by Greg Egan
Sam said, “If Andrea’s right, all you need to be is different: from her, and from the automaton that replaced her. So long as Jarrod’s friends can see that there’s a brand new Emmy in town, that ought to be enough to bring him back.”
#
Sagreda smoothed the woolen shawl around her shoulders as she approached the pale stone building. It stood on a corner block between two streets that met at a crazily acute angle; the would-be sharp corner had been sliced off to make a narrow wall that bore the entrance to the Café Central. High above the doorway, four white statues – three women, and a man in a robe – gazed serenely into the moonlight. If they were meant to be recognizable figures to the citizens of Vienna, they evoked nothing at all in her own contributors’ pool of shared knowledge.
When she entered the café, there were three German officers seated at a table straight ahead of her, looking relaxed and jovial. Her overlays marked them as automata, which lessened her urge to walk over and rip the eagle-and-swastika emblems off their jackets, but she still had to struggle to turn away from them, past the tables of genteelly chatting extras toward her gathered “friends.”
“Emmy!” the Moritz automaton called out warmly. Sagreda smiled and approached the table, trying to act as if these people were trusted comrades for whom she would have lain down her life … and whose survival depended on maintaining the facade that their shared experiences amounted to nothing more than abstruse academic discussions and a spot of restrained carousing.
Moritz pulled out a chair for her, and she greeted everyone in turn. Along with Moritz and his wife Blanche, Jarrod’s four friends were playing their usual roles: Karl Menger, Rudolf Carnap, Alfred Tarski and Van Quine. Tarski was Polish and Quine was American, but everyone was speaking German, with the software whispering an English translation in Sagreda’s skull. Three of the customers were having much the same experience as she was – speaking and hearing English, with the German as a kind of background music – but the man playing Carnap seemed to be fluent in his character’s native tongue.
Andrea had said she’d woken into the game as a monolingual Anglophone, like most comps, but had eventually picked up enough from the running translation to start speaking German herself. The SludgeNet might have valiantly tried to convince her that German was her true first language and she was merely fluent in English – as the real Noether had been – but Sagreda didn’t have to put up with any of these clumsy machinations. She dialed down the German to a faint guttural mutter, leaving her with at least some chance of understanding what remained.
“I think I’ve found a way to generalize one of Kurt’s favorite tricks!” Carnap enthused, as a waitress brought coffee and cake for Sagreda. “Suppose you have a formal language with a list of axioms and rules of deduction that capture the usual properties of the natural numbers. Any statement in this system can be converted into a number, using Kurt’s scheme – its ‘Gödel number,’ if you will. What I want to show is that any formula F with one free variable has a kind of fixed point: a statement G whose Gödel number, when fed into F, turns F into a statement equivalent to G!”
He turned to Sagreda, as if she were the final arbiter as to whether the topic would be of interest to the gathering. “That sounds intriguing,” she said in her puppeteer’s body, then her puppet’s lips moved in synch with the translation.
Carnap needed no more encouragement. “Think about the function Q that takes the Gödel number of any formula, A, with one free variable, and gives you the Gödel number of that formula with the free variable replaced by the Gödel number of the original formula. If the system is powerful enough to represent that function, there will be a formula, B, with two free variables, which can be proved equivalent to asserting that the second variable equals the result of applying the function Q to the first variable. Are you with me so far?”
Sagreda willed him not to look her way as she struggled with the oddly convoluted construction. Why talk about this thing B, instead of Q itself? Ah … because Q might be a perfectly well-defined function, but that didn’t mean the language would let you write “Q(x)” as shorthand for its value at x. The language was only assumed to be strong enough to express the idea that some candidate number, y, passed a series of tests to confirm that it equaled Q(x). B(x,y) couldn’t tell you the answer, Q(x), directly, but it would tell you whether or not your guess, y, was correct.
“We’re with you,” Tarski replied impatiently.
Carnap said, “Remember our formula F, the target of the whole business? We use it to define a formula C, with one free variable, x. C asserts that for all values of y, B holding true for x and y implies F is true for y.”
Menger took a pencil from his waistcoat pocket and started making neat, sparse notes on a napkin. Sagreda thought: Okay, this is the logician’s way of saying what slobs like me would write as “C(x) is true if and only if F(Q(x)) is true” … even though the language won’t let me write Q(x) explicitly.
“Now let’s feed C its own Gödel number, and see where that takes us.” Carnap took on the air of a stage magician who was about to pull a big hat out of a much smaller one. “Given what our system can prove about B and Q, it can also prove that C, fed its own Gödel number, is equivalent to F with its free variable replaced by Q evaluated at C’s Gödel number – and Q evaluates to the Gödel number of C fed its own Gödel number. So, C fed its own Gödel number is equivalent to F fed the Gödel number of C fed its own Gödel number. And that’s exactly what we wanted: G, the fixed point, is C fed its own Gödel number. Feed the Gödel number of G to F and the result is equivalent to G itself!”
Tarski leaned back in his chair and stretched his arms above his head, smiling appreciatively. “That really is quite beautiful!”
Sagreda snuck a peek at Menger’s notes, to be sure she had the whole thing clear in her head. It all sounded impossibly abstract at first, but it wasn’t hard to bring it down to Earth with a simple example. F might assert that the number you fed it was the sum of two integers squared. Then Carnap’s argument showed there was a statement G that could be proved equivalent to the claim that its own Gödel number was the sum of two squares. For any property the language could discuss, you could write down a statement that claimed, rightly or wrongly, that its own Gödel number had that property.
And to recapture Gödel’s own famous result, you’d choose F to assert that its variable was the Gödel number of a statement that could not be proved within the system. Then the corresponding G would be equivalent to the claim that G itself had no proof … so it either had to be a falsehood that the system “proved,” or a truth beyond the powers of the system to validate.
“You must tell Kurt all of this!” she urged Carnap.
“Kurt’s still unwell,” Quine replied.
“Really?” Sagreda frowned. “I’m beginning to worry about him.”
“I wouldn’t be too concerned,” Menger replied. “We all know he can be a bit of a hypochondriac.”
Sagreda didn’t push it; if she pleaded for Gödel to return to the café, the game might decide to fill the role with an automaton.
She said, “Well, in his absence at least I can confess one thing I would never admit in his presence.”
Tarski’s smile grew impish. “We’re all ears.”
“What he means is: we’re your discreet confidants,” Menger assured her.
“I’d expect discretion from my fellow transgressors,” Sagreda replied, hoping she was treading the right line between jest and sincerity. “Let’s be honest: who among us isn’t just a little jealous of Kurt’s achievements? To do what he did at any age … but at 25!” She grimaced with mock anguish. “A mere youth, leaving Russell and Hilbert awe-struck?”
Carnap said, “He’s not the only person I can think of whose prowess left an impression on Hilbert.”
Sagreda had her puppet blush a little. “Professor Hilbert has been inordinately kind to me, but I can promise you that at the age of 25, I did not deserve pra
ise from anyone! When I look back on my thesis now, I can see it was just a jungle of equations. Hundreds of invariants of ternary biquadratic forms, all scribbled out like some inky butterfly collection! There’s nothing elegant in that. It was manure.”
This assessment seemed to leave her colleagues dumbstruck, though Sagreda was just paraphrasing the real woman’s sentiments. Andrea had never said anything like this; the game had offered her no cues in that direction, and she’d been in no position to take character notes from Noether’s biography.
“I’m sure there are times in all of our careers that we look back on and wince,” Quine said. “But if I start listing all the work that you ought to look back on with pride, I’ll just sound like an obsequious flatterer. Kurt’s unique, there’s no doubt about that, but let’s be clear: you have no grounds for jealousy.”
Sagreda lowered her gaze and stared into her half-eaten slice of Schwarzwälder Kirschtorte, hoping she hadn’t swerved so far from Andrea’s precedents that she was making the customers uncomfortable. Who wanted to go Nazi-hunting with a woman who’d suddenly turned neurotically self-deprecating?
“Professor … E did declare that my work on the symmetries of Lagrangian actions had impressed him,” she conceded. She had almost spoken the Jewish name out loud in public; that really would have given her colleagues whiplash.
“So it’s settled,” Carnap declared cheerfully. “No room, and no need, for jealousy.”
They toasted that, with coffee. Sagreda tried not to imagine the customers’ rigs squirting flavors into their mouths. Couldn’t they have just conference-Skyped each other, with real refreshments on hand, while they chatted about the upheavals in mathematical philosophy in the 1930s?
But then they might have had to skip the next part.
#
The logicians left the café and exchanged loud farewells that echoed down the empty streets, but though they set off in different directions, no one actually went far before spiraling back. Menger had sketched the routes they should follow on the back of his Carnapian napkin, making the street plan look like some kind of esoteric fractal.
Sagreda arrived at a corner with a view of the front of the café. Around eleven o’clock, the three officers emerged, and two of them departed in a staff car that had been waiting a short way down the street. The third, though, as Menger had predicted, set out on foot. Apparently he was in the habit of visiting a mistress who he could not be seen with in public … inasmuch as a claim like this had any meaning, when the game classified the man as an automaton and there was no reason for his lover to exist at all.
Sagreda heard Tarski cough quietly up ahead, so she stepped out of the shadows and started walking, ten or fifteen paces ahead of the officer. There was no one else in sight. When Tarski emerged from the alley and grabbed her roughly by the shoulders, she was tempted to ramp up her puppet’s strength and just toss him aside, but she restrained herself. She grunted affrontedly as they struggled, but she did not call out for help; the last thing they wanted to do was attract witnesses.
“Take my necklace,” she whispered. It was the only reason she’d worn it.
“I’m trying!” he complained. Apparently she was putting up such a convincing fight that even though she really couldn’t gouge his eyes out, he was afraid to let go of her with one hand to snatch at the jewelry.
“You there, step away!” the officer shouted, drawing his side-arm. Tarski clung to Sagreda defiantly, and dragged her in front of him so she was shielding most of his body from direct fire. These customers hadn’t ever caused Andrea serious harm, but Andrea had not been the game’s first Noether.
The officer strode toward them, almost apoplectic at this ungentlemanly behavior. Carnap and Quine emerged from their hiding place and seized him from behind; Quine snapped his wrist and the gun dropped to the ground.
As the officer cried out in pain, Carnap stuffed a handkerchief into his mouth while Quine got a leather strap around his neck and began to tighten it. Sagreda’s heart was pounding, and her puppet took its cue from her; as Tarski disentangled himself from her, he squeezed her arm in a consoling gesture. Did he care what the comp who he thought might actually die in this encounter was feeling, or was it all just theater to him?
Further down the street, Moritz and Blanche laughed loudly: someone was coming, but it was a civilian, not a second Nazi they’d be willing to dispatch. The four of them quickly dragged the half-strangled officer into the alley, where Menger was waiting in a doorway.
They followed him into a storeroom that was like an expressionist film set, full of silhouettes and shadows, lit by a single lamp on a shelf. As Sagreda squeezed past the beer barrels and dodged the cured meat hanging from the ceiling, Menger picked up a long metal skewer and the other three men held the officer still.
Menger turned to Sagreda. “Do you want to finish this yourself?”
She shook her head.
“Not even for your brother? The Russians might have shot him, but it was the Nazis who forced him to flee.”
Sagreda took the skewer. Andrea had declined to bloody her hands, even though she’d understood not only that the “Nazis” here were innocent of any crime, they were also as insentient as the clockwork figurines that marched across the Ankeruhr. But her instinct had always been that it was not what the customers expected of her character.
Sagreda looked the struggling automaton in the eye; there was nothing of a tin man about his bulging veins or the horror the software was painting on his features. So far as she knew, these particular customers had never killed a comp, but would they have cared about the difference? If she laughed in Menger’s face and declared herself a Nazi spy, would they all go along with the plot twist, or would the fact that there were no Nazis here, just a woman they’d sat talking and joking with for hours as they all pretended to be smarter than they were, give them pause before they turned on her?
She got a hold of herself. This was not the time for sub-Milgramian sociological experiments; all that mattered was Jarrod and his graphics card. If they wanted a new Emmy, she’d give them a new Emmy.
She plunged the skewer between the automaton’s ribs, granting her puppet precisely the strength it needed to succeed without faltering, ignoring the fake blood and the thing’s muffled death cries. Then she stepped back and turned away. Whatever she put on her puppet’s face was unlikely to convince anyone that she truly believed she was a middle-class German woman, hiding her Jewish heritage behind forged papers, who had just taken a human life for the very first time to avenge the death of her brother. If the SludgeNet had wanted Meryl Streep, they really should have been willing to pay more.
Her comrades’ response to the unexpectedly dark turn she’d taken was to lower their voices and tiptoe around her as they cleaned up the scene of the crime. She heard them sliding the body into a sack that Moritz and Blanche would put in the trunk of their car, to dump somewhere far away after a long drive. Customers never ended up with the tedious jobs.
“Emmy?” Tarski spoke tentatively, from some distance behind her. “Are you all right?”
She turned to face him. “I’ll be fine.” She couldn’t really make out his features, but from his body language she could have sworn he was taking the whole situation far more seriously than she was. Maybe he believed that she believed she’d just killed a man … and he felt worse about that particular deception than he did about the much larger one that made it possible in the first place.
Or maybe she was overthinking it, and he was just empathizing with “Emmy” as he might empathize with any fictional character, in the moment.
“Will we see you back in the café next week?”
She said, “I wouldn’t miss it for the world.”
6
Sagreda sat on the barstool in her kitchen, swiveling back and forth with her palms behind her on the countertop, trying not to look up at the clock. With her eyes forced to sweep the room instead, she felt herself slipping into jamais vu. Like m
ost of the residents of Arrietville who’d had better things to do than play architect, she’d just cloned a bungalow from Close to Heaven, a turn-of-the-millennium melodrama about upper-middle-class families in a fictional Californian suburb. The place had always felt a bit soulless to her, but now she was edging toward the more alarming sense that she’d woken from a drunken blackout to find that she’d broken into a wealthy neighbor’s house, and was sure to be discovered at any moment.
At two p.m. precisely, the doorbell rang. Sam had arrived to give her moral support – probably teleported straight to her porch from a bubble bath, magically dry and fully clothed because that’s what the alarm he’d set had specified. She greeted him warmly anyway; that his presence was effortless made it no less thoughtful, and that his breezy digital agility gave her vertigo made it no less honest.
“No sign of the elusive Herr Gödel?” Sam asked, as they walked down the hall to the dining room, where Sagreda’s laptop sat on the table.
She shook her head. “None of them have logged on yet.” She gestured for Sam to take a seat, then joined him. “I hope I didn’t scare them off.”
“By doing what they do themselves?”
“I’m not suggesting they’re squeamish, but I might have overstepped some boundary.”
“It’s still barely after eight in their time zone,” Sam noted.
The laptop beeped. Sagreda couldn’t bring herself to look, but Sam leaned over and peered at the screen. “It seems we have Herr Menger, famous for his Menger sponge-cake—” It beeped again. “And we also have Herr Carnap, famous for his cake-forkability theorem.”
“Stop reminding me that I haven’t done my homework,” Sagreda moaned. She’d been too busy writing software to find time to study up on her fellow assassins’ work.