Simon

by Alexander Masters

  Conway hates to lose. When I visited him at his house in Princeton he pointed out a photograph of himself on the bookcase, freshly fled from Cambridge and Group Theory, in his new office in America. It was taken a quarter of a century ago. Balanced at the top of a computer screen in the picture is a number, 15.92.

  It’s his fastest time—in seconds—for calculating the days of the week for ten random dates. “I’m the best in the world at it,” he pronounced. “There was a period when I was briefly ill, when I became second-best in the world and somebody overtook me. But that was only for about six months. Then I got better and was the fastest again! What’s your date of birth, Alex? Do you mind if I call you Alex? Ah, Tuesday.”

  It was difficult to keep him on the subject of Simon.

  “Simon? I dunno, the life of an aging prodigy is not easy. Inevitably, you compare yourself with what you were. I mean, we all suffer from it. We slowly substitute, you know, our knowledge and memory for our native wit. I mean, I was very bright, I wasn’t as natively bright as Simon…or…maybe I was! Er, you see, OK, so let me go off about myself again…”

  Conway is crippled by alimony payments.

  Mathematically also, Simon and Conway are poles apart. Simon is meticulous, never makes mistakes; he focuses on one particular problem to the point of burying himself in a rut over it; he’s a supreme manipulator of equations. Conway makes frequent schoolboyish blunders, such as putting in a minus where he should have had a plus— a small oversight, known as a “sign error,” but it instantly flings an answer from one side of the universe to the other, out of quiet correctness into blazing absurdity. This does not make him a bad mathematician; he spots the outrage soon enough.

  Richard Parker (the “Parker” of Conway, Curtis, Parker, Wilson and Simon) remembers one day in the Common Room watching Conway come back from the refreshment counter carrying a full mug of coffee. Slowly, Conway turned the mug upside down and spilled the contents onto the floor. Then he turned the mug the right way up, completed the walk to the table and sat down.

  “Conway!” Richard cried. “What are you doing? What just happened?”

  “I made a sign error,” admitted Conway. “Instead of keeping the mug up, I accidentally put in a minus sign and turned it upside down.”

  Parker is a third sort of theorist. There is no one type of mathematician. They are as varied as lettuces. Conway, colorful, broad-ranging, flighty but also profound—radicchio; Simon, narrow, unforgettable, perfect in a thin field—endive; Parker, unfocused, watery, not at the same level as the first two but now and again precisely right—iceberg.

  I miss Parker terribly [says Conway]. He’s an ideas man. He would come in in the morning and say, “I think I can cure what’s wrong with last night’s maths argument—perhaps it’s like this…” And I’d say, “Richard, that’s really stupid,” and prove that it was just nonsense. Then he’d go away and say, “OK, how about this?” So I’d say, “Give me ten minutes and I’ll come up with a counterexample.” But he was never daunted. And every 100th time…Once, he’d taken three cases in which something corresponded to something else and said that maybe there’s a one-to-one correspondence between these things, and I said, “Richard, you’re so stupid.” To base a conjecture on three cases was ridiculous. So we took a fourth case, and a fifth…I was getting a plane to New Jersey the next morning, and Neil Sloane [an eminent mathematician] met me at the airport because we had some project that we were going to work on. I said, “Drop all that, this idea of Richard Parker’s is absolutely fantastic.” I lived off that paper for the next five years.

  Parker was like that. One in a hundred of his ideas was OK, and there were ten a day at least!

  This type of mathematician—enthusiastic, spotting shadows everywhere, occasionally putting better calculators or theoreticians on the right scent—has been vital to the history of Group Theory, and in particular the Monster.

  Simon is different [says Conway]. He has this tremendous understanding, and he cannot rest if there’s some possible falsehood or contradiction lying around. Even if it’s somebody on a train saying something trivial. He’s like a dog gnawing at a bone. He has to find out what the truth is. I mean, I sort of know whether something I’m doing is deep and subtle and valuable mathematics or whether it’s just a frivolity, but I refuse to be intimidated by it now. I’m prepared to do whatever I like. But Simon to some extent doesn’t know. He really regards all things as equal.

  Conway’s mathematical strength is in his flamboyant irreverence. He discovered:

  The Game of Life. A game in which simple patterns made on a checkerboard “evolve” according to three extremely simple rules and (in certain variants) explain the coordinated movements of flocks of starlings, expose the organizing principles of termites building up woodchips, and reveal the rationale behind the behavior of human crowds.

  An entirely new type of number called “Surreal Numbers.” He discovered them while sitting at a table in a café in Cambridge, playing Chinese checkers. The way Chinese checkers progresses as a game was, he realized, a code for a previously unknown counting system.

  The Conway Groups in the twenty-fourth dimension (though Simon’s been unable to explain to me why Conway was looking for symmetries up there).

  As a young man he wanted to “make some absolutely outstanding contribution and be ‘Conway, who’s the best mathematician in the world.’”

  Well, I don’t necessarily mean the best, actually, just a certain standard…which is easiest to describe by saying “the best.”

  Other mathematicians in the Cambridge faculty would come in at 11:30, spot Conway, Parker, Larissa Queen and Simon playing backgammon in the Common Room, and scowl. When these po-faced types staggered out of their offices again at 3 p.m. for a cup of tea, the four wastrels would still be playing.

  There would always be a crowd of people standing round the board “kibitzing,” which meant offering advice [says Larissa]. Simon was a very, very active “kibitzer,” and the first time you play backgammon with seven or eight “kibitzers” it’s impossible, your brain goes into a stupor and you can’t play because you hear eight different suggestions. The next stage comes when you can ignore them, and in the ultimate stage you can hear everybody. But it takes a long time to arrive there, lots of practice. You have to abandon your work for a while and concentrate on the backgammon. There was a certain vocabulary, a shorthand, you know. They would say, “Come screaming out,” meaning to get out of your home table. In particular, there comes a time when no matter what you throw, the outcome is determined, and then people would say, “The dice are no longer,” meaning the dice are no longer needed. And if Simon said, “The dice are no longer,” people took it very seriously and stopped. They’d say, “Oh well, fine, start a new game,” because on the few occasions when somebody computed, Simon was never wrong.

  After his second marriage broke up, Conway tried to kill himself, and spent a week in the hospital worrying about how he was going to face his students. Everyone in the faculty knew about the attempt. He knew they were all thinking about it. They knew that he knew that they knew; and so he wondered to himself, “What would Conway do? What would be a typically Conway-ish way to deal with this embarrassing moment?”

  It would never cross Simon’s mind to think like this—to care so much about what his audience thought, or to talk about himself in the third person, as though he existed both as a human body and, like numbers themselves, a Platonic Ideal.

  Conway solved the problem with combative brilliance. He borrowed a T-shirt from a friend who’d climbed a notorious granite outcrop in California, and appeared at the lectern wearing it—the logo read “Suicide Rock.”

  * * *

  Conway once invented a pair of spectacles to see in four dimensions:

  Drawing by John Conway.

  He used a flying helmet to hold the spectacles in place, and spent a few hours stumbling round the mulberry tree in Sidney Sussex Fellows’ Garden. His brain a
djusted, he walked around Cambridge with perfect ease, the first man in history to see hyperspheres, portholes to distant universes and observe the spirit world.

  “No, no, Alexander. Obviously my glasses were not going to let you see an extra dimension on top of the three we can already see. That would be ridiculous. Simon told me you were not a mathematician,” concluded Conway worriedly, “but you do know that we can see only in three dimensions, don’t you?”

  * * *

  In the Atlantis office, Conway, Curtis, Parker, Wilson and Simon kept work on the atoms of symmetry that was finished or in immediate progress in a sweaty folder that sat on a chair, growing plump, bloated, disgustingly distended; finally, obese. Occasionally, it burst.

  If any of the five men had a new idea about how to solve a problem, the chair plus the folder of flab was given a forceful kick, and rattled across to him to sort it out.

  Whenever it seemed the room would certainly explode with all the gathered information about symmetry, one of the men—usually Parker, sometimes Conway, never Simon—would sit down at the office’s orange electric typewriter and type out the latest confirmed results on pages that were colored—with deliberate nicety—Atlantic blue.

  The most troublesome Groups were draped over the tables of the Common Room outside the door. Chairs and low tables ran down the middle of this dank, rectangular space, and ended in a “refreshments” counter that was kept locked except at elevenses and mid-afternoon: 2p for a biscuit; sugar from a bowl; self-serve tea and coffee 5p a mug (mug not provided).

  Anybody interested could come in, grab a cup or mug from the dirty crockery by the corner sink, splosh out a hot drink, perch over one of these sheets of crosshatched paper bearing the latest state of knowledge about the symmetries of the universe, and add their tuppenceworth. Spotted a mistake in the top right-hand-corner entries of J4? Don’t think the Harada-Norton Group is quite up to scratch in the 133rd dimension? Pop in your solution, let’s see…in this unscribbled-on scrap of margin here. See if Simon approves your suggestion when he gets back tomorrow from his bus trips to Bean (Kent) and Leek (Staffordshire).

  This Common Room was the focal point of Cambridge mathematical life and, like a hospital, was open twenty-four hours a day. Even at 4 a.m. it was obscurely busy: slippered noises of people pacing back and forth in the surrounding offices; bursts of typewriter clatter; the hum of electrical fittings; a kettle boiling; the clack of dice and backgammon counters; a cry!; an office chair flung back; equations refusing to cooperate; shouts, swearing, feet pounding down concrete steps to street level.

  Left to right: obese folder, Conway, Atlas.

  The squeak, swing, squeak of the front door and a small returning rush of fresh air.

  Simon’s time of Great Silence from the biographer’s point of view was his time of great noise in the hyperdimensional universe of mathematics.

  Simon worked on half a dozen other projects during this period—not just the Atlas: surreal numbers, the Game of Life, Y monograms, the biMonster. He “masterminded” the existence proof of the Harada-Norton Group.

  But it is his work on the Atlas that keeps his name alive.

  After fifteen years of constipation—

  (“I think pregnancy is a better metaphor,” mumbles Simon.)

  —Simon’s most famous joint mathematical publication at Cambridge, the Atlas of Finite Groups, was excreted.

  (“Born,” hurries in Simon. “In 1985,” he adds.)

  The mathematical equivalent of a global malaria-eradication program (the counterpart to malaria being “intellectual despair about Symmetry Groups”), the full title of the publication is the Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. “Groups” because the book is investigating symmetries, similar to and ; “Subgroups” because that’s always the way forward in this subject; “Finite” because the associated Group Tables of these symmetries have a countable number of entries; “Atlas” because it attempts to cover all the types of foundational finite symmetry known to exist, and in a certain sense the tables can be regarded as a type of map; and “Simple” because a) these Groups are all atoms of symmetry and b) the contributors are lying through their teeth.

  The book is imbecilic with complexity—and littered with errors. Even with five of the best mathematicians in the university (two of them, Conway and Simon, among the best in the world) working for a decade and a half, there was too much information to master. Much of Simon’s work since this date has been to do with correcting the mistakes and filling out the gaps of understanding that still remain in the Atlas. The introduction to the Atlas, written by Conway, includes the following salaam to his brilliance:

  Simon Norton constructed the tables for a large number of extensions, including some particularly complicated ones. He has throughout acted as “troubleshooter”—any difficult problem was automatically referred to him in the confident expectation that it would speedily be solved.

  Simon (after a haircut), Rob Wilson (another of the editors) and Conway at an Atlas reunion meeting in the King’s Street Run pub, Cambridge. The event was organized by Siobhan Roberts, Conway’s biographer. “There are so many biographers here,” grumbled Conway, “we can hardly move.”

  As the case of Triangle and three-pieces-of-garbage-kicked around-Simon’s-Excavation showed, symmetries can be categorized into types. Those two symmetries (which appeared at first to have nothing to do with each other) turned out to be identical. The job of the Atlas of Finite Groups was to catalogue all the fundamental types of symmetry. That’s why they’re called the Simple Groups. They are the atoms of the subject. There can be other symmetries apart from these, but they will always be composed of one or more of the basic forms found in the Atlas, in the same way that a molecule is always composed of atoms from the periodic table. The Simple Groups catalogued in the Atlas are the elements, atoms or building blocks of Finite Group Theory. Every Finite Group can be broken down to a collection of one of these fundamentals, just as any molecule can be broken down into a combination of the elements in the periodic table, or any building can be fragmented into doors, windows, bricks, electrical wires, bent pipes and plaster dust. The Atlas is therefore the periodic table or builders’ catalogue for Group Theory.

  This classification of Groups into their simplest components was a project of Victorian grandeur: a taxonomy (to use a third metaphor) of all the insects of Finite Symmetry—every mahogany drawer in the entomology halls of mathematics flung open, every tiny beetle sent in from a vicar’s garden plucked up and investigated to the last follicle.

  The largest “sporadic” atom in the Atlas—the biggest specialist brick, the most gargantuan and armor-plated exotic insect—is the Monster. It is the largest Finite Simple Group in the universe. There is nothing beyond it, except silence.

  As soon as possible after the manuscript was published, Conway emigrated to a professorship in America, desperate never to look at a Group again.

  The mathematician Benedict Gross has said that if his house was on fire, the one thing he’d battle back through the roaring flames and crashing timbers to rescue, leaping across cavernous molten stairwells, dodging the exploding gas boiler—would be the Atlas.

  “Which is quite unnecessary,” points out Conway, “because you can get it online.”

  Simon’s copy of this immortal book—torn to halfway down the ring binding, three blobs of biryani and Ferns’ brinjal pickle staining the top right-hand corner—has moved from the place where we first met it, in Chapter 4. Simon consults the Atlas frequently. Now, it rests on top of the twizzle-legged table, under a love letter to his mother and a new paperback murder mystery called Zombies of the Gene Pool.

  And abruptly we are here: the critical moment, the biographical climax of Simon’s story.

  It’s so important to get this right—to get the pacing correct, to understand its immediacy, its horror.

  I’ll use the present tense. It’s a moment of resonance for everyon
e, not just geniuses.

  We’re back in the Atlantis office—or perhaps it was the Common Room. I’ll make an autocratic biographer’s decision, and call it the office. It is winter. The metal-framed windows ache with cold. Three people critical to the history of Group Theory are in the room, occasionally kicking a chair at each other between the towers of paper: Conway, Parker and Simon. They are discussing J4, the fourth Janko Group, page 188 of the Atlas.

  Simon is sitting by the window. He blows his nose into a hand towel. He has a powerful interest in the Group J4. He’s been chasing it down in 112 dimensions, and has almost caught it.

  Conway—we’ll put him at the desk with the orange typewriter, one hand delicately positioning the page, the other stabbing at the keys. With a Group Table that’s 86,775,571,046,077,562,880 columns wide, even the shorthand versions of the fourth Janko Group flop across two pages and require supreme secretarial skills.

  “Simon,” says Conway, turning round, “if 2(1+12).3.(M22.2) is an involution centralizer of J4, what is…?”

  Today, no one can remember what the question was—something “comparatively trivial,” recollects Parker.

  Simon answered Conway instantly, of course.

  Conway, satisfied, returned to his typing…then he stopped and gasped—Parker uses that word, “gasped”—and turned back to Simon. The unbelievable had happened.