Katabasis — Chapter no 23

Chapter Twenty-Three

Why didn’t you ever tell me?” asked Alice.

And Peter gave the only correct answer, which was the true answer, which was just the question flipped. “Why didn’t you ever tell me?”

All this time, thought Alice. All this time they’d both been drowning, and thinking the other was gloating at them from the shore. She remembered that set theory paper; remembered how proudly Professor Grimes had spoken about it. I’m going to revolutionize how they think about categories, he’d told her. This is the closest we’ve ever come to solving Russell’s Paradox. They’re going to be shitting their pants. Never once had he mentioned Peter.

She pushed her palm against her cheek. “I can’t believe it.”

Peter stiffened. “Why not?”

“I’m not calling you a liar,” she said. “I believe you, I’m just—I mean, I can’t imagine why he would need to do that. He’s Grimes, for heaven’s sake —he has a million projects going on all the time, he shouldn’t have to steal

—”

“You’re doing it again,” said Peter.

“Doing what?”

“Valorizing him. Defending him. You always make him out to be this—

this great, larger-than-life genius—”

“Well, because he was—”

“He’s just an asshole, Law.”

“No he isn’t.” Her voice hitched. “Don’t you understand? He can’t have been. Otherwise we let some—some person jerk us around.”

“And it makes it better if he was a genius?”

“Not better. But—worth it.” Alice splayed her hands. “Don’t you know what I mean? If there was some method to the madness, then at least— right?”

Peter stared at her a long while, then sighed. “I do, yeah.”

“And because—well, he just didn’t seem all that bad, do you know what I mean? He wasn’t like the rest. A bad actor.”

“I know,” said Peter.

This was how they’d consoled themselves for years. At least Professor Grimes wasn’t like those other professors, the toxic ones, the ones who screamed abuses at their lab assistants and called them stupid to their faces, spittle flying out of their mouths. Wasn’t like the anthropology professors who took their students out on field trips in South America and went mad, hurling cups and plates and putting their students’ lives in danger. He wasn’t a bully, wasn’t a tyrant. After all, what all those abuses usually boiled down to was insecurity and incompetence, and Professor Grimes was neither. All he ever did was utter a stern word. He only ever held them to the same exacting standards to which he held himself. Being upset with Grimes was synonymous with being bad at your job. And even now, in the pits of Hell, Alice could not shake the conviction that if she had gotten into a tizzy about it, then it was all her own fault.

“And I just thought, if it wasn’t working for me, then that meant there was something wrong with me,” she said. “After all, you were doing just fine.”

“Funny,” said Peter. “That’s how I felt about you.”

They blinked at one another.

“It’s embarrassing in retrospect,” said Peter. “He was so good at pitting

us against each other.”

“It felt like that for you, too?”

She had always assumed theirs was a one-sided rivalry. She was the mess, and Peter Murdoch was the unreachable yardstick; the standard against which she was always measured. Peter could have done this in his sleep. He gave Peter the Cooke, after all; meanwhile poor Alice was so hopelessly behind that she needed an illegal, permanent pentagram to keep pace.

“Oh, yes. Every time.” Peter affected a growl. “‘You haven’t got Alice’s creativity, Alice’s drive. Alice shows up first and leaves last and she’s the only one between you who will get ahead, she’s got what it takes. Alice is a true scholar. Alice will leave her mark on history, and you will only ever be a dilettante.’”

“He didn’t,” said Alice. Hearing this made her feel better in the stupidest way. “There’s no way he said that.”

“Are you blushing?”

She pressed her hands against her cheeks. “I am not.”

“You’ll take the compliment. Even now.” He shoved at her shoulder.

“Jesus, he really did a number on us.”

“But it wasn’t all bad,” she said. “He made us good magicians.” He made me perfect. Her tattoo twitched, and even then, she could not bring herself to consider it more a curse than a blessing.

“I don’t know, Law.” Peter pulled his legs to his chest and rested his chin atop his knees. “I’ve been wondering this myself. Whether we really needed Grimes to become who we did. Because, honestly, I think anyone could have made us good magicians. He just convinced us we had to suffer for it. Just had me thinking, even when I was on the bathroom floor, that I wasn’t tough enough. That if I just wanted it enough, I’d be all better.” He snorted. “Stupid.”

“So how . . .” Alice glanced at Peter’s midriff, then back up at his face.

“How are you now?”

“Surgery helped,” said Peter. “I’m in remission.”

“Until?”

He shrugged. “Until I’m not.”

“I’m sorry.”

“It’s all right,” he said. “So it goes. Doesn’t matter now.”

THE DECLINE WAS VERY QUICK AFTER that.

Really it was merciful, how rapidly the Escher trap drained their energy.

The air got hotter. Their mouths were like sandpaper, their tongues flat, rough stones. They sat side by side, but as time trickled on, their heads and shoulders drooped, like they were toys running out of battery, until they were slumped, Alice lying on Peter’s lap, Peter lying atop her.

Alice was fairly unbothered by it all. Even if she was upset, her mind and body were too numb to register much; it took energy to grieve and she had none anymore. Her ears buzzed in a way that was almost pleasant. She closed her eyes, and thought only of cool rivers, velvety darkness, blanketing waters. Was this so terrible? All she had to do was go to sleep.

She heard the scratch of pencil against paper. She opened her eyes.

Peter was scribbling on her notebook.

She lifted her head. “What are you doing?”

“Trying to figure a way out.”

“We’ve tried everything,” she mumbled. “There’s nothing else.”

“Oh, there’ll be something.”

“How do you know?”

“Because of Gödel’s Incompleteness Theorem.”

“Of what?”

“It’s a theorem in mathematics.” Peter sounded bizarrely chipper. “I learned about it when I was a child. Basically, it says no theory of mathematics can ever be complete, because for any reasonable mathematical system there will always be truths that the system cannot prove. Math has its limits. There’s always something we don’t know. Some people think Gödel’s theorem proves the existence of God.”

“But it doesn’t prove anything at all.”

“It does, though. It proves there’s always another option. It proves no system is ever closed.”

“Jesus, Peter.” Alice blinked fast, but the dots would not clear from her vision. “That doesn’t prove anything. Sometimes maths is just maths.”

Peter’s pen spun frenetically in his hand. “Well, consider this: In book four of The Inferno, Dante asks Virgil if any soul that was not saved by Christ had ever been rescued from eternal Limbo. And indeed, Adam and Abel and Noah, among others, had been blessed and raised to Heaven. God

broke his rules, just for them.”

“No one takes book four seriously.”

“I mean, I too have some bones to pick with Dante. But the point is, even Dante’s vision of Hell includes exceptions. The Underworld yields and bends. It is unpredictable—it follows no order but its own. It is just as Borges said—the certainty that everything has already been written annuls us, renders us phantasmal—and yet we are not phantasmal, not annulled, because nothing is fully written! There is no coherent set of axioms that explains it in full. Just like maths. Ergo, there will be some way out. And I will find it. I must.”

“That isn’t how logic works,” said Alice. “I am sure this proof is missing quite a lot of steps.”

Peter shrugged but said nothing. He only continued writing.

It exhausted her to watch him. How silly this was, she thought. How silly this all was—not just his scribbling, but the entirety of their efforts.

Their situation in the Escher trap was just a microcosm of their entire lives at Cambridge: endless scribbling in an attempt to prove they could be the golden exception when in truth there was nothing exceptional about them at

all; they were only following the scripts laid out for them from the beginning. And clearly the only thing to do was to get off the wheel, to quit, and refuse playing the game. Really the only victory here was death. How

could she convince him?

She tugged on his sleeve. “Peter.”

He paused. “Yes?”

“We don’t have to be afraid,” she whispered. “It’s just as Elspeth said.

We’re living souls in Hell. We won’t go anywhere. We won’t become anything. It’ll just be over—everything, all of it, done.”

“But I don’t want it to be done.”

“Oh, hush.” She patted his knee. “It will be fine, I promise.”

“Don’t say that.”

She only patted him more firmly, as if he were a crying babe, as if he only needed to stop making such a fuss. “It’ll all be silent. It will be all right.”

She was so tired. Her eyesight had gone blurry. She couldn’t see Peter’s face then. She saw his lips moving but heard nothing, and then all she saw was the outline of a shape, growing smaller and smaller in her view, until all receded to black.

SNAP.

Her eyes blinked open. Peter’s fingers hovered just under her nose. He snapped again, and she startled awake, ears ringing.

“Wake up,” he said briskly. “I’ve figured it out.”

“Hm?” Alice raised her head. Her whole body felt fuzzy. She didn’t remember when she’d drifted off to sleep, nor could she tell how long she’d been out. The low red sky burned above, just the same as always.

Peter sat hunched over an array of papers, still scribbling. He’d filled out eight sheets at least since she’d drifted off—all covered in crossed-out,

circled, and shaded-in algorithms.

“Our way out,” he said. “I’ve got it.”

Reluctantly she sat up. “What?”

“It should have been obvious.” He spoke at a quick, robotic clip, a voice Alice knew quite well. It was the voice he always took on in lab when he’d figured something out; when he needed to get it all out at once but couldn’t get his mouth to keep up with his thoughts. “Something I know from

introductory logic. A silly game, really. It’s got to do with types of knowledge and probabilities. But perhaps I shouldn’t tell you, it’ll weaken

the paradox—”

“Murdoch.”

“Okay. Listen.” He drew a neat circle between them, just large enough for two to stand side by side. “Would you say that when the Kripkes come, it will be a surprise?”

Alice couldn’t determine the significance of this statement, but neither could she find a problem with it. “Ah, sure.”

“And would you say that they must come sometime between now and when we starve to death five days from now, otherwise our blood will go

bad?”

“I suppose that’s what I would do.”

“Excellent.” He pressed his palms together. “Then the conditions are set.”

Her head hurt. “I still don’t understand.”

“Just listen,” said Peter. “Now, suppose a prisoner is awaiting his moment at the gallows. The hangman tells him he’ll be hanged sometime in the week, but otherwise he is not to know the date of his execution. Which day of the week can we rule out, then?”

Alice pondered this, and then ventured, “Sunday?”

“Good. Why?”

“Because it’s the last day of the week. So if he hasn’t been hung on any of the previous days, then he’ll know it’s happening Sunday. Only then it won’t be a surprise.”

“Very good,” said Peter. “So Sunday’s out. What happens, then, if he still hasn’t been hung by Saturday?”

“I suppose Saturday’s out too.” Alice’s thoughts churned sluggishly.

“Because Sunday’s out, which makes Saturday the last day it could happen, but if it hasn’t happened by Friday, then Saturday isn’t a surprise either . . .

Oh.” Something clicked in her mind. “But then Friday is the last day it could happen . . .”

“You see,” Peter said gleefully. “It’s recursive.”

Alice saw now. “So then the prisoner can never be hung, because none of the days will be a surprise. And they have to let him go.”

“Precisely!” Peter beamed. “And are not the conditions perfect for this paradox? We know we will expire in five days. We know the Kripkes

always come when least expected. But if we map you onto the Hangman’s Paradox, then you are invincible—they can never get you at all. So my hypothesis is if we write this into an algorithm, it releases you from the trap —that, or constructs some shroud of invulnerability, so that even if they come they can’t hurt you—”

“Fine,” Alice sighed. Stupid, ineradicable hope crept back into her chest; stupid, exhausting feeling. Things had been so much better when she’d only been numb. “Give that here.”

Peter slid her notebook across the sand.

She traced her finger down the lines, willing herself to focus. Slowly she made sense of his scrawls. “You’ve messed it up.”

“How do you mean?”

“You’ve written it for only one.” She tossed the pages back into his lap.

“It’s no good. You have to do the whole thing over for two.”

“Oh,” said Peter. “No, I’d noticed that. That was intentional.”

It took her a moment to register what he meant by this. “Murdoch . . .”

“It’s not a very strong paradox,” said Peter. “It’s not like Sorites. There’s a couple of obvious solutions, and I know them far too well. I can’t suspend my disbelief long enough for it to work.”

“All paradoxes have solutions.” Alice fought a rising swell of panic.

“That’s why they’re temporary.”

“Very temporary,” said Peter. “I’m afraid this one is particularly flimsy.

And against something like an Escher trap, there’s really no room for doubt.”

“Then how do you know it’ll work on me?”

He cast her a soft smile. “Because you’re not a very good logician.”

“Fuck you, Murdoch.”

“Get in.” He gestured. “Let’s send you out.”

“I am not leaving you.”

“And I’m not letting you die,” said Peter. “We can’t have come all this way for nothing. You said it yourself. This all has to be worth something.”

“Then we go in together.”

“Not sure that’ll work.”

“Well, we’ve got to try.” She smacked the notebook. “Rewrite it. Set it for two.”

“It won’t work for me,” Peter insisted. “And when it doesn’t work, you’ll be a corrupted subject as well, and then you won’t get out either—

you’ll die—”

“I’d rather die.” She meant it. She’d never meant anything so much in her life. A week ago she hadn’t been able to say she’d save Peter’s life with certainty—but now, she knew. She didn’t want to live. Didn’t want the future, this stupid goal they’d been chasing. “We both live, or we both die, there’s no third option.”

“The third option is you live. You deserve to live—”

“I don’t deserve anything.” Alice meant this, too. What had she done since they’d come to Hell? Lied, betrayed Peter, betrayed Elspeth, landed them all in this sorry mess. It was about time she closed the book on this pathetic story. She was so very tired—she only wanted now to expire quietly in the dark, but Peter wouldn’t even let her do that. “I don’t, I really don’t—oh, gods, Peter, just let me die.”

“Can’t. Won’t.”

“Why are you being so noble?” She would have beat him with her fists, if she could muster the strength. “Stop being so noble.”

“You’re the only one with an algorithm to bring him back. Mine doesn’t work. I get out, and the only person we send back up top is Grimes. You get out, and at least you get home alive.”

Violently she shook her head. “I don’t know if the Erichtho spell even works—”

“Well, there’s a chance it does. You’re the only one with a positive outcome here, Law. It’s just how the numbers fall.”

“I don’t care about the numbers!”

“Anyhow, you’re the only one who didn’t try to kill him—”

“You didn’t kill him. I did. I remember I didn’t close the loop. I remember, I don’t forget—”

“I double-checked the submission after. You didn’t close the loop because there was no closure. I never put it in.”

“But still I should have known.” Alice did not make mistakes; she couldn’t. She saw everything; every detail was seared in her brain. And the only way she could overlook a thing like the Ant Test was if some part of her intended it. “I’ve been over this a thousand times. I saw the gap, I knew —”

“Stop it.” Peter flung up his hands. “Just stop. We are not fighting over who gets credit for his murder. Who cares about the details—”

“The details matter,” Alice insisted. “They matter because you think you deserve to die when you don’t, when it’s not your fault at all, and you did nothing wrong, and you shouldn’t even be down here—”

“The fact remains.” Peter raised his voice, spoke right over her. “I can get this paradox to work on you. I am absolutely certain I can make it work.

I have almost no certainty about the two of us. So that’s just basic decision

theory, Law. Maximum expected outcome.”

“Shut up.”

“It’s just maths. I’m sorry if you don’t like it.”

“But you can’t die here.” She swallowed. “Not when—not when I’ve just—”

There was something wild, desperate in Peter’s eyes. “When you’ve just what?”

What did she want to say? Alice didn’t know. She didn’t have the words for this pit of feeling, dark and gnawing and delirious. She wanted to hurl herself into his unknown; wanted an intimacy she couldn’t describe. She wanted him alive; near; beside her. The words that came to mind were clumsy and insufficient, but they were all she had. “When we’ve just

learned not to hate each other.”

Something closed in Peter’s face.

They stared at each other, a chasm yawning between them.

Oh, why was this so hard? Alice wondered desperately. Why couldn’t she ever tell Peter what she thought? Always they had been bodies hurtling just out of one another’s orbit, when all it would have ever taken was an honest word. But that was precisely what magicians lacked; there were no honest words, only puns and illusions and constructions of reality so convoluted that you couldn’t keep track anymore of what was real and what wasn’t. Everyone was always trying so hard to pretend they were somebody else.

If only they had caught one another, looked at each other, forced their ways across the gap.

But it was too late now, too late for everything.

Peter drew out a blade.

“What are you doing?”

“I’m deciding for both of us.”

“You can’t.”

“It’s the only rational solution,” he said. “Please, Alice. Don’t be an idiot.”

She lunged for the blade. He raised it up out of her reach. She tried to shove him over—she tried with all the strength she had, but Peter could not be budged from the pentagram. She smacked his arms, scratched and pulled. But he was taller, heavier, and stronger; all he had to do was wave her aside.

“I hate you,” she cried. “You’re so—you’re such a—”

“Logician.” He gave her a sad smile. “I know.”

He drew the blade across his arm. Blood flowed thick and fast across his skin, dripped down his fingers and onto the chalk, suffused the pentagram like spreading ink until the whole thing glowed crimson. Peter began to chant. Alice wailed. She hit him, she flailed, she railed against his grip—but he was so strong, and nothing she did could break his rhythm. On and on he went, sonorous as ever, confident until the last.

“The rucksack, Alice.” He patted her shoulder. “Don’t forget the rucksack.”

Then, with a flick of chalk, he closed the circle.

Alice screamed, but he didn’t hear. In an instant the sand rose up around her, threw her out and blocked Peter in. The Escher trap vanished before her, cuckoo bird, boulders, and all. Then all she saw was silt, an endless flat plane, under a constant, dying sun.

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“Lines that are parallel

meet at Infinity!”

Euclid repeatedly,

heatedly,

urged

Until he died.

and so reached that vicinity:

in it he

found that the damned things

diverged. —PIET HEIN, “PARALLELISM”

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On Paradoxes

The reason why paradoxes trouble us is not because their conclusions are true. The donkey does not starve. The world does not consist of unending staircases. Of course Achilles could outrun a turtle, of course the arrow hits its mark, of course the heap runs out. The principle we must accept if we want to go on with our lives is that no paradox makes the world stop functioning as it should. The laws of the universe get their say. Things always snap back to how they should be, and a paradox always eventually runs out its charge. The only reasons why paradoxes perpetuate for as long as they do is because we let them.

No—the reason paradoxes trouble us is because their absurd conclusions make us rethink all of our premises. A paradox is like a staircase, in which each step leads inexorably to the destination. But you get to the top, and the destination is impossible; you’ve stepped off into empty air. So one of the steps must be folly. Because the conclusion must necessarily be false, because we cannot live in a world where logic does not work, one of our premises must necessarily be flawed. This is the source of our unease. A paradox means that somewhere along the path, we have gotten something deeply, terribly wrong.

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