Note: If we cut out rows in this high-handed fashion, we must also cut out the corresponding columns. What we are removing is the entrance of a certain symmetry operation into the Table, either through its row or its column. What displeases Simon (and the Closure requirement) is if, having removed all the possible ways a symmetry operation can get into a Table, its ugly mug still pops up inside.
We could have cut back the big Group Table in another way. Say, like this:
If we press together what’s left over in this case, and tidy up, we get this result:
This again is correct as far as the arithmetic goes. It’s been cut out of the larger Table, so it must be:
plus still equals .
But as a table it’s not aesthetically pleasing in Simon’s eyes. Given that all we put into the table was and , the in the bottom right-hand corner looks out of place and ugly and therefore contravenes Closure, the mathematical definition of a Group that’s been specifically designed to prevent this sort of outrage. Group Tables should have an artistic appearance of balance, just like the operations they’re describing, which is why Simon and so many other mathematicians are prepared to dedicate their lives to them.
This is a fundamental difference between the Group Table for the rotations of Square and the one for the rotations of Triangle: Square’s Group Table has a Subgroup; Triangle’s doesn’t. As Group Tables get larger there are usually more and more Subgroups hidden away inside them, but not always. Some Groups preserve their purity.
The French schoolchild who discovered Subgroups was Évariste Galois, one of the great heroes of mathematics. Galois, born in 1811, nearly demented by genius and the failure of people to understand what he was talking about, once threw a blackboard eraser at his math teacher because the man was too slow; later escaped from his school by clambering over the playground wall; then marched through revolutionary Paris with a loaded rifle and was thrown in prison for waving a knife around at a party while making threatening remarks about the king. After being released, he was banged up again for some other absurdity. For several years he seemed on the brink of madness.
He died, “pierced through and through” in a gun duel over a barmaid. (“No, Alex, she was of higher status”—Simon.) The night before, he had written in the margin of his scribbled papers that revealed his astounding discoveries in Group Theory, “I have no time.”
(“In addition to a misquotation that is a romanticization.”)
He was twenty-one.
(“You have at least got that right.”)
* * *
With the discovery of Subgroups, there’s something constructive you can now do with a Group Table rather than simply use it as a tracking device for a triangle or square. The closest analogy I can think of is to a car—that’s the equivalent of a Group. It’s a nice object…
(“No, it is not”—Simon.)
…does useful things…
(“Such as destroy the world?”)
…but if we smash this car apart…
(“Good riddance.”)
…you have two things: an engine (the Subgroup) and wreckage (all the other bits left over from the Group Table once you’ve extracted the Subgroup). Two things (engine and wreckage) instead of one thing (just the car) give you possibilities. You can now be creative about how you put them back together. Taking bits from the wreckage, you can bolt them onto the engine in new ways, to make a lawnmower, a microlight airplane, a go-cart…
(“If you continue to talk about cars and their petrol-driven variants, I’m leaving the book.”)
…or a giant egg whisk.
(“That’s more like it.”)
Group Theorists spend a large portion of their time tinkering with Subgroups and their wreckage. The fellow with an old engine and boxes of oily bits at the bottom of the garden calls his workplace a shed. Simon calls his the Centre for Mathematical Sciences, Cambridge.
26
Simon has no oil on his feathers at all. He’s one of the most completely honest people you’ll encounter, and he takes it for granted that everyone is as honest as he is, so he’s prey for every rogue painter and builder. He’s the only landlord in Cambridge who dropped the rent when the poll tax came in.
Harriet Snape, former tenant
This book—my book—is changing Simon. New sounds have been massing under my floorboards.
Zweeeeeppppth-duB-thrruppPH!—something rolled but abruptly stopped.
Trrr$$$$chchcx—plastic tearing.
Wheeee, thunK! Wheeee, thunK!—items flung about.
XxxccchuuuuhXXxxcchuuhXXXXcchuhxxccchuuuuh
XXxxcchuuhXXXXcchuhhXXXc
Some of these noises have managed to squeeze through my carpet weave and let me know what’s going on. “Wwwshshs-thlump-thlump-duBUMp”: an object, very heavy, being dragged past the booby-trapped stairs. “K-psheeow, KAH-posh, kluumpff”: rubble and wood shavings kicked aside. “Eee-urrgghg,” the cautious opening of the Excavation front door.
Simon emerges with a rubbish bag.
Shocked by descriptions of the Excavation, he has started to clean up the g— (five letters)
“No!”
…the m— (six letters, archaic, think earth closet)
“No!”
…the s— (six letters, found at the bottom of lakes and coffee)
“No! No! NO!”
The mess that is carefully arranged into plastic bags in the Excavation.
“I am ashamed of it,” he admits, dipping and shaking his head like a horse, clomping around the debris with a black dustbin bag. “Wait! What’s that you just dropped in? Take it out! Oh dear, oh dear! King Alfred’s Winchester Bus Schedule, 1994. No! No! I need to keep that. I refer to that often!”
By the end of this book it’s likely I shall be writing about someone entirely different from the man with whom I began.
The reason he’s ashamed of the mess is because “it’s dysfunctional.”
“What are you talking about? What’s dysfunctional?”
“It.”
“What’s ‘it’?”
“The mess.”
“You mean the state of these rooms suggests that you are dysfunctional.”
“No,” he retorts, “it is dysfunctional. It affects how my life works. It functions badly.”
The objects on the rut that leads to his bed are starting to compress into cardboard stone.
“What about if,” I suggest, “instead of papers, it was rubble down here? No other inconvenience, no dysfunction, an occasional sneeze or two, because of the dust, that’s all—would you mind that?”
“Yes. Sneezing is a functional issue.”
“What about if it were vacuumed rubble? And a few carpets laid along the paths so you wouldn’t hurt your feet?”
“Rubble that I can’t move?” asks Simon, intrigued.
“If instead of plastic bags, it was rocks and pebbles, like at the seaside. If that table was a fat boulder. Those drawers filled with shingle. Everything just as it is, chaos and mayhem, but made of stone.”
“I wouldn’t mind that,” says Simon, relaxing.
The whole process of cleaning up leaves Simon riven with anxiety: the refuse men might step through his patio ivy to complain that he’s overfilled his wheeliebin so the lid pops open. They’ll bang on the window, gesticulate at a colleague writhing on the pavement because of a slipped disk, city housing officials will demand words.
Because of the cleaning putsch there have been fresh discoveries:
Under an Asda Supermarket bag wave, two feet from his clothes cupboard, a cache of postcards from 1969. They are from teenage Romanian girls: Marieta, who signs her name, in brackets, like a hurried whisper; “Christina” (“I am at the sea-side and I send you a thought of friendship”); Lucia (“Maybe you will not know who is Lucia Mitrofan who writes you this view card”); Camelia (“When I saw your name, I hurried to write to you…”).
Camelia’s postcard has an efficient pencil tick (by Simon) in one corner. Why
a tick? Because he’s written back. A second card from Camelia has been found a yard away in a saucepan.
Dear Simon,
You asked me if I was in Czechoslovakia on August 20 and 21 [the anniversary of the invasion by the Soviet Union]. Yes I was. I think your question concerns the demonstrations of the people there. I didn’t see myself the incidents, because in Karlovy-Vary all happened in the night when I was inside, but I feeled the tear gas in Prague.
Other finds include more newspaper clippings (collected by his mother) about Simon’s startling boyhood triumphs in competitions (Daily Sketch). Glowing features in the Times Educational Supplement, the Daily Express, the Daily Telegraph, the Sun, the Star (Johannesburg) and (his greatest supporting newspaper) the Daily Mail applaud his brilliance at the International Mathematics Olympiad.
The Sunday Times noted that while “the Russians and Hungarians had special coaching,” the British team “were given no special preparation and seemed to treat the International Olympiad partly as a holiday.” The French “explained away their dismal performance by suggesting that the questions were not ‘avant garde’ enough. ‘They are the questions of Grandpapa,’ said one of their officials.” The only dramatic note “was struck two days before the competition ended. Professor Hanus Weinhart, from Potsdam, leader of the East German team, put on a rucksack and said he was going climbing in the mountains. He has not been seen since. The Jugoslav police say they believe he may have fled to Italy.”
The clean-up has also turned up a heavy blue plastic wallet on a shelf in the back room of the Excavation, embossed to look like leather. Although covered in oily dust, the wallet smells fresh, like a child’s seaside shovel; the shelf varnish cracked as I prised it off.
Inside, there’s a nest of other folders made out of expensive textured paper. The logo on the largest shows a cartoon figure sitting on the letter α with, in place of a head, a ball surrounded by three electron paths. “Moscow VII•1968” reads the label: July 1968, a month before the Soviet invasion of Czechoslovakia. The midwinter of the Cold War.
Along the opening edge is a broken security seal.
Inside this, a folded sheet of paper with red writing. It has been made out to “Simon Noμton” (Simon Noshton) and is signed “ΠρoκoφbεB”: P-r-o-k-o-v-ie-v.
And if you have to ask what that means, you’re too junior to know. Report to Prokoviev for execution.
The folder contains Simon’s Olympiad answer sheets. He was allowed to keep them as a memento of the first victorious year, in which he scored 100 percent. What makes his work beautiful to read is again not its complexity but its simplicity: without drafts or false starts, he lays down his pellucid solutions to questions involving imaginary numbers, infinity and the distribution of primes, with the grace of a ballerina unfolding her hands.
I’ve cleaned up all these articles. In life, they are orange-brown from sunlight, because his mother kept them displayed in a photograph album on top of her piano in London. They are toasted with boasting.
Simon is starting to change me too, says my girlfriend, tartly.
I’ve taken to working in bed, in squalor; at fancy-dress events I appear as a number with arms and legs.
My doctor—I can hardly bear to admit this—insists that the pain in my right foot is g— I can’t say the word. I have lost interest in my hair:
As Simon gets tidier to avoid me, I’m becoming him. (“He,” corrects Simon. “Accusative case.”)
Answer to Fundamental Biographical Question Number 74, subsection b, namely, Why write a book about Simon? Because he is to biography what the Monster is to the mathematics of Group Theory: an intractable problem who nevertheless represents an atomic type of being, a building block for convoluted characters. I think Nature makes up our personalities by mixing together a finite number of fundamental different types—a pinch of adventurer, two splashes of monomaniac, a dash and a half of Pompous Dad: that would be a starting point for a banker. Simon is another of these elemental types: the obsessive logician, untainted by self-regard, sentimentality or any feel for romance. He’s done everything in his power to exclude the fact he’s an emotional human, except on the subject of buses and trains. Add a pinch of Simon to the banker mix above and you might have a health-insurance broker from Swindon, if you use too little; a Dr. Strangelove madman, insanely compounding new breeds of atomic weapon, if too much; the most charming uncle, a champion bridge player in the mold of Omar Sharif, if just the right amount.
But how do you write a biography of a man who is pure Simon? It’s like trying to write the biography of a hedge.
His face gets between me and my sleep.
Some days Simon loosens up.
“I once got a postcard from Julie Christie,” he exclaims, popping his head round my study door.
“Really? Why? What did it say? When did you get it?”
“Huunh, uugh…I don’t know. Excuse me, I need to fill out this voluntary questionnaire for landlords, from the City Council Housing Department.”
Or, meeting me in the hallway as he’s staring at the letterbox, waiting for the post:
“When I was on the bus from Cambridge to London, I was sick, but fortunately there was a sick bag on the bus, and I got rid of it at Trumpington, and the bus driver said, are you all right, and I said yes, and I was sick again. But that was when I got off.”
“Thank you, Simon.”
“Ah, at last, here’s the delivery: 12½ minutes late again!”
Occasionally he lets me in on his latest mathematical discovery:
“As I say, if you have three types of socks in your drawer, and the chance of taking out two of the same type is exactly half, what can you say about the total number of socks in there? I’ve found if you categorize the solutions you get interesting mathematical structures called quadratic forms, which are used to understand the Monster.”
This is very encouraging! This biography is going to be 700 pages long! Math, the universe, genius, the Monster, socks as the source of symmetry in Quantum Field Theory! All this and Julie Christie too! The manuscript will have to be shipped to Waterstone’s bookshop under armed escort and printed on cigarette paper.
Then abruptly he vanishes, mid-sentence: focus disappears, eyes glaze over. Animation, sucked back in like a reverse- action movie. Whoooo-up! No Simon left; just skin and absence.
A flutter of extra grin appears over his mesmerized face, and slowly fades off.
“What are you thinking?” I demand.
“Nothing.”
“Just then, when you smiled like that—what were you thinking?”
“I can’t remember.”
“You must be able to remember! It was two seconds ago. You had a thought about something. Some story came to mind.”
“No. I don’t think it was anything.”
“Then why did you suddenly look like that?”
“I don’t know what you mean. It was just my face.”
“Exactly! And your face showed you were thinking something interesting. Something I might be able to use in the book. What was it?”
“I don’t know what…”
“Think! What was it?” I press desperately.
“Hnnh, aah, oh dear, please!” Simon gasps, dropping his duffel, thrusting up his head and gulping air like one of his mackerel. “Please! I’m not feeling very well. Let me explain in my own time. Oh dear!”
The book will be fifty pages long, enjoy a limited circulation among friends, and be printed at a photocopy shop in the Business Park.
I’ll die a drunk in Deal.
To take my mind off unhappy thoughts, I return to my favorite subject. “Biography, you see,” I announce, sitting cautiously on the edge of his food-stained mattress, while he drags dustbin bags of maps and timetables up and down the Excavation, “biography is about how an author chooses to present what he takes to be the facts: how he opens the story; what mood he picks for his closing line; where he places his adjectives, his quotation marks, his s
ilences. Every biography is therefore a portrait of a relationship between author and subject, not just about the subject. You agree that mathematics is not about numbers per se but about the relationships between numbers?”
Simon squints at this fact, well accepted by every mathematician in the world, with a puzzled look. He never likes to agree to anything when it comes as a bald statement.
He never likes to agree to anything at all when it comes from me.
Biographies of famous people are different. Famous people are in the ring punching and jabbing every day already; you’re just another little scrap on the way. It takes only a month or two for a book about a famous person to feel as if it sagged onto the shelves a decade ago.
But the unknown, I think to myself, feeling suddenly that my argument has become too precise and clever, you lock them in your image, and out of their own, for life.
You’ve got to be careful with a responsibility like that.
“Grief!” I shout. “You say you suffer grief? What are you talking about? Excuse me, Mr. Four-Story-House-Without-a-Mortgage, Mr. No-Need-for-a-Job, Mr. I-Spend-All-Day-Going-on-Holiday-Jaunts-and-Pretend-It-Counts-as-Campaigning-for-Buses-and-Trains: what exactly is it that you have to grieve about?”
It’s a cliché that mathematicians are over the hill by their mid-thirties, but often it’s not loss of mathematical intelligence that weakens their ability but loss of focus. They meet someone luscious in the university canteen, get married, shackle themselves with debt, find they’re suddenly pushing a stroller containing twins; or they decide (surprisingly common among aging genius mathematicians) they haven’t done enough for the health of the world and stumble off to fight American Imperialism. Love, financial worry, a surge of community spirit: the mathematical concentration is shot, and before these ex-geniuses can get it back, it’s time to retire and die themselves.