Vacuum Diagrams

by Stephen Baxter

  Chen watched scrawny little Bayliss passing her bony hands over the data desks, scrolling graphics reflected in her augmented eyes. Bayliss had been called out here for this assignment from some university on Mars, where she had tenure. The woman looked as if she was actually enjoying this. As if she was intrigued.

  Chen wondered if she envied Bayliss her scientific curiosity.

  Maybe, she thought at last. It would be nice to feel detached, unengaged by this. On the other hand, she didn't envy Bayliss's evident lack of humanity.

  With gloved hands and her small kit of imaging and diagnostic gear — trying to ignore the lumpy feel of fatty flesh, the vague, unwashed smell of a man too used to living alone — Chen worked at the body.

  The implant at the top of the skull had some kind of link to the center of the brain: to the corpus callosum, the fleshy bundle of nerve fibers between the hemispheres. She probed at the glowing implant, the crown of her own scalp crawling in sympathy.

  After an hour Hassan called them together. Chen pulled her helmet up around her chin and sucked syrup from a nipple; she savored its apple-juice flavor, trying to drown out Marsden's stink. She wished she was back up at the rudimentary colony gathering around the wormhole Interface, encased in a hot shower-bag.

  Construction work. Building things. That was why she had come out here — why she'd fled the teeming cities of the inner System, her endless, shabby, depressing experience of humanity from the point of view of a policeman.

  But her cop's skills were too valuable to be ignored.

  Hassan rested his back against a data desk and folded his arms; the dull silver of his suit cast curving highlights. "How did he die?"

  "Breakdown of the synaptic functions. There was a massive electrical discharge, which flooded most of the higher centers." She pointed to Marsden's implant. "Caused by that thing." She sniffed. "As far as I could tell. I'm not qualified to perform an autopsy. And—"

  "I don't intend to ask you to," Hassan said sharply.

  "It couldn't be murder." Bayliss's voice was dry. Amused. "He was alone on this moon. A million miles from the nearest soul. It would be a marvelous locked-room mystery."

  Hassan's head swiveled towards Chen. "Do you think it was murder, Susan?"

  "That's up to the police."

  Hassan sighed, theatrically tired. "Tell me what you think."

  "No. I don't think it was murder. How could it be? Nobody even knew what he was doing here, it seems."

  "Suicide, then?" Bayliss asked. "After all we are here to tell Marsden that a wormhole highway is shortly to bring millions of new colonists here from the teeming inner System — that his long solitude is over."

  "He didn't know we were coming, remember?" Hassan said. "And besides—" He looked around, taking in the unmade bed, the drab dome, the unkempt corpse. "This was not a man who cared much for himself — or rather, about himself. But, from what we see here, he was—" he hesitated "—stable. Yes? We see evidence of much work, dedicated, careful. He lived for his work. And Bayliss will tell us that such investigations are never completed. One would not wish to die, too early — if at all." He looked at Bayliss. "Am I correct?"

  Bayliss frowned. Her augmented eyes were blank, reflecting the washed-out light as she considered. "An accident, then? But Marsden was no fool. Whatever he was up to with this clumsy implant in his scalp, I cannot believe he would be so careless as to let it kill him."

  "What was he 'up to'?" Chen asked sourly. "Have you figured that out yet?"

  Bayliss rubbed the bridge of her small, flat nose. "There is an immense amount of data here. Much of it not indexed. I've sent data-mining authorized-sentience algorithms into the main stores, to establish the structure."

  "Your preliminary thoughts?" Hassan demanded.

  "Metamathematics."

  Hassan looked blank. "What?"

  "And many experimental results on quantum nonlinearity, which—"

  "Tell me about metamathematics," Hassan said.

  The patches of woven metal over Bayliss's corneas glimmered; Chen wondered if there was any sentience in those augmentations. Probably. Such devices had been banned on Earth since the passing of the first sentience laws, but they could still be found easily enough on Mars. Bayliss said, "Marsden's data stores contain a fragmented catalogue of mathematical variants. All founded on the postulates of arithmetic, but differing in their resolution of undecidable hypotheses."

  "Undecidability. You're talking about the incompleteness theorems," Chen said.

  "Right. No logical system rich enough to contain the axioms of simple arithmetic can ever be made complete. It is always possible to construct statements which can be neither disproved nor proved by deduction from the axioms; instead the logical system must be enriched by incorporating the truth or falsehood of such statements as additional axioms..."

  The Continuum Hypothesis was an example.

  There were several orders of infinity. There were "more" real numbers, scattered like dust in the interval between zero and one, than there were integers. Was there an order of infinity between the reals and the integers? This was undecidable, within logically simpler systems like set theory; additional assumptions had to be made.

  Hassan poked at the corpse with his booted toe. "So one can generate many versions of mathematics, by adding these true-false axioms."

  "And then searching on, seeking out statements which are undecidable in the new system. Yes." Icons scrolled upwards over Bayliss's eyes. "Because of incompleteness, there is an infinite number of such mathematical variants, spreading like the branches of a tree..."

  "Poetry," Hassan said; he sounded lazily amused.

  "Some variants would be logically rich, with many elegant theorems flowing from a few axioms — while others would be thin, over-specified, sterile. It seems that Marsden has been compiling an immense catalogue of increasingly complete logical systems."

  Silence fell; again Chen was aware of the sour stink of the body at her feet. "Why? Why come here to do it? Why the implant? And how did he die?"

  Hassan murmured, "Bayliss said the catalogue was fragmented. This — metamathematical data — was stored carelessly. Casually." He looked to Bayliss for confirmation; the little woman nodded grudgingly.

  "So?" Chen asked.

  "So, Susan, perhaps this metamathematical experiment was not Marsden's primary concern. It was a byproduct of his core research."

  "Which was what? Quantum nonlinearity?" She glanced around the anonymous data desks. How would Marsden go about investigating quantum nonlinearity? With the glowing floor, the first-sized cylinder at its center?

  Hassan dropped to his knees. He pulled off his gloves and passed his hands over the glowing disc area of floor. "This is warm," he said.

  Chen looked at the disc, the writhing worms of light within. "It looks as if it's grown a little, while we've been here." The irregularity of the boundary made it hard to be sure.

  Hassan patted the small cylindrical box at the center of the light pool. It was featureless, seamless. "Bayliss, what's the purpose of this?"

  "I don't know yet. But it's linked to the nanobots in the pool somehow. I think it's the switch that controls their rate of progress."

  Hassan straightened up, suit material rustling over his knees. "Let's carry on; we haven't enough data, yet, for me to make my report."

  Still he grew, devouring postulates furiously, stripping out their logical essence to plate over his own mathematical bones. Brothers, enfeebled, fell away around him, staring at him with disappointed echoes of his own consciousness.

  It did not matter. The Sky — curving, implacable — was close.

  After another couple of hours Hassan called them together again.

  At Chen's insistence, they gathered close to the dome port — away from the glowing disc, Marsden's sprawled corpse. Hassan looked tired, Bayliss excited, eager to speak.

  Hassan eyed Chen. "Squeamish, Susan?"

  "You're a fool, Hassan," she sa
id. "Why do you waste your breath on these taunts?" She indicated the disc of light, the sharpening shadows it cast on the ribbed ceiling. "I don't know what's going on in that pool. Those writhing forms... but I can see there's more activity. I don't trust it."

  He returned her stare coolly. "Nor I, fully. But I do understand some of it. Susan, I've been studying those structures of light. I believe they are sentient. Living things — artificial — inhabiting the bucky tube lattice, living and dying in that hemisphere of transmuted regolith." He looked puzzled. "But I can't understand their purpose. And they're linked, somehow—"

  Bayliss broke in, her voice even but taut. "Linked, like the branches of a tree, to a common root. Yes?"

  Hassan studied her. "What do you know, Bayliss?"

  "I'm starting to understand. I think I see where the metamathematical catalogue has come from. Hassan, I believe the creatures in there are creatures of mathematics — swimming in a Gödelian pool of logic, growing, splitting off from one another like amoebae as they absorb undecidable postulates. Do you see?"

  Chen struggled to imagine it. "You're saying that they are — living — logical structures?"

  Bayliss grinned at her; her teeth were neat and sharp. "A form of natural selection must dominate, based on logical richness — it's really a fascinating idea, a charming mathematical laboratory."

  Chen stared at the pool of light. "Charming? Maybe. But how does it feel, to be a sentient structure with bones of axioms, sinews of logic? What does the world look like to them?"

  "Now poetry from the policewoman," Hassan said dryly. "Perhaps not so different from ourselves, Susan. Perhaps we too are creatures of mathematics, self-conscious observers within a greater Platonic formalism, islands of awareness in a sea of logic..."

  "Marsden might have been able to tell us," Bayliss said.

  Hassan looked puzzled.

  "The implant in his head." Bayliss turned to Chen. "It was linked to the logic pool. Wasn't it, Chen?"

  Chen nodded. She said to Hassan, "The crazy bastard was taking reports — uh, biographies — from these logic trees, dumped direct from the logic pool, into his corpus callosum."

  "So that's how the metamathematics got out," Hassan said. "Until he blew his mind out with some stupid accident."

  "But I think you were right," Bayliss said in her thin, clear voice.

  "What?"

  "That the metamathematical catalogue was only a byproduct of Marsden's true research. The logic pool with its sentient trees was only a — a culture dish for his real study. The catalogue was a curiosity — a way of recording results, perhaps. Of measuring the limits of growth."

  "Tell us about the cylinder at the hub," Hassan said.

  "It is a simple quantum system," Bayliss said. A remote animation entered her voice. "An isolated nucleus of boron is suspended in a magnetic field. The apparatus is set up to detect variations in the spin axis of the nucleus — tips, precession."

  Chen couldn't see the significance of this. "So what?"

  Bayliss dipped her head, evidently fighting impatience. "According to conventional quantum mechanics, the spin axis is not influenced by the magnetic field."

  "Conventional?"

  The ancient theory of quantum mechanics described the world as a mesh of probability waves, spreading through space-time. The "height" of an electron's wave described the chance of finding the electron there, at that moment, moving in such-and-such a way.

  The waves could combine, like spreading ripples on an ocean, reinforcing and canceling each other. But the waves combined linearly — the combination could not cause the waves to change their form or to break; the component waves could only pass on smoothly through each other.

  "That's the standard theory," Bayliss said. "But what if the waves combine nonlinearly? What if there is some contribution proportional to the product of the amplitudes, not just the sum—"

  "Wouldn't such effects have been detected by now?" Chen asked.

  Bayliss blinked. "Our experiments have shown that any nonlinearity must be tiny... less than a billion billion billionth part... but haven't eliminated the possibility. Any coupling of Marsden's magnetic field and nuclear spin would be a nonlinear effect." She rubbed her nose. "Marsden was studying this simple system intensively. Poking it with changes in the magnetic field to gauge its response, seeking out nonlinearity.

  "The small nonlinear effects — if any — are magnified into macroscopic features of the logic pool, which—"

  "He's using the tipping nucleus as a switch to control the pool."

  "Yes. As I suggested. The spin of the nucleus directs the nanobots in their extension of the pool further through the structure of the moon. And—"

  Uncharacteristically, she hesitated.

  "Yes?"

  "And the spin is used to reinitialize the logic trees."

  "These poor trees are like Schrödinger's cat," Hassan said, sounding amused. "Schrödinger's trees!"

  Reinitialize?

  "Lethe," Chen said. "The trees are being culled. Arbitrarily, almost at random, by a quantum system — that's against the sentience laws, damn it." She stared at the fist-sized quantum device with loathing.

  "We are far from Earth," Hassan said sharply. "Has Marsden found his quantum nonlinearity?"

  "I can't tell." Bayliss gazed at the data desks, longing shining through her artificial eyes. "I must complete my data mining."

  "What's the point?" Hassan asked. "If the nonlinearity is such a tiny effect, even if it exists—"

  "We could construct chaotic quantum systems," Bayliss said dryly. "And if you're familiar with the Einstein-Podolsky-Rosen paradox—"

  "Get to the point," Hassan said wearily.

  "Nonlinear quantum systems could violate special relativity. Instantaneous communication, Hassan."

  Chen stared at the floor uneasily. The thrashing of the trees in the logic pool was becoming more intense.

  The Sky was close, a tangible presence above him. He devoured statements, barely registering their logical content, budding ferociously. Diminished brothers fell away from him, failed copies of himself, urging him on.

  He remembered how — last time, before the Cull — he had struck at that vast, forbidding Interface — lashed through it in the instant before he had fallen back. How he had pushed into something soft, receptive, yielding. How good it had felt.

  The Sky neared. He reached up—

  "I think the trees killed Marsden."

  Hassan laughed. "That's absurd."

  She thought it through again. "No," she said, her voice measured. "Remember they are sentient. Motivated, by whatever they see as their goals. Growth, I suppose, and survival. The culling, if they are aware of it, must create murderous fury—"

  "But they can't have been aware of Marsden, as if he were some huge god outside their logic pool."

  "Perhaps not. But they might be aware of something beyond the boundary of their world. Something they could strike at..."

  Bayliss was no longer with them.

  Chen stepped away from Hassan and scanned the dome rapidly. The glowing logic pool was becoming more irregular in outline, spreading under the floor like some liquid. And Bayliss was working at the data desks, setting up transmit functions, plugging in data cubes.

  Chen took two strides across to her and grabbed her arm. For a moment Bayliss tried to keep working, feverishly; only slowly did she become aware of Chen's hand, restraining her.

  She looked up at Chen, her face working, abstracted. "What do you want?"

  "I don't believe it. You're continuing with your data mining, aren't you?"

  Bayliss looked as if she couldn't understand Chen's language. "Of course I am."

  "But this data has been gained illegally. Immorally. Can't you see that? It's—"

  Bayliss tipped back her head; her augmented cornea shone. "Tainted? Is that what you're trying to say? Stained with the blood of these artificial creatures, Chen?"

  "Artificial or not, th
ey are sentient. We have to recognize the rights of all—"

  "Data is data, Susan Chen. Whatever its source. I am a scientist; I do not accept your—" for a moment the small, precise mouth worked "—your medieval morality."

  "I'm not going to let you take this data out of here," Chen said calmly.

  "Susan." Hassan was standing close to her; with a surprisingly strong grasp he lifted her hands from Bayliss's arm.

  "Keep out of this."

  "You must let her finish her work."

  "Why? For science?"

  "No. For commerce. And perhaps," he said dryly, "for the future of the race. If she is right about non-local communication—"

  "I'm going to stop her."

  "No." His hand moved minutely; it was resting against the butt of a laser pistol.

  With automatic reflex, she let her muscles relax, began the ancient calculation of relative times and distances, of skills and physical conditions.

  She could take him. And—

  Bayliss cried out; it was a high-pitched, oddly girlish yelp. There was a clatter as she dropped some piece of equipment.

  Chen's confrontation with Hassan broke up instantly. They turned, ran to Bayliss; Chen's steps were springy, unnatural in the tiny gravity.

  "What is it?"

  "Look at the floor."

  The Sky resisted for an instant. Then it crumbled, melting away like ancient doubts.

  He surged through the break, strong, exultant, still growing.

  He was outside the Sky. He saw arrays of new postulate fruits, virgin, waiting for him. And there was no further Sky; the Pool went on forever, infinite, endlessly rich.

  He roared outwards, devouring, budding; behind him a tree of brothers sprouted explosively.

  The pool surged, in an instant, across the floor and out beyond the dome. The light, squirming with logic trees, rippled beneath Chen's dark, booted feet; she wanted, absurdly, to get away, to jump onto a data desk.

  "The quantum switch." Bayliss's voice was tight, angry; she was squatting beside the switch, in the middle of the swamped light pool.