beautiful,” Capdeboscq says. She got “hooked” on working on the second-generation proof after Solomon asked for her help in figuring out some of the missing pieces that would eventually become part of the sixth volume. Streamlining the proof, she says, lets mathematicians look for more straightforward approaches to difficult problems.
Capdeboscq likens the effort to refining a rough draft. Gorenstein, Lyons and Solomon laid out the plan, but she says it is her job, and the job of a few other youngsters, to see all the pieces fall into place: “We have the road map, and if we follow it, at the end the proof should come out.
Four Enormous Families
Symmetries can be broken down into basic pieces. Called finite simple groups, they function like elements, coming together in different combinations to form larger, more complicated symmetries.
The Enormous Theorem organizes these groups into four families. Although its proof is huge, the theorem itself is just one sentence that lists all four: “Every finite simple group is cyclic of prime order, an alternating group, a finite simple group of Lie type, or one of the twenty-six sporadic finite simple groups.”
Here is a brief rundown of those families:
Cyclic groups were among the first building blocks to be categorized. Turn a regular pentagon through one fifth of a circle, or 72 degrees, and it looks unchanged. Turn it five times, and you are back at the beginning. Cyclic groups repeat themselves. The cyclic finite simple groups each have a prime number of members. Cyclic groups with more than two even numbers of members can be broken down further, so they are not simple.
Alternating groups come from switching around the members of a set. A full group of symmetries contains all the permutations, or switches. But an alternating group contains only half of them—the ones that have an even number of switches. For example, let us say you had a set of three numbers: 1, 2 and 3. There are six different ways to write that set: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). The alternating group contains three of those. In terms of symmetry, each of these arrangements might correspond to a sequence of symmetries (that is, turn the cube up, then on its side, and so on).
Lie-type groups, named for 19th-century mathematician Sophus Lie, start to get more complicated. They are related to things called infinite Lie groups. The infinite groups include the rotations of a space itself that do not change the volume. For example, there are infinitely many ways to spin a doughnut without changing the doughnut itself. The finite analogues of these infinite groups are the Lie-type groups—in other words, the doughnut in a Lie-type group permits only a finite number of rotations. Most finite simple groups fall into this family. Neither infinite Lie groups nor Lie-type groups are limited to our pedestrian three dimensions. Ready to talk about the symmetries that arise in 15-dimensional space? Then look to these groups.
Sporadic groups make up the family of rogues. They include 26 outliers that do not line up neatly in the other families. (Imagine if the periodic table of elements had a column for “miscreants.”) The largest of these sporadic groups, called the Monster, has more than 1053 elements and can be faithfully represented in 196,883 dimensions.* It is baffling and bizarre, and no one really knows what it means but it is tantalizing to think about. “I have a sneaking hope, a hope unsupported by any facts or any evidence,” physicist Freeman Dyson wrote in 1983, “that sometime in the twenty-first century physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the universe.” —S. O.
*Editor's Note (6/19/15): Because of a formatting error, the figure 1053, describing the number of elements in the finite simple group called The Monster, originally appeared as 1053 in the online version of this story.