Collected short fiction.., p.81
Collected Short Fiction of Greg Egan, page 81
Here he launches an intrepid attack on the most abstract realms of Higher Mathematics with a computer made entirely of light—with potentially disastrous results for the entire universe when those abstract realms start to strike back…
* * * * *
I woke disoriented, unsure why. I knew I was lying on the narrow, lumpy single bed in Room 22 of the Hotel Fleapit; after almost a month in Shanghai, the topography of the mattress was depressingly familiar. But there was something wrong with the way I was lying; every muscle in my neck and shoulders was protesting that nobody could end up in this position from natural causes, however badly they’d slept.
And I could smell blood.
I opened my eyes. A woman I’d never seen before was kneeling over me, slicing into my left tricep with a disposable scalpel. I was lying on my side, facing the wall, one hand and one ankle cuffed to the head and foot of the bed.
Something cut short the surge of visceral panic before I could start stupidly thrashing about, instinctively trying to break free. Maybe an even more ancient response—catatonia in the face of danger—took on the adrenaline and won. Or maybe I just decided that I had no right to panic when I’d been expecting something like this for weeks.
I spoke softly, in English. “What you’re in the process of hacking out of me is a necrotrap. One heartbeat without oxygenated blood, and the cargo gets fried.”
My amateur surgeon was compact, muscular, with short black hair. Not Chinese: Indonesian, maybe. If she was surprised that I’d woken prematurely, she didn’t show it. The gene-tailored hepatocytes I’d acquired in Hanoi could degrade almost anything from morphine to curare; it was a good thing the local anaesthetic was beyond their reach.
Without taking her eyes off her work, she said, “Look on the table next to the bed.”
I twisted my head around. She’d set up a loop of plastic tubing full of blood—mine, presumably—circulated and aerated by a small pump. The stem of a large funnel fed into the loop, the intersection controlled by a valve of some kind. Wires trailed from the pump to a sensor taped to the inside of my elbow, synchronizing the artificial pulse with the real. I had no doubt that she could tear the trap from my vein and insert it into this substitute without missing a beat.
I cleared my throat and swallowed. “Not good enough. The trap knows my blood pressure profile exactly. A generic heartbeat won’t fool it.”
“You’re bluffing .” But she hesitated, scalpel raised. The handheld MRI scanner she’d used to find the trap would have revealed its basic configuration, but few fine details of the engineering—and nothing at all about the software.
“I’m telling you the truth.” I looked her squarely in the eye, which wasn’t easy given our awkward geometry. “It’s new, it’s Swedish. You anchor it in a vein forty-eight hours in advance, put yourself through a range of typical activities so it can memorize the rhythms … then you inject the cargo into the trap. Simple, foolproof, effective.” Blood trickled down across my chest onto the sheet. I was suddenly very glad that I hadn’t buried the thing deeper, after all.
“So how do you retrieve the cargo, yourself?”
“That would be telling.”
“Then tell me now, and save yourself some trouble.” She rotated the scalpel between thumb and forefinger impatiently. My skin did a cold burn all over, nerve ends jangling, capillaries closing down as blood dived for cover.
I said, “Trouble gives me hypertension.”
She smiled down at me thinly, conceding the stalemate—then peeled off one stained surgical glove, took out her notepad, and made a call to a medical equipment supplier. She listed some devices which would get around the problem—a blood pressure probe, a more sophisticated pump, a suitable computerized interface—arguing heatedly in fluent Mandarin to extract a promise of a speedy delivery.
Then she put down the notepad and placed her ungloved hand on my shoulder. “You can relax now. We won’t have long to wait.”
I squirmed, as if angrily shrugging off her hand—and succeeded in getting some blood on her skin. She didn’t say a word, but she must have realized at once how careless she’d been; she climbed off the bed and headed for the washbasin, and I heard the water running.
Then she started retching.
I called out cheerfulIy, “Let me know when you’re ready for the antidote.”
I heard her approach, and I turned to face her. She was ashen, her face contorted with nausea, eyes and nose streaming mucus and tears.
“Tell me where it is!”
“Uncuff me, and I’ll get it for you.”
“No! No deals!”
“Fine. Then you’d better start looking, yourself.”
She picked up the scalpel and brandished it in my face. “Screw the cargo. I’ll do it!” She was shivering like a feverish child, uselessly trying to stem the flood from her nostrils with the back of her hand.
I said coldly, “If you cut me again, you’ll lose more than the cargo.”
She turned away and vomited; it was thin and gray, blood-streaked. The toxin was persuading cells in her stomach lining to commit suicide en masse.
“Uncuff me. It’ll kill you. It doesn’t take long.”
She wiped her mouth, steeled herself, made as if to speak—then started puking again. I knew, first hand, exactly how bad she was feeling. Keeping it down was like trying to swallow a mixture of shit and sulphuric acid. Bringing it up was like evisceration.
I said, “In thirty seconds, you’ll be too weak to help yourself—even if I told you where to look. So if I’m not free…”
She produced a gun and a set of keys, uncuffed me, then stood by the foot of the bed, shaking badly but keeping me targeted. I dressed quickly, ignoring her threats, bandaging my arm with a miraculously clean spare sock before putting on a T-shirt and a jacket. She sagged to her knees, still aiming the gun more or less in my direction—but her eyes were swollen half-shut brimming with yellow fluid. I thought about trying to disarm her, but it didn’t seem worth the risk.
I packed my remaining clothes, then glanced around the room as if I might have left something behind. But everything that really mattered was in my veins; Alison had taught me that that was the only way to travel.
I turned to the burglar. “There is no antidote. But the toxin won’t kill you. You’ll just wish it would, for the next twelve hours. Goodbye.”
As I headed for the door, hairs rose suddenly on the back of my neck. It occurred to me that she might not take me at my word—and might fire a parting shot, believing she had nothing to lose.
Turning the handle, without looking back, I said, “But if you come after me—next time, I'll kill you.”
That was a lie, but it seemed to do the trick. As I pulled the door shut behind me, I heard her drop the gun and start vomiting again.
Halfway down the stairs, the euphoria of escape began to give way to a bleaker perspective. If one careless bounty hunter could find me, her more methodical colleagues couldn’t be far behind. Industrial Algebra was closing in on us. If Alison didn’t gain access to Luminous soon, we’d have no choice but to destroy the map. And even that would only be buying time.
I paid the desk clerk for the room until the next morning, stressing that my companion should not be disturbed, and added a suitable tip to compensate for the mess the cleaners would find. The toxin denatured in air; the bloodstains would be harmless in a matter of hours. The clerk eyed me suspiciously, but said nothing.
Outside, it was a mild, cloudless summer morning. It was barely six o’clock, but Kongjiang Lu was already crowded with pedestrians, cyclists buses—and a few ostentatious chauffeured limousines, ploughing through the traffic at about ten kph. It looked like the night shift had just emerged from the Intel factory down the road; most of the passing cyclists were wearing the orange, logo-emblazoned overalls.
Two blocks from the hotel I stopped dead, my legs almost giving way beneath me. It wasn’t just shock—a delayed reaction, a belated acceptance of how close I’d come to being slaughtered. The burglar’s clinical violence was chilling enough—but what it implied was infinitely more disturbing.
Industrial Algebra was paying big money, violating international law, taking serious risks with their corporate and personal futures. The arcane abstraction of the defect was being dragged into the world of blood and dust, boardrooms and assassins, power and pragmatism.
And the closest thing to certainty humanity had ever known was in danger of dissolving into quicksand.
· · · · ·
It had all started out as a joke. Argument for argument’s sake. Alison and her infuriating heresies.
“A mathematical theorem,” she’d proclaimed, “only becomes true when a physical system tests it out: when the system’s behavior depends in some way on the theorem being true or false.”
It was June 1994. We were sitting in a small paved courtyard, having just emerged yawning and blinking into the winter sunlight from the final lecture in a one-semester course on the philosophy of mathematics—a bit of light relief from the hard grind of the real stuff. We had fifteen minutes to kill before meeting some friends for lunch. It was a social conversation—verging on mild flirtation—nothing more. Maybe there were demented academics lurking in dark crypts somewhere, who held views on the nature of mathematical that they were willing to die for. But we were twenty years old, and we knew it was all angels on the head of a pin.
I said, “Physical systems don’t create mathematics. Nothing creates mathematics—it’s timeless. All of number theory would still be exactly the same, even if the universe contained nothing but a single electron.”
Alison snorted. “Yes, because even one electron, plus a spacetime to put it in, needs all of quantum mechanics and all of general relativity—and all the mathematical infrastructure they entail. One particle floating in a quantum vacuum needs half the major results of group theory, functional analysis, differential geometry—”
“Okay, okay! I get the point. But if that’s the case … the events in the first picosecond after the Big Bang would have ‘constructed’ every last mathematical truth required by any physical system, all the way to the Big Crunch. Once you’ve got the mathematics that underpins the Theory of Everything … that’s it, that’s all you ever need. End of story.”
“But it’s not. To apply the Theory of Everything to a particular system, you still need all the mathematics for dealing with that system—which could include results far beyond the mathematics that the TOE itself requires. I mean, fifteen billion years after the Big Bang, someone can still come along and prove, say … Fermat’s Last Theorem.” Andrew Wiles at Princeton had recently announced a proof of the famous conjecture, although his work was still being scrutinized by his colleagues, and the final verdict wasn’t yet in. “Physics never needed that before.”
I protested, “What do you mean, ‘before’? Fermat’s Last Theorem never has—and never will—have anything to do with any branch of physics.”
Alison smiled sneakily. “No branch—no. But only because the class of physical systems whose behavior depends on it is so ludicrously specific: the brains of mathematicians who are trying to validate the Wiles proof.
“Think about it. Once you start trying to prove a theorem, then even if the mathematics is so ‘pure’ that it has no relevance to any other object in the universe … you’ve just made it relevant to yourself. You have to choose some physical process to test the theorem—whether you use a computer, or a pen and paper … or just close your eyes and shuffle neurotransmitters. There’s no such thing as a proof that doesn’t rely on physical events—and whether they’re inside or outside your skull doesn’t make them any less real.”
“Fair enough,” I conceded warily. “But that doesn’t mean—”
“And maybe Andrew Wiles’s brain—and body, and notepaper—comprised the first physical system whose behavior depended on the theorem being true or false. But I don’t think human actions have any special role … and if some swarm of quarks had done the same thing blindly, fifteen billion years before—executed some purely random interaction that just happened to test the conjecture in some way—then those quarks would have constructed FLT long before Wiles. We’ll never know.”
I opened my mouth to complain that no swarm of quarks could have tested the infinite number of cases encompassed by the theorem—but I caught myself just in time. That was true—but it hadn’t stopped Wiles. A finite sequence of logical steps linked the axioms of number theory—which included some simple generalities about all numbers—to Fermat’s own sweeping assertion. And if a mathematician could test those logical steps by manipulating a finite number of physical objects for a finite amount of time—whether they were pencil marks on paper, or neurotransmitters in his or her brain—then all kinds of physical systems could, in theory, mimic the structure of the proof … with or without any awareness of what it was they were “proving.”
I leant back on the bench and mimed tearing out hair. “If I wasn’t a die-hard Platonist before, you’re forcing me into it! Fermat’s Last Theorem didn’t need to be proved by anyone—or stumbled on by any random swarm of quarks. If it’s true, it was always true. Everything implied by a given set of axioms is logically connected to them, timelessly, eternally … even if the links couldn’t be traced by people—or quarks—in the lifetime of the universe.”
Alison was having none of this; every mention of timeless and eternal truths brought a faint smile to the corner of her mouth, as if I was affirming my belief in Santa Claus. She said, “So who, or what, pushed the consequences of ‘There exists an entity called zero’ and ‘Every X has a successor,’ et cetera, all the way to Fermat’s Last Theorem and beyond, before the universe had a chance to test out any of it?”
I stood my ground. “What’s joined by logic is just … joined. Nothing has to happen—consequences don’t have to be pushed’ into existence by anyone, or anything. Or do you imagine that the first events after the Big Bang, the first wild jitters of the quark-gluon-plasma, stopped to fill in all the logical gaps? You think the quarks reasoned: well, so far we’ve done A and B and C—but now we mustn't do D, because D would be logically inconsistent with the other mathematics we’ve ‘invented’ so far … even if it would take a five-hundred-thousand-page proof to spell out the inconsistency?”
Alison thought it over. ‘‘No. But what if event D took place, regardless? What if the mathematics it implied was logically inconsistent with the rest—but it went ahead and happened anyway … because the universe was too young to have computed the fact that there was any discrepancy?”
I must have sat and stared at her, open-mouthed, for about ten seconds. Given the orthodoxies we’d spent the last two-and-a-half years absorbing, this was a seriously outrageous statement.
“You’re claiming that … mathematics might be strewn with primordial defects in consistency? Like space might be strewn with cosmic strings?”
“Exactly.” She stared back at me, feigning nonchalance. “If spacetime doesn’t join up with itself smoothly, everywhere … why should mathematical logic?”
I almost choked. “Where do I begin? What happens—now—when some physical system tries to link theorems across the defect? If theorem D has been rendered ‘true’ by some over-eager quarks, what happens when we program a computer to disprove it? When the software goes through all the logical steps that link A, B, and C—which the quarks have also made true— to the contradiction, the dreaded not-D … does it succeed, or doesn’t it?”
Alison sidestepped the question. “Suppose they’re both true: D and not-D. Sounds like the end of mathematics, doesn’t it? The whole system falls apart, instantly. From D and not-D together you can prove anything you like: one equals zero, day equals night. But that’s just the boring-old-fart Platonist view—where logic travels faster than light, and computation takes no time at all. People live with omega-inconsistent theories, don’t they?”
Omega-inconsistent number theories were nonstandard versions of arithmetic, based on axioms that “almost” contradicted each other—their saving grace being that the contradictions could only show up in “infinitely long proofs” (which were formally disallowed, quite apart from being physically impossible). That was perfectly respectable modern mathematics—but Alison seemed prepared to replace “infinitely long” with just plain “long”—as if the difference hardly mattered, in practice.
I said, “Let me get this straight. What you’re talking about is taking ordinary arithmetic—no weird counter-intuitive axioms, just the stuff every ten-year-old knows is true—and proving that it’s inconsistent, in a finite number of steps?”
She nodded blithely. “Finite, but large. So the contradiction would rarely have any physical manifestation—it would be ‘computationally distant’ from everyday calculations, and everyday physical events. I mean … one cosmic string, somewhere out there, doesn’t destroy the universe, does it? It does no harm to anyone.”
I laughed drily. “So long as you don’t get too close. So long as you don’t tow it back to the solar system and let it twitch around slicing up planets.”
“Exactly.”
I glanced at my watch. “Time to come down to Earth, I think. You know we’re meeting Julia and Ramesh—?”
