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 said. “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—”