by Tom Pollock
“Sure something did,” I say.
Ingrid looks at me quizzically. “What?”
“Me.”
Recursion: 2 Years,
9 months Ago
The first thing I noticed when I woke up was something hard and sharp lodged in the skin on the back of my hand. The second was my incredible thirst. I opened my eyes and the world blurred for a moment before resolving into a beige wall with seven red-jacketed clockwork soldiers painted on it. My tongue was Velcroed to the roof of my mouth and I croaked for water. My brain felt like a rock in my head.
I tasted fear on the back of my tongue. I didn’t recognise the soldiers. I didn’t recognise the wall. I didn’t recognise the lumpy mechanised bed I found myself in, or the mouthwash-green smock draped over me. My left leg felt huge. I sat up, tried to move it, and screamed.
It felt as if the bone was pulling itself apart from the tendons. I collapsed back on the bed, gasping and sobbing. Hospital. I was in hospital. What had happened to me? The door to the little room opened and a familiar figure hurtled in, looking back over her shoulder to snap, “Of course I’ve sterilised, you cretin. My dandruff has a better working knowledge of microbiology than you do.”
Way to charm the people responsible for my pain relief, Mum.
“Peter.” She hovered over me. “How are you feeling?”
“My leg . . .”
“You broke it. You fell off a roof.”
“My head . . .”
“You broke that too, and you should be grateful.”
“Grate . . . ?” I watched Mum break about sixty medical practice guidelines by plugging a syringe into her own son’s IV. Instantly my thoughts slowed. My head started to feel like a snow globe, a storm of random ideas drifting through a liquid suspension. I looked down at my hand; it was swollen and scraped raw, the other end of the plastic tube feeding out of it. Ah. Morphine. Good. As the proud owner of the world’s most habit-forming personality, it would be just like me to come to hospital with a broken femur and leave with an opiate addiction.
“You cracked your skull,” Mum said. “But not on the pavement, thank god. There were bits of bark in your scalp. Best guess, you clobbered your head on a tree branch on the way down and it spun you around. If it hadn’t, you’d have taken the full force of the impact on your neck. You’re going to have one hell of a scar.”
Lightly, she brushed the bandage wrapped around my forehead. I could feel blood soaking through; head wounds bleed like a bitch.
“But you’re incredibly lucky, Peter.”
“Yeah.” My stomach curdled with disappointment as I remembered the wet car park rushing up towards me. “Lucky me.”
“What were you even doing on that roof?” Mum worried at her thumb cuticle with her teeth. I’d never seen her do that before.
“Exploring,” I lied automatically. “I didn’t know where the door led to. The slates were slick and I slipped.”
“Really?”
“Yeah. Come on, Mum. You know what I’m like with heights. You don’t think I’d have gone up there on purpose, do you?”
Her face became a little less grey.
“Well, then I suppose we just have to be grateful it wasn’t—”
“Mum?” Bel’s voice interrupted her and I started. My sister must have been sitting quietly in the corner of the room this whole time. Watching me wake, hearing me scream, not saying a word.
“Can I talk to Pete alone?”
Mum hesitated, shrugged, smiled, and withdrew. Only when the door clunked shut did Bel approach the bed. She perched by my feet, her face hidden by a curtain of red curls, and she didn’t meet my eye as she spoke.
“How did you fall off the roof, Pete?”
I felt myself go still.
“You were here just now. You heard . . .”
“I heard what you’re going to tell everyone else.” She stared into her cupped hands. “What are you going to tell me?”
I swallowed and made a decision.
“I jumped.”
I was expecting Bel to swear, to yell, to hug me or hit me, but she just nodded and asked, “Why?”
“I panicked.”
“Why?
“I found out there are unsolvable problems in mathematics.”
It hung there in the air between us, for six full seconds, and then Bel started, helplessly, to laugh.
“Bel!”
“I’m sorry, Pete, but that is just . . . so fucking you. The only kid in the world who nearly died from maths.”
I had no right to be surprised, but somehow I was disappointed anyway.
“You don’t understand.”
“No shit.”
“Okay, look. There are seventeen—”
“Oh Christ, Pete, no. Please, not more numbers.”
“There are seventeen,” I insisted, and she must have seen the tears in my eyes because she shut up. “Seventeen elementary particles. They make up everything in the universe from black holes to brain cells. When you get right down it, everything’s made of the same building blocks. What makes the difference is how many, the pattern of how they’re arranged; what makes the difference, always, is numbers.”
I twisted the bedsheet, deliberately hurting my grazed hands as I spoke.
“The difference between 495 and 620 micrometres on the length of a light wave is the difference between red and blue. The difference between 54 and 56 kilos of uranium is the difference between a very toxic paperweight and a nuclear explosion. You think I would have been the first kid to die of maths, Bel? Everyone who’s ever died, died of maths.”
She stared at me. I’d never felt so far away from her. It was as though we no longer shared the same language, but I had no choice but to keep jabbering on in my own, trying to make her understand.
“‘Why are you so scared, Pete?’ All my life people have been asking me that. So I went looking for an answer. I went looking for the number that’s the difference between a normal, healthy, brave brain and mine.”
Bel nodded; that at least made sense to her.
“I believed there was no question maths couldn’t answer, if only I understood it well enough.” I breathed out hard. “But I was wrong. Maths is incomplete. There are questions, questions about numbers, even, that it can’t answer, equations that it can’t decide are true or false. So I’m fucked. Gödel proved it in the thirties. It just took me this long to find out.”
Bel was slowly working the knuckle of her left hand into the palm of her right. Now she looked up, and asked quietly, “How?”
“What?”
“How did he prove it?”
“You really want to know?” I was taken aback.
“Do I want to understand how a German geek who’s been dead fifty years managed to get my little brother who’s petrified of heights to jump off a fifty-foot building?” Her tone was kindling-dry. “You could say I was curious.”
“Eight minutes older,” I grumbled. “And he was Austrian.”
“Whatever. Spill it.” Her gaze was intent. “Make me understand.”
“Okay,” I said, and took a deep, painful breath. “Okay, I’ll try.”
“The first thing you have to get,” I said, “is that to know that anything in maths is true, you have to prove it, and not with the kind of proof you can get looking through a microscope. You need certainty, not just evidence.”
“But how can you prove anything without evidence?”
“With logic,” I told her. “There are foundational beliefs that we just accept without proof because it . . . hurts to doubt them. Stuff like one equals one.” I smile at her, and it feels good. “In maths we call them axioms.”
She smiled back. She liked that.
“To prove a theorem, you have to find the chain of bulletproof logic that shows that it follows from th
ose axioms,” I went on. “That chain of logic? That’s your proof.
“For two thousand years of blissful ignorance, we thought that every equation had one. The true theorems had proofs vindicating them, and the false ones had proofs demonstrating their falsehood. We believed in completeness. That any question you could ask in maths could be answered with it too.
“But there was a flaw. It was absolute, and being absolute it took only one counter example—one equation maths couldn’t decide—to shatter that belief.
“And Gödel”—yesterday I’d only vaguely heard of him; today his name was like bleach in my mouth—“he had a candidate in mind.”
I groped around until I found the pen by my medical chart. With it, on the pale blue sheet of my hospital bed I wrote:
This statement is a lie.
And then crossed out the last two words and added:
This statement is a lie unprovable.
“That,” I spat, slumping back against the pillows. “You can’t prove it, because proving it true proves it false. So it must be true, even though you can never prove it. It’s undecidable, forever poised, like a coin that only ever lands on the edge. If Gödel could find an equation that said that, then maths itself would be fundamentally limited.”
“And did he?” Bel’s gaze was intent.
I nodded.
“He did it in three steps.
“Step one: Encrypt. He created a code that changed equations—theorems and the proofs that proved them—into numbers, and so turned proof—the relationship between theorems and proofs, into an arithmetic relationship between numbers.
“Step two: Invert. He defined the opposite of that relationship. The relation of ‘not being proven by.’ He hypothesised an equation that, when encoded, had that relationship with all numbers: an equation for which a proof was impossible.
“Step three: Recursion, Recoil. He defined the unprovable equation as the equation that stated that the unprovable equation had no proof. It calls on itself. Eats its own tail. And it’s so simple, so elegant.”
On the sheet, underneath This statement is a lie unprovable, I wrote:
F ⊢ GF ↔ ¬ProvF(⌈GF⌉)
“That’s it,” I said. “The unprovable equation. And that is a total fucking disaster, because if it’s possible for one equation to be unprovable, then any equation could be. Any sum you try to solve could stall you for the rest of your life. Maths can’t justify itself. It doesn’t always work, and there’s nothing outside of it that can tell you when it will.”
I tailed off, gulping at air. Speaking this aloud made me nauseated.
Three steps, I thought: Encrypt, Invert, Recoil, and just like that you’ve destroyed the world.
Bel hadn’t said a word the whole time, but finally she seemed to feel the need to clarify.
“And that’s why you jumped off a building?”
“Yes.”
There was another long silence in which all I could hear was my own laboured breath.
“You dickhead!”
I hadn’t even realised there were flowers in the room until a vase came sailing over my head and shattered against the wall. Soil and pottery fragments showered down the back of my neck.
“You massive, weeping, septic penis!”
I began to laugh; I couldn’t help it. The jerking sent little spikes of pain down my collarbones. But then she was standing right beside me and my laughter dried up. She wasn’t joking around; her eyes were red and streaming.
“You tried to leave,” she said. “You tried to abandon us.”
At that word abandon, it felt like someone had jumped on my chest.
“I—I—I—” I stammered and pawed at the ink on the sheet, smearing it, desperate to say something, anything to erase that look of betrayal on her face. “I was just so tired, tired of being scared, tired of running, I . . .”
Her demeanour changed abruptly. She went very still and her eyes locked on mine.
“Running from who, Pete?”
I swallowed.
“No, it’s not like that.”
“Running from who?”
“Bel, I—”
“Were you alone up on that roof?”
“No, but . . .”
“Who was it? Who was up there with you?”
I stared at her, feeling the warmth of connection, of being understood, fading like a dying fire.
“Who?” she insisted. And I got it; she wanted a culprit, someone to blame, someone to hate for me. Flesh and blood, that’s what she understood, not symbols on a hospital sheet. She sat on the edge of the bed, took my hand, careful to avoid the needle lodged in my skin, and asked, “Who did this to you, Pete?”
And in that instant, it was so easy to give one to her, even though I knew who was irrelevant; if it hadn’t been Ben Rigby, it would have been someone else: Gödel had proved that. But it was so easy to interlace fingers with my sister against the old enemy, to take all my frustration and fear and loneliness and crush it into a bullet and fire it at him.
“Ben Rigby,” I said, and then added quietly, “I wish he was dead.”
NOW
Winchester Rise is deserted. From the corner of the street I can see my house, the shiny paint of the front door peeking above the holly bush. We crouch with our backs pressed against a low garden wall and watch through billows of our own breath, steaming white under the moon.
Twenty-four windows, I think, twenty-four windows overlook the pavement between here and my front door. I shrink back from the watchers who I’ve imagined behind every pane of glass. A breeze picks up, and for an instant, I think I hear radio static in the rattle of brittle branches. Then the wind dies and the street is silent again, still as a trap before it’s sprung.
“That one.” Ingrid points at a beaten-up hatchback parked directly across the road from my house, its dirty white paint job turned a sour yellow by the streetlights.
“How can you tell?” I whisper. “Special aerial for the radio? Artificially lowered suspension belying its piece-of-crap appearance?”
“No.”
“Then how?”
Ingrid looks at me. “Peter, how long have you lived on this street?”
“Fourteen years, since we were three.”
“And how many times have you walked up and down this street?”
“Thousands.”
“And have you ever, ever, in all your trips down the length and breadth of this pavement, seen that car before?”
There’s a long silence.
“Oh,” I say, crestfallen. “Being a spy’s just common sense, isn’t it?”
The glare intensifies.
“No, it’s very specific, well-trained sense.”
I scan the other cars, trying to remember which I’ve seen before.
“Just that one?” I ask hopefully.
“Just that one,” Ingrid confirms.
“So . . . it worked.”
It was a shot-to-nothing from yesterday afternoon. Using a phone and a credit card we pinched from a bagpiper’s rucksack on the Royal Mile, I’d booked two plane tickets leaving Edinburgh for Marrakesh that same night, in the names of Maggie Case and Benjamin Rigby. “You’re right,” I’d told her. “57 must know I’m using Ben’s name by now. And after Bel’s”—I’d faltered, remembering the blood droplets clinging to my sister’s hair like leftover dye—“efforts back at school, we know they’re stretched. With a bit of luck, they’ll take the bait and pull some folks off watching my house to come and shove black bags over our faces at the departure gate.”
As an afterthought, I’d added a third alias, Beth Bradley, to the booking.
“Why the third name?” Ingrid had asked, peering over my shoulder.
“For Bel,” I’d replied. I’m betting on you, Sis. “If they haven’t captured her, they mig
ht think she’s with us. Then they’ll have to send their best.”
“Shit.” Ingrid’s mutter drags me back to the present. She’s staring intently at the car.
“What?”
“Only one seat in that car is occupied.”
“Isn’t that a good thing?”
“It is hugely not. We use teams of two for watching, no exceptions. There must be someone else. And if he’s not in the car, then he’s inside the house.”
“Can we take them separately?”
Ingrid’s expression turns incredulous.
“I’m sorry,” she hisses. “I’ve been a bit short on sleep recently, so I must have drifted off for a couple of years there while you trained as a ninja.”
There is a hurt silence.
“No need to be snide about it.”
“They’ll have an open radio link. Whoever we try to hit second will be screaming for reinforcements the minute we so much as tickle his partner.” She exhales noisily in frustration and closes her eyes, tilting her head this way and that as she considers scenarios. Judging by her pallor, none of them are good.
“We need to leave,” she says at last. “This is seventeen kinds of suicidal. We could go anywhere: Tokyo, Mumbai, Mombasa. We’re still twenty-four hours ahead of them—one full day’s lead that I could make last a lifetime—but not if we do this.” She opens her eyes, and in the darkness they’re very pale. “Please, Pete. Don’t. If we do this, we throw everything away.”
I leave the plea hanging in the air and turn back to the car.
“If I deal with him,” I ask, “can you handle the one in the house at the same time?”
“Pete?”
“Can you? Just answer.”
She shrugs helplessly.
“Depends on who they’ve got in there. I had two months at knuckle school, same as any other field agent, but my grades”—she rubs her throat as if she’s soothing a remembered injury—“were middling.”
I consider my options.
I could walk away now: Tokyo, Mumbai, Mombasa, car horns, exhaust fumes, the anonymity of teeming people in a new city; new name, new life, new language, get a job, start a family, always with the past I’ve tried to forget splintering my sleep, jumping at the sound of every footstep on the stair. And worst of all, never knowing.